Geometric Quantum Thermodynamics

Abstract

Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We develop both microcanonical and canonical ensembles, introducing continuous mixed states as distributions on the manifold of quantum states. We call out the experimental consequences for a gas of qudits. We define quantum heat and work in an intrinsic way, including single-trajectory work, and reformulate thermodynamic entropy in a way that accords with classical, quantum, and information-theoretic entropies. We give both the First and Second Laws of Thermodynamics and Jarzynki's Fluctuation Theorem. The result is a more transparent physics, than conventionally available, in which the mathematical structure and physical intuitions underlying classical and quantum dynamics are seen to be closely aligned.

Figures (2)

• Figure 1: Alternate ensembles in the geometric and standard settings: Differences are plainly evident. Canonical probability distributions on a qubit's state manifold $\mathbb{C}P^1$ with coordinates $Z = (Z^0,Z^1) = (\sqrt{1-q},\sqrt{q}e^{i\chi})$ where $q \in [0,1]$ and $\chi \in [-\pi,\pi]$. $\mathbb{C}P^1$ discretized using a $100$-by-$100$ grid on the $(q,\chi)$ coordinates exploiting the fact that, with these coordinates, the Fubini-Study measure is directly proportional to the Cartesian volume element $dV_{FS} = dq d\chi / 2$. The Hamiltonian is $H = \sigma_x + \sigma_y + \sigma_z$, with $\hbar=1$ and inverse temperature $\beta = 5$ ($k_\text{B} = 1$). (Right) Gibbs ensemble, where the measure is concentrated around coordinates of the respective eigenvectors $(q(\ket{E_0}),\chi(\ket{E_0})) = (0.789,-2.356)$ and $(q(\ket{E_1}),\chi(\ket{E_1})) = (0.211,0.785)$. (Left) Geometric Canonical Ensemble. Notice the difference in scale, due to the fact that the Geometric Canonical Ensemble has continuous support on the quantum state space, not just on single points (energy eigenstates).
• Figure 2: Comparing time-aggregated data of a single trajectory generated by Eq. (\ref{['eq:stoch_model']})'s stochastic model (left) to the fit to a geometric canonical ensemble with functional form as in Eq. (\ref{['eq:Canonical']}) (right). Here, $h(Z) = E(p,\phi) =\delta \left\langle \sigma_x \right\rangle + \epsilon \left\langle \sigma_z \right\rangle=\delta 2 \sqrt{p(1-p)}\cos \phi + \epsilon (1-2p)$, with $\delta=\epsilon=1$. The excellent agreement is visually clear, and it is quantified by a total variation distance between the two distributions of $f \approx 5.6 \times 10^{-4}$.