Fully Quantum Fluctuation Theorems

Systems that are driven out of thermal equilibrium typically dissipate random quantities of energy on microscopic scales. Crooks fluctuation theorem relates the distribution of these random work costs to the corresponding distribution for the reverse process. By an analysis that explicitly incorporates the energy reservoir that donates the energy and the control system that implements the dynamic, we obtain a quantum generalization of Crooks theorem that not only includes the energy changes in the reservoir but also the full description of its evolution, including coherences. Moreover, this approach opens up the possibility for generalizations of the concept of fluctuation relations. Here, we introduce “conditional” fluctuation relations that are applicable to nonequilibrium systems, as well as approximate fluctuation relations that allow for the analysis of autonomous evolution generated by global time-independent Hamiltonians. We furthermore extend these notions to Markovian master equations, implicitly modeling the influence of the heat bath.

Popular Summary

What we perceive as temperature, heat, and friction is the effect of microscopic random motions. This thermal randomness influences the dynamics inside living cells, and with progress in technological miniaturization, it is increasingly relevant for understanding the behavior of microscopic devices. Thermal fluctuations also pose intriguing theoretical challenges since they temporarily may decrease the disorder, or “entropy,” of the Universe, seemingly contradicting the second law of thermodynamics. This has led to refined statistical formulations of the second law, often referred to as “fluctuation theorems,” which characterize the nature of the thermal randomness. Miniaturization is currently moving further towards the level of single molecules and atoms, which forces us to take into account the notoriously counterintuitive quantum regime. Here, we show how to obtain fluctuation theorems that combine both thermal and quantum phenomena.

The starting point is Crooks fluctuation theorem, which characterizes the randomness of the work needed for driving thermodynamic processes. A key observation in our investigation is that the energy source that delivers the work can also be used to probe the process. By analyzing the quantum evolution of the energy source, we can formulate generalizations of Crooks theorem that take into account the full quantum dynamics and can be used to describe general nonequilibrium states, coherences, and entanglement.

Apart from allowing for systematic studies of the interplay between thermal and quantum effects, this result also bridges the notion of fluctuation theorems with recent approaches to the foundations of quantum thermodynamics inspired by quantum information theory.

Figure 1

The systems. The model explicitly includes the “system” $S$ on which we operate, the heat bath $B$, the control $C$ that implements the change of the Hamiltonian on $SB$, and the energy reservoir $E$ that donates or accepts the energy required to drive the processes. Most of our fluctuation relations are expressed in terms of the dynamics induced on the energy reservoir $E$. For the main part of this investigation (Secs. 2 and 3), we assume that the total system is described by a time-independent Hamiltonian that is noninteracting between $SBC$ and $E$, i.e., is of the form $H=H_{SBC}\otimes {\widehat{1}}_E+{\widehat{1}}_{SBC}\otimes H_E$. We moreover model the evolution by energy-conserving unitary operators $V$, i.e., $[H,V]=0$. (We go beyond these assumptions when we consider approximate fluctuation relations in Sec. 6.) For most of the derivations, it is not necessary to make any distinction between $S$ and $B$, and for this reason, we often bundle them together into an extended system $S^{\prime }=SB$. This “rationalization” can be taken one step further to $\tilde{S}=S^{\prime }C=SBC$ when we turn to the conditional fluctuation theorems in Sec. 3.

