Shortcut to Equilibration of an Open Quantum System
aa r X i v : . [ qu a n t - ph ] F e b Shortcut to Equilibration of an Open Quantum System
Roie Dann,
1, 2, ∗ Ander Tobalina,
3, 2, † and Ronnie Kosloff
1, 2, ‡ The Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apdo 644, Bilbao, Spain (Dated: February 26, 2019)We present a procedure to accelerate the relaxation of an open quantum system towards itsequilibrium state. The control protocol, termed
Shortcut to Equilibration , is obtained by reverse-engineering the non-adiabatic master equation. This is a non-unitary control task aimed at rapidlychanging the entropy of the system. Such a protocol serves as a shortcut to an abrupt change in theHamiltonian, i.e., a quench . As an example, we study the thermalization of a particle in a harmonicwell. We observe that for short protocols there is a three orders of magnitude improvement inaccuracy.
PACS numbers: 03.65.w,03.65.Yz,32.80.Qk,03.65.Fd
Introduction
Equilibration is a natural process, de-scribing the return of a perturbed system back to a ther-mal state. The relaxation to equilibrium is present inboth the classical [1–3] and quantum [14] regimes. Gain-ing control over the relaxation rate of quantum systemsis crucial for enhancing the performance of quantum heatdevices [5, 7–9]. In addition, fast relaxation is beneficialfor quantum state preparation [10, 11] and open systemcontrol [12–17]. To address these issues, we present ascheme to accelerate the equilibration of an open quan-tum system, serving as a shortcut to the natural relax-ation time τ R . The protocol is termed Shortcut To Equi-libration (STE).This control problem is embedded in the theory ofopen quantum systems [14]. The framework of the the-ory assumes a composite system, partitioned into a sys-tem and an external bath. The Hamiltonian describ-ing the evolution of the composite system reads ˆ H ( t ) =ˆ H S ( t ) + ˆ H B + ˆ H I , where ˆ H S ( t ) is the system Hamilto-nian, ˆ H B is the bath Hamiltonian and ˆ H I is the system-bath interaction term. When the system depends explic-itly on time, the driving protocol influences the system-bath coupling operators and consequently, the relaxationtime.Quantum control in open systems has been addressedin the past utilizing measurement and feedback [18–23]. Typically, the effect of non-adiabatic driving onthe dissipative dynamics was ignored [24–27]. Here, wepresent a comprehensive theory that incorporates thenon-adiabatic effects. The formalism is based on the re-cent derivation of the Non Adiabatic Master Equation(NAME) [11]. This master equation is of the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form, guaran-teeing a complete positive trace-preserving dynamicalmap [29–31]. A further prerequisite is the inertial the-orem [4]. This theorem allows extending the validity ofthe NAME for processes with small ‘acceleration’ of theexternal driving.We consider a driven quantum system, the Hamilto- nian of which varies from ˆ H S (0) to a final Hamiltonianˆ H S ( t f ), while coupled to a thermal bath (see Fig. 1).Our aim is to exploit the non-adiabatic effects of the driv-ing to accelerate the systems return to equilibrium. Byreverse-engineering the NAME, we find a protocol thattransforms the thermal state of ˆ H S (0) at temperature T to the corresponding thermal state of ˆ H S ( t f ). Thisprocedure serves as a shortcut for the natural relaxationtime τ R .Controlling the equilibration rate differs from the con-trol tasks treated by shortcuts to adiabaticity [33–42].The latter protocols generate an entropy-preserving uni-tary transformation, which is effectively the identity mapbetween initial and final diagonal states in the energyrepresentation. Conversely, the STE procedure is anon-unitary transformation, which is designed to rapidlychange the entropy of the system. Quench
STE
FIG. 1: Scheme of the
Shortcut To Equilibration (STE) protocol(curved red line) and the quench protocol (blue step line), trans-forming an initial thermal state at temperature T and frequency ω i to a final thermal state with an equivalent temperature andfrequency ω f . System dynamics
