Overview of Quantum Memory Protection and Adiabaticity Induction by Fast-Signal Control
aa r X i v : . [ qu a n t - ph ] D ec Science Bulletin manuscript No. (will be inserted by the editor)
Overview of Quantum Memory Protection andAdiabaticity Induction by Fast-Signal Control
Jun Jing · Lian-Ao Wu
Received: date / Accepted: date
Abstract
A quantum memory or information processing device is subject to dis-turbance from its surrounding environment or inevitable leakage due to its contactwith other systems. To tackle these problems, several control protocols have beenproposed for quantum memory or storage. Among them, the fast-signal controlor dynamical decoupling based on external pulse sequences provides a prevailingstrategy aimed at suppressing decoherence and preventing the target systems fromthe leakage or diffusion process. In this paper, we review the applications of thisprotocol in protecting quantum memory under the non-Markovian dissipative noiseand maintaining systems on finite speed adiabatic passages without leakage there-from. We analyze leakage and control perturbative and nonperturbative dynamicalequations including second-order master equation, quantum-state-diffusion equa-tion, and one-component master equation derived from Feshbach PQ-partitioningtechnique. It turns out that the quality of fast-modulated signal control is insensi-tive to configurations of the applied pulse sequences. Specifically, decoherence andleakage will be greatly suppressed as long as the control sequence is able to effec-tively shift the system beyond the bath cutoff frequency, almost independent ofthe details of the control sequences which could be ideal pulses, regular rectangularpulses, random pulses and even noisy pulses.
Keywords
Decoherence · Fast-signal control · Quantum noise · Quantum-state-diffusion equation
PACS · · · Jun JingInstitute of Atomic and Molecular Physics and Provincial Key Laboratory of Applied Atomicand Molecular Spectroscopy, Jilin University, Chuangchun 130012, Jilin, ChinaE-mail: [email protected] WuDepartment of Theoretical Physics and History of Science, The Basque Country University(EHU/UPV), PO Box 644, 48080 Bilbao, SpainIkerbasque, Basque Foundation for Science, 48011 BilbaoE-mail: [email protected]. Author to whom any correspondence should be addressed. Jun Jing, Lian-Ao Wu
Control of quantum processes [1] such as quantum storage in open quantum sys-tem [2] collects a number of separate concepts, and has a variety of manifestationsin different areas of physics [3, 4]. Quantum storage is concerned with a quantumstate stored in open systems in presence of leakage [5] and diffusion processes in-duced by either the system-environment interaction or inner-coupling between thetarget storage subspace and the rest of the system space. It is intimately connectedwith the phenomena of decoherence and disentanglement in quantum mechanics.As a consequence quantum memory or storage may be regarded as an intersectionof quantum information theory [3] and quantum control theory [1]. The latter in-vestigates the ability of using an external field to store, transfer, and manipulateinformation in correlated systems. Since it is important to protect quantum in-formation from degradation, quantum storage protocols are naturally and closelyrelated to the decoherence suppression [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] andthe enhancement of adiabaticity of quantum states [19, 20, 21]. Despite its longhistory, quantum decoherence and adiabaticity still challenge our comprehension,and continue to provoke our curiosity, as they are involved with various processesin quantum mechanics. Obviously, we have to be very selective in the topics thatwe will discuss here, and deal with many important aspects of these topics onlybriefly.In quantum information processing, a paramount importance to control ofopen quantum systems is the quest for combating decoherence. Consider a systemembedded in its bath consisted of multiple modes with an arbitrary spectrumdistribution (spectral density matrix), for example, an atomic or molecular systemin electromagnetic fields, the most probable microscopic process is the resonantexchange of quanta between the system and a