A fluctuation theorem for Floquet quantum master equations

The Gallavotti-Cohen (GC) fluctuation theorem states that the ratio of the probability distribution $p(\sigma)$ of having an average total entropy production rate $\sigma$ to that of having $-\sigma$ approaches $\exp(t\sigma)$ as the time interval $t$ increases; that is,

where the Boltzmann constant $k_{B}$ is set to 1 throughout this paper, and the symbol $\asymp$ denotes asymptotic change as $t$$\rightarrow$$\infty$ Touchette (2008). The original GC fluctuation theorem was inspired by a relationship between the probabilities of fluctuations in the shear stress of fluids in nonequilibrium steady states Evans et al. (1993) and was proved in modern dynamical system theory Gallavotti and Cohen (1995, ). Complicated techniques were used; see the latest review Gallavotti (2019). In contrast, its proof in stochastic systems, e.g., in classical Langevin systems or discrete jump systems, is simple Kurchan (1998); Lebowitz and Spohn (1999); Maes (1999); Esposito et al. (2009). Hence, stochastic dynamical systems are suitable to explore new fluctuation theorems.

The motivation of this paper is as follows. Let us imagine that a system contacts a heat bath having an inverse temperature $\beta$ and is in a nonequilibrium steady state due to some external force. The GC theorem can be reexpressed in terms of the probability distribution of the total heat current $j$, the total released heat averaged over the time interval $t$. Then, the exponent on the right-hand side of Eq. (1) is replaced by $t\beta j$. We know that the total heat is composed of positive and negative parts. Accordingly, the total heat current $j$ can be divided into $j_{+}$ and $j_{-}$. Apparently, the individual current does not satisfy the GC fluctuation theorem. However, is the same true of the joined currents? Here, a positive answer is presented, at least for the driven quantum systems described by Floquet quantum master equations Grifoni and Hänggi (1998); Kohn (2001); Szczygielski et al. (2013); Gasparinetti et al. (2014); Cuetara et al. (2015).

The rest of this paper is organized as follows. In Sec. (II), we review the Floquet quantum master equation and its stochastic thermodynamics. In Sec. (III), we prove a fluctuation theorem. In Sec. (IV), a two-level quantum system is used to concretely verify this theorem. Section (V) concludes the paper.

Given the Hamiltonian of a quantum system driven by periodic external forces, denoted as $H(t)$, we have

where ${\cal T}$$=$$2\pi/\Omega$ is the periodicity and $\Omega$ is the driving frequency. According to the Floquet theorem Zeldovich (1967); Shirley (1965), this periodic Hamiltonian satisfies an eigenvalue equation:

where $\epsilon_{n}$ and $|u_{n}(t)\rangle$ ($n$$=$$1,\cdots,N$) are quasi-energies and Floquet bases, respectively, and we set $\hbar$$=$$1$. Note that the Floquet bases are orthonormal and periodic. In addition, we emphasize that these quasi-energies are restricted in a zone with a size of $\Omega$. The heat bath that interacts with the quantum system has an inverse temperature $\beta$. Under the weak system-bath coupling conditions and time scale separation assumptions, the evolution of the reduced density matrix of the quantum system $\rho(t)$ can be described by the Floquet quantum master equation Grifoni and Hänggi (1998); Breuer and Petruccione (1997); Alicki et al. (2006):

The $D$-term represents the dissipation induced by the interaction between the system and the heat bath and is

In the above equation, $\omega$ are Bohr frequencies and are equal to $\epsilon_{n}-\epsilon_{m}+q\Omega$, where $q$ are certain integer numbers. The numbers may be positive or negative but always appear in pairs. Additionally, in the same equation, $A(\omega,t)$ and $A^{\dagger}(\omega,t)$ are called the Lindblad operators and are related by

The interaction operator of the quantum system and the heat bath is given as $A$$\otimes$$B$, where $A$ and $B$ are the system and heat bath components, respectively. These Lindblad operators are obtained by performing a Fourier-like expansion of the interaction-picture operator of $A$ Breuer and Petruccione (1997):

where $\delta$ is the Kronecker symbol, and the time-independent coefficient $\langle\langle u_{m}|A|u_{n}\rangle\rangle_{q}$ is the $q$-th harmonic of the transition amplitude $\langle u_{m}(t)|A|u_{n}(t)\rangle$; that is,