Figure 2

An intermediate quantum Crooks relation. Our quantum Crooks theorem is based on the idealized concept of a perfect control mechanism. This assumes that the energy-conserving unitary evolution $V$ on the joint system $S^{\prime }CE$ turns the control state $|c_i⟩$ into $|c_f⟩$ with certainty, thus implementing the change of the Hamiltonian $H_{S^{\prime }}^i$ to $H_{S^{\prime }}^f$ perfectly. To model the forward process of the Crooks relation, we assume that $S^{\prime }C$ starts in the “conditional” equilibrium state $G(H_{S^{\prime }}^i)\otimes |c_i⟩⟨c_i|$. The subsequent evolution under $V$ induces a channel $\mathcal{F}$ on the energy reservoir $E$. As an intermediate step towards the quantum Crooks relation (2), we assume that the reverse process is given by the globally reversed evolution $V^{†}$. With $S^{\prime }C$ in the initial state $G(H_{S^{\prime }}^f)\otimes |c_f⟩⟨c_f|$, this results in a channel $\mathcal{R}$ on $E$. The preliminary Crooks relation in Eq. (6) relates the channels $\mathcal{F}$ and $\mathcal{R}$. This result does not rely on any additional assumptions on how $V$ comes about or on the nature of the dynamics at intermediate times. However, if one so wishes, it is possible to explicitly model the path $H(x)$ in the space of Hamiltonians. This path can be discretized as $H_l$ for $l=0,\dots ,L$ with $H_0=H_{S^{\prime }}^i$ and $H_L=H_{S^{\prime }}^f$. We associate states $|c_l⟩$ in the control $C$ to these Hamiltonians and construct the joint Hamiltonian $H_{S^{\prime }C}=\sum _{l=0}^L{H}_l\otimes |c_l⟩⟨c_l|$. Hence, if the evolution traverses the family of control states, then we progress along the path of Hamiltonians. An alternative method to model evolving Hamiltonians is discussed in Sec. 6.

Figure 3

A quantum Crooks relation. Derivations of fluctuation relations often rely on time reversals and time-reversal symmetry in some form [3, 4, 5, 6, 8, 9, 10]. In our case, time-reversal symmetry enables us to, in effect, let the evolution run “backwards” by swapping the control states from a “forward track” to a “reverse track.” By this method, we amend for the overambitious global reversal of the evolution that we employed for the intermediate fluctuation relation and obtain a quantum counterpart to the reversal of the control parameter in the classical Crooks relation. Analogous to the setup in Fig. 2, the initial control state $|c_{i+}⟩$ yields an evolution where the initial Hamiltonian $H_{S^{\prime }}^i$ is transformed to the final Hamiltonian $H_{S^{\prime }}^f$ when the control state $|c_{f+}⟩$ is reached. This process induces a channel ${\mathcal{F}}_{+}$ on the energy reservoir $E$. For each of these control states, there exists a time-reversed partner starting with $|c_{f-}⟩$ and ending with $|c_{i-}⟩$, thus bringing the Hamiltonian $H_{S^{\prime }}^f$ back to $H_{S^{\prime }}^i$. This process induces a channel ${\mathcal{F}}_{-}$ on the reservoir. By assuming time-reversal symmetry, the channels ${\mathcal{F}}_{+}$ and ${\mathcal{F}}_{-}$ can be related via the quantum Crooks relation in Eq. (2).

Figure 4

Diagonal and off-diagonal Crooks relations. Because of energy conservation in combination with the particular class of initial states, it turns out that the dynamics of the energy reservoir decouples along the modes of coherence [35]. One can think of these as the collection of diagonals (main and off diagonals) of the density operator represented in an energy eigenbasis. Because of the decoupling, it follows that the quantum fluctuation relation (2) can be separated into individual Crooks relations for each mode of coherence. As an example, suppose that the spectrum of the reservoir includes the energy levels $E_1=0$, $E_2=s$, $E_3=3s$, $E_4=4s$ for some $s>0$. The dynamics on the main diagonal, $\delta =0$, satisfies the relation in Eq. (11). Each of the diagonals with offsets $\delta =\pm s$, $\pm 2s$, $\pm 3s$, $\pm 4s$ satisfies a Crooks relation as in Eq. (12). For a more concrete example of the decoupling, see Fig. 9.

