Geometrical bounds on irreversibility in squeezed thermal bath

Irreversible entropy production (IEP) plays an important role in quantum thermodynamic processes. Here we investigate the geometrical bounds of IEP in nonequilibrium thermodynamics by exemplifying a system coupled to a squeezed thermal bath subject to dissipation and dephasing, respectively. We find that the geometrical bounds of the IEP always shift in contrary way under dissipation and dephasing, where the lower and upper bounds turning to be tighter occurs in the situation of dephasing and dissipation, respectively. However, either under dissipation or under dephasing, we may reduce both the critical time of the IEP itself and the critical time of the bounds for reaching an equilibrium by harvesting the benefits of squeezing effects, in which the values of the IEP, quantifying the degree of thermodynamic irreversibility, also becomes smaller. Therefore, due to the nonequilibrium nature of the squeezed thermal bath, the system-bath interaction energy brings prominent impact on the IEP, leading to tightness of its bounds. Our results are not contradictory with the second law of thermodynamics by involving squeezing of the bath as an available resource, which can improve the performance of quantum thermodynamic devices.


I. INTRODUCTION
Over the past two decades, the nonequilibrium phenomena and thermodynamic irreversibility quantified by irreversible entropy production (IEP) have drawn much attention, since this fundamental concept is one of the cornerstones of classical and quantum thermodynamics [1][2][3][4][5][6][7][8][9][10]. As is well known, the positivity of entropy production has been universally captured by the conventional second law of thermodynamics (SLT) [9], which quantitatively characterize the interplay between the exchange of energy and the irreversibility by introducing the state function entropy [11,12]. Also, through specifying a lower bound for the irreversible entropy change, the related Clausius inequality provides a fundamental feature of irreversible phenomena. Noted that this lower bound (zero) is trivially independent of how far from equilibrium a process operates [6].
Recently, with restriction to a specific class of nonequilibrium phenomena, such as thermal relaxation process, rich features of thermodynamic irreversibility have been found successively [2]. Such a lower bound for classical, nearequilibrium transformation processes has been derived by mean of a geometric approach [13]. S. Deffner and E. Lutz treated the system's Hilbert space as a Riemannian manifold and extended the classical case [13] to the nonequilibrium closed quantum system. The obtained generalized Clausius inequality states that the thermodynamic irreversibility is bounded in terms of the Bures length between the final state and the corresponding equilibrium state, by using information geometry [6]. Soon afterwards, they broadened the closed quantum system further to the weakly coupled open quantum system, then obtained the exact microscopic expressions for the nonequilibrium entropy production [7]. Along this direction, reference [11] theoretically and experimentally determined a sharper geometrical bound for a qubit thermalization process and obtained a tighter version of the Clausius inequality following a similar approach.
In many cases of interest, however, having a sharper lower bound is essential. Therefore, theses above-mentioned progress hence stimulated successive studies on the tightness of the geometrical bounds on irreversibility in open quantum system [6][7][8][9][10][11]. These recent publications tried to develop theories to further understand the thermodynamic irreversibility inherent to nonequilibrium processes. In particular, based on the variational principle and time-reversed map, the authors in reference [10] obtained a information-theoretical bound for entropy production in a relaxation process by a geometrical distance on the Riemannian manifold [14], which was experimentally validated by a single ultracold trapped ion 40 Ca [15].
T. V. Vu and Y. Hasegawa strengthened the Clausius inequality and proved that IEP is bounded from below by a modified Wasserstein distance (quantum generalization of the Wasserstein metric) between the initial and final states [8]. Thereafter, they extended this single-bath case to the case of multiple-bath, and refined the SLT in a quantum regime, through deriving the fundamental bound on irreversibility for thermal relaxation processes of Markovian open quantum systems [9].
FIG. 1: Schematic diagram for two-level system (excited state |e and ground state |g ) interacting with a squeezed thermal bath at temperature T with squeezing parameter s. The transition frequency between the two levels is ω0. The evolution induced by the system-bath interaction produces irreversible entropy.
