The STIRAP-based unitary decelerating and accelerating processes of a single free atom
aa r X i v : . [ qu a n t - ph ] M a y The STIRAP-based unitary deceleratingand accelerating processes of a single freeatom
Xijia MiaoSomerville, MassachusettsDate: May 2007Abstract
The STIRAP-based unitary decelerating and accelerating processes havebeen proposed to realize the time- and space-compressing processes in the quan-tum control process to simulate the reversible and unitary state-insensitive halt-ing protocol (Arxiv: quant-ph/0607144). A standard three-state STIRAP pulsesequence may act as a basic unitary decelerating sequence or a basic unitaryaccelerating sequence. A STIRAP-based unitary decelerating (accelerating) pro-cess then consists of a train of these basic STIRAP unitary decelerating (acceler-ating) sequences. The present work is focused on investigating analytically andquantitatively how the momentum distribution of a momentum superpositionstate of a pure-state quantum system such as a momentum Gaussian wave-packet state of a single freely moving atom affects the STIRAP state transferin these decelerating and accelerating processes. The complete STIRAP statetransfer and the unitarity of these processes are stressed highly in the investi-gation. It has been shown that the momentum distribution has an importantinfluence upon the STIRAP state-transfer efficiency. In the ideal adiabatic con-dition these unitary decelerating and accelerating processes for a freely movingatom are studied in detail, and it is shown that they can be used to manipulateand control in time and space the center-of-mass position and momentum ofa Gaussian wave-packet motional state of a free atom. Two general adiabaticconditions for the basic STIRAP decelerating and accelerating processes arederived analytically. They are strict and accurate. They can be used to setup a conventional STIRAP state-transfer experiment and also the basic STI-RAP decelerating and accelerating processes. With the help of the STIRAPtheory and the unitary quantum dynamics it confirms theoretically that thetime- and space-compressing processes of the quantum control process (Arxiv:quant-ph/0607144) can be realized almost perfectly by the STIRAP-based uni-tary decelerating and accelerating processes in the ideal or nearly ideal adiabaticcondition.
1. Introduction
The stimulated Raman adiabatic passage (STIRAP) processes are very im-portant coherent double-photon processes [1, 2]. The STIRAP method has beenextensively applied to the complete population transfer in energy levels of atoms1nd molecules in the laser spectroscopy [3, 4], the laser cooling in neutral atomicensembles [5, 6, 7, 8, 9] (here may include the conventional Raman adiabaticprocesses), and the atomic quantum interference experiments [10, 11, 12, 13, 14]as well as other science research fields. A standard three-state STIRAP pulsesequence [3, 4, 15, 17, 18] consists of a pair of Raman laser light beams whichare selectively applied to three chosen energy levels of an atomic or molecularsystem. The largest advantage of the STIRAP method is that the completepopulation transfer or state transfer in an atomic or molecular system may beachieved by the STIRAP pulse sequence with delayed and overlapping Ramanlaser light beams [15, 16, 17], and the STIRAP method is tolerant to the ex-perimental imperfections. The basic theory of the STIRAP method has beenwell set up [16, 17, 18]. The experimental confirmation for the STIRAP methodhas been carried out first in the atomic and molecular laser spectroscopy [3, 15]and then in diverse other science research fields. The basic STIRAP theoret-ical and experimental methods have been first extended to study and designthe quantum interference experiments in cold atomic ensembles [10, 11, 12, 13,14], which involve the atomic motional momentum transfer. The theory alsohas been developed to study the Raman laser cooling processes in a neutralatom ensemble [20, 21] by combining the velocity-selective coherent populationtrapping [5, 19]. In these two types of experiments [5, 10, 11, 12, 13, 14, 19,20] the Raman laser pulse sequence such as the STIRAP pulse sequence usuallyconsists of a pair of counterpropagating Raman laser light beams.It has been shown that the dynamical state-locking pulse field plays a keyrole in constructing the reversible and unitary state-insensitive halting protocol[22] and solving efficiently the quantum search problem [22, 36]. A generalstate-locking pulse field [22] consists of a sequence of the time- and space-dependent electromagnetic pulse fields and could also contain the time- andspace-dependent potential fields which could be generated by the external elec-tric and / or magnetic field. A unitary decelerating (or accelerating) laser lightpulse sequence that is used to decelerate (or accelerate) a moving free atomcould be thought of as the component of a dynamical state-locking pulse field.The standard three-state STIRAP pulse sequence may act as either a basic uni-tary decelerating sequence or a basic unitary accelerating sequence, which isdependent upon the parameter settings of the two counterpropagating Ramanadiabatic laser light beams of the STIRAP pulse sequence. The STIRAP-baseddecelerating (accelerating) process then consists of a train of the basic STIRAPunitary decelerating (accelerating) sequences which are applied to a movingatom consecutively. It has been proposed that the STIRAP-based unitary decel-erating and / or accelerating processes may be used to coherently manipulate thehalting-qubit atom in the quantum control process to simulate the reversible andunitary state-insensitive halting protocol [22]. Thus, the STIRAP-based unitarydecelerating and accelerating processes are the important building blocks of thereversible and unitary state-insensitive halting protocol and the efficient quan-tum search process based on the unitary quantum dynamics in time and space.A unitary decelerating process is much like the conventional laser cooling pro-cess in an atomic ensemble. The essential difference between the two processes2s that the unitary decelerating process is reversible and unitary, while the lasercooling process usually is irreversible. A unitary accelerating process could bemore like the momentum transfer process in the quantum interference experi-ments of an atomic ensemble. The atomic momentum transfer process in thesequantum interference experiments is generally transverse with respect to theinitial atomic moving direction, while here the unitary accelerating process islongitudinal. The unitary decelerating and accelerating processes stress theirunitarity and complete state transfer in the quantum control process.A quantum computation tends to avoid using a space-dependent unitary op-eration as its basic building block, since such a unitary operation usually is morecomplicated and inconvenient to manipulate and control in a quantum systemwith respect to the conventional quantum-gate operations. It is necessary tomanipulate and control at will the internal motion of an atom (or atomic ion)in quantum computation when the specific internal states of the atom such asthe hyperfine ground electronic states or the nuclear spin states are taken as aquantum bit, but at the same time the atomic center-of-mass motion tends to bekept unchanged simply (or to be constrained simply) so that it does not affectthese quantum-gate operations and hence in quantum computational theory itis usually not considered explicitly. However, as pointed out in the previouspaper [22], it is of crucial importance to unitarily manipulate and control atwill in time and space the center-of-mass motion, the internal motion, and themutual cooperation and coupling of the two motions of the halting-qubit atomin order to realize the quantum control process to simulate the reversible andunitary state-insensitive halting protocol. The unitary manipulation and con-trol in time and space for the atom is also a key step toward the realizationto solve efficiently the quantum search problem. An electromagnetic wave fieldsuch as a laser light field can manipulate and control in time and space notonly the center-of-mass motion and the internal electronic (or spin) motion ofan atom separately, but also it can create and control the coupling betweenthe center-of-mass motion and the internal motion of an atomic system. Thisis the theoretical fundament for the laser cooling of a neutral atomic ensemble[21, 23, 24] and the unitary decelerating and accelerating of a free atom [22].One large advantage to use a laser light field to manipulate an atom is thatthe space-selective and / or the internal-state-selective unitary operations ofthe atom can be realized easily. The STIRAP-based unitary decelerating andaccelerating processes could be a very useful double-photon method to coher-ently manipulate and control in time and space the center-of-mass motion, theinternal motion, and the coupling of both the motions of a moving free atom. Itusually uses a pair of Raman adiabatic laser light beams to couple the atomiccenter-of-mass motional state and internal electronic states (or spin polarizationstates) to realize the coupling between the two motions. The STIRAP-basedunitary decelerating and accelerating processes have been proposed to realizethe unitary time- and space-compressing processes which are necessary compo-nents of the quantum control process to simulate efficiently the state-insensitivereversible and unitary halting protocol [22]. On the other hand, as far as theGaussian wave-packet state of a free atom is concerned, the STIRAP-based3nitary decelerating and accelerating processes could generate a type of time-and space-dependent unitary propagators which can manipulate and controlthe center-of-mass position and momentum of the atomic Gaussian wave-packetstate. This important result will be shown in the paper.The basic STIRAP theory generally does not consider explicitly the effect ofthe center-of-mass motional momentum distribution of an atomic or molecularsystem on the STIRAP population transfer in the laser spectroscopy [16, 17,18], although the Doppler effect in these atomic and molecular systems has alsobeen considered suitably in the STIRAP experiments [3, 15]. This could bedue to that the center-of-mass motion of an atom or molecule is generally muchslower than the internal electronic motion of the atom or molecule and could beneglected in the basic STIRAP theory, and the STIRAP experimental settings[3, 15] are also favorable for the experiments to minimize the center-of-massmotional effect, for example, the STIRAP experiments may use a pair of co-propagating Raman laser light beams. However, a pure-state quantum systemsuch as a single freely moving atom may be in a superposition of the center-of-mass motional momentum states which may has a broad momentum distribu-tion. It is generally hard to realize the complete state transfer in a quantumsystem with a broad momentum distribution by the standard STIRAP method.One therefore must consider the effect of the momentum distribution of a su-perposition state of the quantum system on the STIRAP population or statetransfer. In concept the momentum distribution of a momentum superpositionstate of a pure-state quantum system is essentially different from the conven-tional momentum distribution of a quantum ensemble, the latter is a statisticaldistribution of momentum of the particles which form the quantum ensemble.But the effect of the momentum distribution on the STIRAP state transfer issimilar for the two types of quantum systems according to the unitary quantumdynamical principle that both a closed quantum system and its ensemble obeythe same unitary quantum dynamics [22] (it seems that a quantum system inthe presence of an external electromagnetic field is not a closed quantum system,but in theory such a quantum system may be treated conditionally as if it is aclosed quantum system, as shown in the section 11 in Ref. [40]). It has beeninvestigated in theory how the momentum distribution of an atomic ensembleaffects the population or state transfer in the atomic laser cooling based on thevelocity-selective coherent population trapping [5, 19, 20] which also uses theRaman laser light beams, the STIRAP-based momentum transfer in the coldatomic interference experiments [10, 12], and the conventional Raman-laser-light-based cold atomic interference experiments [11], but these investigationsare usually either qualitative or numerical. It is important to investigate ana-lytically and quantitatively how a superposition of the momentum states affectsthe STIRAP state transfer in a pure-state quantum system when the STIRAPmethod is used to perform the state transfer or the unitary operation in quan-tum computation. This paper is devoted to such a theoretic investigation: howa superposition of the momentum states affects the STIRAP state transfer ina single freely moving atom. According to quantum mechanics a freely movingatom in the presence of the STIRAP pulse sequence may be described by the4omplete set of the product states of both the center-of-mass motional statesand the internal states (the electronic states or spin polarization states) of theatom. The atomic center-of-mass motional state may be a superposition of theatomic momentum eigenstates. When the atom is transferred from one internalstate to another by a STIRAP pulse sequence, the transfer efficiency is gen-erally dependent upon the atomic center-of-mass motional state. The purposeto investigate this dependence is to understand how the momentum distribu-tion of the center-of-mass motional state affects the transfer efficiency and thendesign a better STIRAP pulse sequence so as to achieve a complete STIRAPstate transfer over the whole effective momentum distribution of the atomiccenter-of-mass motional state.In order to manipulate and control the halting-qubit atom in time and spacein the quantum control process [22] it is necessary to investigate the time evo-lution process of the halting-qubit atom in the STIRAP-based unitary decel-erating and accelerating processes. In order to investigate quantitatively howa superposition of the momentum states affects the STIRAP state transfer fora single freely moving atom in the STIRAP-based decelerating or acceleratingprocess it is also necessary to calculate the time evolution process of the atomin the presence of the Raman laser light field. For example, it needs to solvethe unitary dynamical equation to set up a general adiabatic condition for thebasic STIRAP decelerating and accelerating processes for the atom. It seemsthat a single freely moving atom in the presence of an external electromagneticfield such as the Raman laser light field is a simple quantum system, but thetime evolution process of the atomic system is not so simple, partly because theinteraction between the atom and the external electromagnetic field is usuallytime-dependent, and on the other hand, because the atomic internal motionand center-of-mass motion as well as the coupling of the two motions inducedby the external electromagnetic field need to be considered explicitly and si-multaneously in a theoretical treatment. The time evolution process of a freeatom in the presence of a laser light field becomes so complex that it is generallydifficult to solve exactly the unitary dynamical equation of the atomic system.It is also quite inconvenient even to use an approximation method to solve theunitary dynamical equation with a high accuracy. The STIRAP-based unitarydecelerating and accelerating processes for a free atom are relatively simple, be-cause these decelerating and accelerating processes are adiabatic and the Ramanlaser light beams of the STIRAP decelerating and accelerating pulse sequencesaffect only the three chosen atomic internal states. These special points maysimplify greatly the investigation of the time evolution process of the atom inthe STIRAP decelerating and accelerating processes. Because the time evolu-tion process of the atom involves only the three chosen internal states of theatom, it may be investigated conveniently in the atomic three-internal-statesubspace. On the other hand, it is well known that a unitary process of a freeatom absorbing or emitting a photon has to obey the energy, momentum, andangular momentum conservation laws. The energy, momentum, and angularmomentum conservation laws put a restriction on the time evolution processfor the free atom during the STIRAP-based unitary decelerating and accelerat-5ng processes. These laws lead to that the atomic motional momentum cannotbe changed arbitrarily by these Raman laser light beams in the STIRAP uni-tary decelerating and accelerating processes, but rather it can be changed onlywithin the Raman-laser-light-induced momentum state subspace [5, 19]. Thenthe time evolution process of a free atom may be investigated conveniently in theRaman-laser-light-induced momentum state subspace in the STIRAP unitarydecelerating and accelerating processes. Here the unitarity of these processes isemphasized again. This greatly simplifies the evaluation for the time evolutionprocess.In this paper the basic STIRAP theory [15, 16, 17, 18] has been developedto study and construct the STIRAP-based unitary decelerating and accelerat-ing processes for a free atom by combining the quantum superposition principle[25] and the energy, momentum, and angular momentum conservation laws forthe atomic photon absorption and emission processes [5, 19]. This research isfocused on investigating analytically and quantitatively the effect of the mo-mentum distribution of a superposition of momentum states of the atom on theSTIRAP state transfer in these processes. Both the ideal and the real adia-batic condition for the basic STIRAP decelerating and accelerating processesare derived analytically. One important result of the paper is to confirm theoret-ically the time- and space-compressing processes of the quantum control process[22], which are realized by the STIRAP unitary decelerating and acceleratingprocesses, with the help of the STIRAP state-transfer theory and the unitaryquantum dynamics. The Hamiltonian for a single atom in the presence of the Ramanlaser light beams
The STIRAP-based laser cooling processes and the unitary decelerating andaccelerating processes are generally involved in the center-of-mass motion, theinternal electronic motion, and the interaction between the two motions in anatomic system. A complete theoretical description for these processes needconsider the atomic center-of-mass motion, the internal electronic motion, andthe coupling of both the motions of the atom. When an atom is irradiated byan externally applied electromagnetic field such as the Raman laser light beamsin the STIRAP experiments, the total Hamiltonian for the physical systemconsisting of the atom and the electromagnetic field may be generally writtenas [25] H = H a + H rad + H int (1)where H a is the atomic Hamiltonian in the absence of the externally appliedelectromagnetic field, H a = X k m k p k + V a , (2) H rad the Hamiltonian for the electromagnetic field which may be written as H rad = 18 π Z d x ( E.E ∗ + B.B ∗ ) , H int the interaction between the atom and the externally applied electro-magnetic field, H int = − X k e k m k c p k .A ( r k , t ) + X k e k m k c A ( r k , t ) . The atomic Hamiltonian H a describes both the atomic center-of-mass motionand the internal electronic (or spin) motion of the atom, while the interac-tion H int induced by the external electromagnetic field creates the couplingbetween the atomic center-of-mass motion and the internal electronic motion.The external electromagnetic field usually is weak in the conventional STIRAPexperiments and the interaction H int between the atom and the electromagneticfield could be considered as a perturbation.It is usually inconvenient to use directly the total Hamiltonian of Eq. (1) todescribe the STIRAP-based decelerating and accelerating processes, althoughit is the most exact to use the total Hamiltonian to treat these processes the-oretically. Without losing generality one may use a simpler form of the totalHamiltonian of Eq. (1) to describe clearly the unitary decelerating and accelerat-ing processes. It is well known that the semiclassical theory of electromagneticradiation has been extensively used to describe the laser spectroscopy in theatomic and molecular systems [1, 16, 17, 18], the atomic coherent laser-coolingprocesses [20, 21, 23], and the atomic quantum interference experiments [10, 11,12]. If the semiclassical theory is also reasonable for the unitary deceleratingand accelerating processes, then the Hamiltonian H rad of the electromagneticfield itself may be omitted from the total Hamiltonian of Eq. (1). Thoughthe semiclassical theory of electromagnetic radiation generally can not describeexactly the spontaneous emission in an atomic system [1, 25] and especially thespontaneous emission in the long-time atomic laser cooling process [26], it maybe suited to describe the STIRAP-based unitary decelerating and acceleratingprocesses as these unitary STIRAP processes can avoid the atomic spontaneousemission by setting the suitable experimental parameters. Furthermore, if theelectric dipole approximation for the atomic system is also reasonable, then onemay use conveniently the electric dipole interaction H d to replace the interac-tion H int to describe the unitary decelerating and accelerating processes. Theelectric dipole interaction H d may be written as H d = − D.E ( x, t ) (3)where D is the atomic electric dipole moment and E ( x, t ) is the electric field ofthe externally applied electromagnetic field, and the coordinate x in the electricfield E ( x, t ) is the center-of-mass position of the atom. The electric dipole ap-proximation is reasonable when the wave length of the external electromagneticfield is much larger than the atomic dimension under study so that the atommay be considered as a point particle — a point electric dipole — in the electro-magnetic field [1]. It has been shown that in the electric dipole approximationthe electric-dipole Hamiltonian H d is really equivalent to the interaction H int up to a gauge transformation which is also a unitary transformation [1, 27].7he electric dipole approximation is very popular in the theoretical descriptionof a variety of laser light and matter interactions. For example, one generallyuses the electric-dipole approximation to deal with theoretically the dynamicalprocess of atomic laser cooling in an atomic ensemble [1, 4, 5, 10, 19, 20, 26].Now in the semiclassical theory of electromagnetic radiation and in the elec-tric dipole approximation the total Hamiltonian (1) of the atom in the externalelectromagnetic field is reduced to the form H = P / (2 M ) + V ( x ) + H ( r ) + H d (4)where the sum of the first three terms is just the atomic Hamiltonian H a ofEq. (2). The center-of-mass Hamiltonian H cm = P / (2 M ) + V ( x ) describes theatomic center-of-mass motion, while the internal Hamiltonian H ( r ) describesthe internal electronic (or spin) motion of the atom. In the center-of-massHamiltonian H cm the term H K = P / (2 M ) with the atomic mass M and mo-mentum P is the atomic kinetic energy and the term V ( x ) the atomic potentialenergy in an external potential field. In the unitary decelerating and acceler-ating processes of a free atom the external potential energy V ( x ) of the atomis zero, i.e., V ( x ) = 0 . If there is an external electric (or magnetic) field duringthese processes, then the external potential energy V ( x ) could not be zero. Forexample, the potential energy V ( x ) = 0 when an atom in a harmonic potentialwell is applied by an external electromagnetic field. Generally, both the center-of-mass kinetic energy operator P / (2 M ) and the internal Hamiltonian H ( r )are not commutable with the electric dipole interaction H d . The Hamiltonian H of Eq. (4) still needs to be further simplified so that it can describe con-veniently the standard three-state STIRAP experiment. This simplification isbased on the facts that the externally applied electromagnetic field in the STI-RAP experiments can only affect some specific internal electronic energy levelsof the atom and the internal electronic states are discrete. In the following onlyone-dimensional center-of-mass motion of the atom is considered.Because there are the center-of-mass motion, the internal electronic motion,and even the coupling between the two motions of an atom in the STIRAPprocesses, one must use simultaneously both the atomic center-of-mass motionalstates and the internal electronic states of the atom to describe exactly theSTIRAP processes. Suppose that the wave functions | ψ P ( x ) i and | ψ j ( r ) i are theeigenstates of the center-of-mass Hamiltonian H cm and the internal Hamiltonian H ( r ), respectively, H cm | ψ P ( x ) i = E P | ψ P ( x ) i , H ( r ) | ψ j ( r ) i = E j | ψ j ( r ) i , where E P and E j are the eigen-energy of the center-of-mass motional state | ψ P ( x ) i and the internal state | ψ j ( r ) i , respectively. For the STIRAP processesof a free atom the eigen-energy E P is equal to the atomic center-of-mass ki-netic energy P / M since the potential energy V ( x ) = 0 and H cm | ψ P ( x ) i = H K | ψ P ( x ) i = P / (2 M ) | ψ P ( x ) i . According to quantum mechanics [25] the com-plete set {| ψ P ( x ) i| ψ j ( r ) i} of the product states of the center-of-mass motionalstates and the internal states of the atom can be used to describe completely8oth the center-of-mass motion and the internal electronic motion as well as thecoupling of the two motions of the atom. Now the total wave function | Ψ( x, r, t ) i of the atom may be generally expressed as a linear combination of these productstates [25], | Ψ( x, r, t ) i = X j,P a ( j, P, t ) | ψ P ( x ) i| ψ j ( r ) i where a ( j, P, t ) is an expansion coefficient. In the decelerating and acceleratingprocesses an atom may be in a given internal state, while its center-of-massmotional state is a superposition state. Then in this case the total wave functionof the atom may be expanded as | Ψ( x, r, t ) i = X j,p a ( j, p, t ) | ψ p ( x, t ) i| ψ j ( r ) i (5)where the wave function | ψ p ( x, t ) i is a superposition of the atomic center-of-massmotional states. For example, the center-of-mass motional state | ψ p ( x, t ) i maybe a wave-packet state of the halting-qubit atom in the quantum control process,which was denoted as | CM, R i in the previous paper [22]. In the deceleratingand accelerating processes of a free atom any center-of-mass motional state | ψ p ( x, t ) i of the atom may be expanded in terms of the complete eigenstate set {| ψ P ( x ) i} [25], | ψ p ( x, t ) i = X P a ( p, P ) exp( − i ℏ P M t ) | ψ P ( x ) i . (6)Note that the state | ψ P ( x ) i is also an eigenstate of the atomic center-of-massmomentum operator P and in one-dimensional case it may be written as | ψ P ( x ) i ≡ | P i = 1 √ π exp( iP x/ ℏ ) . (7)The momentum wave function | P i represents that the atom moves along thedirection + x with the center-of-mass motional velocity P/M .In what follows it is supposed that the halting-qubit atom has a three-level Λconfiguration for the three-state STIRAP experiments. In the STIRAP experi-ment the external electromagnetic field can have a real effect only on the specificthree-state subspace {| ψ ( r ) i , | ψ ( r ) i , | ψ ( r ) i} of the internal electronic statesof the atom. This three-state subspace will further simplify the Hamiltonianof Eq. (4) as the time evolution process of the internal states of the atom inthe STIRAP experiment is confined in the three-state subspace. Those internalstates of the atom outside the three-state subspace will not be affected by theexternal electromagnetic field and are not considered in the STIRAP experi-ment. Thus, it is sufficient to use the internal Hamiltonian ( H ( r )) projectiononto the three-state subspace to describe the three-state STIRAP experiment.Since the internal states {| ψ j ( r ) i} are the eigenstates of the internal Hamilto-nian H ( r ), this projection Hamiltonian onto the three-state subspace may be9iven by H ( r ) = E | ψ ( r ) ih ψ ( r ) | + E | ψ ( r ) ih ψ ( r ) | + E | ψ ( r ) ih ψ ( r ) | . (8)On the other hand, the electric dipole interaction H d of Eq. (3) can also besimplified further in the three-state subspace. In the STIRAP-based decelerat-ing and accelerating processes the total external electromagnetic field generallyconsists of a pair of counterpropagating electromagnetic fields, which may beamplitude- and phase-modulation Raman laser light beams. The total electricfield for the two counterpropagating linearly polarized Raman laser light beamsmay be expressed as E ( x, t ) = 12 E L ( t ) exp[ i ( − k L .x − ω L t )]+ 12 E L ( t ) exp[ i ( k L .x − ω L t )] + C.C. (9)where
C.C. stands for the complex (or hermite) conjugate of the first two terms.Notice that the three internal states {| ψ k ( r ) i} have quite different energy eigen-values. The first Raman laser light beam with the electric field E L ( t ) couplesonly the two internal states | ψ ( r ) i and | ψ ( r ) i of the atom, while the secondwith the electric field E L ( t ) connects only the two internal states | ψ ( r ) i and | ψ ( r ) i . The frequency difference | ω L − ω L | should be near the resonancefrequency of the two atomic internal energy levels | ψ ( r ) i and | ψ ( r ) i . The firstRaman laser light beam usually is named the pumping laser pulse and the sec-ond the Stokes laser pulse in the laser spectroscopy [15]. If among the threeinternal states the two states | ψ ( r ) i and | ψ ( r ) i which are usually the groundstates have the same energy eigenvalues, then one may use a pair of σ + and σ − circularly polarized laser light beams [19] to replace the present two Ramanlaser light beams, one circularly polarized laser light beam coupling only thetwo internal states | ψ ( r ) i and | ψ ( r ) i and another connecting only the twointernal states | ψ ( r ) i and | ψ ( r ) i . Then in the three-state (internal) subspacethe electric dipole interaction H d of Eq. (3) may be written as, in the rotatingwave approximation [1], H d = ℏ Ω ( t ) exp { i ( − k L .x − ω L t ) }| ψ ( r ) ih ψ ( r ) | + ℏ Ω ( t ) exp { i ( k L .x − ω L t ) }| ψ ( r ) ih ψ ( r ) | + C.C. (10)where the Rabi frequencies for the two Raman laser light beams are defined asΩ ( t ) = − h ψ ( r ) | D.E L ( t ) | ψ ( r ) i , Ω ( t ) = − h ψ ( r ) | D.E L ( t ) | ψ ( r ) i . The electromagnetic field of the Raman laser light beams is usually weak inthe conventional STIRAP experiments, so that the effect of the electromagnetic10eld on the atom could be considered as a perturbation. Hence the rotatingwave approximation is reasonable. On the other hand, if each Raman laserlight beams in the STIRAP experiment is replaced with a pair of the laserlight beams with the orthogonal electric field vectors and the suitable phases[38] or one circularly polarized laser light beam [19], then the rotating-waveapproximation may be eliminated and hence the electric dipole interaction (10)may be constructed exactly. The total Hamiltonian of Eq. (4) associated withthe electric dipole interaction H d of Eq. (10) and the internal Hamiltonian H ( r )of Eq. (8) and the product basis set {| ψ P ( x ) i| ψ j ( r ) i} may be used convenientlyto describe the STIRAP-based unitary decelerating and accelerating processesof a free atom. The transition matrix elements of the electric dipole interaction H d can be calculated in the product basis set, W ( j ′ , P ′ ; j, P ) = h ψ j ′ ( r ) |h ψ P ′ ( x ) | H d | ψ P ( x ) i| ψ j ( r ) i . These matrix elements are not zero only when both the internal states | ψ j ( r ) i and | ψ j ′ ( r ) i are in the three-state (internal) subspace. They are also subjectedto the constraint of the energy, momentum, and angular momentum conserva-tion laws for the atomic photon absorption and emission process in the STIRAPdecelerating and accelerating processes. This is an instance of the velocity-selective rules which have been used in the atomic laser cooling processes [5,19]. Below it is shown how the energy and momentum conservative laws have aconstraint on the electric dipole transition matrix elements { W ( j ′ , P ′ ; j, P ) } inthe three-state STIRAP experiments of the STIRAP-based unitary deceleratingand accelerating processes.In the STIRAP experiments the internal states | ψ ( r ) i and | ψ ( r ) i of thethree-state subspace usually are taken as the hyperfine ground electronic states | g i and | g i of an atom, while the internal state | ψ ( r ) i may be taken as someexcited state | e i of the atom. For example, the two internal states | g i and | g i may be the hyperfine ground electronic states 3 S / ( F = 1) and 3 S / ( F = 2) of sodium atom ( N a ), respectively, while the excited state | e i may bethe excited electronic state 3 P / ( F = 2) of the sodium atom. Suppose that atthe initial time in the STIRAP experiment the atom is in the ground internalstate | g i and the center-of-mass momentum state | P ′ i = ( √ π ) − exp( iP ′ x/ ℏ ) , which also means that the atom is in the product state | P ′ i| g i and it travelsalong the direction + x with the velocity v ′ = P ′ /M . Now a laser light fieldpropagating along the direction − x is applied to the atom. Then the movingatom may absorb a photon from the laser light field if the frequency ( ω = k c )of the laser light field is just equal to the transition frequency of the atom inmotion between the ground internal state | g i and the excited state | e i after theDoppler effect is taken into account. After the atom absorbs a photon from thelaser light field, the atomic motional momentum becomes P ′ − ℏ k according tothe momentum conservation law and hence the atom is decelerated by ℏ k /M .Since the atomic internal energy levels are discrete, the momentum change ofthe moving atom is also discrete after the atom absorbs a photon. This meansthat when the atom is excited to the internal state | e i , its motional momentumcan not take an arbitrary value but it has to be P ′ − ℏ k due to the fact11hat the atomic optical absorption process obeys the energy and momentumconservation laws [19]. After the atom absorbs a photon it jumps to the excitedstate | e i from the ground state | g i and its initial motional state | P ′ i is changedto | P ′ − ℏ k i and hence the atom is in the product excited state | P ′ − ℏ k i| e i . This is just the atomic decelerating process based on the optical absorptionmechanism [23, 24]. The atom in the product excited state | P ′ − ℏ k i| e i maybe further decelerated by another laser light field. This laser light field travelsalong the same direction + x as the atom and its frequency ( ω = k c, k = k )is equal to the transition frequency of the moving atom between the groundinternal state | g i and the excited state | e i after the Doppler effect is takeninto account. Thus, this laser light field may stimulate the atom in the excitedinternal state | e i to jump to the ground internal state | g i . Because the energydifference between the two internal states | g i and | e i is quite different from thatone between the two internal states | g i and | e i , this laser light field will notaffect the transition between the two internal states | g i and | e i . Likewise, thefirst laser light field does not affect the transition between the two internal states | g i and | e i . Different from the first decelerating process this atomic deceleratingprocess is based on the