Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution

A universal scheme is introduced to speed up the dynamics of a driven open quantum system along a prescribed trajectory of interest. Shortcuts to adiabaticity designed in this fashion can be implemented in two alternative scenarios: one is characterized by the presence of balanced gain and loss, the second involves non-Markovian dynamics with time-dependent Lindblad operators. As an illustration we engineer superadiabatic cooling, heating, and isothermal strokes for a two-level system.

Introduction.-Shortcuts to adiabaticity (STA) allow one to control the evolution of a quantum system without the requirement of slow driving [1].The controlled speedup of quantum processes is broadly recognized as a necessity for the advance of quantum technologies, and STA have found a variety of applications, including phase-space preserving cooling [2], population transfer [3,4], and friction suppression in finitetime thermodynamics [5][6][7], to name some relevant examples.To date, STA have been demonstrated in the laboratory using ultracold gases [8][9][10][11][12][13], nitrogen-vacancy centers [14,15], trapped ions [16], superconducting qubits [17,18], and other systems [1].Despite this remarkable progress, the use of STA has been predominantly restricted to tailor the dynamics of isolated driven systems.However, any physical system is embedded in a surrounding environment with which it can interact and exchange energy, particles, etc.In such a setting, the dynamics of the system is no longer-described by a Hamiltonian and is associated with a master equation [19].A notable exception concerns the dynamics of an isolated system conditional to a given subspace of interest.The dynamics can then be described in terms of a non-Hermitian Hamiltonian, that generates loss and gain when the system leaves the subspace of interest and returns to it, respectively [20].Scenarios characterized by a balance of gain and loss arise naturally, e.g., in the presence of a non-Hermitian potential that breaks timereversal symmetry but preserves parity-time-reversal symmetry, i.e., in PT -symmetric quantum mechanics [21][22][23][24][25][26].
The use of STA to speed up open quantum processes is expected to make possible a wide range of applications such as design of novel cooling techniques, information erasure [27], or the engineering of superadiabatic quantum machines [28].The engineering of fast control protocols for Markovian processes was first presented in Ref. [29].More recently, the reverse engineering of a non-adiabatic Markovian master equation has been proposed for the fast thermalization of a harmonic oscillator [30].A related study has shown the possibility of speeding up the thermalization of a system oscillator locally coupled to a harmonic bath [31].
In this Letter we introduce a general scheme to engineer STA in arbitrary open quantum systems.We consider the evolution of a quantum system described by a mixed state along a prescribed trajectory of interest.We then find the equation of motion that generates the desired dynamics.The latter can be recast in terms of the nonlinear evolution of a system in the presence of balanced gain and loss.Alternatively, the dynamics can be associated with a non-Markovian master equation with time-dependent Lindblad operators whose explicit form is determined by the prescribed trajectory.
STA by counterdiabatic driving.-Consider a quantum evolution of interest described by the mixed state where r = rank( ).We pose the problem of enforcing the evolution of the system through this trajectory.
Under unitary dynamics, eigenvalues of the density matrix remain constant, λ n (t) = λ n (0)-denoted briefly as λ n .The equation of motion for the density matrix in this case reads and can be recast as a Liouville-von Neumann equation, ∂ t (t) = −i[H 1 (t), (t)] (with = 1), whenever the dynamics is generated by the Hamiltonian This Hamiltonian generates parallel transport along each of the eigenstates |n(t) and is often used in proofs of the adiabatic theorem [32,33].
In the context of control theory, the derivation of H 1 (t) can be systematically achieved by the so-called counterdiabatic (CD) driving technique, also known as transitionless quantum driving [3,4,34].Specifically, CD assumes that |n(t) are the eigenstates of a reference system H 0 (t) that can be controlled by the auxiliary field H 1 (t) so that the full dynamics is actually generated by H 0 (t) + H 1 (t).Yet, in the most general setting, the instantaneous eigenstates used in the specification arXiv:1907.07460v1[quant-ph] 17 Jul 2019 of the trajectory (1) need not be the eigenstates of the physical Hamiltonian of the system.To identify a reference Hamiltonian in this case, we choose (t) to evolve as a thermal state, where Z 0 (t) = Tr[e −βH0(t) ] denotes the partition function, and β is the inverse temperature (assuming k B = 1).With this definition, the spectral decomposition of the reference Hamiltonian reads where the eigenvalues E n = −β −1 log(Z 0 λ n ) are timeindependent, and so is the partition function.By construction [H 0 (t), (t)] = 0, and the state (t) is a solution of where CD driving of open quantum systems.
