Quantum Fluctuation Theorems for Arbitrary Environments: Adiabatic and Nonadiabatic Entropy Production

We analyze the production of entropy along nonequilibrium processes in quantum systems coupled to generic environments. First, we show that the entropy production due to final measurements and the loss of correlations obeys a fluctuation theorem in detailed and integral forms. Second, we discuss the decomposition of the entropy production into two positive contributions, adiabatic and nonadiabatic, based on the existence of invariant states of the local dynamics. Fluctuation theorems for both contributions hold only for evolutions verifying a specific condition of quantum origin. We illustrate our results with three relevant examples of quantum thermodynamic processes far from equilibrium.

Popular Summary

Thermodynamics can be extended far beyond the large macroscopic systems of our daily lives. However, stochastic fluctuations in thermodynamic quantities—like work, heat, and entropy—become essential to the thermodynamics of small systems. Nevertheless, these fluctuations turn out to be more than random noise: Their statistics are imprinted by the time-reversal symmetry of the underlying dynamics, satisfying a class of universal relations called “fluctuation theorems.”

Fluctuation theorems have been developed for small quantum systems, mostly focusing on systems coupled with ideal thermal equilibrium environments. Recent research has focused on lifting this assumption of equilibrium because of the effective finite size of realistic thermal baths and the possibility of engineering nonthermal reservoirs with quantum properties. These generalized environments provide new sources of free energy and may pave the way for heat engines surpassing traditional limits on performance. In this study, we demonstrate the existence of fluctuation theorems for a variety of classical and quantum environments, and we show that, in some situations, the total entropy can be decomposed into two different contributions: adiabatic and nonadiabatic entropy production, accounting for different sources of irreversibility.

Our results may have profound implications for studies of small quantum devices performing thermodynamic tasks, such as extracting work or refrigerating, and may help to clarify the role of “quantumness” in thermodynamics.

Figure 1

Schematic picture of the forward process presented in the main text. The system and environment start from an uncorrelated state ${\rho }_S\otimes {\rho }_E$. A local measurement of observables with projectors $\{{\mathcal{P}}_n,{\mathcal{Q}}_{\nu }\}$ is carried out, which does not alter the density matrix in the average evolution but selects a pure state $|{\psi }_n⟩\otimes |{\varphi }_{\nu }⟩$ at the trajectory level. The system and environment then interact with each other according to the unitary evolution $U_{\mathrm{\Lambda }}$, ending in an entangled state ${\rho }_{SE}^{\prime }$. Finally, we measure again, now using projectors $\{{\mathcal{P}}_m^{*},{\mathcal{Q}}_{\mu }^{*}.\}$. In the last measurement, quantum correlations in state ${\rho }_{SE}^{\prime }$ are erased, while the final state ${\rho }_{SE}^{*}$ may still have, in general, nonzero classical correlations. The reduced evolution of the system conditioned to the measurement in the environment is described through the quantum operation ${\mathcal{E}}_{\mu \nu }$ (shaded area).

Figure 2

(a) Schematic diagram of a trajectory generated by the map concatenation. Projective measurements on the system are only performed at the beginning and at the end of the concatenation. (b) Any operation ${\mathcal{E}}_{{\mu }_l,{\nu }_l}^{(l)}$ in the concatenation consists in the interaction of the system with an environmental ancilla in the state ${\rho }_E^{(l)}$ via the unitary ${\widehat{U}}_{\mathrm{\Lambda }}^{(l)}$ depending on the protocol ${\mathrm{\Lambda }}_l$. The ancilla is measured before and after interaction, generating outcomes ${\nu }_l$ and ${\mu }_l$, respectively.

Figure 3

Probability distribution $P({\mathrm{\Delta }}_{\text{i}}{s}_{\gamma })$ of the entropy production per trajectory for noninclusive (blue solid line) and inclusive (purple dashed line) cases. Initial states of the system and environment correspond to parameters $\alpha =0.8$ and $\beta \epsilon =2.5$. Inset: Plot of the different versions of the average entropy production as a function of $\beta$ ($\epsilon =1$ and $\alpha =0.8$).

Figure 4

Schematic diagram of a three-level thermal machine acting as a refrigerator. The three transitions of the machine are weakly coupled to thermal reservoirs at temperatures ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$, inducing jumps between the machine energy levels (double arrows). In a refrigeration cycle, the machine performs a sequence of three jumps $|g⟩\to |e_A⟩\to |e_B⟩\to |g⟩$, where it absorbs a quantum of energy $\hslash {\omega }_1$ from the cold reservoir, together with a quantum $\hslash {\omega }_2$ from the hot one, while emitting a quantum $\hslash {\omega }_3$ into the reservoir at intermediate temperature.

Figure 5

Comparison between the inverse effective (or virtual) temperatures ${\beta }_r^{\prime }$ (solid lines) and the real inverse temperatures of the reservoirs ${\beta }_r$ (dashed lines) for $r=1$, 2, 3 (blue, red, orange), as a function of ${\beta }_1$ when $\hslash {\omega }_1=1$ and $\hslash {\omega }_2=1.5$. In the refrigerator regime, the transition $g\leftrightarrow e_A$ is, at an effective temperature, colder than the coldest reservoir, ${\beta }_1^{\prime }\ge {\beta }_1$, inducing heat extraction from it, while the other transitions induce dissipation of heat to the reservoir at intermediate temperature, ${\beta }_2\ge {\beta }_2^{\prime }$, and absorption of heat in the hotter one, ${\beta }_2^{\prime }\ge {\beta }_2$. In the heat pump regime, the three heat flows change directions as the previous inequalities become inverted.

Figure 6

Schematic picture of the setup. The intracavity mode $H_0$ is externally driven by a resonant laser field $V_S(t)$ while in weak contact with the environment at inverse temperature $\beta$, producing the emission and absorption of photons.

Figure 7

Time evolution of (a) adiabatic (${\dot{S}}_{\text{a}}$), nonadiabatic (${\dot{S}}_{\mathrm{na}}$), and total (${\dot{S}}_{\text{i}}$) entropy production rates represented by solid lines, and (b) input power ($\dot{W}$), rate at which the cavity mode absorbs energy ($\dot{U}$), and displacement rate (${\dot{X}}_{\phi }$). The cavity mode starts in equilibrium with the thermal reservoir, ${\rho }_0=e^{-\beta {\widehat{H}}_0}Z$, and the laser driving is switched on at $t=0$. The dashed line in (a) corresponds to $\beta {\dot{W}}_{\mathrm{ss}}$, and we use vertical dotted lines to highlight the instant of time at which the adiabatic entropy production rate changes its sign ($t_n$). We use parameters $\epsilon =0.02\omega$, ${\gamma }_0=0.01\omega$, and temperature $kT=10\hslash \omega$, for $\hslash \omega =1$ units.