particular field mode. Probability ofthis process is much higher than the Raman scatterings including Stokes and anti-Stokes effects, when the electromagnetic fields are not strong. Thus the leakagerate of the system can be measured by the overlap between the spectra of thepulse-modulated system and that of the bath. In other words, the control quality isdetermined by the effective gap between the system energy level and the bath cutofffrequency. The system leakage is subject to the accumulation effect during thehistory that the system exchanges quanta with the environment. For a memorylessor structure-less Markov environment, the open system contacts different modesat different instant, which yields a one-way flow of the quantum information andleaves no chance to memorize the information of a quantum state. On the contrary,a finite environmental memory time allows the information flow to go back to thesystem to some extend, which will be considerably helpful to the control effort.Unified approaches that treat the free dynamical evolution and dynamical de-coupling [6, 22, 23, 24, 25, 26, 27] on an equal footing are still under development. Itis mostly due to the absence of an exact convolutionless differential equation [2]for the system density matrix, which accommodates arbitrary coupling strengthbetween system and environment. In previous literatures, it seems that in the moreconvenient way one treats the time evolution of the system, the more unphysical(or unpractical) perturbative method one has to adopt in dealing with the controlprocess. For example, if the control Hamiltonian for a two-level system is describedby H c = J σ x and the original total Hamiltonian for system and environment is H tot , then in order to flip the system in a short time δ , the evolution operator itle Suppressed Due to Excessive Length 3 combining the control mechanism U ( δ ) = − iσ x ≈ exp( − iH tot δ − iJ δσ x ) requires J δ = π/ δ → H eff = H tot + H c withthe negligible perturbation parameter δ . Besides the difficulty encountered in ex-periments, the ideal pulse forces one to ignore the evolution of a strongly-drivensystem in the finite duration time of the pulse.The dynamical decoupling and the dynamics of the open quantum systemconfront with the same difficulty: how to describe the effect from the externalenvironment as well as control on the system dynamics in an exact way? Mi-croscopically, the system-environment interaction and the environmental statis-tical property should be carefully taken into the master equation or stochasticSchr¨odinger equation. On the other hand, configurations of the control sequencesshould be also reflected in the time-dependent system Hamiltonian. One of thefirst trials along this way was a second-order master equation, with respect tothe square of the system-environment coupling strength, targeting on the leakagecontrol of open systems in terms of fast signal control [5]. Recently, we proposeda nonperturbative dynamical decoupling [16, 29, 30, 31] protocol by employing thequantum-state-diffusion (QSD) equation [32, 33, 34, 35, 36], where we can treat thefree evolution and control of the system in a united way. Combined with the Fesh-bach PQ-partitioning technique [5], a one-component differential equation [37] hasbeen derived to address the dynamics of one target instantaneous eigenstate of thesystem, by which the fast signals control on leakage of quantum memory can beused in manipulating adiabatic condition. Under fast signals control, adiabaticitycan be established even when the original Hamiltonian lives in a nonadiabaticregime.In what follows, we would present a comprehensive revisit to our fast signalcontrol protocol based on the perturbative and nonperturbative dynamical equa-tions. This short review distinguishes itself from any control based on the artificialdynamical decoupling method using the ideal pulse sequence consisted of multipledelta functions in time course. It is clear that this approach is able to eliminatethe decoherence and leakage more efficiently with less or optimized control pulses.
