Non-hermitian quantum thermodynamics

Thermodynamics is the phenomenological theory of heat and work. Here we analyze to what extent quantum thermodynamic relations are immune to the underlying mathematical formulation of quantum mechanics. As a main result, we show that the Jarzynski equality holds true for all non-hermitian quantum systems with real spectrum. This equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium. In the quasistatic limit however, the second law leads to the Carnot bound which is fulfilled even if some eigenenergies are complex provided they appear in conjugate pairs. Furthermore, we propose two setups to test our predictions, namely with strongly interacting excitons and photons in a semiconductor microcavity and in the non-hermitian tight-binding model.

Scientific RepoRts | 6:23408 | DOI: 10.1038/srep23408 slow process. The angular brackets denote the average over an ensemble of finite-time realizations of the process characterized by their nonequilibrium work W.
The present study is dedicated to an even more fundamental question. In the following we will analyze to what extent quantum thermodynamic relations are immune to the underlying mathematical formulation of quantum mechanics. Contrary to different studies (see e.g. 11 ) conducted on a similar subject we present the broadest possible class of non-hermitian systems that still allows a thermodynamic theory in the "conventional" sense.
As a main result we will prove that equilibrium as well as non-equilibrium identities of quantum thermodynamics hold, without modification also for quantum systems described by pseudo-hermitian Hamiltonians 15 . Those systems have either entirely real spectrum or complex eigenvalues appear in complex conjugate pairs. In particular, we will show that the Carnot statement of the second law of thermodynamics holds for any such system and that the quantum Jarzynski equality is not violated as long as the eigenvalue spectrum is real. If the two-time energy measurement could be realized e.g. in a microcavity 6 , then the Jarzynski equality for pseudo-hermitian systems could be put into a test (see Discussion).

Fundamentals of Pseudo-Hermitian Quantum Mechanics
To address physical properties of recent experiments 4,6 we start by briefly reviewing the mathematical foundations of pseudo-hermitian quantum mechanics 8 . Let H be a general, non-hermitian Hamiltonian of a physical system, and we assume for the sake of simplicity that the spectrum of H, {E n }, is discrete (possibly degenerate). Such a Hamiltonian is of physical relevance only if it is measurable, i.e., if a representation of the eigenbasis α E n, is experimentally accessible. Then H is diagonal in this basis. Here n is the quantum number and α counts possible degeneracy. Diagonalizability of H is equivalent to the existence of biorthonormal set of left, φ α n, , and right, ψ α n, , eigenvectors 16 . In general, the energy eigenvalues are complex, and the eigenvalue problem reads 15  17 . However, when heat is exchanged the two-time energy measurement can no longer describe the work done during a thermodynamic process. Therefore we shall not focus on such cases here. Another interesting class relates to systems that interact with environments, but do not exchange heat. This phenomenon is called dephasing (loss of information) 18 . For such systems, work can still be determined by the two-time energy measurement and the Jarzynski equality holds as well [19][20][21] . Condition (2) assures that H is, in fact, hermitian however with respect to a new inner product, namely g Note that g always exists such that 〈 ψ|φ〉 g is positive-definite (this is a genuine inner product), and it can be found if and only if the spectrum of H is real. To make a consistent definition of work for a quantum system within the two-time energy measurement paradigm its spectrum has to be real. Therefore, unless stated otherwise, we shall always assume this to be the case. Then, Eq. (2) can be fulfilled by the following positive-definite operators (g is a proper metric operator) 22 Often, g fulfilling (2) can be deduced easily from physical properties such as the parity reflection or time reversal 23 . Nevertheless, only Eq. (4) assures that 〈 ψ|ψ〉 g > 0 for all states ψ ≠ 0. This means that the proper metric may reflect "symmetries" that are hidden from the observer 24,25 . For instance, if a rotation V exists such that V −1 HV is diagonal in an orthonormal basis, then g = V † V. This follows directly from Eq. (4). The last formula is especially useful in practice. It allows one to find the metric by analyzing an experimental setup (e.g. inspecting the orientation of the axis, etc.). In the following we only consider cases where changes of the Hamiltonian are induced by a time-dependent thermodynamic process λ t , that is to say H t = H(λ t ). If such changes occur then the metric operator satisfying Eq. (2) is time-dependent. Nevertheless, the dynamics is still governed by a time-dependent Schrödinger equation. However, a slight modification becomes necessary to preserve unitarity 26,27 , Above, ∂ t denotes the derivative with respect to time t. The Schrödinger equation (5) can also be rewritten in the standard form, that is, with H t being the generator. Indeed, it is sufficient to replace ∂ t with a covariant derivative = ∂ + ∂ t t t t t 1 28 . By construction the unique solution to Eq. (5) obeys the relation This relation can be viewed as the corresponding unitarity condition similar to the "standard" one, i.e., . For pseudo-hermitian systems an average value of a non-hermitian observable A, tr{A}, can be computed as Formally, this suggests one to use the following Dirac correspondence between bra and ket vectors ψ ψ ↔ g 16 .

