Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution
SShortcuts to Adiabaticity in Driven Open Quantum Systems:Balanced Gain and Loss and Non-Markovian Evolution
S. Alipour, A. Chenu,
2, 3
A. T. Rezakhani, and A. del Campo
2, 3, 5 QTF Center of Excellence, Department of Applied Physics,Aalto University, P. O. Box 11000, FI-00076 Aalto, Espoo, Finland Donostia International Physics Center, E-20018 San Sebasti´an, Spain IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain Department of Physics, Sharif University of Technology, Tehran 14588, Iran Department of Physics, University of Massachusetts, Boston, MA 02125, USA
A universal scheme is introduced to speed up the dynamics of a driven open quantum system along a pre-scribed trajectory of interest. Shortcuts to adiabaticity designed in this fashion can be implemented in twoalternative 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 superadiabaticcooling, heating, and isothermal strokes for a two-level system.
Introduction.—
Shortcuts to adiabaticity (STA) allow one tocontrol the evolution of a quantum system without the require-ment of slow driving [1]. The controlled speedup of quan-tum processes is broadly recognized as a necessity for the ad-vance of quantum technologies, and STA have found a varietyof applications, including phase-space preserving cooling [2],population transfer [3, 4], and friction suppression in finite-time thermodynamics [5–7], to name some relevant examples.To date, STA have been demonstrated in the laboratory us-ing ultracold gases [8–13], nitrogen-vacancy centers [14, 15],trapped ions [16], superconducting qubits [17, 18], and othersystems [1].Despite this remarkable progress, the use of STA has beenpredominantly restricted to tailor the dynamics of isolateddriven systems. However, any physical system is embeddedin a surrounding environment with which it can interact andexchange energy, particles, etc. In such a setting, the dynam-ics of the system is no longer-described by a Hamiltonian andis associated with a master equation [19]. A notable excep-tion concerns the dynamics of an isolated system conditionalto a given subspace of interest. The dynamics can then bedescribed in terms of a non-Hermitian Hamiltonian, that gen-erates loss and gain when the system leaves the subspace ofinterest and returns to it, respectively [20]. Scenarios char-acterized by a balance of gain and loss arise naturally, e.g.,in the presence of a non-Hermitian potential that breaks time-reversal symmetry but preserves parity-time-reversal symme-try, i.e., in PT -symmetric quantum mechanics [21–26].The use of STA to speed up open quantum processes is ex-pected to make possible a wide range of applications such asdesign of novel cooling techniques, information erasure [27],or the engineering of superadiabatic quantum machines [28].The engineering of fast control protocols for Markovian pro-cesses was first presented in Ref. [29]. More recently, the re-verse engineering of a non-adiabatic Markovian master equa-tion has been proposed for the fast thermalization of a har-monic oscillator [30]. A related study has shown the possi-bility of speeding up the thermalization of a system oscillatorlocally coupled to a harmonic bath [31]. In this Letter we introduce a general scheme to engineerSTA in arbitrary open