Dynamical decoupling is also termed as bang-bang control in its early days [22].As an open-loop method, it is a close cousin of the spin-echo effect [38]. Thedecoherence-countering strategies rely on the ability to apply strong and fastpulses. Suppose one needs to eliminate the decoherence effect induced by an op-erator X in the system-environment interaction Hamiltonian H I , one can inserta system operator A satisfying { A, X } = 0 into the time evolution operator U ≡ exp( − iH I t ). Under the short-time approximation, e − i ( A + X ) t ≈ e − iAt e − iXt ac-cording to the Trotter formula [28]. It is straightforward to realize a gate R = e − iAt generated by A , which will remove the undesired unitary evolution U X = e − iXt via RU X R = U † X . Therefore, ideal dynamical decoupling or bang-bang controlis realized by using the parity-kick cycle. While proposals to control decoherenceby realistic (nonideal) pulses have to invoke techniques resulting in master equa-tion up to the second order in the system-bath coupling or stochastic Schr¨odinger Jun Jing, Lian-Ao Wu equation, which allows to present the shapes of nonideal pulse sequence. Thatconstitutes the main content of this section.2.1 Second-order Master EquationThe most general total Hamiltonian in the framework of open quantum system isrepresented by the summation of system, bath and their interaction H tot = H S + H b + H I = H + H I , (1)where the system-bath interaction term can be always decomposed into H I = P j A j B j . A j ’s ( B j ’s) are operators in the space of system (environment). Based onthe Nakajima-Zwanzigs projection [10, 39], P [ ρ ( t )] = Tr b [ ρ ( t )] ⊗ ρ b ( t ), the masterequation for the entire system density matrix in the interaction representation(setting ~ = 1 in the whole review) is ∂ t ρ S ( t ) = − (X mn Z t dt ′ C mn ( − t ′ )[ A m ( t ) , A n ( t − t ′ ) ρ S ( t )] ) , (2)to the second-order with respect to H I , where C mn ( − t ) ≡ Tr b [ ρ b B m B n ( − t )] con-stitutes the matrix of the bath correlation functions for multi-term system-bathinteractions. Here X ( − t ) ≡ e − iH t Xe iH t , X = A j or B j . The associated super-operator P can be redefined as P [ · ] = P ρ S ( t ) P ⊗ ρ b , where P denotes a projectionoperator onto a desired subspace of the entire system space, in order to study thedynamics and control of the P subspace, then the well-known master equation (2)becomes ∂ t P ρ S ( t ) P = − (X mn Z t dt ′ C mn ( − t ′ ) P [ A m ( t ) , A n ( t − t ′ ) P ρ S ( t ) P ] P ) . (3)Suppose the system starts from a pure state | φ i , which is one of the orthonormalbasis elements constituting the subspace of P ≡ | φ ih φ | . Then the probability thatthe system occupies the P -subspace b ( t ) satisfies˙ b ( t ) = − b ( t )Re "X mn Z t dt ′ C mn ( − t ′ ) A mn ( t, t − t ′ ) , (4)where A mn ( t, s ) ≡ h A m ( t ) A n ( s ) i φ − h A m ( t ) i φ h A n ( s ) i φ .Note that b (0) = 1. Thus, as a functional of H S , b ( t ) is equivalent to thefidelity of the system during the leakage process. Inside the integral of Eq. (4),the only term that could be under control is A mn ( t, s ) or A m ( t ). Within theframework of the second-order master equation, one can seek a function H c ( t ) inthe system space, to minimize | ˙ b ( t ) | under realistic constraints on pulse energy andwidth. Physically, Stark shifted by the alternating field might be a straightforwardoption, e.g. H S ⇒ H S + c ( t ) H S . Consider the extreme case that c ( t ) = c >
0, when the enhanced system frequency is larger than the cutoff frequency ofthe bath, A mn ( t, t − s ) oscillates faster than the alternating rate of the bathcorrelation function C mn ( − s ). The integral of the product of these two functions itle Suppressed Due to Excessive Length 5 would oscillates around zero so that | ˙ b ( t ) | vanishes. The goal of the leakage controlthen can be achieved.Equation (4) or more explicitly, the fidelity obtained by the second-order mas-ter equation F ( t ) = exp ( − "X mn Z t dt ′ C mn ( − t ′ ) A mn ( t, t − t ′ ) (5)is a significant improvement in quantum control methods, in comparison withthe “standard” bang-bang control, towards an open-loop control method basedon the quantum microscopic model. It is a consistent approach, by which theenvironmental statistical property, i.e. the correlation function and an