Irreversible entropy production: From classical to quantum

Entropy production is a key quantity in any finite-time thermodynamic process. It is intimately tied with the fundamental laws of thermodynamics, embodying a tool to extend thermodynamic considerations all the way to nonequilibrium processes. It is also often used in attempts to provide the quantitative characterization of logical and thermodynamic irreversibility, stemming from processes in physics, chemistry, and biology. Notwithstanding its fundamental character, a unifying theory of entropy production valid for general processes, both classical and quantum, has not yet been formulated. Developments pivoting around the frameworks of stochastic thermodynamics, open quantum systems, and quantum information theory have led to substantial progress in such endeavors. This has culminated in the unlocking of a new generation of experiments able to address stochastic thermodynamic processes and the impact of entropy production on them. This review aims to provide a compendium on the current framework for the description, assessment, and manipulation of entropy production. Formal aspects of its formulation and the implications stemming from the potential quantum nature of a given process, including a detailed survey of recent experiments, are both presented.

Figure 1

General schematics for the forward and backward trajectories for the fluctuation theorems. The solid red line shows the forward trajectory, while the dashed blue curve shows the backward one.

Figure 2

Typical nonequilibrium lag scenario. A system initially prepared in a thermal state ${\rho }_i^{\mathrm{th}}$ is driven unitarily by a work protocol to a nonequilibrium state ${\rho }^{\prime }$. After the protocol, the system is allowed to thermalize from ${\rho }^{\prime }$ to ${\rho }_f$.

Figure 3

(a) Diagramatic illustration of the collisional model in Eq. (107). (b) Scheme for studying NESSs obtained when a system is coupled to multiple reservoirs. In this case the ancillas cycle through an alphabet of states ${\rho }_A$, ${\rho }_B$, ${\rho }_C$, ${\rho }_A$, ${\rho }_B$, $\dots$, so the system never reaches equilibrium, even when ${\mathcal{U}}_n=1$ (no unitary strokes).

Figure 4

Thermomajorization condition determining when two states are interconvertible using thermal operations. (a) Procedure for constructing the thermomajorization curve is given in the main text and summarized in Eq. (195). (b) This curve is then used to compare different states. The state ${\rho }_1$ thermomajorizes ${\rho }_2$, ${\rho }_3$, and ${\rho }_{\beta }$. Conversely, ${\rho }_2$ and ${\rho }_3$ do not thermomajorize each other. Adapted from [108].

Figure 5

Illustration of the cyclic protocol based on catalytic thermal operations used by [24] to generalize the second law of thermodynamics to the microscopic quantum domain. (a) System $S$, catalyst $C$, and reservoir $E$, each endowed with their respective Hamiltonians [$H_j(j=S,C,E)$], are prepared in an uncorrelated state. The initial state of the reservoir consists of the tensor product of $n$ copies of the same thermal state ${\rho }_{\mathrm{th},E}$. (b) Parties involved in the protocol evolve jointly via the catalytic thermal operation $U$ such that $[U,{\sum }_j{H}_j]=0$. (c) The process is such that, at the end of the evolution, the catalyst is brought back to its initial state, while the system $S$ ends up in a state ${\rho }_S^{\prime }$.

Figure 6

(a) The two strokes of a swap engine. (b) Average thermodynamic quantities and entropy production as a function of ${\epsilon }_b/{\epsilon }_a$ for $T_b/T_a=1/2$. All energetic quantities are plotted in units of ${\epsilon }_a=1$. (c) Operation regimes of the swap engine.

Figure 7

Circuit diagram of a four-stroke heat engine composed of two unitaries $V_1$ and $V_3$ interlaced with two dissipative interactions with a hot and a cold bath, respectively.

Figure 8

Examples of dissipative transition. (a) Average photon number for the Kerr bistability model (292), with $\kappa =1/2$, $\mathrm{\Delta }=-2$, and $U=1/N$, where $N$ is a parameter used to tune the thermodynamic limit. The various curves correspond to $N=1$, 5, and 20 (long-dashed green curve, short-dashed red curve, and solid black curve, respectively) and were computed by numerically finding the steady state of Eq. (292). We also show as a dashed light gray curve the semiclassical result expected for optical bistability, showing that there is a region where there are two possible solutions. (b) Steady-state magnetization of the macrospin model (293) computed for $S=10$, 50, and 100 (dashed, dotted, and solid curves, respectively).

