Quantum thermodynamics

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https://en.wikipedia.org/wiki/Quantum_thermodynamics

Quantum thermodynamics is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics. The two independent theories address the physical phenomena of light and matter. In 1905, Albert Einstein argued that the requirement of consistency between thermodynamics and electromagnetism leads to the conclusion that light is quantized, obtaining the relation ${\displaystyle E=h\nu }$. This paper is the dawn of quantum theory. In a few decades quantum theory became established with an independent set of rules. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. In addition, there is a quest for the theory to be relevant for a single individual quantum system.

Dynamical view

There is an intimate connection of quantum thermodynamics with the theory of open quantum systems. Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics. The main assumption is that the entire world is a large closed system, and therefore, time evolution is governed by a unitary transformation generated by a global Hamiltonian. For the combined system bath scenario, the global Hamiltonian can be decomposed into:

${\displaystyle H=H_{\mathrm{S}}+H_{\mathrm{B}}+H_{\mathrm{S}\mathrm{B}}}$

where ${\displaystyle H_{\mathrm{S}}}$ is the system Hamiltonian, ${\displaystyle H_{\mathrm{B}}}$ is the bath Hamiltonian and ${\displaystyle H_{\mathrm{S}\mathrm{B}}}$ is the system-bath interaction.

The state of the system is obtained from a partial trace over the combined system and bath:

${\displaystyle {\rho }_{\mathrm{S}}(t)={\mathrm{T}\mathrm{r}}_{\mathrm{B}}({\rho }_{\mathrm{S}\mathrm{B}}(t))}$.

Reduced dynamics is an equivalent description of the system dynamics utilizing only system operators. Assuming Markov property for the dynamics the basic equation of motion for an open quantum system is the Lindblad equation (GKLS):

${\displaystyle {\dot{\rho }}_{\mathrm{S}}=-\frac{i}{\hslash }[H_{\mathrm{S}},{\rho }_{\mathrm{S}}]+L_{\mathrm{D}}({\rho }_{\mathrm{S}})}$

${\displaystyle H_{\mathrm{S}}}$ is a (Hermitian) Hamiltonian part and ${\displaystyle L_{\mathrm{D}}}$:

${\displaystyle L_{\mathrm{D}}({\rho }_{\mathrm{S}})=\sum _n\left(V_n{\rho }_{\mathrm{S}}{V}_n^{†}-\frac{1}{2}\left({\rho }_{\mathrm{S}}{V}_n^{†}{V}_n+V_n^{†}{V}_n{\rho }_{\mathrm{S}}\right)\right)}$

is the dissipative part describing implicitly through system operators ${\displaystyle V_n}$ the influence of the bath on the system. The Markov property imposes that the system and bath are uncorrelated at all times ${\displaystyle {\rho }_{\mathrm{S}\mathrm{B}}={\rho }_s\otimes {\rho }_{\mathrm{B}}}$. The L-GKS equation is unidirectional and leads any initial state ${\displaystyle {\rho }_{\mathrm{S}}}$ to a steady state solution which is an invariant of the equation of motion ${\displaystyle {\dot{\rho }}_{\mathrm{S}}(t\to \mathrm{\infty })=0}$.

The Heisenberg picture supplies a direct link to quantum thermodynamic observables. The dynamics of a system observable represented by the operator, ${\displaystyle O}$, has the form:

${\displaystyle \frac{dO}{dt}=\frac{i}{\hslash }[H_{\mathrm{S}},O]+L_{\mathrm{D}}^{\ast }(O)+\frac{\mathrm{\partial }O}{\mathrm{\partial }t}}$

where the possibility that the operator, ${\displaystyle O}$ is explicitly time-dependent, is included.

Emergence of time derivative of first law of thermodynamics

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

${\displaystyle \frac{dE}{dt}=⟨\frac{\mathrm{\partial }{H}_{\mathrm{S}}}{\mathrm{\partial }t}⟩+⟨L_{\mathrm{D}}^{\ast }(H_{\mathrm{S}})⟩}$

where power is interpreted as

${\displaystyle P=⟨\frac{\mathrm{\partial }{H}_{\mathrm{S}}}{\mathrm{\partial }t}⟩}$

and the heat current

${\displaystyle J=⟨L_{\mathrm{D}}^{\ast }(H_{\mathrm{S}})⟩}$.