accessiblecontrol pulse sequence, have been taken account into consideration. However, evenwith the Born approximation (the weak-coupling condition), it is still farawayfrom satisfaction because it is merely accurate to the second order of system-bathinteraction.2.2 Quantum-state-diffusion EquationBeyond Born and Markov approximations, quantum-state-diffusion equation iscapable of dealing with the strong coupling strength and the arbitrary correlationfunction of the environment. It is also a useful tool in deriving the exact masterequation. In addition, the QSD equation obtained with the approximation onthe integral over environmental noises may includes the high-order contributionsbeyond the second-order non-Markovian master equation. It has been found thatthe QSD equation naturally serves as a useful tool in studying control theories.Consider a total Hamiltonian describing a quantum system coupled to a bathof bosonic modes: H tot = H S + X k ( g ∗ k La † k + g k L † a k ) + X k ω k a † k a k , (6)where L and a k ( a † k ) are the coupling operator and annihilation (creation) operatorfor the k -th mode of the bath, respectively. The stochastic wave-function of thesystem is governed by QSD equation: ∂ t ψ t ( z ∗ ) = − iH eff ψ t ( z ∗ ) = [ − iH S + Lz ∗ t − L † ¯ O ( t, z ∗ )] ψ t ( z ∗ ) , (7)where z ∗ t ≡ − i P k g ∗ k z ∗ k e iω k t and z ∗ k ’s are individual Gaussian-distributed complexrandom numbers. The ensemble average of z ∗ t is M [ z t z ∗ s ] = P k | g k | e − iω k ( t − s ) ,which is equivalent to α ( t, s ), the environmental correlation function, at low tem-perature limit. Note that the ans¨atz ¯ O ( t, z ∗ ) ψ t ≡ R t dsM [ z t z ∗ s ] O ( t, s, z ∗ ) ψ t is apolynomial function of operators acting on the system Hilbert space, which is de-termined by ∂ t O ( t, s, z ∗ ) = [ − iH eff , O ( t, s, z ∗ )] − L † δ ¯ O ( t,z ∗ ) δz ∗ s . For a given model,a time-local exact QSD equation can be obtained once the exact O-operator isattained. More importantly for control, the formal ans¨atz of QSD equation is ir-respective of the Stark shift in the system Hamiltonian H S ( t ), which means that Jun Jing, Lian-Ao Wu the free or control dynamics can be treated on an equal footing. By using Novikovtheorem, the QSD equation becomes the following exact form ∂ t ρ S ( t ) = − i [ H S , ρ S ( t )] + [ L, M [ | ψ t ( z ∗ ) ih ψ t ( z ∗ ) | ¯ O † ( t, z ∗ )]] + h.c. (8)It will be cast into a convolutionless form in case of O ( t, s, z ∗ ) = O ( t, s ): ∂ t ρ S ( t ) = − i [ H S , ρ S ( t )] + [ L, ρ S ¯ O † ] + [ ¯ Oρ S , L † ] . (9)To determine the survival probability of the initial state | φ i , the fidelity is F ( t ) = h φ | ρ S ( t ) | φ i = M [ h φ | ψ t ( z ∗ ) ih ψ t ( z ∗ ) | φ i ] . (10)Different from the general fidelity (5), the fidelity based on QSD equation hasto be derived case by case. Yet a compensable advantage of QSD equation isthat the “bath” here (usually referred as a thermal bath with infinite number ofmodes) can be composed of arbitrary number of modes, which makes it a genuine“environment”. For example, in the dissipative process of a two-level system H S = E ( t )2 ω z , when the initial state is chosen as | φ i = µ | i + ν | i , | µ | + | ν | = 1, thefidelity is found to be F ( t ) = 1 −| µ | − ( | µ | − | µ | ) e − R t ds Re[ F ( s )] +2( | µ | −| µ | )Re[ e − R t dsF ( s ) ] , (11)where F ( t ) ≡ R t dsM [ z t z ∗ s ] f ( t, s ) satisfies F (0) = 0 and ∂ t f ( t, s ) = [ iE ( t ) + F ( t )] f ( t, s ). In the controlled dynamics, E ( t ) = ω + c ( t ), where ω is the barefrequency of the system and c ( t ) is the control function.2.3 Numerical ResultsBased on the second-order master equation and QSD equation, this subsectionpresents the fidelity dynamics of an open two-level (qubit) system under the fastsignal control. A logic extension to the ideal pulse sequence is the rectangular pulse,periodical or non-periodical, where the period, duration time and the strength foreach pulse are finite and experimentally feasible. Moreover, we also extend theregular pulse sequences to those with random pulse and even noisy pulse [30,31]. In what follows, numerical simulations are performed and the results fromrandom, chaotic [40] or noisy pulse [41, 