stimulated optical emission mechanism [23, 24]. An atomin an excited state may jump to the ground state when it is stimulated by anexternal laser light field [25]. When the atom in the excited state | e i jumps to theground state | g i , it may emit coherently a photon to the laser light field. Sincethe atomic motion direction is the same as the propagation direction of the laserlight field, the atom really sends part of its motional momentum to the laserlight field and hence is decelerated in the stimulated transition process from theexcited state | e i to the ground state | g i . This part of motional momentum isjust ℏ k according to the momentum conservation law and accordingly the atomis decelerated by ℏ k /M. Thus, after the stimulated optical emission process theatom is in the ground internal state | g i and has to be in the motional state | P ′ − ℏ k − ℏ k i , that is, the atom is in the product state | P ′ − ℏ k − ℏ k i| g i . Boththe atomic optical absorption and emission processes are required to be unitaryhere, as pointed out before [22]. The unitary decelerating process based on thethree-state STIRAP process just consists of the reversible optical absorption andemission processes mentioned above, where the two Raman laser light beams areadiabatic and usually counterpropagating. This basic STIRAP-based unitarydecelerating process may be expressed in an intuitive form | P + ℏ k i| g i → | P i| e i → | P − ℏ k i| g i (11)Here for convenience in the later discussion the atomic motion momentum P ′ isdenoted as P + ℏ k . Thus, a conventional three-state STIRAP pulse sequencemay be really used as a basic unitary decelerating sequence if the two Ramanlaser light beams of the STIRAP pulse sequence are arranged suitably suchthat the moving atom is decelerated consecutively by the two Raman laser lightbeams. Obviously, the inverse process of the STIRAP-based unitary deceler-ating process (11) may be used to accelerate the atom in motion. However, itis more convenient to use directly the reversible optical absorption and emis-sion processes to accelerate an atom in motion and this may be achieved by12etting suitably the parameters of the two Raman laser light beams of the STI-RAP pulse sequence. In the STIRAP-based unitary accelerating process thefirst laser light field that induces the optical absorption process travels alongthe motional direction (+ x ) of the atom, while the second laser light field thatstimulates the atomic optical emission process propagates along the oppositedirection ( − x ) to the moving atom. When the atom in the initial product state | P ′ i| g i is excited to the product state | P ′ + ℏ k ′ i| e i by the first laser lightfield, it absorbs a photon from the laser light field and is accelerated by ℏ k ′ /M. When the atom in the excited state | P ′ + ℏ k ′ i| e i jumps to the ground state | P ′ + ℏ k ′ + ℏ k ′ i| g i under the stimulation of the second laser light field, it emitsa photon to the laser light field and is accelerated further by ℏ k ′ /M. Therefore,the basic STIRAP-based unitary accelerating process may be expressed in anintuitive form ( P ′ = P − ℏ k ′ ) | P − ℏ k ′ i| g i → | P i| e i → | P + ℏ k ′ i| g i . (11a)In what follows only the basic STIRAP-based unitary decelerating process (11)is treated explicitly. In an analogous way, one can also deal with the basicSTIRAP-based unitary accelerating process (11a).It follows from the basic decelerating sequence (11) that the time evolutionprocess of the atom in the unitary decelerating process (11) is restricted withinthe three-state (product state) subspace {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} fora given atomic motion momentum P . Then during the STIRAP-based unitarydecelerating process (11) the total wave function | Ψ( x, r, t ) i of the atom at anyinstant of time t can be expanded in the three-state (product state) subspace [5,10, 12, 19, 20], according to the superposition principle in quantum mechanics[25], | Ψ( x, r, t ) i = X P ρ ( P ) { A ( P, t ) | P + ℏ k i| g i + A ( P, t ) | P i| e i + A ( P, t ) | P − ℏ k i| g i} . (12)where the sum over the momentum P is due to the fact that the atom may bein a superposition of momentum states, as can be seen in Eq. (6), ρ ( P ) is thetime-independent amplitude which has the physical meaning that | ρ ( P ) | is theprobability in the superposition to find the atom in the three-state subspace {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} labelled by the momentum P, and thetime-dependent amplitudes { A k ( P, t ) } satisfies the normalization condition: | A ( P, t ) | + | A ( P, t ) | + | A ( P, t ) | = 1 . (13)The amplitude ρ ( P ) is time-independent because the two Raman laser lightbeams induce a change only within the three-state subspace {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} for each given momentum P during the unitary decelerat-ing process [19]. If at the initial time t the atom is in the ground internal state | g i and in a wave-packet motional state, then the initial wave packet state ofthe atom may be expanded as | Ψ( x, r, t ) i = X P ′ ρ ( P ′ ) | P ′ i| g i . (12a)13bviously, here the coefficients A ( P, t ) = 1 and A ( P, t ) = A ( P, t ) = 0 , which can be deduced from Eqs. (12) and (12a) with P ′ = P + ℏ k , while | ρ ( P ′ ) | is the probability to find the atom in the three-state subspace {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} . As an example, the initial state | Ψ( x, r, t ) i may betaken as the Gaussian wave-packet motional state of the halting-qubit atom inthe right-hand potential well of the double-well potential field in the quantumcontrol process [22]. The three-state (product state) subspace and the basic de-celerating process (11) show that only special dipole transition matrix elements { W ( j ′ , P ′ ; j, P ) } can take nonzero values, that is, for a given momentum P there are only four matrix elements to take nonzero value: W ( e, P ; g , P + ℏ k ), W ( e, P ; g , P − ℏ k ) , W ( g , P + ℏ k ; e, P ), and W ( g , P − ℏ k ; e, P ) . For the ba-sic STIRAP decelerating process (11) the total electric field of the two Ramanlaser light beams can be explicitly obtained from equation (9) by setting theparameter sets: ( E L ( t ) , k L , ω L ) = ( E ( t ) , k , ω ) and ( E L ( t ) , k L , ω L ) =( E ( t ) , k , ω ), where the first Raman laser light beam ( E ( t ) , k , ω ) couplesthe two internal states | g i and | e i and its propagating direction is opposite tothe motional direction of the atom, while the the second beam ( E ( t ) , k , ω )connects the two internal states | g i and | e i and it travels along the motionaldirection (+ x ) of the atom. Here suppose that the energy difference (measuredin frequency unit) between the two ground internal states | g i and | g i is muchlarger than the detunings of the two Raman laser light beams. Now these fournonzero electric-dipole-transition matrix elements for the basic decelerating pro-cess (11) can be obtained with the help of the total electric field of Eq. (9) withthese parameter settings, the electric dipole interaction H d of Eq. (10), and themomentum eigenstates of Eq. (7) as well as their orthogonalizations, W ( t ) = W ∗ ( t ) = h e |h P | H d | P + ℏ k i| g i = ℏ Ω ( t ) exp( − iω t ) ,W ( t ) = h g |h P + ℏ k | H d | P i| e i ,W ( t ) = W ∗ ( t ) = h e |h P | H d | P − ℏ k i| g i = ℏ Ω ( t ) exp( − iω t ) ,W ( t ) = h g |h P − ℏ k | H d | P i| e i . Here the star ⋆ stands for the complex conjugate and the Rabi frequencies Ω ( t )and Ω ( t ) are defined in Eq. (10) with the states: | ψ ( r ) i = | g i , | ψ ( r ) i = | g i , and | ψ ( r ) i = | e i .Now the time evolution process of the atom in the presence of the Ramanlaser light beams in the basic decelerating process (11) is described by the time-dependent Schr¨odinger equation: i ℏ ∂∂t Ψ( x, r, t ) = H ( t )Ψ( x, r, t ) . (14)Here the total Hamiltonian H ( t ) is given by Eq. (4), in which V ( x ) = 0 and H ( r ) and H d are given by Eq. (8) and (10), respectively, while the wave func-tion | Ψ( x, r, t ) i is given by Eq. (12). By using the four nonzero electric-dipole-transition matrix elements and the orthonormalization of the momentum eigen-states of Eq. (7) the Schr¨odinger equation (14) can be reduced to a three-state14chr¨odinger equation for a given momentum P, which may be written in thematrix form i ℏ ∂∂t A ( P, t ) A ( P, t ) A ( P, t ) = ˆ H ( P, t ) A ( P, t ) A ( P, t ) A ( P, t ) (14a)where the three-state vector ( A ( P, t ) , A ( P, t ) , A ( P, t )) T (here T stands forthe vector transpose) satisfies the normalization of Eq. (13) and the reducedHamiltonian ˆ H ( P, t ) is a 3 × − dimensional Hermitian matrix,ˆ H ( P, t ) = ( P + ℏ k ) M + E W ∗ ( t ) 0 W ( t ) P M + E W ( t )0 W ∗ ( t ) ( P − ℏ k ) M + E . Here the three basis vectors | i = (1 , , T , | i = (0 , , T , and | i = (0 , , T of the three-state vector space { ( A ( P, t ) , A ( P, t ) , A ( P, t )) T } stand for thethree basis product states | P + ℏ k i| g i , | P i| e i , and | P − ℏ k i| g i of the originalthree-state subspace {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} , respectively. Thisreduced three-state Schr¨odinger equation can describe completely the three-state ST IRAP experiments just like the original Schr¨odinger equation (14).By making a unitary transformation on the three-state vector in Eq. (14a) [20,12, 18, 25]: ¯ A ( P, t ) = exp[ i ℏ ( ( P + ℏ k ) M + E ) t ] A ( P, t ) , (15a)¯ A ( P, t ) = exp[ i ℏ ( P M + E ) t ] A ( P, t ) , (15b)¯ A ( P, t ) = exp[ i ℏ ( ( P − ℏ k ) M + E ) t ] A ( P, t ) , (15c)the Schr¨odinger equation (14a) is further reduced to the form i ℏ ∂∂t ¯ A ( P, t )¯ A ( P, t )¯ A ( P, t ) = H ( P, t ) ¯ A ( P, t )¯ A ( P, t )¯ A ( P, t ) . (16)Now the Hamiltonian H ( P, t ) is a traceless Hermitian matrix, H ( P, t ) = W ∗ ( P, t ) 0¯ W ( P, t ) 0 ¯ W ( P, t )0 ¯ W ∗ ( P, t ) 0 , where the time- and momentum-dependent complex parameters ¯ W ( P, t ) and¯ W ( P, t ) are given respectively by¯ W ( P, t ) = ℏ Ω ( t ) exp {− i [ 2 P k + ℏ k M − ( ω − ω )] t } , ¯ W ( P, t ) = ℏ Ω ( t ) exp {− i [ − P k + ℏ k M − ( ω − ω )] t } , ℏ ω = E − E and ℏ ω = E − E . Suppose that the Raman laser lightbeams are amplitude- and phase-modulating. Then the Rabi frequencies for thetwo Raman laser light beams (the pumping pulse (Ω p ( t )) and the Stokes pulse(Ω s ( t ))) are written asΩ ( t ) = Ω p ( t ) exp[ − iφ ( t )] , Ω ( t ) = Ω s ( t ) exp[ − iφ ( t )] , and the parameters in the Hamiltonian H ( P, t ) therefore are given by¯ W ( P, t ) = ℏ Ω p ( t ) exp[ − iα p ( P, t )] , ¯ W ( P, t ) = ℏ Ω s ( t ) exp[ − iα s ( P, t )] , where the phases α p ( P, t ) and α s ( P, t ) are dependent upon the momentum P , α p ( P, t ) = [ 2
P k + ℏ k M − ( ω − ω )] t + φ ( t ) ,α s ( P, t ) = [ − P k + ℏ k M − ( ω − ω )] t + φ ( t ) . Now the Hamiltonian H ( P, t ) can be rewritten in the explicit form H ( P, t ) = ℏ p ( t ) e iα p ( P,t ) p ( t ) e − iα p ( P,t ) s ( t ) e − iα s ( P,t ) s ( t ) e iα s ( P,t ) (17)This type of Hamiltonians often have been met in the three-state STIRAP ex-periments in the laser spectroscopy [4, 15, 17, 18] and in the atomic interferenceexperiments [12]. The three-state Schr¨odinger equation (16) and the tracelessHamiltonian (17) are the theoretical basis to design the Raman adiabatic pulsesof the STIRAP-based decelerating and accelerating processes. There is a specialpoint in the STIRAP-based unitary decelerating and accelerating processes thatthe Hamiltonian (17) is dependent upon the center-of-mass momentum of theatom besides the frequency offsets { ω k − ω k } ( k = 0 and 1). This is similar tothe situations of the STIRAP-based atomic laser cooling [20] and quantum inter-ference experiments [10, 12]. The effect of the frequency offsets on the STIRAPpopulation transfer has been examined in detail in the laser spectroscopy [18b].Though here considers only the pure-state quantum system of a single atominstead of an atomic ensemble, the atomic momentum distribution could havea great effect on the population transfer of the atom from an internal state toanother in the STIRAP decelerating and accelerating processes. This is becausethe atom may be in a superposition of the momentum eigenstates and hencehas a momentum distribution. For example, though a freely moving atom is ina Gaussian wave-packet state in coordinate space, it is also in a superposition ofthe momentum eigenstates of the atom in momentum space. Here the positionof the momentum P in the unitary decelerating and accelerating processes issimilar to that one of the frequency offsets in the STIRAP experiments of theconventional laser spectroscopy [18]. The frequency offsets (with respect to the16ransition frequencies of given atomic internal energy levels) could be set at willfor a single atom, since they are the parameters of the Raman laser light beams,while the superposition of momentum eigenstates of the atom (i.e. the atomicmomentum distribution) is an inherent property of the atomic motional state.Whether or not the Raman adiabatic pulses obtained from the Schr¨odingerequation (16) and the Hamiltonian (17) are suitable for the decelerating andaccelerating processes are dependent upon the atomic momentum distribution.The excitation bandwidth for a good STIRAP pulse sequence must be muchlarger than the effective spreading of the atomic momentum distribution.In the quantum control process [22] to simulate the reversible and unitaryhalting protocol and the quantum search process the halting-qubit atom needsto be decelerated and accelerated by the STIRAP-based unitary deceleratingand accelerating processes, respectively. As required by the quantum controlprocess, if the halting-qubit atom is completely in the ground internal state | g i ( | g i ) at the initial time in the decelerating or accelerating process, thenit must be converted completely into another ground internal state | g i ( | g i )at the end of the process. Because the conversion efficiency from the initialstate to the end state in these processes has a great effect on the performanceof the quantum control process, it is of crucial importance to achieve a highenough conversion efficiency in these processes. This is the first guidance todesign the STIRAP pulse sequences for the unitary decelerating and acceleratingprocesses. On the other hand, in order that a high conversion efficiency isachieved in the STIRAP experiments one must also consider the decoherenceeffect due to the atomic spontaneous emission when the atom is in the excitedinternal state. The atomic spontaneous emission becomes an important factorto cause the decoherence effect when the atom is in a short-lifetime excitedinternal state in the STIRAP experiments. Therefore, the atom should avoidbeing in the excited internal state during the STIRAP experiments. This maybe realized by setting the favorable detunings for the two Raman laser lightbeams. Since the three product states | P + ℏ k i| g i , | P i| e i , and | P − ℏ k i| g i are the eigenstates of the total atomic Hamiltonian H a of Eq. (2) and haveeigenenergy: ( P + ℏ k ) / (2 M ) + E , P / (2 M ) + E , and ( P − ℏ k ) / (2 M ) + E , respectively. Then the energy difference between the ground state | P + ℏ k i| g i and the excited state | P i| e i is given by ( E − E ) − P ℏ k /M − ℏ k / (2 M ) andthe energy difference between the ground state | P − ℏ k i| g i and the excitedstate | P i| e i is ( E − E ) + P ℏ k /M − ℏ k / (2 M ) . Since the Raman laser lightbeam with the carrier frequency ω couples the ground state | P + ℏ k i| g i andthe excited state | P i| e i , the detuning ∆ p of the beam is given by∆ p ( P ) = ( ω − ω ) − ( k M ) P − ℏ k M , while the detuning ∆ s of another Raman laser light beam with the carrierfrequency ω is ∆ s ( P ) = ( ω − ω ) + ( k M ) P − ℏ k M .
By using the two formulae one may set conveniently the detunings ∆ p and ∆ s
3. The basic differential equations for the STIRAP-based decel-erating and accelerating processes
The three-state STIRAP experiments have been studied extensively both intheory and experiment in the laser spectroscopy [15, 16, 17, 18]. These studiestell ones which conditions the complete population transfer can be achieved bythe STIRAP method among the energy levels of atomic and molecular systems.From the point of view of theory an important feature of the three-state STI-RAP experiments is that there is the special eigenstate of the Hamiltonian (17)that corresponds to the atomic trapping state [28]. This special eigenstate isindependent of the intermediate state [17], which is usually taken as the excitedinternal state of the atom in the STIRAP experiments. Under the adiabatic con-dition the complete state or population transfer through the special eigenstatecan occur from the initial state directly to the final state, while the intermedi-ate state is bypassed in the transfer process. Such an adiabatic state-transferprocess is particularly favorable for the STIRAP-based decelerating and accel-erating processes. This is because the atomic spontaneous emission could beavoided in the decelerating and accelerating processes when the excited state ofthe atom is bypassed in the STIRAP state-transfer processes. This also showsthat the semiclassical theory of electromagnetic radiation is suited to treat theSTIRAP experiments. On the other hand, from the experimental point of viewthe three-state STIRAP experiment needs to set suitably the experimental pa-rameters for the Raman laser light beams. The important thing in experiment isthe settings for the Rabi frequencies of the two Raman laser light beams and forthe pulse delay between the two Raman laser light beams [16]. The atomic sys-tem should first interact with the Stokes pulse and then with the pumping pulse,and an appropriate overlapping between the two Raman laser light beams is alsorequired in experiment [3, 4, 15, 16, 17, 18]. These requirements are generallyrelated to the adiabatic condition of the three-state STIRAP experiment.The special point for the STIRAP decelerating and accelerating processesis that one must consider the atomic momentum distribution when a generaladiabatic condition is set up for these processes. According to the adiabatictheorem and the adiabatic approximation method in quantum mechanics [29,30] one should first calculate the three eigenvectors and eigenvalues of the in-18tantaneous Hamiltonian H ( P, t ) of Eq. (17), H ( P, t ) | g ( P, t ) i = E ( P, t ) | g ( P, t ) i . (18)The three eigenvectors {| g ( P, t ) i} and their eigenvalues { E ( P, t ) } of the Hamil-tonian H ( P, t ) are usually named the adiabatic eigenvectors and eigenvalues,respectively. Notice that the three basis vectors of the three-state vector space { ( A ( P, t ) , A ( P, t ) , A ( P, t )) T } are | i = (1 , , T , | i = (0 , , T , and | i =(0 , , T , respectively. The three basis vectors really stand for the three basisproduct states | P + ℏ k i| g i , | P i| e i , and | P − ℏ k i| g i of the original three-statesubspace for each given momentum P , respectively. Any one of the three eigen-vectors of the Hamiltonian H ( P, t ) may be expanded in terms of the three basisvectors. By the explicit Hamiltonian of Eq. (17) one can obtain one of the threeeigenvectors [15, 17, 18], | g ( P, t ) i = exp[ − iγ ( t )] { cos θ ( t ) | i − sin θ ( t ) exp[ − i ( α p ( P, t ) − α s ( P, t ))] | i} (19a)where the phase γ ( t ) is a global phase, the mixing angle θ ( t ) and the Rabifrequency Ω( t ) are defined respectively bycos θ ( t ) = Ω s ( t )Ω( t ) , sin θ ( t ) = Ω p ( t )Ω( t ) , and Ω( t ) = q Ω p ( t ) + Ω s ( t ) . The eigenvalue corresponding to the eigenvector | g ( P, t ) i is E ≡ ℏ ω = 0 . The eigenvector | g ( P, t ) i is special in that it does not contain the intermediateeigenvector | i which contains the excited internal state | e i of the atom. It isthe so-called atomic trapping state [28]. The adiabatic population transfer isachieved through the eigenvector | g ( P, t ) i in all the three-state STIRAP ex-periments [15, 17]. Therefore, when the initial state | i is transferred to thefinal state | i through the eigenvector | g ( P, t ) i in the STIRAP state-transferprocess, the intermediate state | i is not involved and hence is bypassed. Sincethe phase difference α p ( P, t ) − α s ( P, t )) in the eigenvector | g ( P, t ) i is depen-dent upon the atomic motional momentum P the adiabatic state transfer in theSTIRAP experiments is really affected by the atomic momentum distribution.In the laser spectroscopy the similar trapping state is obtained [4, 15, 17, 18]but that state is not dependent upon the momentum P . Thus, there is not anytheoretical problem involved in the effect of the momentum distribution on theadiabatic state transfer in the laser spectroscopy. Of course, in the laser spec-troscopy the Doppler effect usually is considered in the STIRAP experimentsof those quantum systems such as an atomic or molecular beam [15]. However,in the atomic laser cooling [5, 19, 20, 21], quantum interference experiments[10, 11, 12, 13, 14], and the atomic decelerating and accelerating processes theatomic momentum distribution generally needs to be considered explicitly. Theother two adiabatic eigenstates of the Hamiltonian of Eq. (17) are given by [15,17, 18] | g ± ( P, t ) i = 1 √ − iδ ( t )] { sin θ ( t ) | i ∓ exp[ − iα p ( P, t )] | i
19 cos θ ( t ) exp[ − i ( α p ( P, t ) − α s ( P, t ))] | i} (19b)and their corresponding eigenvalues are E ± ≡ ℏ ω ± = ∓ ℏ Ω( t ) . Here the phase δ ( t ) is also a global phase. The phase difference α p ( P, t ) − α s ( P, t ) in the adia-batic eigenvectors is given by α p ( P, t ) − α s ( P, t ) = φ ( t ) − φ ( t )+ {− [ ω − ω ] + [ ω − ω ] + P ( k + k ) M + ℏ k − ℏ k M } t, (20)and there is a relation between the phase difference and the detunings: α p ( P, t ) − α s ( P, t ) = [ φ ( t ) − φ ( t )] + [∆ s ( P ) − ∆ p ( P )] t. The phase difference is dependent upon both the frequency offsets ( ω − ω )and ( ω − ω ) and also the momentum P. It could be considered that the term( k + k ) P/M in the phase difference is generated by the Doppler effect. A largeDoppler effect is not favorable for the decelerating and accelerating processes.Assume that the atomic motional state is a wave-packet state with a finite wave-packet spreading. Then in order to minimize the effect of the Doppler-effect term( k + k ) P/M on the STIRAP state transfer in the decelerating and acceler-ating processes one must choose suitably the frequency offsets ( ω − ω ) and( ω − ω ) for the two Raman laser light beams. Now P and ∆ P M are denotedas the central position and the effective momentum bandwidth of the atomicmomentum wave-packet state , respectively, and ∆ P = P − P as the deviation ofthe position P of the momentum wave-packet state from the central position P . For simplicity, hereafter assume that the momentum P is always much greaterthan ∆ P M / x in the decelerating process. Supposethat the absolute amplitude at the position P of the momentum wave-packetstate decays exponentially as the deviation ∆ P . A typical example of suchwave-packet states is Gaussian wave-packet states [25, 31]. Then the amplitudeof the position P outside the effective momentum region [ P ] = [ P − ∆ P M / ,P + ∆ P M /
2] in the momentum wave-packet state is almost zero and the prob-ability to find that the atom is not in the effective momentum region [ P ] is sosmall that it can be negligible. This is just the definition of the effective mo-mentum bandwidth ∆ P M of the atomic momentum wave-packet state. Thenit is sufficient to consider only the effective momentum region [ P ] of the mo-mentum wave-packet state when the adiabatic condition is investigated for thethree-state STIRAP experiments of the decelerating and accelerating processes.Here the carrier frequencies ω and ω of the two Raman laser light beams arenot determined until the adiabatic condition for the STIRAP experiments isexamined later.The unitary decelerating and accelerating processes of the quantum controlprocess [22] require that the atom in the initial internal state | g i ( | g i ) be com-pletely transferred to another internal state | g i ( | g i ) and at the same time theinitial atomic wave-packet motional state be completely converted into another20ave-packet motional state. It is well known that in an ideal condition the STI-RAP population transfer process can achieve 100% transfer efficiency from oneatomic internal state to another [3, 4, 15, 17, 18]. The adiabatic state-transferchannel for the three-state STIRAP experiments is formed generally through thespecial adiabatic eigenstate | g ( P, t ) i of the Hamiltonian H ( P, t ) [17] . Therefore,if the unitary decelerating and accelerating processes want to achieve their maingoal, they had better make full use of this adiabatic state-transfer channel. Ifnow the atom is prepared in the adiabatic eigenstate | g ( P, t ) i of the Hamilto-nian H ( P, t ) at the initial time t , then according as the adiabatic theorem [25,30] the atom will be in the adiabatic eigenstate | g ( P, t ) i of the Hamiltonian H ( P, t ) at any instant of time t in the adiabatic process for t ≤ t ≤ t + T if theadiabatic condition is met, that is, if the time period T is infinitely large or moregenerally if the eigenvectors of the Hamiltonian H ( P, t ) vary infinitely slowly[30]. However, in practice the time period T can not be infinitely large and ro-tating of the eigenvectors of the Hamiltonian are not infinitely slow. Thus, theideal adiabatic condition can not met perfectly in practice. In fact, the quan-tum control process does not allow the time period T of the adiabatic processto take an infinitely large value. Obviously, if the time interval T is taken asa finite value but large enough or the eigenvectors of the Hamiltonian H ( P, t )rotate sufficiently slowly, then the adiabatic theorem still holds approximatelyand the real adiabatic state at the end time t + T of the adiabatic process willbe very close to the ideal adiabatic eigenstate | g ( P, t + T ) i . Since the realadiabatic state at the end of the adiabatic process is close to the ideal adiabaticeigenstate | g ( P, t + T ) i , one may calculate the real wave-packet state of theatom at the end of the STIRAP-based unitary decelerating or accelerating pro-cess with the help of the ideal adiabatic eigenstate | g ( P, t + T ) i . Then thissimplifies greatly the calculation for the real wave-packet state of the atom atthe end of the decelerating or accelerating process, although such a calculationcould generate an error for the real wave-packet state of the atom. Similarly, ifthe real transfer efficiency for the ST IRAP process is calculated with the helpof the ideal adiabatic eigenstate | g ( P, t + T ) i instead of the real adiabatic stateat the final time of the decelerating or accelerating process, then there existscertainly an error for the real transfer efficiency. However, these errors may beestimated. In fact, if one finds the real adiabatic condition for the ST IRAP process, one may estimate these errors, as shown below. Therefore, for the STI-RAP process satisfying the real adiabatic condition there is a simple scheme tocalculate the real transfer efficiency and the real wave-packet state of the atomat the end of the STIRAP process: one may use the ideal adiabatic eigenstate | g ( P, t + T ) i of the Hamiltonian H ( P, t + T ) to simplify the calculation ofthe real transfer efficiency and the real wave-packet state of the atom, thenevaluate the generated errors and control these errors to be within the desiredupper bound by setting suitably the experimental parameters for the STIRAPprocess. This is really the procedure to design the STIRAP pulse sequences forthe unitary decelerating and accelerating processes. Thus, the problem to besolved below is how to design the STIRAP pulse sequence such that in the realadiabatic condition the final adiabatic state for the real adiabatic process is still21ery close to the ideal adiabatic eigenstate | g ( P, t + T ) i even if the time period T takes a finite value.In the STIRAP-based decelerating and accelerating processes the adiabaticevolution process of the atom could occur simultaneously and in a parallelform in these three-state subspaces {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} forall the given momentum P within the effective momentum region [ P ] if theadiabatic condition is met in the effective momentum region [ P ] . It is suffi-cient to examine the adiabatic evolution process of the atom in a three-statesubspace {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} with a given momentum P ofthe effective momentum region [ P ] to set up the adiabatic condition for theSTIRAP experiments of the decelerating and accelerating processes. Obvi-ously, any atomic state in the three-state subspace during the adiabatic evo-lution process can be expanded in terms of the three basis vectors of the sub-space. If now the three basis vectors of the three-state subspace {| P + ℏ k i| g i , | P i| e i , | P − ℏ k i| g i} are taken as the three orthonormal adiabatic eigenvec-tors {| g ( P, t ) i , | g ± ( P, t ) i} of the Hamiltonian H ( P, t ) , then the atomic three-state vector | Φ( P, t ) i = ( ¯ A ( P, t ) , ¯ A ( P, t ) , ¯ A ( P, t )) T in the Schr¨odinger equa-tion (16) at any instant of time t in the adiabatic evolution process may beexpanded as [4, 5, 10, 12, 15, 17, 18, 19, 20, 25] | Φ( P, t ) i = a ( P, t ) | g ( P, t ) i + a + ( P, t ) | g + ( P, t ) i exp[ i Z tt dt ′ Ω( t ′ )]+ a − ( P, t ) | g − ( P, t ) i exp[ − i Z tt dt ′ Ω( t ′ )] . (21)By substituting the adiabatic eigenstates of Eqs. (19a) and (19b) into the state | Φ( P, t ) i of Eq. (21) one can find that the coefficients { a ( P, t ) , a ± ( P, t ) } in Eq.(21) are related to those { ¯ A l ( P, t ) , l = 0 , , } in Eq. (16) by¯ A ( P, t ) = a ( P, t ) cos θ ( t ) exp[ − iγ ( t )]+ 1 √ a + ( P, t ) sin θ ( t ) exp[ − iδ ( t )] exp[ i Z tt dt ′ Ω( t ′ )]+ 1 √ a − ( P, t ) sin θ ( t ) exp[ − iδ ( t )] exp[ − i Z tt dt ′ Ω( t ′ )] , (22a)¯ A ( P, t ) = 1 √ − iα p ( P, t )] exp[ − iδ ( t )] {− a + ( P, t ) exp[ i Z tt dt ′ Ω( t ′ )]+ a − ( P, t ) exp[ − i Z tt dt ′ Ω( t ′ )] } (22b)¯ A ( P, t ) = exp[ − i ( α p ( P, t ) − α s ( P, t ))] {− a ( P, t ) sin θ ( t ) exp[ − iγ ( t )]+ 1 √ a + ( P, t ) cos θ ( t ) exp[ − iδ ( t )] exp[ i Z tt dt ′ Ω( t ′ )]22 1 √ a − ( P, t ) cos θ ( t ) exp[ − iδ ( t )] exp[ − i Z tt dt ′ Ω( t ′ )] } . (22c)Thus, the atomic three-state vector | Φ( P, t ) i may be determined if one knows thecoefficients a ( P, t ) and a ± ( P, t ) . Assume that at the initial time t the atomis in the adiabatic eigenstate | g ( P, t ) i . Then the adiabatic theorem showsthat the atom is kept in the adiabatic eigenstate | g ( P, t ) i of the Hamiltonian H ( P, t ) at any instant of time t during the adiabatic process if the ideal adiabaticcondition is met. Then it follows from the atomic state | Φ( P, t ) i of Eq. (21) thatif the ideal adiabatic condition is met, the time-dependent coefficient | a ( P, t ) | should be kept almost unchanged and is very close to unity over the wholeadiabatic process, while two other coefficients {| a ± ( P, t ) |} should be very close tozero. Generally, these coefficients may be evaluated by solving the Schr¨odingerequation (16) that the state | Φ( P, t ) i of Eq. (21) obeys. Inserting the state | Φ( P, t ) i of Eq. (21) into the Schr¨odinger equation (16) one obtains a set of thethree differential equations with the variables a ( P, t ) and a ± ( P, t ) ,∂∂t a ( P, t ) + iω ( P, t ) a ( P, t ) + a + ( P, t ) ω , + ( P, t ) exp[ i Z tt dt ′ Ω( t ′ )]+ a − ( P, t ) ω , − ( P, t ) exp[ − i Z tt dt ′ Ω( t ′ )] = 0 , (23a) ∂∂t a ± ( P, t ) + iω ± ( P, t ) a ± ( P, t ) + a ∓ ( P, t ) ω ± , ∓ ( P, t ) exp[ ∓ i Z tt dt ′ Ω( t ′ )]+ a ( P, t ) ω ± , ( P, t ) exp[ ∓ i Z tt dt ′ Ω( t ′ )] = 0 , (23b)where the coefficients { ω k ( P, t ) , ω k,l ( P, t ) } are defined as iω k ( P, t ) = h g k ( P, t ) | ∂g k ( P, t ) /∂t i ( k = 0 , ± ) ,ω k,l ( P, t ) = h g k ( P, t ) | ∂∂t | g l ( P, t ) i ( k = l, k, l = 0 , ± ) . These coefficients can be obtained explicitly from the adiabatic eigenstates ofEqs. (19), ω ( P, t ) = − ∂∂t γ ( t ) − sin θ ( t ) ∂∂t [ α p ( P, t ) − α s ( P, t )] , (24a) ω + ( P, t ) = ω − ( P, t ) = − ∂∂t δ ( t ) − ∂∂t α p ( P, t ) −
12 cos θ ( t ) ∂∂t [ α p ( P, t ) − α s ( P, t )] (24b)and ω , ± ( P, t ) = − ω ± , ( P, t ) ∗ = 1 √ i ( γ ( t ) − δ ( t ))]23 { ∂∂t θ ( t ) + i sin θ ( t ) cos θ ( t ) ∂∂t [ α p ( P, t ) − α s ( P, t )] } , (24c) ω + , − ( P, t ) = ω − , + ( P, t ) = i ∂∂t α p ( P, t ) − i
12 cos θ ( t ) ∂∂t [ α p ( P, t ) − α s ( P, t )] . (24d)Though these coefficients contain the global phases γ ( t ) and δ ( t ), the final re-sults of the adiabatic process obtained from these coefficients will not be affectedby these global phase factors [18b], as can be seen later. In order to solve con-veniently the equations (23) it could be better to make variable transformations[25, 30]: a k ( P, t ) = b k ( P, t ) exp[ − i Z tt dt ′ ω k ( P, t ′ )] ( k = 0 , ± ) . (25)Then with the new variables { b k ( P, t ) } these equations (23) are changed to ∂∂t b ( P, t ) = exp[ i ( γ ( t ) − δ ( t ))] √ P, t ) ∗ × { b + ( P, t ) exp[ i Z tt dt ′ Ω + ( P, t ′ )] + b − ( P, t ) exp[ − i Z tt dt ′ Ω − ( P, t ′ )] } , (26a) ∂∂t b ± ( P, t ) = − i b ∓ ( P, t )Γ(
P, t ) exp[ ∓ i Z tt dt ′ t ′ )] − exp[ − i ( γ ( t ) − δ ( t ))] √ b ( P, t )Θ(
P, t ) exp[ ∓ i Z tt dt ′ Ω ± ( P, t ′ )] . (26b)The coefficients in Eqs. (26) are obtained from those in Eq. (24):Ω ± ( P, t ) = Ω( t ) ± { [ ˙ α s ( P, t ) −
12 ˙ α p ( P, t )] sin θ ( t )+ [ ˙ α p ( P, t ) −
12 ˙ α s ( P, t )] cos θ ( t ) } , (27a)Θ( P, t ) = − ˙ θ ( t ) + i
12 sin 2 θ ( t )[ ˙ α p ( P, t ) − ˙ α s ( P, t )] , (27b)Γ( P, t ) = sin θ ( t ) ˙ α p ( P, t ) + cos θ ( t ) ˙ α s ( P, t ) , (27c)where ˙ θ ( t ) = ddt θ ( t ) , ˙ α s ( P, t ) = ∂∂t α s ( P, t ) , and so on. The equations (26)are the basic differential equations to describe completely the STIRAP-baseddecelerating and accelerating processes. The set of basic equations (26) is ageneralization of the basic equations to describe the conventional three-stateSTIRAP experiments [17, 16, 18] when the effect of a momentum distributionis taken into account on the STIRAP experiments. The basic equations (26)may be used to set up a general adiabatic condition for the three-state STIRAP-based decelerating and accelerating processes.24he basic differential equations (26) may be rewritten in the matrix form i ∂∂t B ( P, t ) = M ( P, t ) B ( P, t ) . (28)Here the normalization three-state vector B ( P, t ) is defined as ( b ( P, t ) , b + ( P, t ) ,b − ( P, t )) T and the 3 × − dimensional hermitian Hamiltonian M ( P, t ) is givenby M ( P, t ) = M M M ∗ M M ∗ M ∗ , in which the matrix elements { M ij } are defined by M = i √ i ( γ ( t ) − δ ( t ))]Θ( P, t ) ∗ exp[ i Z tt dt ′ Ω + ( P, t ′ )] ,M = i √ i ( γ ( t ) − δ ( t ))]Θ( P, t ) ∗ exp[ − i Z tt dt ′ Ω − ( P, t ′ )] ,M = 12 Γ( P, t ) exp[ − i Z tt dt ′ t ′ )] . Notice that the momentum P in the hermitian Hamiltonian M ( P, t ) is a pa-rameter instead of an operator. The three-state Schr¨odinger equation (28) mayhave the formal solution: B ( P, t ) = T exp {− i Z tt dt ′ M ( P, t ′ ) } B ( P, t )= { i ) Z tt dt M ( P, t ) + ( 1 i ) Z tt Z t t dt dt M ( P, t ) M ( P, t ) + ... } B ( P, t ) . (29)The Dyson series solution (29) may be useful to set up a general adiabaticcondition for the STIRAP-based decelerating and accelerating processes. Thedetailed discussion will appear in the section seven of the paper.