On the other hand, quantum bath engineering techniques are powerful tools that enable the realization of arbitrary thermal and nonthermal environments [16][17][18][19][20][21][22][23]. Additionally, due to the unique vantage of the quantum control, bath engineering has aroused widespread interest in the context of quantum thermodynamical processes. Various strategies have attempted to improve the the performance of thermodynamic devices [24][25][26][27][28][29][30][31][32], whose efficiency is usually reduced by the presence of IEP [7]. For instance, the use of a squeezed thermal bath allows us to operate thermodynamic devices beyond the classical bound [23,24]. In particular, the experiment in reference [30] demonstrated that the efficiency of the quantum heat engine may go beyond the standard Carnot efficiency by employing a squeezed thermal bath.
However, the influence of squeezed thermal bath on the bounds of IEP in the quantum thermodynamics has been largely unexplored. A precise characterization of the IEP in such an unchartered domain, and a general framework providing a deeper understanding of the associated quantum thermodynamic phenomena therefore appear necessary. Therefore, it is instructive to look into the role of squeezed thermal bath playing during the process of thermodynamic irreversibility.
In this work, we study and quantify the geometrical bounds on irreversibility of a quantum system contacting with a squeezed thermal bath. Here we consider two distinct situations including the dissipation model and dephasing model, respectively. Starting with Born-Markovian quantum master equation, we derived analytical expressions for time-dependent reduced density matrix of system, in which the squeezing parameter of bath is involved. Then through quantifying the degree of irreversibility by the IEP, we find that the geometrical bounds of IEP decrease (increase) with the growth of degree of squeezing in the case of dissipation (dephasing) model, respectively. Additionally, the common feature is that the critical times of the IEP itself and that of IEP's bounds reaching equilibrium, as well as the values of IEP quantifying the degree of thermodynamic irreversibility are reduced due to the presence of squeezing effect, for these two models. Therefore, due to the nonequilibrium nature of squeezed thermal bath, the interaction energy between the system and bath brought important impact on the irreversibility, as well as the tightness of its bounds. As expected, our finding obeys the principles of thermodynamics and reveals richer features of thermodynamics relaxation process, and the presence of a quantum property, such as the squeezing effects included in the bath, could be used to serve as an available resource to improve the performance of the quantum thermodynamic devices.
The paper is organized as follows. We give a brief account of the method about geometrical bounds on irreversibility in open quantum system (Section II). We evaluate the geometrical bounds on irreversibility for squeezed thermal bath in the dissipation model (Section III) and dephasing model (Section IV), respectively, and we conclude and give prospects for future developments in Section V.

II. GEOMETRICAL BOUNDS OF IRREVERSIBLE ENTROPY PRODUCTION
Considering an arbitrary quantum system with the Hamiltonian coupled to a thermal bath. Usually, the quantum system is initialized in a given state ρ 0 , then interacts with a bath at temperature T . The evolution induced by the interactions brings the system in a state ρ(t) and produces irreversible entropy. Furthermore, the system will thermalize with the bath and then asymptotically reaching the unique canonical equilibrium state ρ th if the system Hamiltonian H maintains constant. The total entropy variation of the system is defined as where S [ρ] = −tr (ρ ln ρ) is the von Neumann entropy. The IEP (irreversible part of the total entropy variation ∆S tot ) is denoted by which is the thermodynamic irreversibility under consideration in the present work [2,11]. S (ρ 1 ρ 2 ) = tr (ρ 1 ln ρ 1 ) − tr (ρ 1 ln ρ 2 ) represents the quantum relative entropy of ρ 1 to ρ 2 . The entropy flow between the system and the environment (reversible part of the total entropy variation ∆S tot ) is ∆S re = ∆Q/T , where ∆Q = tr (Hρ(t))−tr (Hρ 0 ) is the heat absorbed by the system [11]. On the other hand, the