Additional conditions have to be imposed on the dissipator ${\displaystyle L_{\mathrm{D}}}$ to be consistent with thermodynamics.

First the invariant ${\displaystyle {\rho }_{\mathrm{S}}(\mathrm{\infty })}$ should become an equilibrium Gibbs state. This implies that the dissipator ${\displaystyle L_{\mathrm{D}}}$ should commute with the unitary part generated by ${\displaystyle H_{\mathrm{S}}}$. 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, ${\displaystyle L_{\mathrm{D}}}$, in the weak system bath coupling limit. 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, ${\displaystyle H_{\mathrm{S}}}$, 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 ${\displaystyle L_{\mathrm{D}}}$. 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.

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

Phenomenological formulations of irreversible quantum dynamics consistent with the second law and implementing the geometric idea of "steepest entropy ascent" or "gradient flow" have been suggested to model relaxation and strong coupling.

Emergence of the second law

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. 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.

Entropy

In thermodynamics, entropy is related to the amount of energy of a system that can be converted into mechanical work in 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.

The entropy of an observable is associated with the complete projective measurement of an observable,${\displaystyle ⟨A⟩}$, where the operator ${\displaystyle A}$ has a spectral decomposition:

${\displaystyle A=\sum _j{\alpha }_j{P}_j,}$ where ${\displaystyle P_j}$ are the projection operators of the eigenvalue ${\displaystyle {\alpha }_j.}$

The probability of outcome ${\displaystyle j}$ is ${\displaystyle p_j=\mathrm{T}\mathrm{r}(\rho P_j).}$ The entropy associated with the observable ${\displaystyle ⟨A⟩}$ 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, ${\displaystyle S_E.}$

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

${\displaystyle S_{\mathrm{v}\mathrm{n}}=-\mathrm{T}\mathrm{r}(\rho \ln \rho ).}$

As a consequence, ${\displaystyle S_A\ge S_{\mathrm{v}\mathrm{n}}}$ for all observables. At thermal equilibrium the energy entropy is equal to the von Neumann entropy: ${\displaystyle S_E=S_{\mathrm{v}\mathrm{n}}.}$

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

${\displaystyle \rho ={\mathrm{\Pi }}_j\otimes {\rho }_j}$

Clausius version of the II-law

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

${\displaystyle \sum _n\frac{J_n}{T_n}\ge 0}$

A dynamical version of the II-law can be proven, based on Spohn's inequality:

${\displaystyle \mathrm{T}\mathrm{r}\left(L_{\mathrm{D}}\rho [\ln \rho (\mathrm{\infty })-\ln \rho ]\right)\ge 0,}$

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

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 thought to violate the second law of thermodynamics. In 2018 has been shown that, by correctly taking into account all work and energy contributions in the full system, local master equations are fully coherent with the second law of thermodynamics.

Quantum and thermodynamic adiabatic conditions and quantum friction

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, ${\displaystyle S_{\mathrm{v}\mathrm{n}}}$ 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: ${\displaystyle [H(t),H(t^{\prime })]=0}$.

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.

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.

Emergence of the dynamical version of the third law of thermodynamics

There are seemingly two independent formulations of the third law of thermodynamics. Both were originally 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

• 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 ${\displaystyle T_{\mathrm{c}}\to 0}$, therefore

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

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 ${\displaystyle \alpha \ge 0}$ the third law imposes ${\displaystyle \alpha >0}$, guaranteeing that at absolute zero the entropy production at the cold bath is zero: ${\displaystyle {\dot{S}}_{\mathrm{c}}=0}$. This requirement leads to the scaling condition of the heat current ${\displaystyle J_{\mathrm{c}}\propto T_{\mathrm{c}}^{\alpha +1}}$.

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

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

The dynamics of the cooling process is governed by the equation:

${\displaystyle J_{\mathrm{c}}(T_{\mathrm{c}}(t))=-c_V(T_{\mathrm{c}}(t))\frac{d{T}_{\mathrm{c}}(t)}{dt}.}$

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

${\displaystyle \frac{d{T}_{\mathrm{c}}(t)}{dt}\propto -T_{\mathrm{c}}^{\zeta },T_{\mathrm{c}}\to 0,\zeta =\alpha -\eta +1}$

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 violation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the Nernst heat theorem.

