Abstract
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 nonhermitian 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 nonhermitian tightbinding model.
Introduction
More and more nonhermitian systems are becoming experimentally accessible^{1}. Therefore, it has become evident that questions concerning foundations of quantum mechanics are no longer only of academic interest. Recent experiments have demonstrated that hermiticity may not be as fundamental as mandated by quantum mechanics^{2,3}. For instance, in^{4} a spontaneous symmetry breaking has been observed indicating a condition weaker than hermiticity (namely ^{5}) being realized in nature. Furthermore, in^{6} exceptional eigenenergies of complex value have been measured challenging the reality of the spectrum imposed by hermiticity.
Conventional quantum mechanics is built upon the Diracvon Neumann axioms^{2,3}. These state that if is a complex Hilbert space of countable, infinite dimension, then (i) observables of a quantum system are defined as hermitian operators O on , (ii) quantum states are unit vectors in and (iii) the expectation value of an observable O in a state is given by the inner product, 〈O〉 = 〈ϕOϕ〉. Interestingly, only axioms (ii) and (iii) are of mathematical necessity needed for a proper probabilistic, physical theory. To demand, however, that any quantum mechanical theory has to be built on hermitian operators is rather mathematically convenient than being fundamentally necessary^{5,7}.
In particular, the restriction to hermitian observables excludes the description of, for instance, quantum field theories with symmetry, cases where the language of quantum mechanics is used for problems within classical statistical mechanics or diffusion in biological systems, or cases where effective complex potentials are introduced to describe interactions at edges^{8}. Particularly striking examples are optical systems with complex index of refraction. Imagine, for instance, polarized light in a stratified, nontransparent, biaxially anisotropic, dielectric medium warped cyclically along the propagation direction. For such systems it has been shown^{9} that not only a nonhermitian description becomes necessary, but also that physical intuition has to be invoked carefully. For instance, Berry highlighted^{9} that adiabatic intuition can be countered dramatically for systems with nonhermitian Hamiltonians.
Very recently, it has become evident that for a special class of nonhermitian systems, namely in symmetric quantum mechanics^{10}, the quantum Jarzynski equality holds without modification^{11}. For isolated quantum systems evolving under unitary dynamics the socalled twotime energy measurement approach has proven to be practical and powerful. In this paradigm, quantum work is determined by projective energy measurements at the beginning and the end of a process induced by an externally controlled Hamiltonian. The Jarzynski equality^{12} together with subsequent Nonequilibrium Work Theorems, such as the Crooks fluctuation theorem^{13}, is undoubtedly among the most important breakthroughs in modern Statistical Physics^{14}. Jarzynski showed that for isothermal processes the second law of thermodynamics can be formulated as an equality, no matter how far from equilibrium the system is driven^{12}, 〈exp(−βW)〉 = exp(−βΔF). Here β is the inverse temperature of the environment and ΔF is the free energy difference, i.e., the work performed during an infinitely slow process. The angular brackets denote the average over an ensemble of finitetime 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 nonhermitian systems that still allows a thermodynamic theory in the “conventional” sense.
As a main result we will prove that equilibrium as well as nonequilibrium identities of quantum thermodynamics hold, without modification also for quantum systems described by pseudohermitian 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 twotime energy measurement could be realized e.g. in a microcavity^{6}, then the Jarzynski equality for pseudohermitian systems could be put into a test (see Discussion).
Fundamentals of PseudoHermitian Quantum Mechanics
To address physical properties of recent experiments^{4,6} we start by briefly reviewing the mathematical foundations of pseudohermitian quantum mechanics^{8}. Let H be a general, nonhermitian 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 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, and right, , eigenvectors^{16}. In general, the energy eigenvalues are complex and the eigenvalue problem reads^{15}
with 〈ψ_{n,α}ϕ_{m,β}〉 = δ_{mn}δ_{αβ} and . A nonhermitian Hamiltonian such as (1) is called pseudohermitian if a g exists such that
It does exist if and only if either all eigenenergies are real or complex ones appear in conjugate pairs with the same degeneracy^{15}. If none of those criteria are met H is generally nonhermitian^{8}; yet it still can be useful, e.g. for an effective description of open quantum systems^{17}. However, when heat is exchanged the twotime 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 twotime 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
Note that g always exists such that 〈ψϕ〉_{g} is positivedefinite (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 positivedefinite 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 〈ψψ〉_{g} > 0 for all states . 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 timedependent. Nevertheless, the dynamics is still governed by a timedependent 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 ^{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 pseudohermitian systems an average value of a nonhermitian observable A, tr{A}, can be computed as
Formally, this suggests one to use the following Dirac correspondence between bra and ket vectors ^{16}.
