1 Introduction

The probabilistic nature of quantum physics is related to a process whose correct interpretation and mathematically sound formulation is still under debate, the so-called collapse of the wave function [1]. The quantum mechanical collapse is of fundamental importance for our common-sense concept of macroscopic reality. In most cases, collapses are invoked in the context of measurements of quantum observables. But extending the applicability range of quantum mechanics beyond the microscopic realm prompts the question whether the discontinuous change of the wave function during the measurement is a physical process or not. Schrödinger’s famous Gedankenexperiment involving a cat demonstrates drastically the consequences of the assumption that the collapse process is merely epistemic and thus concerns only the knowledge of the observer [2]. Schrödinger’s goal was to demonstrate that a paradox arises if the collapse is not considered as a microscopic process taking place independently from the presence of an observer. To explain the absence of macroscopic superpositions without the assumption of a physical collapse, the “decoherence” interpretation considers not individual systems but the density matrix of an ensemble which becomes mixed after tracing over unobserved environmental degrees of freedom [3,4,5]. Although this avoids the need for a “measurement apparatus”, the tracing operation is not physical but epistemic. Without a physical mechanism triggering a real collapse event as assumed e.g. in [6], the decoherence interpretation is equivalent to the many-world hypothesis [7].

Already in Schrödinger’s cat example, the transition from unitary and deterministic to probabilistic evolution is tied to a microscopic, unpredictable event, the decay of a radioactive nucleus. There are strong arguments from a fundamental perspective supporting the occurence of spatio-temporally localized events as objective and observer-independent equivalent of collapse processes [8].

These random events, called “quantum jumps” in the early debate between Schrödinger and Bohr [9], are generally thought to underlie the statistical character of the emission and absorption of light quanta by atoms. Einstein used arguments from the theory of classical gases to derive Planck’s formula by assuming detailed balance between the atoms and the radiation in thermal equilibrium. His derivation did not require a microscopic Hamiltonian [10]. Nevertheless, it is easy to derive the corresponding rate equations for the time-dependent occupancy \(\langle n\rangle (t)\) of a light mode with frequency \(\Omega\) coupled resonantly to M two-level systems (TLS) within quantum mechanics. The transition probabilities follow from the microscopic interaction Hamiltonian by employing Fermi’s golden rule [11, 12], which tacitly incorporates the collapse event by replacing the unitary time evolution by a stochastic process.

The interaction Hamiltonian is well known [13, 14]:

$$\begin{aligned} H_{\text {int}}=g\sum _{l=1}^M\left( a\sigma ^+_l+a^{\dagger }\sigma ^-_l\right) , \end{aligned}$$

where \(a,a^{\dagger }\) denote the annihilation/creation operators of the radiation mode and \(n=a^{\dagger }a\). The Pauli lowering/raising operators \(\sigma ^-_l,\sigma ^+_l\) describe the l-th two-level system with \(H_{\text {TLS}}=\hbar \Omega (\sum _l\sigma _l^+\sigma _l^--1/2)\). The probabilities for a transition between the upper and lower state of a TLS accompanied by the emission or absorption of a photon with frequency \(\Omega\) are computed with the golden rule (see “Derivation of the Rate Equations (2) and (3)”) to obtain the rate equations

$$\begin{aligned} \frac{\text {d}\langle n\rangle }{\text {d}t}&= \gamma \left[ \langle m\rangle (\langle n\rangle +1)- (M-\langle m\rangle )\langle n\rangle \right] , \end{aligned}$$
$$\begin{aligned} \frac{\text {d}\langle m\rangle }{\text {d}t}&= \gamma '\left[ (M-\langle m\rangle )\langle n\rangle -\langle m\rangle (\langle n\rangle +1)\right] . \end{aligned}$$

Here, \(\langle m(t)\rangle\) denotes the time-dependent average number of excited TLS. The rate constants \(\gamma ,\gamma '\) depend on the coupling g and the density of states of the radiation continuum around \(\Omega\) (see below). These equations describe the irreversible change of average quantities and thus use the ensemble picture of statistical mechanics [15]. Nevertheless they account for the temporal behavior of a single system as well, because a typical trajectory will exhibit a fraction m(t)/M of excited TLS close to \(\langle m(t)\rangle /M\) for sufficiently large M [16,17,18].