The last ingredient of the Floquet quantum master equation is the assumption that the heat bath is always in a thermal state. Then, $r(\omega)$, the Fourier transformation of the correlation function of the operator $B$, satisfies the important Kubo-Martin-Schwinger (KMS) condition Breuer et al. (2000); Rivas and Huelga (2012):

Stochastic thermodynamics can be established for the Floquet quantum master equation Liu and Xi (2016); Liu (2018); Cuetara et al. (2015). Roughly, Eq. (4) is unraveled into the dynamics of individual quantum systems Carmichael (1993); Plenio and Knight (1998); Breuer and Petruccione (2002); Wiseman and Milburn (2010). The evolution of each system is composed of a continuous process alternating with discrete random jumps. Assume that these jumps occur at time points $t_{i}$ ($i$$=$$1,\cdots$). Each jump indicates that a quantum $\omega_{i}$ is released to the heat bath Breuer (2003); Roeck and Maes (2006); Crooks (2008); Horowitz (2012); Hekking and Pekola (2013); Liu and Xi (2016); Liu (2018). The subscript $i$ represents the time points of these energy exchanges. When the evolution of a quantum system ends at time $t$, a quantum jump trajectory is generated and is marked as $\{\overrightarrow{\omega}\}$$=$$\{\omega_{1},\cdots\}$. If the density matrixes of these individual quantum systems are $\widetilde{\rho}(\{\overrightarrow{\omega}\},t)$, their average weighted by the probabilities of all possible quantum jump trajectories is just the reduced density matrix $\rho(t)$ of Eq. (4). From a thermodynamic point of view, the quanta are the heat released to the heat bath. Hence, given a quantum jump trajectory $\overrightarrow{\omega}$, we define the total heat along it as

In the second equation, we specifically define the positive and negative heat, $Q_{+}\{\vec{\omega}\}$ and $Q_{-}\{\vec{\omega}\}$, respectively; clearly, they are simply equal to the sums of positive and negative Bohr frequencies.

Because the occurrences of quantum jump trajectories and time points of quantum jumps are random events, all three types of heat are stochastic quantities. Let the joint probability distribution of the positive and negative heat, $Q_{+}$ and $Q_{-}$, respectively, be $p(Q_{+},Q_{-})$. We can construct its histogram by directly simulating quantum jump trajectories Liu and Xi (2016); Liu (2018). Because we are interested in the statistics over long time limits, a more practical approach is to compute its moment generation function,

We introduce a characteristic operator $\hat{\rho}(t)$ in the second equation above and find that it satisfies an evolution equation:

where the super-operator

If $\chi_{\pm}$$=$$0$, Eq. (13) reduces to Eq. (4). This result comes from a simple extension of the previous equation (Eq. (19) in Ref. Liu and Xi (2016)), which concerned the moment-generating function of the total heat, and we can reobtain it by letting $\chi_{+}$$=$$\chi_{-}$ in Eq. (13). Because the derivation is the same, we do not repeat it here.

The abstract Eq. (12) is not the most convenient to use in analyses. According to Eq. (11), $\Phi(\chi_{+},\chi_{-})$ is equal to a sum of diagonal elements of $\hat{\rho}(t)$. Hence, we write the evolution equations for $P_{n}(t)$$=$$\langle u_{n}(t)|\hat{\rho}(t)|u_{n}(t)\rangle$ in the Floquet bases:

where the vectors ${\textbf{P}}(t)$$=$$(P_{1}(t),\cdots,P_{N}(t))^{T}$ and $T$ represents the transpose and the nondiagonal matrix elements

($m\neq n$), and the diagonal elements

Note that Eq. (7) reminds us that $\textbf{R}(\lambda_{+},\lambda_{-})$ is in fact a constant matrix. We can easily prove that this matrix possesses symmetry:

Using this property, we obtain a fluctuation theorem by simply following a standard procedure, e.g., that presented by Lebowitz and Spohn Lebowitz and Spohn (1999). First, because the transpose matrix has the same eigenvalues as the original matrix, Eq. (17) implies that the scaled cumulant generating function Touchette (2008), $\phi(\chi_{+},\chi_{-})$, or the maximal eigenvalue of the R-matrix has the same symmetry:

Given the large deviation function $I(j_{+},j_{-})$ of the distribution $p(j_{+},j_{-})$ for the positive heat current $j_{+}$$=$$Q_{+}/t$ and the negative heat current $j_{-}$$=$$Q_{-}/t$, and because the function is a Legendre transform of the scaled cumulant generating function, Eq. (18) immediately leads to

Then, the probability distribution for these two heat currents satisfies the fluctuation theorem

The conventional GC fluctuation theorem (1) can be easily derived from Eq. (20).