42] are obtained by ensemble average.In order to distinguish effects of the different fast signal sequences, we choosethe same environmental spectral density function: α ( t, s ) = Γ γ e − γ | t − s | , which istermed as Ornstein-Uhlenbeck noise. The Ornstein-Uhlenbeck process is a usefulapproach to modeling noisy relaxation with a finite environmental memory timescale 1 /γ . When γ → ∞ , the environment memory time approaches to zero and α ( t, s ) reduces to Γ δ ( t − s ), corresponding to the Markov limit.Figure 1 demonstrates differences arising from the above two dynamical equa-tions, which decreases with the intense of fast signals. It is shown that under amoderate non-Markovian environment, γ = 0 . ω , the second-order master equa-tion cannot precisely describe the control dynamics of the open system. Therefore,the higher-order contribution cannot be omitted in such an environment. Indepen-dent of the chosen control parameters, there is always an abnormal interval in thetime domain, where the fidelity yields a revival pattern. With both approaches, itle Suppressed Due to Excessive Length 7 ω t F i de li t y ∆ / τ =0.25, ME ∆ / τ =0.25,QSD ∆ / τ =0.50, ME ∆ / τ =0.50, QSD ∆ / τ =0.75, ME ∆ / τ =0.75, QSD Fig. 1 (Color online) Fidelity dynamics of a qubit system under regular periodic pulse eval-uated with the second-order master equation (ME) and QSD equation. Here we employ therectangular pulse with period τ , duration time ∆ and strength Ψ/∆ , i.e. c ( t ) = Ψ/∆ for regions nτ − ∆ < t ≤ nτ , n ≥ c ( t ) = 0. The parameters are Ψ = 0 . ω , τ = 0 . ωt , Γ = ω , and γ = 0 . ω . The initial state of qubit satisfies | µ | = 0 . ω t F i de li t y γ =3, ∆ / τ =0.25 γ =3, ∆ / τ =0.50 γ =1, ∆ / τ =0.25 γ =1, ∆ / τ =0.50 γ =0.3, ∆ / τ =0.25 γ =0.3, ∆ / τ =0.50 Fig. 2 (Color online) Fidelity dynamics of a qubit system under random pulse sequenceevaluated with QSD equation. Here the random pulse sequence modifies the regular pulseby X ′ = X + D X Rand( − , X = τ, ∆, Ψ , respectively, D X ’s are their individualdeviation scales and Rand( − ,
1) denotes a random number uniformly distributed between − Ψ = 0 . ω , τ = 0 . ωt , D τ = D ∆ = 0 . τ , D Ψ = 0 . Ψ , and Γ = ω .The result has been averaged over ensemble as well as different initial states. Jun Jing, Lian-Ao Wu ω t F i de li t y ∆ / τ =0.25, Random ∆ / τ =0.75, Random ∆ / τ =0.25, Chaos ∆ / τ =0.75, ChaosJ=5 ω , PoissonJ=15 ω , Poisson Fig. 3 (Color online) Fidelity dynamics of a qubit system under different shapes of pulsesequence, including random pulses, chaotic pulses and Poisson white shot noise, evaluated withQSD equation. The random pulse and chaotic pulse are obtained by modifying the regular pulsewith strengthes randomly distributed within [0 , Ψ ] and multiplied with chaotic dimensionlessintensity L n , which constitutes a logistic map L n +1 = µ ( L n − L n ) with µ = 3 .
9, respectively.The Poisson noise c ( J, W, t ) satisfies M [ c ( J, W, t )] = JW , where J is the noise strength and W measures the average frequency of noise shots. The parameters are Ψ = 0 . ω , τ = 0 . ωt , and Γ = ω . The result has been averaged over ensemble as well as different initial states. the control quality under rectangular periodic sequence is steadily enhanced byincreasing the ratio of duration time and period of pulse. According to the exactresult given by QSD equation, when ∆/τ ≤ .
50, the fast pulse can maintain thestate up to F ( ωt = 10) > .
95 (see the red dashed and dot-dashed lines).In the fast signal control protocol with practical pulse sequence, regular pulsescan be replaced by random pules, meaning that all the three parameters of thepulses are allowed to stochastically fluctuate around their average values. Physi-cally, it represents the influence from the out-of-control factors or noise resourcesin laboratory. In Fig. 2, the fluctuation amplitude for both period and durationtime is assumed to be 20% and that for the strength is relaxed up to 90%. It isfound that under such a remarkable fluctuation, the random control still works aswell as regular control, especially under a strong non-Markovian environment. In anear-Markovian environment, γ = 3, the time course that the fidelity is preservedabove 0 .