4. The STIRAP state-transfer process in the ideal adiabatic condi-tion
Consider first the special case: the ideal adiabatic condition. The idealadiabatic condition usually means that the time interval T of the adiabaticprocess is infinitely large or the adiabatic eigenstates of the Hamiltonian (17)rotate infinitely slowly. Here the ideal adiabatic condition means that for anyinstant of time of the adiabatic process the integrations of the basic differentialequations (26) approach zero for any given momentum P within the effectivemomentum region [ P ] of the atomic wave-packet motional state. The idealadiabatic condition may be expressed as Z tt dt ′ [ ∂∂t ′ b l ( P, t ′ )] → , l = 0 , ± ; t ≤ t ≤ t + T ; P ∈ [ P ] . (30)25he ideal adiabatic condition (30) shows that the basic equations (26) havethe solution b l ( P, t ) → b l ( P, t ) ( l = 0 , ± ) for any instant of time t of theadiabatic process and any given momentum P within the effective momentumregion [ P ] . If at the initial time t the atom is prepared in the adiabatic eigen-state | g ( P, t ) i , which is the initial atomic state | Φ( P, t ) i of Eq. (21) with a ( P, t ) = 1 and a ± ( P, t ) = 0 , then according to the adiabatic theorem [25,30] one should find that at any instant of time t of the adiabatic process theatom is in the adiabatic eigenstate | g ( P, t ) i of the Hamiltonian H ( P, t ) . In fact,by the equations (25) and (26) one can obtain | a l ( P, t ) | → | a l ( P, t ) | ( l = 0 , ± )in the ideal adiabatic condition (30). Hence | a ( P, t ) | → | a ( P, t ) | = 1 and a ± ( P, t ) → a ± ( P, t ) = 0 for the atomic state | Φ( P, t ) i at the instant of time t of the ideal adiabatic process, while this atomic state | Φ( P, t ) i is just theadiabatic eigenstate | g ( P, t ) i of the Hamiltonian H ( P, t ) , as can be seen fromEq. (21). The basic equations (26) show that the ideal adiabatic condition (30)could be achieved if the coefficients Θ( P, t ) and Γ(
P, t ) approach zero sufficientlyand the Rabi frequencies Ω( t ) and Ω ± ( P, t ) are sufficiently large. The Rabi fre-quencies Ω( t ) and Ω ± ( P, t ) can have a great effect on the adiabatic conditionof the STIRAP experiments. This point is very important for the quantumcontrol process, since a sufficiently long time period T of the adiabatic processis not accepted for the quantum control process. Then one may likely make theadiabatic condition to be met by setting suitably the Rabi frequencies of theRaman laser light beams, although the adiabatic condition is met usually bymaking the time interval T sufficiently large. A general adiabatic condition willbe discussed in detail later.In order to optimize the adiabatic condition the carrier frequencies ω and ω of the Raman laser light beams could be chosen suitably such that α p ( P, t ) = [ 2
P k + ℏ k M − ( ω − ω )] t + φ ( t ) = ∆ PM k t + ϕ ( t ) (31)and α s ( P, t ) = [ − P k + ℏ k M − ( ω − ω )] t + φ ( t ) = − ∆ PM k t + ϕ ( t ) (32)where the momentum difference ∆ P = P − P and the momentum P is just thecentral position of the effective momentum region [ P ] of a general momentumwave-packet state. When the carrier frequencies are chosen according to Eq.(31) and (32), the maximum momentum difference value within the effectivemomentum region [ P ] is minimum. These equations (31) and (32) can be sat-isfied when the carrier frequencies { ω l } ( l = 0 ,
1) are determined from the twoequations: ℏ k M − ( ω − ω ) + c = − P M k (33a)and ℏ k M − ( ω − ω ) + c = P M k (33b)26here the wave number k l = ω l /c , c = ∆ p ( P ), c = ∆ s ( P ), and ddt φ l ( t ) = c l + ddt ϕ l ( t ) . On the other hand, if the carrier frequencies are given in advance,then the detunings c = ∆ p ( P ) and c = ∆ s ( P ) may also be determined fromthese two equations (33) , respectively. In the decelerating and acceleratingprocesses the momentum value P is generally different for each basic STIRAP-based decelerating or accelerating process. Then one may adjust the carrierfrequencies { ω l } or the detunings { c l } so that the equations (33) can be satisfied.Now using the phase α p ( P, t ) of Eq. (31) and α s ( P, t ) of Eq. (32) one can rewritethe phase difference of Eq. (20) in the simple form α p ( P, t ) − α s ( P, t ) = ∆ PM ( k + k ) t + ϕ ( t ) − ϕ ( t ) . (34)If in the STIRAP experiments at the initial time t the Rabi frequency Ω s ( t )of the Stokes pulse is much greater than the one Ω p ( t ) of the pumping pulse,then at the initial time the mixing angle θ ( t ) in the adiabatic eigenstates of Eqs.(19) satisfies [15, 16, 17, 18]cos θ ( t ) = Ω s ( t )Ω( t ) → θ ( t ) → . (35)In practice both the Rabi frequencies Ω s ( t ) and Ω p ( t ) usually could be rela-tively small at the initial time, but the Rabi frequency Ω p ( t ) of the pumpingpulse is much less than Ω s ( t ) of the Stokes pulse. The initial mixing angle θ ( t ) → H ( P, t ) ofEq. (17) to the asymptotic forms | g ( P, t ) i → exp[ − iγ ( t )] | i , (36a) | g ± ( P, t ) i → √ − iδ ( t )] {∓ exp[ − iα p ( P, t )] | i + exp[ − i ( α p ( P, t ) − α s ( P, t ))] | i} . (36b)Notice that the three-state basis vectors | i , | i , and | i stand for the basis vec-tors | P + ℏ k i| g i , | P i| e i , and | P − ℏ k i| g i , respectively. The adiabatic eigen-state | g ( P, t ) i of the Hamiltonian H ( P, t ) is much simpler than | g ± ( P, t ) i , the latter is more complex in that their expansion coefficients are dependenton the momentum. In general, at the initial time an atomic system may beprepared more easily in the adiabatic eigenstate | g ( P, t ) i up to a global phasefactor. An important example is that at the initial time the atom is completelyin the ground internal state | g i and in the Gaussian wave-packet motional stateor in a superposition of the momentum states. Now consider that the initialstate of the atom is prepared in the superposition state | Ψ( x, r, t ) i given by Eq.(12a). By comparing Eq. (12) with Eq. (12a) with P ′ = P + ℏ k one sees thatthe coefficients of the state | Ψ( x, r, t ) i of Eq. (12) are given by A ( P, t ) = 1 ,A ( P, t ) = 0 , and A ( P, t ) = 0 at the initial time t . Then it follows fromEqs. (15) that the coefficients { ¯ A l ( P, t ) } can be obtained from { A l ( P, t ) } :¯ A ( P, t ) = exp[ i ℏ ( ( P + ℏ k ) M + E ) t ] , ¯ A ( P, t ) = ¯ A ( P, t ) = 0 . (37)27hese coefficients associated with the initial mixing angle θ ( t ) = 0 (here θ ( t )is so small that it can be taken as zero without losing generality) are insertedinto Eqs. (22) one obtains, by solving the equations (22), a ( P, t ) = exp[ iγ ( t )] exp[ i ℏ ( ( P + ℏ k ) M + E ) t ] , a + ( P, t ) = a − ( P, t ) = 0 . (38)Indeed, at the initial time the atom is completely in the adiabatic eigenstate | g ( P, t ) i of the Hamiltonian H ( P, t ) of (17) because both the coefficients a + ( P, t ) and a − ( P, t ) are zero in the initial atomic state | Φ( P, t ) i . It followsfrom Eq. (25) that at the initial time t the coefficient a l ( P, t ) = b l ( P, t )( l = 0 , ± ). Then at the initial time t the variables { b l ( P, t ) } of the basicequations (26) can be obtained from those coefficients of Eq. (38), b ( P, t ) = exp[ iγ ( t )] exp[ i ℏ ( ( P + ℏ k ) M + E ) t ] , b + ( P, t ) = b − ( P, t ) = 0 . (39)These coefficients { b l ( P, t ) } of Eq. (39) provide the basic equations (26) withthe initial values.Now the atom with the initial wave-packet state | Ψ( x, r, t ) i of Eq. (12a)undergoes the STIRAP-based decelerating process (11) in the ideal adiabaticcondition (30). Then one can calculate the atomic state at the end time t f = t + T of the ideal adiabatic process (11). This atomic state | Ψ( x, r, t ) i isstill given by Eq. (12), but the coefficients in Eq. (12) need to be calculatedexplicitly. By integrating the basic equations (26) and using the ideal adiabaticcondition (30) the solution to the basic equations (26) is given by b l ( P, t ) = b l ( P, t ) , l = 0 , ± ; t ≤ t ≤ t + T ; P ∈ [ P ] . (40)Here the initial values { b l ( P, t ) } are given by Eq. (39). Once the coefficients { b l ( P, t ) } are obtained from Eqs. (40) one may use equation (25) to calculatethe coefficients { a l ( P, t ) } : a ( P, t ) = b ( P, t ) exp[ i ( γ ( t ) − γ ( t ))] × exp { i Z tt dt ′ sin θ ( t ′ ) ∂∂t ′ [ α p ( P, t ′ ) − α s ( P, t ′ )] } , (41a) a + ( P, t ) = a − ( P, t ) = 0 . (41b)Here the frequency parameter ω ( P, t ) of Eq. (24a) has been used. The factthat the coefficients a + ( P, t ) = a − ( P, t ) = 0 shows that the atom in the adi-abatic eigenstate | g ( P, t ) i of the Hamiltonian H ( P, t ) at the initial time t is transferred adiabatically to the adiabatic eigenstate | g ( P, t ) i of the Hamil-tonian H ( P, t ) and finally to the desired adiabatic eigenstate | g ( P, t f ) i of theHamiltonian H ( P, t f ) at the end of the adiabatic process. This is just consistentwith the prediction of the adiabatic theorem in quantum mechanics [25, 30]. Bysubstituting the coefficients { a l ( P, t ) , l = 0 , ±} of Eqs. (41) into Eqs. (22) oneobtains ¯ A ( P, t ) = exp[ − iγ ( t )] a ( P, t ) cos θ ( t ) , (42a)28 A ( P, t ) = 0 , (42b)¯ A ( P, t ) = − a ( P, t ) exp[ − iγ ( t )] exp[ − i ( α p ( P, t ) − α s ( P, t ))] sin θ ( t ) . (42c)The coefficient ¯ A ( P, t ) = 0 shows clearly that in the ideal adiabatic conditionthe atomic excited internal state | e i indeed does not appear during the STIRAPstate-transfer process. Though one has the coefficients { a l ( P, t ) , l = 0 , ±} ofEqs. (41) at hand, it is not sufficient from Eqs. (42) to determine uniquely thecoefficients { ¯ A k ( P, t ) } if one does not know the mixing angle θ ( t ) at the endtime t f = t + T of the ideal adiabatic process. Suppose that at the end of theideal adiabatic process the Rabi frequency Ω p ( t ) for the pumping pulse is muchgreater than the one Ω s ( t ) of the Stokes pulse. Then the mixing angle θ ( t ) atthe end of the ideal adiabatic process takes the asymptotic form [15, 17]sin θ ( t f ) = Ω p ( t f )Ω( t f ) → θ ( t f ) → π/ . Now inserting the mixing angle θ ( t f ) = π/ A ( P, t f ) = ¯ A ( P, t f ) = 0 , (43a)¯ A ( P, t f ) = − a ( P, t f ) exp[ − iγ ( t f )] exp[ − i ( α p ( P, t f ) − α s ( P, t f ))] . (43b)Thus, with the aid of Eqs. (15) and (43) the atomic three-state vector { A ( P, t ) ,A ( P, t ) , A ( P, t ) } T at the end time t f of the ideal adiabatic process is deter-mined by A ( P, t f ) = A ( P, t f ) = 0 , (44a) A ( P, t f ) = − exp[ i ℏ ( ( P + ℏ k ) M + E ) t ] exp[ − i ℏ ( ( P − ℏ k ) M + E ) t f ] × exp[ − i ( α p ( P, t f ) − α s ( P, t f ))] exp { i Z t f t dt ′ sin θ ( t ′ ) ∂∂t ′ [ α p ( P, t ′ ) − α s ( P, t ′ )] } . (44b)The coefficients A ( P, t f ) = A ( P, t f ) = 0 show that at the end of the idealadiabatic process the atom is transferred completely to the ground internalstate | g i from the initial internal state | g i by the basic STIRAP deceleratingprocess (11). This is just the desired result of the ideal adiabatic process (11).By using the phase difference in Eq. (34) one can further express the coefficient A ( P, t f ) as A ( P, t f ) = exp[ iβ ( t f , t )] exp[ i ℏ ( ( P + ℏ k ) M + E ) t ] × exp[ − i ℏ ( ( P − ℏ k ) M + E ) t f ] × exp {− i ∆ PM ( k + k )[ t + Z t f t dt ′ cos θ ( t ′ )] } (44c)29here the global phase factor is given byexp[ iβ ( t f , t )] = − exp[ − i ( ϕ ( t f ) − ϕ ( t f ))] × exp { i Z t f t dt ′ sin θ ( t ′ )[ ˙ ϕ ( t ′ ) − ˙ ϕ ( t ′ )] } . (45)Once the atomic three-state vector { A ( P, t ) , A ( P, t ) , A ( P, t ) } T is obtained atthe end time t f of the basic STIRAP decelerating process (11), through theequation (12) one can calculate the motional state of the atom at the end ofthe ideal adiabatic process. In order to calculate conveniently the wave-packetmotional state of the atom one may use the momentum P ′ = P + ℏ k as variableto express the three-state vector { A ( P ′ , t ) , A ( P ′ , t ) , A ( P ′ , t ) } T , and then thevector at the end time t f can be determined by A ( P ′ , t f ) = A ( P ′ , t f ) = 0 , (46a) A ( P ′ , t f ) = exp[ iβ ′ ( t f , t )] exp[ i ℏ ( P ′ M + E ) t ] × exp[ − i ℏ ( ( P ′ − ℏ k − ℏ k ) M + E ) t f ] × exp {− i ∆ P ′ M ( k + k )[ t + Z t f t dt ′ cos θ ( t ′ )] } . (46b)Note that here the momentum difference ∆ P ′ still represents the deviation ofthe momentum point P ′ from the central point of the effective momentum region[ P ] in the initial wave-packet motional state.For the STIRAP-based accelerating process (11a) in the ideal adiabatic con-dition the three-state vector { A ( P ′ , t ) , A ( P ′ , t ) , A ( P ′ , t ) } T at the end of theideal adiabatic process (11a) should be determined by A ( P ′ , t f ) = A ( P ′ , t f ) = 0 , (47a) A ( P ′ , t f ) = exp[ iβ ′ ( t f , t )] exp[ i ℏ ( P ′ M + E ) t ] × exp[ − i ℏ ( ( P ′ + ℏ k + ℏ k ) M + E ) t f ] × exp { i ∆ P ′ M ( k + k )[ t + Z t f t dt ′ cos θ ( t ′ )] } . (47b)This is because the propagating directions of the Raman laser light beams inthe accelerating process are just opposite to those in the decelerating process,respectively.
5. The decelerating and accelerating processes of a Gaussian wave-packet state in the ideal adiabatic condition
30s a typical example, consider that an atom with a Gaussian wave-packetstate is decelerated by the basic STIRAP-based decelerating sequence (11). Forsimplicity, here consider the simple situation that the STIRAP deceleratingprocess (11) satisfies the ideal adiabatic condition (30). The theory developedin previous sections can determine the wave-packet motional state of the atom atthe end of the basic decelerating process (11). Suppose that the initial motionalstate of the atom is a standard Gaussian wave-packet state [25, 31] in one-dimensional coordinate space:Ψ ( x, t ) = exp( iϕ )[ (∆ x ) π ] / s x ) + i ( ℏ T M ) × exp {−
14 ( x − z ) (∆ x ) + i ( ℏ T M ) } exp { ip x/ ℏ } . (48)Here the complex linewidth of the Gaussian wave-packet state Ψ ( x, t ) is de-fined as W ( T ) = (∆ x ) + i ( ℏ T M ) . The probability density of the state Ψ ( x, t ) is given by | Ψ ( x, t ) | = 1 √ π s
12 1(∆ x ) + ( ℏ T M (∆ x ) ) exp {−
12 ( x − z ) (∆ x ) + ( ℏ T M (∆ x ) ) } . By comparing | Ψ ( x, t ) | with the standard Gaussian function G ( x ) = [ ε √ π ] − × exp[ − ( x − x ) /ε ] one sees that the center-of-mass position and the wave-packet spreading of the state Ψ ( x, t ) are z and ε = q x ) + ( ℏ T M (∆ x ) ) ] , respectively. If the atom is a free particle, the Gaussian wave-packet stateΨ ( x, t ) has an explicit physical meaning that the atom with the Gaussianwave-packet state moves along the direction + x with the velocity p /M. Onemay expand the coordinate-space Gaussian wave-packet state Ψ ( x, t ) in termsof the momentum eigenstates {| p i} of Eq. (7),Ψ ( x, t ) = X p ρ ( p, t ) | p i . (49)The expansion coefficient or the amplitude ρ ( p, t ) is determined by ρ ( p, t ) = Z dx Ψ ( x, t ) ψ p ( x ) ∗ (49a)where ψ p ( x ) = √ π exp( ipx/ ℏ ) is just the momentum eigenstate | ψ p ( x ) i or | p i of Eq. (7). By substituting the wave-packet state Ψ ( x, t ) of Eq. (48) andthe momentum eigenstate | ψ p ( x ) i of Eq. (7) into the amplitude ρ ( p, t ) of Eq.(49a) one obtains, by a complex calculation, ρ ( p, t ) = exp( iϕ )[ 2(∆ x ) π ] / exp {− (∆ x ) ( p − p ℏ ) } exp {− i ( ℏ T M )( p − p ℏ ) } exp {− i ( p − p ℏ ) z } . (50)The amplitude ρ ( p, t ) is really the Fourier transform state of the coordinate-space Gaussian wave-packet state Ψ ( x, t ). It also has a Gaussian shape andthis can be seen more clearly from its absolute square: | ρ ( p, t ) | = [ 2(∆ x ) π ] / exp {− x ) ( p − p ℏ ) } . (50a)This is a standard Gaussian function with the propagation-vector variable k = p/ ℏ . Therefore, it satisfies the normalization, Z + ∞−∞ dk | ρ ( k, t ) | = Z ∞−∞ dx | Ψ ( x, t ) | = 1 . (51)The center-of-mass position of the Gaussian function is p for the momen-tum variable or p / ℏ for the propagation-vector variable. The wave-packetspreading of the Gaussian function for the momentum p is determined by(∆ p ) = ℏ / [ √ x )] and for the propagation-vector variable k is given by ∆ k =[ √ x )] − . The state ρ ( p, t ) is called the momentum-space Gaussian wave-packet state of the atom. There is a Gaussian decay factor exp[ − (∆ x ) ( p − p ℏ ) ]in the momentum-space wave-packet state ρ ( p, t ) , which decides the effec-tive momentum region [ P ] of the momentum wave-packet state. Obviously, theprobability density | ρ ( p, t ) | approaches zero rapidly (exponentially) when themomentum p takes a value such that the deviation | p − p | is greater than thewave-packet spreading (∆ p ). Equation (49) really shows that the coordinate-space Gaussian wave-packet state Ψ ( x, t ) can be expanded in terms of themomentum-space Gaussian wave-packet states { ρ ( p, t ) } .Consider the superposition of the momentum wave-packet states { ρ ( p, t ) } :Φ ( x, t ) = X | p − p |≤ ∆ P M / ρ ( p, t ) | p i , (52)where ∆ P M is just the bandwidth of the effective momentum region [ P ] of themomentum wave-packet state ρ ( p, t ). When the bandwidth ∆ P M → ∞ , thesuperposition state Φ ( x, t ) is just the Gaussian wave-packet state Ψ ( x, t ) , as can be seen from Eqs. (49) and (52). The deviation of the state Φ ( x, t )from the state Ψ ( x, t ) may be measured by the probability P { Ψ ( x, t ) − Φ ( x, t ) } = | X | p − p | > ∆ P M / ρ ( p, t ) | p i| = 2 Z ∞ [∆ P M / p ] / ℏ dk | ρ ( k, t ) | where the momentum eigenstates | p i of Eq. (7) and their orthonormal relationsand the momentum wave-packet state ρ ( p, t ) of Eq. (50) have been used. Nowusing the probability density | ρ ( p, t ) | of Eq. (50a) one has P { Ψ ( x, t ) − Φ ( x, t ) } = 2 √ π Z ∞ y M dy exp( − y ) (53)32here the lower integral limit is y M = (∆ P M )(∆ x ) √ ℏ . The probability P { Ψ ( x, t ) − Φ ( x, t ) } is bounded on by [32]2 √ π exp( − y M ) y M + p y M + 2 < P { Ψ ( x, t ) − Φ ( x, t ) } ≤ √ π exp( − y M ) y M + p y M + 4 /π . The important thing is that the probability P { Ψ ( x, t ) − Φ ( x, t ) } decays ex-ponentially with the number y M . If the bandwidth ∆ P M of effective momentumregion [ P ] of the momentum wave-packet state ρ ( p, t ) is chosen such that thenumber y M >> , then the probability P { Ψ ( x, t ) − Φ ( x, t ) } is almost zero.As a result, the superposition state Φ ( x, t ) is almost equal to the Gaussianwave-packet state Ψ ( x, t ) . If an atomic system is in a momentum superposition state which spreadsfrom −∞ to + ∞ in momentum space , then it is generally hard to achieve acomplete STIRAP state transfer in the atomic system by the basic decelerat-ing process (11), since the adiabatic condition can not be met over the wholemomentum space ( −∞ , + ∞ ) . On the other hand, an almost complete STIRAPstate transfer could be achieved for an atomic wave-packet motional state witha finite wave-packet spreading by the basic decelerating process (11). This canbe illustrated through the momentum wave-packet state Φ ( x, t ) of Eq. (52).The state Φ ( x, t ) is also a superposition of the momentum eigenstates of Eq.(7). All the momentum components of the state Φ ( x, t ) are within the ef-fective momentum region [ P ] = [ p − ∆ P M / , p + ∆ P M / P M of the effective momentum region[ P ] is finite and satisfies p − ∆ P M / > ℏ k + ℏ k , a complete STIRAP statetransfer could be achieved for the state Φ ( x, t ) within the effective momentumregion [ P ] by the STIRAP decelerating process (11) if the ideal adiabatic condi-tion (30) is met within the effective momentum region [ P ] for the deceleratingprocess (11). Now using the same STIRAP pulse sequence (11) one can makean almost complete STIRAP state transfer for the Gaussian wave-packet stateΨ ( x, t ) as the state Ψ ( x, t ) is almost equal to the state Φ ( x, t ) when thenumber y M >> . Hereafter suppose that the number y M >> ( x, t ) can be replaced with the state Φ ( x, t )and vice versa without generating a significant error in evaluating the unitarydecelerating and accelerating processes.Now suppose that at the initial time t the atom is in the Gaussian wave-packet motional state Ψ ( x, t ) of (48) and in the internal state | g i . Then thetotal product state of the atom at the initial time is given byΨ ( x, r, t ) = Ψ ( x, t ) | g i = X p ρ ( p, t ) | p i| g i . (54)By comparing the product state Ψ ( x, r, t ) with that state of Eq. (12a) onecan see that the amplitude ρ ( p, t ) in the state Ψ ( x, r, t ) just corresponds to33he amplitude ρ ( P ′ ) in the state of Eq. (12a). This means that at the initialtime t the probability to find the atom in the three-state subspace {| p i| g i , | p − ℏ k i| e i , | p − ℏ k − ℏ k i| g i} is given by | ρ ( p, t ) | . Note that during theSTIRAP decelerating process (11) this probability is not changed with time.Obviously, the coefficients A ( p, t ) = 1 , A ( p, t ) = 0 , and A ( p, t ) = 0 at theinitial time, as can be seen in Eq. (54). Now the initial atomic product stateΨ ( x, r, t ) of (54) undergoes the basic STIRAP decelerating process (11). Thenat the end time t f = t + T of the decelerating process (11) the total productstate of the atom is given by Eq. (12),Ψ ( x, r, t f ) = X p ρ ( p, t ) { A ( p, t f ) | p i| g i + A ( p, t f ) | p − ℏ k i| e i + A ( p, t f ) | p − ℏ k − ℏ k i| g i} (55)where in the ideal adiabatic condition (30) the coefficients { A ( p, t f ) , l = 0 , , } are given by Eqs. (46) with the parameter settings P ′ = p and ∆ P ′ = p − p .Though in Eq. (55) the sum for the momentum p runs over only the effectivemomentum region [ P ], it will not generate a significant error if the sum reallyruns over the whole momentum region ( −∞ , + ∞ ) , as pointed out before. Sincein the ideal adiabatic condition (30) the coefficients A ( p, t f ) = A ( p, t f ) = 0 , the total product state (55) is reduced to the simple formΨ ( x, r, t f ) = X p ρ ( p, t ) A ( p, t f ) | p − ℏ k − ℏ k i| g i . (56)The product state Ψ ( x, r, t f ) shows that in the ideal adiabatic condition (30)at the end of the decelerating process (11) the atom is completely in the groundinternal state | g i and also in the wave-packet motional state:Ψ ( x, t f ) = X p ρ ( p, t ) A ( p, t f ) | p − ℏ k − ℏ k i . (57)The initial product state of Eq. (54) and the final product state of Eq. (56)show that indeed, the atom is transferred completely from the initial internalstate | g i and the Gaussian wave-packet motional state Ψ ( x, t ) of (48) to thefinal internal state | g i and the motional state Ψ ( x, t f ) of (57) , respectively, bythe STIRAP decelerating sequence (11) in the ideal adiabatic condition (30). Itcan turn out that the motional state Ψ ( x, t f ) of Eq. (57) is still a Gaussianwave-packet state. By the new variable q = ( p − p ) / ℏ the coefficient A ( p, t f )of Eq. (46b) with P ′ = p and ∆ P ′ = p − p and the amplitude ρ ( p, t ) of Eq.(50) are respectively rewritten as A ( p, t f ) = exp[ iβ ( t f , t )] exp[ i ℏ ( p M + E ) t ] × exp[ − i ℏ ( ( p − ℏ k − ℏ k ) M + E ) t f ] exp[ − iq ℏ ( t f − t )2 M ]34 exp[ − iq ( p − ℏ k − ℏ k ) M ( t f − t )] exp {− iq ℏ ( k + k ) M Z t f t dt ′ cos θ ( t ′ ) } and ρ ( p, t ) = exp( iϕ )[ 2(∆ x ) π ] / exp {− q [(∆ x ) + i ( ℏ T M )] } exp( − iqz ) . Inserting the two coefficients and the momentum eigenstate | p − ℏ k − ℏ k i ofEq. (7) into Eq. (57) one obtainsΨ ( x, t f ) = exp( iϕ )[ 2(∆ x ) π ] / exp[ iβ ( t f , t )] exp[ i ℏ ( p M + E ) t ] × exp[ − i ℏ ( ( p − ℏ k − ℏ k ) M + E ) t f ] exp[ i ( p − ℏ k − ℏ k ) x/ ℏ ] × √ π Z ∞−∞ dq exp( − aq + bq )where the sum P p has been replaced with the integral as the momentum p iscontinuous for a free atom, and the coefficients a and b are given by a = (∆ x ) + i ℏ ( T + t f − t )2 M ,b = i [ x − z − ( p − ℏ k − ℏ k ) M ( t f − t ) − ℏ ( k + k ) M Z t f t dt ′ cos θ ( t ′ )] . The Gaussian integral in the state Ψ ( x, t f ) can be calculated by Z ∞−∞ dq exp( − aq + bq ) = r πa exp( b a ) . (58)Now by a simple calculation one obtains the final state:Ψ ( x, t f ) = exp[ iϕ ( t f , t )][ (∆ x ) π ] / s x ) + i ℏ ( T + t f − t )2 M × exp {−
