Squeezing-enhanced parameter estimation with a hybrid spin-oscillator interferometer

Inspired by the recent work [Nature (London) 521, 336 (2015)], we propose a scheme to suppress the backaction-induced dephasing of the spin state in a spin-oscillator system by squeezing the quantum fluctuation of the two-mode oscillator. We show specifically that squeezing either on a single mode individually or on both modes simultaneously could be used to effectively suppress spin dephasing. Therefore, with a spin-oscillator interferometer input by the spin-dependent squeezing cat state, the sensitivity of the rotational parameter estimation can be significantly enhanced. Given that various squeezings on the two-mode vibrations of the oscillator have been experimentally demonstrated, the proposal could provide a possibility to implement the sensitive parameter estimation, particularly in the ion-trap system.

Figure 1

The proposed spin-oscillator interferometer, wherein a two-mode oscillator (as a probe) is coupled to a spin and the parameter $\mathrm{\Omega }$ of the two-mode oscillator is estimated by reading out the populations of the spin states. In the figure, $a$ and $b$ represent the two mode vibrations of the oscillator, respectively. ${\widehat{S}}_k\left({\xi }_k\right)$ $(k=a,b)$ is the single-mode squeezing operator in mode $a$ or mode $b$. ${\widehat{D}}_b\left(\beta \right)$ and ${\widehat{D}}_b^{†}\left(\beta \right)$ are the displacement operators applied in mode $b$. ${\widehat{D}}_a\left(i\alpha {\sigma }_z\right)$ and ${\widehat{D}}_a^{†}\left(i\alpha {\sigma }_z\right)$ are the spin-dependent displacement operators applied in mode $a$, the amplitudes of which are conditional on the spin state. These two spin-dependent displacement operations are used for first entangling and then disentangling the spin and the two-mode oscillator, respectively. Finally, $H$ is the Hadmard gate operator applied to the spin, and $D$ is the detector for measuring the populations of the spin states.

Figure 2

The value of the $|R_1|$ parameter changes with $\theta$ for various squeezing configurations with the selected squeezing strengths: no squeezing (red solid), individual SMS on modes $a$ (green) and $b$ (magenta), and simultaneously squeezing on both modes $a$ and $b$ (blue). Here, the displacement amplitudes are set as $\alpha =2$ and $\beta =10$.

Figure 3

The value of the $|R_1|$ parameter changes with the $\theta$ parameter: (a) for different squeezing strengths: $r=0,1,2,3$ with $\alpha =2$ and $\beta =10$, and (b) for $\alpha =2,5,10$ with $r=2.3$ and $\beta =10$.

Figure 4

The value of the populations $P_{\downarrow }$ of the spin state $|\downarrow \rangle$ changes with the $\theta$ parameter for the non squeezing (NS) (red), a single SMS on mode $a$ (green) andmode $b$ (magenta), and simultaneously squeezing on the two modes (blue), respectively. Here, the displacement amplitudes are set as $\alpha =2,\beta =10$.

Figure 5

The sensitivity $\mathrm{\Delta }{\mathrm{\Omega }}_1$ changes with the $\theta$ paramter for the cases of NS, simultaneously squeezing on modes $a$ and $b$ with two individual SMSs, respectively. The squeezing strength and displacement amplitudes are set as $r_a=r_b=2.3$ and $\alpha =2,\beta =10$. Here, we set $\mathrm{\Omega }=1$.