Figure 5

Conditional fluctuation theorems. Similar to our previous fluctuation theorems, here we relate the dynamics induced on the energy reservoir $E$ by the forward and reverse processes but conditioned on successful control measurements on $\tilde{S}=SBC$. The forward process is characterized by a pair of positive semidefinite operators $Q_{\tilde{S}}^{i+},Q_{\tilde{S}}^{f+}$ on $\tilde{S}$, such that the initial state is given via the Gibbs map ${\mathcal{G}}_{\beta H_{\tilde{S}}}(Q_{\tilde{S}}^{i+})$, and the control measurement is represented by $Q_{\tilde{S}}^{f+}$. More precisely, $Q_{\tilde{S}}^{f+}$ corresponds to the “successful” outcome in the POVM $\{Q_{\tilde{S}}^{f+},{\widehat{1}}_{\tilde{S}}-Q_{\tilde{S}}^{f+}\}$ and results in the induced CPM ${\tilde{\mathcal{F}}}_{+}$ on the reservoir. One should keep in mind that the mapping $Q_{\tilde{S}}^{i+}\mapsto {\mathcal{G}}_{\beta H_{\tilde{S}}}(Q_{\tilde{S}}^{i+})$ only serves as a convenient description of the initial state and should not be interpreted as representing a particular physical process or measurement. The reverse process, resulting in the CPM ${\tilde{\mathcal{F}}}_{-}$, is correspondingly characterized by the pair $Q_{\tilde{S}}^{f-}={\mathcal{T}}_{\tilde{S}}(Q_{\tilde{S}}^{f+})$, $Q_{\tilde{S}}^{i-}={\mathcal{T}}_{\tilde{S}}(Q_{\tilde{S}}^{i+})$, where ${\mathcal{G}}_{\beta H_{\tilde{S}}}(Q_{\tilde{S}}^{f-})$ is the initial state, and $Q_{\tilde{S}}^{i-}$ gives the control measurement. The CPMs ${\tilde{\mathcal{F}}}_{\pm }$ are related via the conditional fluctuation theorem in Eq. (26). We are free to choose the operators $Q_{\tilde{S}}^{i+},Q_{\tilde{S}}^{f+}$ in any way we wish, and in the finite-dimensional case, we can obtain arbitrary initial states. Hence, we are not restricted to initial equilibrium states. The price that we pay for this additional freedom is that nonequilibrium initial states are translated to nontrivial control measurements in the reverse process.

Figure 6

A control particle. One way to “quantize” the control parameter $x$ of the classical Crooks relation would be to regard it as the position of a quantum particle. The propagation of the particle would approximately implement the time-dependent Hamiltonian for suitable wave packets. A particularly clean example is obtained if $E$ and $S^{\prime }$ only interact when the control particle is in an “interaction region” $[x_i,x_f]$ (cf. the construction in Ref. [116]). Since the global Hamiltonian contains interactions between $E$ and $S^{\prime }$, it does not fit with our previous classes of models, and in particular, it does not satisfy the factorization property (30). However, for suitable choices of measurement operators $Q_C^{i+}$ and $Q_C^{f+}$ that are localized outside the interaction region, the systems becomes approximately noninteracting, and the factorization property (30) is approximately satisfied. This makes it possible to derive an approximate conditional fluctuation relation on the energy reservoir.

Figure 7

A control particle that also serves as an energy reservoir. An alternative to the setup in Fig. 6 is a particle that simultaneously serves as both a control system and an energy reservoir. Hence, it is the kinetic energy of the control particle that drives the nonequilibrium process (if the other systems start in equilibrium). This is another example of a system that generally would not satisfy the factorization property (30), but for measurement operators $Q_{CE}^{i+}$ and $Q_{CE}^{f+}$ that are localized outside the interaction region, the systems become approximately noninteracting. For this setup, one can obtain an approximate fluctuation relation in terms of the transition probabilities introduced in Sec. 4.