It is crucial that the rate equations (2) and (3) satisfy the detailed balance condition, which implies that they lead from arbitrary initial values \(\langle n\rangle (0), \langle m(0)\rangle\) to a unique steady state characterized by

$$\begin{aligned} \langle m\rangle (\langle n\rangle +1) = (M-\langle m\rangle )\langle n\rangle . \end{aligned}$$

We have the relations

$$\begin{aligned}&P^e=\langle m\rangle /M,\quad P^g=(M-\langle m\rangle )/M, \nonumber \\&P_{e\rightarrow g}=\gamma '(\langle n\rangle +1),\quad P_{g\rightarrow e}=\gamma '\langle n\rangle , \end{aligned}$$

for the probabilities \(P^g\) (\(P^e\)) for a TLS to be in its ground (excited) state and the probabilities for emission \(P_{e\rightarrow g}\) and absorption \(P_{g\rightarrow e}\). Equation (4) can therefore be written as

$$\begin{aligned} P^eP_{e\rightarrow g} = P^gP_{g\rightarrow e} \end{aligned}$$

which is the definition of detailed balance [15, 19]. Equation (4) entails Planck’s formula for thermal equilibrium between radiation and matter. If one considers (3) as an equation of motion for the probability \(P^e(t)\), even a microscopic system consisting of a single TLS will thermally equilibrate with the surrounding continuum of radiation.

This surprising result rests on the fact that the TLS does not interact with the light mode exactly on resonance only but with all modes in a frequency interval of width \(\Delta\) around \(\Omega\) with a similar strength g [13, 14]. The energy uncertainty \(\hbar \Delta\) allows for a radiation event occuring during a short time span \(\tau _c\sim \Delta ^{-1}\), whereas the process itself is energy conserving (see [13], p. 419). The coupling to a continuum of modes leads therefore to real and irreversible microscopic processes, the emission or absorption of light quanta, although no macroscopic measurement apparatus is involved. Such a microscopic collapse process is tacitly assumed whenever the golden rule is employed. If, however, the TLS is embedded into a cavity and coupled only to a single radiation mode, the collapse cannot take place; the system shows the Rabi oscillations of an unitarily evolving state instead, the TLS being entangled with the bosonic mode. To this case, the golden rule cannot be applied. Therefore, the golden rule cannot be taken as an approximation to the full unitary time development given by solving the Schrödinger equation, although it corresponds formally to a perturbative computation of the unitary dynamics for short times [20]. The very concept of a transition rate implies that the deterministic evolution of the state vector is replaced with a probabilistic description of events. The central element in the computation is the overlap \(\langle \psi _{\text {final}}|H_\text {int}|\psi _{\text {initial}}\rangle\), which, according to the Born rule, determines the probability for the transition from \(|\psi _{\text {initial}}\rangle\) to \(|\psi _{\text {final}}\rangle\). The Born rule is invoked here although the process is not a macroscopic measurement. The “macroscopic” element is provided by the continuum of radiation modes. The presence of this continuum causes the necessity of a statistical description [13].

Within the decoherence interpretation, one would require that the continuum of radiation modes acts as an environment for the TLS, the “system”. The environment is then traced out to yield the irreversible dynamics of the system alone. But if the system consists of the walls of a hohlraum, it excercises a strong influence on the “environment”, the enclosed radiation, such that they equilibrate together. Therefore, it is not possible to separate the environment from the system to explain the microscopic collapse processes driving the compound system towards thermal equilibrium.

The rate equations neglect completely the coherences of the TLS. The irrelevance of the coherences follows naturally from the interpretation of the radiation process as a collapse: each such event projects the TLS into their energy eigenbasis, just as a macroscopic measurement projects any quantum system into the basis entangled with the eigenbasis of the measurement device [21]. A macroscopic measurement device is not needed here because both subsystems, the TLS and the radiative continuum, contain a macroscopic number of degrees of freedom. This alone seems to justify a statistical description as in classical gas theory, although the interaction Hamiltonian (1) has no classical limit. Indeed, in our case the Born rule replaces the assumption of “molecular chaos”, needed in Boltzmann’s derivation of the H-theorem [15, 19]. Therefore, it seems almost natural that the rate equations (2), (3) lead to thermal equilibrium from a non-equilibrium initial state although the radiation and the collection of atoms are both treated as ideal gases. If \(\langle m(0)\rangle\) and \(\langle n\rangle (0)\) correspond to equilibrium ensembles with different temperatures at \(t=0\),

$$\begin{aligned} T_{\text {TLS}}(0)=\frac{\hbar \Omega }{k_B\ln \left( \frac{M}{\langle m(0)\rangle }-1\right) }, \quad T_{\text {rad}}(0)= \frac{\hbar \Omega }{k_B\ln \left( \frac{1}{\langle n\rangle (0)}+1\right) }, \end{aligned}$$

the rate equations derived from the quantum mechanical interaction Hamiltonian (1) together with the golden rule entail thermal equilibration according to Clausius’ formulation of the second law of thermodynamics: the two gases exchange heat which flows from the hotter to the colder subsystem until a uniform temperature and maximum entropy of the compound system is reached [15, 19].