Pseudo-hermitian Jarzynski equality.
Having analyzed the mathematical structure of pseudo-hermitian quantum systems, we turn to the physical description to analyze the Jarzynski equality. Without loss of generality and to simplify our notation we assume the spectrum to be non-degenerate.
For an isolated quantum system, the work done during a thermodynamic process λ t of duration τ is commonly determined by a two-time energy measurement 29 . At t = 0 a projective energy measurement is performed. Next, the system evolves unitarily under the generalized time-dependent Schrödinger equation (5) only to be measured again at t = τ. By averaging over an ensemble of realizations of such processes one can reconstruct the distribution of work values 30,31 , n is the corresponding work done during this transition. It is important to stress that this work is associated with H t rather than H t + G t as G t is a gauge field, and hence it can have no influence on physical observables 32 .
The transition probability p nm can be seen as the joint probability that the first measurement will yield the energy value E n given the system has been initially prepared in a state ρ 0 , and the probability that the outcome of the second measurement will be τ E m given the initial state ψ n . Therefore, where U τ denotes the evolution operator generated by H t + G t at time t = τ, whereas Π n = 〈 ψ n , g 0 ⋅ 〉 ψ n is the projector into the space spanned by the nth eigenstate. Since Π n is not hermitian the formula for probabilities p nm accounts for the metric g, and hence differs from the one usually adopted for hermitian systems 31 .
Assume the system is initially in a Gibbs state, that is nm E m n m 0 1 n To obtain the last expression for p nm we have also invoked the unitarity condition (6). Now, the average exponentiated work can be expressed as The last equation shows that the Jarzynski equality holds also for non-hermitian systems that admit real spectrum. This is our first main result. Jarzynski has shown that the second law of thermodynamics for isothermal processes can be expressed as an equality arbitrarily far from equilibrium. Our analysis has shown that his result is true for all non-hermitian systems with real spectrum.

Carnot bound.
In the preceding section we argued that if the two-time energy measurement can be performed on a non-hermitian quantum system, then the Jarzynski equality holds as long as the eigenenergies are real. Now, we will prove that the Carnot statement of the second law is also true for all pseudo-hermitian systems. Consider a generic system that operates between two heat reservoirs with hot, T h , and cold, T c , temperatures, respectively. Then, the Carnot engine consists of two isothermal processes during which the system absorbs or exhausts heat and two thermodynamically adiabatic, that is, isentropic strokes while the extensive control parameter λ is varied 33,34 . It is well established that the maximum efficiency η for classical systems, attained in the quasistatic limit, is given by the Carnot bound [35][36][37] : Recent years have witnessed an abundance of research [38][39][40] investigating whether quantum correlations can be harnessed to break this limit. Recently, the Carnot limit has been proven to be universal within the usual Scientific RepoRts | 6:23408 | DOI: 10.1038/srep23408 framework 33 . This limit can be seen as yet another formulation of the second law of thermodynamics for quasistatic processes. We will show that it holds for all pseudo-hermitian systems whether their spectrum is real or not.
We begin by proving that both the energy E = tr{ρH} and entropy S are real in our present framework. Indeed, from (2) it immediately follows that  showing that if E n is in the spectrum of H so is ⁎ E n . Moreover g −1 maps the subspace spanned by all eigenvectors belonging to E n to that belonging to ⁎ E n . Since g −1 is invertible, the mapping is one-to-one, and the multiplicity of both E n and ⁎ E n is the same. An interesting realization of such systems is the non-hermitian tight-binding model 41 . The result (14) can also be obtained directly, that is, without invoking the metric g explicitly. Indeed, we have In the present case, the thermodynamic entropy is given by the von Neumann entropy 42 . The latter can be further simplified and it takes the well known form S = β(E − F) 33 . Since the partition function Z is real so is the free energy F. Hence, we conclude that the entropy S is real.
According to the first law of thermodynamics 43 , dE = δQ + δW, there are two forms of energy: heat δQ is the change of internal energy associated with a change of entropy, whereas work δW is the change of internal energy due to the change of an extensive parameter, i.e., change of the Hamiltonian of the system. To identify those contributions we write 33 In the quasistatic regime, the second law of thermodynamics for isothermal processes states that dS = βδQ.
Combining the latter with (17) proves that (i) δQ and thus δW are real and (ii) the intuitive definitions of heat and work introduced in Ref. 44 apply also to pseudo-hermitian systems. After completing a cycle, a quantum pseudo-hermitian heat engine has performed work 〈 W〉 = 〈 Q h 〉 − 〈 Q c 〉 and exhausted a portion of heat 〈 Q c 〉 to the cold reservoir. Therefore, the efficiency of such a device is given by 33 In conclusion, we have shown that the Carnot bound, which expresses the second law of thermodynamics for quasistatic processes, holds for all pseudo-hermitian systems. In contrast, the second law for arbitrarily fast processes encoded in the Jarzynski equality (12), only holds for all non-hermitian systems with real spectrum.

