The Jaynes Cummings model as an autonomous Maxwell demon

TL;DR

This work analyzes a Jaynes–Cummings qubit–cavity system with the cavity initialized in a displaced thermal state to reveal three thermodynamic regimes: an initial unitary drive acting as a quasi-ideal work source, a subsequent autonomous measurement, and a final measurement-dependent feedback that purifies the qubit. Using a framework that treats work and heat in autonomous quantum machines and accounts for mutual information as a thermodynamic resource, the authors show the cavity functions as an autonomous Maxwell demon, converting mutual information into cooling power and reaching Landauer-like limits in the large displacement limit. The results highlight a fundamental link between quantum measurement, feedback, and thermodynamics, and illustrate how information flows can drive reversible energy transformations in closed quantum systems. The study also outlines experimental feasibility in circuit QED setups and suggests directions to optimize purification and assess robustness against imperfections.

Abstract

We revisit the Jaynes-Cummings model as an autonomous thermodynamic machine, where a qubit is driven by a cavity containing initially a large coherent field. Our analysis reveals a transition between the expected behavior of ideal-work source of the cavity at short times, and a long-time dynamics where the cavity autonomously measures the qubit and exerts a result-dependent drive. This autonomous feedback then purifies the qubit irrespective of its initial state. We show that the cavity functions thermodynamically as an autonomous Maxwell demon, trading mutual information for cooling power.

Figures (4)

• Figure 1: (a) Entropy variation $\Delta S_\text{Q}$ of the qubit over a long time scale for different initial qubit states $| \psi_1 \rangle = | e \rangle\langle e |$, $| \psi_2 \rangle =\frac{1}{\sqrt{2}} (| e \rangle + e^{i\frac{\pi}{4}} | g \rangle)$, and $| \psi_3 \rangle = | +_y \rangle= \frac{1}{\sqrt{2}} (| e \rangle + i | g \rangle)$. (b) zoom in the early times of (a), in semilog scale. (c) Excited state population $P_e=\langle e |\hat{\rho}_\text{Q}(t)| e \rangle$ (in blue) and coherence $C_{eg}=\langle e |\hat{\rho}_\text{Q}(t)| g \rangle$ (in orange and red) of the qubit when it is initialized in $\hat{\rho}_\text{Q}(0) = | e \rangle\langle e |$. The dots correspond to expansions up to second order in $1/n_0$ and the solid lines are obtained from the numerical evaluation of the exact expressions, truncating the cavity Hilbert space to $N_\text{ph} = 500$. Parameters: $n_0=100$, $\bar{n}=1$.
• Figure 2: Heat $Q_\text{C}$ (red), work $W_\text{C}$ (green) and internal energy variation of the cavity $\Delta E_\text{C}$ (blue) as a function of $\theta=2gt\sqrt{n_0}$, for $\hat{\rho}_\text{Q}(0) = | e \rangle\langle e |$, $n_0=500$ and $\bar{n}=1$. Dots are obtained from the second order expansion in $1/n_0$, and exact expressions with truncated cavity Hilbert space are displayed in solid black (not distinguishable). For smaller values of $n_0$, the heat provided by the cavity is non-negligible, as shown by the dashed black line for $n_0=10$.
• Figure 3: Autonomous feedback mechanism. (a) Trajectory of the qubit on the equator of the Bloch sphere starting from states $| +_y \rangle$ (blue) and $| -_y \rangle$ (green) and (b) of the average field amplitude. The red circle locates the qubit state reached at $t_\text{min}$. The yellow circle locates the initial cavity displacement, for $\phi_0=0$.
• Figure 4: (a) von Neumann entropy of the cavity $S_\text{C}$ and of the qubit $S_\text{Q}$, and mutual information between the qubit and cavity $I_\text{QC}$. (b) Cavity heat $Q_\text{C}$, work $W_\text{C}(t)$ and internal energy variation $\Delta E_\text{C}(t)$associated to the time interval $[0,t]$. For both (a) and (b), $n_0=50$, $\bar{n}=1$. Plotted quantities are computed from exact calculation with truncated cavity Hilbert space.