quantum systems. We consider the evo-lution of a quantum system described by a mixed state alonga prescribed trajectory of interest. We then find the equationof motion that generates the desired dynamics. The latter canbe recast in terms of the nonlinear evolution of a system in thepresence of balanced gain and loss. Alternatively, the dynam-ics can be associated with a non-Markovian master equationwith time-dependent Lindblad operators whose explicit formis determined by the prescribed trajectory. STA by counterdiabatic driving.—
Consider a quantum evo-lution of interest described by the mixed state (cid:37) ( t ) = (cid:80) rn =1 λ n ( t ) | n ( t ) (cid:105)(cid:104) n ( t ) | , (1)where r = rank( (cid:37) ) . We pose the problem of enforcing theevolution of the system through this trajectory.Under unitary dynamics, eigenvalues of the density matrixremain constant, λ n ( t ) = λ n (0) —denoted briefly as λ n . Theequation of motion for the density matrix in this case reads ∂ t (cid:37) ( t ) = (cid:80) n λ n ( | ∂ t n ( t ) (cid:105)(cid:104) n ( t ) | + | n ( t ) (cid:105)(cid:104) ∂ t n ( t ) | ) , (2)and can be recast as a Liouville-von Neumann equation, ∂ t (cid:37) ( t ) = − i [ H ( t ) , (cid:37) ( t )] (with (cid:126) = 1 ), whenever the dy-namics is generated by the Hamiltonian H ( t ) = i (cid:80) n ( | ∂ t n ( t ) (cid:105)(cid:104) n ( t ) | − (cid:104) n ( t ) | ∂ t n ( t ) (cid:105)| n ( t ) (cid:105)(cid:104) n ( t ) | ) . (3)This Hamiltonian generates parallel transport along each ofthe eigenstates | n ( t ) (cid:105) and is often used in proofs of the adia-batic theorem [32, 33].In the context of control theory, the derivation of H ( t ) canbe systematically achieved by the so-called counterdiabatic(CD) driving technique, also known as transitionless quantumdriving [3, 4, 34]. Specifically, CD assumes that | n ( t ) (cid:105) are theeigenstates of a reference system H ( t ) that can be controlledby the auxiliary field H ( t ) so that the full dynamics is ac-tually generated by H ( t ) + H ( t ) . Yet, in the most generalsetting, the instantaneous eigenstates used in the specification a r X i v : . [ qu a n t - ph ] J u l of the trajectory (1) need not be the eigenstates of the physicalHamiltonian of the system. To identify a reference Hamilto-nian in this case, we choose (cid:37) ( t ) to evolve as a thermal state, (cid:37) ( t ) = (1 /Z ( t )) e − βH ( t ) , (4)where Z ( t ) = Tr[ e − βH ( t ) ] denotes the partition function,and β is the inverse temperature (assuming k B = 1 ). With thisdefinition, the spectral decomposition of the reference Hamil-tonian reads H ( t ) = (cid:80) n E n | n ( t ) (cid:105)(cid:104) n ( t ) | , (5)where the eigenvalues E n = − β − log( Z λ n ) are time-independent, and so is the partition function. By construction [ H ( t ) , (cid:37) ( t )] = 0 , and the state (cid:37) ( t ) is a solution of ∂ t (cid:37) ( t ) = − i [ H CD ( t ) , (cid:37) ( t )] , (6)where H CD ( t ) = H ( t ) + H ( t ) . CD driving of open quantum systems.—