90 is less than ωt = 3 even with an intensive fast signal, ∆/τ = 0 .
50 (seethe red solid line). Yet when γ = 0 .
3, a weak signal control, ∆/τ = 0 .
25, can evenmaintain the fidelity above 0 .
90 until ωt = 10 (see the blue dot-dashed line).Figures 1 and 2 suggest that the key element of nonideal pulse influencing thefidelity of the open system is the pulse intensity. This result can also be justifiedby Fig. 3, where we compare the random control (solid lines), the chaotic con-trol (dashed lines) and even the noisy control (dot-dashed lines). Specifically, thestrengthes per period along the pulse sequence are no longer identical but withrandom, chaotic and noisy distributions, respectively. It is shown that as long asthe fast signal is sufficiently intensive, ∆/τ ≥ .
75, the fidelity remains 0 .
95 for itle Suppressed Due to Excessive Length 9 all the three control sequences at least during the time course ωt ≤
10. There isno obvious difference among these dynamics. This conclusion not only relaxes therequirement in practical experiments, but also indicates that one should recon-sider the underlying reasons for some working and popular dynamical decouplingschemes.
Although the adiabatic principle had been proposed at the very beginning of thequantum theory, adiabatic passage [43, 44, 45, 46] has been reinforced in recent de-velopments in quantum information processing and quantum control. However,in the open quantum system framework, adiabaticity will be often modified andeven ruined by the environment noise. In this section, we show that the fast signalcontrol can be used to enhance adiabaticity even induce adiabaticity from a nona-diabatic regime. Contrary to intuition, the transition occurring between differenteigenstates can be suppressed not only by an ordered pulse sequence, but also bythe chaotic and noisy signals under conditions. To put the protocol into perspec-tive, we present the adiabatic dynamics of one target instantaneous eigenstate bya one-component integro-differential equation based on the following Feshbach PQpartitioning technique.In general, the wave-function and the effective Hamiltonian in the Schr¨odingeror stochastic Schr¨odinger equation can be always partitioned into, | ψ ( t ) i = (cid:20) PQ (cid:21) , H eff = (cid:18) h RW D (cid:19) , (12)according to the interested subspace indicated by P and the irrelevant part Q ,where the system is prepared at P -subspace, i.e. P (0) = 1 and Q (0) = 0. InEq. (12), h and D correspond to the self-Hamiltonians living in the P subspaceand the Q subspace, respectively; and R and W are their mutual correlation terms.For the closed system, R = W † . Consequently, we have i∂ t P = hP + RQ, i∂ t Q = W P + DQ. (13)The formal expression for P ( t ) can be rewritten as ∂ t P ( t ) = − ih ( t ) P ( t ) − Z t dsR ( t ) G ( t, s ) W ( s ) P ( s ) , (14)where g ( t, s ) incorporates the influence from the remain subspace of the systemand external control field and G ( t, s ) = T ← { exp[ − i R ts D ( s ′ ) ds ′ ] } is a time-orderedevolution operator. The merit of Eq. (14) is that it addresses the dynamics of onetarget component rather than multiple variables. Meanwhile h ( t ) can be exploitedfor state control.Before using Eq. (14), we now rewrite the Schr¨odinger equation i∂ t | ψ ( t ) i = H ( t ) | ψ ( t ) i into the adiabatic representation. The instantaneous eigenequation is H ( t ) | E n ( t ) i = E n ( t ) | E n ( t ) i , where E n ( t )’s and | E n i ’s are instantaneous eigenval-ues and non-degenerate eigenvectors, respectively. A state at time t can then beexpressed as | ψ ( t ) i = P n ψ n ( t ) e iθ n ( t ) | E n ( t ) i , where θ n ( t ) ≡ − R t E n ( s ) ds is the dynamical phase. Substituting them into the Schr¨odinger equation, we obtain thefollowing differential equation, ∂ t ψ m = −h E m | ˙ E m i ψ m − X n = m h E m | ˙ E n i e i ( θ n − θ m ) ψ n . (15)Without loss of generality, the target component can be chosen as ψ , the ampli-tude of the target eigenstate | E ( t ) i of H ( t ). Equation (15) can be