We consider a quantum particle incontact with a thermal bath while confined by a time-dependent harmonic trap. The system Hamiltonian readsˆ H S ( t ) = ˆ P m + 12 mω ( t ) ˆ Q , (1)where ˆ Q and ˆ P are the position and momentum oper-ators, respectively, m is the particle mass and ω ( t ) isthe time-dependent oscillator frequency. We assume aBosonic bath with 1D Ohmic spectral density and an in-teraction Hamiltonian of the form ˆ H I = − ˆ D ⊗ ˆ B , whereˆ D = d ˆ a + d ∗ ˆ a † ( a and a † are the annihilation and cre-ation operators of the oscillator, respectively), d is theinteraction strength and ˆ B is the bath interaction oper-ator. Throughout the paper, we choose units related tothe minimum frequency ω min , time 2 π/ω min and energy ~ ω min with ~ = 1.At initial time, the open quantum system is in equilib-rium with the bath, and the state is of a Gibbs canonicalform ˆ ρ S (0) = Z − e − ˆ H S (0) /k B T , where Z is the parti-tion function, k B is the Boltzmann constant and T isthe temperature of the bath. We search for a protocolthat varies the Hamiltonian toward ˆ H S ( t f ) with a targetthermal state ˆ ρ T hS ( t f ) = Z − e − ˆ H S ( t f ) /k B T . This proce-dure serves as a shortcut to an isothermal process. Theaccuracy of this transformation can be quantified usingthe fidelity F , which is a measure of the distance be-tween the final state ˆ ρ S ( t f ) of the protocol and ˆ ρ T hS ( t f )[9, 10, 45]. A classical analogous problem has been ad-dressed by Martinez et al. [3].The most straightforward protocol is a quench proto-col. ’Quench’ means abruptly changing the Hamiltonianfrom H S (0) to H S ( t f ), and then letting the system equi-librate with the bath, Cf. Supplemental Material (SM)III. When ˆ H S (0) and ˆ H S ( t f ) do not commute, whichis the case for a non-rigid harmonic oscillator, such asudden change generates coherence in the energy basis,leading to deviations from equilibrium. The quenchedsystem relaxes at an exponential rate toward equilibrium,which leads to an asymptotic exponential convergence ofthe fidelity toward unity 1 − F ( t ) ∝ e − kt , for t/k > k = k ↓ − k ↑ , where k ↓ and k ↑ are decay rates, Cf.SM IIIA. We use the quench protocol as a benchmark toassess the STE protocol’s performance.To describe the reduced dynamics under the STE, wefollow the derivation presented in Refs. [4, 11]. First, weobtain a solution for the unitary propagator ˆ U S ( t,
0) fora protocol determined by a constant adiabatic parameter µ = ˙ ω/ω . The closed-form solution of ˆ U S ( t,
0) allowsconstructing a master equation that includes the bath’sinfluence on the reduced dynamics. Then, by utilizingthe inertial theorem , we extend the description to proto-cols where µ varies slowly ( dµ/dt ≪ dµ/dt ≪ t f > f · max s (cid:18) ω q ω ′′ ( s )2 ω − µ (cid:19) , where s = t/t f and f < f = 0 .
05, the lowerbound is t f > .
38 (2 π/ω min ), where ω min = 5 a.u.The range of validity of the NAME sets a number ofconditions: (i) weak coupling between system and bath,which also allows for a reduced description of the system’sdynamics in terms of ˆ ρ S [14]; (ii) Markovianity [46]; (iii)large Bohr frequencies relative to the relaxation rate τ R ;(iv) slow driving relative to the decay of the bath corre-lations. In the following, we consider a regime where theNAME and inertial theorem are valid.The dynamics of the externally driven open quantumsystem, in the interaction representation, is described by ddt ˜ ρ S ( t ) = k ↓ ( t ) (cid:18) ˆ b ˜ ρ S ( t ) ˆ b † − { ˆ b † ˆ b, ˜ ρ S ( t ) } (cid:19) + k ↑ ( t ) (cid:18) ˆ b † ˜ ρ S ( t ) ˆ b − { ˆ b ˆ b † , ˜ ρ S ( t ) } (cid:19) . (2)Here, the interaction picture density operator reads˜ ρ S ( t ) = ˆ U ( t,
0) ˆ ρ S ( t ) ˆ U † ( t, A to describe operators in an interaction picture relative tothe system Hamiltonian. For an ohmic Bosonic bath, thedecay rates are k ↓ ( t ) = k ↑ ( t ) e α ( t ) /k B T = α ( t ) | ~d | πε ~ c (1 + N ( α ( t ))) , (3)where N is the occupation number of the Bose-Einsteindistribution and α is a modified frequency, determinedby the non-adiabatic driving protocol [11]. In terms ofthe oscillator frequency, the modified frequency is givenby α ( t ) = s − (cid:18) ˙ ω ( t ) ω ( t ) (cid:19) ω ( t ) . (4)The Lindblad jump operators become ˆ b ≡ ˆ b (0) = q mω (0)2 ~ ( κ + iµ ) κ (cid:16) ˆ Q (0) + µ + iκ mω (0) ˆ P (0) (cid:17) where κ = p − µ .In the interaction representation the Lindblad