In what follows weshall focus on the case where the eigenvalues of the densitymatrix are time-dependent. The von Neumann entropy of thestate is then a function of time, and the dynamics is generallyopen and nonunitary. Indeed, for an arbitrary change of λ n sthe dynamics is generally non-trace-preserving.For a given time-dependence of λ n ( t ) , the equation of mo-tion for the trajectory (cid:37) ( t ) can be analogously derived as ∂ t (cid:37) ( t ) = − i [ H CD ( t ) , (cid:37) ( t )] + (cid:80) n ∂ t λ n ( t ) | n ( t ) (cid:105)(cid:104) n ( t ) | . (7)This equation admits several physical interpretations that wediscuss below. (i) Mixed evolution under balanced gain and loss.— Theadditional term in Eq. (7) can be associated with the anti-Hermitian operator − i Γ( t ) = i (cid:80) n [ ∂ t λ n ( t ) /λ n ( t )] | n ( t ) (cid:105)(cid:104) n ( t ) | . (8)The equation of motion for (cid:37) ( t ) is then generated by the fullnon-Hermitian Hamiltonian H ( t ) = H CD ( t ) − i Γ( t ) , i.e., ∂ t (cid:37) ( t ) = − i (cid:0) H ( t ) (cid:37) ( t ) − (cid:37) ( t ) H † ( t ) (cid:1) = − i (cid:2) H CD ( t ) , (cid:37) ( t ) (cid:3) − (cid:8) Γ( t ) , (cid:37) ( t ) (cid:9) . (9)This evolution is not necessarily norm-preserving, with a lossof norm occurring at a rate ∂ t Tr[ (cid:37) ( t )] = − t ) (cid:37) ( t )] = (cid:80) n ∂ t λ n ( t ) . (10)A norm-preserving evolution through the trajectory (cid:37) ( t ) isgoverned by the modified equation of motion ∂ t (cid:37) = − i ( H(cid:37) − (cid:37)H † ) − ∂ t Tr[ (cid:37) ] (cid:37) = − i (cid:2) H CD , (cid:37) (cid:3) + (cid:0) (cid:104) Γ (cid:105) (cid:37) − (cid:8) Γ , (cid:37) (cid:9)(cid:1) , (11)where (cid:104) Γ (cid:105) = Tr[Γ (cid:37) ] and the time-dependence of all termshas been dropped for brevity. Note that the resulting equa-tion is nonlinear in the quantum state (cid:37) . This dynamics thustakes the form of a mixed-state evolution in the presence ofbalanced 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 de-scribe a variety of experiments [22–26]. (ii) Lindblad-like form.— Considering the prescribed trajec-tory (1) and its derivative (7), one can recast the incoherentpart D CD ( (cid:37) ) = (cid:80) n ∂ t λ n ( t ) | n ( t ) (cid:105)(cid:104) n ( t ) | (12)as an auxiliary CD dissipator in a Lindblad-like form for atrace-preserving trajectory. Assuming a trace-preserving evo-lution, (cid:80) n ∂ t λ n ( t ) = 0 , we find the time-dependent Lindbladoperators and rates as [37] L mn ( t ) = | m ( t ) (cid:105)(cid:104) n ( t ) | , (13) γ mn ( t ) = ∂ t λ m ( t ) / [ rλ n ( t )] , (14)that are determined by (the spectral resolution of) (cid:37) ( t ) —andthus state-dependent. The resulting master equation ∂ t (cid:37) = − i [ H CD , (cid:37) ] + (cid:80) mn γ mn (cid:0) L mn (cid:37)L † mn − { L † mn L mn , (cid:37) } (cid:1) (15)is non-Markovian. We remark that existence of a Lindblad-like master equation for an arbitrary dynamics has recentlybeen proven in Ref. [38].The equivalence of Eqs. (11) and (15) shows that the non-linear evolution of a mixed state under balanced gain andloss can be represented by a nonlinear and generally non-Markovian master equation with time-dependent Lindblad op-erators, determined by choice of the trajectory (1).We note that the time-evolution operator generated by theCD Hamiltonian takes the form [34] U CD ( t,