Fluctuation theorem (20) has a time-reversal explanation. Analogous to that of classical stochastic processes Lebowitz and Spohn (1999), the ratio of the probability distribution ${\cal P}\{\overrightarrow{\omega}\}$ of observing a quantum jump trajectory $\overrightarrow{\omega}$$=$$\{\omega_{1},\cdots,\omega_{M}\}$ to ${\cal P}\{\overleftarrow{\omega}\}$ of observing its reversed trajectory $\overleftarrow{\omega}$$=$$\{-\omega_{M},\cdots,-\omega_{1}\}$ approaches $\exp(\beta Q\{\overrightarrow{\omega}\})$ as the time interval $t$ increases Liu (2018). Note that not only is the time order of the quantum jumps reversed in the revered trajectory, but the signs of these Bohr frequencies are reversed, we obtain $Q_{\pm}\{\overleftarrow{\omega}\}$$=$$-Q_{\mp}\{\overrightarrow{\omega}\}$ and

Some unimportant constants are ignored here. This rough proof explains why the positions of $j_{+}$ and $j_{-}$ in the probability distribution on the right-hand side are exchanged and minus signs are added simultaneously; they are just the consequences of the time-reversal. Eqs. (19) and (20) are the central results of this paper. In the next section, we use a two-level quantum system to concretely show these results.

Consider the Hamiltonian of a two-level quantum system Breuer and Petruccione (1997); Szczygielski et al. (2013); Langemeyer and Holthaus (2014); Gasparinetti et al. (2014); Cuetara et al. (2015)

where $\omega_{0}$ is the transition frequency of the bare system, $\Omega_{R}$ is the Rabi frequency, and $\Omega$ is the frequency of the periodic external field. The Floquet bases and the quasi-energies of this system are

and $\epsilon_{\pm}$$=$$(\Omega\pm\Omega^{\prime})/2$, respectively, where $\Omega^{\prime}$$=$$\sqrt{\delta^{2}+\Omega_{R}^{2}}$ and the detuning parameter $\delta$$=$$\omega_{0}$$-$$\Omega$. Here, we additionally set $\Omega$$>$$\Omega^{\prime}$. Assume that the coupling between the two-level system and the heat bath is $\sigma_{x}$-coupling. There are six Lindblad operators, and the Bohr frequencies are $\pm\Omega$, $\pm(\Omega-\Omega^{\prime})$, and $\pm(\Omega+\Omega^{\prime})$. By performing some simple derivations, the R-matrix is obtained:

where we do not explicitly write $\chi_{\pm}$ on the left-hand side and the coefficients are

In the matrix, symmetry (17) is apparent. Because this is a simple $2$$\times$$2$ matrix, the analytical expression of the scaled cumulant generating function is

where

We clearly see that symmetry (17) is inherited in Eq. (28).

We compute the large deviation function $I(j_{+},j_{-})$ by numerically performing a Legendre transformation on Eq. (28) and depict it in Fig. (1). Eq. (19) can be easily verified (data not shown). On the other hand, it is interesting to approximately compute the large deviation function by simulating quantum jump trajectories with finite times; see the spheres in the same figure. We find that even if the simulation time is short, the simulated data roughly exhibit the profile of the function. Although longer simulation times lead to better results, the regimes of the sampled data decrease around the mean currents $(\overline{j}_{+},\overline{j}_{-})$. This observation reminds us that simply simulating quantum jump trajectories is not enough to solve a large deviation function.

In this paper, we present a detailed GC fluctuation theorem for the quantum systems described by the Floquet quantum master equations. Although this theorem is proved for systems that interact with one heat bath, its generalization to the case of multiple heat baths is straightforward. For instance, for the case of two heat baths, we have

where $j_{k}^{+}$ and $j_{k}^{-}$ are the positive and negative heat currents of the quantum system released to the heat bath with the inverse temperature $\beta_{k}$ ($k$$=$$1,2$). Finally, if a quantum system contacts two heat baths at two different temperatures and can be described by a Lindblad quantum master equation, the system is able to evolve into a nonequilibrium steady state without external driving fields. In such a situation, by carrying out the same argument presented here, we can prove that the fluctuation theorem (30) is still true.