# Non-Markovian quantum thermodynamics: second law and fluctuation theorems

###### Abstract

This work brings together Keldysh non-equilibrium quantum theory and thermodynamics, by showing that a real-time diagrammatic technique is an equivalent of stochastic thermodynamics for non-Markovian quantum machines (heat engines, refrigerators, etc). Symmetries are found between quantum trajectories and their time-reverses on the Keldysh contour, for any interacting quantum system coupled to ideal reservoirs of electrons, phonons or photons. These lead to quantum fluctuation theorems the same as the well-known classical ones (Jarzynski and Crooks equalities, non-equilibrium partition identity, etc), whether the system’s dynamics are Markovian or not. Some of these are also shown to hold for non-factorized initial states. We identify a family of approximations, suitable for concrete calculations of a machine’s power and efficiency, which respect the symmetries that ensure fluctuation theorems. In all cases (exact and approximate) and all initial states, the second law of thermodynamics is proven to hold on average, with fluctuations violating it.

###### pacs:

73.63.-b, 05.30.-d, 05.70.Ln, 05.10.Gg, 72.15.Jf, 84.60.RbIntroduction. The laws of thermodynamics were derived for macroscopic machines, where entropy-reducing fluctuations (e.g. a gas spontaneously drifting into one corner of its container) are so rare that they have been called “thermodynamic miracles” Thermdynamic-Miracles . In microscopic systems on short timescales, these “miracles” are rather common, and we now know they obey fluctuation theorems Searles-Review-2008 ; Campisi-review2011 ; qu-thermo-review-Anders ; qu-thermo-review-Millen . Stochastic thermodynamics Seifert-PRL2005 ; Schmiedl-Seifert2007 ; Seifert-review2012 ; van-Broeck-review2015 ; Seifert-PRL2016 ; our-review-2016 is a unifying theory of such theorems in classical systems; it gives the Jarzynski Jarzynski and Crooks Crooks1999 ; Tasaki equalities in the relevant limits. It was used to show Seifert-PRL2005 that any classical system with Markovian dynamics obeys the non-equilibrium partition identity Yamada-Kawasaki1967 ; Morriss1985 ; Carberry2005 , (called the integral fluctuation theorem in Seifert-PRL2005 ; Schmiedl-Seifert2007 ; Seifert-review2012 ; van-Broeck-review2015 ),

(1) |

where the average is over all possible thermal fluctuations Footnote:units . Eq. (1) tells us that the second law of thermodynamics is obeyed on average, . Yet Eq. (1) also tells us that fluctuation with must occur (even if rarely), otherwise would be less than one.

This work shows that a Keldysh theory — the real-time diagrammatic technique Schoeller-Schon1994 ; Konig96 ; Konig97 ; Schoeller-review1997 — provides an equivalent of stochastic thermodynamics for any quantum system coupled to reservoirs (Fig. 1), whether that system’s dynamics are Markovian or not. It makes the connection between the contribution of a double-trajectory, , on the Keldysh contour and the contribution of its time-reverse, (Fig. 3a). This is enough to show that such systems respect the same fluctuation theorems as classical Markovian systems, and so obey the second law of thermodynamics on average. For the second law, our proof goes beyond those for Markovian quantum systems Kosloff-review2013 , those for systems with mean-field interactions Nenciu2007 ; 2012w-2ndlaw , and Keldysh treatments for non-interacting systems (quadratic Hamiltonians) Sanchez-2ndlaw2014 ; Esposito-2ndlaw ; Bruch-2016 or adiabatic driving Sanchez-2ndlaw-2016 . This connection between fluctuation theorems Campisi-review2011 ; qu-thermo-review-Anders ; qu-thermo-review-Millen and the Keldysh theory for transport through interacting systems Schoeller-Schon1994 ; Konig96 ; Konig97 ; Schoeller-review1997 ; Schoeller2009 ; Wegewijs2014 ; Sothmann2014 ; Schulenborg2016 , provides a powerful tool for modelling energy production and refrigeration at the nanoscale. In this context, significant currents and power outputs require significant system-reservoir coupling. However, only systems in the weak coupling limit have Markovian dynamics Davies74 ; Davies76 , thus non-Markovian systems are of great interest.

