Quantum thermodynamics

When ${\displaystyle O=H_{S}}$ the first law of thermodynamics emerges:

${\displaystyle {\frac {dE}{dt}}=\left\langle {\frac {\partial H_{S}}{\partial t}}\right\rangle +\langle L_{D}^{*}(H_{S})\rangle }$

where power is interpreted as ${\displaystyle P=\left\langle {\frac {\partial H_{S}}{\partial t}}\right\rangle }$ and the heat current ${\displaystyle J=\langle L_{D}^{*}(H_{S})\rangle }$.[8][9][10]

Additional conditions have to be imposed on the dissipator ${\displaystyle L_{D}}$ to be consistent with thermodynamics. First the invariant ${\displaystyle \rho _{S}(\infty )}$ should become an equilibrium Gibbs state. This implies that the dissipator ${\displaystyle L_{D}}$ should commute with the unitary part generated by ${\displaystyle H_{S}}$.[5] In addition an equilibrium state is stationary and stable. This assumption is used to derive the Kubo-Martin-Schwinger stability criterion for thermal equilibrium i.e. KMS state.

A unique and consistent approach is obtained by deriving the generator, ${\displaystyle L_{D}}$, in the weak system bath coupling limit.[11] In this limit, the interaction energy can be neglected. This approach represents a thermodynamic idealization: it allows energy transfer, while keeping a tensor product separation between the system and bath, i.e., a quantum version of an isothermal partition.

Markovian behavior involves a rather complicated cooperation between system and bath dynamics. This means that in phenomenological treatments, one cannot combine arbitrary system Hamiltonians, ${\displaystyle H_{S}}$, with a given L-GKS generator. This observation is particularly important in the context of quantum thermodynamics, where it is tempting to study Markovian dynamics with an arbitrary control Hamiltonian. Erroneous derivations of the quantum master equation can easily lead to a violation of the laws of thermodynamics.

An external perturbation modifying the Hamiltonian of the system will also modify the heat flow. As a result, the L-GKS generator has to be renormalized. For a slow change, one can adopt the adiabatic approach and use the instantaneous system’s Hamiltonian to derive ${\displaystyle L_{D}}$. An important class of problems in quantum thermodynamics is periodically driven systems. Periodic quantum heat engines and power-driven refrigerators fall into this class.

A reexamination of the time-dependent heat current expression using quantum transport techniques has been proposed.[12]

A derivation of consistent dynamics beyond the weak coupling limit has been suggested.[13]

The second law of thermodynamics is a statement on the irreversibility of dynamics or, the breakup of time reversal symmetry (T-symmetry). This should be consistent with the empirical direct definition: heat will flow spontaneously from a hot source to a cold sink.

From a static viewpoint, for a closed quantum system, the 2nd law of thermodynamics is a consequence of the unitary evolution.[14] In this approach, one accounts for the entropy change before and after a change in the entire system. A dynamical viewpoint is based on local accounting for the entropy changes in the subsystems and the entropy generated in the baths.

In thermodynamics, entropy is related to a concrete process. In quantum mechanics, this translates to the ability to measure and manipulate the system based on the information gathered by measurement. An example is the case of Maxwell’s demon, which has been resolved by Leó Szilárd.[15][16][17]

The entropy of an observable is associated with the complete projective measurement of an observable, ${\displaystyle \langle A\rangle }$, where the operator ${\displaystyle A}$ has a spectral decomposition: ${\displaystyle A=\sum _{j}\alpha _{i}P_{j}}$ where ${\displaystyle P_{j}}$ is the projection operators of the eigenvalue ${\displaystyle \alpha _{j}}$. The probability of outcome j is ${\displaystyle p_{j}=Tr(\rho P_{j})}$ The entropy associated with the observable ${\displaystyle \langle A\rangle }$ is the Shannon entropy with respect to the possible outcomes:

${\displaystyle S_{A}=-\sum _{j}p_{j}\ln p_{j}}$

The most significant observable in thermodynamics is the energy represented by the Hamiltonian operator ${\displaystyle H}$, and its associated energy entropy, ${\displaystyle S_{E}}$.[18]

John von Neumann suggested to single out the most informative observable to characterize the entropy of the system. This invariant is obtained by minimizing the entropy with respect to all possible observables. The most informative observable operator commutes with the state of the system. The entropy of this observable is termed the Von Neumann entropy and is equal to:

${\displaystyle S_{vn}=-Tr(\rho \ln \rho )}$

As a consequence, ${\displaystyle S_{A}\geq S_{vn}}$ for all observables. At thermal equilibrium the energy entropy is equal to the von Neumann entropy: ${\displaystyle S_{E}=S_{vn}}$.