Figure 8

Free-energy difference. The goal is to determine the free-energy difference between two basins $D_0$ and $D_1$ of states, embedded in a larger collection of configurations $D_2$. Within each such collection, there is a fast thermalization process, while there is a slow global thermalization. An external energy reservoir $E$, in the form of a single two-level system, is in resonance with the transition between the global ground state $|0⟩$ in basin $D_0$ and the local ground state $|n^{*}⟩$ in basin $D_1$. The energy reservoir acts as the counterpart to the external control in the classical EFRs. The transition probability ${\mathcal{P}}^{+}$ (red solid curve) of the forward process and the transition probability ${\mathcal{P}}^{-}$ (blue solid curve) of the reverse process, as defined by Eq. (48), are plotted as functions of time, here expressed in a unit-free manner via $t/(\beta \hslash )$. The forward process is defined by $Q_E^{i+}=|1⟩⟨1|$ and $Q_E^{f+}={\widehat{1}}_E$. The transition probabilities ${\mathcal{P}}^{+}$ and ${\mathcal{P}}^{-}$ can be estimated by repeated experiments, and they determine, via Eq. (47), the quotient $Z_1/Z_0$ of the partition functions $Z_0$ and $Z_1$ of $D_0$ and $D_1$, respectively, and thus also the desired free-energy difference. The dotted lines correspond to the transition probabilities in the limit of infinite evolution times, where the system has reached the fixpoint of the master equation.

Figure 9

Decoupling of the modes of coherence in a dissipative Jaynes-Cummings model. A harmonic oscillator, with energy eigenbasis $|n⟩$, serves as the energy reservoir in interaction with an open two-level system. As the initial state of the energy reservoir, we choose the (rather arbitrary) superposition of number states $|\psi ⟩=(|9⟩+|15⟩+|42⟩+|47⟩+|50⟩+|67⟩+|79⟩)/\sqrt{7}$. The red dots represent the nonzero matrix elements of the density operator $|\psi ⟩⟨\psi |$ with respect to the number basis. The conditional evolution ${\tilde{\mathcal{F}}}_{+}$ on the energy reservoir, defined in Eq. (42), is calculated for the evolution time $t/(\beta \hslash )=1.5$, and the dark colors correspond to the values $|⟨n|{\tilde{\mathcal{F}}}_{+}(|\psi ⟩⟨\psi |)|n^{\prime }⟩|$. The upper left corner, $n=n^{\prime }=0$, corresponds to the probability of finding the oscillator in the ground state, and the main diagonal $n=n^{\prime }$ is the probability for the excited states. The off-diagonal elements, $n\ne n^{\prime }$, represent the coherences between the energy eigenstates. One can recognize the leakage of energy out of the oscillator, as well as the decay of coherence. It is also clearly visible how each initial off-diagonal element only evolves and spreads along the particular diagonal that it belongs to, thus illustrating the decoupling of the modes of coherence.

Figure 10

Off-diagonal fluctuation relation. The fluctuation theorem (50) relates the evolution of the coherences carried by two off-diagonal elements along a displaced diagonal. The plot on the right displays $|q_{+}|$ (red curve) and $|q_{-}|$ (blue curve) plotted as functions of time. The plot on the left depicts the trajectories that $q_{+}$ (red curve) and $q_{-}$ (blue curve) sweep in the complex plane during the same time interval. As predicted by Eq. (50), $|q_{+}|$ and $|q_{-}|$ are proportional to each other, and the phase factors of $q_{+}$ and $q_{-}$ are identical. Analogous to how the classical Crooks relation relates the probability distribution of the work, and thus the change of energy in the reservoir, of the forward and reverse processes, the off-diagonal Crooks relation relates the changes of coherence.