Previous proofs of fluctuation theorems in non-Markovian quantum systems exist Campisi-review2011 , but rely on treating the system and reservoirs together as a single isolated quantum system. This is elegant, but is not amenable to calculating a given machine’s power or efficiency, except in the rare cases where the full Hamiltonian (system plus reservoirs) is exactly soluble. It gives no indication of what approximations allow calculations of this power or efficiency, without an unphysical violation of fluctuation theorems and of the second law of thermodynamics. This work finds a microscopic symmetry which underlies the fluctuation theorems, beyond the Markovian quantum systems considered in Ref. Maxime+Alexia2016 . This enables one to identify a family of approximations that allow tractable calculations of machine power and efficiency, with no risk of violating the second law or fluctuation theorems.

Hamiltonian. This work considers a time-dependent system Hamiltonian, , including interaction effects. Each term in contains one creation operator for a system electronic state, , for every annihilation operator, . This system is coupled to multiple reservoirs of non-interacting fermions (electrons) via couplings , or non-interacting bosons (photons or phonons) via couplings . The total Hamiltonian is

(2) |

The sums are over electron (el) and photon/phonon (ph) reservoirs. For el reservoirs, , for reservoir ’s state with energy, creation and annihilation operators , and . The tunnel coupling, , where and contain only system operators, and may be time-dependent. The change in the system state when an electron is added from reservoir ’s state is given by . The reverse process is given by . The simplest case has , however if the coupling depends on the system state, then contains extra factors . For bosonic reservoirs, one replaces the fermionic operators and by bosonic ones. The simplest case has , meaning the system goes from to when a boson is absorbed from reservoir ’s state .

We write all system operators as matrices acting on the basis of many-body system states. We go to an interaction representation (indicated by caligraphic symbols), where system operators evolve under a matrix , with indicating time-ordering. Hence,

(3) |

Reservoir operators evolve under , so we have and .

The initial condition (at time ) is an arbitrary system state in a product state with the reservoirs. Each reservoir is in its local equilibrium with temperature and chemical potential ( for reservoirs of photons or phonons). We treat exactly, and keep the reservoir’s effect on the system finite, in the limit of vanishing reservoir level-spacing. This requires taking the system’s coupling to each reservoir mode to zero, as the density of such modes goes to infinity, so this coupling can be treated at lowest-order (second-order) Caldiera-Leggett-1983a ; Caldiera-Leggett-1983b ; Leggett-et-al ; Schoeller-review1997 . None the less, the system may interact with any number reservoir modes at one time (all orders of co-tunnelling events), and these interactions do not commute. Upon tracing out the reservoirs, the resulting system dynamics are highly non-Markovian. This dynamics is represented in terms of a Keldysh double-trajectory, as in Fig. 1b, where each second-order interaction with a given reservoir mode is represented by a pair of interactions joined by a dashed line.

Assumption of no Maxwell demons. This work assumes that the machine operates without knowing microscopic details of the reservoirs. In other words, Eq. (2) cannot be varied in response to the detection of the system state or of individual particles in a reservoir. Hence, work cannot be extracted from these details. This rules out ”Maxwell demon” physics, which occur when a detection of microscopic information is fed back on the system. Just as in classical mechanics, assuming no Maxwell demons is crucial in the emergence of the second law from the underlying theory. This assumption makes all classical correlations and quantum entanglement between system and reservoirs at the end of the evolution irrelevant, since the system cannot extract work from them. Hence, one can trace out the reservoirs when calculating system properties. Further, even though the system pushes certain reservoir modes out of equilibrium, it is assumed that this information is inaccessible, so no more work can be extracted from the reservoir than if it were in a thermal state with the same energy.

Trajectories. Consider a trajectory on the Keldysh contour, whose upper-line goes from system’s many-body state at time to at time , and lower line goes from to (see examples in Fig. 3a). Matrix elements for transitions are time ordered on the upper line and reverse-time ordered on the lower line. Each transition (each dashed-line in ) has a weight determined by whether it is , or in Fig. 2 (see below). Real transition correspond to in Fig. 2 and virtual transitions to and . The trajectory’s weight, , is the product of all of these factors of , multiplied by a factor of for each crossing of dashed-lines Schoeller-review1997 . The probability to go from one system state to another in time , is simply the sum of the weights of all trajectories between those states.