Certainly, the rate equations do not correspond to the exact quantum dynamics of the system, which is non-integrable in the quantum sense if the interaction (1) is generalized to a continuum of bosonic modes, rendering it equivalent to the spin-boson model [22]. To obtain the exact evolution equation for the density matrix of both the TLS and the radiative modes, one would have to solve the full many-body problem. The golden rule is then seen as a method to approximate the time-dependent expectation values \(\langle n\rangle (t)\), \(\langle m(t)\rangle\), which is justified by a large body of experimental evidence, but not through an analytical proof of equivalence between both approaches. It may even be that the golden rule provides a phenomenological description of microscopic collapse processes whose actual dynamics is not yet known. In this case, the full quantum mechanical calculation would not yield (2), (3), although they account correctly for the experimentally observed dynamics. The rate equations are derived from the interaction Hamiltonian (1) and the golden rule (see  “Derivation of the Rate Equations (2) and (3)” and “Derivation of the Rate Equations (17)–(19) for the Open System”), thereby demonstrating its applicability in the situation considered. This corroboration of the golden rule by experimental results is independent of any theoretical justification of its validity.

The interaction Hamiltonian (1) satisfies the detailed balance condition which is crucial for the physically expected behavior. We shall demonstrate in the next section that this condition does not follow from elementary principles like time reversal invariance or hermiticity as (1) seems to suggest. On the contrary, there are feasible experimental setups violating the detailed balance condition while satisfying all other prerequisites for the application of the golden rule. The consequence of this violation is a macroscopic disagreement with the second law of thermodynamics.

2 The Gedanken Experiment

In the Gedankenexperiment, we consider two identical cavities A and B supporting a quasi-continuous mode spectrum described by bosonic annihilation operators \(a_{Aj}\), \(a_{Bj}\) and frequencies \(\omega _j\) (Fig. 1). They are coupled bilinearly to the right- and left-moving modes \(a_{1k}\), \(a_{2k}\) of an open-ended waveguide which form a quasi-continuum like the cavity modes [23]. The loss processes through the open ends of the waveguide are caused formally by a heat bath at \(T=0\), leading to a reduction of the system entropy through heat transport. This coupling to the outside world is one argument for the applicability of the golden rule, the second is the already discussed quasi-continuum of modes. Even if the full continuum of radiation modes in the cavities is treated as part of the “system”, which would then be subject to purely unitary time evolution, the waveguide would still couple to a decohering “environment”, justifying the statistical description even if one denies that real collapse events take place in the system itself. Here we study a model which is commonly used in quantum optics to describe unidirectional loss processes [24]. In this way, the generally accepted arguments substantiating irreversible evolution equations can be transferred to the present situation.

Fig. 1
figure 1

Layout of the Gedankenexperiment. Two reservoirs A and B containing black-body radiation are coupled via a non-reciprocal, open waveguide to a collection of two-level systems (TLS) and to each other. The right-, respectively left-moving modes in channels 1 and 2 couple with different parameters \(g_1\) and \(g_2\) to the two-level systems

The modes \(a_{1k}\) and \(a_{2k}\) of the waveguide are coupled to a collection of M two-level systems located at the center of the waveguide (see Fig. 1). The total Hamiltonian is given by

$$\begin{aligned} H=H_A + H_B + H_{wg} + H_{\text {TLS}} + H^1_{\text {int}} + H^2_{\text {int}}. \end{aligned}$$

\(H_q\) denotes the Hamiltonian in cavity q for \(q=A,B\),

$$\begin{aligned} H_q=\hbar \sum _{j}\omega _j a_{qj}^\dagger a_{qj}. \end{aligned}$$

The Hamiltonian of the waveguide reads

$$\begin{aligned} H_{wg}=\hbar \sum _{k}\omega _k \left( a_{1k}^\dagger a_{1k} + a_{2k}^\dagger a_{2k}\right) , \end{aligned}$$

where the modes 1 and 2 belong to waves traveling to the right and to the left, respectively. For the TLS we have \(H_{\text {TLS}}=(\hbar \Omega /2)\sum _{l=1}^M\sigma ^z_l\), with the Pauli matrix \(\sigma ^z\). The coupling between the reservoirs and the modes 1 and 2 of the waveguide is bilinear,

$$\begin{aligned} H^1_{\text {int}}=\sum _{q=A,B}\sum _{j,k} h_{jk}\left( a_{qj}^\dagger [a_{1k} + a_{2k}] +\text {h.c.}\right) . \end{aligned}$$

Using the rotating wave approximation, the interaction with the TLS has the standard form [13, 14] which is equivalent to (1),

$$\begin{aligned} H^2_{\text {int}}=\sum _{l=1}^M \left( \sum _{k} g_{1k}a_{1k} +g_{2k}a_{2k}\right) \sigma ^+_l + \text {h.c.}, \end{aligned}$$

where \(\sigma ^+_l\) denotes the raising operator of the lth TLS. We consider in the following the (time-dependent) average occupancy per mode j, \(\langle n_q\rangle (t)\) for reservoir \(q=A,B\) in an energy interval around the TLS energy, \(\Omega -\Delta /2<\omega _j<\Omega +\Delta /2\), where \(\Delta\) is much larger than the natural linewidth of spontaneous emission from an excited TLS into the waveguide. The occupancy does not depend on j if the couplings \(h_{jk}\), \(g_{(1,2)k}\), the density of states \(\rho _q(\hbar \omega _j)\) of the reservoirs and \(\rho _{1,2}(\hbar \omega _k)\) of the waveguide are constant in the frequency interval of width \(\Delta\) around \(\Omega\).

