Local quantum coherence with intersource interactions at nonzero temperature

TL;DR

This work analyzes autonomous local quantum coherence generated in a target two-level system by coupling it to a finite environment of N interacting source TLSs with Ising-type intersource interactions. The authors derive an exact solution for the composite system at thermal equilibrium, obtaining a closed-form expression for the target’s coherence C(T) and showing that C can be enhanced by increasing the intersource coupling J, the environmental gap ωa, and the source–target coupling γ. A central finding is that C(T) can exhibit nonmonotonic temperature dependence and signatures of a zero-temperature quantum phase transition between ferromagnetic and antiferromagnetic ground states, with distinct behavior for even vs odd N. The results provide upper and lower bounds on C, reveal noncommuting limits at high temperature and strong inter-environment coupling, and offer practical guidance for optimizing autonomous coherence in realistic, thermally noisy settings, with implications for quantum thermodynamics and experimental platforms such as quantum dots, NV centers, and superconducting qubits.

Abstract

Local quantum coherence in a two-level system (TLS) is typically generated via time-dependent driving. However, it can also emerge autonomously from symmetry-breaking interactions between the TLS and its surrounding environment at a low temperature. Although such environments often consist of interacting atoms or spins, the role of interactions within the environment in generating the autonomous local coherence has remained unexplored. Here, we address this gap by analyzing an exactly solvable model, which comprises a target TLS coupled to $N$ interacting source TLSs that represent the environment, with the whole system being in thermal equilibrium. We show that the local coherence not only persists but can be enhanced at finite temperatures of the environment compared to the case of no inter-source interactions. The temperature dependence of the coherence bears signatures of a quantum phase transition, and our analytical results suggest strategies for its optimization. Our findings reveal generic properties of the autonomously generated quantum coherence and point to viable routes for observing the coherence at nonzero temperatures.

Figures (3)

• Figure 1: a) Coherence $C$ [Eq. \ref{['eq:C-R']}] as a function of temperature $T$ for different values of the interaction strength $J\geq 0$. Curves for finite $J>0$ lie between the upper bound \ref{['eq:C-upperbound']} ($J\to\infty$, dashed black line) and $C$ for $J=0$ (solid blue line). b) The lowest-order $O(T^{-2})$ term in the asymptotic approximation \ref{['eq:C-highT']} deviates from the exact result \ref{['eq:C-R']} as $T\to\infty$. c) Limits $T\to\infty$ and $J\to\infty$ do not commute. Parameters used: $\omega_{0}=10$, $\omega_{\rm a}=2$, $\gamma=3$, $N=8$, and in b) and c) $J=250$.
• Figure 2: Temperature dependence of coherence $C$ [Eq. \ref{['eq:C-R']}] for $J>0$ and various values of a) interaction strength $\gamma$, b) source TLSs' gap $\omega_{\rm a}$, and c) number $N$ of source TLSs. a) Increasing $\gamma$ slows down the decay of $C$ with $T$ and increases $C$ near $T=0$. The black dashed line represents the $\gamma\to\infty$ limit \ref{['eq:C-highgamma']}. b) Increasing $\omega_{\rm a}$ yields higher coherence at finite $T$ up to the upper bound given in Eq. \ref{['eq:C-highwa']} (black dashed line). $C(0)$ does not change with $\omega_{\rm a}$, see Eq. \ref{['eq:C0-result']}. c) $C$ at any $T$ is an increasing function of $N$. Parameters used: $\omega_{0}=10$, $J=4$, and a) $\omega_{\rm a}=2$, $N=6$, b) $\gamma=3$, $N=7$, c) $\omega_{\rm a}=2$, $\gamma=3$.
• Figure 3: a) The phase diagram for fixed $\omega_0=20$, $\gamma=3$, and even $N=8$. In the gray (upper) region, the ground state is ferromagnetic with $C_0>0$, while in the blue region it is antiferromagnetic and $C_0=0$. The phase transition line follows the equation shown within the white region (bottom). Coordinates $(J,\omega_{\rm a}/2)$ of the colored circles on a horizontal dashed line correspond to the parameter values used to draw the curves of the respective color in b). The circles on the vertical line correspond to the curves in c). b) $C(T)$ for different values of the interaction strength $J$, illustrating typical behavior of $C(T)$ for the two phases and on the phase transition line. c) $C(T)$ for different values of the source TLSs gap $\omega_{\rm a}$. d) $C(T)$ for different values of the coupling constant $\gamma$ and for $\omega_{\rm a}=12$, $J=-6.7$, $N=8$. Increasing $\omega_{\rm a}$ has a qualitatively similar effect to increasing $\gamma$: it slows the decrease of $C(T)$ and raises its maximum value.