Clausius inequality ∆S ir ≥ 0 putting forward the lower limit of IEP is always non-negative. In order to deepen the understanding of how much energy in the irreversible process is consumed, it is essential to search a sharper or tighter bound on irreversibility. By treating the Hilbert space of the system as a Riemannian manifold, the relationship between IEP and the geodesic distance D corresponding to the metric that is contractive under complete positive and trace preserving maps can be directly established, and a generalized form of Clausius inequality can be obtained by deriving the Wigner-Yanase length between the initial and final states of the system. Note that the only cases in which an analytical expression for the geodesic distance is known are the Wigner-Yanase metric D W Y (ρ 1 , ρ 2 ) = arccos tr √ ρ 1 √ ρ 2 and the quantum Fisher information metric D QF (ρ 1 , ρ 2 ) = arccos tr √ ρ 1 ρ 2 √ ρ 1 [11]. Based on these analytical expressions, one can obtain the geometrical LB of IEP as and the geometrical UB is From the above two relations one can define the related bound gap as and the deviation of the IEP from the LB or UB is given by respectively. In this paper, we say a LB (UB) is relative tighter if the LB (UB) takes a larger (smaller) value compared with the case of conventional thermal bath.

III. GEOMETRICAL BOUNDS ON IRREVERSIBILITY IN THE DISSIPATION MODEL
Here we consider the dissipation model taking into account the effect of the squeezed thermal bath with temperature T , in the case of single-excitation. The total Hamiltonian is (in units of = 1) where H S , H B , and H SB stand for the Hamiltonians of the system, bath, and system-bath interaction, respectively. σ + (σ − ) = |e g| (|g e|) and ω 0 are the inversion operator and transition frequency of the system with |e and |g being the excited and ground states.b † k (b k ) are the creation (annihilation) operators of the k-th mode of the bath. The coupling strength between the system and the bath is denoted by g k .
The master equation, in the interaction picture, is given by the following Lindblad form [33,34] ρ The spontaneous emission rate of system is γ and N = N th cosh 2 (s) + sinh 2 (s) + sinh 2 (s), and M = − sinh (2s) e iφ (2N th + 1) /2 with s and φ the bath squeezing strength and phase, respectively Rewriting the density matrix as ρ s (t) = (I + r (t) · σ) /2 by mean of the Bloch vector r (t) = T r[ σρ s (t)] with the Identity matrix I, we can transform the master equation (9) into the Bloch equation with and m = (0, 0, , −γ 0 ) T . Hereγ = γ (2N + 1) is the total transition rate. Assuming the system to be initially in the ground state |g , a straightforward calculation yields the analytical solution Here σ ± ss and σ z ss are the stationary solutions of differential equation (10). µ = {γ s − e (−4γ+γs)t/4 [γ s cos(γ s t/4)+ (γ s + γM ) sin(γ s t/4)]}/γ s and ν =γe (−4γ+γs)t/4 [cos(γ s t/4) − sin(γ s t/4)]/γ s , where γ s =γ + 2γM . The time dependence of the IEP, and its geometrical bounds are explored by numerically calculating the quantities (equations (2) and (3), (4)) and plotted in figure 2. It is clear from figure 2 that IEP and its bounds (UB and LB) increase monotonically with time toward the corresponding equilibrium values for a squeezed thermal bath. For the non-squeezing case (s = 0), our results based on quantum master equation are fully consistent with that for the thermal bath in reference [11] using the method of Kraus operators. In the case of squeezed thermal bath (s > 0), with increase in degree of squeezing the UB becomes higher in the early stage of evolution t 0.15, and then becomes reduced with the growth of s when t 0.45, compared with the thermal bath (s = 0), as shown in figure 2(a). Also, the dynamical behavior of LB shares similar features with that of UB. Another common feature between UB and LB is that their equilibrium times decrease with the growing degree of squeezing, for instance, the equilibrium times for UB (LB) are 4.5 (7.5), 2 (1.4) and 0.15 (0.2) when s = 0, 1, and 2, respectively. We conclude that the summarized overall trends of geometrical bounds (LB and UB) found in reference [11] also hold for squeezed thermal bath, while only the UB becomes tighter in the long-time limit, and the LB exhibits subtle tightness in the early stage of evolution, compared with the traditional thermal bath [11].