The dashed-lines have the following weights,

(4) | |||||

(5) | |||||

(6) |

where and . The factor is the number of particles in state of reservoir ; it is with for fermionic reservoirs and for bosonic reservoirs. Here Footnote:units ,

(7) |

which is the entropy change of reservoir when a particle is added to state . The weight of (for ) is given by the Hermitian conjugate of (so and ) with replaced by . Here is the number of ways one can add a particle to state of reservoir . For any reservoir (fermionic, bosonic or other) in internal equilibrium,

(8) |

which is know as local detailled balance or micro-reversibility. For fermion or boson distributions, this is guaranteed by the fact that with for fermions, and for bosons. Physically, removes a particle from the system and adds it to state of reservoir ; this adds a work of and heat of to reservoir . Thus, involves a change of reservoir ’s entropy of in Eq. (7). The reverse process, , removes such a particle from reservoir , changing the reservoir’s entropy by . Contributions and do not change the number of particles in the reservoirs, and so involve no reservoir entropy change.

Total Entropy. The assumption of no Maxwell demons implies that entanglement between system and reservoir cannot be used to produce work. Then the correct definition of the total entropy production, , is the sum of that for the system (sys) and reservoirs (res),

(9) |

with no term related to system-reservoir entanglement. The change in reservoir ’s entropy, , for a trajectory is taken to be the sum of the entropy changes associated with each of the transitions in .

For arbitrary system states, is taken to be the difference between the entropies of the system’s initial and final states. To get this entropy for the system’s initial density matrix (at time ), we write it as , so is the probability to find the system in state of its diagonal basis. A Boltzmann entropy weight of is associated with state , so the average over all gives the von Neumann entropy, . The final system state’s entropy (at time ) is calculated in the same way by rotating to the diagonal-basis of the final system density matrix, given by for unitary , and assigning an entropy of to the th state in this diagonal basis.

Time-reversed set-up. As in classical systems, one derives fluctuation theorems by comparing two different set-ups (A and B), where the Hamiltonian in set-up B is the time-reverse of the Hamiltonian in set-up A over the time-window from to . If A’s Hamiltonian (system+reservoir) is in Eq. (2), then B’s is , where is the time-reverse operator in Messiah’s texbook Messiah-chapt15 . For any trajectory, , on the Keldysh contour in set-up A, one can define a trajectory in set-up B which is the time-reverse of . More precisely, is defined by rotating by in the plane of the page, and replacing all states by their time reverse, see Fig. 3a. The time-reverse of state is .

To make the connection between the weight of and , we chose the basis of final system states such that , then a little algebra gives , Combining this with Eq. (8) one gets (see Supplementary Material),

(10a) | |||||

(10b) |

where . One then observes that if contains a -factor on the right hand side of one of the equality in Eq. (10), then then contains a the -factor on the left hand side of the same equality, and vice-versa. The weights of trajectory in set-up A and in set-up B are given by products of the factors of that form each of them, this results in the central observation of this work (shown graphically in Fig. 3b),

(11) |

where is the weight of double-trajectory in set-up A, and is that of in set-up B. The reservoir entropy change, , is the sum of the for all transitions in .

Fluctuation theorems. Now consider a double-trajectory, , which goes from the th state in the diagonal basis of to the th state in the diagonal basis of (see Fig. 1b). The subscript “d” is to indicate that it goes from diagonal basis to diagonal basis. Let us define its weight as , this equals multiplied by a factor of to transformation out of the diagonal basis at time , and a factor of to go to the diagonal basis at time . The unitarity of the transformations and means they do not change or , so one has . This relation is the same as in Markovian stochastic thermodynamics of classical systems, so from here on the proofs of fluctuation theorems follows exactly as in such systems Schmiedl-Seifert2007 , for reviews see Refs. Seifert-review2012 ; van-Broeck-review2015 ; our-review-2016 . For completeness, they are given in the Supplementary Material. In particular, any such system obeys the non-equilibrium partition identity, , which means it satisfies the second law of thermodynamics on average.

