Vacuum spin squeezing

We investigate the generation of entanglement (spin squeezing) in an optical-transition atomic clock through the coupling to an optical cavity in its vacuum state. We show that if each atom is prepared in a superposition of the ground state and a long-lived electronic excited state, and viewed as a spin-1/2 system, then the collective vacuum light shift entangles the atoms, resulting in a squeezed distribution of the ensemble collective spin, without any light applied. This scheme reveals that even an electromagnetic vacuum can constitute a useful resource for entanglement and quantum manipulation. By rotating the spin direction while coupling to the vacuum, the scheme can be extended to implement two-axis twisting resulting in stronger squeezing.

Figure 1

Principle of vacuum spin squeezing on an optical clock transition. $N$ two-level atoms are trapped in a high-finesse optical cavity. The cavity enhanced vacuum is coupling the atomic transition, $|\uparrow \rangle \to |\downarrow \rangle$, with a detuning $\mathrm{\Delta }$. After the atomic ensemble is initialized into a coherent spin state near the equator, the atomic spin distribution begins to spontaneously squeeze without the application of any light. The cavity coupling is detuned from the atomic resonance by $\mathrm{\Delta }$, so that virtual photon emission into the cavity and reabsorption leads to a light shift of the atomic levels. Since the coupling to the cavity for different Dicke states varies with $S_z$, each Dicke state experiences a different nonlinear light shift. This results in an $S_z$-dependent state rotation that is equivalent to one-axis twisting [19].

Figure 2

Illustration of the exponential squeezing obtained by combining spin squeezing and rotation [46]. When viewed as a discretized process, each step of squeezing is followed by a rotation. The rotation increases the lever arm $\sqrt{\langle S_z^2\rangle }$ for the next squeezing step, resulting in exponential squeezing with time. When the step size approaches zero, the system is described by the Hamiltonian $H_2/\hslash =\mathrm{\Omega }S{\widehat{S}}_x+\mathrm{\Omega }{\widehat{S}}_z^2$.

Figure 3

(a) Time evolution of the squeezing parameter $\xi \left(t\right)$ for ${10}^4{}^{171}\mathrm{Yb}$ atoms $[\mathrm{\Gamma }/(2\pi )=7$ mHz] contained in an optical cavity with $\kappa /\left(2\pi \right)=100$ kHz and cooperativity $\eta =10$ (red solid line) or 1 (red dashed line). The detuning is set to $\mathrm{\Delta }/\left(2\pi \right)=11.2$ MHz for $\eta =10$ [or $\mathrm{\Delta }/\left(2\pi \right)=3.5$ MHz for $\eta =1]$. When $\eta =10$ and $t=0.46$ s, $\xi \left(t\right)$ reaches the minimal value, corresponding to 9 dB of squeezing beyond the standard quantum limit. (b) Comparison of one-axis twisting (solid red) and two-axis twisting (dashed blue) when $\eta =10$. The latter is accomplished by appropriate state rotation during the squeezing (see text) and calculated by numerical method directly. Here we include both noise terms, cavity leakage and atomic spontaneous decay. The dotted line shows the evolution of two-axis twisting for a perfect cavity without any decay.