Quantum Isothermal Processes are Not Isoenergetic

Published May 16, 2020

In a classical isothermal process the internal energy of the system remains invariant because for a classical ideal gas, the internal energy is proportional to the temperature of the gas¹. However, if we define a quantum analogue of the isothermal process, this is not necessarily true. With all the weird things that happen in quantum systems, this is not really surprising, but it is definitely interesting.

In this post, I start by building some elementary quantum thermodynamics: we shall look at the quantum version of the first law of thermodynamics, some general facts about quantum thermodynamic processes, and then define an effective temperature for quantum systems and a quantum analogue of the isothermal process. Once we have the definition of an isothermal process, we shall see with an easy computation that when the working substance is a quantum two-level system, the internal energy is not invariant during the process.

The Quantum First Law of Thermodynamics

If the energy eigenstates of our quantum system are labelled by \(|n\rangle\) with corresponding eigenenergies \(E_n\), the Hamiltonian (in the energy basis) can be written as

It is reasonable to identify the internal energy \(U\), of the quantum system with the expectation value of the Hamiltonian, hence

where \(P_n = \langle n | \hat{\rho} |n \rangle\) is the occupation probability for the \(n\)th eigenstate, and \(\hat{\rho}\) is the density matrix.

We recall the usual expression of the first law: \(dU = dQ + dW\), and take the exterior derivative of \(\eqref{internal-energy}\) to get a quantum analogue²

In order to identify the analogues of heat exchanged and work done, we recall the Gibbs formula for entropy: \(S = - \sum_n P_n \ln P_n\), and because the heat exchanged is \(dQ = TdS\), we identify

Consequently, the work done is identified as

In heat engines based on quantum systems, a change in energy levels is associated with work done by the engine, and a change in occupation probabilities is associated with the heat exchanged between the engine and the heat bath.

Effective Temperature for Quantum Systems and the Quantum Isothermal Process

Statistical mechanics tells us that when a quantum system is in equilibrium with a heat bath at temperature \(T = 1/k_B \beta\), the density matrix is given by

where \(Z(\beta) = \text{Tr}(e^{-\beta \hat{H}})\) is the partition function. Occupation probabilities \(P_n\) can be obtained from the diagonal elements of the density matrix,

and we note that when the system is at a fixed temperature \(T\), the occupation probabilities must satisfy the above distribution.

The effective temperature of a quantum system is defined by ‘inverting’ \(\eqref{canonical-distribution}\). In order to understand what this means, we look at a system with only two states \(|g\rangle\) and \(|e\rangle\) with energies \(E_g\) and \(E_e\) respectively. If the system is in equilibrium with a heat bath, the occupation probabilities satisfy

and the relation can easily be inverted to get the temperature in terms of the occupation probabilities

Based on the above expression, we can obtain an effective temperature even when the system is not in equilibrium with a heat bath: \(k_B T_{eff} = (E_e - E_g) / \ln (P_g/P_e)\).

We see that for a two-level system, the effective temperature is uniquely defined for all occupation probabilities \(P_g\) and \(P_e\). However, this might not be so when the system has more than two levels. In general, a unique effective temperature is defined only when the occupation probabilities satisfy the canonical distribution in \(\eqref{canonical-distribution}\).

Once we know what a temperature means for an arbitrary quantum system, we can define a quantum isothermal process in the obvious way. In a quantum isothermal process, energy levels and occupation probabilities must change simultaneously to always satisfy the canonical distribution for a fixed temperature.

Before looking at what happens to a two-level system in an isothermal process, we shall note a fact that will make certain computations easier: the work done \(dW\), heat exchanged \(dQ\), and therefore the change in internal energy \(dU\), are invariant under a uniform shift of all energy levels.

If we assume that all energy levels shift uniformly: \(E_n' = E_n + \delta\), the first thing we note is that that \(dE_n' = dE_n\) because \(\delta\) is a constant. Next, we consider the occupation probabilities,

and observe that \(dP_n' = dP_n\). Finally, from the quantum analogues of heat \(\eqref{quantum-heat}\), work \(\eqref{quantum-work}\); and the first law \(\eqref{quantum-first-law}\), the result follows.

In particular we note that, for the two-level system, assuming \(E_g = 0\) has no effect on \(dU.\)

The Two-Level System in an Isothermal Process

In an isothermal process \(i \to f\), we can assume that \(E_g^f = E_g^i = 0\) and \(E_e^f = \zeta E_e^i\), where \(\zeta > 0\) is some constant that can be used to parametrize the evolution of the system under the process. For our final trick, we consider the internal energy of the two-level system

and its derivative with respect to \(\zeta\)

which, in general, is non-zero. Thus, we have shown that the internal energy is not invariant³ in a quantum isothermal process, and I have fulfilled my promise.

This blog post was inspired by an appendix in Quantum Thermodynamic Cycles and Quantum Heat Engines (PhysRevE 76.031105, arXiv:quant-ph/0611275) by Quan et al. In the article, the authors have done a more general computation which can also be applied to a quantum harmonic oscillator and a particle in an infinite square well, but they assume (without any justification) that all the energy levels change in the same ratio in an isothermal process. This is not a problem for the two-level system because there are only two energy levels.