Non-factorized initial conditions. Eq. (1) also applies for the evolution from time to time , when the state at time is an arbitrary entangled state of system and reservoirs. This follows from Eq. (1) being obeyed by evolution from a time (in the distant past) to and by evolution from to , see the supplementary material. Similarly, a non-factorized steady-state obeys the Evans-Searles fluctuation relation Evans-Searles1994 , if the Hamiltonian in Eq. (2) is invariant under time-reversal.

Approximate theories. This work connects fluctuation theorems to a microscopic symmetry of the system-reservoirs interactions, going beyond Ref. Campisi-review2011 . This can be used to identify a family of approximations which are guaranteed to satisfy fluctuation theorems. These approximations must contains a trajectory for every trajectory , and individual transitions must satisfy local-detailed balance, thereby satisfying Eqs. (10). Then the above arguments apply, so Eq. (11) is recovered, which leads to all the usual fluctuation theorems, which means they will always obey the second law on average.

The first approximation is the Born approximation for weak system-reservoir coupling, also called Bloch-Redfield Bloch57 ; redfield or sequential tunnelling approximation Schoeller-review1997 , see also Refs. Nakajima58 ; Zwanzig60 ; Davies74 ; Davies76 or textbooks Bellac-qm-book ; Atom-Photon-Interactions-book ; Blum-book . This neglects trajectories where the system interacts with multiple reservoir modes at the same time, which is reasonable when the coupling is weak on the scale of the reservoir’s memory time. The approximation has a trajectory for every , and individual transitions satisfy local-detailed balance, which is enough to proof that it obeys all the usual fluctuation theorems. For strictly vanishing memory time (Markovian dynamics), this reduces to a Lindblad equation lindblad ; Davies74 ; whitney2008 , for which a different proof exists Maxime+Alexia2016 . However, our proof applies equally to systems with short (but non-zero) memory times.

Next is the co-tunnelling approximation Schoeller-review1997 , in which the system can interact with two reservoir modes at the same time. This is a used in Coulomb-blockaded quantum dots, where it can dominate the transport in certain regimes Schoeller-review1997 . Since this approximation obeys the conditions discussed above, this constitutes a proof that the co-tunnelling approximation obeys all the usual fluctuation theorems. Similarly, by allowing up to simultaneous interactions with reservoir modes (for different ), one gets a family approximations which all obey the fluctuation theorems.

Conclusions. This work uses a real-time Keldysh theory for the quantum stochastic thermodynamics of arbitrary systems coupled to ideal reservoirs. By finding the symmetry between trajectories on the Keldysh contour in Eq. (11), it shows that Eq. (1) holds for all non-Markovian system dynamics, including non-factorized initial conditions, so these dynamics obey the second law on average. It gives other fluctuation theorems, such as Jarzynski or Crooks, in the right conditions. A family of approximations is identified which satisfies Eq. (11), and so fulfil the fluctuation theorems. This provides a powerful tool to analyse nanoscale energy-harvesting and refrigeration beyond weak-coupling.

###### Acknowledgements.

Acknowledgments. In fond memory of Maxime Clusel, whose ideas on quantum fluctuation theorems stimulated this work. I thank A. Auffeves, D. Basko, M. Campisi, A. Crepieux, C. Elouard, M. Esposito, E. Jussiau and F. Michelini for useful comments or discussions. I acknowledges the financial support of the COST Action MP1209 “Thermodynamics in the quantum regime” and the CNRS PEPS grant “ICARE”.## References

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This supplementary material contains five sections. Section A recalls the aspects of quantum mechanics we need in this work; this is textbook material, but we regroup it here for convenience. Section B gives more details on the derivation of this work’s central observation, Eq. (11), than would fit the body of the text. Section C gives a concrete example of a trajectory on the Keldysh contour, and shows that it obeys Eq. (11). Section D derives a variety of well-known fluctuation theorems from Eq. (11), such as the non-equilibrium partition identity (or integral fluctuation theorem) in Eq. (1), the Jarzynski equality and Crooks equation. A reader familiar with the stochastic thermodynamics of classical rate equations Schmiedl-Seifert2007 ; van-Broeck-review2015 ; our-review-2016 will note that the derivations are almost identical as those for such classical rate equations, even though the nature of the trajectories themselves is very different. Finally, Section E shows how to generalize the method to non-factorized initial conditions, and to show that Eq. (1) holds under these conditions, as does the Evans-Searles steady-state fluctuation theorem Evans-Searles1994 .