Discussion
Example 1a. We begin with a model for localization effects in solid state physics 41 . The general form of its Hamiltonian in one dimension reads Therefore, since [V(x), e 2ξx ] = 0, we conclude that H is pseudo-hermitian. The metric g = e 2ξx is positive definite and thus the spectrum of (19) is real. Further, we assume that the corresponding classical potential V c (x) has a non-vanishing second derivative, and a minimum at x = 0 (e.g. ′ = V (0) 0 c ). Then Scientific RepoRts | 6:23408 | DOI: 10.1038/srep23408 where H n (x) are the Hermite polynomials. Now we assume that the size of this harmonic trap (e.g. ω) is changed, and thus g does not depend on time. Experimentally, harmonic traps are sensitive to initial excitations resulting for a discontinuity of the protocol itself at the beginning 45 . The most common way to minimize this effect, while quenching between ω i , and ω f , is to use functions smooth enough at the "edges", for instance, where erf(⋅ ) denotes the error function, τ is a time scale, and N is an integer emulating infinity. The transition probabilities (9) can be expressed via the following integral where the partition function Z 0 = 1/sinh(βω i /2) has been calculated exactly; and ψ m (x, Nτ) = U Nτ ψ n (x) is the solution of Eq. (5), with the initial condition given by (22), at t = Nτ. Although ψ m (x, Nτ) cannot be obtained analytically, a closed form expressed in terms of a solution to the corresponding classical equation of motion can be found (see e.g. Ref. 46). Figure 1 (Left panel) shows the average exponentiated work 〈 e −βW 〉 (blue curve) as a function of the number of terms N max included in the summation (11). This function quickly converges to e −βΔF proving that the Jarzynski equality (12) holds. On the right panel we have depicted the irreversible work 〈 W irr 〉 = 〈 W〉 − ΔF (blue curve) as a function of τ which determines the speed at which the energy is supplied to the system. When τ → ∞ the system enters its quasistatic regime and the irreversible work becomes negligible, that is 〈 W irr 〉 → 0 47,48 . The inset (red curve) shows the irreversible work calculated for a linear protocol, ω(t) = ω i + (ω f − ω i )t/τ. As we can see, it takes longer for the system to reach its quasistatic regime. Moreover, the oscillatory behavior is a signature of the initial excitation which dominates for fast quenches (small τ). Example 1b. Another class of systems that is used to explain localization effects relates to non-hermitian tight-binding models 49,50 . For example where, † a x and a x are bosonic creation and annihilation operators respectively, e ν are the unit lattice vectors, and t is the hopping parameter, and V x denotes the on-site potential. Interestingly, the complex eigenvectors appear in conjugate pairs (see Eq. (2) in Ref. 41 and the discussion that follows). Therefore, this model provides another example for a building block of a non-hermitian Carnot engine.