Figure 9

Example of behavior of $\dot{\mathrm{\Sigma }}$ across a continuous dissipative transition. (a) Mean-field majority vote model given by [177], where $f$ is the so-called misalignment parameter. (b) Two-bath Ising model of [252], where the transition is driven by the temperature $T$. In both cases $\dot{\mathrm{\Sigma }}$ is continuous across the transition but has a kink at the critical point, implying that the derivative of $\dot{\mathrm{\Sigma }}$ is discontinuous. It is also possible, as shown in (b), that the derivative presents a logarithmic discontinuity.

Figure 10

Similar to Fig. 9 but exemplifying $\dot{\mathrm{\Sigma }}$ across discontinuous transitions. (a) Majority vote model. This is the same model used in Fig. 9, which can be tuned from a continuous to a discontinuous transition depending on the parameter range. From [177]. (b) Ising model subject to an oscillating field. Adapted from [264].

Figure 11

The two contributions in Eq. (294) to the entropy production for the Kerr bistability model (292). The points correspond to different values of $N$ and have been plotted so as to yield a data collapse; see the text for more details.

Figure 12

The two contributions in Eq. (294) to the entropy production for the driven-dissipative Dicke model in Eq. (295). The critical point occurs at ${\lambda }_c=\sqrt{{\omega }_0({\kappa }^2+{\omega }^2)/\omega }$.

Figure 13

(a) [(b)] Evolution of the Bloch vector of the forward (backward) spin-$1/2$ state ${\rho }_t^F$ (${\rho }_{\tau -t}^B$) during a quench of the transverse magnetic field in the experiment reported on by [13] that was obtained via quantum state tomography. A sampling of 21 intermediate steps was used. The initial magnetization (gray arrow) is parallel to the external driven rf field, aligned along the positive $x$ ($y$) axis for the forward (backward) process. The final state is represented as a red (blue) arrow pointing nearly to the north pole of the sphere. (c) Polar projection (indicating only the magnetization direction) of the Bloch sphere with the trajectories of the spin. Green lines represent the path followed in a quasistatic ($\tau \to \infty$) process. From [13].

Figure 14

Distribution of irreversible entropy production. Black dots represent the measured negative and positive values of the entropy production $\mathrm{\Sigma }$ of the spin-$1/2$ system after a quench of the transverse magnetic field of duration $\tau =100\hspace{0.17em}\hspace{0.17em}\mu \mathrm{s}$. The mean entropy production (dashed red arrow) is positive, which is in agreement with the second law. From [13].

Figure 15

Scheme of principle of the experiment reported on by [103]. A two-level system whose states $\{|\uparrow ⟩,|\downarrow ⟩\}$ differ in frequency by ${\omega }_q$ is coupled through the dispersive Hamiltonian $H_{\mathrm{int}}$ to a harmonic oscillator of frequency ${\omega }_c\ne {\omega }_q$. The interaction term correlates the states of the two-level system to a quadrature of the oscillator. The state of the two-level system is measured by probing the oscillator resonance with a microwave pulse that acquires a phase shift that depends on whether the two-level system is prepared in $|\uparrow ⟩$ or $|\downarrow ⟩$.

Figure 16

Scheme of the 1-bit erasure process reported on by [20]. An overamped colloidal particle (a silica bead $2\mu \mathrm{m}$ in diameter) is trapped at the focus of a laser beam by an optical tweezer. The laser is focused at two distinct but closely spaced places, alternately and at a high switching rate. This provides the effective double-well potential into which the bead moves. Initially, due to thermal fluctuations the bead is equally likely to be in either one of the two wells. The erasure process always takes the particle to the rightmost well, which corresponds to the logical state 1 of a classical bit. The initial entropy of the system is thus $S_i=\ln 2$. The erasure process where the particle is moved from the left to the right well is displayed. (a) The barrier is initially high , (b),(c) then lowered and tilted to push the particle to the right well, thus switching the bit to the logical 1 state, which erases the memory. Raising the barrier again completes the erasure process. The particle is now in the right well with certainty, so the initial side that it was on has been irreversibly erased. As the process occurs in a finite time, it is stochastic in nature and the heat dissipated along a given trajectory $x(t)$ [with $x(t)$ the instantaneous position of the particle in the potential] is given by $Q=-{\int }_0^{{\tau }_{\text{cycle}}}dt\dot{x}(t)\partial U(x,t)/\partial x$, where $U(x,t)$ is the analytical form of the trapping potential and ${\tau }_{\text{cycle}}$ is the time taken to close an erasure cycle. The average dissipated heat is obtained by averaging $Q$ over 600 cycles, with each started by randomly choosing the initial configuration.