14 ( x − z ) [(∆ x ) + i ℏ ( T + t f − t )2 M ] } exp { ip x/ ℏ } . (59)Indeed, the motional state Ψ ( x, t f ) is also a Gaussian wave-packet state justlike the initial motional state Ψ ( x, t ) of Eq. (48). Here the Gaussian wave-packet state Ψ ( x, t f ) has the center-of-mass position z = z + ( p − ℏ k − ℏ k ) M ( t f − t ) + ℏ ( k + k ) M Z t f t dt ′ cos θ ( t ′ ) , (60)the wave-packet spreading ε = s x ) + 2[ ℏ ( T + t f − t )2 M (∆ x ) ] , (61)35he mean motional momentum p = p − ℏ k − ℏ k , and the global phase factorexp[ iϕ ( t f , t )] = exp( iϕ ) exp[ iβ ( t f , t )] × exp { i ℏ ( p M + E ) t } exp {− i ℏ ( ( p − ℏ k − ℏ k ) M + E ) t f } . It is interesting to compare the final motional state Ψ ( x, t f ) with the initialstate Ψ ( x, t ) of the decelerating process (11). It needs only three parameters tocharacterize completely a Gaussian wave-packet motional state of a free particle:the center-of-mass position, the mean motional momentum (or velocity), and thecomplex linewidth. Here the wave-packet spreading is determined completely bythe complex linewidth. For the first point, the atom is decelerated by ℏ k /M + ℏ k /M by the STIRAP pulse sequence (11) as expected, because the averagemotional momentum p = p − ℏ k − ℏ k of the final state Ψ ( x, t f ) is smallerthan p of the initial state Ψ ( x, t ) and their difference is ( ℏ k + ℏ k ) . Heresuppose that the initial velocity for the moving atom p /M >> ℏ ( k + k ) /M. Note that ℏ k /M and ℏ k /M are the atomic recoil velocities in the two Ramanlaser light beams with the wave numbers k and k , respectively. For the secondpoint, the atom moves a distance ( z − z ) along the direction + x during thedecelerating process. If a free atom moved along the direction + x with thevelocities p /M and p /M, respectively, then in the time interval T = t f − t the atom would move distances equal to T × p /M and T × p /M, respectively.Here the velocities p /M and p /M are the atomic moving velocities before andafter the decelerating process, respectively. One can expect that the distance( z − z ) should lie in between the distances T × p /M and T × p /M, thatis, T × p /M < z − z < T × p /M. Indeed, the equation (60) shows thispoint. For the third point, the wave-packet spreading of the atom at the endof the decelerating process is larger than the initial one. If one compares thewave-packet spreading of Eq. (61) with the free-atom wave-packet spreading(see section 6 below), one can see that though the atom is irradiated by theRaman laser light beams, the wave-packet spreading of the atom during thedecelerating process is not really affected by the Raman laser light beams andis just the same as that one of the atom in the absence of the Raman laserlight beams. This point is important as the wave-packet spreading of the atomafter the decelerating (or accelerating) process can be calculated easily. Notethat the wave-packet spreading of a free atom becomes larger and larger as timeincreases, as can be seen in section 6 below.From the experimental viewpoint one does not expect that after the de-celerating process (11) the wave-packet spreading of the atomic momentumwave-packet state could become larger, because this may make a trouble for thedesign of the STIRAP pulse sequence (11). Fortunately, it can turn out that inthe ideal adiabatic condition (30) the wave-packet spreading of the momentum36ave-packet state is not really affected by the STIRAP pulse sequence (11).One can expand the wave-packet motional state of Eq. (59) in terms of themomentum eigenstates {| p i} of Eq. (7): Ψ ( x, t f ) = P p ρ ( p, t f ) | p i just likethe expansion (49), and just like the momentum wave-packet state ρ ( p, t ) themomentum wave-packet state ρ ( p, t f ) can be calculated from Eq. (49a), ρ ( p, t f ) = exp[ iϕ ( t f , t )][ 2(∆ x ) π ] / exp {− (∆ x ) ( p − p ℏ ) }× exp {− i ℏ ( T + t f − t )2 M ( p − p ℏ ) } exp[ − i ( p − p ) ℏ z ] . (62)Indeed, the probability density | ρ ( p, t f ) | is also a Gaussian function and itis really equal to | ρ ( p, t ) | of the initial state Ψ ( x, t ) if the initial momen-tum p is replaced with p . Thus, the wave-packet spreading of the Gaussianfunction | ρ ( p, t f ) | is equal to that one of | ρ ( p, t ) | and is also given by(∆ p ) = ℏ / [ √ x )] . This shows that the effective momentum region [ P ] ofthe initial momentum wave-packet state ρ ( p, t ) is not changed after the atomis decelerated by the STIRAP pulse sequence (11), although the center-of-massposition of the momentum wave-packet state is changed to p = p − ℏ k − ℏ k from the initial one p after the atom is decelerated.In the quantum control process the halting-qubit atom usually needs to bedecelerated continuously [22] because each STIRAP pulse sequence (11) usuallycan decelerate the atom only by a small velocity value. As shown above, afterthe STIRAP decelerating process (11) the decelerated atom is in the productstate Ψ ( x, r, t f ) of Eq. (56), that is, the atom is in the internal state | g i andthe Gaussian wave-packet state Ψ ( x, t f ) of Eq. (59). Now the atom needs tobe decelerated further by another STIRAP pulse sequence. This basic STIRAP-based decelerating process may be expressed as | P + ℏ l i| g i → | P i| e i → | P − ℏ l i| g i . (63)In this decelerating process the atom is changed from the internal state | g i to | g i . This is opposite to the previous decelerating process (11). Therefore, theexperimental parameters for the STIRAP pulse sequence (63) needs to be setsuitably. Now the two Raman laser light beams in Eq. (9) for the STIRAPpulse sequence (63) should have the parameter settings: ( E L ( t ) , k L , ω L ) =( E l ( t ) , l , ω l ) and ( E L ( t ) , k L , ω L ) = ( E l ( t ) , l , ω l ), where the first Ramanlaser light beam ( E l ( t ) , l , ω l ) (the pumping pulse) couples the two internalstates | g i and | e i and it propagates along the opposite motional direction tothe atom, while the second beam ( E l ( t ) , l , ω l ) (the Stokes pulse) connects thetwo internal states | g i and | e i and it travels along the motional direction of theatom. The atomic three-state subspace {| ψ k ( r ) i} in the STIRAP deceleratingprocess (63) should be set by | ψ ( r ) i = | g i , | ψ ( r ) i = | g i , and | ψ ( r ) i = | e i , and the transition frequencies between the atomic internal energy levels in thedecelerating process (63) should be defined by ω l = ω = ( E − E ) / ℏ and ω l = ω = ( E − E ) / ℏ , respectively. If one makes an exchange E ↔ E in37ll those results obtained in the previous decelerating process (11), then theseresults can be adopted in the present decelerating process (63).Now suppose that at the initial time t the atom is in the product stateΨ ( x, r, t ) of Eq. (54). The atom first undergoes the STIRAP deceleratingprocess (11) and hence the product state Ψ ( x, r, t ) is completely transferredto the product state Ψ ( x, r, t f ) of Eq. (56) at the end time t f = t + T ofthe decelerating process (11). Then the atom undergoes the second STIRAPdecelerating process (63). Now one wants to calculate the atomic wave-packetproduct state at the end of the second decelerating process (63). At the initialtime t = t f of the second decelerating process (63) the atomic product stateis given by Ψ ( x, r, t ) = Ψ ( x, t ) | g i . Obviously, this product state is justthe product state of the atom at the end of the first decelerating process (11).Thus, the initial motional state Ψ ( x, t ) is the Gaussian wave-packet state ofEq. (59): Ψ ( x, t ) = Ψ ( x, t f ) . Then in the ideal adiabatic condition (30) atthe end time t ′ f = t + T of the second decelerating process (63) the atom iscompletely in the product state:Ψ ( x, r, t ′ f ) = Ψ ( x, t ′ f ) | g i (64)where the wave-packet motional state Ψ ( x, t ′ f ) can be calculated byΨ ( x, t ′ f ) = X p ρ ( p, t ) A ( p, t ′ f ) | p − ℏ l − ℏ l i . (65)Here the amplitude ρ ( p, t ) is given by Eq. (62) with the time t f = t , themomentum eigenstate | p − ℏ l − ℏ l i is still given by Eq. (7) , and the coefficient A ( p, t ′ f ) with the center-of-mass momentum P = p and ∆ p = p − p is writtenas A ( p, t ′ f ) = exp[ iβ l ( t ′ f , t )] exp[ i ℏ ( p M + E ) t ] × exp[ − i ℏ ( ( p − ℏ l − ℏ l ) M + E ) t ′ f ] × exp {− i ∆ pM ( l + l )[ t + Z t ′ f t dt ′ cos θ l ( t ′ )] } , (66)where the global phase β l ( t ′ f , t ) is still calculated by Eq. (45) with the relatedparameter settings such as the mixing angle θ l ( t ) and the phase modulationfunctions ϕ l ( t ) and ϕ l ( t ) of the present STIRAP pulse sequence (63). Thenby a complex calculation one can obtain from Eq. (65) the Gaussian wave-packetmotional state at the end of the decelerating process (63):Ψ ( x, t ′ f ) = exp[ iϕ ( t ′ f , t )][ (∆ x ) π ] / s x ) + i ℏ ( T +( t − t )+( t ′ f − t ))2 M × exp {−
14 ( x − z ) (∆ x ) + i ℏ ( T +( t − t )+( t ′ f − t ))2 M } exp[ ip x/ ℏ ] (67)38here the center-of-mass position z is given by z = z + ( p − ℏ l − ℏ l ) M ( t ′ f − t ) + ℏ ( l + l ) M Z t ′ f t dt ′ cos θ l ( t ′ ) , (68)the wave-packet spreading by ε = s x ) + 2[ ℏ ( T + ( t − t ) + ( t ′ f − t ))2 M (∆ x ) ] , (69)the mean momentum by p = p − ℏ l − ℏ l , and the global phase factor byexp[ iϕ ( t ′ f , t )] = exp[ iϕ ( t , t )] exp[ iβ l ( t ′ f , t )] × exp[ i ℏ ( p M + E ) t ] exp[ − i ℏ ( ( p − ℏ l − ℏ l ) M + E ) t ′ f ] . Here both the basic decelerating sequences (11) and (63) are studied in detailas they are the basic STIRAP-based decelerating processes. Any unitary decel-erating process in the quantum control process [22] may be constructed with atrain of these two basic decelerating sequences.When the atom is in the Gaussian wave-packet motional state Ψ ( x, t ) ofEq. (48) at the initial time t , the complex linewidth of the motional stateis W ( T ) = (∆ x ) + i ℏ T / (2 M ) . After the first basic STIRAP deceleratingprocess (11) the atom is in the Gaussian wave-packet motional state Ψ ( x, t )( t = t + T ) of Eq. (59) and the state has the complex linewidth W ( t − t + T ) = (∆ x ) + i ℏ ( T + t − t ) / (2 M ) . Then after the second basic STIRAPdecelerating process (63) the atom is in the Gaussian wave-packet motional stateΨ ( x, t ′ f ) ( t ′ f = t + T ) of Eq. (67) and the complex linewidth of the state is W ( t ′ f − t + T ) = (∆ x ) + i ℏ [ T + ( t − t ) + ( t ′ f − t )] / (2 M ) . Thus, one cansee that the real part of the complex linewidth of the Gaussian wave-packetmotional state of the atom keeps unchanged during these decelerating processes(11) and (63), while the imaginary part increases linearly with the time periodsof these decelerating processes. This result is found not only in the deceleratingprocesses but also in the accelerating processes and the free-particle motionalprocess.In a general case a unitary decelerating process may consist of a train of thetwo basic STIRAP decelerating processes (11) and (63). For convenience, hereeach basic STIRAP decelerating process is set to have the same time period T = t d and suppose that at the initial time of the unitary decelerating processthe atom is in the internal state | g i and has a large motional momentum suchthat the atom still moves along the initial direction + x even after the unitarydecelerating process. The basic decelerating sequences (11) and (63) may forma basic STIRAP decelerating cycle in such a way that first the decelerating39equence (11) and then the sequence (63) is applied to the decelerated atom.The unitary decelerating process may consist of many these basic STIRAP de-celerating cycles. Denote U d (11) and U d (63) as the unitary propagators of thebasic decelerating processes (11) and (63), respectively. Then a general unitarydecelerating process may be expressed as U D (2 N ) = U d N (63) U d N − (11) ...U d n (63) U d n − (11) ...U d (63) U d (11) (70a)or U D (2 N −
1) = U d N − (11) ...U d n (63) U d n − (11) ...U d (63) U d (11) (70b)where U d n − (11) is the propagator of the (2 n − − th basic decelerating unitfor n = 1 , , ..., N in the unitary decelerating process U D (2 N ) or U D (2 N − U d n (63) is the propagator of the 2 n − th basic decelerating unit. Each basicdecelerating unit with an even index 2 n in the unitary decelerating process istaken as the basic decelerating process (63), while that with an odd index (2 n − U d n − (11) = U d (11)and U d n (63) = U d (63) for n = 1 , , ..., N . In particular, U D (0) = E (theunit operator), U D (1) = U d (11) , and U D (2) = U d (63) U d (11). The unitarydecelerating processes U D (1) and U D (2) have been investigated in detail inthe preceding paragraphs. Obviously, the unitary decelerating process U D (2 N )consists of 2 N basic decelerating processes (11) and (63) alternately or N basicdecelerating cycles, while U D (2 N −
1) consists of N basic decelerating process(11) and N − U D (2 N ) or U D (2 N −
1) can be calculated exactly in the idealadiabatic condition (30). For the simplest cases N = 1 and 2 the time evolutionprocesses of the atom have already calculated in the ideal adiabatic condition(30) in the previous paragraphs. In order to calculate the time evolution pro-cess of the atom in a general unitary decelerating process one may first set upthe recursive relation between the two atomic product states at the end of theunitary decelerating processes U D (2 n −
1) and U D (2 n ) for n = 1 , , ..., N. Sincethe initial internal state of the atom is | g i in both the unitary decelerating pro-cesses U D (2 N ) and U D (2 N − , after the unitary decelerating process U D (2 n )(or U D (2 n − ≤ n ≤ N ) the final internal state of the atom is clearly | g i (or | g i ). Then the initial internal states of the atom in the basic deceleratingprocesses U d n +1 (11) and U d n (63) should be | g i and | g i , respectively. It isknown that at the initial time t the atom is in the Gaussian wave-packet mo-tional state Ψ ( x, t ) of Eq. (48) and the product state Ψ ( x, r, t ) of Eq. (54).It is also known that after the unitary decelerating sequences U D (1) and U D (2)act on the initial product state Ψ ( x, r, t ) of Eq. (54) the initial motional stateΨ ( x, t ) of Eq. (48) is converted into the Gaussian wave-packet motional statesΨ ( x, t f ) of Eq. (59) and Ψ ( x, t ′ f ) of Eq. (67), respectively. This means thatthe unitary decelerating sequences U D (1) and U D (2) do not change the Gaus-sian shape of the atomic motional state. Therefore, it is reasonable to deducethat after the unitary decelerating process U D (2 n ) for n = 0 , , , ..., N the atom40s in the Gaussian wave-packet motional state:Ψ d n ( x, t e n − ) = exp( iϕ d n )[ (∆ x ) π ] / s x ) + i ℏ ( T d +2 nt d )2 M × exp {−
14 ( x − z d n ) (∆ x ) + i ℏ ( T d +2 nt d )2 M } exp { iP d n x/ ℏ } (71)and also in the atomic wave-packet product state:Ψ d n ( x, r, t e n − ) = Ψ d n ( x, t e n − ) | g i = X p ρ ( p, t e n − ) | p i| g i . (72)where t e n − is the end time of the unitary decelerating process U D (2 n ) . In ananalogous way to calculating the amplitude ρ ( p, t ) via the equation (49a) onecan calculate the amplitude ρ ( p, t e n − ) of Eq. (72) from the motional stateΨ d n ( x, t e n − ) of Eq. (71). The result is ρ ( p, t e n − ) = exp( iϕ d n )[ 2(∆ x ) π ] / exp {− (∆ x ) ( p − P d n ℏ ) }× exp {− i ( ℏ ( T d + 2 nt d )2 M )( p − P d n ℏ ) } exp {− i ( p − P d n ℏ ) z d n } . (73)It will prove below that the states Ψ d n ( x, t e n − ) of (71) and Ψ d n ( x, r, t e n − ) of(72) are indeed the wave-packet motional state and product state of the atomat the end of the unitary decelerating process U D (2 n ) , respectively.First of all, the product state Ψ ( x, r, t ) of Eq. (54) and the motional stateΨ ( x, t ) of Eq. (48) are the initial product state and motional state of the atomin the presence of the unitary decelerating process U D (2 n ) (or U D (2 n − U D (0) = E (the unity operator), respectively.This means that the atomic wave-packet product state Ψ d ( x, r, t e − ) of Eq. (72)and the motional state Ψ d ( x, t e − ) of Eq. (71) with n = 0 should be equal toΨ ( x, r, t ) of Eq. (54) and Ψ ( x, t ) of Eq. (48), respectively,Ψ d ( x, r, t e − ) = Ψ ( x, r, t ) , Ψ d ( x, t e − ) = Ψ ( x, t ) , while the momentum wave-packet state ρ ( p, t e − ) of Eq. (73) thus is just ρ ( p, t )of Eq. (50). Indeed, these equations (71), (72), and (73) show this point, if inEqs. (71), (72), and (73) one sets the parameters: n = 0 , t e − = t d = t ,ϕ d = ϕ , T d = T , z d = z , P d = p , and t d = T, where T is the time periodof the basic decelerating sequence (11). Next, the product state Ψ ( x, r, t ′ f ) ofEq. (64) and the motional state Ψ ( x, t ′ f ) of Eq. (67) with the time t ′ f = t e arejust the product state Ψ d ( x, r, t e ) of Eq. (72) and the motional state Ψ d ( x, t e )of Eq. (71) with n = 1 at the end time t e of the unitary decelerating process U D (2), respectively,Ψ d ( x, r, t e ) = Ψ ( x, r, t ′ f ) , Ψ d ( x, t e ) = Ψ ( x, t ′ f ) . t d = t f = t = t + t d , t e = t ′ f = t d + t d = t + 2 t d , ϕ d = ϕ ( t e , t d ) , z d = z , and P d = p , One therefore shows that the motional state Ψ d n ( x, t e n − ) of Eq.(71) and the product state Ψ d n ( x, r, t e n − ) of Eq. (72) are correct for both theunitary decelerating processes U D (0) ( n = 0) and U D (2) ( n = 1) . It will provebelow that both the motional state Ψ d n ( x, t e n − ) of Eq. (71) and the productstate Ψ d n ( x, r, t e n − ) of Eq. (72) are also correct for the unitary deceleratingprocess U D (2 n ) with n = 0 , , ..., N .Suppose that the states Ψ d n ( x, t e n − ) of Eq. (71) and Ψ d n ( x, r, t e n − ) ofEq. (72) are correct for the unitary decelerating process U D (2 n ). It is knownthat the internal state of the atom is | g i at the end of the unitary deceleratingprocess U D (2( n + 1)) . Then one needs only to prove that the motional state ofEq. (71) is also correct for the unitary decelerating process U D (2( n + 1)). Thepropagator of the unitary decelerating process U D (2( n + 1)) can be written as U D (2( n + 1)) = U d n +2 (63) U D (2 n + 1) = U d n +2 (63) U d n +1 (11) U D (2 n ) . According to the assumption the motional state Ψ d n ( x, t e n − ) is just the finalmotional state of the atom when the atom is acted on by the unitary propagator U D (2 n ). Obviously, the motional state Ψ d n ( x, t d n ) = Ψ d n ( x, t e n − ) is also theinitial motional state of the (2 n + 1) − th basic decelerating process (11) with thepropagator U d n +1 (11) in the unitary decelerating process U D (2( n + 1)), wherethe initial time is denoted as t d n = t e n − and there are the recursive relations: t e − = t d = t , t e = t d = t d + t d , t e = t d = t d + 2 t d , and t d n = t d n − + t d ( n > U d n +1 (11) isgiven by Ψ d n ( x, r, t d n ) = Ψ d n ( x, r, t e n − ) of Eq. (72). Now the initial productstate Ψ d n ( x, r, t d n ) is acted on by the unitary propagator U d n +1 (11) . Then itcan turn out that at the end time t e n = t d n + t d of the basic decelerating process U d n +1 (11) the initial motional state Ψ d n ( x, t d n ) and product state Ψ d n ( x, r, t d n )are respectively transferred into the motional state:Ψ d n +1 ( x, t e n ) = exp( iϕ d n +1 )[ (∆ x ) π ] / s x ) + i ℏ ( T d +(2 n +1) t d )2 M × exp {−
14 ( x − z d n +1 ) (∆ x ) + i ℏ ( T d +(2 n +1) t d )2 M } exp { iP d n +1 x/ ℏ } (74)and the product state:Ψ d n +1 ( x, r, t e n ) = Ψ d n +1 ( x, t e n ) | g i (75)where P d n +1 = P d n − ℏ k − ℏ k , (76) z d n +1 = z d n + P d n +1 M t d + ℏ ( k + k ) M Z t e n t d n dt ′ cos θ ( t ′ ) , (77)42xp( iϕ d n +1 ) = exp( iϕ d n ) exp[ iβ ( t e n , t d n )] exp { i ℏ ( ( P d n ) M + E ) t d n }× exp {− i ℏ ( ( P d n +1 ) M + E ) t e n } . (78)The computational process from the initial state Ψ d n ( x, t e n − ) to the final stateΨ d n +1 ( x, t e n ) is the same as the previous one from the initial state Ψ ( x, t ) of(48) to the final state Ψ ( x, t f ) of (59). There are the relations:Ψ d n +1 ( x, r, t e n ) = U d n +1 (11)Ψ d n ( x, r, t d n ) = U D (2 n + 1)Ψ ( x, r, t ) . These relations show that both the states Ψ d n +1 ( x, r, t e n ) and Ψ d n +1 ( x, t e n )are also the product state and the motional state of the atom at the end time t e n = t d n + t d of the unitary decelerating process U D (2 n + 1) , respectively.The atomic product state Ψ d n +1 ( x, r, t e n ) of Eq. (75) at the end of theunitary decelerating process U D (2 n + 1) will be used to calculate the atomicproduct state at the end of the unitary decelerating process U D (2 n + 2) . Thiscomputational process is the same as the previous one from the initial stateΨ ( x, t ) = Ψ ( x, t f ) of (59) to the final state Ψ ( x, t ′ f ) of (67). There are therelations:Ψ d n +2 ( x, r, t e n +1 ) = U d n +2 (63)Ψ d n +1 ( x, r, t e n ) = U D (2 n + 2)Ψ ( x, r, t ) .. These relations show that the atomic product state Ψ d n +2 ( x, r, t e n +1 ) at the endof the unitary decelerating process U D (2 n + 2) can be obtained by applying thepropagator U d n +2 (63) to the atomic product state Ψ d n +1 ( x, r, t e n ) of Eq. (75).For convenient calculation, the atomic motional state Ψ d n +1 ( x, t e n ) of Eq. (74)is rewritten as ( n ′ = n + 1)Ψ d n ′ − ( x, t d n ′ − ) = exp( iϕ d n ′ − )[ (∆ x ) π ] / s x ) + i ℏ ( T d +(2 n ′ − t d )2 M × exp {−
14 ( x − z d n ′ − ) (∆ x ) + i ℏ ( T d +(2 n ′ − t d )2 M } exp { iP d n ′ − x/ ℏ } . (79)Then the atomic product state of Eq. (75) can be rewritten asΨ d n ′ − ( x, r, t d n ′ − ) = Ψ d n ′ − ( x, t d n ′ − ) | g i = X p ρ ( p, t d n ′ − ) | p i| g i (80)where the amplitude ρ ( p, t d n ′ − ) can be calculated from the motional state ofEq. (79) and is given by ρ ( p, t d n ′ − ) = exp( iϕ d n ′ − )[ 2(∆ x ) π ] / exp {− (∆ x ) ( p − P d n ′ − ℏ ) }× exp {− i ℏ ( T d + (2 n ′ − t d )2 M ( p − P d n ′ − ℏ ) } exp[ − i ( p − P d n ′ − ) ℏ z d n ′ − ] . (81)43ow the atomic product state Ψ d n ′ − ( x, r, t d n ′ − ) of Eq. (80) is applied by theunitary propagator U d n ′ (63) ( n ′ = n + 1). Then it can turn out that at the endtime t e n ′ − = t d n ′ − + t d of the unitary decelerating process U D (2 n + 2) theatomic wave-packet motional state takes the formΨ d n ′ ( x, t e n ′ − ) = exp( iϕ d n ′ )[ (∆ x ) π ] / s x ) + i ℏ ( T d +2 n ′ t d )2 M × exp {−
14 ( x − z d n ′ ) (∆ x ) + i ℏ ( T d +2 n ′ t d )2 M } exp { iP d n ′ x/ ℏ } . (82)and the atomic product state isΨ d n ′ ( x, r, t e n ′ − ) = Ψ d n ′ ( x, t e n ′ − ) | g i , (83)where P d n ′ = P d n ′ − − ℏ l − ℏ l , (84) z d n ′ = z d n ′ − + P d n ′ M t d + ℏ ( l + l ) M Z t e n ′− t d n ′− dt ′ cos θ l ( t ′ ) , (85)exp( iϕ d n ′ ) = exp( iϕ d n ′ − ) exp[ iβ l ( t e n ′ − , t d n ′ − )] × exp[ i ℏ ( ( P d n ′ − ) M + E ) t d n ′ − ] exp[ − i ℏ ( ( P d n ′ ) M + E ) t e n ′ − ] . (86)Now by comparing the motional state Ψ d n ′ ( x, t e n ′ − ) of Eq. (82) with themotional state Ψ d n ( x, t e n − ) of Eq. (71) and the product state Ψ d n ′ ( x, r, t e n ′ − )of Eq. (83) with the product state Ψ d n ( x, r, t e n − ) of Eq. (72) one can concludeby the mathematical principle of induction that the motional state Ψ d n ( x, t e n − )of Eq. (71) and the product state Ψ d n ( x, r, t e n − ) of Eq. (72) are indeed thestates of the atom at the end of the unitary decelerating process U D (2 n ) for n = 0 , , , ..., N . In an analogous way, one can prove that the product stateΨ d n +1 ( x, r, t d n +1 ) of Eq. (75) and the motional state Ψ d n +1 ( x, t d n +1 ) of Eq.(74) are the states of the atom at the end of the unitary decelerating process U D (2 n + 1) for n = 0 , , , ..., N − P d n +1 of Eq. (76)and P d n ′ of Eq. (84) are given by, respectively, P d n +1 = P d − ( n + 1)( ℏ k + ℏ k ) − n ( ℏ l + ℏ l ) ,P d n ′ = P d − n ′ ( ℏ k + ℏ k ) − n ′ ( ℏ l + ℏ l ) . It is known that the recursive relations for the atomic motional momentumare given by P d n +1 = P d n − ℏ k − ℏ k for n = 0 , , ..., N − , and P d n ′ = P d n ′ − − ℏ l − ℏ l for n ′ = 1 , , ..., N, which are obtained from Eqs. (76) and(84), respectively. The two recursive relations together can lead directly to thetwo formulae for the atomic motional momentum P d n +1 and P d n ′ .44ne also can calculate exactly the time evolution process of the atom inthe unitary STIRAP-based accelerating process in the ideal adiabatic condition(30). There are also two basic STIRAP-based accelerating sequences whichcorrespond to the basic decelerating sequences (11) and (63), respectively. Oneof which is already expressed as (11a). The basic accelerating sequence (11a)corresponds to the basic decelerating sequence (11). Another may be expressedin an intuitive form | P − ℏ l i| g i → | P i| e i → | P + ℏ l i| g i . (63a)This basic accelerating sequence corresponds to the basic decelerating sequence(63). In an analogous way to constructing the unitary decelerating processes U D (2 N ) and U D (2 N −
1) one may build up the unitary accelerating processes U A (2 N ) and U A (2 N −
1) out of the basic accelerating sequences (11a) and (63a), U A (2 N ) = U a N (63 a ) U a N − (11 a ) ...U a n (63 a ) U a n − (11 a ) ...U a (63 a ) U a (11 a )(87a)or U A (2 N −
1) = U a N − (11 a ) ...U a n (63 a ) U a n − (11 a ) ...U a (63 a ) U a (11 a ) (87b)where U a n − (11 a ) and U a n (63 a ) for n = 1 , , ..., N are the unitary propagators ofthe (2 n − − th basic accelerating sequence (11a) and (2 n ) − th basic acceleratingsequence (63a), respectively. Here also suppose that the atom is in the internalstate | g i at the initial time in both the unitary accelerating processes U A (2 N )and U A (2 N − . The time evolution process of the atom in the unitary accelerating process U A (2 N ) (and U A (2 N − U D (2 N ) (and U D (2 N − . Actually, the recursive relations (71)–(78) and (79)–(86) of the unitary decelerating process U D (2 N ) or U D (2 N −
1) can be used aswell for the unitary accelerating process U A (2 N ) or U A (2 N −
1) if one makestransformations: k → − k a , k → − k a , and θ ( t ) → θ a ( t ) in those recursiveequations (71)–(78) and l → − l a , l → − l a , and θ l ( t ) → θ la ( t ) in those recur-sive equations (79)–(86). As an example, suppose that the initial wave-packetmotional state for the atom in the unitary accelerating process U A (2 n ) is givenby Ψ a ( x, t a ) = exp( iϕ a )[ (∆ x ) π ] / s x ) + i ℏ T a M ] × exp {−
14 [ x − z a ] [(∆ x ) + i ℏ T a M ] } exp[ iP a x/ ℏ ] (88)and the atomic wave-packet product state byΨ a ( x, r, t a ) = Ψ a ( x, t a ) | g i = X p ρ ( p, t a ) | p i| g i . (89)45hen it can turn out that the momentum wave-packet state ρ ( p, t a ) of themotional state Ψ a ( x, t a ) can be written as ρ ( p, t a ) = exp( iϕ a )[ 2(∆ x ) π ] / exp {− (∆ x ) ( p − P a ℏ ) }× exp {− i ( ℏ T a M )( p − P a ℏ ) } exp {− i ( p − P a ℏ ) z a } . (90)Now the initial wave-packet product state Ψ a ( x, r, t a ) of the atom undergoes theunitary accelerating process U A (2 n ). Then it can be proved that at the end ofthe unitary accelerating process U A (2 n ) the atomic wave-packet motional stateis given byΨ a n ( x, t e n − ) = exp( iϕ a n )[ (∆ x ) π ] / s x ) + i ℏ ( T a +2 nt a )2 M × exp {−
14 [ x − z a n ] (∆ x ) + i ℏ ( T a +2 nt a )2 M } exp { iP a n x/ ℏ } (91)and the atomic wave-packet product state byΨ a n ( x, r, t a n ) = Ψ a n ( x, t e n − ) | g i (92)where the end time of the unitary accelerating process U A (2 n ) is t e n − = t a n = t a + 2 nt a for n = 0 , , , ...N, the atomic motional momentum P a n is given by P a n = P a + n ( ℏ k a + ℏ k a ) + n ( ℏ l a + ℏ l a ) , (93)and the center-of-mass position z a n is determined from these recursive relations: z a k − = z a k − + P a k − M t a − ℏ ( k a + k a ) M Z t a +(2 k − t a t a +(2 k − t a dt ′ cos θ a ( t ′ ) , (94a) z a k = z a k − + P a k M t a − ℏ ( l a + l a ) M Z t a +2 kt a t a +(2 k − t a dt ′ cos θ la ( t ′ ) , (94b) P a k − = P a k − + ℏ ( k a + k a ) , P a k = P a k − + ℏ ( l a + l a ) , where the initial center-of-mass position and momentum are z a and P a , respec-tively, the index 1 ≤ k ≤ n, both the basic accelerating sequence (11a) and(63a) have the same duration t a , and the global phase factor exp( iϕ a n ) can alsobe calculated through the recursive relations similar to Eq. (78) and (86). Inan analogous way, one also can calculate exactly the time evolution process ofthe atom in the unitary accelerating process U A (2 n −
1) in the ideal adiabaticcondition (30).