oper-ators are time-independent. This property provides anexplicit solution in terms of the second-order moments B = { ˆ b † ˆ b, ˆ b , ˆ b † } [4, 11], Cf. SM I, which, together withthe identity operator, form a closed Lie algebra. The so-lution is given by a generalized canonical state, which hasa Gaussian form in terms of B . Such states are canoni-cal invariant under the dynamics described by Eq. (2),implying that the system can be described by the gen-eralized canonical state throughout the entire evolution[1–3, 50]. The system state is given by˜ ρ S ( t ) = Z − e γ ( t )˜ b e β ( t )˜ b † ˜ b e γ ∗ ( t )˜ b † , (5)which is completely defined by the time-dependent co-efficients γ and β and the driving protocol. The parti-tion function reads Z ( β, γ ) = e − β ( e − β − √ − | γ | / ( e − β − .In the adiabatic limit, the adiabatic parameter µ ap-proaches zero, the state follows the adiabatic solution,and ˆ b † ˆ b → ˆ a † ˆ a .Substituting ˜ ρ S ( t ) into the master equation, Eq. (2),multiplying by ˜ ρ − S from the right and comparing theterms proportionate to the operators ˜ b † ˜ b , ˜ b and ˜ b † leadsto ˙ β = k ↓ (cid:0) e β − (cid:1) + k ↑ (cid:0) e − β − e β | γ | (cid:1) , ˙ γ = ( k ↓ + k ↑ ) γ − k ↓ γe − β . (6)These equations describe the evolution of the system forany initial squeezed thermal state. Here, we assume thatthe system is in a thermal state at the initial time, whichinfers γ (0) = 0. This simplifies the expression of the stateto ˜ ρ S ( β ( t ) , µ ( t )) = Z − e β ˆ b † ˆ b ( µ ) , (7)and consequently the system dynamics are described bya single non-linear differential equation˙ β = k ↓ ( t ) (cid:0) e β − (cid:1) + k ↑ ( t ) (cid:0) e − β − (cid:1) , (8)with initial conditions β (0) = − ~ ω (0) k B T and µ (0) = 0.Equation (8) constitutes the basis for the suggested con-trol scheme. Control
The control target is to transform a thermalstate, defined by frequency ω i , to a thermal state of fre-quency ω f , while interacting with a bath at temperature T . The control utilizes the fact that at all times, the stateis fully defined by µ ( t ) and β ( t ). This property implies β (0) = − ~ ω i k B T , β ( t f ) = − ~ ω f k B T and µ (0) = µ ( t f ) = 0.The initial and final β are connected through Eq. (8),where the protocol defines the rates k ↑ ( t ) and k ↓ ( t ).These rates are determined by the parameter α ( t ) in Eq.(3), which in turn is completely defined by the controlparameter ω ( t ) in Eq. (4). Furthermore, µ ( t ) is deter-mined by ω ( t ), and therefore ω ( t ) fully determines thestate of the system at all times.The strategy to solve the control equation is based ona reverse-engineering approach, and the protocol is de-noted by Shortcut To Equilibration (STE). The methodproceeds as follows: we define a new variable y = e β , andpropose an ansatz for y that satisfies the boundary con-ditions. Then we solve for α ( t ), and from α ( t ) determine ω ( t ).The initial and final thermal states determine theboundary conditions of µ ( t ), which implies that the stateis stationary at initial and final times. This leads to ad-ditional boundary conditions ˙ β (0) = ˙ β ( t f ) = 0.A third-degree polynomial is sufficient to obey all ofthe constraints. Introducing s = t/t f , the solution reads y ( s ) = y (0) + 3∆ s − s , (9) where ∆ = y ( t f ) − y (0). In principle, more complicatedsolutions for Eq. (8) exist; however, here we restrict theanalysis to a polynomial solution [51]. The implicit equa-tion for α ( t ) becomes t f dds y ( s ) = k ↓ ( α ( s )) y ( s ) − y ( s ) ( k ↓ ( α ( s )) + k ↑ ( α ( s ))) + k ↑ ( α ( s )) . (10)Solving the equation by numerical means generates α ( s ).This solution is substituted into Eq. (4) and the con-trol ω ( t ) is obtained by an iterative numerical procedure.The protocol satisfies the inertial condition on µ , infer-ring that the derivation is self-consistent.The solution of the STE incorporates the adiabatic re-sult in the limit of slow driving. For large protocol timeduration ( t f → ∞ ), the system’s instantaneous state is athermal state at temperature T with frequency ω ( t ), seeSM IV.We compare the STE protocol to a quench protocol in-volving a sudden change from ω (0) = ω i to ω ( t f ) = ω f [52]. Two cases are studied, a compression of the