${\displaystyle S_{vn}}$ is invariant to a unitary transformation changing the state. The Von Neumann entropy ${\displaystyle S_{vn}}$ is additive only for a system state that is composed of a tensor product of its subsystems:

${\displaystyle \rho =\Pi _{j}\otimes \rho _{j}}$

No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.

This statement for N-coupled heat baths in steady state becomes:

${\displaystyle \sum _{n}{\frac {J_{n}}{T_{n}}}\geq 0}$

A dynamical version of the II-law can be proven, based on Spohn’s inequality[19]

${\displaystyle Tr\left(L_{D}\rho [\ln \rho (\infty )-\ln \rho ]\right)\geq 0}$

which is valid for any L-GKS generator, with a stationary state, ${\displaystyle \rho (\infty )}$.[5]

Consistency with thermodynamics can be employed to verify quantum dynamical models of transport. For example, local models for networks where local L-GKS equations are connected through weak links have been shown to violate the second law of thermodynamics.[20]

Thermodynamic adiabatic processes have no entropy change. Typically, an external control modifies the state. A quantum version of an adiabatic process can be modeled by an externally controlled time dependent Hamiltonian ${\displaystyle H(t)}$. If the system is isolated, the dynamics are unitary, and therefore, ${\displaystyle S_{vn}}$ is a constant. A quantum adiabatic process is defined by the energy entropy ${\displaystyle S_{E}}$ being constant. The quantum adiabatic condition is therefore equivalent to no net change in the population of the instantaneous energy levels. This implies that the Hamiltonian should commute with itself at different times: ${\displaystyle [H(t),H(t')]=0}$.

When the adiabatic conditions are not fulfilled, additional work is required to reach the final control value. For an isolated system, this work is recoverable, since the dynamics is unitary and can be reversed. In this case, quantum friction can be suppressed using shortcuts to adiabaticity as demonstrated in the laboratory using a unitary Fermi gas in a time-dependent trap.[21] The coherence stored in the off-diagonal elements of the density operator carry the required information to recover the extra energy cost and reverse the dynamics. Typically, this energy is not recoverable, due to interaction with a bath that causes energy dephasing. The bath, in this case, acts like a measuring apparatus of energy. This lost energy is the quantum version of friction.[22][23]

There are seemingly two independent formulations of the third law of thermodynamics both originally were stated by Walther Nernst. The first formulation is known as the Nernst heat theorem, and can be phrased as:

• The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.

The second formulation is dynamical, known as the unattainability principle[24]

• It is impossible by any procedure, no matter how idealized, to reduce any assembly to absolute zero temperature in a finite number of operations.

At steady state the second law of thermodynamics implies that the total entropy production is non-negative. When the cold bath approaches the absolute zero temperature, it is necessary to eliminate the entropy production divergence at the cold side when ${\displaystyle T_{c}\rightarrow 0}$, therefore

${\displaystyle {\dot {S}}_{c}\propto -T_{c}^{\alpha }~~~,~~~~\alpha \geq 0~~.}$

For ${\displaystyle \alpha =0}$ the fulfillment of the second law depends on the entropy production of the other baths, which should compensate for the negative entropy production of the cold bath. The first formulation of the third law modifies this restriction. Instead of ${\displaystyle \alpha \geq 0}$ the third law imposes ${\displaystyle \alpha >0}$, guaranteeing that at absolute zero the entropy production at the cold bath is zero: ${\displaystyle {\dot {S}}_{c}=0}$. This requirement leads to the scaling condition of the heat current ${\displaystyle {J}_{c}\propto T_{c}^{\alpha +1}}$.

The second formulation, known as the unattainability principle can be rephrased as;[25]

• No refrigerator can cool a system to absolute zero temperature at finite time.

The dynamics of the cooling process is governed by the equation

${\displaystyle {J}_{c}(T_{c}(t))=-c_{V}(T_{c}(t)){\frac {dT_{c}(t)}{dt}}~~.}$

where ${\displaystyle c_{V}(T_{c})}$ is the heat capacity of the bath. Taking ${\displaystyle {J}_{c}\propto T_{c}^{\alpha +1}}$ and ${\displaystyle c_{V}\sim T_{c}^{\eta }}$ with ${\displaystyle {\eta }\geq 0}$, we can quantify this formulation by evaluating the characteristic exponent ${\displaystyle \zeta }$ of the cooling process,

${\displaystyle {\frac {dT_{c}(t)}{dt}}\propto -T_{c}^{\zeta },~~~~~T_{c}\rightarrow 0,~~~~~{\zeta =\alpha -\eta +1}}$

This equation introduces the relation between the characteristic exponents ${\displaystyle \zeta }$ and ${\displaystyle \alpha }$. When ${\displaystyle \zeta <0}$ then the bath is cooled to zero temperature in a finite time, which implies a valuation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the Nernst heat theorem.