It is crucial that \(g_{1k}\ne g_{2k}\), which specifies that the TLS couple with unequal strengths to the right- and left-moving photons in the waveguide. Such an unequal coupling is a hallmark of chiral quantum optics [25]. Although the interaction Hamiltonian (12) appears to break time-reversal invariance, as the time-reversal operator maps left-moving to right-moving modes, this is actually not the case because the effective interaction term (12) does not contain the polarization degree of freedom. The angular momentum selection rules for light-matter interaction lead naturally to a dependence of the coupling strength on the propagation direction in engineered geometries [26], especially if the spin-momentum locking of propagating modes in nanofibers is employed [27, 28]. The unwanted coupling to non-guided modes can be effectively eliminated, leading to large coupling differences \(|g_{1k}-g_{2k}|\) [29, 30].

We shall now study the temporal behavior of the two cavities, assuming at time \(t=0\) separate thermal equilibria in A, B and the TLS system, all at the same temperature T. The probability \(\langle m(0)\rangle /M\) for a TLS to be excited obeys the Boltzmann distribution

$$\begin{aligned} \frac{\langle m(0)\rangle }{M}=\frac{1}{e^{\hbar \Omega /k_BT}+1}. \end{aligned}$$

Considering the M TLS as independent classical objects, their Gibbs entropy is given by

$$\begin{aligned} S_M=-k_BM\big (p_e\ln p_e +(1-p_e)\ln (1-p_e)\big ), \end{aligned}$$

with \(p_e(t)=\langle m(t)\rangle /M\). This approach is justified by the quick relaxation of the two-level systems by non-radiative processes, which decohere them on time scales much shorter than the time scale of spontaneous emission and quickly quench finite coherences of the TLS [31]. This argument for a classical description of the TLS is independent from the general justification of the golden rule via the mode continuum discussed above. Equation (14) provides an upper bound for the entropy of the TLS subsystem [32]. Analogously, the entropy of the compound system reads

$$\begin{aligned} S^{\text {sys}}=S_A +S_B + S_M, \end{aligned}$$

where \(S_q\) denotes the v. Neumann entropy of the radiation in cavity q for \(q=A,B\). The average occupation number \(\langle n_q(\omega )\rangle\) per mode at frequency \(\omega\) follows from the Bose distribution

$$\begin{aligned} \langle n_q(\omega )\rangle (0)=\frac{1}{e^{\hbar \omega /k_BT}-1}. \end{aligned}$$

Because the temperature depends on \(\langle n_q\rangle\) as described by (16), we can define effective temperatures \(T_q(t)\) by \(\langle n_q\rangle (t)\) for each reservoir q under the assumption that the photons in each reservoir thermalize in the usual way quickly as a non-interacting Bose gas. Furthermore, we consider the case that \(\langle n_{(1,2)k}\rangle (t)=0\) for the occupancy of the modes in the waveguide, i.e., the waveguide is populated through the sufficiently weak coupling to the reservoirs and its modes appear only as intermediate states (see “Derivation of the Rate Equations (17)–(19) for the Open System”). Neglecting the waveguide, the system is thus composed of three subsystems, each in thermal equilibrium at any time \(t>0\), with locally assigned time-dependent temperatures. The subsystems interact through random emission and absorption processes which do not lead to entanglement, because in each such process the wave function undergoes a collapse towards a product state. This description is in obvious accord with the derivation of Planck’s law given by Einstein [10]. Note that the compound system is not coupled to several thermal baths which have different temperatures. In such a case, a description with separate master equations for the subsystems is inconsistent if the interaction between subsystems is still treated quantum mechanically. Using such a description, a violation of the second law has been deduced [33, 34], which is only apparent and caused by the inconsistent computation [34]. Our system differs from those models because the system dynamics is not described by a unitary evolution as in [33, 34], but by a random process, with all subsystems coupled to the same bath (the open waveguide).