## Appendix A Background

### a.1 Time-reversal in quantum mechanics

Here we recall the results that we will need related to time-reversal in quantum mechanics, which can be found in Messiah’s famous textbook Messiah-chapt15 . Firstly, the time-inversion of a quantum state is defined as

(S1) |

where is the time-inversion operator. In the absence of spins, time-inversion of a wavefunction is just taking its complex conjugate; thus , where is the complex-conjugation operator. To understand the role of for a single particle problem, one notes that position states are invariant under time-inversion, and so if one writes the system wavefunction as a vector of position states, then is the operator which takes the complex conjugate of all elements of the vector. For a many body problem, the same is also true, if one writes the system state as a vector of many-body position states (with a position for each particle). Defining such that , one has for any matrix written in a basis of many-body position states.

In the presence of spin-halves, the time-inversion operator also flips the spins about the y-axis, so

(S2) |

The time-inversions of a position operator, , a momentum operator, , and a Paulli spin operator are

(S3a) | |||||

(S3b) | |||||

(S3c) |

The time reverse of a Hamiltonian in the time-window to as sketched in Fig. S1 is

(S4) | |||||

where the dependence of on external fields, , and Paulli spin-matrices, , is explicitly shown to recall how they transform under time-reversal. The evolution operator from time to time under such a time-dependent Hamiltonian (the solid part of the curve in Fig S1a) is given by the usual time-ordered integral

(S5) |

where is the time-ordering operator. Similarly, the evolution operator from time to time (the dashed part of the curve in Fig S1a) is

(S6) |

If one now compares this to the evolution operator from time to time in the system with the time-reversed Hamiltonian (the solid part of the curve in Fig S1b)

(S7) |

where one should recall that Then it is straight-forward to show that

(S8) |

### a.2 No time-reversal of reservoir states

If one takes any initial state, evolves it under any Hamiltonian for a time , time-reverses the state, evolves it under the time-reversed Hamiltonian for a time , and then one time-reverses the state, the final state will coincide with the initial state. The dynamics in the second part of the evolution will look like the dynamics in the first part of the evolution, but going backwards.

However, this work considers a different situation, in which the set-up is divided into a system and reservoirs. We assume the reservoirs are infinite, and that it is beyond our power to time-reverse the infinite number of modes that make up the reservoir state. Thus, even if we have enough control to time-reverse the system state and to time-reverse the system and reservoir Hamiltonians (typically time-reversing reservoir Hamiltonians only requires interchanging spin-up and spin-down reservoirs), we will not see time-reversed dynamics.

To see the difference, suppose we start with the system in a pure state and the reservoirs in a thermal state, and we let it evolve. The system will become correlated and/or entangled with individual reservoir modes. If we measure the system state alone, it will look like it is decohering and decaying towards a thermal state, while if we measure individual reservoir modes we will see that an infinitesimal proportion of them are acquiring a non-thermal state. If we time-reverse the Hamiltonian and all states (without measuring), then the system will evolve back towards its initial pure state and the reservoir modes back to their initial thermal state. However, if we time-reverse everything except the reservoirs modes, then the system will continue to become entangled with more reservoir modes. The result being that a measurement of the system will indicate that it continues to decohere and decay towards a thermal state.

### a.3 The interaction representation

This work considers the Hamiltonian in Eq. (2) and treating the interactions between the system and the reservoirs as a perturbation which is treated to all orders. The bare system Hamiltonian is written as a matrix in a many-body basis as (the hat is dropped to indicate that this is a matrix rather than an operator), while the bare Hamiltonian for the electronic reservoir is , and that for photonic or phononic reservoir is . If the system consists of electronic states, each of which can be occupied by zero or one electron, so it has many-body states and will be a matrix. The th element of this matrix will be

(S9) |

where is one of the states in the many-body basis of position states discussed below Eq. (S1). The system operators in the system-reservoir coupling written as matrices are .