6. The space- and time-compressing processes based on the uni-tary decelerating and accelerating processes n d basic STIRAP decelerating sequences (11) and (63)alternately, which is given by U D (2 n d ) of Eq. (70a), and the ideal adiabaticcondition (30) is met for all these basic decelerating sequences. According tothe quantum control process the unitary decelerating sequence is applied selec-tively in the given spatial region [ D L , D R ] in the right-hand potential well of thedouble-well potential field, where D L and D R are the left- and right-boundarypositions of the spatial region in the coordinate axis, respectively. The halting-qubit atom can be decelerated by the unitary decelerating sequence U D (2 n d )only when the atom enters into the spatial region [ D L , D R ]. Thus, the spatialregion [ D L , D R ] may be called the decelerating spatial region. The deceleratingspatial region must cover sufficiently the whole wave-packet motional state ofthe atom during the whole unitary decelerating process when the atom is de-celerated in the decelerating region. The decelerating region is so wide that forthe wave-packet motional state of the atom the Raman laser light beams of theunitary decelerating sequence U D (2 n d ) can be thought of as infinite plane-waveelectromagnetic fields. Suppose that the halting-qubit atom is in the productstate Ψ ( x, r, t ) of Eq. (54) (or in the motional state Ψ ( x, t ) of Eq. (48) andthe internal state | g i ) and in the decelerating region [ D L , D R ] when the unitarydecelerating sequence U D (2 n d ) is turned on at the initial time t . Here the spa-tial position of an atom is defined as the center-of-mass position of the atomicwave-packet motional state. Now the center-of-mass position and wave-packetspread of the initial motional state Ψ ( x, t ) are z and ε , respectively. Sincethe halting-qubit atom is in the decelerating region [ D L , D R ] at the time t , thatis, z ∈ [ D L , D R ], both the distances ( z − D L ) and ( D R − z ) must be muchgreater than the wave-packet spreading ε , that is, ( D R − z ) > ( z − D L ) >> ε , meaning that the decelerating region [ D L , D R ] covers sufficiently the whole ini-tial wave-packet motional state Ψ ( x, t ) . The halting-qubit atom starts to bedecelerated by the unitary decelerating process U D (2 n d ) at the initial time t and in the decelerating region [ D L , D R ]. With the help of the recursive relations(71)–(78) and (79)–(86) one can prove that at the end time t d n d = t + 2 n d t d ofthe unitary decelerating process U D (2 n d ) the motional state of the halting-qubitatom is given byΨ d n d ( x, t d n d ) = exp( iϕ d n d )[ (∆ x ) π ] / s x ) + i ℏ ( T d +2 n d t d )2 M × exp {−
14 ( x − z d n d ) (∆ x ) + i ℏ ( T d +2 n d t d )2 M } exp { iP d n d x/ ℏ } (95)and the atomic product state byΨ d n d ( x, r, t d n d ) = Ψ d n d ( x, t d n d ) | g i , (96)where the atomic motional momentum P d n d is given by P d n d = p − n d ( ℏ k + ℏ k ) − n d ( ℏ l + ℏ l ) , (97)47nd the center-of-mass position z d n d can be calculated by the recursive relations(77) and (85), z d n − = z d n − + P d n − M t d + ℏ ( k + k ) M Z t e n − t d n − dt ′ cos θ ( t ′ ) , (98a) z d n = z d n − + P d n M t d + ℏ ( l + l ) M Z t e n − t d n − dt ′ cos θ l ( t ′ ) , (98b) P d n − = P d n − − ( ℏ k + ℏ k ) , P d n = P d n − − ( ℏ l + ℏ l ) , where 1 ≤ n ≤ n d ; z d = z , P d = p , T d = T ; t d = t , t dk +1 = t dk + t d , and t ek = t dk +1 for 0 ≤ k ≤ n d − , and the global phase factor exp( iϕ d n d ) canbe calculated by Eq. (78) and (86) with the initial phase ϕ d = ϕ . Both theinitial atomic product state Ψ ( x, r, t ) of Eq. (54) and the final product stateΨ d n d ( x, r, t d n d ) of Eq. (96) show that before and after the unitary deceleratingprocess U D (2 n d ) the halting-qubit atom is in the same internal state | g i , whileits initial motional state Ψ ( x, t ) of Eq. (48) is changed to the motional stateΨ d n d ( x, t d n d ) of Eq. (95) after the unitary decelerating process. The finalmotional state Ψ d n d ( x, t d n d ) of Eq. (95) has the center-of-mass position z d n d and the wave-packet spreading ε = s x ) + 2[ ℏ ( T d + 2 n d t d )2 M (∆ x ) ] . Though the atom moves a distance ( z d n d − z ) during the unitary deceleratingprocess, the center-of-mass position z d n d ∈ [ D L , D R ] as the atom is still in thedecelerating region [ D L , D R ] at the end of the unitary decelerating process.Then both the distances ( z d n d − D L ) and ( D R − z d n d ) must be much greaterthan the wave-packet spreading ε, that is, ( z d n d − D L ) > ( D R − z d n d ) >> ε. Thismeans that the decelerating region [ D L , D R ] also covers sufficiently the wholefinal motional state Ψ d n d ( x, t d n d ) . There are two extra constraint conditions onthe decelerating region [ D L , D R ]. If the atom has not yet entered into thedecelerating region when the unitary decelerating sequence is switched on or itleaves the decelerating region after the unitary decelerating process is switchedoff, then it will not be affected by the unitary decelerating sequence or by nextunitary decelerating sequences. The two constraint conditions are stated below.In the quantum control process [22] the halting-qubit atom may enter intothe right-hand potential well from the left-hand one in any i − th cycle of thequantum program for i = 1 , , ..., m r . The i − th possible wave-packet motionalstate of the halting-qubit atom is just defined as the atomic motional statewhen the atom enters into the right-hand potential well in the i − th cycle ofthe quantum program. Thus, there is a different time for any possible atomicmotional state such as the i − th wave-packet motional state to enter into theright-hand potential well. If the time period of each cycle of the quantumprogram is ∆ T, then the time difference between the i − th and j − th ( i < j )48ave-packet motional states to enter into the right-hand potential well is givenby ∆ T ( i, j ) = ( j − i )∆ T for i < j and i, j = 1 , , ..., m r . This time differenceresults in a center-of-mass distance in space between the two wave-packet mo-tional states. If the halting-qubit atom moves along the direction + x with thevelocity p /M , then the distance is given by ∆ L ( i, j ) = ( j − i )∆ T ( p /M ) . Nowexamine two consecutive possible wave-packet motional states: the i − th and( i + 1) − th wave-packet motional states. Here for convenience the i − th wave-packet motional state is set to the motional state Ψ ( x, t ) of Eq. (48). Thenat the time t the i − th wave-packet motional state is in the decelerating region[ D L , D R ] and its center-of-mass position is z , while the center-of-mass positionof the ( i + 1) − th wave-packet motional state is clearly [ z − ∆ T ( p /M )]. It isknown that the total duration for the unitary decelerating sequence U D (2 n d )is 2 n d t d . Then at the end time t + 2 n d t d of the unitary decelerating pro-cess the center-of-mass position of the ( i + 1) − th wave-packet motional statebecomes [ z − (∆ T − n d t d )( p /M )] . Obviously, the distance between this center-of-mass position and the left-end position of the decelerating region [ D L , D R ]is D L − [ z − (∆ T − n d t d )( p /M )] . Denote ε i +1 ( t + 2 n d t d ) as the wave-packet spreading of the ( i + 1) − th wave-packet motional state at the time t + 2 n d t d . The wave-packet spreading ε i +1 ( t + 2 n d t d ) may be calculated withthe help of the i − th wave-packet state Ψ ( x, t ) and the free-particle propaga-tor. Then this distance must be much greater than ε i +1 ( t + 2 n d t d ) , that is, D L − [ z − (∆ T − n d t d )( p /M )] >> ε i +1 ( t + 2 n d t d ) , so that the ( i + 1) − thwave-packet motional state is not affected by the unitary decelerating sequence U D (2 n d ) during the whole unitary decelerating process. This is a constraintcondition on the decelerating region [ D L , D R ] . It is known that at the end time t + 2 n d t d of the unitary decelerating pro-cess U D (2 n d ) the i − th wave-packet motional state is the state Ψ d n d ( x, t d n d )of Eq. (95), which has the center-of-mass position z d n d and the motional mo-mentum P d n d . After the unitary decelerating process the i − th wave-packetmotional state (i.e., the halting-qubit atom) moves along the direction + x withthe velocity P d n d /M . Since the atom is usually decelerated greatly by thedecelerating sequence U D (2 n d ) the atomic velocity P d n d /M is much less thanthe original velocity p /M . Obviously, the i − th wave-packet motional statemoves to the position z d n d + (∆ T − n d t d ) P d n d /M when next unitary decel-erating sequence starts to work at the time t + ∆ T. Then the distance be-tween this position and the right-end position of the decelerating region [ D L ,D R ] is given by z d n d + (∆ T − n d t d ) P d n d /M − D R . Denote ε i ( t + ∆ T ) asthe wave-packet spreading of the i − th wave-packet motional state at the time t + ∆ T. The wave-packet spreading ε i ( t + ∆ T ) can be calculated with thehelp of the motional state Ψ d n d ( x, t d n d ) of Eq. (95) and the free-particle prop-agator. Then this distance must be much greater than ε i ( t + ∆ T ) , that is, z d n d + (∆ T − n d t d ) P d n d /M − D R >> ε i ( t + ∆ T ) , so that, from the time t + ∆ T on, the i − th wave-packet motional state is no longer affected by theunitary decelerating sequences. This is another constraint condition on thedecelerating region [ D L , D R ] . i − th wave-packet motional state is decelerated from the initial time t to the end time t + 2 n d t d by the unitary decelerating process U D (2 n d ) in thedecelerating region [ D L , D R ]. It moves a distance z d n d − z along the direction+ x and spends the time 2 n d t d and it is decelerated down to P d n d /M from theinitial velocity p /M during the unitary decelerating process. According to thequantum control process [22] the ( i + 1) − th wave-packet motional state arrivesat the position z in the decelerating region [ D L , D R ] at the time t + ∆ T. Thenthe ( i + 1) − th wave-packet motional state at the time t + ∆ T is really equal tothe i − th wave-packet motional state at the time t up to a global phase factor,indicating that the ( i + 1) − th wave-packet motional state at the time t + ∆ T isalso equal to the motional state Ψ ( x, t ) of Eq. (48) up to a global phase factor.Generally, according to the quantum control process each of these m r possiblewave packet motional states is really equal to the motional state Ψ ( x, t ) ofEq. (48) up to a global phase factor when the wave-packet motional statearrives at the same position z in the decelerating region [ D L , D R ]. Just likethe i − th wave-packet motional state at the time t the ( i + 1) − th wave-packetmotional state at the time t + ∆ T is decelerated by the unitary deceleratingprocess U D (2 n d ). It also moves the distance z d n d − z along the direction + x andspends the time 2 n d t d and it is also decelerated down to P d n d /M from the initialvelocity p /M during the unitary decelerating process. Generally, each of these m r possible wave-packet motional states moves the same distance z d n d − z alongthe direction + x and also spends the same time 2 n d t d and it is also decelerateddown to the same velocity P d n d /M from the same initial velocity p /M duringthe unitary decelerating process. The difference among these m r possible wave-packet motional states is that the starting time is different to decelerate eachone of these wave-packet motional states by the unitary decelerating process U D (2 n d ).In the quantum control process [22] the unitary decelerating sequence isused to decelerate the halting-qubit atom so that the center-of-mass distancesbetween these m r possible wave-packet motional states of the atom can benarrowed greatly. Thus, the unitary decelerating process U D (2 n d ) is really aspace-compressing process for these possible wave-packet motional states. Sinceeach one of these m r possible wave-packet motional states spends the sametime 2 n d t d in the unitary decelerating process U D (2 n d ) , the time difference∆ T ( i, j ) = ( j − i )∆ T between the i − th and j − th wave-packet motional states( i < j ; i, j = 1 , , ..., m r ) does not change before and after the unitary decel-erating process. It is known that each possible wave-packet motional state hasthe initial moving velocity p /M before the unitary decelerating process and thefinal moving velocity P d n d /M > P d n d /M can be obtained from Eq. (97), P d n d /M = [ p − n d ( ℏ k + ℏ k ) − n d ( ℏ l + ℏ l )] /M. (99)If the number n d of the unitary decelerating process U D (2 n d ) is chosen suitably,then the velocity P d n d /M can be much less than the initial one p /M. Beforethe unitary decelerating process the distance between the i − th and j − th wave-50acket motional states is ∆ L ( i, j ) = ( j − i )∆ T ( p /M ), since the velocity is p /M and the time difference is ∆ T ( i, j ) = ( j − i )∆ T before the unitary de-celerating process. After the unitary decelerating process the atomic movingvelocity is P d n d /M and the time difference is still ∆ T ( i, j ) = ( j − i )∆ T . Thenafter the unitary decelerating process the distance between the i − th and j − th( i < j ) wave-packet motional states is equal to∆ L ( i, j ) = ( j − i )∆ T ( P d n d /M ) , (100)where i < j ; i, j = 1 , , ..., m r . Since the velocity ( P d n d /M ) << p /M, thedistance ∆ L ( i, j ) << ∆ L ( i, j ) , indicating that the spatial region to cover allthese m r possible wave-packet motional states is greatly compressed after theunitary decelerating process. The distance ∆ L ( i, j ) of Eq. (100) has beenobtained in the previous paper [22], where the atomic velocity ( P d n d /M ) isdenoted as v after the unitary decelerating process. Then the ratio of the twodistances ∆ L ( i, j ) and ∆ L ( i, j ) is the space-compressing factor for these wave-packet motional states after and before the unitary decelerating process, whichcan be calculated by R s = ∆ L ( i, j )∆ L ( i, j ) = [ p − n d ( ℏ k + ℏ k ) − n d ( ℏ l + ℏ l )] p . (101)The space-compressing factor is not dependent upon the indices i and j , sincethe time difference ∆ T ( i, j ) does not change before and after the unitary decel-erating process and since all these possible wave-packet motional states have thesame initial motional momentum p and also the same final motional momen-tum P d n d after each of these possible wave-packet motional states undergoesthe same unitary decelerating process U D (2 n d ) in the same decelerating region[ D L , D R ].Before the unitary accelerating process comes to making a real action on thehalting-qubit atom, the atom needs to stay in the right-hand potential well fora time period to wait for the quantum program running to the end accordingto the quantum control process [22]. The time period during which the halting-qubit atom stays in the right-hand potential well is different and dependentupon how early the halting-qubit atom enters into the right-hand potential wellfrom the left-hand one. When the atom enters into the right-hand potentialwell at an earlier time, it will stay in the right-hand potential well for a longertime. Denote that T s ( i ) = T s − ( i − T with the index i = 1 , , ..., m r isthe time period during which the atom moves freely along the direction + x in the right-hand potential well after the atom is decelerated by the unitarydecelerating sequence U D (2 n d ) and before the atom starts to be accelerated atthe end time of the quantum program. The index i indicates that the halting-qubit atom enters into the right-hand potential well from the left-hand one inthe i − th cycle of the quantum program. Here suppose that the last unitarydecelerating process U D (2 n d ) is turned off before the quantum program comesto the end. The calculation for the time evolution process of the halting-qubitatom moving freely during the time period T s ( i ) needs to use the free-particle51nitary propagator. Now the unitary propagator of a free particle is written as[25] G ( x ′ , t ′ ; x, t ) = s M πi ℏ ( t ′ − t ) exp[ iM ( x ′ − x ) ℏ ( t ′ − t ) ] . (102)Then the time evolution process of an atom in a free-particle motion with thetime period T = t ′ − t can be calculated byΨ( x ′ , t ′ ) = Z dxG ( x ′ , t ′ ; x, t )Ψ( x, t ) . (103)It is known that the i − th wave-packet motional state of the atom is givenby the motional state Ψ d n d ( x, t d n d ) of Eq. (95) at the end time t d n d = t +2 n d t d of the unitary decelerating process. When the wave-packet motional stateΨ d n d ( x, t d n d ) moves freely along the direction + x for the time period T s ( i )from the time t d n d to the time t d n d + T s ( i ) , it will change to another Gaussianwave-packet motional state. This Gaussian wave-packet motional state canbe calculated from the equation (103) by taking the initial state Ψ( x, t ) asΨ d n d ( x, t d n d ) of Eq. (95), using the free-particle propagator G ( x ′ , t ′ ; x, t ) of Eq.(102), and denoting t ′ = t d n d + T s ( i ) and t = t d n d . By a complex calculation, inwhich the Gaussian integral (58) has been used, the final Gaussian wave-packetstate Ψ( x ′ , t ′ ) can be obtained explicitly, which now is renamed Ψ Fi ( x, t d n d + T s ( i )), Ψ Fi ( x, t d n d + T s ( i )) = exp( iϕ d n d ) exp {− i ( P d n d ) T s ( i )2 ℏ M }× [ (∆ x ) π ] / s x ) + i ℏ ( T d +2 n d t d + T s ( i ))2 M ] × exp {−
14 [ x − z d n d − ( P d n d /M ) T s ( i )] [(∆ x ) + i ℏ ( T d +2 n d t d + T s ( i ))2 M ] } exp { iP d n d x/ ℏ } . (104)On the other hand, the atomic internal state | g i and the motional momentum P d n d keep unchanged during the free-particle motion of the atom. Therefore, be-fore the unitary accelerating process starts at the end of the quantum program,these m r possible wave-packet motional states are given by Ψ Fi ( x, t d n d + T s ( i ))of Eq. (104) for i = 1 , , ..., m r and each of them has a different center-of-massposition: z d n d + ( P d n d /M ) T s ( i ) , a different global phase factor:exp( iϕ d n d ) exp {− i ( P d n d ) T s ( i ) / (2 ℏ M ) } , and a different complex linewidth: W i ( T d + 2 n d t d + T s ( i )) = (∆ x ) + i [ ℏ ( T d + 2 n d t d + T s ( i ))2 M ] . (105)An important fact is that the imaginary part of the complex linewidth of themotional state Ψ Fi ( x, t d n d + T s ( i )) increases linearly with the time period T s ( i ) , m r possiblewave-packet motional states has the same wave-packet spreading and the samecomplex linewidth W ( T d +2 n d t d ) = (∆ x ) + i ℏ ( T d +2 n d t d )2 M before the free-particlemotion, as can seen from the state Ψ d n d ( x, t d n d ) of Eq. (95), each possible wave-packet motional state has a larger wave-packet spreading and a different com-plex linewidth W i ( T d + 2 n d t d + T s ( i )) when the unitary accelerating sequencestarts to act on the halting-qubit atom at the end of the quantum program.Obviously, the first wave-packet motional state Ψ F ( x, t d n d + T s ) has the largestwave-packet spreading, while the last motional state Ψ Fm r ( x, t d n d + T s ( m r )) hasthe least one. These show that the free-particle motion of the halting-qubitatom leads to the difference among the wave-packet spreads of these m r possi-ble wave-packet motional states and makes these wave-packet motional statesbroader. This difference may have a significant impact on the quantum controlprocess [22]. On the other hand, the free-particle motion of the halting-qubitatom does not change the time differences and the distances in space betweenthese m r possible wave-packet motional states. This is because the motionalmomentum P d n d is the same for all these m r possible wave-packet motionalstates and keeps unchanged during the free-particle motion. Thus, the distancebetween the i − th and j − th ( i < j ) wave-packet motional states is still given by∆ L ( i, j ) of Eq. (100) and their time difference by ∆ T ( i, j ) = ( j − i )∆ T . Par-ticularly, the distance between two nearest wave-packet motional states is givenby ∆ T ( P d n d /M ) . Obviously, the halting-qubit atom moves a distance equalto ( P d n d /M ) T s ( i ) along the direction + x in the time period T s ( i ) of the free-particle motion. This distance is dependent upon the index i . The first wave-packet motional state Ψ F ( x, t d n d + T s ) moves the largest distance ( P d n d /M ) T s which decides mainly the dimensional size of the right-hand potential well, whilethe last wave-packet motional state Ψ Fm r ( x, t d n d + T s ( m r )) moves the shortestdistance ( P d n d /M )[ T s − ( m r − T ].The atomic wave-packet states { Ψ Fi ( x, t d n d + T s ( i )) } of Eq. (104) showthat just before the unitary accelerating sequence is switched on, all these m r possible wave-packet states of Eq. (104) are in the spatial region [ x ( m r ) − ε d ( m r ) , x (1) + ε d (1)] , where x ( j ) and ε d ( j ) ( j = 1 , , ..., m r ) are the center-of-mass position and the wave-packet spreading of the j − th wave-packet stateΨ Fj ( x, t d n d + T s ( j )) , respectively. Suppose that all these m r possible wave-packet states are accelerated uniformly by the unitary accelerating sequenceand each possible wave-packet state moves the same distance L A during theunitary accelerating process. The distance L A will be obtained later. Obviously,after the unitary accelerating process all these m r possible wave-packet motionalstates are in the spatial region [ x ( m r )+ L A − ε a ( m r ) , x (1)+ L A + ε a (1)] , where ε a ( j ) ( j = 1 , , ..., m r ) is the wave-packet spreading of the j − th wave-packetmotional state of the atom after the unitary accelerating process. Therefore,during the unitary accelerating process any possible wave-packet motional stateof the halting-qubit atom is within the effective spatial region:[ A L , A R ] = [ x ( m r ) − ε d , x (1) + L A + ε a ]53here ε a >> ε a (1) and ε d >> ε d ( m r ). The effective spatial region [ A L , A R ]covers all these m r possible wave-packet motional states of the atom duringthe whole unitary accelerating process. Now the spatial region of the unitaryaccelerating sequence must encompass sufficiently the whole effective spatial re-gion [ A L , A R ] , so that for all these m r possible wave-packet motional states theRaman laser light beams of the unitary accelerating sequence can be thoughtof as infinite plane-wave electromagnetic fields, and the most important is thatthe unitary accelerating sequence can act on all these m r possible wave-packetmotional states simultaneously and uniformly during the whole unitary accel-erating process. The spatial region [ A L , A R ] may be called the acceleratingspatial region.According to the quantum control process [22] the halting-qubit atom is ac-celerated by a unitary accelerating sequence at the end time of the quantumprogram. Here the unitary accelerating sequence may be given by U A (2 n a ) ofEq. (87a), which consists of n a pairs of the basic STIRAP accelerating sequences(11a) and (63a) in an alternate form and each basic accelerating sequence hasthe same time period t a . The unitary accelerating process U A (2 n a ) has a totaltime period 2 n a t a . The ideal adiabatic condition (30) is also met in the uni-tary accelerating process. Now one may use the recursive relations (88)—(94)to obtain the final wave-packet motional state of the halting-qubit atom afterthe atom is accelerated by the unitary accelerating sequence U A (2 n a ). Herethe starting time of the unitary accelerating process is the end time t m r of thequantum program. At the initial time t m r each possible wave-packet motionalstate of the halting-qubit atom is given by Ψ Fj ( x, t d n d + T s ( j )) of Eq. (104)for j = 1 , , ..., m r . All these m r possible wave-packet motional states startto undergo the same unitary accelerating process U A (2 n a ) at the initial time t m r simultaneously. In order to use the recursive relations (88)–(94) the initialmotional state Ψ a ( x, t a ) of Eq. (88) needs first to be obtained from the stateΨ Fj ( x, t d n d + T s ( j )) of Eq. (104). By comparing the initial state Ψ a ( x, t a ) of Eq.(88) with Ψ Fj ( x, t d n d + T s ( j )) of Eq. (104) one can see that at the initial time t a = t m r the center-of-mass position, momentum, and global phase factor of theinitial state Ψ a ( x, t a ) are given by z a ≡ z a ( j ) = z d n d + ( P d n d /M ) T s ( j ) , P a = P d n d , and exp[ iϕ a ] ≡ exp[ iϕ a ( j )] = exp( iϕ d n d ) exp[ − i ( P d n d ) T s ( j ) / (2 ℏ M )] , re-spectively, and in the complex linewidth W ( T a ) of the initial state Ψ a ( x, t a )the time interval T a ≡ T a ( j ) = T d + 2 n d t d + T s ( j ) . It is known that the initialinternal state is | g i . The initial atomic wave-packet product state then is givenby Ψ a ( x, r, t a ) = Ψ Fj ( x, t d n d + T s ( j )) | g i . After the unitary accelerating process U A (2 n a ) the wave-packet motional state of the halting-qubit atom will take theform, according to the recursive relations (88)–(94),Ψ a n a ,j ( x, t a n a ) = exp[ iϕ a n a ( j )][ (∆ x ) π ] / s x ) + i ℏ ( T a ( j )+2 n a t a )2 M × exp {−
14 [ x − z a n a ( j )] (∆ x ) + i ℏ ( T a ( j )+2 n a t a )2 M } exp { iP a n a x/ ℏ } (106)54nd the atomic wave-packet product state is given byΨ a n a ,j ( x, r, t a n a ) = Ψ a n a ,j ( x, t a n a ) | g i , (107)where the end time of the unitary accelerating process is t a n a = t m r + 2 n a t a , and the atomic motional momentum is given by P a n a = P d n d + n a ( ℏ k a + ℏ k a ) + n a ( ℏ l a + ℏ l a ) , (108)and the center-of-mass position z a n a ( j ) can be determined from the recursiverelations: z a k − ( j ) = z a k − ( j )+ P a k − M t a − ℏ ( k a + k a ) M Z t a +(2 k − t a t a +(2 k − t a dt ′ cos θ a ( t ′ ) , (109a) z a k ( j ) = z a k − ( j ) + P a k M t a − ℏ ( l a + l a ) M Z t a +2 kt a t a +(2 k − t a dt ′ cos θ la ( t ′ ) , (109b) P a k − = P a k − + ℏ ( k a + k a ) , P a k = P a k − + ℏ ( l a + l a ) , where 1 ≤ k ≤ n a . The global phase factor exp[ iϕ a n a ( j )] in Eq. (106) can alsobe obtained from the recursive relations similar to Eq. (78) and (86). Nowone can find from the final motional state Ψ a n a ,j ( x, t a n a ) of Eq. (106) that themoving distance L A of the halting-qubit atom is L A = z a n a ( j ) − z a ( j ) duringthe unitary accelerating process, which appears in the accelerating region [ A L ,A R ] above. Note that the distance L A is the same for each one of these m r possible wave-packet motional states.The unitary accelerating process tells ones some facts. For the first point,the halting-qubit atom indeed is accelerated by n a ( ℏ k a + ℏ k a )+ n a ( ℏ l a + ℏ l a ) andthis accelerating process is uniform, that is, the accelerating process is the samefor each one of these m r possible wave-packet motional states { Ψ Fj ( x, t d n d + T s ( j )) } . Thus, after the unitary accelerating process the atom is accelerated tothe velocity ( P a n a /M ) . For the second point, it can be seen from the motionalstates { Ψ a n a ,j ( x, t a n a ) } of Eq. (106) that in the complex linewidth the imaginarypart increases linearly with the time period of the unitary accelerating processand is increased by ℏ (2 n a t a ) / (2 M ) , which is also independent of any index value j , while the real part keeps unchanged in the unitary accelerating process. Forthe third point, the distances between these m r possible wave-packet motionalstates keep unchanged during the unitary accelerating process. This fact canbe deduced from the recursive relations (109a) and (109b) because the motionalmomentum P al ( l = 0 , , , ..., n a ) , the mixing angles θ a ( t ) and θ la ( t ) , and thewave numbers ( k a + k a ) and ( l a + l a ) all are independent of the index value j . This means that each one of these m r possible wave-packet motional statesmoves the same spatial distance during the unitary accelerating process. Sincethe distance ∆ L ( i, j ) between the i − th and j − th ( i < j ) wave-packet motionalstates is still given by Eq. (100) and the atomic moving velocity is ( P a n a /M )after the unitary accelerating process, the time difference between the two wave-packet states Ψ a n a ,i ( x, t a n a ) and Ψ a n a ,j ( x, t a n a ) then is given by∆ T ( i, j ) = ∆ L ( i, j ) / ( P a n a /M ) = ( j − i )∆ T ( P d n d /P a n a ) , (110)55here i < j ; i, j = 1 , , ..., m r . It is known that the time difference ∆ T ( i, j ) =( j − i )∆ T before the unitary accelerating process. Since the atomic veloc-ity ( P a n a /M ) after the accelerating process is much greater than the veloc-ity ( P d n d /M ) before the accelerating process, the time difference ∆ T ( i, j ) << ∆ T ( i, j ) , indicating that the time differences are compressed greatly for these m r possible wave-packet motional states after the unitary accelerating process.Then the time-compressing factor for these possible wave-packet motional statesafter and before the unitary accelerating process U A (2 n a ) can be calculated by R t = ∆ T ( i, j )∆ T ( i, j ) = P d n d P d n d + n a ( ℏ k a + ℏ k a ) + n a ( ℏ l a + ℏ l a ) . (111)The time-compressing factor R t is independent of the indices i and j . Thus,the time-compressing process is uniform. The time-compressing factor R t hasbeen obtained in the previous paper [22], where R t = ( v /v ) and v and v aredenoted as the atomic moving velocities P d n d /M and P a n a /M before and afterthe unitary accelerating process, respectively.
