A fast scheme for the implementation of the quantum Rabi model with trapped ions
aa r X i v : . [ qu a n t - ph ] A ug A fast scheme for the implementation of thequantum Rabi model with trapped ions
H´ector M Moya-Cessa
Instituto Nacional de Astrof´ısica, ´Optica y Electr´onicaCalle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico
Abstract.
We show how to produce a fast quantum Rabi model with trapped ions.Its importance resides not only in the acceleration of the phenomena that may beachieved with these systems, from quantum gates to the generation of nonclassicalstates of the vibrational motion of the ion, but also in reducing unwanted effects suchas the decay of coherences that may appear in such systems.PACS numbers: fast scheme for the implementation of the quantum Rabi model with trapped ions
1. Introduction
Trapped ions are considered one of the best candidates to perform quantum informationprocessing. By interacting them with laser beams they are, somehow, easy tomanipulate, which makes them an excellent choice for the production of nonclassicalstates of their vibrational motion.The trapping of individual ions also offers many possibilities in spectroscopy [1], inthe research of frequency standards [2, 3], in the study of quantum jumps [4], and in thegeneration of nonclassical vibrational states of the ion [5], to name some applications.To make the ions more stable in the trap, increasing the time of confinement, and alsoto avoid undesirable random motions, it is needed that the ion be in its vibrationalground state which may be accomplished by means of an adequate use of lasers.Because of the high nonlinearities of the ion-laser interaction its theoreticaltreatment is a nontrivial problem [6–10]. Even in the simplest cases of interactionone has to employ physically motivated approximations in order to find a solution. Awell-known example is the
Lamb-Dicke approximation, in which the ion is considered tobe confined within a region much smaller than the laser wavelength. Other examplesare optical and vibrational rotating wave approximations that are usually performed inorder to find simpler Hamiltonians.Many treatments also assume a weak coupling approximation, such that, by tuningthe laser frequency to integer multiples of the trap frequency results in effective(nonlinear) Hamiltonians of the Jaynes-Cummings type [11, 12], in which the centre-of-mass of the trapped ion plays the role of the field mode in cavity QED.Recently it was shown that the quantum Rabi model could be engineered via theinteraction of two laser beams with a trapped ion [13]. Pedernales et al. did it by slightlydetuning both laser beams from the blue and red side bands, allowing them to constructa Hamiltonian of the Rabi type and reaching all the possible regimes. However, becausethe parameters involved are much smaller than the vibrational frequency of the ion, ν , the ion can suffer losses that lead to the decay of Rabi oscillations [14, 15]. Therehave been attempts to explain such loss of coherences via laser intensity and phasefluctuations [16].We will show here two approaches in which we can engineer a fast Quantum Rabimodel (QRM), fast in the sense that the parameters involved in the interaction may beof the order of ν . Instead of two off-resonant lasers [13], we use only one resonant beam.
2. Lamb-Dicke regime
We can write the Hamiltonian of the trapped ion as H = H vib + H at + H int , (1)where H vib is the ion’s center of mass vibrational energy, H at is the ion internal energy,and H int is the interaction energy between the ion and the laser. The vibrational motioncan be approximated by a harmonic oscillator. Internally, the ion will be modelled by a fast scheme for the implementation of the quantum Rabi model with trapped ions − e~r · ~E , where − e~r is the dipolar momentum of the ion and ~E is the electric field of the laser, that willbe considered a plane wave. Thus, we write the Hamiltonian, after an optical rotatingwave approximations as H = ν ˆ n + ω σ z + Ω (cid:2) e i ( kx − ω l t + φ l ) σ + + e − i ( kx − ω l t + φ l ) σ − (cid:3) . (2)The first term in the Hamiltonian is the ion vibrational energy; in the ion vibrationalenergy, the operator ˆ n = ˆ a † ˆ a is the number operator, and the ladder operators ˆ a and ˆ a † are given by the expressionsˆ a = r ν x + i ˆ p √ ν , ˆ a † = r ν x − i ˆ p √ ν , (3)where we have set the ion mass equal to one. Also, for simplicity, we have displacedthe vibrational Hamiltonian by ν/
2, the vacuum energy, that without loss of generalitymay be disregarded.The second term in the Hamiltonian corresponds to the ion internal energy; the matrices σ z , σ + , and σ − are the Pauli matrices, and obey the commutation relations[ σ z , σ ± ] = ± σ ± , [ σ + , σ − ] = σ z , (4)and ω is the transition frequency between the ground state and the excited state of theion. By considering the resonant condition, ω = ω l , and transforming to a picturerotating at ω l we obtain the Hamiltonian H = ν ˆ n + Ω h e iφ l ˆ D ( iη ) σ + + e − iφ l σ − ˆ D † ( iη ) i , (5)where we have defined the so-called Lamb-Dicke parameter η = k r mν (6)that is a measure of the amplitude of the oscillations of the ion with respect to thewavelength of the laser field represented by its wave vector k .If we consider the condition η √ ¯ n ≪
1, where ¯ n is the average number of vibrationalquanta, we can expand the Glauber displacement operator [17] in Taylor seriesˆ D ( iη ) ≈ iη ˆ a † + iη ˆ a, (7)such that the Hamiltonian (2) reads H ≈ ν ˆ n + Ω (cid:2) e iφ l σ + + e − iφ l σ − (cid:3) + iη Ω(ˆ a † + ˆ a ) (cid:2) e iφ l σ + − e − iφ l σ − (cid:3) . (8)By setting φ l = π and making now a rotation around the Y axis (by means of thetransformation exp( i π σ y )), with σ y = iσ − − σ + , we obtain the usual form of the RabiHamiltonian H = ν ˆ n − Ω σ z − iη Ω(ˆ a † + ˆ a )( σ + − σ − ) (9) fast scheme for the implementation of the quantum Rabi model with trapped ions ν = − H = − iη Ω (cid:0) ˆ aσ − − σ + ˆ a † (cid:1) . (10)On the other hand, if we set φ l = 0 and follow the same procedure we obtain H = ν ˆ n − Ω σ z − iη Ω(ˆ a † + ˆ a )( σ + − σ − ) (11)that, by taking ν = 2Ω, and using the rotating wave approximation now reduces to theJaynes-Cummings (JC) interaction Hamiltonian H = iη Ω (cid:0) ˆ aσ + − σ − ˆ a † (cid:1) . (12)Up to here we have been able to construct the Rabi interaction, equation (8), with aset of parameters that do not allow all the regimes because η ≪ et al. [13] as Ω is the order of ν .