The golden rule applied to the Hamiltonian (8) yields the rate equations for \(\langle n_q\rangle (\Omega ,t)\) and \(\langle m(t)\rangle\),

$$\begin{aligned} \frac{\text {d}\langle n_{A}\rangle }{\text {d}t}&= -2\gamma _\text {dec}\langle n_{A}\rangle + \gamma _0(-\langle n_{A}\rangle +\langle n_{B}\rangle )- \gamma _1(M-\langle m\rangle )\langle n_{A}\rangle \nonumber \\&\quad + \gamma _2\langle m\rangle (\langle n_{A}\rangle +1), \end{aligned}$$
$$\begin{aligned} \frac{\text {d}\langle n_{B}\rangle }{\text {d}t}&= -2\gamma _\text {dec}\langle n_{B}\rangle + \gamma _0(-\langle n_{B}\rangle +\langle n_{A}\rangle )- \gamma _2(M-\langle m\rangle )\langle n_{B}\rangle \nonumber \\&\quad + \gamma _1\langle m\rangle (\langle n_{B}\rangle +1) , \end{aligned}$$
$$\begin{aligned} \frac{\text {d}\langle m\rangle }{\text {d}t}&= -[\tilde{\gamma }_{11}(\Omega ) +\tilde{\gamma }_{12}(\Omega )]\langle m\rangle + \tilde{\gamma }_1(\Omega )\left[ \langle n_{A}\rangle (M-\langle m\rangle )-(\langle n_{B}\rangle +1)\langle m\rangle \right] \nonumber \\&\quad +\tilde{\gamma }_2(\Omega )\left[ \langle n_{B}\rangle (M-\langle m\rangle )-(\langle n_{A}\rangle +1)\langle m\rangle \right] . \end{aligned}$$

The first terms on the right hand side of (17)–(19) describe the loss of photons through the open ends of the waveguide. These terms are of first order in \(|h_{jk}|^2\), resp. \(|g_{1k}|^2\), \(|g_{2k}|^2\). The following terms correspond to coherent processes of second order in the couplings. The effective rates \(\gamma _{\text {dec},0,1,2}\) and \(\tilde{\gamma }_{1r},\tilde{\gamma }_{1,2}\) used for the numerical solution of (17)–(19) shown in Figs. 2 and 3 belong to the strong coupling regime of the TLS and the waveguide with values accessible within a cavity QED framework [35]. The chiral nature of the coupling, \(g_{1k}\ne g_{2k}\), entails \(\gamma _1\ne \gamma _2\). In our example we have assumed \(\tilde{\gamma }_2=\gamma _2=\tilde{\gamma }_{12}=0\), i.e., channel 2 is not coupled to the TLS. One sees from (17)–(19) that the chiral coupling leads to a breakdown of the detailed balance condition in second-order processes because absorption is no longer balanced by stimulated and spontaneous emission. The radiation processes generate an effective transfer of photons from reservoir A to B on a time scale \(\tau _{\text {char}}\) given by the strong coupling between channel 1 and the TLS, \(\tau _{\text {char}}\sim \tilde{\gamma }^{-1}=0.1\;\mu\)s. This corresponds to a difference in the local temperatures calculated via (16). Figure 2 shows the temporal behavior of the temperatures of reservoirs A, B and the TLS for intermediate times.

Fig. 2
figure 2

Solutions of the rate equations (17)–(19) as function of time, starting from initial thermal equilibrium. The effective temperatures of the reservoirs A and B deviate. The temperature drop of the TLS parallels that of A. Parameters used are \(\gamma _\text {dec}=\gamma _0=10\) kHz, \(\tilde{\gamma }_{11}=\tilde{\gamma }_1=10\) MHz, \(\gamma _1=100\) kHz; \(\tilde{\gamma }_{12}=\tilde{\gamma }_2=\gamma _2=0\) and \(\hbar \Omega /kT(0)=1\). \(\Omega\) corresponds to a wavelength of 10 \(\mu\)m

Although both reservoirs A and B loose photons through the open waveguide, the ratio between \(\langle n_{A}\rangle\) and \(\langle n_{B}\rangle\) attains a constant value for \(t\rightarrow \infty\), which is depicted in Fig. 3. This figure reveals that the losses of reservoir B are characterized by a much larger time scale than the spontaneous population of B through reservoir A and the TLS.

Fig. 3
figure 3

The occupations of reservoirs A and B for long times, plotted on a logarithmic scale. The ratio \(\langle n_{A}\rangle /\langle n_{B}\rangle\) is asymptotically time-independent. The losses through the open waveguide are much slower than the breakdown of local equilibrium caused by the non-reciprocal interaction with the TLS

3 Conflict with the Second Law of Thermodynamics

Due to the steady loss of photons, our system is always out of equilibrium and the only steady state solution of the Eqs. (17)–(19) corresponds to empty cavities and all TLS in their ground state. It is clear that the system entropyFootnote 1 diminishes at any time due to the outgoing heat flow. The question arises how to apply the second law of thermodynamics to this situation. The second law has been formulated in several versions (see, e.g., [36,37,38]). It is not the purpose of this paper to discuss these in detail. Three representative formulations provide exemplary definitions of the second law [15, 19]:

  1. (1)

    The entropy of the universe always increases.

  2. (2)

    The entropy of a completely isolated system stays either constant or increases.