7. General adiabatic conditions and the error estimation for thedecelerating and accelerating processes
The starting point to set up a general adiabatic condition for a basic STI-RAP decelerating or accelerating process is to solve the basic equations (23) tofind the coefficients { a k ( P, t ) } or to solve the basic equations (26) to obtain thecoefficients { b k ( P, t ) } . Then it is to seek under what experimental conditionsa real adiabatic condition for the basic STIRAP decelerating or acceleratingprocess can be sufficiently close to the ideal adiabatic condition (30). This is aroutine procedure in quantum mechanics [25]. There are three basic parametersto affect the real adiabatic condition of a STIRAP experiment: the time periodof the STIRAP experiment, the Rabi frequencies and the phase-modulationfunctions of the Raman laser light beams. From the point of view of quan-tum computation one usually does not expect the quantum control process toconsume a long time. However, a long time period of the STIRAP experimentusually can lead to that the adiabatic condition for the STIRAP experimentis met better [30]. If the time period of each basic STIRAP pulse sequence inthe STIRAP-based unitary decelerating and accelerating processes is not longenough, then the adiabatic condition could not be met well. Then in this situ-ation one may use jointly the time period, the Rabi frequencies, and even thephase-modulation functions to achieve a better adiabatic condition for thesedecelerating and accelerating processes. Actually, the Rabi frequencies of theRaman laser light beams are very important to achieve a better adiabatic con-dition for the STIRAP experiment [15, 18b]. Without losing generality heretake the basic STIRAP decelerating sequence (11) as an example to discuss ageneral adiabatic condition. The obtained results can be used as well for otherbasic STIRAP decelerating and accelerating processes. The STIRAP adiabaticconditions have been discussed in detail in many references [15, 16, 17, 18] in theconventional STIRAP experiments without considering explicitly the atomic or56olecular momentum distribution. The conventional adiabatic conditions [4,15, 17, 18] usually are based on the first-order approximation solution to thebasic equations similar to the present basic differential equations (26). Theseadiabatic conditions are usually a qualitative and approximate description tothe adiabatic theorem. In the following two strict and different general adia-batic conditions are derived analytically. They are a quantitative descriptionto the adiabatic theorem. The first general adiabatic condition is based on theDyson series solution (29) of the basic differential equations (26). The secondis based on a new method to solve the basic differential equations (26). Thisnew method uses the equivalent transformations to solve the basic differentialequations (26). That is, by making repeatedly the equivalent transformationsthe three basic differential equations (26) are transformed to the three equiva-lent linear algebra equations. Though the final solution to the basic differentialequations (26) obtained with the new method is approximate, the truncation er-ror of the approximation solution can be controlled as desired. The two generaladiabatic conditions may be used to set up the conventional STIRAP experi-ments. Thus, they may be used to design the STIRAP pulse sequence to realizethe perfect state (or population) transfer for a quantum ensemble of the atomsor molecules. But their more important application is that they may be usedto set up the basic STIRAP unitary decelerating and accelerating processes fora free atom and an atomic or molecular ensemble.The basic differential equations (26) or their matrix form (28) can be in-tegrated formally. The formal solution to the basic equations (28) may beexpressed as the Dyson series (29). Here one needs to use the initial condi-tion of the basic STIRAP decelerating sequence (11). At the initial time t ofthe basic STIRAP decelerating sequence (11) the three-state vector B ( P, t ) =( b ( P, t ) , b + ( P, t ) , b − ( P, t )) T is given by Eq. (39). The initial condition (39)has been used to set up the ideal adiabatic condition (30). For a real adiabaticcondition the initial condition may be generally given in (130) below. At firstthe formal solution (29) may be rewritten as B ( P, t ) = B ( P, t ) + E r ( P, t ) (112)where t ≤ t ≤ t + T and T is the time period of the basic STIRAP deceleratingprocess, and the error term E r ( P, t ) measures the deviation of a real adiabaticcondition from the ideal adiabatic condition and it may be expressed as E r ( P, t ) = { ( 1 i ) Z tt dt M ( P, t ) + ( 1 i ) Z tt Z t t dt dt M ( P, t ) M ( P, t )+ ( 1 i ) Z tt Z t t Z t t dt dt dt M ( P, t ) M ( P, t ) M ( P, t ) + ... } B ( P, t ) . (113)The upper bound of the error term is evaluated accurately below. Denote themaximum norm of the hermitian matrix M ( P, t n ) which is given in (28) in thetime region [ t , t n − ] as || M ( P, t n ) || max = max t ≤ t n ≤ t n − {|| M ( P, t n ) ||} , t < ... < t n < t n − < ... < t < t + T. Obviously, there are the followingrelations for the maximum norms {|| M ( P, t n ) || max } : || M ( P, t ) || max ≤ ... ≤ || M ( P, t n ) || max ≤ || M ( P, t n − ) || max ≤ || M ( P, t ) || max ≤ || M ( P, t ) || max = || M ( P, t ) || max . (114)Here the maximum norm || M ( P, t ) || max is defined as || M ( P, t ) || max = max t ≤ t ≤ t + T {|| M ( P, t ) ||} . (115)Then with the help of (113) and (114) it can turn out that the upper bound ofthe deviation E r ( P, t ) may be determined from || E r ( P, t ) || ≤ exp[( || M ( P, t ) || max ) T ] × || B (1) ( P, t ) || max , (116)where the first-order approximation solution B (1) ( P, t ) to the basic differentialequations (26) is given by B (1) ( P, t ) = ( 1 i ) Z tt dt M ( P, t ) B ( P, t ) , ( t ≤ t ≤ t + T ) , (117)while || B (1) ( P, t ) || max is the maximum norm of the solution B (1) ( P, t ) in thetime region t ≤ t ≤ t + T . This norm || B (1) ( P, t ) || max is written as || B (1) ( P, t ) || max = q ( | b (1)0 ( P, t ) | + | b (1)+ ( P, t ) | + | b (1) − ( P, t ) | ) max . (118)On the other hand, it follows from the matrix M ( P, t ) in (28) that the maximumnorm || M ( P, t ) || max is bounded by || M ( P, t ) || max ≤ sX i,j | M ij ( P, t ) | ≤ √ { | Θ( P, t ) | + | Γ( P, t ) |} max . (119)The adiabatic condition (116) is strict because it is required that at any instantof time in the whole STIRAP decelerating or accelerating process the deviationfrom the ideal adiabatic condition (30) be limited within a given small value,that is, the upper bound of the error term E r ( P, t ) is less than a given smallvalue at any instant of time. Notice that in theory at the initial time t theatom is prepared to be in the trapped state | g ( P, t ) i of (19a) completely. Ifthe error term E r ( P, t ) is large, then this will mean that during the STIRAPdecelerating or accelerating process there is a large probability for the atom tobe excited to the two eigenstates | g ± ( P, t ) i of the instantaneous Hamiltonian H ( P, t ) of (17). It is known from (19b) that any one of the two eigenstates | g ± ( P, t ) i contains the excited internal state of the atom. Then the atom couldbe easily affected due to the atomic spontaneous emission if it is in any oneof the eigenstates | g ± ( P, t ) i . On the other hand, a high probability for theatom to stay in the trapped state | g ( P, t ) i may lead to that the atom is not58asily affected by environment and may avoid the spontaneous emission. Theadiabatic condition (116) indicates that the probability for the atom to leavethe trapped state | g ( P, t ) i may be limited to a small value as desired duringthe STIRAP decelerating or accelerating process. Therefore, it ensures that theatom is almost completely in the trapped state | g ( P, t ) i during the STIRAPdecelerating or accelerating process. The adiabatic condition (116) is moresevere than those in the conventional STIRAP experiments [15, 17, 18]. Thelatter usually require that the probability for the atoms or molecules underinvestigation in the two eigenstates | g ± ( P, t ) i be much smaller than one. Thisis a qualitative description for the adiabatic theorem. The present adiabaticcondition (116) is closely related to the requirement that Gaussian shape of theGaussian wave-packet motional state of the decelerated or accelerated atom keepunchanged before and after the basic STIRAP decelerating and acceleratingprocesses. It measures the deviation of a real adiabatic condition from theideal adiabatic condition (30), while the deviation may occur not only in thetwo eigenstates | g ± ( P, t ) i but also in the trapped state | g ( P, t ) i . The presentadiabatic condition (116) limits the upper bound of the deviation to a givensmall value. This is a quantitative description for the adiabatic theorem. Thisresults in that the present adiabatic condition (116) is more severe.When the adiabatic condition (116) is met, the error term || E r ( P, t + T ) || of the final state at the time t = t + T is clearly not more than the upperbound (116) and the real error term || E r ( P, t + T ) || could be much less thanthe upper bound (116). It may be required in theory that the real error term || E r ( P, t + T ) || of the final state be less than some given value which is muchless than the upper bound (116). This requirement is not severe in theory withrespect to the adiabatic condition (116). It may be met by setting the suitableexperimental parameters at the final time t = t + T for the STIRAP deceleratingor accelerating process. However, in practice the lower bound of the error term E r ( P, t + T ) of the final state is generally affected by the adiabatic condition(116). If the upper bound (116) is large, then the lower bound of the error term E r ( P, t + T ) usually is large too.According to the superposition principle in quantum mechanics in a realadiabatic condition the atomic product state at any instant of time t ( t ≤ t ≤ t + T ) in the basic STIRAP decelerating process (11) may be calculated fromEq. (12) ( P = P ′ − ℏ k ) , | Ψ r ( x, r, t ) i = X P ρ ( P ) { [ A i ( P, t ) + δ A ( P, t )] | P + ℏ k i| g i + [ A i ( P, t ) + δ A ( P, t )] | P i| e i + [ A i ( P, t ) + δ A ( P, t )] | P − ℏ k i| g i} (120)where the coefficients { A ik ( P, t ) } for k = 0 , , { δ Ak ( P, t ) } measure the deviation of thereal adiabatic condition from the ideal one. The product state (120) may berewritten as | Ψ r ( x, r, t ) i = | Ψ i ( x, r, t ) i + E r ( x, r, t )59here the wave-packet state | Ψ i ( x, r, t ) i is the atomic state at the time t in thebasic STIRAP decelerating process (11) in the ideal adiabatic condition and itmay be written as | Ψ i ( x, r, t ) i = X P ρ ( P ) { A i ( P, t ) | P + ℏ k i| g i + A i ( P, t ) | P i| e i + A i ( P, t ) | P − ℏ k i| g i} , (121)and the error term E r ( x, r, t ) is given by E r ( x, r, t ) = X P ρ ( P ) { δ A ( P, t ) | P + ℏ k i| g i + δ A ( P, t ) | P i| e i + δ A ( P, t ) | P − ℏ k i| g i} . (122)It turns out in the preceding section 5 that the final state | Ψ i ( x, r, t ) i with t = t + T in the ideal adiabatic condition is a perfect Gaussian wave-packet stateif the initial state of the basic STIRAP decelerating process (11) is a Gaussianwave-packet state. Obviously, it follows from (122) that the probability for theerror term E r ( x, r, t ) at any time t may be calculated by || E r ( x, r, t ) || = X P | ρ ( P ) | {| δ A ( P, t ) | + | δ A ( P, t ) | + | δ A ( P, t ) | } . (123)(Notice that the error probability (121) in the previous versions of this paperwhich is denoted as E r ( P, t ) is just equal to 2 || E r ( x, r, t ) || of (123)). In or-der to use directly the solution to the basic differential equations (26) or theirmatrix form (28) to calculate the upper bound of the error term E r ( x, r, t ) onemay use the coefficients b ( P, t ) and b ± ( P, t ) to express the error probability || E r ( x, r, t ) || of (123). Notice that there is the unitary transformation betweenthe two three-state vectors ( b ( P, t ) , b + ( P, t ) , b − ( P, t )) T and ( A ( P, t ) , A ( P, t ) ,A ( P, t )) T . The three-state vector ( A ( P, t ) , A ( P, t ) , A ( P, t )) T is first con-verted into the three-state vector ( ¯ A ( P, t ) , ¯ A ( P, t ) , ¯ A ( P, t )) T by the unitarytransformation of (15a)-(15c), then into the three-state vector ( a ( P, t ) , a + ( P, t ) ,a − ( P, t )) T by the unitary transformation of (22a)-(22c), and finally into thethree-state vector ( b ( P, t ) , b + ( P, t ) , b − ( P, t )) T by the unitary transformation(25). Thus, under these unitary transformations there is the relation:( A ( P, t ) , A ( P, t ) , A ( P, t )) T = U Ab ( b ( P, t ) , b + ( P, t ) , b − ( P, t )) T (124)where U Ab is the unitary transformation between the two three-state vectors( b ( P, t ) , b + ( P, t ) , b − ( P, t )) T and ( A ( P, t ) , A ( P, t ) , A ( P, t )) T . If now the so-lution to the basic equations (26) in the ideal adiabatic condition is givenby ( b i ( P, t ) , b i + ( P, t ) , b i − ( P, t )) T , then after the unitary transformation U Ab oneobtains the three-state vector ( A i ( P, t ) , A i ( P, t ) , A i ( P, t )) T of the ideal adi-abatic condition and then the state | Ψ i ( x, r, t ) i can be calculated from Eq.(121) by using the three-state vector. If the solution to the basic equations6026) in a real adiabatic condition is given by ( b ( P, t ) , b + ( P, t ) , b − ( P, t )) T with b k ( P, t ) = b ik ( P, t ) + δ bk ( P, t ) for k = 0 , + , − , then after the unitary transforma-tion U Ab one obtains the three-state vector ( A ( P, t ) , A ( P, t ) , A ( P, t )) T of thereal adiabatic condition, where A k ( P, t ) = A ik ( P, t ) + δ Ak ( P, t ) . Thus, there is theunitary transformation between the two three-state deviation vectors:( δ A ( P, t ) , δ A ( P, t ) , δ A ( P, t )) T = U Ab ( δ b ( P, t ) , δ b + ( P, t ) , δ b − ( P, t )) T . (125)It is well known that the unitary transformation U Ab does not change the normof the three-state deviation vector ( δ b ( P, t ) , δ b + ( P, t ) , δ b − ( P, t )) T . This indicatesthat there is the relation: | δ A ( P, t ) | + | δ A ( P, t ) | + | δ A ( P, t ) | = | δ b ( P, t ) | + | δ b + ( P, t ) | + | δ b − ( P, t ) | . This relation leads to that the error probability || E r ( x, r, t ) || of (123) may beexpressed as || E r ( x, r, t ) || = X P | ρ ( P ) | {| δ b ( P, t ) | + | δ b + ( P, t ) | + | δ b − ( P, t ) | } . (126)It is convenient to calculate the error upper bound || E r ( x, r, t ) || by using theequation (126), since the deviation vector ( δ b ( P, t ) , δ b + ( P, t ) , δ b − ( P, t )) T can beobtained conveniently by solving the basic differential equations (26). Thus, anaccurate error upper bound || E r ( x, r, t ) || could be obtained directly by comput-ing the equation (126) by using the deviation vector ( δ b ( P, t ) , δ b + ( P, t ) , δ b − ( P, t )) T for the basic STIRAP decelerating or accelerating process. Obviously, the three-state deviation vector ( δ b ( P, t ) , δ b + ( P, t ) , δ b − ( P, t )) T has the maximum norm orlength over the effective momentum distribution region [ P ] and in the timeperiod t ≤ t ≤ t + T, q | δ b ( P, t ) | + | δ b + ( P, t ) | + | δ b − ( P, t ) | ≤ q ( | δ b ( P, t ) | + | δ b + ( P, t ) | + | δ b − ( P, t ) | ) max , for P ∈ [ P ] and t ≤ t ≤ t + T. Then the upper bound of the error term E r ( x, r, t ) may be determined from || E r ( x, r, t ) || ≤ q ( | δ b ( P, t ) | + | δ b + ( P, t ) | + | δ b − ( P, t ) | ) max , (127)where the normalization relation P P | ρ ( P ) | = 1 is used and the truncationerror is neglected for any momentum components outside the effective momen-tum region [ P ]. The inequality (127) is a general adiabatic condition for thebasic STIRAP decelerating or accelerating process of a free atom in a wave-packet motional state. There is also a simpler method to obtain the errorupper bound || E r ( x, r, t ) || , as stated below. It uses the general adiabatic condi-tion (116). It is known that the formal solution to the basic differential equa-tions (26) or their matrix form (28) may be expressed as (112), where thesolution in the ideal adiabatic condition (30) is given by B ( P, t ) = B ( P, t ) ,
61s shown in Eq. (40) in the previous section 4. Then the equation (112)shows that the three-state deviation vector is just E r ( P, t ) and hence one has E r ( P, t ) = ( δ b ( P, t ) , δ b + ( P, t ) , δ b − ( P, t )) T . Furthermore, the adiabatic condition(116) and the equation (126) show that there are the relations: || E r ( x, r, t ) || = { X P | ρ ( P ) | || E r ( P, t ) || } / ≤ { X P | ρ ( P ) | exp[2( || M ( P, t ) || max ) T ] × || B (1) ( P, t ) || } / ≤ exp[( || ˆ M ( P, t ) || max ) T ] × || ˆ B (1) ( P, t ) || max , (128)where the relation P P | ρ ( P ) | = 1 is used and the truncation error has been ne-glected for any momentum components outside the effective momentum region[ P ], and the maximum norms || ˆ M ( P, t ) || max and || ˆ B (1) ( P, t ) || are respectivelydefined as ( || ˆ M ( P, t ) || max ) = max P ∈ [ P ] ( || M ( P, t ) || max ) , || ˆ B (1) ( P, t ) || max = max P ∈ [ P ] || B (1) ( P, t ) || max . The last inequality in (128) is a real adiabatic condition of the basic STIRAPdecelerating or accelerating process for a free atom in a wave-packet motionalstate. It could be useful to design the basic STIRAP decelerating or acceleratingprocess.At first the adiabatic condition (128) requires one to calculate the norm( || M ( P, t ) || max ) and the first-order approximation solution B (1) ( P, t ) . It is easyto calculate the first-order approximation solution to the basic differential equa-tions (26). Actually, the first-order approximation solution may be obtainedfrom the equation (117). Here for convenience setting the global phases γ ( t ) = δ ( t ) = 0 in the basic equations (26) and the phase-modulation functions of theRaman laser light beams to be ϕ ( t ) = 0 in Eq. (31) and ϕ ( t ) = 0 in Eq. (32).It should be pointed out that the following methods are available as well for thephase-modulation Raman laser light beams. It follows from Eqs. (31) and (32)that ddt α p ( P, t ) = ∆
PM k , ddt α s ( P, t ) = − ∆ PM k . By inserting these two equations into Eqs. (27a)–(27c) one obtainsΩ ± ( P, t ) = Ω( t ) ± K ( t )∆ P, (129a)Θ( P, t ) = − ˙ θ ( t ) + iK ( t )∆ P, Γ( P, t ) = k M ∆ P − K ( t )∆ P, (129b)where K ( t ) = 14 M [( k − k ) + 3( k + k ) cos 2 θ ( t )] ,K ( t ) = k + k M sin 2 θ ( t ) , K ( t ) = ( k + k ) M cos θ ( t ) . θ ( t ) is very small ( θ ( t ) <<
1) but not equal to zero, then theinitial condition (39) may be changed to the general form b ( P, t ) = exp[ i ℏ ( ( P + ℏ k ) M + E ) t ] cos θ ( t ) , (130a) b + ( P, t ) = b − ( P, t ) = 1 √ i ℏ ( ( P + ℏ k ) M + E ) t ] sin θ ( t ) . (130b)Then in the initial condition (130) the first-order approximation solution to thebasic equations (26) for the coefficient b ( P, t ) may be written as, by integratingby parts the integral (117), b (1)0 ( P, t ) = b ( P, t ) − b ( P, t ) = C (1)0 ( P, t ) + C (2)0 T ( P, t ) , (131a)where the main term C (1)0 ( P, t ) that is proportional to Θ(
P, t ) ∗ / Ω( t ) is writtenas C (1)0 ( P, t ) = √ b + ( P, t ) Θ( P, t ) ∗ Ω( t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] sin[ i Z tt dt ′ Ω( t ′ )] , (131b)and the secondary term C (2)0 T ( P, t ) is given by C (2)0 T ( P, t ) = i √ b + ( P, t ) Z tt dt { [ i ∂∂t ( Θ( P, t ) ∗ Ω( t ) ) − K ( t )Θ( P, t ) ∗ Ω( t ) ∆ P ] × exp[ i ∆ P Z t t dt ′ K ( t ′ )] sin[ i Z t t dt ′ Ω( t ′ )] } . (131c)The first-order solution for the coefficients b ± ( P, t ) is given by b (1) ± ( P, t ) = b ± ( P, t ) − b ± ( P, t ) = C (1) ± ( P, t ) + C (2) ± T ( P, t ) + F (1) ± ( P, t ) + F (2) ± T ( P, t )(132a)where the main terms C (1) ± ( P, t ) are given by C (1) ± ( P, t ) = ∓ i √ b ( P, t ) Θ( P, t )Ω( t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ ∓ i Z tt dt ′ Ω( t ′ )] ± i √ b ( P, t ) Θ( P, t )Ω( t ) , (132b)and the secondary terms are C (2) ± T ( P, t ) = ± √ b ( P, t ) Z tt dt { [ i ∂∂t ( Θ( P, t )Ω( t ) ) + K ( t )Θ( P, t )Ω( t ) ∆ P ] × exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ ∓ i Z t t dt ′ Ω( t ′ )] } , (132c)63 (1) ± ( P, t ) = ± b ∓ ( P, t ) Γ( P, t )Ω( t ) exp[ ∓ i Z tt dt ′ t ′ )] ∓ b ∓ ( P, t ) Γ( P, t )Ω( t ) , (132d) F (2) ± T ( P, t ) = ∓ b ∓ ( P, t ) Z tt dt { [ ∂∂t ( Γ( P, t )Ω( t ) )] exp[ ∓ i Z t t dt ′ t ′ )] } . (132e)The dominating terms in the first-order approximation solution of (131a) and(132a) are C (1) ± ( P, t ) , which are proportional to Θ( P, t ) / Ω( t ) and b ( P, t ) . It canturn out by integrating by parts the integral (132c) that the terms C (2) ± T ( P, t )are proportional to Ω( t ) − . Thus, the terms C (2) ± T ( P, t ) are secondary in the first-order solution for a large Rabi frequency Ω( t ) . Similarly, it can turn out that C (2)0 T ( P, t ) ∝ b + ( P, t )Ω( t ) − and F (2) ± T ( P, t ) ∝ b + ( P, t )Ω( t ) − , by integrating byparts the integrals (131c) and (132e), respectively. Thus, these terms C (2)0 T ( P, t )and F (2) ± T ( P, t ) are secondary with respect to the terms C (1)0 ( P, t ) and F (1) ± ( P, t ) , respectively. On the other hand, the imperfection for the initial conditions b + ( P, t ) = b − ( P, t ) = 0 could mainly affect C (1)0 ( P, t ) and F (1) ± ( P, t ) . Its mag-nitude is approximately proportional to the factors | b ± ( P, t ) || Θ( P, t ) | / Ω( t ) or | b ± ( P, t ) || Γ( P, t ) | / Ω( t ) . Since | b ± ( P, t ) | << | b ( P, t ) | , these terms C (1)0 ( P, t )and F (1) ± ( P, t ) are secondary with respect to the main terms C (1) ± ( P, t ) . Thus, theerror upper bound || ˆ B (1) ( P, t ) || max in (128) may be determined from the mainterms C (1) ± ( P, t ) . Now by inserting b (1)0 ( P, t ) of (131a) and b (1) ± ( P, t ) of (132a)into (118) it can be found that the norm || B (1) ( P, t ) || is bounded by || B (1) ( P, t ) || ≤ | Θ( P, t ) | Ω( t ) + | Θ( P, t ) | Ω( t )= q ˙ θ ( t ) + K ( t ) | ∆ P | Ω( t ) + q ˙ θ ( t ) + K ( t ) | ∆ P | Ω( t ) , (133)where those secondary terms of the first-order approximation solution are ne-glected and only the main terms C (1) ± ( P, t ) of (132b) are used and | b ( P, t ) | ≤ θ ( t ) → , as can be seenin (35). Note that K ( t ) = ( k + k ) sin 2 θ ( t ) / (2 M ) and the time derivative ˙ θ ( t )of the mixing angle is given by˙ θ ( t ) = ˙Ω p ( t ) cos θ ( t ) − ˙Ω s ( t ) sin θ ( t )Ω( t ) . If the Rabi frequencies Ω p ( t ) and Ω s ( t ) are chosen suitably in experiment suchthat at the initial and final times the mixing angle satisfies the relations:tan θ ( t ) = Ω p ( t ) / Ω s ( t ) ≈ ˙Ω p ( t ) / ˙Ω s ( t ) → , ˙Ω p ( t + T ) cos θ ( t + T ) − ˙Ω s ( t + T ) sin θ ( t + T ) → , θ ( t ) ≈ θ ( t + T ) ≈ . On the other hand, themomentum distribution satisfies | ∆ P | ≤ ∆ P M / P M . Therefore, at the initialtime | K ( t )∆ P | ≤ | K ( t ) | ∆ P M / P M ( k + k ) | sin 2 θ ( t ) | / (4 M ) ≈ || ˆ B (1) ( P, t ) || max may be determined from, byneglecting the second term on the rightest side of (133), || B (1) ( P, t ) || ≤ { q ˙ θ ( t ) + (∆ P M ) ( k + k ) sin θ ( t ) / (16 M )Ω( t ) } max . (134)Here the subscript ′ max ′ means that the function on the right-hand side of(134) is taken as the maximum value in the time period t ≤ t ≤ t + T of the STIRAP process. The first-order approximation adiabatic condition isthat the maximum value of the function on the right-hand side of (134) iscontrolled to be smaller than some desired small value. (Notice that this first-order adiabatic condition (134) is slightly different from that one (128a) in theprevious versions of this paper). As shown below, in the initial and final timeperiods of the STIRAP process the adiabatic condition (134) still may be meteven if the Rabi frequency Ω( t ) is small in these time periods. This is a globaladiabatic condition, since it is involved in the whole time period of the STIRAPprocess. Here the global adiabatic condition has a different definition from theconventional one in Ref. [4]. In the conventional STIRAP experiments [4, 15,17, 18] the (local) adiabatic condition is defined as that at any instant of timeof the STIRAP process the population or probability in the two eigenstates | g ± ( P, t ) i that contain the excited internal state is much smaller unity. Thisis approximately equivalent to the first inequality of (133) for the first-orderapproximation, where the inequality symbol ′ ≤ ′ is replaced with ′ << ′ . Onthe other hand, according to (119) the maximum matrix norm ( || ˆ M ( P, t ) || max )is determined from || M ( P, t ) || ≤ √ { | Θ( P, t ) | + | Γ( P, t ) |}≤ √ { q | ˙ θ ( t ) | + ( k + k ) (∆ P M ) / (16 M ) + (∆ P M )2 M max( k , k ) } (135)where | ˙ θ ( t ) | max is the maximum value of | ˙ θ ( t ) | in the whole STIRAP deceleratingor accelerating process. After the upper bound || ˆ B (1) ( P, t ) || max and maximumnorm ( || ˆ M ( P, t ) || max ) are determined from (134) and (135), respectively, onemay determine the upper bound of the error term E r ( x, r, t ) from (128). It canbe seen from (134), (135), and (128) that the adiabatic condition (128) maybe better satisfied for a small time derivative ˙ θ ( t ) , a short time period T , alarge Rabi frequency Ω( t ), and a narrow momentum wave-packet state (∆ P M issmall). If the time derivative ˙ θ ( t ) is smaller, then the time period T usually islarger. Thus, there is a compromise between the settings of the time derivative65 θ ( t ) and the time period T in experiment. Obviously, one has the relation forany basic STIRAP decelerating or accelerating process: θ ( t + T ) − θ ( t ) = Z t + Tt dt ′ ˙ θ ( t ′ ) = π/ . This relation may be used to determine the time period T if one knows the timederivative ˙ θ ( t ) of the mixing angle.It seems that the adiabatic conditions (116) and (128) could not be bettersatisfied in the initial and final time periods, since the Rabi frequency Ω( t )takes a smaller value in these time periods. It is well known that any STIRAPexperiment requires that at the initial and final time the mixing angle θ ( t ) satisfythe constraint conditions: θ ( t ) → , θ ( t + T ) → π/ . The two constraint conditions are compatible with the adiabatic conditions (116)and (128). This can be seen from (134). At the initial time period the Rabifrequency Ω( t ) takes generally a smaller value, but at the same time the mixingangle θ ( t ) → θ ( t ) may be set to a value close tozero, leading to that the value || B (1) ( P, t ) || may be kept at a smaller value atthe initial time period. At the final time the mixing angle θ ( t + T ) → π/ θ ( t + T ) → θ ( t + T ) also may be set to a valueclose to zero. Then the value || B (1) ( P, t ) || still may be kept at a smaller value,although the Rabi frequency Ω( t ) takes a smaller value at the final time period.These results show that in theory the adiabatic conditions (116) and (128) stillmay be met in the initial and final time periods as long as the Raman laserlight beams of the STIRAP experiment are suitably designed. The adiabaticconditions (116) and (128) could be better used for a conventional three-stateSTIRAP experiment that uses a pair of copropagating Raman laser light beamsand those STIRAP-based decelerating and accelerating processes of the atomicor molecular systems with a narrow momentum distribution.The deviation E r ( P, t ) of (113) and its upper bound (116) are obtained fora single basic STIRAP decelerating sequence (11). If a unitary deceleratingprocess consists of n d pairs of the basic STIRAP decelerating sequences (11)and (63), then the total deviation generated in the unitary decelerating processis bounded by || E r ( P, t ) || ≤ n d exp[( || ˆ M k ( P, t ) || max ) T ] × || ˆ B (1) k ( P, t ) || max + n d exp[( || ˆ M l ( P, t ) || max ) T ] × || ˆ B (1) l ( P, t ) || max , (136)where the subscript k marks the basic STIRAP decelerating sequence (11) thatuses a pair of the Raman laser light beams with the Rabi frequency Ω( t ) , themixing angle θ ( t ), the wave numbers k and k , and the carrier frequencies ω and ω , while the subscript l denotes the basic STIRAP decelerating sequence(63) that may use another pair of the Raman laser light beams with the Rabi66requency Ω l ( t ) , the mixing angle θ l ( t ) , the wave numbers k l and k l , and thecarrier frequencies ω l and ω l . Both the basic STIRAP decelerating processeshave the same time period T . The upper bound of the total deviation E r ( P, t )on the right-hand side of the inequality (136) for the unitary decelerating oraccelerating process may be controlled by setting suitably the experimentalparameters of the Raman laser light beams, which include the Rabi frequenciesΩ( t ) and Ω l ( t ) , the mixing angles θ ( t ) and θ l ( t ) , and the time derivatives ˙ θ ( t )and ˙ θ l ( t ), and so on.The basic STIRAP decelerating or accelerating process is a time-dependentunitary quantum dynamical problem from the viewpoint of quantum mechan-ics. The three-state basic STIRAP decelerating or accelerating process is a quitesimple unitary dynamical process, but it can describe completely the complexSTIRAP-based unitary decelerating and accelerating processes of a free atom.It is relatively simple to solve approximately such a unitary dynamical problemas the three-state basic STIRAP decelerating or accelerating process in quan-tum mechanics, although this problem is time-dependent and it is difficult tosolve exactly the basic differential equations (26) except for some special cases[39]. For example, one may use the conventional methods of successive approx-imations [33] to solve these basic differential equations approximately when theRabi frequencies Ω( t ) is large. As mentioned before, the Dyson series solution(29) to the basic differential equations (26) or (28) may be used to calculatethe deviation of a real adiabatic condition from the ideal adiabatic condition.The problem to be answered is that one needs to calculate how many leadingterms in the Dyson series (29) so that the result obtained is enough accurate.The leading term number is dependent upon the desired error value and themaximum norm of the matrix M ( P, t ) in (28). If one uses the first n termson the right-hand side of (113) to calculate the error term E r ( P, t ) , then theresidual term may be given by R ( P, t ) = { ( 1 i ) n +1 Z tt Z t t ... Z t n t dt dt ...dt n +1 M ( P, t ) M ( P, t ) ...M ( P, t n +1 )+( 1 i ) n +2 Z tt Z t t ... Z t n +1 t dt dt ...dt n +2 M ( P, t ) M ( P, t ) ...M ( P, t n +2 )+ ...... } B ( P, t ) . (137)Then the residual term R ( P, t ) is bounded by || R ( P, t ) || ≤ ∞ X k = n +1 ( t − t ) k k ! ( || ˆ M ( P, t ) || max ) k ≤ ( || ˆ M ( P, t ) || max ( t − t )) n +1 ( n + 1)! exp[( || ˆ M ( P, t ) || max )( t − t )] (138)where || B ( P, t ) || = 1 is used. Now ( n + 1)! ≈ p π ( n + 1)[( n + 1) /e ] n +1 . Then67he upper bound of the residual term is determined from || R ( P, t ) || ≤ { || ˆ M ( P,t ) || max ( t − t )( n +1) /e exp[ ( || ˆ M ( P,t ) || max )( t − t ) n +1 ] } n +1 p π ( n + 1) . (139)If ( || ˆ M ( P, t ) || max )( t − t ) < ( n + 1) , then exp[ ( || ˆ M ( P,t ) || max )( t − t ) n +1 ] < e. Thus,if ( n + 1) /e > || ˆ M ( P, t ) || max ( t − t ) exp[ ( || ˆ M ( P,t ) || max )( t − t ) n +1 ] , then the residualterm R ( P, t ) is exponentially small. Suppose that the upper bound of the errorterm E r ( P, t ) is set to a given value ε r : || E r ( P, t ) || ≤ ε r . Then this requires theresidual term to satisfy || R ( P, t ) || << ε r . The condition || R ( P, t ) || << ε r canbe easily met by setting a minimum integer n such that1 p π ( n + 1) { || ˆ M ( P, t ) || max T ( n + 1) /e exp[ ( || ˆ M ( P, t ) || max ) Tn + 1 ] } n +1 << ε r . (140)Once the minimum integer n is determined, one may use the first n terms onthe right-hand side of (113) to calculate the error term E r ( P, t ) and its upperbound, while the residual term R ( P, t ) does not affect significantly the finalresult.It can be seen from (133) that the first-order approximation solution showsthat the error upper bound || B (1) ( P, t ) || is mainly dependent upon the parame-ter | Θ( P, t ) | / Ω( t ) and almost independent of the parameters Γ( P, t ) and K ( t ) . Actually, in the first-order approximation solution the parameter Γ(