Typicality as a source of emergence of thermodynamic phenomena

The basic idea of quantum typicality is that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. This is meant to apply to Schrödinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average.

Quantum ergodic theorem originated by John von Neumann is a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of termed normal typicality, i.e. the statement that, for typical large systems, every initial wave function ${\displaystyle {\psi }_0}$ from an energy shell is ‘normal’: it evolves in such a way that ${\displaystyle {\psi }_t}$ for most t, is macroscopically equivalent to the micro-canonical density matrix.

Resource theory

The second law of thermodynamics can be interpreted as quantifying state transformations which are statistically unlikely so that they become effectively forbidden. The second law typically applies to systems composed of many particles interacting; Quantum thermodynamics resource theory is a formulation of thermodynamics in the regime where it can be applied to a small number of particles interacting with a heat bath. For processes which are cyclic or very close to cyclic, the second law for microscopic systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints. These second laws are not only relevant for small systems, but also apply to individual macroscopic systems interacting via long-range interactions, which only satisfy the ordinary second law on average. By making precise the definition of thermal operations, the laws of thermodynamics take on a form with the first law defining the class of thermal operations, the zeroth law emerging as a unique condition ensuring the theory is nontrivial, and the remaining laws being a monotonicity property of generalised free energies.

Engineered reservoirs

Nanoscale allows for the preparation of quantum systems in physical states without classical analogs. There, complex out-of-equilibrium scenarios may be produced by the initial preparation of either the working substance or the reservoirs of quantum particles, the latter dubbed as "engineered reservoirs".

There are different forms of engineered reservoirs. Some of them involve subtle quantum coherence or correlation effects, while others rely solely on nonthermal classical probability distribution functions. Interesting phenomena may emerge from the use of engineered reservoirs such as efficiencies greater than the Otto limit, violations of Clausius inequalities, or simultaneous extraction of heat and work from the reservoirs.

Quantum statistical mechanics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_statistical_mechanics

Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics.

Expectation

See also: Expectation value (quantum mechanics) and Density matrix § Measurement

From classical probability theory, we know that the expectation of a random variable X is defined by its distribution D_X by

${\displaystyle \mathbb{E}(X)={\int }_{\mathbb{R}}d\lambda {\mathrm{D}}_X(\lambda )}$

assuming, of course, that the random variable is integrable or that the random variable is non-negative. Similarly, let A be an observable of a quantum mechanical system. A is given by a densely defined self-adjoint operator on H. The spectral measure of A defined by

${\displaystyle {\mathrm{E}}_A(U)={\int }_{U}d\lambda \mathrm{E}(\lambda ),}$

uniquely determines A and conversely, is uniquely determined by A. E_A is a Boolean homomorphism from the Borel subsets of R into the lattice Q of self-adjoint projections of H. In analogy with probability theory, given a state S, we introduce the distribution of A under S which is the probability measure defined on the Borel subsets of R by

${\displaystyle {\mathrm{D}}_A(U)=\mathrm{Tr}({\mathrm{E}}_A(U)S).}$

Similarly, the expected value of A is defined in terms of the probability distribution D_A by

${\displaystyle \mathbb{E}(A)={\int }_{\mathbb{R}}d\lambda {\mathrm{D}}_A(\lambda ).}$

Note that this expectation is relative to the mixed state S which is used in the definition of D_A.

Remark. For technical reasons, one needs to consider separately the positive and negative parts of A defined by the Borel functional calculus for unbounded operators.

One can easily show:

${\displaystyle \mathbb{E}(A)=\mathrm{Tr}(AS)=\mathrm{Tr}(SA).}$

The trace of an operator A is written as follows:

${\displaystyle \mathrm{Tr}(A)=\sum _m⟨m|A|m⟩.}$

Note that if S is a pure state corresponding to the vector ${\displaystyle \psi }$, then:

${\displaystyle \mathbb{E}(A)=⟨\psi |A|\psi ⟩.}$

Von Neumann entropy

Main article: Von Neumann entropy

Of particular significance for describing randomness of a state is the von Neumann entropy of S formally defined by

${\displaystyle \mathrm{H}(S)=-\mathrm{Tr}(S{\log }_{2}S)}$.

