Delta-Kick Squeezing

We explore the possibility to overcome the standard quantum limit (SQL) in a free-fall atom interferometer using a Bose-Einstein condensate (BEC) in either of the two relevant cases of Bragg or Raman scattering light pulses. The generation of entanglement in the BEC is dramatically enhanced by amplifying the atom-atom interactions via the rapid action of an external trap, focusing the matter waves to significantly increase the atomic densities during a preparation stage—a technique we refer to as delta-kick squeezing (DKS). The action of a second DKS operation at the end of the interferometry sequence allows one to implement a nonlinear readout scheme, making the sub-SQL sensitivity highly robust against imperfect atom counting detection. We predict more than 30 dB of sensitivity gain beyond the SQL for the variance, assuming realistic parameters and ${10}^6\mathrm{atoms}$.

Figure 1

Complete operation of a DKS-enhanced free-fall atom interferometer. State preparation (a) consists of (i) a first expansion of the BEC $T_0$, (ii) a beam splitter (BS1), (iii) a DKS of duration $\mathrm{\Delta }{t}_1$ focusing the matter waves (inset), and (iv) two mirror pulses ($M$). The interferometer sequence (b) starts with the beam splitter (BS2) followed by a mirror ($M$) and ending with a recombining beam splitter (BS3). The pulses are equally separated in time by $T_{\theta }$. In a linear detection scheme, the phase is evaluated by counting the number of atoms at the two output ports. In a nonlinear readout case (c), one applies operations analog to the state preparation. Right before the first mirror pulse, a second DKS of duration $\mathrm{\Delta }{t}_2$ is applied and generates an “untwisting” dynamics. The phase is finally evaluated by counting the number of atoms at the two output ports.

Figure 2

DKS engineering. (a) Thomas-Fermi (TF) Radii along the $z$ ($R_z$) and transverse directions ($R_{\perp }$) as a function of time during state preparation [67]. The different laser pulses are highlighted by the vertical black lines with $T_{\tau }=T_{\tau }^{\prime }=5.25\mathrm{ms}$ and the DKS time is fixed to $\mathrm{\Delta }t=0.25\mathrm{ms}$ (vertical blue lines). (b) Corresponding nonlinear coefficient $\chi (t)$ for Raman and Bragg scattering. (c) Effective nonlinear coefficient $\tau$ as a function of the DKS duration $\mathrm{\Delta }t$, during the state preparation (solid lines) and during the interferometer sequence (dashed). The vertical line denotes the case shown in (a) and (b). (d),(e) The sensitivity gain $\mathcal{G}$ (color scale) for the Bragg ($d$) and Raman ($e$) configurations as a function of the preexpansion time and DKS duration. The white lines denote ${\tau }^B=0$ and we distinguish regions with ${\tau }^B>0$ and ${\tau }^B<0$. Here, $\mathcal{G}=40$ corresponds to a variance of 32 dB below the SQL.

Figure 3

Nonlinear readout. (a) Sensitivity gain for linear $\mathcal{G}(\tau /{\tau }_{\mathrm{opt}})$ and nonlinear $\mathcal{Q}(\tau /{\tau }_{\mathrm{opt}})$ readout as a function of the effective nonlinear coefficient, $\tau ={\tau }_1=-{\tau }_2$. The vertical lines denote $0.1{\tau }_{\mathrm{opt}}$ and ${\tau }_{\mathrm{opt}}$. (b) Sensitivity gain as a function of the detection noise $\mathrm{\Delta }n$ for linear and nonlinear readout, $\tau ={\tau }_{\mathrm{opt}}$ (solid) and $\tau =0.1{\tau }_{\mathrm{opt}}$ (dashed). The number of atoms is $N={10}^4$. (c),(d) The nonlinear coefficient as a function of the second DKS duration $\mathrm{\Delta }{t}_2$ (solid blue line). The size of the BECs are assumed to be constant after BS2 in (c) while they continue to expand after BS2 in (d). For specific values of $\mathrm{\Delta }{t}_2$, the nonlinear coefficient matches $-{\tau }_1\approx -0.1{\tau }_{\mathrm{opt}}$ [70].

Figure 4

Robustness of the nonlinear readout. Interferometer sensitivity gain as a function of ${\tau }_1$ and ${\tau }_2$ for linear (${\mathcal{Q}}^L$) (a) and nonlinear (${\mathcal{Q}}^{\mathrm{NL}}$) interferometer sequence with ${\tau }_{\mathrm{AI}}=10\%{\tau }_1$ (b) and ${\tau }_{\mathrm{AI}}=-10\%{\tau }_1$ (c). Here $\mathcal{Q}=10(3)$ correspond to a variance of 20 (9.5) dB below the SQL. (d) Sensitivity gain as a function of $\tau ={\tau }_1=-{\tau }_2$. The case $\tau =0.1{\tau }_{\mathrm{opt}}$ corresponds to the circles in (a)–(c) and is highlighted by the vertical line. (e) Sensitivity gain as a function of the detection noise parameter $\mathrm{\Delta }n$ and for $\tau =0.1{\tau }_{\mathrm{opt}}$. In all $N={10}^3$.