po-tential, which corresponds to the transition ω (0) = 5 → ω ( t f ) = 10, and a reversed expansion, associated withthe transition ω (0) = 10 → ω ( t f ) = 5. Both pro-tocols for each process are presented in Fig. 2 panels(a) and (b). We add, as a reference, an adiabatic pro-cess obtained in the limit t f → ∞ . The initial stage ofthe quench protocol is effectively isolated, as the changein frequency is rapid relative to the relaxation rate to-ward equilibrium. As a result, the state stays constantwhile the Hamiltonian abruptly transforms to ˆ H S ( t f ).Coherence is generated with respect to ˆ H ( t f ), because h ˆ H (0) , ˆ H ( t f ) i = 0. After the initial stage energy is ex-changed with the bath and the coherence dissipates.In figure 3, we compare the fidelity with respect to thetarget thermal state of the expansion and compressionprotocols, for increasing stage times t f . The STE proto-col transfers the system to the target thermal state withfidelities close to unity F ≈
1, while the quench targethas lower fidelity due to the slow relaxation. Therefore,the STE protocol equilibrates the system faster and withhigher accuracy than the quench protocol. For a givenfidelity, the STE achieves the target state up to five timesfaster than the quench protocol.Figure 2 panels (c) and (d) presents a comparison ofthe quantum state’s energy for the STE, quench and adi-abatic protocols. During the quench protocol, there is asudden change in the energy, which is followed by a slowexponential decay toward the thermal energy. The adi-abatic and STE protocols are characterized by an over-shoot beyond the final thermal energy. In the final stageof the STE protocol, the energy rapidly converges to thedesired thermal energy, whereas the quenched system re-mains far from equilibrium (see insets in Fig. 2 panels(c) and (d)).
FIG. 2: Control protocols as a function of the scaled time t/t f :(a,b) the oscillator frequency ω and (c,d) energy for the STE (redline), quench (dashed blue line) and adiabatic (dot-dashed greenline) protocols. (a,c) Expansion, (b,d) compression protocols. Thedynamics of the STE and quenched systems are shown for t f =8 a.u, and the adiabatic dynamics are obtained in the limit t f → ∞ .(c,d) Inset: details of the final approach to the target state. Modelparameters (atomic units): ω (0) /ω f = 5 /
10 for the compression,and reverse for the expansion and bath temperature T = 2.FIG. 3: The fidelity of the final state relative to the target ther-mal state for the short-cut to equilibration (red) and quench (blue)protocols. (a) Expansion protocol, (b) compression protocol. Theinset shows the accuracy A = − log (1 − F ), highlighting the 3-digit accuracy of the STE protocol. Model parameters are the sameas in Fig. 2 Energy and entropy cost
A control task can be evalu-ated by the work and entropy cost required to implementthe control. Restrictions on the cost can be connectedto quantum friction [8, 53], which implies that quickertransformations are accompanied by a higher energy cost[38, 54–57]. Moreover, in any externally controlled pro-cess there is an additional cost in energy and entropy togenerate faster driving [58, 59]. The work cost for theSTE protocol with a duration time t is defined by theintegral form W ( t ) = Z t tr ˆ ρ S ( t ′ ) ∂ ˆ H ( t ′ ) ∂t ′ ! dt ′ . (11)For the quench protocol, the sudden transition occurson a much faster timescale than the exchange rate ofenergy with the bath. This implies that the change ininternal energy is equal to the work cost. For the ex-pansion stroke (Fig. 4) the work generated during the FIG. 4: Work required to perform the driving protocol as a functionof the normalized time. Model parameters are identical to Fig. 3.Upper part: compression, lower part: expansion.
STE protocol exceeds the quenched system result, yetremains below the adiabatic limit. When the system iscompressed, the STE and quench protocols require addi-tional work compared to the adiabatic process. We candefine the efficiency of the process relative to the adia-batic work W adi ( η comp = W adi /W for compression and η exp = W/W adi for the expansion). For the studied case,the efficiency of the STE protocol exceeds that of the quench , η quenchcomp ≈ . η quenchexp ≈ .
7, while η ST Ecomp > . η ST Eexp > .