All system operators are now transformed to the interaction representation (indicated by calligraphic letters), in this representation the system operators in the system-reservoir couplings become

(S10) |

where is evolution under given by

(S11) |

with indicating time-ordering. To simplify the analysis, it is assumed that a complete solution of the dynamics under exists, then the final state of the system (its state at given time ) can always be written in a basis chosen such that . Then, the unitary of the matrix means that

(S12) |

which will be necessary in the derivation that follows.

Now, let us turn to the operators of reservoir that appear in the system-reservoir coupling. In the interaction representation, their time-dependence is given by their evolution under those reservoir’s Hamiltonians. Thus

(S13) | |||||

(S14) | |||||

where is for electronic reservoirs or for photonic or phonic reservoirs.

### a.4 The Keldysh contour

To understand the trajectories on the Keldysh contour presented in this work, we have to first recall how to write a perturbation expansion of a propagator in term of Feynman diagrams. In the interaction representation, nothing happens to the system except when the perturbation acts. Thus the evolution operator that takes the system’s wavefunction at time to the wavefunction at time is

(S15) | |||||

where is the time-ordering operator, and is the perturbation written in the interaction representation. This time ordering means that operators at earlier times are always to the right of those at latter times. Diagrammatically we can write this as

(S16) |

where indicates . Each of these diagrams can be thought of as a perturbative trajectory for the evolution of the wavefunction. Then the wavefunction is the sum over all such trajectories (sum over all orders and all intermediate states, and integrating over all intermediate times).

However, to treat initial states which are not pure (such as reservoirs starting in thermal states), one must consider the evolution of the density matrix from time to , rather than consider the wavefunction alone. The propagation of the density matrix from time to (in the interaction representation) is given by . Thus the th element of the final density matrix is given by

(S17) |

To write this in terms of a perturbation, one needs as well as , this is given by

(S18) | |||||

where the fact that is Hermitian, means that . This has inverse time-ordering, so that operators at earlier times are always to the left of those at latter times (cf. Eq. (S15) when they were to the right). Diagrammatically this is

(S19) |

where the arrows from larger to smaller times are to remind us that the s are inverse time-ordered. This means that a typical contribution to the evolution from the density matrix element at time to the density matrix element at time involves two diagrams; one taken from Eq. (S16) and one taken from Eq. (S19). Writing these two diagrams one above the other, we get a trajectory on the Keldysh contour for the evolution of the density matrix. For example, if one takes the second-order term from Eq. (S16) and the first-order term from Eq. (S19), one gets a trajectory on the Keldysh contour of the form

(S20) |

In contrast, if one takes the sixth-order term from Eq. (S16) and the fourth-order term from Eq. (S19), one gets a trajectory on the Keldysh contour of the form

(S21) |

Up to this point this perturbative treatment is completely general, now we turn to Hamiltonians of the form in Eq. (2) in the limit where each reservoir is infinite. Following Calidiera and Leggett Caldiera-Leggett-1983a ; Caldiera-Leggett-1983b ; Leggett-et-al , we imagine a large reservoir whose size is then taken to infinity, i.e. whose level-spacing is taken to zero. However, the total effect of the system-reservoir coupling on the system’s dynamics should remain finite, as the number of reservoir modes diverges. The only way to do this is to take the coupling to each reservoir mode to zero. Thus the treatment of each reservoir mode, can stop at the leading term in the perturbation expansion. The first-order term drops out so, the leading order term is the second-order term. The first-order term drops out because it always takes the initial thermal state of a reservoir mode to an off-diagonal state (e.g. to a state for ), and this does not contribute when we measure the system state at the end of the evolution, since such a measurement is a trace over environment modes (for which only those with contribute). Thus, in the limit of an infinite reservoir, all contributions to a system’s dynamics involve either a second-order coupling to a given reservoir mode, or no coupling to that reservoir mode, This observation has appeared many times independently in the literature; for example, it is crucial in the Nakajima-Zwanzig equation, Nakajima58 ; Zwanzig60 , the Caldiera-Leggett analysis of open systems Caldiera-Leggett-1983a ; Caldiera-Leggett-1983b ; Leggett-et-al , and real-time transport theory Schoeller-review1997 . It is at the root of the Bloch-Redfield method Bloch57 ; redfield and mathematical derivations of Lindblad equations from system-reservoir Hamiltonians Davies74 ; Davies76 , even if these works make additional weak-coupling approximations.