  3. (3)

    The entropy of a system thermally coupled to the environment satisfies Clausius’ inequality: \(\Delta S \ge \Delta Q/T\).

Variant (1) is not subject to our Gedankenexperiment, because the entropy production including the environment is formally infinite, as the external bath has zero temperature. Variant (2) follows from variant (3) because the heat transfer \(\Delta Q\) vanishes. Variant (3) applies to the present system: the heat transfer \(\Delta Q\) is negative and therefore also \(\Delta S\) may be negative. However, the second law in the form of Clausius’ inequality forbids the case \(\Delta S \le \Delta Q/T\): The local entropy production \(\sigma\) in the system must be non-negative [19].

The change of system entropy \(S^{\text {sys}}(t)\) can be written as [39]

$$\begin{aligned} \frac{\text {d}S^{\text {sys}}}{\text {d}t} = -\int \text {d}{\varvec{o}}\cdot {\varvec{J}}_{\text {sys}} +\sigma , \end{aligned}$$

where the surface integral of the entropy current \({\varvec{J}}_{\text {sys}}\) accounts for the heat transfer to the environment, characterized by the rates \(\gamma _\text {dec}\) and \(\tilde{\gamma }_{11},\tilde{\gamma }_{12}\). We find for \(\sigma (t)\) (see “Derivation of the Entropy Production (22)–(24)”)

$$\begin{aligned} \sigma (t)= \sigma _{A,B}(t) + \sigma _{A,\text {TLS}}(t) + \sigma _{B,\text {TLS}}(t), \end{aligned}$$


$$\begin{aligned} \sigma _{A,B}&=k_B\mathcal{N}\gamma _0(\langle n_{B}\rangle - \langle n_{A}\rangle )\big (\ln (\langle n_{B}\rangle [\langle n_{A}\rangle +1])-\ln (\langle n_{A}\rangle [\langle n_{B}\rangle +1])\big ), \end{aligned}$$
$$\begin{aligned} \sigma _{A,\text {TLS}}&=k_B\big (\tilde{\gamma }_2\langle m\rangle (\langle n_{A}\rangle +1)-\tilde{\gamma }_1\langle n_{A}\rangle (M-\langle m\rangle )\big )\big (\ln (\langle m\rangle [\langle n_{A}\rangle +1]) \nonumber \\&\quad -\ln ([M-\langle m\rangle ]\langle n_{A}\rangle )\big ), \end{aligned}$$
$$\begin{aligned} \sigma _{B,\text {TLS}}&=k_B\big (\tilde{\gamma }_1\langle m\rangle (\langle n_{B}\rangle +1)-\tilde{\gamma }_2\langle n_{B}\rangle (M-\langle m\rangle )\big )\big (\ln (\langle m\rangle [\langle n_{B}\rangle +1]) \nonumber \\&\quad -\ln ([M-\langle m\rangle ]\langle n_{B}\rangle )\big ), \end{aligned}$$

with \(\mathcal{N}=\tilde{\gamma }_1/\gamma _1=\tilde{\gamma }_2/\gamma _2\). The three contributions result from the heat exchange between the three subsystems. Only \(\sigma _{A,B}\) is always non-negative, because it has the form \((x-y)(\ln (x)-\ln (y))\) characteristic for systems satisfying the detailed balance condition. Because \(\tilde{\gamma }_1\ne \tilde{\gamma }_2\), the two other contributions are not necessarily non-negative. Figure 4 shows \(\sigma (t)\) calculated using the parameters of Fig. 2. The entropy production is negative for \(t<0.75\;\mu\)s. The time span during which the entropy is reduced beyond the loss to the environment is almost an order of magnitude longer than the characteristic time scale \(\tau _{\text {char}}= 0.1\;\mu\)s. The especially notable point is the entropy reduction which occurs although the dynamical evolution started with thermal equilibrium between the subsystems.

During the exchange of heat with the other subsystems, each subsystem remains in local thermal equilibrium. In principle, the exchange of heat between subsystems gives only a lower bound to the entropy production for irreversible processes and the actual \(\sigma (t)\) could be larger than the value given in (21), which contains the contributions from the mutual heat exchange. However, the entropy change in each subsystem can be computed directly via the formulae (14) and (96), (97) as well. Doing so, one finds that no additional entropy production besides the mutual heat exchange occurs in the process described by the rate equations (17)–(19). This, obviously, is due to the fact that during this process each subsystem remains in local thermal equilibrium. It follows that the entropy change of the full system would even be negative if the initial entropy of the TLS subsystem was lower than the upper bound given in (14), because the entropy change is caused solely by mutual heat transfer between subsystems if the condition of local equilibrium is satisfied.