3. Fast Rabi Hamiltonian . We turn out attention again to the Hamiltonian given in equation (8) and set φ l = 0 H = ν ˆ n + Ω (cid:16) σ + ˆ D ( iη ) + σ − ˆ D † ( iη ) (cid:17) , (13)we rewrite equation (13) in a notation where operators acting on the internal ionic levelsare represented explicitly in terms of their matrix elements, as H = ν ˆ n Ω ˆ D ( iη )Ω ˆ D † ( iη ) ν ˆ n ! (14)and consider now the unitary operator [18, 19] T = 1 √ ˆ D † ( iη/
2) ˆ D ( iη/ − ˆ D † ( iη/
2) ˆ D ( iη/ ! . (15)It is possible to check after some algebra that H QRM =
T HT † = ν ˆ n + Ω + νη ιην (cid:0) ˆ a − ˆ a † (cid:1) + δ ιην (cid:0) ˆ a − ˆ a † (cid:1) + δ ν ˆ n − Ω + νη ! , (16)that, after returning to matrix notation, reads H QRM = ν ˆ n + Ω σ z + ιην σ + + σ − ) (cid:0) ˆ a − ˆ a † (cid:1) + νη , (17)that is nothing but the quantum Rabi Hamiltonian plus a constant term that can bedisregarded. A solution for this model has been given recently by Braak [20]It should be stressed now that in the above Hamiltonian we have not made anyassumptions on the parameters Ω and η . fast scheme for the implementation of the quantum Rabi model with trapped ions Now we show how to produce a fast dispersive Hamiltonian. Pedernales et al. [13]showed that it is possible to build such a Hamiltonian by using two slightly of resonantlaser beams tuned almost to the blue and red sidebands. However, as the parametersthey used are in general much smaller than ν , the dispersive interaction constant, maybe very small. Here, we take advantage of the fact that the Hamiltonian given in (17)has not been approximated and therefore there are no restriction on the values of theirparameters. By transforming the Hamiltonian (17) with the unitary operators [23]ˆ U = e ǫ (ˆ a † ˆ σ + − ˆ a ˆ σ − ) , ˆ U = e ǫ (ˆ a ˆ σ + − ˆ a † ˆ σ − ) ; (18)with ǫ , ǫ ≪ H eff = ˆ U ˆ U ˆ H QRM ˆ U † ˆ U † , (19)and setting ǫ = ην ν + 2Ω) ǫ = ην − ν ) , (20)remaining up to first order in the expansion e ǫA Be − ǫA = B + ǫ [ A, B ]+ ǫ [ A, [ A, B ]]+ ... ≈ B + ǫ [ A, B ], i.e. doing a small rotation [23], we obtain the so-called dispersiveHamiltonian ˆ H eff = ν ˆ a † ˆ a + Ωˆ σ z − χ QRM ˆ σ z (ˆ a † ˆ a + 12 ) (21)where the effective interaction constant has the form χ QRM = 2 η ν Ω4Ω − ν . (22)
4. Conclusions
Note that most regimes may be achieved with this fast treatment: Jaynes-Cummingsand anti Jaynes-Cummings were produced with the first method ( η ≪
1) and mayalso be produced with the last one, where the decoupling regime may also be achieved ν ≫ ην/ ≫ ην/ < ν, , | − ν | , @2Ω + ν | may be considered, etc. It should again be stressed that, because decay actually happensin ion-laser interactions [14, 16] it is of great importance to have fast interactions [22] in fast scheme for the implementation of the quantum Rabi model with trapped ions [1] Itano W M, Bergquist J C, Hulet R G and Wineland D J 1988 Precise optical spectroscopy withion traps. Phys. Scr. T Phys. Rev. A R6797-R6800[3] Bollinger J J, Prestage JD, Itano W M and Wineland D J 1985 Laser-Cooled-Atomic FrequencyStandard
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