Example 2.
The remainder of the present work is dedicated to a careful study of a second, experimentally relevant example 6 . Consider a two level system described by the Hamiltonian showing that the Jarzynski equality (12) holds. Right panel: 〈 W irr 〉 = 〈 W〉 − ΔF as a function of τ which relates to the speed at which the energy is supplied to the system. The irreversible work 〈 W irr 〉 → 0 as τ approaches the quasistatic regime. The inset (red curve) shows the irreversible work calculated for a linear protocol, ω(t) = ω i + (ω f − ω i )t/τ. We see that it takes longer for the system to reach its quasistatic regime. Parameters used in the numerical simulations are: w i = 0.2, w f = 0.6, Nτ = 1.5 (left panel) and Nτ = 3. (right panel); the remaining parameters were set to 1.
Scientific RepoRts | 6:23408 | DOI: 10.1038/srep23408 where λ t is a complex control parameter, and γ is a complex constant, whereas σ + and σ − are the raising and lowering fermionic operators. This simple model (26) has been extensively studied in the literature 11,51,52 , and it has been also realized experimentally both in optics 4 and semiconductor microcavities 6 .
To make the spectrum of (26) real we set λ t to be purely imaginary (λ t → iλ t ); and without any loss of generality we choose γ = 1. This corresponds to the following parameters E 1,2 = 0, Γ 1,2 = ± λ t , and q = γ = 1 for the hybrid light-matter system of quasiparticles investigated in Ref. 6. Such systems are formed as a result of a strong interaction between excitons and photons in a semiconductor microcavity 53 . They are commonly referred to as exciton-polaritons 54 .
A simple calculation shows that where σ x is the Pauli matrix in x direction. Thus H t is indeed pseudo-hermitian. However, the corresponding σ x is not a metric. For instance 〈 e 1 , σ x e 1 〉 = 0, where = e (1, 0) t 1 . Nevertheless, we can easily find one by rewriting H t in its diagonal form, Note, both E t 1,2 are real as long as λ t ≤ 1, otherwise = ⁎ E E t t 1 2 . Therefore, the Carnot bound (13) holds in both these regimes, whereas the Jarzynski equality (12) only in the first one. Now, the proper metric can be defined via the similarity transformation V t To investigate the dynamics of (26) we assume that λ t changes on a time scale τ in a linear manner, that is The relaxation time diverges as λ f approaches the critical point at λ = 1. Similar behavior has been observed for the irreversible work 〈 W irr 〉 := 〈 W〉 − ΔF in PT -symmetric systems 11 . The critical point separates the unbroken domain, where energies are real, from the broken one characterized by complex energy values. The energetic cost associated with a potential crossover between those two regimes becomes infinite, and the system "freezes out" before even having a chance to cross to the other regime 56,57 .
In the broken regime, Eq. (28) no longer reflects pseudo-hermiticity of the system, that is V t does not fulfill Eq. (4). In fact, all operators g for which the latter equation is true, σ x being an example (see Fig. 2, Right panel), lead to indefinite inner product spaces. Note that in Fig. 2 (Right panel) the norm can be both positive and negative. Therefore, the evolution within those spaces cannot be unitary and the two-time energy measurement paradigm can no longer be applied 58 . In the quasistatic limit, however, quantum work can still be defined, and we have shown that the second law still holds for all pseudo-hermitian systems.

Conclusions
In summary, we have carefully studied thermodynamic properties of quantum systems that do not satisfy one of the basic requirements imposed on them by the axiom of quantum mechanics -hermiticity. We have shown that if quantum work can be determined by the two-time projective energy measurements, then the Jarzynski equality still holds for non-hermitian systems with real spectrum. Note, this equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium. , as a function of the final value λ f for the linear quench λ t = λ i + (λ f − λ i )t/τ. Parameters are λ i = 0, β = ħ = τ = 1. Inset: numerical confirmation of the Jarzynski equality (12). Right panel: In the broken regime quantum work can no longer be determined by the two-time energy measurement as 〈 ψ, gψ〉 can be both positive and negative. To construct the plot we set g = σ x . States ψ(n) have been chosen randomly; and n is an integer that has been assigned to them.
Scientific RepoRts | 6:23408 | DOI: 10.1038/srep23408 We have also argued that the Carnot bound is attained for all pseudo-hermitian systems in the quasistatic limit. Furthermore, we have also proposed an experimental setup to test our predictions. As elaborated in the previous section, the system in question consists of strongly interacting excitons and photons in a semiconductor microcavity 6 . Moreover, we have investigated two non-hermitian models that were originally introduced to explain localization effects in solid state physics 41 . The first one, a non-hermitian harmonic oscillator that admits real spectrum was used to demonstrate the Jarzynski equality. The second one, the so called non-hermitian tight-binding model was given as an example of a quantum system having complex eigenenergies that appear in conjugate pairs. This model provides another example of a building block of a non-hermitian Carnot engine.