Figure 17

Scheme of the experimental assessment of Landauer principle reported on by [123]. (a) A 200 nm fluorescent particle moves in an aqueous solution (blue-shaded area) while (b) being monitored by a camera. A computer (c) reconstructs the position $\overline{x}$ from the images and (d) generates a feedback electric force $F(\overline{x})$, applied via two electrodes. The force is chosen so as to create a virtual potential $V(\overline{x})$ rather than an actual one, as in Fig. 16, imposed by a computer algorithm and calculated at the estimated position $\overline{x}$ rather than $x$ itself. This is not a limiting feature of this implementation since, for feedback updates that are fast enough, the dynamics in such virtual potentials is known to converge asymptotically to the corresponding actual one ([122]). In the experiment reported on by [123], the virtual potential and the erasure process are both along the lines of Fig. 16; see [63].

Figure 18

(a) Quantum circuit for the reconstruction of the heat probability distribution in Eq. (311); see also Eq. (82). Here $h=({\sigma }_z+{\sigma }_x)/\sqrt{2}$ is the Hadamard gate on the ancilla $A$, while $v_t=\exp [i{H}_{E}t]$ is the free evolution of the environment $E$. Finally, $U$ embodies the $S+E$ unitary governing the heat-dissipation process. (b) 3D chemical structure of the trifluoroiodoethylene molecule (${\mathrm{C}}_2{\mathrm{F}}_3\mathrm{I}$) accommodating the nuclear spins encoding the ancilla, system, and environment. Suitably arranged rf pulses are employed to prepare and manipulate the state of such a tripartite system.

Figure 19

Schematic diagram of the system used for the verification of the information-theoretic contributions to quantum erasure. A ${{}^{40}\mathrm{Ca}}^{+}$ ion is confined in a linear Paul trap that provides an axial $z$ pseudopotential of frequency ${\omega }_z$. The ion is subjected to a magentic field that Zeeman splits the $4{{}^{2}S}_{1/2}$ and $3{{}^{2}D}_{5/2}$ atomic states into manifolds of hyperfine levels. Among them, the $|4{{}^{2}S}_{1/2},1/2⟩$ and $|3{{}^{2}D}_{5/2},3/2⟩$ ones are chosen to encode the logical $|\downarrow {⟩}_S$ and $|\uparrow {⟩}_S$ pseudospin states of a two-level system embodying $S$ in the erasure protocol. The vibrational $z$ mode of the ion is used to encode the environment $E$, which is thus an infinite-dimensional system. From [263].

Figure 20

(a) Optomechanical setup. A micromechanical oscillator ($\delta {\widehat{q}}_b$) is coupled to the field mode of an optical Fabry-Perot cavity ($\delta {\widehat{q}}_a$). For this setup only the cavity is pumped. (b) Cavity-BEC setup. The external degree of freedom of a BEC ($\delta {\widehat{q}}_b$) is coupled to the field mode of a cavity ($\delta {\widehat{q}}_a$). For this setup only the atoms are pumped. Red and blue jagged lines indicate heating or cooling of the subsystems via coupling to the baths. In both setups the number of excitations in the optical bath is 0, i.e., $n_{T_a}=0$. (c) Both systems can be modeled as two quantum harmonic oscillators at frequencies ${\omega }_a$ and ${\omega }_b$, linearly coupled with a strength $g_{ab}$. Each oscillator is coupled to independent local baths at temperatures $T_a$ and $T_b$, respectively. The corresponding coupling rates are ${\kappa }_a$ and ${\gamma }_b$. The oscillators can be pumped by an external field (purple and orange arrows). From [33].

Figure 21

Experimental assessment of the irreversible entropy production rate ${\mathrm{\Pi }}_s$ at the nonequilibrium steady state for (a) the cavity optomechanical system and (b) the cavity-BEC system. For the cavity-BEC setup, $g_{ab}^{\mathrm{cr}}=\sqrt{({\kappa }_a^2+{\omega }_a^2){\omega }_b/4{\omega }_a}$ is the critical parameter of the model. Insets: behavior of ${\mu }_b$. The solid black lines show the theoretical predictions based on parameters extracted from experiment ([33]). The blue and red dots show the experimental data for the optomechanical and cavity-BEC experiments, respectively. In (a), the vertical error bars represent statistical errors extracted from the fit, while the horizontal ones show the experimental error on the values of the parameter. In (b), the vertical and horizontal error bars represent the statistical errors from the fit and the determination of the critical point, respectively ([134]). From [33].