We conclude that the initial thermal equilibrium between reservoirs and the TLS is unstable and variant (3) of the second law is violated. This is true although the total entropy of the system plus the environment always increases. The second law, applied to the system alone, demands a non-negative local entropy production for any process driven by the coupling to the bath at \(T=0\) [19, 39]. Such processes may generate local temperature gradients between the reservoirs, but \(\sigma (t)\) must always be larger or equal zero to satisfy Clausius’ inequality. The violation of this inequality in our Gedankenexperiment shows clearly that the chiral coupling between TLS and the waveguide generates radiation processes which are in conflict with thermodynamics if they are treated statistically in the same way as black body radiation.

Fig. 4
figure 4

The total entropy production \(\sigma (t)\) within the system calculated for \(\mathcal{N}\gamma _0=1\) MHz, \(\tilde{\gamma }_1=10\) MHz, \(\tilde{\gamma }_2=0\). For these parameters, the entropy production is negative until \(t'\sim 0.75\;\mu\)s, when it turns positive and stays so. This behavior entails that for \(t<t'\) the second law of thermodynamics is not satisfied. Note that \(t'\) is appreciably larger than \(\tau _{\text {char}}\)

4 Discussion and Conclusions

In discussing the experiment we first note that it is based on phenomena taking place in the realm where the quantum world interfaces classical physics. The three subsystems are clearly macroscopic, but their interaction Hamiltonian (11, 12) is purely quantum mechanical and cannot be treated in a (semi-)classical approximation. A unique quantum feature of the interaction is given by the fact that the emission rate of the TLS depends on the occupation of the final states. In “Detailed Balance for a System with an Embedded Cavity”, we demonstrate that this counter-intuitive effect leads to the restoration of detailed balance in a cavity system without non-reciprocal elements. The non-unitary, probabilistic state development of the device can neither be achieved in classical Hamiltonian dynamics nor in the unitary pure quantum regime described by the Schrödinger equation. The collapse processes that link the classical world and the quantum regime [40] are the cause of the thermal imbalance between the otherwise equivalent reservoirs A and B. Our Gedankenexperiment therefore reveals a clear conflict between thermodynamics and the probabilistic description of quantum phenomena on a macroscopic scale. The corresponding rate equations (17)–(19) do not satisfy the condition for detailed balance. Instead they predict a time-dependent state that violates the Clausius inequality, i.e. the original formulation of the second law of thermodynamics.

According to the quantum description of the statistical absorption and emission processes, the chirally coupled cavities are expected to develop unequal occupation numbers. This imbalance creates a temperature gradient between them, although no work is done on the system, which is coupled to the environment through the open waveguide only. This coupling to a heat bath at temperature zero leads to heat flow out of the system which is usually accompanied by a positive local entropy production. But in our case the entropy production within the system is negative during a well defined time interval. This interval is larger than the time characterizing the coupling between the subsystems. The violation of the second law is temporary. Because this violation is described by the rate equations, it is not caused by a statistical fluctuation. Such a fluctuation may occur in the stochastic evolution of the state vector of a single system, even when the initial state is typical [41], but cannot appear in the fully deterministic equations for averages.

The rate equations (17)–(19) have been derived under the assumption that the two channels of the waveguide are fed by A and B through the emission of wavepackets which in turn interact with the TLS in a causal fashion. The emission and absorption of single photons by the TLS are considered thus as probabilistic processes taking place within a finite time span, due to the quasi-continuum of modes available in the waveguide and the reservoirs. They therefore satisfy causality: It is not possible for a right-moving photon in channel 1 to be emitted by the TLS and be subsequently absorbed by reservoir A. The photons entering A are either generated by a fluctuation in channel 1 of the open waveguide or arrive through channel 2. In the latter case, they may come from the outside, from a TLS or from reservoir B. Pure scattering events at the TLS are neglected in this approximation because they are of higher order in the coupling constants. Their inclusion cannot restore the detailed balance broken by the chiral coupling.

Our reasoning is based on the assumption that the interaction of the TLS with the radiation continuum leads to real events [8] which must be described statistically, and are therefore caused by a collapse process. The physical mechanism of this “real” collapse plays no role in these considerations because no hypothesis beyond the golden rule enters the derivation of the rate equations. Of course, we have also assumed that the macroscopic nature of the radiation and the collection of TLS removes any detectable entanglement between the subsystems. It entails the statistical independence of the radiation processes and therefore Markovian dynamicsFootnote 2. This assumption is corroborated by all available experimental evidence up to now. In case the second law of thermodynamics would be correct and therefore the presented statistical analysis wrong, the validity of the second law would be tantamount to the actual realization of a macroscopic superposition of states in the cavity system, although it is coupled to an unobserved environment, the open ends of the waveguide.