75, and improves for increasing protocolduration. This result is in accordance with thermody-namic principles, as any rapid driving will induce irre-versible dynamics, which in turn leads to sub-optimalperformance. For long times, the work of the STE proce-dure approaches the adiabatic result according to a t − scaling law. At this limit, the global entropy productionapproaches zero. For shorter times, the system entropychange, for the STE procedure, is almost independent ofprotocol duration as a result of the accurate control. Theprice for shorter protocols is an increase in irreversibility,manifested by larger global entropy production (see SMV). Discussion
Quantum control is achieved by manipu-lating the system Hamiltonian via a change of an exter-nal control parameter. In turn, the change in the systemHamiltonian influences the system-bath interaction andthe equation of motion. Hence, manipulating the Hamil-tonian indirectly controls the dissipation rate.The control procedure employs a closed Lie algebraof system operators. The algebra is used to describe theHamiltonian, system-bath interaction term and the state.The state is described by a generalized canonical form,Eq. (5); this state is the maximum entropy state con-strained by the expectation values of the operators inthe algebra. For moderate acceleration of the driving,the inertial theorem can be employed to obtain the non-adiabatic master equation, Eq. (2) [11], for which thegeneralized canonical form of the system state is pre-served.Substituting the generalized canonical form in theequation of motion, Eq. (2), leads to a set of couplednon-linear differential equations of the state parameters, γ and β , which define the generalized canonical state, Eq.(5). These equations completely describe the system dy-namics and implicitly depend on the control parameter.They are the basis for the control procedure.To solve the control problem, we insert a functionalform for the state parameters which obeys the cor-rect boundary conditions. Specifically, the parametersare associated with the initial and final thermal states, β (0) = − ~ ω /k B T and β ( t f ) = − ~ ω f /k B T , with van-ishing derivatives at the boundaries. The consideredfunctional form is a third-order polynomial, the coeffi-cients of which are determined by the boundary condi-tions. This leads to an implicit equation in terms of thecontrol parameter ω ( t ).At first glance, it would seem that the quench protocolis optimal, since the approach to equilibrium is exponen-tially fast. However, a superior solution is obtained bythe STE protocol. The advantage of the latter is that itincorporates both the dissipative and unitary parts of thedynamics, changing the rates and engineering the statesimultaneously.The related work cost, required for compression of theharmonic potential or obtained for expansion, is in accor-dance with thermodynamic principles. These infer thatsudden or fast driving of the system increases the poweroutput at the expense of wasted resources and entropygeneration.The STE protocol can be generalized beyond theisothermal example studied here, for three different kindsof scenarios: (i) the temperature of the initial state differsfrom the bath temperature; (ii) the case of varying bathtemperature (with the help of Eq. (3)); (iii) squeezed ini-tial and final states. These general control tasks shouldbe approached by reverse-engineering of both β and γ inEq. (6). Furthermore, once a non-adiabatic master equa-tion is obtained [4, 11], the method can be generalized tosystems characterized by a closed Lie algebra.To conclude, the STE result demonstrates the feasibil-ity of controlling the entropy of an open quantum system.Such control can be combined with fast unitary trans-formations to obtain a broad class of states within thesystem algebra. This will pave the way to faster high-precision quantum control, altering the state’s entropy. Acknowledgement
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Shortcut to Equilibration of an Open Quantum System: supplementary material
Roie Dann, , Ander Tobalina, , Ronnie Kosloff, , The Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apdo 644, Bilbao, Spain
Canonical invarience and representation of thesystem in terms of the generalized Gibbs statecoefficients.