Figure 11

Structure of the Hamiltonian. The Hamiltonian of the extended system $S^{\prime }=SB$ and the control has the form $H_{S^{\prime }C}=H_{S^{\prime }}^i\otimes |c_i⟩⟨c_i|+H_{S^{\prime }}^f\otimes |c_f⟩⟨c_f|+H^{\perp }$. Here, $H_{S^{\prime }}^i$ and $H_{S^{\prime }}^f$ are the initial and final Hamiltonians, respectively, and $|c_i⟩,|c_f⟩$ the corresponding orthonormal control states. If the control is in state $|c_i⟩$, the Hamiltonian of $S^{\prime }$ is $H_{S^{\prime }}^i$, while it is $H_{S^{\prime }}^f$ if the control is in state $|c_i⟩$. The Hamiltonian $H^{\perp }$, which has orthogonal support to $H_{S^{\prime }}^i\otimes |c_i⟩⟨c_i|$ and $H_{S^{\prime }}^f\otimes |c_f⟩⟨c_f|$, corresponds to possible intermediate stages. In the proofs of our fluctuation theorems, $H^{\perp }$ plays no particular role. However, it can be used to simulate a path of Hamiltonians $H_{S^{\prime }}(x)$, e.g., via a discretization [see Appendix pp7-s3b].

Figure 12

Structure of the Hamiltonian. The Hamiltonian $S^{\prime }C$ is of the form $H_{S^{\prime }C}=H_{S^{\prime }}^i\otimes P_C^i+H_{S^{\prime }}^f\otimes P_C^f+H^{\perp }$. Hence, whether the state of the control is in subspace ${\mathcal{H}}_C^i$ onto which $P_C^i$ projects or in ${\mathcal{H}}_C^f$ onto which $P_C^f$ projects, this determines the Hamiltonian of $S^{\prime }$. The Hamiltonian $H^{\perp }$ corresponds to any possible intermediate stages. The initial control space ${\mathcal{H}}_C^i$ is spanned by two states $|c_{i+}⟩$ and $|c_{i-}⟩$, which are the time reversals of each other. Analogously, the final control space ${\mathcal{H}}_C^f$ is spanned by $|c_{f+}⟩$ and $|c_{f-}⟩$. The global evolution $V$ is such that it brings control state $|c_{i+}⟩$ into $|c_{f+}⟩$, while it brings $|c_{f-}⟩$ into $|c_{i-}⟩$, thus implementing both the forward and the reverse process.

Figure 13

Approximate factorization. To assess the quality of the approximate factorization, we evaluate $d_{\widehat{K},E_0{\sigma }_z/2}^H({\widehat{1}}_{S^{\prime }}\otimes |\alpha ⟩⟨\alpha |)$ (red curve), $d_{\widehat{K},E_0{\sigma }_x/4}^H({\widehat{1}}_{S^{\prime }}\otimes |\alpha ⟩⟨\alpha |)$ (green curve), and $d_{\widehat{K},E_0\overline{n}(r{y}_0)·\overline{\sigma }/2}^H({\widehat{1}}_{S^{\prime }}\otimes |\alpha ⟩⟨\alpha |)$ (blue curve) for $\alpha ≔r+2i$, with $r\in [-10,10]$. The dotted black lines correspond to the borders of the interaction region. The red circles are the positions of the coherent states that give the measurement operators $Q_{CE}^{i+}=|-4+2i⟩⟨-4+2i|$ and $Q_{CE}^{f+}=|4+2i⟩⟨4+2i|$.

Figure 14

The forward and reverse processes. The density operators of the combined control and energy reservoir particle in the position representation (the absolute values of the matrix elements) at the end of the forward (left panel) and reverse (right panel) processes. The pairs of horizontal and vertical lines show the borders of the interaction region. The red circles indicate the positions of the initial states ${\mathcal{G}}_{\beta \widehat{K}}(Q_{CE}^{i+})$ and ${\mathcal{G}}_{\beta \widehat{K}}(Q_{CE}^{f+})$ for the initial and final measurement operators $Q_{CE}^{i+}=|-4+2i⟩⟨-4+2i|$, $Q_{CE}^{f+}=|4+2i⟩⟨4+2i|$. The error in the approximate fluctuation relation is small (approximately $1.6\times {10}^{-8}$), which corresponds to the fact that the measurement operators are well separated from the interaction region. One should compare this with the final wave packets, where one can clearly see that these have significant weights within the interaction region. This illustrates the fact that the properties of the measurement operators, rather than the final states, are important for the quality of the approximation. One can also note that the final states are not mirror images of each other. Hence, the symmetry discussed in Appendix pp9-s1 does not imply that the wave packets of the forward and reverse processes have to be symmetric images of each other.