A second-order coupling involves two s somewhere on the Keldysh contour associated with the same reservoir mode. To indicate that these two are for the same reservoir mode, we connect them two with a dashed line. There are three possibilities; both s on the upper line, one on each line, or both s on the lower line. These correspond to the three types of contribution in Fig. 2. Thus Eq. (S20) with its odd number of s gives no contribution, but Eq. (S21) (with its even number of s) can give multiple contributions, in which the s are paired up by dashed lines in all possible ways.

Next, one should note the form of the system-reservoir coupling in Eq. (2) always contain two terms, one for adding a particle to the system from a reservoir, and one for taking a particle to the system from a reservoir, these operators are and . Then in trajectories like Eq. (S21) can be for or for , as in the figures in this work. On the upper line in the Keldysh contour it is the -operator that removes a particle from mode of reservoir and adds it to the system, while the -operator does the reverse. Thus, for example

(S22) |

where indicates that there are particles in mode of reservoir , and is the system state that one gets when a particle is added to system state . However, since operators are given by their Hermitian conjugates on the lower line of the Keldysh contour, the role of and are interchanged on this line, thus acting on an initial state removes a particle from mode of reservoir and adds it to the system giving a final state .

Once one has replaced all by or in a trajectory like Eq. (S21), one can note that the only non-zero contributions to system properties are those where each second-order term is a dashed line between a and a (i.e. ). This is because these are the only possibilities that generate a diagonal final state for the reservoir mode . if the reservoir mode starts in state , and both and are on the same line of the Keldysh contour (upper or lower), then the final reservoir mode state will be , and if they are on different lines the final reservoir mode state will be (with plus-sign when is on the upper line and minus-sign when it is on the lower line. In contrast, if one has two s on the upper line then the final state will be , which is clearly off-diagonal. One can easily check for oneself that two always give an off-diagonal final state for the mode regardless of whether each is on the upper or lower line.

To summarize, the only trajectories that contribute to the final state of the system are those of the type shown in the figures of this work, which have each connected by a dashed line to an , where the dashed line indicates that these operators both act on the same reservoir.

The final ingredient is the weight of due to the pair of reservoir operators. As we are interested in system dynamics, we trace over reservoir operators, so the contribution of the reservoir operators to in Fig. 2 is

(S23) | |||||

where is the initial thermal state for mode . Here the partition function , where the sum is over n=0,1 for fermions, and over for bosons. Thus is a Fermi factor for a reservoir of electrons, and a Bose factor for a reservoir of photons or phonons; these factors are

(S24) |

with for fermions and for bosons. The contribution of the reservoir operators to and in Fig. 2 are respectively

(S25) |

By performing a cyclic permutation inside the trace, one sees that both these quantities equal Eq. (S23). This explains the results in Eq. (4-6).

One can do the same thing for the three other possible contributions, which have and interchanged; these contributions are , and . The reservoir operators contribute the same in all cases up to cyclic permutations inside the trace, and so all of them are the following

(S26) | |||||

where is the number of ways one can add a particle to state of reservoir . For any reservoir (fermionic, bosonic or other) in internal equilibrium,

(S27) |

which is know as local detailed balance or micro-reversibility. For fermion or boson distributions, this is guaranteed by the fact that with for fermions, and for bosons.

## Appendix B Derivation of Eq. (11)

### b.1 Two set-ups as time-reverses of each other

This work is based on considering two different set-ups; set-up A is a system and its reservoirs that is described by a given Hamiltonian of the form in Eq. (2), while set-up B is one described by the time-reverse of set-up A’s Hamiltonian. Our objective is to derive fluctuation relations be comparing the dynamics in these two different set-ups. In the special case of a time-independent Hamiltonian without external magnetic fields or spins, the two set-ups are identical, but otherwise they are not.

If set-up A has a given time-dependent system Hamiltonian, , with given system-reservoir couplings, , then set-up B is chosen to have the system Hamiltonian and system-reservoir couplings

(S28) | |||||

(S29) |

where the bar above a symbol means that it is in set-up B, while the bar’s absence means in it is in set-up A.

These equations are cast in terms of matrix elements by inserting them between and , then

(S30) | |||||