We demonstrate how Lie algebra properties and canon-ical invariance can be utilized to obtain an alternativerepresentation of the system dynamics. For a closed Liealgebra, the system state can be represented in a prod-uct form [1, 2]. For the closed set B = { ˆ b † ˆ b, ˆ b , ˆ b † } thestate is given by a generalized Gibbs state, presented inequation (5) of the Main Part (MP). Next, we calcu-late d ˜ ρ S ( t ) /dt (˜ ρ S ( t )) − explicitly, using the generalizedGibbs state ∂ ˜ ρ S ( t ) ∂t (˜ ρ S ( t )) − = ˙ γ ˆ b + ˙ˆ βe γ ˆ b ˆ b † ˆ be − γ ˆ b + ˙ γ ∗ e γ ˆ b e β ˆ b † ˆ b ˆ b † e − β ˆ b † ˆ b e − γ ˆ b . (12)Introducing the BakerCampbellHausdorff relation, leadsto ∂ ˜ ρ S ( t ) ∂t (˜ ρ S ( t )) − = ˆ b (cid:16) βγ + ˙ γ + 4 γ ˙ γ ∗ e β (cid:17) + ˆ b † ˆ b (cid:16) ˙ β + 4 γ ˙ γ ∗ e β (cid:17) + ˆ b † ˙ γ ∗ e β + ˆ I γ ˙ γ ∗ e β . (13)By substituting Eq. (5) MP into the master equation, Eq.(2) MP, we obtain an expansion of d ˜ ρ S ( t ) /dt (˜ ρ S ( t )) − in terms of the operators of B . Such a property is termedcanonical invariance, it implies that an initial state thatbelongs to the class of canonical states, will remain inthis class throughout the evolution [3]. The expressionreads ∂ ˜ ρ S ( t ) ∂t (˜ ρ S ( t )) − = ˆ b A + ˆ b † ˆ bB + ˆ b † C + ˆ ID , (14)
TABLE I coefficient value coefficient value a γe β c − γ ∗ e β a γ (cid:0) − | γ | e β (cid:1) c − γ ∗ e β b e β d e β b − | γ | e β d − | γ | e β b e − β − | γ | e β d − | γ | e β + 1 where A = k ↓ (cid:18) a − a (cid:19) − k ↑ a (15) B = k ↓ (cid:18) b − b − (cid:19) + k ↑ (cid:18) b − b − (cid:19) (16) C = −
12 ( k ↓ + k ↑ ) c + k ↑ c D = k ↓ (cid:18) d − d (cid:19) + k ↑ (cid:18) −
12 ( d + 1) (cid:19) . The values of of the coefficients are summarized in TableI. To satisfy both Eq. (13) and (14) the coefficients mul-tiplying each operator must be equal. Comparing terms,leads to four coupled differential equations2 ˙ βγ + ˙ γ + 4 γ ˙ γ ∗ e β = γ (cid:0)(cid:0) e β − (cid:1) k ↓ + 4 e β ( k ↓ + k ↑ ) | γ | − k ↑ (cid:1) (17)˙ β + 4 γ ˙ γ ∗ e β = e − β (cid:0)(cid:0) e β − (cid:1) (cid:0) e β k ↓ − k ↑ (cid:1) +4 e β | γ | (cid:0) e β ( k ↓ + k ↑ ) − k ↑ (cid:1)(cid:1) (18)˙ γ ∗ e β = k ↓ e β γ ∗ + k ↑ γ ∗ (cid:0) e β − e β (cid:1) (19)2 γ ˙ γ ∗ e β = k ↓ (cid:0) e β (cid:0) e β | γ | + 1 (cid:1)(cid:1) + k ↑ (cid:0) e β | γ | − (cid:1) (20)After some algebraic manipulations we obtain the sim-plified form˙ β = k ↓ (cid:0) e β − (cid:1) + k ↑ (cid:0) e − β − e β | γ | (cid:1) (21)˙ γ = ( k ↓ + k ↑ ) γ − k ↓ γe − β , These coupled differential equations completely deter-mine the system’s dynamics.
Lower bound of the protocol duration
The validity of the inertial theorem is quantified bythe inertial parameter Υ [4]. When Υ ≪ ∼ µ ′ ( θ )(2 κ ) ,( θ = θ ( t ) = R t ω ( t ′ ) dt ′ and κ = p − µ ), which explic-itly becomes Υ = 1(2 κ ) (cid:18) ¨ ωω − µ (cid:19) . (22)Transforming variables to the dimensionless parameter s = t/t f and constraining the inertial parameter byΥ < f ≪
1, introduces a lower bound for the protocolduration: t f > f · max s ω s ω ′′ ( s )2 ω ( s ) 18 − ( µ ( s )) ! . (23) Quench protocol
When the parametric quantum harmonic oscillator isin a Gaussian state [2], it is convenient to analyze thesystem in terms of three time-dependent operators, theHamiltonian Eq. (1) MP, Lagrangian ˆ L ( t ) = ˆ P m − mω ( t ) ˆ Q , and the position-momentum correlation op-erator ˆ C ( t ) = ω ( t )2 (cid:16) ˆ Q ˆ P + ˆ P ˆ Q (cid:17) . The quench protocol in-cludes an initial abrupt shift in frequency from ω (0) = ω i to ω ( t f ) = ω f . The sudden transformation is approxi-mately isolated, as the bath’s influence on the systemoccurs on a much longer timescale. Moreover, for a sud-den quench the system state remains unchanged. Hence,time-independent