Pseudo–hermitian Jarzynski equality
Having analyzed the mathematical structure of pseudohermitian 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 nondegenerate.
For an isolated quantum system, the work done during a thermodynamic process λ_{t} of duration τ is commonly determined by a twotime energy measurement^{29}. At t = 0 a projective energy measurement is performed. Next, the system evolves unitarily under the generalized timedependent 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},
Above, p_{nm} denotes a probability that a specific transition will occur, whereas 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 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 ρ_{0} = exp(−βH_{0})/Z_{0} with Z_{0} = tr{exp(−βH_{0})} being the partition function, then
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
Finally, summing out all projectors Π_{n} and taking into account that we arrive at
where F = (−1/β) ln(Z) is the system’s free energy.
The last equation shows that the Jarzynski equality holds also for nonhermitian 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 nonhermitian systems with real spectrum.
Carnot bound
In the preceding section we argued that if the twotime energy measurement can be performed on a nonhermitian 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 pseudohermitian 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 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 pseudohermitian 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
with ρ being a Gibbs thermal state. Interestingly, this result holds true even if some of the eigenvalues E_{n} are complex. Note, in that case g exists but is not positive definite and thus cannot be expressed like in Eq. (4).
To understand why Eq. (14) holds when complex eigenvalues appear in conjugate pairs note that and consider
showing that if E_{n} is in the spectrum of H so is . Moreover g^{−1} maps the subspace spanned by all eigenvectors belonging to E_{n} to that belonging to . Since g^{−1} is invertible, the mapping is onetoone and the multiplicity of both E_{n} and is the same. An interesting realization of such systems is the nonhermitian tightbinding 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 pseudohermitian systems.
After completing a cycle, a quantum pseudohermitian 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 pseudohermitian systems. In contrast, the second law for arbitrarily fast processes encoded in the Jarzynski equality (12), only holds for all nonhermitian 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
where V(x) is a confining potential and p and x are the momentum and position operators respectively. They obey the canonical commutation relation [x, p] = iħ. Real parameter ξ expresses an external magnetic field and m is the mass. Using the BakerCampbellHausdorff formula one can verify that
Therefore, since [V(x), e^{2ξx}] = 0, we conclude that H is pseudohermitian. 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 nonvanishing second derivative and a minimum at x = 0 (e.g. ). Then
where has been introduced. After quantization, the eigenvalues and eigenvectors of this nonhermitian harmonic oscillator read (for the sake of simplicity we set m = ħ = 1 throughout)
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 nonhermitian tightbinding models^{49,50}. For example
where, 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 onsite 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 nonhermitian 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
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 pseudohermitian. However, the corresponding σ_{x} is not a metric. For instance 〈e_{1}, σ_{x}e_{1}〉 = 0, where . Nevertheless, we can easily find one by rewriting H_{t} in its diagonal form,
Note, both are real as long as λ_{t} ≤ 1, otherwise . 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 λ_{t} = λ_{i} + (λ_{f} − λ_{i})t/τ. The linearity does not pose any restriction on our analysis as the Jarzynski equality holds for all protocols λ_{t}^{11}. Figure 2 (Left panel) depicts the relaxation time T_{r} = Δ^{−1}, where , as a function of the final value λ_{f}^{55}. 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 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 pseudohermiticity 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 twotime 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 pseudohermitian 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 twotime projective energy measurements, then the Jarzynski equality still holds for nonhermitian systems with real spectrum. Note, this equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium.
We have also argued that the Carnot bound is attained for all pseudohermitian 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 nonhermitian models that were originally introduced to explain localization effects in solid state physics^{41}. The first one, a nonhermitian harmonic oscillator that admits real spectrum was used to demonstrate the Jarzynski equality. The second one, the so called nonhermitian tightbinding 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 nonhermitian Carnot engine.
Additional Information
How to cite this article: Gardas, B. et al. Nonhermitian quantum thermodynamics. Sci. Rep. 6, 23408; doi: 10.1038/srep23408 (2016).
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Acknowledgements
This work was supported by the Polish Ministry of Science and Higher Education under project Mobility Plus 1060/MOB/2013/0 (B.G.); S.D. acknowledges financial support from the U.S. Department of Energy through a LANL Director’s Funded Fellowship.
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B.G., S.D. and A.S. developed ideas and derived the main results. B.G. prepared Figures 1 and 2. B.G., S.D. and A.S. wrote and reviewed the manuscript.
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Gardas, B., Deffner, S. & Saxena, A. Nonhermitian quantum thermodynamics. Sci Rep 6, 23408 (2016). https://doi.org/10.1038/srep23408
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