Cavity-Assisted Single-Mode and Two-Mode Spin-Squeezed States via Phase-Locked Atom-Photon Coupling

We propose a scheme to realize the two-axis countertwisting spin-squeezing Hamiltonian inside an optical cavity with the aid of phase-locked atom-photon coupling. By careful analysis and extensive simulation, we demonstrate that our scheme is robust against dissipation caused by cavity loss and atomic spontaneous emission, and it can achieve significantly higher squeezing than one-axis twisting. We further show how our idea can be extended to generate two-mode spin-squeezed states in two coupled cavities. Because of its easy implementation and high tunability, our scheme is experimentally realizable with current technologies.

Figure 1

Atomic energy levels and transitions between them. The complex Rabi frequencies ${\tilde{\mathrm{\Omega }}}_1{e}^{-i\phi }$, ${\tilde{\mathrm{\Omega }}}_2{e}^{i\phi },{\mathrm{\Omega }}_1{e}^{-i[\phi -(\pi /2)]}$, and ${\mathrm{\Omega }}_2{e}^{i[\phi -(\pi /2)]}$ are associated with four phase-locked driving lasers. $g_{1,2}$ is the coupling strength between the atom and the cavity mode. ${\mathrm{\Delta }}_{1,2}$, ${\delta }_{1,2}$, and ${\gamma }_{1,2}$ are detunings.

Figure 2

(a), (b) Time evolution of ${\xi }^2$ and $F$ with $H$ in Eq. (1) and $H_{\mathrm{eff}}$ in Eq. (5) without dissipation for $N=6$, 8, and 10. (c), (d) Time evolution of ${\xi }^2$ and $F$ under the original Hamiltonian $H$ in Eq. (1) with dissipation for $N=8$.

Figure 3

(a) Comparison of the maximum achievable squeezing in our system with strong atomic and cavity dissipation rates $\kappa ={\gamma }_d=5\times 10^6{\mathrm{s}}^{-1}$ and in a dissipation-free OAT system $H_{\mathrm{OAT}}=\chi S_x^2$. Other parameters are the same as in Fig. 2, except that $\mathrm{\Omega }=2\times 10^7{\mathrm{s}}^{-1}$. (b) Time evolution of ${\xi }^2$ in our system with $N=10^3–10^5$ atoms and no dissipation. (Inset) The maximum achievable squeezing in our system and in OAT, both without dissipation. The relevant parameters are ${\mathrm{\Delta }}_1=-{\mathrm{\Delta }}_2=1.88\times 10^{10}{\mathrm{s}}^{-1}$, $\gamma =2\delta =1.26\times 10^{10}{\mathrm{s}}^{-1}$, and $A=B/\sqrt{2}=0.4\delta /N$, where $N=500$, 1000, 2000, 4000, and $10^5$.

Figure 4

Time evolution of squeezing, entanglement, and quantum Fisher information of two-mode spin-squeezed states with the same total spin number for the two modes. (a) ${\mathrm{\Delta }}^{\prime }$ vs $t$ while $S=N/2=5$, and $J=0$, 0.03, 0.05, 0.1. (b) ${\mathrm{\Delta }}^{\prime }$ vs $t$ while $J=0.1$, and $S=5$, 15, 25. (c) Dependence of the von Neumann entropy $E({\rho }_L)$ on time for $J=0.05$, 0.1 when $S=5$. (d) The quantum Fisher information for $|\mathrm{\Psi }(t)⟩$ at $J=0.05$, 0.1 when $S=5$.

Figure 5

Time evolution of squeezing and entanglement of two-mode spin-squeezed states with different total spin numbers for the two modes. (a) ${\mathrm{\Delta }}^{\prime }$ and (b) $E({\rho }_L)$ vs $t$ for a fixed $S_L=N_L/2=25$ and $J=0.1$. $S_R$ varies from 20 to 30.