operators, such as ˆ Q (0) and ˆ P (0) donot vary. This property allows expressing the operatorsafter the sudden quench ˆ H ′ , ˆ L ′ and ˆ C ′ in terms of theoperators at initial time (for the sudden quench)ˆ H ′ = h ˆ H (0) (cid:16) ω ( t ) ω (0) (cid:17) + ˆ L (0) (cid:16) − ω ( t ) ω (0) (cid:17)i ˆ L ′ = h ˆ H (0) (cid:16) − ω ( t ) ω (0) (cid:17) + ˆ L (0) (cid:16) ω ( t ) ω (0) (cid:17)i ˆ C ′ = ω ( t ) ω (0) ˆ C (0) . (24)The sudden change in frequency generates coherence,which is manifested by non-vanishing values of h ˆ L ( t ) i and h ˆ C ( t ) i [5].Once the system is quenched, the frequency remainsconstant and the system relaxes towards equilibrium.Such dynamics were derived in Ref. [5], where the state’sevolution is expressed as a matrix vector multiplication ~v ( t ) = U S ( t, ~v ′ , with ~v ( t ) = { ˆ H ( t ) , ˆ L ( t ) , ˆ C ( t ) , ˆ I } , ~v ′ = { ˆ H ′ , ˆ L ′ , ˆ C ′ , ˆ I } , ˆ I is the identity operator and U S ( t,
0) = R h ˆ H i eq (1 − R )0 Rc − Rs Rs Rc
00 0 0 1 . (25)Here, R = e − Γ t with Γ = k ↓ − k ↑ , c = cos ( ω f t ), s =sin ( ω f t ) and h ˆ H i eq = ~ ω f coth (cid:16) ~ ω f k B T (cid:17) . Utilizing Eq.(24) and (25), the evolution of the quenched system iscompletely defined. Asymptotic behaviour of the fidelity of the quenchprocedure
The fidelity is a measure of the similarity between twoquantum states. It was introduced by Uhlmann as themaximal quantum-mechanical transition probability be-tween the two states’ purifications in an enlarged Hilbertspace [6–8]. For two displaced squeezed thermal statesthe fidelity obtains the form [9, 10] F (ˆ ρ , ˆ ρ ) = 2 √ ∆ + δ − √ δ exp h − ~u T ( A + A ) − ~u i , (26)where ∆ = det ( A + A ), δ =(det ( A ) −
1) (det ( A ) − A i = 2 (cid:20) σ iQQ σ iP Q / ~ σ iP Q / ~ σ iP P / ~ (cid:21) , (27)where σ QQ , σ P P and σ P Q are the variances and covari-ance of the position and momentum operators. The vec-tor ~u is given by, ~u = {h ˆ Q i − h ˆ Q i , h ˆ P i − h ˆ P i } T . Inthe equilibration process the system’s state remains cen-tered at the origin (for all the considered protocols) andis compared to a thermal state. For such a case the cal-culation of the fidelity is greatly simplified, namely ~u = 0and the fidelity obtains the form, F (ˆ ρ S ( t f ) , ˆ ρ th ) = 2 √ ∆ + δ − √ δ . (28)For the quench procedure, once the sudden transition tothe final frequency ω f the system relaxes to a thermalstate, following an exponential decay rate. For a time-independent Hamiltonian Eq. (2) MP, [11], reduces tothe standard master equation for the harmonic oscillator[12–14]: ddt ˆ ρ S ( t ) = − iω (cid:2) ˆ a † ˆ a, ˆ ρ S ( t ) (cid:3) + k ↓ (cid:18) ˆ aρ S ( t ) ˆ a † − { ˆ a † ˆ a, ˆ ρ S ( t ) } (cid:19) + k ↑ (cid:18) ˆ a † ˆ ρ S ( t ) ˆ a − { ˆ a ˆ a † , ˆ ρ S ( t ) } (cid:19) , (29)where the annihilation operator is given by ˆ a = (cid:18)q mω (0)2 ~ ˆ Q + i √ ~ mω (0) ˆ P (cid:19) . The solution of the masterequation can be represented in the Heisenberg picture,obtaining the formˆ a ( t ) = ˆ a (0) e − iω − k/ t (30)ˆ a † ˆ a ( t ) = ˆ a † ˆ a (0) e − kt + N (cid:0) − e − kt (cid:1) . (31)Next, we write the elements of A i in terms of the creationannihilation operators, and neglect terms that decay witha rate 2 k . This leads to a simplified form for the fidelity F (ˆ ρ S ( t f ) , ˆ ρ th ) ≈ p c + c e − kt − p c + c e − kt , (32)where k ≡ k ↓ − k ↑ c = (cid:0) σ thP P σ thQQ / ~ + 1 (cid:1) (33) c = 4 (cid:0) σ thP P σ thQQ / ~ + 1 (cid:1) (cid:0) c P P σ thQQ + c QQ σ thP P / ~ (cid:1) c = (cid:0) σ thP P σ thQQ / ~ − (cid:1) c = 4 (cid:0) σ thP P σ thQQ / ~ − (cid:1) (cid:0) σ thP P c QQ / ~ + σ thQQ c P P (cid:1) . Here, c P P and c QQ are time-independent parameters, de-fined by the evolution of the system’s variances σ QQ = c QQ e − kt + σ thQQ (34) σ P P = c P P e − kt + σ thP P . In the asymptotic limit, equation (32) can be expandedin orders of e − kt , and the fidelity’s asymptotic behaviourreads: 1 − F ( t ) ∝ e − kt . Adiabatic limit