P, t ) ap-pears in the secondary terms F (1) ± ( P, t ) and F (2) ± T ( P, t ) and K ( t ) in the sec-ondary terms C (2) ± T ( P, t ) (these parameters may appear in the phase factors,and if so, they do not make a contribution to the error upper bound). It isknown that the parameter Θ(
P, t ) = − ˙ θ ( t ) + i (∆ P/M )( k + k ) sin 2 θ ( t ) . Ifthe two Raman laser light beams are copropagating, then the wave-number sum( k + k ) will be changed to the wave-number difference ( k − k ) in the param-eter Θ( P, t ) . Then the effect of the momentum distribution on the error upperbound || B (1) ( P, t ) || will be greatly weakened and the momentum distributionwill become a higher-order effect on the STIRAP state transfer. Therefore, inthis sense the copropagating Raman laser light beams used to construct theSTIRAP pulse sequence may be better than the counterpropagating ones to re-alize the perfect STIRAP state transfer. Unlike the parameter Θ( P, t ) these twoparameters Γ(
P, t ) and K ( t ) are dependent on both the wave-number sum anddifference. That the momentum distribution affects the STIRAP state trans-fer is mainly through the parameters Γ( P, t ) and K ( t ) , which appear in thehigher-order terms in the Dyson series (29), if the copropagating Raman laserlight beams are used in the STIRAP state transfer. In fact, the maximum norm( || ˆ M ( P, t ) || max ) determined from (135) shows that it is proportional to | Γ( P, t ) | . These parameters make an important effect for the momentum distribution onthe STIRAP state transfer. Thus, that the conventional three-state STIRAPexperiments [4, 15] use the copropagating Raman laser light beams is favorablefor the perfect state transfer and may minimize the effect of the momentum68istribution of the atomic and molecular systems under investigation on theperfect state transfer. However, in the basic STIRAP decelerating or accel-erating process the counterpropagating Raman laser light beams are generallyused so that the fast moving atom can be decelerated or accelerated more ef-ficiently. On the other hand, the first-order solution (117) could not exactlyaccount for the momentum distribution in any case that either copropagatingor counterpropagating laser light beams is used in the STIRAP experiments.The adiabatic conditions (116) and (128) are accurate, but due to that there isan exponential correction factor in (116) and (128) they could not be met for abroad momentum distribution. A broad momentum distribution is often met inan atomic or molecular quantum ensemble. It is necessary to consider the effectof the momentum distribution when these physical ensembles are decelerated(or accelerated) by the STIRAP decelerating (or accelerating) pulse sequence.Thus, it is necessary to find a more useful adiabatic condition that can accountfor the effect of a broad momentum distribution on the STIRAP state transfer.It is still complex to use the Dyson series solution (29) to calculate the errorterm E r ( P, t ) of (113) and its upper bound. In the following an equivalenttransformation method based on the integration by parts to solve the basicdifferential equations (26) is proposed so that the error term E r ( P, t ) of (113)and its upper bound can be obtained conveniently in a high accuracy. On theother hand, by solving the basic differential equations (26) to obtain an enoughaccurate solution one may further use the solution to calculate convenientlythe time evolution process for the basic STIRAP decelerating and acceleratingprocesses for a free atom. Generally, it is quite inconvenient to calculate the timeevolution process of an atomic decelerating or accelerating process by directlysolving the Schr¨odinger equation. The present scheme is convenient to calculatethe time evolution process of the basic STIRAP decelerating or acceleratingprocess because it does not solve directly the original Schr¨odinger equationbut solves the three first-order differential equations (26) that are equivalentto and much simpler than the original Schr¨odinger equation. The equivalenttransformation method to solve the basic equations (26) is based on the fact thatthe Rabi frequency Ω( t ) may be set to a large value in experiment. Though thesolution to the basic equations (26) obtained by this method is approximate, thetruncation error of the solution can be controlled as expected. The procedureto solve the basic equations (26) with the equivalent transformation methodmay be described below. By integrating the basic differential equations (26)one obtains the equivalent integral equations: b ( P, t ) − b ( P, t ) = 1 √ Z tt dt { b + ( P, t )Θ( P, t ) ∗ × exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ i Z t t dt ′ Ω( t ′ )] } + 1 √ Z tt dt { b − ( P, t )Θ( P, t ) ∗ exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] } , (141a)69 ± ( P, t ) − b ± ( P, t ) = − i Z tt dt { b ∓ ( P, t )Γ( P, t ) exp[ ∓ i Z t t dt ′ t ′ )] }− √ Z tt dt { b ( P, t )Θ( P, t ) exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ ∓ i Z t t dt ′ Ω( t ′ )] } . (141b)Hereafter the initial condition (39) is used for convenience. The equivalent trans-formation method is that by the integration by parts and another transformation(see below) the integral equations (141) may be approximately reduced to thethree linear algebra equations. These three linear algebra equations are equiva-lent to the original integral equations (141) if the initial condition (39) is takeninto account and when the truncation error can be neglected. At the first stepof the equivalent transformation method the integrals on the right-hand sidesof (141) are calculated by the integration by parts. Then the initial condition(39) is used to simplify the calculated results If there are the time derivativesof the variables b ( P, t ) and b ± ( P, t ) in the integrands after the integration byparts, then one may use the basic differential equations (26) to replace thesetime derivatives. As an example, by integrating by parts the equation (141b)for the variable b + ( P, t ) and then using the basic differential equations (26) andthe initial condition (39), one can obtain the following equation: b ( P, t ) ≡ b + ( P, t ) = b − ( P, t )Γ − +1 ( P, t ) exp[ − i Z tt dt ′ t ′ )]+ ib ( P, t )Γ ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )] − ib ( P, t )Γ ( P, t ) + i Z tt dt { b + ( P, t )Γ +1 ( P, t ) } + i Z tt dt { b − ( P, t )Θ − +1 ( P, t ) exp[ − i Z t t dt ′ t ′ )] } + Z tt dt { b ( P, t )Θ ( P, t ) exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] } (142)where the five amplitudes are given byΓ − +1 ( P, t ) = 14 Γ(
P, t )Ω( t ) , Γ ( P, t ) = − √ P, t )Ω( t ) , Γ +1 ( P, t ) = 18 Γ(
P, t ) + 4 | Θ( P, t ) | Ω( t ) , Θ − +1 ( P, t ) = 12 { i ∂∂t ( Γ( P, t )Ω( t ) ) + | Θ( P, t ) | Ω( t ) } , Θ ( P, t ) = 1 √ { i ∂∂t ( Θ( P, t )Ω( t ) ) + 14 Γ( P, t )Θ(
P, t )Ω( t ) + K ( t )Θ( P, t )Ω( t ) ∆ P } . b + ( P, t ) . The unique difference between the two equa-tions is that the equation (142) uses the initial condition (39), while the originalequation does not. The first four terms on the right-hand side of (142) maybe considered as the main terms, since these terms have a greater contribu-tion to the solution b ( P, t ) of (142). On the other hand, each one of thelast two integrals on the right-hand side of (142) contains the integrands withthe largely oscillatory phase factor exp[ − i R t t dt ′ Ω( t ′ )] or exp[ − i R t t dt ′ t ′ )] . These largely oscillatory phase factors make the two integrals secondary in thesolution (142). This can be seen by integrating by parts the two integrals onceagain. If these two integrals are neglected, then one obtains the first-order ap-proximation solution to the original equation (141b) for the variable b + ( P, t ) , since all the five amplitudes including Γ − +1 ( P, t ) , Γ ( P, t ) , etc., appearing inthe equation (142) are inversely proportional to the Rabi frequency Ω( t ).One may further obtain a better approximation solution than the first-orderone. This can be done by integrating by parts the last two integrals on theright-hand side of (142) again. However, the fourth term (or the first integral)contains the solution b + ( P, t ) itself on the right-hand side of (142). While onemay substitute the first-order approximation solution of b ( P, t ) into the integralto obtain a better approximation, the calculation process becomes so complexthat one can only obtain a lower-order approximation solution. In order toavoid this complex one may make a transformation on the solution b + ( P, t )to cancel the integral before integrating by parts the last two integrals. Thistransformation is given byˆ b ( P, t ) = b ( P, t ) exp[ − i Z tt dt Γ +1 ( P, t )] . (143)This transformation is the key point to the present equivalent transformationmethod to solve the basic differential equations (26). By this transformationand the initial condition (39) the transformed solution ˆ b ( P, t ) may be writtenas ˆ b ( P, t ) = b − ( P, t )ˆΓ − +1 ( P, t ) exp[ − i Z tt dt ′ t ′ )]+ ib ( P, t )ˆΓ ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )] − ib ( P, t )ˆΓ ( P, t ) + i Z tt dt { b − ( P, t ) ˆΘ − +1 ( P, t ) exp[ − i Z t t dt ′ t ′ )] } + Z tt dt { b ( P, t ) ˆΘ ( P, t ) exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] } . (144)Now there is not any integral containing the solution b ( P, t ) itself on the right-hand side of (144). The amplitudes of the solution ˆ b ( P, t ) are related to those71mplitudes of the solution b ( P, t ) of (142) by the recursive relations:ˆΓ − +1 ( P, t ) = Γ − +1 ( P, t ) exp[ − i Z tt dt Γ +1 ( P, t )] , (145a)ˆΓ ( P, t ) = Γ ( P, t ) exp[ − i Z tt dt Γ +1 ( P, t )] , (145b)ˆΘ − +1 ( P, t ) = [Θ − +1 ( P, t ) + Γ − +1 ( P, t )Γ +1 ( P, t )] exp[ − i Z tt dt Γ +1 ( P, t )] , (145c)ˆΘ ( P, t ) = [Θ ( P, t ) − Γ ( P, t )Γ +1 ( P, t )] exp[ − i Z tt dt Γ +1 ( P, t )] . (145d)The transformation (143) and the integration by parts may be called the equiv-alent transformations as they does not generate any error term in these trans-formation processes. The transformation (143) does not improve essentially thefirst-order approximation solution, but it does simplify greatly the calculationprocess to further obtain a higher-order approximation solution. Now by inte-grating by parts the last two integrals on the right-hand side of (144) and usingthe basic equations (26) and the initial condition (39) the solution ˆ b ( P, t ) maybe written as b ( P, t ) ≡ ˆ b ( P, t ) = b − ( P, t )Γ − +2 ( P, t ) exp[ − i Z tt dt ′ t ′ )]+ ib ( P, t )Γ ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )] − ib ( P, t )Γ ( P, t ) + i Z tt dt { b ( P, t )Γ +2 ( P, t ) } + i Z tt dt { b − ( P, t )Θ − +2 ( P, t ) exp[ − i Z t t dt ′ t ′ )] } + Z tt dt { b ( P, t )Θ ( P, t ) exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] } (146)where the amplitudes satisfy the recursive relations:Γ − +2 ( P, t ) = ˆΓ − +1 ( P, t ) −
12 ˆΘ − +1 ( P, t )Ω( t ) , Γ ( P, t ) = ˆΓ ( P, t )+ ˆΘ ( P, t )Ω( t ) , (147a)Γ +2 ( P, t ) = − [ 14 Γ( P, t ) ˆΘ − +1 ( P, t )Ω( t ) + 1 √ P, t ) ∗ ˆΘ ( P, t )Ω( t ) ] exp[ i Z tt dt Γ +1 ( P, t )] , (147b)Θ − +2 ( P, t ) = − i ∂∂t ( ˆΘ − +1 ( P, t )Ω( t ) ) − √ P, t ) ∗ ˆΘ ( P, t )Ω( t ) , (147c)72 ( P, t ) = − √ P, t ) ˆΘ − +1 ( P, t )Ω( t ) − i ∂∂t ( ˆΘ ( P, t )Ω( t ) ) − K ( t ) ˆΘ ( P, t )Ω( t ) ∆ P. (147d)The unique difference between the equation (146) and the original equation(141b) for the variable b + ( P, t ) is that the initial condition (39) has been usedin (146), while it is not used in (141b). Now the amplitudes Γ +2 ( P, t ) , Θ − +2 ( P, t ) , and Θ ( P, t ) of the last three integrals on the right-hand side of (146) areinversely proportional to Ω( t ) . Thus, these three integrals are secondary withrespect to the first three terms on the right-hand side of (146). Moreover, thelast two integrals contain the integrands with the largely oscillatory phase factorexp[ − i R t t dt ′ Ω( t ′ )] or exp[ − i R t t dt ′ t ′ )] . Then the integration by parts showsthat these two integrals are secondary with respect to the other four terms onthe right-hand side of (146). Thus, by the integration by parts the last twointegrals becomes less important in the solution than before.The above equivalent transformations can be repeated many times that thesolution b k + ( P, t ) ( k = 1 , , ..., ) is transformed to the solution ˆ b k + ( P, t ) by theequivalent transformation similar to (143) and then the solution ˆ b k + ( P, t ) ischanged to the solution b k +1+ ( P, t ) by integrating by parts the last two inte-grals of the solution ˆ b k + ( P, t ) . The equivalent transformation from the solution b k + ( P, t ) to the transformed solution ˆ b k + ( P, t ) may be generally given byˆ b k + ( P, t ) = b k + ( P, t ) exp[ − i Z tt dt Γ + k ( P, t )] , k = 1 , , .... (148)The solutions ˆ b k + ( P, t ) and b k + ( P, t ) are called the k − order exact solutions to theequation (141b) for the variable b + ( P, t ) . From the solution b k + ( P, t ) to the trans-formed solution ˆ b k + ( P, t ) the amplitudes { Γ α + k ( P, t ) , Θ α + k ( P, t ) } ( α = 0 , − ) of thesolution b k + ( P, t ) are transformed to the amplitudes { ˆΓ α + k ( P, t ) , ˆΘ α + k ( P, t ) } of thesolution ˆ b k + ( P, t ) according to the recursive equations (145) if in the recursiverelations (145) one makes the following replacements: ˆΓ α +1 ( P, t ) ↔ ˆΓ α + k ( P, t ),ˆΘ α +1 ( P, t ) ↔ ˆΘ α + k ( P, t ) , Γ α +1 ( P, t ) ↔ Γ α + k ( P, t ) , Θ α +1 ( P, t ) ↔ Θ α + k ( P, t ) , for α = 0 , − , and Γ +1 ( P, t ) ↔ Γ + k ( P, t ) . On the other hand, from the solution ˆ b k + ( P, t )to the solution b k +1+ ( P, t ) the recursive relations for their amplitudes are stillgiven by (147) except the relation (147b), in which one needs to make the follow-ing replacements: ˆΓ α +1 ( P, t ) ↔ ˆΓ α + k ( P, t ), ˆΘ α +1 ( P, t ) ↔ ˆΘ α + k ( P, t ) , Γ α +2 ( P, t ) ↔ Γ α + k +1 ( P, t ) , Θ α +2 ( P, t ) ↔ Θ α + k +1 ( P, t ) , for α = 0 , − , and Γ +1 ( P, t ) ↔ Γ + k ( P, t )and Γ +2 ( P, t ) ↔ Γ + k +1 ( P, t ) . The relation (147b) is modified to the formΓ + k +1 ( P, t ) = − [ 14 Γ( P, t ) ˆΘ − + k ( P, t )Ω( t ) + 1 √ P, t ) ∗ ˆΘ k ( P, t )Ω( t ) ] × exp[ i Z tt dt Γ + k ( P, t )] exp[ i Z tt dt Γ + k − ( P, t )] ... exp[ i Z tt dt Γ +1 ( P, t )] .
73t can be found that after integrating by parts the last two integrals of thesolution ˆ b k + ( P, t ) many times, the two integrals become less and less importantin the solution. Actually, it can turn out that the amplitudes ˆΘ − + k ( P, t ) andˆΘ k ( P, t ) of the solution ˆ b k + ( P, t ) is inversely proportional to the k − th power ofthe Rabi frequency Ω( t ) , that is, ˆΘ − + k ( P, t ) ∝ Ω( t ) − k and ˆΘ k ( P, t ) ∝ Ω( t ) − k . Thus, by making only a few equivalent transformations of the integration byparts one can obtain a highly accurate solution to the original equation (141b)for the variable b + ( P, t ) even if the last two integrals are neglected.Now it is easy to obtain the first-order approximation solution to the orig-inal equation (141b) for the variable b + ( P, t ) from the equation (144) by ne-glecting the last two integrals on the right-hand side of (144). The first-orderapproximation solution is a linear algebra equation with the three variables { b ( P, t ) − b ( P, t ) , b ± ( P, t ) } and is given by b + ( P, t ) ≡ b ( P, t ) = b − ( P, t )Γ − +1 ( P, t ) exp[ − i Z tt dt ′ t ′ )]+ i ( b ( P, t ) − b ( P, t ))Γ ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )]+ ib ( P, t )Γ ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )] − ib ( P, t )Γ ( P, t ) exp[ i Z tt dt Γ +1 ( P, t )] , (149)while the truncation error is just the last two integrals: E + r ( P, t ) = exp[ i Z tt dt Γ +1 ( P, t )] ×{ i Z tt dt { b − ( P, t ) ˆΘ − +1 ( P, t ) exp[ − i Z t t dt ′ t ′ )] } + Z tt dt { b ( P, t ) ˆΘ ( P, t ) exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] }} . By the integration by parts it can turn out that the error term E + r ( P, t ) isbounded by | E + r ( P, t ) | ≤ | ˆΘ − +1 ( P, t ) | Ω( t ) + | ˆΘ ( P, t ) | Ω( t ) + | ˆΘ ( P, t ) | Ω( t )+ Z tt dt {| Γ +2 ( P, t ) | + | Θ − +2 ( P, t ) | + | Θ ( P, t ) |} . (150)74t is clear that this error upper bound is inversely proportional to Ω( t ) . Thesecond-order approximation solution is given by b ( P, t ) = b − ( P, t )Γ − +2 ( P, t ) exp[ − i Z tt dt ′ t ′ )]+ i ( b ( P, t ) − b ( P, t ))Γ ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )]+ ib ( P, t )Γ ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )] − ib ( P, t )Γ ( P, t ) exp[ i Z tt dt Γ +2 ( P, t )] (151)and the truncation error is bounded by | E + r ( P, t ) | ≤ | ˆΘ − +2 ( P, t ) | Ω( t ) + | ˆΘ ( P, t ) | Ω( t ) + | ˆΘ ( P, t ) | Ω( t )+ Z tt dt {| Γ +3 ( P, t ) | + | Θ − +3 ( P, t ) | + | Θ ( P, t ) |} . (152)Obviously, this error upper bound is inversely proportional to Ω( t ) . The first-and second-order approximation solutions may be used to set up the adiabaticcondition.By using the similar equivalent transformations mentioned above one mayobtain the k − order exact solution b k − ( P, t ) ( k = 1 , , ... ) from the equation(141b) for the variable b − ( P, t ), b k − ( P, t ) = b + ( P, t )Γ + − k ( P, t ) exp[ i Z tt dt ′ t ′ )]+ ib ( P, t )Γ − k ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )] − ib ( P, t )Γ − k ( P, t ) + i Z tt dt { b k − ( P, t )Γ − k ( P, t ) } + i Z tt dt { b + ( P, t )Θ + − k ( P, t ) exp[ i Z t t dt ′ t ′ )] } + Z tt dt { b ( P, t )Θ − k ( P, t ) exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ i Z t t dt ′ Ω( t ′ )] } . (153)The equation (153) is equivalent to the original equation (141b) for the variable b − ( P, t ) if the initial condition (39) is taken into account. In particular, the75rst-order exact solution b − ( P, t ) ≡ b − ( P, t ) and its five amplitudes are givenby Γ + − ( P, t ) = −
14 Γ(
P, t )Ω( t ) , Γ − ( P, t ) = 1 √ P, t )Ω( t ) , (154a)Γ − ( P, t ) = −
18 Γ(
P, t ) + 4 | Θ( P, t ) | Ω( t ) , (154b)Θ + − ( P, t ) = − { i ∂∂t ( Γ( P, t )Ω( t ) ) + | Θ( P, t ) | Ω( t ) } , (154c)Θ − ( P, t ) = 1 √ { i ∂∂t ( Θ( P, t )Ω( t ) ) + 14 Γ( P, t )Θ(
P, t )Ω( t ) + K ( t )Θ( P, t )Ω( t ) ∆ P } . (154d)The equivalent transformation from the solution b k − ( P, t ) of (153) to the solutionˆ b k − ( P, t ) is given byˆ b k − ( P, t ) = b k − ( P, t ) exp[ − i Z tt dt Γ − k ( P, t )] . (155)By the transformation (155) and the initial condition (39) the fourth term onthe right-hand side of (153) is cancelled and the solution b k − ( P, t ) is changed tothe transformed solution ˆ b k − ( P, t ):ˆ b k − ( P, t ) = b + ( P, t )ˆΓ + − k ( P, t ) exp[ i Z tt dt ′ t ′ )]+ ib ( P, t )ˆΓ − k ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )] − ib ( P, t )ˆΓ − k ( P, t ) + i Z tt dt { b + ( P, t ) ˆΘ + − k ( P, t ) exp[ i Z t t dt ′ t ′ )] } + Z tt dt { b ( P, t ) ˆΘ − k ( P, t ) exp[ − i ∆ P Z t t dt ′ K ( t ′ )] exp[ i Z t t dt ′ Ω( t ′ )] } . (156)The recursive relations for the amplitudes of both the solutions ˆ b k − ( P, t ) and b k − ( P, t ) are given byˆΓ + − k ( P, t ) = Γ + − k ( P, t ) exp[ − i Z tt dt Γ − k ( P, t )] , (157a)ˆΓ − k ( P, t ) = Γ − k ( P, t ) exp[ − i Z tt dt Γ − k ( P, t )] , (157b)ˆΘ + − k ( P, t ) = [Θ + − k ( P, t ) + Γ + − k ( P, t )Γ − k ( P, t )] exp[ − i Z tt dt Γ − k ( P, t )] , (157c)76Θ − k ( P, t ) = [Θ − k ( P, t ) − Γ − k ( P, t )Γ − k ( P, t )] exp[ − i Z tt dt Γ − k ( P, t )] . (157d)After making the integration by parts on the last two integrals on the right-hand side of (156) the solution ˆ b k − ( P, t ) is changed to the ( k + 1) − order exactsolution b k +1 − ( P, t ) which is also given by (153). The recursive relations betweenthe amplitudes of both the solutions ˆ b k − ( P, t ) and b k +1 − ( P, t ) are generally givenby Γ + − ( k +1) ( P, t ) = ˆΓ + − k ( P, t ) + 12 ˆΘ + − k ( P, t )Ω( t ) , (158a)Γ − ( k +1) ( P, t ) = ˆΓ − k ( P, t ) − ˆΘ − k ( P, t )Ω( t ) , (158b)Γ − k +1 ( P, t ) = [ 14 Γ(
P, t ) ˆΘ + − k ( P, t )Ω( t ) + 1 √ P, t ) ∗ ˆΘ − k ( P, t )Ω( t ) ] × exp[ i Z tt dt Γ + k ( P, t )] exp[ i Z tt dt Γ + k − ( P, t )] ... exp[ i Z tt dt Γ +1 ( P, t )] , (158c)Θ + − ( k +1) ( P, t ) = i ∂∂t ( ˆΘ + − k ( P, t )Ω( t ) ) + 1 √ P, t ) ∗ ˆΘ − k ( P, t )Ω( t ) , (158d)Θ − ( k +1) ( P, t ) = 12 √ P, t ) ˆΘ + − k ( P, t )Ω( t ) + i ∂∂t ( ˆΘ − k ( P, t )Ω( t ) ) + K ( t ) ˆΘ − k ( P, t )Ω( t ) ∆ P. (158e)The recursive relations (157c), (157d), (158d), and (158e) show that in the lasttwo integrals of the k − order exact solution ˆ b k − ( P, t ) of (156) the amplitudesˆΘ + − k ( P, t ) and ˆΘ − k ( P, t ) are inversely proportional to the k − th power of theRabi frequency Ω( t ) , that is, ˆΘ + − k ( P, t ) ∝ Ω( t ) − k and ˆΘ − k ( P, t ) ∝ Ω( t ) − k . Thus, by making a few equivalent transformations one may obtain a high-orderapproximation solution from the exact solution b k − ( P, t ) or ˆ b k − ( P, t ) . The first-order approximation solution may be obtained from the exact solution ˆ b − ( P, t )by neglecting the last two integrals on the right-hand side of (156) ( k = 1), b − ( P, t ) ≡ b − ( P, t ) = b + ( P, t )Γ + − ( P, t ) exp[ i Z tt dt ′ t ′ )]+ i ( b ( P, t ) − b ( P, t ))Γ − ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )]+ ib ( P, t )Γ − ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )] − ib ( P, t )Γ − ( P, t ) exp[ i Z tt dt Γ − ( P, t )] , (159)77nd the truncation error is just given by these last two integrals neglected, andby the integration by parts it can turn out that the truncation error is boundedby | E − r ( P, t ) | ≤ | ˆΘ + − ( P, t ) | Ω( t ) + | ˆΘ − ( P, t ) | Ω( t ) + | ˆΘ − ( P, t ) | Ω( t )+ Z tt dt {| Γ − ( P, t ) | + | Θ + − ( P, t ) | + | Θ − ( P, t ) |} . (160)The error upper bound is proportional to Ω( t ) − . The second-order approxima-tion solution is obtained from the exact solution ˆ b − ( P, t ) by neglecting the lasttwo integrals on the right-hand side of (156) ( k = 2), b − ( P, t ) = b + ( P, t )Γ + − ( P, t ) exp[ i Z tt dt ′ t ′ )]+ ib ( P, t )Γ − ( P, t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )] − ib ( P, t )Γ − ( P, t ) exp[ i Z tt dt Γ − ( P, t )] (161)and the truncation error is bounded by | E − r ( P, t ) | ≤ | ˆΘ + − ( P, t ) | Ω( t ) + | ˆΘ − ( P, t ) | Ω( t ) + | ˆΘ − ( P, t ) | Ω( t )+ Z tt dt {| Γ − ( P, t ) | + | Θ + − ( P, t ) | + | Θ − ( P, t ) |} . (162)The error upper bound is proportional to Ω( t ) − . The first- and second-orderapproximation solutions may be used to set up the adiabatic condition below.With the help of the equivalent transformations similar to those used aboveone may obtain the k − order exact solution for the variable b ( P, t ) from theequation (141a). At first the first-order exact solution may be given by b ( P, t ) − b ( P, t ) = ib + ( P, t )Γ +01 ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )]+ ib − ( P, t )Γ − ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )]+ Z tt dt { b + ( P, t )Θ +01 ( P, t ) exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ i Z t t dt ′ Ω( t ′ )] } + Z tt dt { b − ( P, t )Θ − ( P, t ) exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] } (163)78here the four amplitudes are given byΓ +01 ( P, t ) = − Γ − ( P, t ) = − √ P, t ) ∗ Ω( t ) , Θ +01 ( P, t ) = − Θ − ( P, t ) = i ∂∂t ( Θ(
P, t ) ∗ Ω( t ) ) −
12 Γ(
P, t )Θ(
P, t ) ∗ Ω( t ) − K ( t )Θ( P, t ) ∗ Ω( t ) ∆ P. A special point in the first-order exact solution (163) is that there is not anyterm containing the solution b ( P, t ) itself on the right-hand side of (163). Thus,one may make directly the integration by parts on the last two integrals on theright-hand side of (163). Here still denote that δ b ( P, t ) = b ( P, t ) − b ( P, t ) and δ b ± ( P, t ) = b ± ( P, t ) − b ± ( P, t ) . Then the second-order exact solution δ b ( P, t )may be expressed as δ b ( P, t ) ≡ δ b ( P, t ) = ib + ( P, t )Γ +02 ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )]+ ib − ( P, t )Γ − ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )]+ ib ( P, t )Γ ( P, t ) + i Z tt dt { δ b ( P, t )Γ ( P, t ) } + Z tt dt { b + ( P, t )Θ +02 ( P, t ) exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ i Z t t dt ′ Ω( t ′ )] } + Z tt dt { b − ( P, t )Θ − ( P, t ) exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] } (164)where the amplitudes are given byΓ +02 ( P, t ) = Γ +01 ( P, t ) − Θ +01 ( P, t )Ω( t ) , Γ − ( P, t ) = Γ − ( P, t ) + Θ − ( P, t )Ω( t ) , Γ ( P, t ) = Z tt dt Γ ( P, t ) , Γ ( P, t ) = −√ P, t )Θ +01 ( P, t )Ω( t ) , Θ +02 ( P, t ) = − Θ − ( P, t ) = i ∂∂t ( Θ +01 ( P, t )Ω( t ) ) − K ( t )Θ +01 ( P, t )Ω( t ) ∆ P −
12 Γ(
P, t )Θ − ( P, t )Ω( t ) . Now the first-order approximation solution may be obtained from the first-orderexact solution (163) by neglecting the last two integrals on the right-hand sideof (163). It is given by δ b ( P, t ) = ib + ( P, t )Γ +01 ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )]79 ib − ( P, t )Γ − ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )] (165)and the truncation error is just the last two integrals on the right-hand side of(163), and it can turn out by the integration by parts that the truncation erroris bounded by | E r ( P, t ) | ≤ | Θ +01 ( P, t ) | Ω( t ) + Z tt dt {| Γ ( P, t ) | + 2 | Θ +02 ( P, t ) |} . (166)This error upper bound is clearly proportional to Ω( t ) − . A higher-order ex-act solution than the second-order one (164) may be obtained by integratingby parts the last two integrals in (164), but it needs first to eliminate thefourth term that contains the solution δ b ( P, t ) itself on the right-hand sideof (164). This can be done by making an equivalent transformation on the so-lution δ b ( P, t ) . In general, for the k − order exact solution δ bk ( P, t ) ( k = 2 , , ... )the equivalent transformation is written asˆ δ bk ( P, t ) = δ bk ( P, t ) exp[ − i Z tt dt Γ k ( P, t )] . (167)Then the k − order transformed solution ˆ δ bk ( P, t ) may be expressed asˆ δ bk ( P, t ) = ib + ( P, t )ˆΓ +0 k ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )]+ ib − ( P, t )ˆΓ − k ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )]+ ib ( P, t )ˆΓ k ( P, t )+ Z tt dt { b + ( P, t ) ˆΘ +0 k ( P, t ) exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ i Z t t dt ′ Ω( t ′ )] } + Z tt dt { b − ( P, t ) ˆΘ − k ( P, t ) exp[ i ∆ P Z t t dt ′ K ( t ′ )] exp[ − i Z t t dt ′ Ω( t ′ )] } (168)where the amplitudes satisfy the recursive relations:ˆΓ +0 k ( P, t ) = Γ +0 k ( P, t ) exp[ − i Z tt dt Γ k ( P, t )] , (169a)ˆΓ − k ( P, t ) = Γ − k ( P, t ) exp[ − i Z tt dt Γ k ( P, t )] , (169b)ˆΓ k ( P, t ) = Z tt dt { [ ∂∂t Γ k ( P, t )] exp[ − i Z t t dt ′ Γ k ( P, t ′ )] } , (169c)ˆΘ +0 k ( P, t ) = [Θ +0 k ( P, t ) − Γ +0 k ( P, t )Γ k ( P, t )] exp[ − i Z tt dt Γ k ( P, t )] , (169d)80Θ − k ( P, t ) = [Θ − k ( P, t ) − Γ − k ( P, t )Γ k ( P, t )] exp[ − i Z tt dt Γ k ( P, t )] . (169e)From the k − order transformed solution ˆ δ bk ( P, t ) to the ( k + 1) − order solution δ bk +10 ( P, t ) there is an equivalent transformation of the integration by parts. The( k + 1) − order exact solution δ bk +10 ( P, t ) is still given by (164) as long as onemakes the replacement: δ b ( P, t ) ↔ δ bk +10 ( P, t ) and the following replacementsfor the amplitudes:Γ α ( P, t ) ↔ Γ α k +1 ( P, t ) , Θ α ( P, t ) ↔ Θ α k +1 ( P, t ) , α = + , − ;Γ ( P, t ) ↔ Γ k +1 ( P, t ) , Γ ( P, t ) ↔ Γ k +1 ( P, t ) . The recursive relations for the amplitudes between the two solutions ˆ δ bk ( P, t )and δ bk +10 ( P, t ) ( k ≥