Let us next examine the time dependence of the IEP, the numerical simulation plotted in figure 2(b), and 2(c) suggests that the values of IEP are well limited in the region between the LB and UB, and the squeezed thermal bath has prominent influences in both the concrete values of IEP and the critical times T c of IEP reaching equilibrium, where the values of T c become less for growing values of s. Another observation in figure 2(b) is that the deviation ∆δ L (equation (6))increases monotonically and gradually coincides with ∆U (equation (5)), whereas the deviation ∆δ U (equation (7)) decreases monotonically and gradually disappears to zero in the case of s = 0. With the increase of squeezing strength, i.e., s = 2, as shown in figure 2(c), the deviation ∆δ L (∆δ U ) increases (decreases) gradually to a fixed value 0.51 (0.542), and the rates of change of deviations ∆δ L and ∆δ U become slower. From the above results we deduce that the actual amount of IEP becomes depart from its UB gradually and approaches its LB with the growth of s.
To get a clear picture of how the IEP evolves in the parameter space of {s, t}, we plot the evolution of IEP in figure 3. It shows that the values of IEP are obviously dependent on the squeezing parameters of bath. Here we provide remarks on the parameter dependence. Although the values of s are not directly related to the systembath interaction, whereas they depend on both correlation time and occupation number of the bath and then affect immensely the relaxation dynamics and steady state of the relevant system during the thermodynamic process. As a result, any change of this key parameter will bring significant influences on the irreversibility, and the inherent squeezing effect stemming from the bath plays a crucial role in understanding the relative tightness of the bounds. As we have derived analytically in the previous paragraphs, the above summarized dependence of bounds and IEP on the squeezing parameter is reflected in the equations (3,4,12).
Traditionally, the IEP could be used to evaluate the performance of thermodynamic devices, such as the availability (or exergy) that can be extracted from a given system, and the maximal useful work, which are usually reduced by the presence of irreversibility [7,35]. It implies that the irreversibility can be restrained by controlling the amount of IEP through adjusting the degree of squeezing effect, as shown in figure 3(b), where the values of IEP become less for growing values of s. From the perspective of quantum bath engineering, employing a squeezed thermal bath is a promising avenue of using squeezing effect as a quantum resource to improve the performance thermodynamic devices [23,24,30].
As we known, squeezing effect that is rooted in Heisenberg's uncertainty principle can be defined as the reduction in the uncertainty of some observable, at the cost of the build-up in the conjugate one [26,36,37]. Physically, the squeezing involved in the bath thereby inevitably modifies the nonunitary relaxation dynamics of system and the relevant irreversibility during thermodynamic process. Compared with the thermal bath, the squeezed thermal bath is taken out of thermodynamics equilibrium through the squeezing operating, with a consequence that its excitation number changing from N th to N = N th (cosh 2 r + sinh 2 r) + sinh 2 r [26,38], which can be seen as an increase in its effective temperature T ef f = ω h /k B ln[1/(N −1 th + 1)] with a higher frequency ω h > ω 0 . Therefore, being purely quantum mechanical fuel in nature, squeezed thermal bath is beneficial in its own way by providing us with more compact energy-storage and higher effective high-temperature bath without being actually too hot [5]. That is to say, the squeezed thermal state has the same entropy as the Gibbs state, but increased mean energy, which is instrumental in the suppression of irreversibility.