In the limit of infinite protocol time duration t f → ∞ the STE result converges to the adiabatic solution. Thiscan be seen by studying the change in y ( t ). Differenti-ating Eq. (9) MP leads to˙ y ( t ) = 6∆ tt f (cid:18) − tt f (cid:19) . (35)Hence, in the limit t f → ∞ , ˙ y vanishes. Moreover, theeffective frequency converges to α ( t ) → ω ( t ) (Eq. (4)MP). Substituting this result into Eq. (10) MP gives theadiabatic solution y = k adi ↑ ( t ) k adi ↓ ( t ) , (36)where the excitation and decay rate, in the adi-abatic limit, are k adi ↓ ( t ) = k adi ↑ ( t ) e ω ( t ) /k B T = ω ( t ) | ~d | πε ~ c (1 + N ( ω ( t ))). Writing Eq. (36) in terms of β ,the system state (Eq. (5)) obtains the form ρ S ( t ) = Z th ( t ) e − ~ ω ( t ) ( ˆ a † ˆ a + ) , (37) with Z th ( t ) = tr (cid:0) exp (cid:0) − ~ ω ( t ) (cid:0) ˆ a † ˆ a + (cid:1)(cid:1)(cid:1) . Thus, in theadiabatic limit the STE solution converges to the adia-batic state. FIG. 5: The efficiency relative to the optimal adiabatic result forthe short-cut to equilibration (red) and quench (blue) protocols, for(a) expansion protocol, (b) compression protocol.FIG. 6: The change in entropy during the (a) expansion and (b)compression processes for the STE (red) and quench (blue) pro-tocols as a function of protocol duration. The change in systementropy ∆ S sys , bath entropy ∆ S B and global entropy ∆ S U areindicated by dashed, dashed-dot, and continuous lines respectively. Entropy calculation
The shortcut to equilibration procedure induces a swiftchange in the system’s entropy ∆ S sys . An expansionprotocol is accompanied by an increase in the systementropy, while a compression is followed by a decreasein entropy, Figure 6. The STE transforms the systemto the target state with high precision, which is almostindependent on the protocol duration. As a result, thechange in system entropy ∆ S sys also remains constant.On the contrary, the global entropy generation dependson the trajectory between initial and final states. Forlarge protocol times the STE approaches the adiabaticlimit and the global entropy generation ∆ S U vanishes.The heat exchange with the bath increases for shorterprotocol duration, which in turn, increases the changein the bath’s entropy, ∆ S B = − Q sys /T B . During theexpansion (compression) process the system energy de-creases (increases), and heat is transferred from (to) thebath, this is accompanied by and decrease (increase) inthe bath entropy see Fig. 6.The change in entropy associated with the quench pro-tocol is composed of a fast isoentropic process, followedby a natural decay towards equilibrium. During the re-laxation stage the coherence decays, leading to a rise in0 TABLE II
Coefficient Value [atomic units]Oscillator mass 1Compression ω (0) 5Compression ω ( t f ) 10Expansion ω (0) 10Expansion ω ( t f ) 5Bath temperature 2Coupling prefactor | ~d | πε ~ c = 0 . the system entropy. For both expansion and compres-sion quench protocols energy flows from the system tothe bath, resulting in an increase in the bath entropyand a global entropy production. For both compressionand expansion the global irreversible entropy generationof the quench exceeds the STE result. Numerical details
The value of ω ( t ) is assessed by a numerical solutionof equations (10) MP and (4) MP, employing a built inMatlab solver. The solution is used to calculate the sys-tem’s evolution according to the inertial solution [4]. Thevalidity of the inertial approximation has been verified,with the inertial parameter obtaining maximum valuesof Υ ≈ . .