Nondestructive optomechanical detection scheme for Bose-Einstein condensates

Abstract

We present a two-tone heterodyne optical readout scheme to extract unequal-time density correlations along an arbitrary stationary interaction path from a pancake-shaped Bose-Einstein condensate, using a modulated laser probe. Analysing the measurement noise both from imprecision and backaction, we identify the standard quantum limit for the signal-extraction scheme, and examine how a class of two-mode squeezed initial states can be used to push beyond this limit. As an application, we show how the readout scheme can be used for an experimentally feasible realisation of acceleration-dependence of quantum-vacuum fluctuations in the system, including the analogue spacetime circular motion Unruh effect. The scheme is adaptable beyond Bose-Einstein condensates, providing nondestructive access to unequal-time correlations in quantum fluids.

Figures (2)

• Figure 1: Optical circuit diagram for sampling BEC density fluctuations along a stationary interaction point trajectory, using a modulated laser probe with two sidebands. The band centred at $\omega_+ = \omega_0+\Omega$ ($\omega_- = \omega_0-\Omega$) is shown in blue (red). Post-interaction, the modulated signal is heterodyned using a two-tone reference beam, in which an acousto-optic modulator has inserted a frequency shift $\Delta_{\mathrm{LO}}$. EOM: electro-optic modulator; BEC: Bose-Einstein condensate; BS: beamsplitter; EOD: electro-optic deflector; LO: local oscillator; AOM: acousto-optic modulator; PS: phase-shifter; PD: photodiode; SA: spectrum analyser.
• Figure 2: Plots of the normalised added noise $\bar{\mathcal{N}}$, defined as \ref{['diffPSDadd']} divided by the SQL value \ref{['diffPSDaddSQL']}, as a function of $\lambda$ (squeezing parameter) and $\mu$ (laser-BEC coupling parameter), for $0 \le \lambda \le 1.5$ and $\mu = k\mu_\lambda$, where $\mu_\lambda$\ref{['mulambda']} is the optimal value of $\mu$. Three selected values of $k$ are shown for both $\nu>0$ and $\nu<0$. The $k=1$ curves are \ref{['diffPSDaddsqueeze']}, and the ranges of $\lambda$ where $\bar{\mathcal{N}} < 1$ are as described in the text. For $k>1$, the ranges of $\lambda$ where $\bar{\mathcal{N}} < 1$ are narrower: this range exists for $1 < k < 3^{-1/2} (2 + 2\sqrt{6} + \sqrt{19 + 8\sqrt{6}}\,)^{1/2} \approx 2.091$ when $\nu>0$, and for $1 < k < 3^{-1/4} (\sqrt{6} -2 + \sqrt{13 - 4\sqrt{6}}\,)^{1/2} \approx 1.137$ when $\nu<0$. For $0<k<1$, replace $k$ by $1/k$. The physical mechanism that makes the added noise more sensitive to $\mu$ for $\nu<0$ than for $\nu>0$ is the same as in the backaction cooling in the measurement of zero point mechanical oscillator in a cavity optomechanical system Khalili2012, as we show in Supplemental Material suppmat.