-In what follows we shall focus on the case where the eigenvalues of the density matrix are time-dependent.The von Neumann entropy of the state is then a function of time, and the dynamics is generally open and nonunitary.Indeed, for an arbitrary change of λ n s the dynamics is generally non-trace-preserving.
For a given time-dependence of λ n (t), the equation of motion for the trajectory (t) can be analogously derived as This equation admits several physical interpretations that we discuss below.
(i) Mixed evolution under balanced gain and loss.-Theadditional term in Eq. ( 7) can be associated with the anti-Hermitian operator The equation of motion for (t) is then generated by the full non-Hermitian Hamiltonian This evolution is not necessarily norm-preserving, with a loss of norm occurring at a rate A norm-preserving evolution through the trajectory (t) is governed by the modified equation of motion where Γ = Tr[Γ ] and the time-dependence of all terms has been dropped for brevity.Note that the resulting equation is nonlinear in the quantum state .This dynamics thus takes the form of a mixed-state evolution in the presence of balanced gain and loss [35] with a time-dependent generator [36].Balanced gain and loss arises naturally in the study of PT -symmetric quantum systems [21], that can be used to describe a variety of experiments [22][23][24][25][26].
(ii) Lindblad-like form.-Considering the prescribed trajectory (1) and its derivative (7), one can recast the incoherent part as an auxiliary CD dissipator in a Lindblad-like form for a trace-preserving trajectory.Assuming a trace-preserving evolution, n ∂ t λ n (t) = 0, we find the time-dependent Lindblad operators and rates as [37] that are determined by (the spectral resolution of) (t)-and thus state-dependent.The resulting master equation is non-Markovian.We remark that existence of a Lindbladlike master equation for an arbitrary dynamics has recently been proven in Ref. [38].
The equivalence of Eqs.(11) and (15) shows that the nonlinear evolution of a mixed state under balanced gain and loss can be represented by a nonlinear and generally non-Markovian master equation with time-dependent Lindblad operators, determined by choice of the trajectory (1).
We note that the time-evolution operator generated by the CD Hamiltonian takes the form [34] U CD (t, 0) = n e iφn(t) |n(t) n(0)|, (16) where the time-dependent phase φ n (t) is the sum of the dynamical and geometric contributions.In the co-moving frame associated to U CD (t, 0), the master equation for (t) = U † CD (t, 0) (t)U CD (t, 0) takes the simple form with L mn = |m(0) n(0)|.As a result, the time-dependent Lindblad operators {L mn } map to the time-independent ones { L mn }, while keeping the same rates γ mn (t).This feature is specific to the superadiabatic driving of open quantum systems and differs from the general case that leads to more complex time-dependent Lindblad operators [19].
Quantum speed limit for STA in open quantum processes.-Speedlimits provide a minimum time for a physical processes to occur in terms of the generator of the evolution and can be used to relate the operation time of a protocol to the amplitude of the required unitary and nonunitary CD terms.The geometric formulation of the quantum speed limit [39] states that τ D( (0), (τ ))/ √ g tt , where g tt is the metric for a given distance D, and The quantum Fisher information F is the metric (with a 1/4 prefactor) associated with the Bures distance , that is defined in terms of the fidelity √ 1 between 1 and 2 [40].From Eq. ( 9), we can identify −2iH as a non-Hermitian symmetric logarithmic derivative, satisfying 2∂ t = L + L † [41], based on which an upper bound on the quantum Fisher information is obtained as As a result, the quantum speed limit reads Alternatively, using the trace distance rather than the Bures distance, the relevant metric is 9) and ( 15) for ∂ t and the triangle inequality, one obtains ∂ t [H CD , ] + {Γ, } for the gain-loss equation and ∂ t [H CD , ] + D CD for the Lindblad-like equation.In all of these bounds, both the CD Hamiltonian and dissipator set the speed of evolution.
Example I: Strokes for a two-level system.-Consider a twolevel system described by a time-dependent Hamiltonian where σ z and σ x are the Pauli matrices.The instantaneous eigenstates read , where θ(t) = arctan(Ω(t)/∆(t)) and the corresponding eigenstates are with σ z |0 = |0 and σ z |1 = −|1 .We consider the system to be described by the time-dependent mixed state (t) = α=± λ α (t)|α(t) α(t)|.Thus, the target trajectory is already diagonal in the eigenbasis of the uncontrolled system Hamiltonian H 0 (t).The auxiliary control term required to guide the dynamics is known to be of the form [3,4,34] so that the full dynamics is generated by The dynamics is open when the eigenvalues λ ± are timedependent.
