Squeezing of collective excitations in spin ensembles

We analyze the possibility of creating two-mode spin squeezed states of two separate spin ensembles by inverting the spins in one ensemble and allowing spin exchange between the ensembles via a near resonant cavity field. We investigate the dynamics of the system using a combination of numerical and analytic calculations, and we obtain squeezing for a wide range of parameters. We also investigate the transfer of the squeezing properties to the cavity field and to an output mode from the cavity. Finally, we investigate how the squeezing is affected by effects of inhomogeneities which would be present in solid-state implementations of the spin ensembles.

Figure 1

Two physical setups for collective spin squeezing of two-level ensembles. (a) Two ensembles $a$ and $b$ of two-level systems are prepared in opposite eigenstates, illustrated with spin directions up and down. The spins are situated in a transmission waveguide cavity which can be tuned into resonance with the spin transition. (b) Two ensembles of atoms or ions with pairs of ground states and optically excited states are coupled to a classical laser field with Rabi frequency $\Omega$ and to a quantized optical cavity field with coupling strength $\chi$, such that Raman transitions between the ground states are accompanied by the creation or annihilation of a photon in the cavity mode. In both physical implementations the idea is to prepare each ensemble in one of the two internal states, enabling correlated transitions in the two ensembles, by exchange of cavity photons.

Figure 2

A schematic picture of the system in which we implement spin squeezing. We have three harmonics oscillators of which one is inverted. The cavity oscillator operates as a bus for the transfer of excitations between the two ensemble oscillators.

Figure 3

The average number of excitations in the two ensembles calculated numerically with the parameters $\Delta ={\omega }^{\prime }-\omega =10\kappa ,g_b=5\kappa$, and with $g_a=0.6g_b,g_a=0.9g_b$, and $g_a=1.2g_b$, in panels (a)–(c), respectively.

Figure 4

Variances of the collective quadrature observables $X_{ab}\left(0\right)$ (red solid line), and $X_{ab}(\pi /2)$ (blue dashed line) for the coupling parameters and detuning, $g_b=\kappa$, $\Delta =5\kappa$, and $g_a=0.5g_b$.

Figure 5

The variance of $X_{ab}\left(0\right)$ for different detuning of the cavity. We have $g_b=5\kappa$ and $g_a=0.9g_b$. The black (solid) line is for $\Delta =75\kappa$. The red (dashed) line is for $\Delta =50\kappa$. The blue (dotted) line is $\Delta =20\kappa$ and the green (dashed dotted) line is $\Delta =5\kappa$.

Figure 6

The minimally achieved variance for $\Delta =g_b$ and different values of $\kappa$ and $g_a$.

Figure 7

The collective quadrature variance, determined by adiabatic elimination of the detuned cavity field, Eq. (20), with $\Delta =75\kappa$ and $g_b=5\kappa$. We vary the coupling strength, while remaining in the adiabatic regime. For the black (solid) line we choose $g_a=0.9g_b$, corresponding to the black line in Fig. 5. For the red (dashed) line we choose $g_a=0.8g_b$, the blue (dotted) line $g_a=0.7g_b$, and the green (dashed dotted) $g_a=0.6g_b$.

Figure 8

The output field quadrature variance with $u\left(t\right)=A{e}^{-\kappa t/2}$ (blue dashed dotted), $u\left(t\right)=B{\left(\langle c^{†}c\rangle \right)}^{1/2}$ (red dashed), and an optimized $u\left(t\right)$ (solid black line) for different values of $\kappa$. We take ${\Delta }_c=0.01g_b$ and $g_a=0.9g_b$.

Figure 9

Non-normalized $u\left(t\right)$. We have $u\left(t\right)=A{e}^{-\kappa t/2}$ (blue dashed dotted), $u\left(t\right)=A{\left(\langle c^{†}c\rangle \right)}^{1/2}$ (red dashed), and optimized $u\left(t\right)$ (solid black line). We take $g_b=1.25\kappa$, $\Delta =0.001\kappa$, and $g_a=0.9g_b$.

Figure 10

The variance of $X_{ab}\left(0\right)$ for different broadening width. Defining $g_b=\sqrt{{\sum }_k{\left|g_{k,b}\right|}^2}$, we have at the top $g_b=5\kappa$, $g_a=\sqrt{{\sum }_k{\left|g_{k,a}\right|}^2}=0.9g_b$ and ${\Delta }_c=75\kappa$ and in the bottom we have the same but ${\Delta }_c=0.5\kappa$. The black (solid) line is for a width of $0g_b$. The red (dashed) line is for $0.05\kappa$. The blue (dotted) line is $0.5\kappa$ and the green (dashed dotted) line is for $5\kappa$.