2) are given byΓ +0 k +1 ( P, t ) = ˆΓ +0 k ( P, t ) − ˆΘ +0 k ( P, t )Ω( t ) , Γ − k +1 ( P, t ) = ˆΓ − k ( P, t )+ ˆΘ − k ( P, t )Ω( t ) , (170a)Γ k +1 ( P, t ) = ˆΓ k ( P, t ) + 1 √ Z tt dt {− Θ( P, t ) ˆΘ +0 k ( P, t )Ω( t ) + Θ( P, t ) ˆΘ − k ( P, t )Ω( t ) } , (170b)Γ k +1 ( P, t ) = [ − √ P, t ) ˆΘ +0 k ( P, t )Ω( t ) + 1 √ P, t ) ˆΘ − k ( P, t )Ω( t ) ] × exp[ i Z tt dt Γ k ( P, t )] exp[ i Z tt dt Γ k − ( P, t )] ... exp[ i Z tt dt Γ ( P, t )] , (170c)Θ +0 k +1 ( P, t ) = i ∂∂t ( ˆΘ +0 k ( P, t )Ω( t ) ) − K ( t ) ˆΘ +0 k ( P, t )Ω( t ) ∆ P −
12 Γ(
P, t ) ˆΘ − k ( P, t )Ω( t ) , (170d)Θ − k +1 ( P, t ) = − i ∂∂t ( ˆΘ − k ( P, t )Ω( t ) ) + K ( t ) ˆΘ − k ( P, t )Ω( t ) ∆ P + 12 Γ( P, t ) ˆΘ +0 k ( P, t )Ω( t ) . (170e)It can turn out that the amplitudes Θ +0 k ( P, t ) and Θ − k ( P, t ) of the k − order exactsolution δ bk ( P, t ) are inversely proportional to Ω( t ) − k . Thus, the last two inte-grals of the k − order exact solution δ bk ( P, t ) have a negligible contribution tothe solution if the Rabi frequency Ω( t ) is large. Now the second-order approxi-mation solution may be obtained from the second-order exact solution ˆ δ b ( P, t )of (168), δ b ( P, t ) = ib + ( P, t )Γ +02 ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ i Z tt dt ′ Ω( t ′ )]+ ib − ( P, t )Γ − ( P, t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] exp[ − i Z tt dt ′ Ω( t ′ )]81 b ( P, t ) { − exp[ i Z tt dt Γ ( P, t )] } , (171)while the truncation error is given by the last two integrals on the right-handside of (168), and the truncation error upper bound is determined from | E r ( P, t ) | ≤ | ˆΘ +02 ( P, t ) | Ω( t ) + | ˆΘ − ( P, t ) | Ω( t )+ Z tt dt {| Γ ( P, t ) | + | Θ +03 ( P, t ) | + | Θ − ( P, t ) |} . (172)This error upper bound is clearly proportional to Ω( t ) − . Now by solving the three first-order approximation solutions (149), (159),and (165), which are all linear algebra equations, one may obtain the first-order approximation solution to the basic equations (141). At first accordingto the first-order approximation solutions (149), (159), and (165) and theirtruncation errors the first-order exact solution to the equations (141) may beformally written as, (this is really the first-order approximation solutions plustheir truncation errors), b + ( P, t ) = α + − b − ( P, t ) + α +0 δ b ( P, t ) + β +0 + E + r ( P, t ) , (173a) b − ( P, t ) = α − + b + ( P, t ) + α − δ b ( P, t ) + β − + E − r ( P, t ) , (173b) δ b ( P, t ) = α b + ( P, t ) + α − b − ( P, t ) + E r ( P, t ) , (173c)where the truncation errors E ± r ( P, t ) and E r ( P, t ) may be considered as smallparameters and any other parameters such as α + − , α − + , α , etc., can be obtaineddirectly from the first-order approximation solutions (149), (159), and (165).These three equations are linear algebra equations as E ± r ( P, t ) and E r ( P, t )are considered as the small parameters. The exact solution to the three linearalgebra equations (173) may be written as δ b ± ( P, t ) = δ ± m ( P, t ) + E ± rt ( P, t ) , δ b ( P, t ) = δ m ( P, t ) + E rt ( P, t ) , (174)where the main terms are written as δ ± m ( P, t ) = ∓ i √ b ( P, t ) F ( P, t ) ×{ Θ( P, t )Ω( t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ ∓ i Z tt dt ′ Ω( t ′ )] − Θ( P, t )Ω( t ) exp[ ± i Z tt dt Γ( P, t ) + 4 | Θ( P, t ) | Ω( t ) ] } , (175a) δ m ( P, t ) = 0 . (175b)82ere the factors F ( P, t ) and G ± ( P, t ) are defined by F ( P, t ) = 1 + | Θ( P,t ) | Ω( t )
116 Γ(
P,t ) Ω( t ) + | Θ( P,t ) | Ω( t ) , G ± ( P, t ) = ±
14 Γ(
P,t )Ω( t ) + | Θ( P,t ) | Ω( t )
116 Γ(
P,t ) Ω( t ) + | Θ( P,t ) | Ω( t ) . The factors G ± ( P, t ) appear in the error terms E ± rt ( P, t ) and E rt ( P, t ) , as canbe seen below. Then the upper bound of the error term that is generated bythe main terms δ ± m ( P, t ) and δ m ( P, t ) is determined from || E (1) r ( P, t ) || = p | δ + m ( P, t ) | + | δ − m ( P, t ) | + | δ m ( P, t ) | ≤ F ( P, t )[ | Θ( P, t ) | Ω( t ) + | Θ( P, t ) | Ω( t ) ] . (176)It is clear that this upper bound is proportional to Ω( t ) − . Notice that the factor F ( P, t ) is unity approximately if | Θ( P, t ) | / Ω( t ) << P, t ) / Ω( t ) << . By comparing the inequality (176) with the inequality (133) one can find thatif the factor F ( P, t ) is unity, then the inequality (176) is really the inequality(133) that leads to the first-order approximation adiabatic condition (134).Now investigate the secondary error terms E ± rt ( P, t ) and E rt ( P, t ) in theexact solution (174). These error terms contain the first-order truncation errors E ± r ( P, t ) and E r ( P, t ) (See: (150), (160), and (166)). They may be written as E ± rt ( P, t ) = E ± rt ( P, t ) + E ± rt ( P, t ) + E ± rt ( P, t ) , (177a) E rt ( P, t ) = E rt ( P, t ) + E rt ( P, t ) + E rt ( P, t ) + E rt ( P, t ) , (177b)where E ± rt ( P, t ), E rt ( P, t ) ∝ Ω( t ) − ; E ± rt ( P, t ) , E rt ( P, t ) ∝ Ω( t ) − ; E ± rt ( P, t ) ,E rt ( P, t ) ∝ Ω( t ) − ; E rt ( P, t ) ∝ Ω( t ) − . It is not difficult to obtain the upperbounds for all these error terms, but one needs only to consider the dominatingterms E ± rt ( P, t ) and E rt ( P, t ) that are proportional to Ω( t ) − , since the trun-cation errors E ± r ( P, t ) and E r ( P, t ) are proportional to Ω( t ) − and the othererror terms E ± krt ( P, t ) and E lrt ( P, t ) ( k = 3 , l = 3 , ,
5) are higher-order andcan be neglected with respect to the dominating terms E ± rt ( P, t ) and E rt ( P, t ).The dominating error terms are given by E ± rt ( P, t ) = E ± r ( P, t ) F ( P, t ) ± √ G ± ( P, t ) exp[ ∓ i Z tt dt ′ t ′ )] ×{ ib ( P, t ) Θ( P, t )Ω( t ) exp[ − i ∆ P Z tt dt ′ K ( t ′ )] exp[ ± i Z tt dt ′ Ω( t ′ )] − ib ( P, t ) Θ( P, t )Ω( t ) exp[ ∓ i Z tt dt Γ( P, t ) + 4 | Θ( P, t ) | Ω( t ) ] } , (178a) E rt ( P, t ) = E r ( P, t ) − b ( P, t ) F ( P, t ) | Θ( P, t ) | Ω( t ) b ( P, t ) F ( P, t ) Θ(
P, t ) ∗ Ω( t ) Θ( P, t )Ω( t ) exp[ i ∆ P Z tt dt ′ K ( t ′ )] × cos { Z tt dt [Ω( t ) + Γ( P, t ) + 4 | Θ( P, t ) | t ) ] } . (178b)The last term on the right-hand sides of (178) for each one of the error terms E ± rt ( P, t ) and E rt ( P, t ) is proportional to the value Θ(
P, t ) / Ω( t ) at the initialtime t of the basic STIRAP decelerating or accelerating process. As discussedbefore (See: (133) and (134)), the initial value | Θ( P, t ) | / Ω( t ) may be controlledto be so small that it can be neglected. Then the norm for the error vector( E rt ( P, t ) , E +2 rt ( P, t ) , E − rt ( P, t )) T is bounded by || E (2) r ( P, t ) || = q | E +2 rt ( P, t ) | + | E − rt ( P, t ) | + | E rt ( P, t ) | ≤ q | E r ( P, t ) | + | F ( P, t ) | ( | E + r ( P, t ) | + | E − r ( P, t ) | )+ | Θ( P, t ) | Ω( t ) s | F ( P, t ) | | Θ( P, t ) | Ω( t ) + 12 | G + ( P, t ) | + 12 | G − ( P, t ) | . (179)This error upper bound is proportional to Ω( t ) − . Now it follows from (174)and (177) and then the inequalities (176) and (179) that the total deviation ofa real STIRAP adiabatic process from the ideal one at any instant of time isbounded by || E r ( P, t ) || = q | δ b + ( P, t ) | + | δ b − ( P, t ) | + | δ b ( P, t ) | ≤ || E (1) r ( P, t ) || + || E (2) r ( P, t ) ||≤ F ( P, t ) | Θ( P, t ) | Ω( t ) + q | E r ( P, t ) | + | F ( P, t ) | ( | E + r ( P, t ) | + | E − r ( P, t ) | )+ | Θ( P, t ) | Ω( t ) vuut | F ( P, t ) | | Θ( P, t ) | Ω( t ) +
116 Γ(
P,t ) Ω( t ) + | Θ( P,t ) | Ω( t ) [1 +
116 Γ(
P,t ) Ω( t ) + | Θ( P,t ) | Ω( t ) ] , (180)where the upper bounds for the truncation errors | E + r ( P, t ) | , | E − r ( P, t ) | , and | E r ( P, t ) | are obtained from (150), (160), and (166), respectively. In order toobtain the global adiabatic condition one needs to limit the maximum value onthe rightest side of (180) not to be more than a desired small value over theeffective momentum region [ P ] and in the whole time period t ≤ t ≤ t + T of the STIRAP decelerating or accelerating process. Denote A d ( P, t ) as thefunction on the rightest side of (180). Then the global adiabatic condition maybe expressed as || E r ( P, t ) || ≤ A d ( P, t ) ≤ max P ∈ [ P ] , t ≤ t ≤ t + T { A d ( P, t ) } ≤ ε r , (181)84here ε r is the desired value and ε r <<
1. Unlike the adiabatic condition(128) there is not an exponential correction factor in the adiabatic condition(180) and (181). The adiabatic condition (180) and (181) consists of the first-order term that is proportional to Ω( t ) − and the second-order terms that areproportional to Ω( t ) − , while the adiabatic condition (128) is the first-orderterm with the factor F ( P, t ) = 1 times the exponential correction factor. Thus,the adiabatic condition (180) and (181) is much less severe than the adiabaticcondition (128), the latter is most severe for a quantum ensemble with a broadmomentum distribution. However, just like the adiabatic condition (128) theadiabatic condition (180) and (181) is also strict and accurate (It is not difficultto include the omitted secondary error terms of (177) in the adiabatic condition(180) and (181)). Therefore, if any one of the two general adiabatic conditionsis met, then a better STIRAP pulse sequence is obtained and the perfect state(or population) transfer may be realized by the STIRAP pulse sequence. Theadiabatic condition (180) and (181) may be more useful in practice to realizethe perfect STIRAP state (or population) transfer in an atomic or molecularensemble with a broad momentum distribution. It may also be used to realizethe STIRAP decelerating and accelerating processes in the laser cooling andthe quantum coherence interference experiments of a cold atomic or molecularensemble.The adiabatic condition (180) and (181) is still slightly severe. This is be-cause it is based on the first-order approximation solution to the basic equations(141), leading to that the upper bounds for the truncation errors | E + r ( P, t ) | , | E − r ( P, t ) | , and | E r ( P, t ) | are not the lowest ones. As known before, these trun-cation errors are proportional to Ω( t ) − . A better adiabatic condition may beset up on the basis of the second-order approximation solution to the basic equa-tions (141), which are given by (151), (161), and (171), and the correspondingtruncation error upper bounds | E + r ( P, t ) | , | E − r ( P, t ) | , and | E r ( P, t ) | are givenby (152), (162), and (172), respectively. These upper bounds are proportionalto Ω( t ) − . Thus, they are the lower ones with respect to those truncation errorupper bounds of the first-order solution, leading to that the adiabatic conditionbased on the second-order approximation solution is better one.It is known that in the adiabatic condition (180) and (181) the truncationerrors | E + r ( P, t ) | , | E − r ( P, t ) | , and | E r ( P, t ) | are proportional to Ω( t ) − andthe factor F ( P, t ) is a function of the ratio Γ(
P, t ) / Ω( t ) . Then the parame-ter Γ(
P, t ) appears only in those terms that are proportional to Ω( t ) − on therightest side of (180). This is different from the case that the parameter Θ( P, t )may appear in the first term on the rightest side of (180) that is proportionalto Ω( t ) − . Thus, the parameter Γ(
P, t ) has a smaller contribution to the adi-abatic condition (180) and (181) than the parameter Θ(
P, t ) . As pointed outbefore, the effect of the momentum distribution is dependent upon whether theRaman laser light beams of a STIRAP experiment are copropagating or coun-terpropagating. Consider that the Raman laser light beams are copropagating.Then the momentum distribution could have a relatively small effect, becausein this case the parameter Θ(
P, t ) has a smaller value. This is correct for thefirst-order approximation adiabatic condition (134). However, it could not be85o simple for the inequality (180) in a quantum ensemble with a broad momen-tum distribution. When the first term on the rightest side of (180) is smallerdue to a small parameter Θ(
P, t ) , the second and third terms could becomemore important and have a dominating contribution to the adiabatic condition(180) and (181). Then in this case the parameters Γ( P, t ) and K ( t ) becomemore important in the adiabatic condition (180) and (181), resulting in that themomentum distribution has a large effect on the adiabatic condition. There-fore, the momentum distribution needs to be considered explicitly even for aconventional STIRAP state (or population) transfer experiment in a quantumensemble with a broad momentum distribution, which uses the copropagatingRaman laser light beams. Obviously, for the STIRAP decelerating and acceler-ating processes in a quantum ensemble with a broad momentum distribution,which use the counterpropagating Raman laser light beams, one needs to con-sider generally the effect of the momentum distribution on the STIRAP state(or population) transfer.The present adiabatic theoretical methods including the equivalent trans-formation method to solve the basic differential equations (26) not only can beused to set up a general adiabatic condition for the basic STIRAP deceleratingand accelerating processes of a single atom or molecule or a quantum ensembleof the atoms or molecules, but also they will have an extensive application inother research fields such as the NMR spectroscopy (See: for example, Ref. [41])and the magnetic resonance image (MRI).
8. Discussion
In the paper the standard three-state STIRAP population transfer theoryin the laser spectroscopy [4, 15, 16, 17, 18] has been developed to describe theo-retically the STIRAP-based unitary decelerating and accelerating processes of asingle freely moving atom by combining the superposition principle in quantummechanics [25] and the energy, momentum, and angular momentum conserva-tion laws for the atomic photon absorption and emission processes [5, 19]. Thereare similar theoretical works or developments to describe the atomic laser cool-ing process [5, 19, 20, 21] in a neutral atom ensemble and the atomic quantuminterference experiments [10, 12] in a cold atomic ensemble. There are also anumber of works to investigate the atomic decelerating and accelerating pro-cesses by the laser light techniques [21, 23, 24]. However, the present workis focused on the analytical and quantitative investigation how the momentumdistribution of a superposition of the momentum states of a pure-state quantumsystem such as a single freely moving atom affects the state-transfer efficiencyin these STIRAP unitary decelerating and accelerating processes. It empha-sizes the complete STIRAP state transfer and the unitarity of these processes.This means that in the present work any decoherence effect of the atomic sys-tem under study is not considered. A main purpose to investigate the effect ofthe momentum distribution on the STIRAP state transfer is to build up bet-ter STIRAP unitary decelerating and accelerating sequences, so that the time-and space-compressing processes of the quantum control process to simulate86he reversible and unitary state-insensitive halting protocol [22] can be realizedthrough these decelerating and accelerating processes. Thus, this is involvedin setting up a general adiabatic condition for the STIRAP unitary decelerat-ing and accelerating processes. In the paper two general adiabatic conditionshave been obtained analytically, one based on the Dyson series solution to thebasic differential equations to govern the STIRAP processes, another based onthe equivalent transformation method to solve the basic differential equations.Both the general adiabatic conditions may be used to set up a conventionalSTIRAP experiment and also the STIRAP-based decelerating and accelerat-ing processes. A complete STIRAP state transfer could be achieved only inthe ideal adiabatic condition. Generally, it is hard to achieve a complete statetransfer for the STIRAP processes when the atomic momentum superpositionstate has a broad momentum distribution. However, in the ideal or nearly idealadiabatic condition an almost complete STIRAP state transfer may be realizedif the superposition of the momentum states has a small effective wave-packetspreading or a narrow momentum distribution. When the initial motional stateof a freely moving atom is a Gaussian wave-packet state, the final motional stateof the atom is still a perfect or almost perfect Gaussian wave-packet state afterthe atom undergoes the STIRAP unitary decelerating (or accelerating) processin the ideal or nearly ideal adiabatic condition. Therefore, in the paper it isshown that the time- and space-compressing processes of the quantum controlprocess [22] can be realized almost perfectly through the STIRAP deceleratingand accelerating processes in the ideal or nearly ideal adiabatic condition. Thisis one of the important results in the paper.The standard STIRAP population transfer theory is generally based on thesemiclassical theory of electromagnetic radiation. In the semiclassical theory theexternally applied electromagnetic fields such as the Raman laser light beamsare considered as the classical electromagnetic fields, while the atomic systemitself and the interaction between the atomic system and the external electro-magnetic fields are treated quantum mechanically. It has been shown that thesemiclassical theory can describe almost perfectly the three-state STIRAP pop-ulation transfer experiments of the atomic and molecular beams in the laserspectroscopy [4, 15, 16, 17, 18]. On the other hand, the semiclassical theoryis also successful to describe the STIRAP-based laser cooling processes in aneutral atomic ensemble [20, 21] and especially the atomic quantum interfer-ence experiments in a cold neutral atom ensemble [10, 12, 13, 14]. In the paperthe semiclassical theory also is directly employed to describe the STIRAP-basedunitary decelerating and accelerating processes of a single atom. The semiclassi-cal theory of electromagnetic radiation generally can not explain reasonably theatomic spontaneous emission [1, 25, 26]. However, it is suited to describe theseSTIRAP unitary decelerating and accelerating processes due to that these STI-RAP processes can avoid the atomic spontaneous emission by setting suitablythe experimental parameters.As far as the Hamiltonian of Eq. (4) to describe the three-state STIRAPexperiment of an atom system is concerned, there are three requirements: ( i )the three-state subspace for the atomic internal states is closed under the Hamil-87onian; ( ii ) the electric-dipole approximation is satisfied; and ( iii ) the rotatingwave approximation ( RW A ) is reasonable. The first requirement can be sat-isfied if one chooses suitably the atom and its three internal states and theexperimental parameters of the Raman laser light beams. Since the size of anatom is generally much less than the wave lengths of the Raman laser lightbeams at the optical frequencies ( ∼ ω and ω ), the second requirement maybe met generally. If the Raman laser light beams are strong, it can turn outthat in the first approximation the strong laser light field may generate a Bloch-Siegert shift to the transition frequency of the atomic internal energy levels [1].When the Rabi frequencies (Ω p ( t ) and Ω s ( t )) of the Raman laser light beamsare much less than the resonance frequencies ( ω and ω ) of the atomic inter-nal energy levels and the detunings for the Raman laser light beams are small,the magnitude of the Bloch-Siegert shifts generated by the Raman laser lightbeams at the optical frequencies is very small and may be negligible and hencethe rotating wave approximation is reasonable [1]. However, it is also conve-nient to correct the Bloch-Siegert shifts in the STIRAP experiment because oneneeds only to add the Bloch-Siegert shifts to the resonance frequencies ( ω and ω ). A simple evaluation for the Bloch-Siegert shifts can be seen in Ref.[1] and a general treatment may use the average Hamiltonian theory [34]. Onthe other hand, it is also possible to apply an extra laser light field for eachone of the Raman laser light beams in the STIRAP experiment to compensatethe rotating-wave approximation. In fact, if each one of the two Raman laserlight beams in the STIRAP experiment is replaced with a pair of the laser lightbeams with the orthogonal electric field vectors and the suitable phases [38],one can eliminate the rotating-wave approximation. Similarly, one may also usethe circularly polarized lights to prepare the dipole interaction Hamiltonian (10)without the rotating wave approximation [19, 38].There is also another condition to be met that the electromagnetic field ofany Raman laser light beam is considered as an infinite plane-wave electromag-netic field when calculating the time evolution process of a freely moving atomunder the STIRAP unitary decelerating and accelerating processes. Here theinfinite plane-wave electromagnetic field has spatially uniform amplitude andphase. This condition can be met only when the electromagnetic field can en-compass sufficiently the whole wave-packet motional state of the moving atom.Note that the electromagnetic field of the Raman laser light beam propagatesalong a direction parallel to the atomic moving direction in one-dimensionalspace. Since the atom is in a Gaussian wave-packet motional state which has afinite wave-packet spreading, then one can set suitably the experimental param-eters for the Raman laser light beam such that the electromagnetic field in spaceis much wider than the effective wave-packet spreading of the atomic motionalstate during the whole decelerating or accelerating process. Then in this casethe electromagnetic field in space can be reasonably considered as an infiniteand uniform plane-wave electromagnetic field for the atomic wave-packet mo-tional state. The condition may be satisfied more easily for a heavy atom asthe wave-packet motional state for such atom has a more narrow wave-packetspreading. If the electromagnetic field in space has a finite bandwidth less than88r comparable to the wave-packet spreading of an atomic motional state, thenit can not be considered as an infinite and uniform plane-wave electromagneticfield and the electric dipole interaction of Eq. (10) with space-independent Rabifrequencies is not suited to describe the STIRAP process, since in this case theRabi frequencies are dependent upon the spatial coordinate [35].As an important result in the paper, it is shown that if a free atom is ina Gaussian wave-packet motional state at the initial time, then it is still in aGaussian wave-packet motional state after it is decelerated (or accelerated) bythe STIRAP-based unitary decelerating (or accelerating) sequence in the idealor nearly ideal adiabatic condition. As far as a Gaussian wave-packet state isconcerned, there are two types of time- and space-dependent unitary evolutionprocesses that do not change the Gaussian wave-packet shape of the atomicmotional state. The first type is that the unitary evolution processes may ma-nipulate and control the center-of-mass position and/or momentum of a Gaus-sian wave-packet state but can not manipulate at will the complex linewidthof a Gaussian wave-packet state. The second type is that the unitary evolu-tion processes may manipulate and control the complex linewidth of a Gaussianwave-packet state. The STIRAP-based unitary decelerating and acceleratingprocesses belong to the first type. This type of the unitary evolution processestend to have the property that in the unitary evolution process the imaginarypart of the complex linewidth of a Gaussian wave-packet state increases linearlywith the time period of the unitary evolution process, while the real part usuallykeeps unchanged, i.e., the wave-packet spreading of the Gaussian wave-packetstate becomes larger and larger early or late as the time period increases. Theseunitary evolution processes which have the property also include the free-particlemotion and the atomic bouncing process off a hard potential wall in the specialcase. Thus, the free-particle motion and the atomic bouncing process may beassigned to the first type. The Hamiltonians of the quantum systems to createthe first type of the time- and space-dependent unitary propagators usually cannot be singly used to generate their inverse unitary propagators without anyhelp of the interactions from outside the quantum systems. Therefore, this typeof unitary evolution processes do not have their own inverse unitary propaga-tors in these quantum systems separated from the outside or their environment.Obviously, these separate quantum systems also include the isolated quantumsystems in the quantum statistical physics and they are described completelyby the Hamiltonians of the quantum systems. These unitary evolution pro-cesses could be considered to be self-irreversible in the sense that there do notexist their own inverse unitary propagators in the same separate quantum sys-tems, although these processes obey the unitary quantum dynamics and theirinverse unitary propagators could be generated with the help of the specificinteractions from outside the quantum systems. Take a free-particle motion asa typical example. A free-particle motion can be described completely by theunitary propagator U ( t ) = exp[ − ip t/ (2 m ℏ )]. Of course, one may also choosethe unitary propagator U ( t ) + = exp[ ip t/ (2 m ℏ )] to describe the free-particlemotion. However, once one chooses one of the two unitary propagators to de-scribe the free-particle motion, another is the inverse propagator of the chosen89nitary propagator. The free particle may move toward the left or the right inthe coordinate axis and both the motions can be described by the same unitarypropagator U ( t ). Now the inverse free-particle motion is described by the uni-tary propagator U ( t ) + . The Hamiltonian H = p / m of the free particle canreally generate only the unitary propagator U ( t ) (or U ( t ) + ) but can not reallygenerate both the unitary propagator U ( t ) and its inverse propagator U ( t ) + simultaneously by the Schr¨odinger equation i ℏ ∂U ( t ) /∂t = H ( t ) U ( t ) . One maytake a hermite conjugate on the Schr¨odinger equation and then could obtain U ( t ) + , but taking a Hermite conjugate is not a real physical process and hencethere does not exist the inverse propagator U ( t ) + in the free-particle quantumsystem. Then the inverse free-particle motion will never really take place if thefree particle is not acted on by any external interaction. In other words, onemay argue that the time reversal process (i.e., the inverse free-particle motion)may be a real physical process [25, 30], but this process can not really takeplace for the free particle without any help of the specific external interactions.Then in this sense the free-particle motion is really self-irreversible, althoughit is governed by the unitary propagator U ( t ). Of course, it is possible to con-struct the inverse unitary propagator of the free particle if the specific externalinteractions are applied to the free particle. How to construct these inversepropagators of the type of time- and space-dependent unitary propagators men-tioned above will be reported in next paper. As pointed out in the previouspapers [22, 36], such a situation that a quantum system that obeys the uni-tary quantum dynamics can not really have both the unitary propagator andits inverse propagator is often met in the quantum systems which are used toimplement a quantum computation. The spontaneously irreversible processesof isolated quantum systems in the quantum statistical physics [40] could berelated to the situation. It has been stressed in the previous papers [22, 36, 37]that these irreversible processes should be understood through the unitary quan-tum dynamics instead of the stochastic process and probability statistics [40],and they could be related to the difference between the unitary evolution pro-cess and its inverse process. On the other hand, there also exist other quantumsystems that have both the unitary propagators and their inverse propagators,where both the types of the unitary propagators can be generated by the sameHamiltonians of the quantum systems. A conventional harmonic oscillator istypically one of such quantum systems. In the quantum statistical physics suchquantum systems obey completely the Poincar´e ′ s recurrence theorem.There are the second type of time- and space-dependent unitary propagatorsthat can manipulate and control the complex linewidth of a Gaussian wave-packet motional state. A general quadratic Hamiltonian can generate a time-and space-dependent unitary propagator that can keep the Gaussian shape un-changed for a Gaussian wave-packet motional state under the action of theunitary propagator. Obviously, the Hamiltonian is different from those of theSTIRAP unitary decelerating and accelerating processes. One needs this typeof unitary propagators to manipulate and control the complex linewidth of aGaussian wave-packet motional state to build up the quantum circuit for thereversible and unitary state-insensitive halting protocol and also needs their in-90erse unitary propagators to realize the efficient quantum search process [22,36]. How to construct these unitary propagators will be reported in next paper(Arxiv: quant-ph/0708.2129). References
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