The first approach we have introduced relies on the presence of gain and loss, for which the dynamics is generally no longer trace-preserving, i.e., λ − + λ + is time-dependent and different from unity.Such evolution is generated by the non-Hermitian Hamiltonian H = H CD − iΓ, where Under balanced gain and loss, the trace-preserving property is restored by the nonlinear equation (11) with this choice of Γ.Alternatively, STA in an open two-level system can be associated with a Lindblad-like master equation with the Lindblad operators The rates are given by γ +− (t) = ∂tλ+ 2λ− and γ −+ (t) = ∂tλ− 2λ+ .Assume that the system is initially prepared in a thermal state at inverse temperature β(0), (0) = α=± λ α (0)|α(0) α(0)|, where λ α = e −β(0)Eα(0) /Z(0) with Z(0) = e −β(0)E−(0) + e −β(0)E+(0) .We focus on description of thermodynamic protocols for which the target trajectory (t) is an instantaneous thermal state with inverse temperature β(t), i.e., One can engineer different processes of interest which are of this type.For example, in a superadiabatic isothermal stroke, the state is always in a thermal form at a given reference inverse temperature β(t) = β(0), regardless of the rate at which H 0 (t) is driven.Nonadiabatic excitations are cancelled by the auxiliary term H 1 in Eq. ( 21), while the thermal form of λ ± (t) is guaranteed by the Lindblad operators and rates.For arbitrary ∆(t) and Ω(t), they read where α, α ∈ {±} and α = α .A typical modulation in time is shown in Fig. 1 for a twolevel system to evolve along STA for an isothermal stroke induced by driving of ∆(t) while keeping Ω constant.Specifically, ∆(t) is chosen as a fifth-order polynomial in time interpolating between the initial and final values.The rates have opposite signs, vanish identically at the avoided crossing, and flip signs during the subsequent evolution.
It is possible to look as well for cooling and heating protocols characterized by a time-dependent inverse temperature β(t) keeping H 0 constant, as required, e.g., in a quantum Otto cycle.In such a case, H 1 vanishes, and the cooling and heating strokes are implemented by time-independent Lindblad operators with time-dependent rates, where α, α ∈ {±} and α = α .The time-dependence of the rates is explicitly illustrated for both cooling and heating processes in Fig. 1, for constant values of ∆ and Ω, and β(t) interpolating between β(0) and β(t f ) again as a fifth-order polynomial.The non-Markovian character of the evolution is manifest given the time-dependence of the Lindblad operators and the opposite sign of the corresponding rates.Beyond these two prominent examples, more general strokes can be considered.The required Lindblad operators in the most general setting are provided in Ref. [37].We also note that in all cases the corresponding operator Γ associated with gain and loss can be conveniently expressed in terms of the rates as In the following, we consider another example in which the real physical dynamics of the system keeps its state always in the Gibbsian form with a time-dependent temperature.
Example II: STA for equilibration of a thermalizing atom.-Consider a two-level atom in a thermal bosonic bath at inverse temperature β B (0).The dynamics of the atom under some conditions can be described by [42][43][44] where j, k ∈ {0, 1}, and Here, n(ω 0 , β B (0)) = (e β B ω0 − 1) −1 is the mean boson number in a mode with frequency ω 0 , and γ is a time-independent constant indicating the strength of the coupling between the atom and the thermal bath.
If the atom is initially in a thermal state S (0) = e −β S (0)H S /Z S (0), its instantaneous state is obtained by solving the above master equation, which gives a Gibbsian thermal state S (t) = e −β S (t)H S /Z S (t), with Here Equation ( 15) suggests another dynamical equation realizing the same trajectory S (t).Since H 0 = H S is timeindependent, H 1 will be zero as well.The Lindblad operators are given in terms of the eigenstates of H S as L mn = 01 < l a t e x i t s h a 1 _ b a s e 6 4 = " + m E k k Z P p h k n t b N a U k P 3 j 3 i N 8 T / W H u 3 + Q r D + 8 h f B 8 X g U P B m N 3 4 8 H e z v t r 9 g g 9 8 k D s k M C s k v 2 y B t y Q C a E E 0 W + k K / k m 1 d 6 3 7 0 f 3 s / z U q / X 9 t w j H f N + / Q W j C O j p < / l a t e x i t >    15) for a thermalizing atom in the case of heating, when βS = 0.1, βB = 0.01, ω0 = 2, and γ = 0.005.The inset corresponds to the cooling case, with βS ↔ βB.With these parameters, the rates for the Lindblad master equation of the subsystem in Eqs. ( 29) and (30) constant; γ01 = 4.95 and γ10 = 5.05.Right: Inverse temperature of the system for the Lindblad master equation (dashed) and for the STA (solid).The dependence in time is shown for the case of heating (see inset for the cooling case) with the same set of parameters as in the left panel.