regarded as theSchr¨odinger equation for the vector | ψ ( t ) i = ( ψ , ψ , ψ , · · · ) ′ with the effective“rotating representation” Hamiltonian with H mn = − i h E m | ˙ E n i e i ( θ n − θ m ) .Using Eq. (14), ψ ( t ) satisfies the following one-dimensional integro-differentialequation, ∂ t ψ ( t ) = −h E | ˙ E i ψ ( t ) − Z t dsg ( t, s ) ψ ( s ) , g ( t, s ) = R ( t ) G ( t, s ) W ( s ) . (16)In this case, R ≡ [ R , R , · · · ] with R m = − i h E | ˙ E m i e i ( θ m − θ ) , and W = R † .The first term on the right-hand side of Eq. (16) is the same as that in Eq. (15),which corresponds to the Berry’s phase [47, 48] that may be switched off in arotating frame. | ψ ( t ) | , the probability of finding the eigenstate | E ( t ) i at time t , is determined by the accumulation history of product of the propagator g ( t, s )and ψ ( s ). | ψ | T=1/ ω T=10/ ω J= ω J=10 ω Fig. 4 (Color online) | ψ | vs dimensionless time t/T for different passage times T , which is thekey parameter in a widely used model in quantum adiabatic algorithms: H ( t ) = ω [ t/T σ x +(1 − t/T ) σ z ]. The adiabatic limit | ψ | ≈ T when the systemfollows an adiabatic path or with the assistance by Poisson white shot noise. With the exact dynamical equation (16), a crucial and general adiabatic con-dition can be cast into the following compact form, Z t ds g ( t, s ) ψ ( s ) = 0 . (17) itle Suppressed Due to Excessive Length 11 The condition is satisfied when g ( t, s ) = 0 or g ( t, s ) is factored into a product ofone rapid oscillating function around zero and one much slowly varying function.Mathematically, it is understood that the integral of the product of the fast-varying g ( t, s ) and the slow-varying ψ ( s ) gives rise to a vanishing result. For a two-levelsystem with frequency difference E ( t ) = E − E , when it is initially prepared asthe eigenstate | E i , the propagator g ( t, s ) is given by, g ( t, s ) = −h E ( t ) | ˙ E ( t ) ih E ( s ) | ˙ E ( s ) i e R ts ( iE −h E | ˙ E i ) ds ′ . (18)If E ( t ) can be manipulated by fast signal, then the exponential function in g ( t, s )will play a crucial role to make the absolute value of the integral in Eqs. (16) or(17) as small as possible.In Fig. 4, we consider an time-dependent Hamiltonian that is of those typicalmodels describing adiabatic passage: H ( t ) = a ( t/T ) H + b ( t/T ) H , where a (0) = 1, b (0) = 0 and a ( T ) = 1, b ( T ) = 0. T is a key element to observe the adiabatic pas-sage time. It is known for these models, when T is sufficient large, the system statecan spontaneously adiabatically evolve from one of the instantaneous eigenvectorof a ( t ) H to the corresponding eigenvector of b ( t/T ) H . Otherwise, the transitionbetween different eigenstates of H ( t ) occurs. The two blue lines in Fig. 4 illustratethe difference between the slow adiabatic and the diabatic passages. The formerpassage is about 10 times as long as the latter one. According to Eq. (16), if | ψ | is maintained as unity, then the adiabatic passage can also be realized. Here thesystem frequency ω is modulated as ω + c ( t ), where c ( t ) = c ( W, J, t ), the Poissonwhite shot noise. The red lines in Fig. 4 show that we can greatly accelerate thepassages with the help of fast signal when it is sufficiently intensive.
In this short review, we systematically report our progress on fast signals controlmethods in quantum storage and adiabatic process. The nonperturbative controlmethod has been developed for decoherence- and leakage-suppression. The resultsbased on the second-order master equation and QSD equation demonstrate thata system dynamics can be stabilized in terms of arbitrary configurations of thefast signals. In particular situations, the environmental dissipative noise can evenbe neutralized by the white noise. The fast signal control can also be used inrealizing a shortcut to adiabaticity and to the suppression of the leakage fromthe target adiabatic passages. Our strategy remarkably relaxes the experimentalrequirements in precisely-engineering control sequences.
