Dicke materials as a resource for quantum squeezing

Abstract

We study magnetic materials whose low energy physics can be effectively described by a Dicke model, which we term Dicke materials. We show how a Dicke model emerges in such materials due to a coexistence of fast-dispersing and slow-dispersing spins, which are strongly coupled. Analogous to the paradigmatic Dicke model describing light-matter interactions, these materials also exhibit signatures of a superradiant phase transition. The ground state near the superradiant phase transition is expected to be squeezed, making Dicke materials a resource for quantum metrology and witnessing entanglement in solid-state systems. However, as an entanglement measure, squeezing can be sensitive to perturbations that are otherwise irrelevant for usual correlation functions and order parameters. Motivated by the prospect of observing squeezing in such Dicke materials, we study the robustness of ground state squeezing under ubiquitous imperfections such as finite temperature, disorder, and local interactions. Using analytical and numerical techniques, we show that the squeezing obtained is perturbatively stable against these imperfections and quantitatively evaluate regimes promising for experimental observation.

Figures (7)

• Figure 1: A schematic of a Dicke material. The two rungs host two spin degrees of freedom colored red and blue. The red (blue) spins have exchange interactions amongst themselves with strength $J_{\text{r}}$ ($J_{\text{b}}$) with excitation frequency $\omega_r$ ($\omega_b$). The red and blue spins have interspecies exchange interactions of strength $J_{\text{rb}}$. In a Dicke material, the $J_{\text{r}}$ is a dominantly large energy, providing a large velocity for excitations in the red leg analogous to the speed of light for photons in the Dicke model, while the $J_{\text{rb}}$ should also be significant enough to influence the physics of the blue spins.
• Figure 2: Squeezing ratio, $\xi$ as a function of $g/\omega$ at resonance ($\omega = \omega_0$) in the ground state of the Dicke model in Eq. \ref{['eq:bosonicdicke']} (solid blue curve) and with the inclusion of the $A^2$ term in the Dicke model in Eq. \ref{['eq:bosonicdickea2']} (dashed red curve) with its numerical coefficient, $D = g^2/\omega$ . The squeezing ratio goes to zero (perfect squeezing) at $g/\omega=1/2$ only when there is no $A^2$ term.
• Figure 3: Ground state variance through exact diagonalization of operators $\tilde{p}_-$ (orange figures) and $\tilde{S}_y$ (blue figures) as a function of number of spins, $N$. Variance below 1 denotes squeezing and the squeezing gets better with increasing $N$. The circle and triangles show calculation with different boson truncation numbers, $N_a=40$ and $N_a=50$ respectively.
• Figure 4: Contour plot of the squeezing ratio, $\xi$, in the normal phase ($g \leq g_c$) of the Dicke model as a function of temperature $T$ and $(g_c-g)$ when (a) $\omega_0=\omega$ and (b) $\omega_0 = 2\omega$. The contours are labeled by values of $\xi$. Both cases are qualitatively similar and display a finite region in the parameter space where $\xi < 1$, indicating squeezing.
• Figure 5: (a) Contour plot of the squeezing ratio, $\xi$, in the normal phase as a function of temperature $T$ and $\omega_0/\omega$ at a fixed value of $g=0.1\omega$. The contours are labeled by values of squeezing ratio, $\xi$. The $T=0$ SRPT critical point is at the bottom left corner where $\omega_0/\omega = 0.04$. (b) Squeezing ratio, $\xi$, as a function of $\omega_0/\omega$ at temperatures ranging from $k_B T/\omega = 0.013$ to $0.025$ in steps of 0.002.