0) = (cid:80) n e iφ n ( t ) | n ( t ) (cid:105)(cid:104) n (0) | , (16)where the time-dependent phase φ n ( t ) is the sum of thedynamical and geometric contributions. In the co-movingframe associated to U CD ( t, , the master equation for (cid:101) (cid:37) ( t ) = U † CD ( t, (cid:37) ( t ) U CD ( t, takes the simple form ∂ t (cid:101) (cid:37) = (cid:80) mn γ mn (cid:0)(cid:101) L mn (cid:101) (cid:37) (cid:101) L † mn − { (cid:101) L † mn (cid:101) L mn , (cid:101) (cid:37) } (cid:1) , (17)with (cid:101) L mn = | m (0) (cid:105)(cid:104) n (0) | . As a result, the time-dependentLindblad operators { L mn } map to the time-independent ones { (cid:101) L mn } , while keeping the same rates γ mn ( t ) . This featureis specific to the superadiabatic driving of open quantum sys-tems and differs from the general case that leads to more com-plex time-dependent Lindblad operators [19]. Quantum speed limit for STA in open quantum processes.—
Speed limits provide a minimum time for a physical pro-cesses to occur in terms of the generator of the evolutionand can be used to relate the operation time of a protocolto the amplitude of the required unitary and nonunitary CDterms. The geometric formulation of the quantum speed limit[39] states that τ (cid:62) D ( (cid:37) (0) , (cid:37) ( τ )) / (cid:104)√ g tt (cid:105) , where g tt is themetric for a given distance D , and (cid:104)√ g tt (cid:105) τ = τ (cid:82) τ dt √ g tt .The quantum Fisher information F is the metric (with a / prefactor) associated with the Bures distance D B ( (cid:37) , (cid:37) ) =[2(1 − F ( (cid:37) , (cid:37) )] / , that is defined in terms of the fidelity F ( (cid:37) , (cid:37) ) = Tr (cid:112) √ (cid:37) (cid:37) √ (cid:37) between (cid:37) and (cid:37) [40]. FromEq. (9), we can identify − iH as a non-Hermitian symmetriclogarithmic derivative, satisfying ∂ t (cid:37) = L (cid:37) + (cid:37) L † [41], basedon which an upper bound on the quantum Fisher informationis obtained as F = Tr[ (cid:37) L ] (cid:54) H(cid:37)H † ] . As a result, thequantum speed limit reads τ (cid:62) D B ( (cid:37) (0) , (cid:37) ( τ ))4 (cid:104) Tr[( H CD − i Γ) (cid:37) ( t )( H CD + i Γ † )] / (cid:105) τ . (18)Alternatively, using the trace distance rather than theBures distance, the relevant metric is g tt = (cid:107) ∂ t (cid:37) (cid:107) ≡ (Tr[ (cid:112) ( ∂ t (cid:37) ) ]) . Using Eqs. (9) and (15) for ∂ t (cid:37) and the tri-angle inequality, one obtains (cid:107) ∂ t (cid:37) (cid:107) (cid:54) (cid:107) [ H CD , (cid:37) ] (cid:107) + (cid:107){ Γ , (cid:37) }(cid:107) for the gain-loss equation and (cid:107) ∂ t (cid:37) (cid:107) (cid:54) (cid:107) [ H CD , (cid:37) ] (cid:107) + (cid:107) D CD (cid:107) for the Lindblad-like equation. In all of these bounds, both theCD Hamiltonian and dissipator set the speed of evolution. Example I: Strokes for a two-level system.—
Consider a two-level system described by a time-dependent Hamiltonian H ( t ) = 12 (∆( t ) σ z + Ω( t ) σ x ) , (19)where σ z and σ x are the Pauli matrices. The instanta-neous eigenstates read E ± ( t ) = ± (cid:112) Ω ( t ) + ∆ ( t ) / ±| Ω( t ) | / (2 sin θ ( t )) , where θ ( t ) = arctan(Ω( t ) / ∆( t )) andthe corresponding eigenstates are | + ( t ) (cid:105) = cos( θ ( t ) / | (cid:105) + sin( θ ( t ) / | (cid:105) , | − ( t ) (cid:105) = sin( θ ( t ) / | (cid:105) − cos( θ ( t ) / | (cid:105) , (20)with σ z | (cid:105) = | (cid:105) and σ z | (cid:105) = −| (cid:105) . We consider thesystem to be described by the time-dependent mixed state (cid:37) ( t ) = (cid:80) α = ± λ α ( t ) | α ( t ) (cid:105)(cid:104) α ( t ) | . Thus, the target trajectory (cid:37) is already diagonal in the eigenbasis of the uncontrolled sys-tem Hamiltonian H ( t ) . The auxiliary control term requiredto guide the dynamics is known to be of the form [3, 4, 34] H = 12 ∂ t ∆ Ω − ∂ t Ω ∆Ω + ∆ σ y , (21)so that the full dynamics is generated by H CD = H + H .The dynamics is open when the eigenvalues λ ± are time-dependent.The first approach we have introduced relies on the pres-ence of gain and loss, for which the dynamics is generally nolonger trace-preserving, i.e., λ − + λ + is time-dependent anddifferent from unity. Such evolution is generated by the non-Hermitian Hamiltonian H = H CD − i Γ , where Γ = ∂ t λ + (cid:16) λ − | − ( t ) (cid:105)(cid:104)− ( t ) | − λ + | + ( t ) (cid:105)(cid:104) +( t ) | (cid:17) . (22)Under balanced gain and loss, the trace-preserving property isrestored by the nonlinear equation (11) with this choice of Γ . ��� ��� ��� ��� ��� ��� - ��� - ������������ t/t f
Consider a two-level atom in a thermal bosonic bath at inversetemperature β B (0) . The dynamics of the atom under someconditions can be described by [42–44] ∂ t (cid:37) S = − i (cid:2) H S , (cid:37) S (cid:3) (29) + (cid:80) j,k | ( j (cid:54) = k ) γ jk ( L jk (cid:37) S L † jk − { L † jk L jk , (cid:37) S } ) , where j, k ∈ { , } , and H S = ω σ z , L = | (cid:105)(cid:104) | = σ − , L = | (cid:105)(cid:104) | = σ + , (30) γ = γ (cid:0) ¯ n ( ω , β B (0)) + 1 (cid:1) , γ = γ ¯ n. Here, ¯ n ( ω , β B (0)) = (e β B ω − − is the mean boson num-ber in a mode with frequency ω , and γ is a time-independentconstant indicating the strength of the coupling between theatom and the thermal bath.If the atom is initially in a thermal state (cid:37) S (0) = e − β S (0) H S /Z S (0) , its instantaneous state is obtained by solv-ing the above master equation, which gives a Gibbsian thermalstate (cid:37) S ( t ) = e − β S ( t ) H S /Z S ( t ) , with β S ( t ) = − ω log 1 − e − (cid:101) γt tanhΘ S + (e − (cid:101) γt − B − (cid:101) γt tanhΘ S − (e − (cid:101) γt − B . (31)Here Θ k = ω β k (0) / ( k ∈ { S, B } ), (cid:101) γ = γ coth Θ B , and Z S ( t ) = Tr[ e − β S ( t ) H S ] .Equation (15) suggests another dynamical equation real-izing the same trajectory (cid:37) S ( t ) . Since H = H S is time-independent, H will be zero as well. The Lindblad opera-tors are given in terms of the eigenstates of H S as L mn =
We have introduced a univer-sal scheme to design shortcuts to adiabaticity in open quantumsystems, interacting with an environment. We first specify atarget trajectory for the evolution of the system, and then findthe required auxiliary Hamiltonian terms and dissipators thatgenerate it. The resulting dynamics can be associated witha driven system in the presence of balanced gain and loss, ascenario that occurs naturally, e.g., in PT -symmetric quan-tum mechanics. Alternatively, it can be implemented via anon-Markovian evolution in which the equation governing thedynamics takes a generalized Lindblad-like form. Our formal-ism thus enables to engineer superadiabatic open processesto speed up, i.e., heating, cooling, and isothermal strokes. Itshould find broad applications in quantum thermodynamics,and more generally in quantum technologies requiring the fastcontrol of an open system embedded in an environment. Acknowledgements.—