Acknowledgements
We acknowledge grant support from the Basque Government (grantIT472-10), the Spanish MICINN (No. FIS2012-36673-C03-03) and the NSFC No. 11175110.
References , 080405 (2009)6. L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. , 2417 (1999)7. G. S. Uhrig, New J. Phys. 10, 083024 (2008)8. G. S. Uhrig, Phys. Rev. Lett. , 120502 (2009)9. J. R.West, D. A. Lidar, B. H. Fong, andM. F. Gyure, Phys. Rev. Lett. , 230503 (2010)10. A. G. Kofman and G. Kurizki, Phys. Rev. Lett. , 130406 (2004)11. M. Mukhtar, W. T. Soh, T. B. Saw, and J. Gong, Phys. Rev. A , 052338 (2010)12. J. Clausen, G. Bensky, and G. Kurizki, Phys. Rev. Lett. , 040401 (2010)13. S. Xue, J. Zhang, R. Wu, C. Li, and T.-J. Tarn, J. Phys. B , 154016 (2011)14. G. Bensky, R. Ams¨uss, J. Majer, D. Petrosyan, J. Schmiedmayer, and G. Kurizki, QuantumInf Process , 214304 (2011)16. J. Jing, L.-A. Wu, J. Q. You, and T. Yu, Phys. Rev. A , 045501 (2012)18. Z.-M. Wang, L.-A. Wu, J. Jing, B. Shao, and T. Yu, Phys. Rev. A , 012114 (2005)22. L. Viola and S. Lloyd, Phys. Rev. A , 2733 (1998)23. H. Y. Carr and E. M. Purcell, Phys. Rev. , 630 (1954)24. G. S. Uhrig, Phys. Rev. Lett. , 100504 (2007)25. W. Yang and R.-B. Liu, Phys. Rev. Lett. , 180403 (2008)26. M. J. Biercuk, A. C. Doherty, and H. Uys, J. Phys. B , 154002 (2011)27. A. Z. Chaudhry and J. Gong, Phys. Rev. A , 207902 (2002)29. J. Jing, L.-A. Wu, J. Q. You, and T. Yu, Phys. Rev. A , 2746 (2013)31. J. Jing, C. A. Bishop, and L.-A. Wu, Sci. Rep. , 6229 (2014)32. L. Di´osi and W. T. Strunz, Phys. Lett. A , 569 (1997)33. L. Di´osi, N. Gisin, and W. T. Strunz, Phys. Rev. A , 1699 (1998)34. W. T. Strunz, L. Di´osi, and N. Gisin, Phys. Rev. Lett. , 1801 (1999)35. T. Yu, L. Di´osi, N. Gisin, and W. T. Strunz, Phys. Rev. A , 91 (1999)36. J. Jing and T. Yu, Phys. Rev. Lett. , 240403 (2010)37. J. Jing, L. -A. Wu, T. Yu, J. Q. You, Z. -M. Wang, and L. Garcia, Phys. Rev. A ,032110 (2014)38. E. L. Hahn, Phys. Rev. , 580 (1950)39. R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A , 062101 (2002)40. C. Grebogi, E. Ott, and J. A. Yorke, Physica D: Nonlinear Phenomena, 7(1):181C200,(1983)41. C. Kim, E. K. Lee, P. Hanggi, and P. Talkner, Phys. Rev. E , 011109 (2007)42. J. Spiechowicz, J. Luczka, and P. Hanggi, J. Stat. Mech.: Theor. Exp. P02044 (2013)43. J. Oreg, F. T. Hioe, and J. H. Eberly, Phys. Rev. A , 690 (1984)44. K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. , 1003 (1998)45. P. Kr´al, I. Thanopulos, and M. Shapiro, Rev. Mod. Phys. , 53 (2007)46. S. Oh, Y-P. Shim, J. Fei, M. Friesen, and X. Hu, Phys. Rev. A , 022332 (2013)47. M. Berry, Proc. R. Soc. London A , 45 (1984)48. F. Wilczek and A. Zee, Phys. Rev. Lett.52