Interestingly, the use of the golden rule can also be justified even in a completely isolated system. If the system is isolated, it could in principle be described by the full unitary dynamics, leading to a trivial reconciliation with the second law in its restricted form: the fine-grained Gibbs/v. Neumann entropy does not change at all. However, also in this case an “environment” is present which consists of the infinitely many degrees of freedom of the photon gas, decohering the dynamics of the TLS, at least according to the opinion of the majority of physicists working in quantum optics [13]. The effective coarse-grained description of the closed system proceeds again via the golden rule (see “Derivation of the Rate Equations for the Closed System”). A similar temperature difference between A and B appears and a new steady state develops from initial thermal equilibrium for \(t\rightarrow \infty\), having a lower entropy than the initial state, thus violating variant (2) of the second law. This is shown in Fig. 5. The temperature difference corresponds to a “sorting” between the reservoirs A and B in the closed system and resembles the action of a Maxwell demon [43]. No information is processed, stored or erased, neither in the TLS nor in the waveguide [44], therefore the usual arguments for positive entropy production based on information theory [45, 46] cannot be applied.

Fig. 5
figure 5

Temporal development of the total entropy for the closed variant of our system shown in the inset. The initial state at \(t=0\) (thermal equilibrium between all subsystems, including the waveguide) maximizes the entropy. The stable steady state for long times has a lower entropy

It has been argued that it is not possible to discern experimentally an interpretation of quantum mechanics based on probabilistic dynamics and real collapse from the decoherence interpretation which replaces the physical collapse by an epistemic operation: the tracing over environmental degrees of freedom in the full density matrix at the final observation time \(t_\text {fin}\) [47]. As mentioned in Sect. 1, it is not known whether the photon densities in the reservoirs at \(t_\text {fin}\), calculated with the tracing procedure, would differ from the results of “Derivation of the Rate Equations (17)–(19) for the Open System” based on the golden rule. If so, our proposed experiment, if performed with an isolated system, would allow to decide between interpretations based on real collapses and epistemic interpretations. Only the latter do not contradict the second law of thermodynamics, provided an actual solution of the full many-body problem would effectively restore the detailed balance condition in the statistical description. Such a solution would also be necessary to identify possible reasons for the breakdown of well-established tools like the golden rule or the Markov approximation in case the system shows the equilibrium state predicted by thermodynamics at all times. In any case, a difference between the full solution and the approximation by the golden rule would entail another mystery: why is the approximation valid for black-body radiation in arbitrary cavities (see “Detailed Balance for a System with an Embedded Cavity”) but not for chiral waveguides?

As a computation of the full quantum dynamics appears out of reach at present, the question can only be decided experimentally. A direct implementation of our model appears feasable with current technology [23, 25, 35].

In case that the experiment reported unequal distributions in A and B for the closed system, one would be forced to conclude that statistical processes such as spontaneous emission and absorption are able to reduce the total entropy for arbitrary large isolated systems and on average, not only for short times and small systems as expected from fluctuations [41]. Then the state of classical thermal equilibrium with maximal entropy is unstable and the system moves to steady states with a lower entropy. In this case, entropy-reducing processes would be expected to occur actually in nature, in structures differing greatly from our deviceFootnote 3.

The contradictions presented are rooted in the still unresolved status of the measurement problem of quantum physics: quantum mechanical probabilities can only be computed under the assumption that a collapse (either real or epistemic) takes place. These stochastic probabilities are formally encoded in the Born rule. To our knowledge, neither the Born rule nor the golden rule have ever been used for a derivation of the second law of thermodynamics [16]. There have been attempts to deduce the second law from quantum mechanics by employing several “coarse-graining” prescriptions. The proof of the H-theorem given by v. Neumann employs assumptions about the density matrix of pure states and macroscopic distinguishability but excludes explicitly collapse or measurement processes from the quantum dynamics. In v. Neumann’s approach the quantum dynamics stays always unitary [52, 53]. This may seem surprising, because the collapse processes underlying the golden rule are inherently irreversible, as noted first also by v. Neumann [21]. To our opinion, the intrinsic probabilistic features of quantum mechanics encoded in the Born rule add an elementary irreversible process to the dynamical laws of nature. This process, the non-unitary collapse of the wavefunction, appears during macroscopic measurements but also as microscopic event, thereby leading to the correct statistics of a photon gas interacting with matter. Therefore, we disagree with the position put forward in [54], that the irreversibility of the measurement is just due to the macroscopic nature of the apparatus and has essentially the same origin as the irreversible behavior of macroscopic variables in classical mechanics which obeys the second law of thermodynamics. We have shown that the presence of microscopic collapse processes may lead under certain circumstances to a conflict with this law.

In conclusion, transition rates of quantum systems are commonly calculated with great success by using Fermi’s golden rule. This approach is widely accepted, as the golden rule directly results from the Born rule. Here, we have introduced practically realizable, open and closed quantum systems of coupled cavities and determined their behavior by applying the golden rule. The predicted behavior of both systems violates the second law of thermodynamics. We therefore conclude that

  1. (1)

    the statistical description of quantum mechanical transitions given by the golden rule is incorrect or

  2. (2)

    the second law of thermodynamics is not universally valid.