IV. GEOMETRICAL BOUNDS ON IRREVERSIBILITY IN THE DEPHASING MODEL
Next we focus on the dephasing model with respect to the squeezed thermal bath, where the bath operator is simply a sum of linear couplings to the coordinates of a continuum of harmonic oscillators described by a spectral density function J(ω) [39][40][41][42][43][44], and the decay of the coherence occurs without a decay of the corresponding populations. Now the total Hamiltonian is The dynamics of the system can be characterized by the reduced density matrix which is obtained by tracing out the degrees of freedom of the bath. In the interaction picture, the reduced density matrix of system can be written as [42][43][44][45][46] where the phase decay behavior of the qubit under the influence of the bath is denoted by the factor [42]. The associated master equation is given bẏ . In the following we consider that the bath starts from a squeezed thermal state [45][46][47][48] where ρ th = e −βHB /Z β is thermal state with Z β the partition function.ς = kŝ k , whereŝ k = e [(s * k e −iφ kb 2 k −s k e iφ k (b † k ) 2 )/2] is the squeezing operator for the boson bath modeb k with s k and φ k being the bath squeezing strength and phase, respectively. In this situation, the function Γ (t) could be evaluated under the summation of the modes of the squeezed thermal bath as [42][43][44][45][46] with γ k (t) = cosh 2s k − sinh 2s k cos (ω k t − ∆φ k ) and ∆φ k is the phase difference between the squeezing phase φ k relative to the phase of the coupling strength g k . Substituting the coupling spectral density J(ω) = 2π k |g k | 2 δ (ω − ω k ) into the equation (17), we can transform the above summation in Γ (t) into an integral for continuous bath modes as In the present work we adopt the Ohmic coupling spectral density J(ω) = ηωe −ω/̟ with ̟ cutoff frequency and η is the coupling strength [49]. Note that such engineering of the spectrum's Ohmicity seems possible when simulating the dephasing model using trapped ultracold atom, as demonstrated in reference [50,51]. In the high-temperature regime, the dephasing process is Markovian, and after straightforward algebra, one finds where the approximation coth(ω/2T ) ≈ 2T /ω has been taken. Expression (equation (19)) is the exact analytic for the time-dependent dephasing rate Γ (t) in the present model.  Special attention was paid to the time dependence of the IEP as well as its geometrical bounds, as shown in figure  4, where one can find that all the quantities exhibit asymptotic behaviors approaching their stationary values. They correspond to relaxations of the system through the dephasing channel due to the system-bath coupling. Regarding the time dependence of the bounds, we encounter another common feature is that the equilibrium times of bounds decrease with the growing degree of squeezing, for instance, the equilibrium times for UB (LB) are 3 (7.3), 2.5 (3.3) and 0.4 (0.6) when s = 0, 1, and 2, respectively. But unlike the dissipation model, both UB and LB increase monotonically as the squeezing character of the bath grows, and only the LB becomes tighter (compared with the thermal bath) during the whole dynamic process in the dephasing model. As a contrast, the tightness of LB only appears in the early stage of evolution for the dissipation model. By comparing the time evolutions of IEP with the two blue solid-lines in panel (b,c) of figure 4, we find that the IEP is well located inside the region between the LB and UB, and reaches its stationary value faster with the increase of the squeezing parameter s, and the values of the IEP is reduced due to the existence of squeezing effect.
In figure 5, we provide numerical estimates of the IEP in the parameter space of {s, t}. The figure 5 tells us that one can precisely control the thermodynamic irreversibility through adjusting the parameters of the bath. As shown in figure 5(b), under the dephasing model, the IPE reaches equilibrium faster as the squeezed parameter increases, and the value of the IEP in long-time limit △S irr (∞) will eventually converge together, irrespective of the values of parameter s. It means that, in the dephasing model, the squeezing effect could not make too much impact on the thermodynamic irreversibility in the long-time limit, although the existence of squeezing drives the system into equilibrium faster. Physically, on a fundamental level, quantum coherence and the related dephasing process could also alter the possible state transitions in thermodynamic processes [52], and may even modify the fluctuation-dissipation relation [53,54] and quantum nonequilibrium work relation [55]. Additionally, when a system relaxes to equilibrium through contacting with a thermal bath, quantum coherences are known to contribute an additional term to the IEP [56][57][58]. Different from the dissipation case where the system can exchange energy with its bath, in the dephasing model, this open system can never exchange energy with its bath. But the information and correlation exchange between the system and the bath are dominant during the dynamics and this exchange also influence the IEP. As a result, any alteration in the von Neumann entropy (basis of IEP and its bound) resulting from the relaxation process (dissipation or dephasing) has contributions not only from the change in population but also from decoherence. In this regard, it was pointed out that the entropy production can be split in two contributions, an incoherent one (stemming from populations) and a coherent one (stemming from quantum coherences) [59][60][61].

