The two-time Leggett-Garg inequalities of a superconducting qubit interacting with thermal photons in a cavity

TL;DR

The paper investigates two-time Leggett-Garg inequalities for a superconducting Josephson-junction qubit interacting with thermal photons in a cavity, using a non-solvable quadratic qubit–photon coupling Hamiltonian implemented in circuit QED. Employing second-order perturbation theory, the authors derive time-evolved correlation functions and LG observables, and analyze how LG violations depend on temperature and coupling strength. They find that LG violations, notably for $\text{LG}_{1,-1}$, weaken with increasing temperature and stronger coupling, following approximate power-law scaling with exponents that vary with $g$. The results illuminate a quantum-to-classical transition in a realistic circuit-QED setting and provide guidance for designing experiments to test LG inequalities in superconducting devices.

Abstract

In this paper, we study the two-time Leggett-Garg (LG) inequalities of a quantum optical model that appears in the Josephson-junction quantum bit (qubit) interacting with an external magnetic flux. This model is a natural extension of an exactly solvable model whose interaction between a qubit and single-mode photons is given by a product of the Pauli $z$ operator of the qubit and a linear combination of annihilation and creation operators of the photons. By contrast, a photon's part of the interaction of our model is given by the square of the linear combination. Because our model is not solvable, we approximately investigate its time evolution up to the second-order perturbation. Our numerical calculations show that violation of the LG inequality diminishes as the temperature increases. Moreover, it exhibits power laws of the temperature, whose exponents vary depending on the coupling constant of the interaction between the qubit and photons. The violation of the LG inequality decreases and becomes less sensitive to the temperature as the coupling constant of the interaction gets larger.

Figures (6)

• Figure 1: A circuit with two Josephson junctions in parallel. The magnetic field pierces the plane of the circuit.
• Figure 2: Plots of $\hbox{LG}_{s_{0},s_{1}}$ for $s_{0},s_{1}\in\{\pm 1\}$ as functions of $T=t_{1}-t_{0}(\geq 0)$. Curves of (a), (b), (c), and (d) represent $\hbox{LG}_{1,1}$, $\hbox{LG}_{1,-1}$, $\hbox{LG}_{-1,1}$, and $\hbox{LG}_{-1,-1}$, respectively. In those plots, we set $\Omega=1$, $g=0.075$, $\omega=0.1$, $a_{x}=a_{z}=1/\sqrt{2}$, and $a_{y}=0$. The thick solid red, thin solid blue, and thin dashed purple curves represent $\beta=10, 1.5$, and $0.8$, respectively, for (a), (b), (c), and (d). Looking at (b), we note that $\hbox{LG}_{1,-1}<0$ at specific times, so that violation of the two-time LG inequality occurs at those times.
• Figure 3: (a) Plots of $-\hbox{LG}_{\hbox{\scriptsize min}}$ as a function of the temperature $\beta^{-1}$, where $\hbox{LG}_{\hbox{\scriptsize min}}$ represents the first local minimum value of $\hbox{LG}_{1,-1}$ as a function of the time $T=t_{1}-t_{0}(\geq 0)$. (b) Plots of $T_{\hbox{\scriptsize min}}$ as a function of the temperature $\beta^{-1}$, where $T_{\hbox{\scriptsize min}}$ represents the time when $\hbox{LG}_{1,-1}$ takes the first local minimum value $\hbox{LG}_{\hbox{\scriptsize min}}$. The thick solid red, thick dashed blue, thin solid red, thin dashed blue, and thin dotted purple curves represent $g=0.26, 0.52, 0.78, 1.04$, and $1.3$, respectively. The other parameters are set as $\Omega=1$, $\omega=0.1$, $a_{x}=a_{z}=1/\sqrt{2}$, and $a_{y}=0$. In graphs (a) and (b), the vertical axis is shown on a logarithmic scale.
• Figure 4: (a) The solid red curves represent plots of $-\hbox{LG}_{\hbox{\scriptsize min}}$ as functions of the temperature $\beta^{-1}$. The dashed blue curves represent the results of fitting approximate functions $a_{1}\beta^{-b_{1}}+c_{1}$ to $\log(-\hbox{LG}_{\hbox{\scriptsize min}})$. (b) The solid red curves represent plots of $T_{\hbox{\scriptsize min}}$ as functions of the temperature $\beta^{-1}$. The dashed blue curves represent the results of fitting approximate functions $a_{2}\beta^{-b_{2}}+c_{2}$ to $\log T_{\hbox{\scriptsize min}}$. For both graphs (a) and (b), the upper solid red and dashed blue curves correspond to the case of $g=0.52$, while the lower solid red and dashed blue curves correspond to the case of $g=1.04$. In the graphs, the vertical axis is shown on a logarithmic scale.
• Figure 5: (a) A plot of the constant $b_{1}$ as a function of $g$, where $b_{1}$ is obtained by the fitting of $\{(\beta^{-1},\log(-\hbox{LG}_{\hbox{\scriptsize min}}))\}$ with the approximate function $a_{1}\beta^{-b_{1}}+c_{1}$. (b) A plot of the constant $b_{2}$ as a function of $g$, where $b_{2}$ is obtained by the fitting of $\{(\beta^{-1},\log T_{\hbox{\scriptsize min}})\}$ with the approximate function $a_{2}\beta^{-b_{2}}+c_{2}$. In both graphs (a) and (b), the red points represent the data obtained from the fitting. We join those points with blue curves.