It is a pleasure to thank Jack J. Mayoand Tapio Ala-Nissila for comments on the manuscript. Sup-port by the Academy of Finland’s Center of Excellence pro-gram QTF Project 312298 (to S.A.) is acknowledged. A.T.R.also acknowledges support by the QTF and also Aalto Univer-sity’s AScI Visiting Professor Fund. [1] E. Torrontegui, S. Ib´a˜nez, S. Mart´ınez-Garaot, M. Modugno,A. del Campo, D. Gu´ery-Odelin, A. Ruschhaupt, X. Chen, andJ. G. Muga, in
Advances in Atomic, Molecular, and OpticalPhysics , Vol. 62, edited by E. Arimondo, P. R. Berman, andC. C. Lin (Academic Press, 2013) pp. 117 – 169.[2] X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Gu´ery-Odelin, and J. G. Muga, Phys. Rev. Lett. , 063002 (2010).[3] M. Demirplak and S. A. Rice, J. Phys. Chem. A , 9937(2003).[4] M. Demirplak and S. A. Rice, J. Phys. Chem. B , 6838(2005).[5] J. Deng, Q.-H. Wang, Z. Liu, P. H¨anggi, and J. Gong, Phys.Rev. E , 062122 (2013).[6] A. del Campo, J. Goold, and M. Paternostro, Sci. Rep. , 6208(2014).[7] K. Funo, J.-N. Zhang, C. Chatou, K. Kim, M. Ueda, and A. delCampo, Phys. Rev. Lett. , 100602 (2017).[8] J.-F. Schaff, X.-L. Song, P. Vignolo, and G. Labeyrie, Phys.Rev. A , 033430 (2010).[9] J.-F. Schaff, X.-L. Song, P. Capuzzi, P. Vignolo, andG. Labeyrie, Europhys. Lett. , 23001 (2011).[10] M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo,R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, Nat.Phys. , 147 (2012).[11] W. Rohringer, D. Fischer, F. Steiner, I. E. Mazets, J. Schmied-mayer, and M. Trupke, Sci. Rep. , 9820 (2015).[12] S. Deng, P. Diao, Q. Yu, A. del Campo, and H. Wu, Phys. Rev.A , 013628 (2018).[13] S. Deng, A. Chenu, P. Diao, F. Li, S. Yu, I. Coulamy, A. delCampo, and H. Wu, Sci. Adv. , eaar5909 (2018).[14] J. Zhang, J. H. Shim, I. Niemeyer, T. Taniguchi, T. Teraji,H. Abe, S. Onoda, T. Yamamoto, T. Ohshima, J. Isoya, andD. Suter, Phys. Rev. Lett. , 240501 (2013).[15] J. K¨olbl, A. Barfuss, M. S. Kasperczyk, L. Thiel, A. A. Clerk,H. Ribeiro, and P. Maletinsky, Phys. Rev. Lett. , 090502(2019).[16] S. An, D. Lv, A. del Campo, and K. Kim, Nat. Commun. ,12999 (2016).[17] T. Wang, Z. Zhang, L. Xiang, Z. Jia, P. Duan, W. Cai, Z. Gong,Z. Zong, M. Wu, J. Wu, L. Sun, Y. Yin, and G. Guo, New J.Phys. , 065003 (2018).[18] T. Wang, Z. Zhang, L. Xiang, Z. Jia, P. Duan, Z. Zong, Z. Sun,Z. Dong, J. Wu, Y. Yin, and G. Guo, Phys. Rev. Applied ,034030 (2019). [19] H.-P. Breuer and P. Petruccione, The Theory of Open QuantumSystems (Oxford University Press, Oxford, 2007).[20] M. B. Plenio and P. L. Knight, Rev. Mod. Phys. , 101 (1998).[21] C. M. Bender and S. Boettcher, Phys. Rev. Lett. , 5243(1998).[22] C. E. R¨uter, K. G. Makris, R. El-Ganainy, D. N.Christodoulides, M. Segev, and D. Kip, Nat. Phys. , 192(2010).[23] A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov,D. N. Christodoulides, and U. Peschel, Nature , 167 (2012).[24] L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira,V. R. Almeida, Y.-F. Chen, and A. Scherer, Nat. Mater. , 108(2012).[25] B. Peng, S. K. ¨Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L.Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Nat. Phys. , 394 (2014).[26] B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick,S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇci´c, Nature ,354 (2015).