Actually, the operator S log₂ S is not necessarily trace-class. However, if S is a non-negative self-adjoint operator not of trace class we define Tr(S) = +∞. Also note that any density operator S can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form

${\displaystyle \left[\begin{array}{ccccc}{\lambda }_1& 0& \cdots & 0& \cdots \\ 0& {\lambda }_2& \cdots & 0& \cdots \\ ⋮& ⋮& \ddots & \\ 0& 0& & {\lambda }_n& \\ ⋮& ⋮& & & \ddots \end{array}\right]}$

and we define

${\displaystyle \mathrm{H}(S)=-\sum _i{\lambda }_i{\log }_2{\lambda }_i.}$

The convention is that ${\displaystyle 0{\log }_{2}0=0}$, since an event with probability zero should not contribute to the entropy. This value is an extended real number (that is in [0, ∞]) and this is clearly a unitary invariant of S.

Remark. It is indeed possible that H(S) = +∞ for some density operator S. In fact T be the diagonal matrix

${\displaystyle T=\left[\begin{array}{ccccc}\frac{1}{2({\log }_{2}2{)}^2}& 0& \cdots & 0& \cdots \\ 0& \frac{1}{3({\log }_{2}3{)}^2}& \cdots & 0& \cdots \\ ⋮& ⋮& \ddots & \\ 0& 0& & \frac{1}{n({\log }_{2}n{)}^2}& \\ ⋮& ⋮& & & \ddots \end{array}\right]}$

T is non-negative trace class and one can show T log₂ T is not trace-class.

Theorem. Entropy is a unitary invariant.

In analogy with classical entropy (notice the similarity in the definitions), H(S) measures the amount of randomness in the state S. The more dispersed the eigenvalues are, the larger the system entropy. For a system in which the space H is finite-dimensional, entropy is maximized for the states S which in diagonal form have the representation

${\displaystyle \left[\begin{array}{cccc}\frac{1}{n}& 0& \cdots & 0\\ 0& \frac{1}{n}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & \frac{1}{n}\end{array}\right]}$

For such an S, H(S) = log₂ n. The state S is called the maximally mixed state.

Recall that a pure state is one of the form

${\displaystyle S=|\psi ⟩⟨\psi |,}$

for ψ a vector of norm 1.

Theorem. H(S) = 0 if and only if S is a pure state.

For S is a pure state if and only if its diagonal form has exactly one non-zero entry which is a 1.

Entropy can be used as a measure of quantum entanglement.

Gibbs canonical ensemble

Main article: canonical ensemble

Consider an ensemble of systems described by a Hamiltonian H with average energy E. If H has pure-point spectrum and the eigenvalues ${\displaystyle E_n}$ of H go to +∞ sufficiently fast, e^{−r H} will be a non-negative trace-class operator for every positive r.

The Gibbs canonical ensemble is described by the state

${\displaystyle S=\frac{{\mathrm{e}}^{-\beta H}}{\mathrm{Tr}({\mathrm{e}}^{-\beta H})}.}$

Where β is such that the ensemble average of energy satisfies

${\displaystyle \mathrm{Tr}(SH)=E}$

and

${\displaystyle \mathrm{Tr}({\mathrm{e}}^{-\beta H})=\sum _n{\mathrm{e}}^{-\beta E_n}=Z(\beta )}$

This is called the partition function; it is the quantum mechanical version of the canonical partition function of classical statistical mechanics. The probability that a system chosen at random from the ensemble will be in a state corresponding to energy eigenvalue ${\displaystyle E_m}$ is

${\displaystyle \mathcal{P}(E_m)=\frac{{\mathrm{e}}^{-\beta E_m}}{\sum _n{\mathrm{e}}^{-\beta E_n}}.}$

Under certain conditions, the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.

Grand canonical ensemble

Main article: grand canonical ensemble

For open systems where the energy and numbers of particles may fluctuate, the system is described by the grand canonical ensemble, described by the density matrix

${\displaystyle \rho =\frac{{\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)}}{\mathrm{Tr}\left({\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)}\right)}.}$

where the N₁, N₂, ... are the particle number operators for the different species of particles that are exchanged with the reservoir. Note that this is a density matrix including many more states (of varying N) compared to the canonical ensemble.

The grand partition function is

${\displaystyle \mathcal{Z}(\beta ,{\mu }_1,{\mu }_2,\cdots )=\mathrm{Tr}({\mathrm{e}}^{\beta (\sum _i{\mu }_i{N}_i-H)})}$