|m n| where m, n ∈ {0, 1}, and the rates are obtained from Eq. ( 14) by considering that λ 0 = e −β S (t)ω0/2 /Z S (t) and λ 1 = e β S (t)ω0/2 /Z S (t) can also be identified simply as 00 and 11 , respectively (see Ref. [37]).While the Lindblad operators here are equal to those in Eq. ( 29), the rates in the Markovian master equation ( 29) are positive constant.By contrast, the rates in Eq. ( 15) are time-dependent and negative for some time intervals, as illustrated in Fig. 2. Nonetheless, in both cases equilibration with the bath at temperature β B takes infinite time.
Summary and conclusions.-Wehave introduced a universal scheme to design shortcuts to adiabaticity in open quantum systems, interacting with an environment.We first specify a target trajectory for the evolution of the system, and then find the required auxiliary Hamiltonian terms and dissipators that generate it.The resulting dynamics can be associated with a driven system in the presence of balanced gain and loss, a scenario that occurs naturally, e.g., in PT -symmetric quantum mechanics.Alternatively, it can be implemented via a non-Markovian evolution in which the equation governing the dynamics takes a generalized Lindblad-like form.Our formalism thus enables to engineer superadiabatic open processes to speed up, i.e., heating, cooling, and isothermal strokes.It should find broad applications in quantum thermodynamics, and more generally in quantum technologies requiring the fast control of an open system embedded in an environment.
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FIG. 1 .
FIG.1.Left: Time-dependence of the rates in an isothermal process at inverse temperature β = 1, keeping Ω constant with initial ∆(0) = 1 and final ∆(t f ) = −1.Right: Time-dependence of the rates for the superadiabatic cooling (blue) and heating (red) of a twolevel system.Taking ∆ = Ω as the unit of frequency, the process corresponds to cooling a thermal state from β(0) = 1 to β(t f ) = 2 and heating a thermal state from β(0) = 2 to β(t f ) = 1.
6 r g e / A j + H n e G n S W M 3 d J C 8 G v v + p w 6 n 4 = < / l a t e x i t > 10 < l a t e x i t s h a 1 _ b a s e 6 4 = " C O g 6 9 k gB z p 1 F l Z N r v m 4 K t 3 P Y w D c = " > A A A C + 3 i c b V J d b 9 M w F H X D 1 y g f 2 + C R F 4 s O a U K l S o r Q e E B o C D T x g h h i 3 S a 1 U e U 4 t 4 0 1 O z H 2 D a J Y + R 2 8 g n h D v P J f 4 N / g Z B V q O o4 U 3 Z P 7 c X x y 4 0 R L Y T E M / 3 S C S 5 e v X L 2 2 c b 1 7 4 + a t 2 5 t b 2 3 e O b V E a D i N e y M K c J s y H 8 g X 8 p V 8 C 6 r g e / A j + H n e G n S W M 3 d J C 8 G v v + p x 6 n 4 = < / l a t e x i t > S < l a t e x i t s h a 1 _ b a s e 6 4 = " f r c + 1 6 F e 4 e W z n v 8 a 9 V p a m J 4 + 3 d I = " > A A A C 9 3 i c b V J b b 9 M w F H b D b Z T b

t
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FIG. 2 .
FIG.2.Left: Time dependence of the rates γ10 (red, dashed) and γ01 (blue, solid) in Eq. (15) for a thermalizing atom in the case of heating, when βS = 0.1, βB = 0.01, ω0 = 2, and γ = 0.005.The inset corresponds to the cooling case, with βS ↔ βB.With these parameters, the rates for the Lindblad master equation of the subsystem in Eqs.(29) and (30) constant; γ01 = 4.95 and γ10 = 5.05.Right: Inverse temperature of the system for the Lindblad master equation (dashed) and for the STA (solid).The dependence in time is shown for the case of heating (see inset for the cooling case) with the same set of parameters as in the left panel.