[27] A. B. Boyd, A. Patra, C. Jarzynski, and J. P. Crutchfield,arXiv:1812.11241.[28] A. del Campo, A. Chenu, S. Deng, and H. Wu, “Friction-free quantum machines,” in Thermodynamics in the QuantumRegime: Fundamental Aspects and New Directions , edited byF. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso(Springer International Publishing, Cham, 2018) pp. 127–148.[29] G. Vacanti, R. Fazio, S. Montangero, G. M. Palma, M. Pater-nostro, and V. Vedral, New J. Phys. , 053017 (2014).[30] R. Dann, A. Tobalina, and R. Kosloff, Phys. Rev. Lett. ,250402 (2019).[31] T. Villazon, A. Polkovnikov, and A. Chandran,arXiv:1902.05964.[32] T. Kato, J. Phys. Soc. Jpn. , 435 (1950).[33] J. E. Avron, R. Seiler, and L. G. Yaffe, Commun. Math. Phys. , 33 (1987).[34] M. V. Berry, J. Phys. A: Math. Theor. , 365303 (2009).[35] D. C. Brody and E.-M. Graefe, Phys. Rev. Lett. , 230405(2012).[36] J. Gong and Q.-H. Wang, J. Phys. A: Math. Theor. , 485302(2013).[37] See Supplemental Material.[38] S. Alipour, A. T. Rezakhani, A. P. Babu, K. Mølmer,M. M¨ott¨onen, and T. Ala-Nissila, arXiv:1903.03861.[39] K. Funo, N. Shiraishi, and K. Saito, New J. Phys. , 013006(2019).[40] A. T. Rezakhani, D. F. Abasto, D. A. Lidar, and P. Zanardi,Phys. Rev. A , 012321 (2010).[41] S. Alipour and A. T. Rezakhani, Phys. Rev. A , 042104(2015).[42] S. Alipour, F. Benatti, F. Bakhshinezhad, M. Afsary, S. Marcan-toni, and A. T. Rezakhani, Sci. Rep. , 35568 (2016).[43] R. W. Rendell and A. K. Rajagopal, Phys. Rev. A , 062110(2003).[44] H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993).
Supplemental Material: Shortcuts to Adiabaticity in Driven Open Quantum Systems: BalancedGain and Loss and Non-Markovian Evolution
LINDBLAD-LIKE MASTER EQUATION
In this section, we verify that the dissipator D ( (cid:37) ) = (cid:88) mn γ mn (cid:0) L mn ( t ) (cid:37)L † mn ( t ) − { L † mn ( t ) L mn ( t ) , (cid:37) } (cid:1) , (S1)with the choice of the Lindblad operators and rates given in the main text satisfies the identity D ( (cid:37) ) = (cid:88) m ∂ t λ m ( t ) | m ( t ) (cid:105)(cid:104) m ( t ) | . (S2)Employing the explicit form of L mn ( t ) in Eq. (S1) one finds D ( (cid:37) ) = (cid:88) mn ∂ t λ m ( t ) r ( | m ( t ) (cid:105)(cid:104) m ( t ) | − | n ( t ) (cid:105)(cid:104) n ( t ) | ) . (S3)Noting that (cid:80) rn =1 r = rank( (cid:37) ) and (cid:80) rm =1 ∂ t λ m ( t ) = ∂ t Tr[ (cid:37) ] , it follows that D ( (cid:37) ) = (cid:88) m ∂ t λ m ( t ) | m ( t ) (cid:105)(cid:104) m ( t ) | − r ∂ t Tr( (cid:37) ) (cid:88) n | n ( t ) (cid:105)(cid:104) n ( t ) | . (S4)As the second term on the right-hand side vanishes identically for a norm-preserving evolution, this completes the proof of Eq.(S2). LINDBLAD OPERATORS FOR ARBITRARY STROKES IN TWO-LEVEL SYSTEMS
Consider the trajectory described by the instantaneous thermal state of a two-level system (cid:37) ( t ) = (cid:88) α = ± e α β √ Ω +∆ β √ Ω + ∆ ] | α ( t ) (cid:105)(cid:104) α ( t ) | , (S5)where β , ∆ , and Ω are time-dependent. The Lindblad operators are L + − = | + (cid:105)(cid:104)−| and L − + = |−(cid:105)(cid:104) + | , as in Eq. (25) in themain text, with rates γ + − ( t ) = ∆ ∂ t β + Ω (Ω ∂ t β + β∂ t Ω) + β ∆ ∂ t ∆2 √ ∆ + Ω (cid:16) e − β √ ∆ +Ω + 1 (cid:17) ,γ − + ( t ) = − ∆ ∂ t β + Ω (Ω ∂ t β + β∂ t Ω) + β ∆ ∂ t ∆2 √ ∆ + Ω (cid:16) e β √ ∆ +Ω + 1 (cid:17) ..