V. DISCUSSION AND CONCLUSIONS
The study of IEP is of importance due to its intimate relation with the arrow of time in classical and quantum systems [62,63], the SLT [64][65][66], thermodynamic operations and thermal machines [67][68][69][70], and quantum and classical speed limits [8,71,72]. Therfore, tightening the bounds of IEP not only deepens our understanding of how much entropy production changes during the thermodynamic process but also provides insights into how to improve the performance of quantum thermodynamic devices.
Interaction with a squeezed thermal bath is not the only generalized process that goes beyond the typical settings in classical thermodynamics. Our findings demonstrate how to utilize the squeezing effect of bath as a resource to control the irreversibility, where the use of nonthermal bath offers more degrees of control and manipulation, such as the amount of squeezing. Note that quantum bath engineering techniques has become powerful tools that enable the realization of arbitrary thermal and nonthermal bath, for instance, experimental realizations of squeezed thermal states range from superconducting circuit QED [73][74][75] to optomechanical mechanical oscillators [76,77]. The key parameters considered in our numerical simulation, such as the inverse temperature β and the degree of squeezing s, could be experimentally controlled using the current technologies demonstrated in the above-mentioned experiments. Additionally, there have been many experiments focus on the assessment of nonequilibrium thermodynamics irreversibility using the technology of quantum trajectories of stochastic dynamics in nuclear magnetic resonance setups [3], superconducting qubit [81], and mechanical resonator [82], respectively. Our results reveal more detailed properties of thermodynamic irreversibility that are stronger than the conventional SLT, for given a restricted class of irreversible processes. Along with other studies addressing squeezing effects in quantum thermodynamics, we hope that our analyses help to unveil the role of squeezing effects in quantum thermodynamics devices.
In summary, we studied the influence of squeezed characteristics of bath on the IEP of open quantum systems. The results show that the equilibrium rates of IEP and its bounds become faster, and the values of IEP are reduced through harvesting the benefits of squeezing effects in the case of both dissipation model and dephasing model. In the dissipation model, the summarized overall trends of geometrical bounds (LB and UB) found in reference [11] also hold for squeezed thermal bath, while only the UB becomes tighter in the long-time limit, and the LB exhibits subtle tightness in the early stage of evolution, compared with the thermal bath. Unlike the dissipation model, both UB and LB increase monotonically as the squeezing character of the bath grows, and only the LB becomes tighter (compared with the thermal bath) during the whole dynamic process in the dephasing model.
Moreover, the concrete amount of thermodynamic bounds greatly depends on the explicit form of the system-bath coupling, whereas the trends of them is independent of the details of the models. Also, the above-summarized trends for the bounds are independent of system size and hold for systems having more degrees of freedom. Our results do not contradict the SLT, which is modified by the inclusion of squeezing as an available resource in the bath. It is worth noting that a general evolution and the associated geometrical bounds of irreversibility of two-level system in thermal bath were theoretically analyzed and experimentally demonstrated in [11]. Here, we further highlight the role of adjustable parameters in bath, such as temperture and squeezing degree, on the reduction of thermodynamic irreversibility. It is expected that the present work helps in developing a better understanding of the irreversibility under ambient conditions.