Figure 15

Classes of channels. Schematic description of the relation between the time-reversal-symmetric thermal operations (TRSTO), the thermal operations (TO), the Gibbs preserving maps (GP), and the set of channels (F) that satisfy the fluctuation relation (l1), for a given $\beta$, $H$, and $\mathcal{T}$ such that $\mathcal{T}(H)=H$. We know from Sec. 7a that $\mathrm{F}\supseteq \mathrm{TRSTO}$, and it is clear from the definitions that $\mathrm{TO}\supseteq \mathrm{TRSTO}$. Moreover, $\mathrm{GP}\supseteq \mathrm{TO}$ [130]. In Lemma 20, it is shown that $\mathrm{GP}\supseteq \mathrm{F}$. For specific choices of $\beta$, $H$, and $\mathcal{T}$, we moreover find explicit examples of channels in the sets TO\F, F\TO, and $\mathrm{GP}\backslash (\mathrm{TO}\cup \mathrm{F})$; i.e., there are cases where these sets are nonempty. The drawing suggests that $(\mathrm{TO}\cap \mathrm{F})\backslash \mathrm{TRSTO}$ would be nonempty. However, it is not clear whether this is the case or not. In Appendix pp12-s1d, it is shown that TRSTO is convex. The set TO is also convex [35], and one can realize that GP and F are convex.

Figure 16

Approximate fluctuation relation with error bound. (a) $f(t)$ is plotted as the red solid line, and $r(t)$ the blue solid line, for the interval $t/(\beta \hslash )\in [0,50]$. In the ideal case, $f$ and $r$ would be identical. In the interval, the maximal error is ${\max }_{t/(\beta \hslash )\in [0,50]}|f(t)-r(t)|\approx 0.0795$. This should be compared with the upper bound in Proposition 12, which is approximately 0.806. The dotted lines correspond to the limits ${\lim }_{t\to +\infty }f(t)$ and ${\lim }_{t\to +\infty }r(t)$, where the system approaches the fixpoint of the master equation. (b) For otherwise fixed settings, the maximum error ${\max }_{t/(\beta \hslash )\in [0,50]}|f(t)-r(t)|$ is plotted (filled circles) against the value of $\lambda \beta$. The empty circles correspond to the error bound in Proposition 12.

Figure 17

Approximate fluctuation relation without bound. (a) Analogous to Fig. 16, the functions $f(t)$ (red solid line) and $r(t)$ (blue solid line) are as defined in Eq. (m13) but for the new generator (m14). These are plotted for the interval $t/(\beta \hslash )\in [0,50]$. (b) The maximum error ${\max }_{t/(\beta \hslash )\in [0,50]}|f(t)-r(t)|$ is plotted (filled circles) against the value of $\lambda \beta$.

Figure 18

Off-diagonal fluctuation relation in the case of no decoupling. The fluctuation theorem (m18) relates the evolution of the coherences carried by two off-diagonal elements along two different displaced diagonals. Hence, as opposed to the case in Fig. 10, the two off-diagonal elements belong to two different modes of coherence. For the setting in Fig. 10, the quantities $d_{\pm }$ defined in Eq. (m18) would be identically zero, while for the new measurement operators (m19), they become nontrivial and satisfy the fluctuation relation in Eq. (m18). The plot on the right displays $|d_{+}|$ (red curve) and $|d_{-}|$ (blue curve), while the left plot depicts the trajectories $d_{+}$ (red curve) and $d_{-}$ (blue curve) in the complex plane. Like in Fig. 10, the proportionality of the absolute values and identical phase factors predicted by Eq. (m18) are visible.