Squeezing on Momentum States for Atom Interferometry

We propose and analyze a method that allows for the production of squeezed states of the atomic center-of-mass motion that can be injected into an atom interferometer. Our scheme employs dispersive probing in a ring resonator on a narrow transition in order to provide a collective measurement of the relative population of two momentum states. We show that this method is applicable to a Bragg diffraction-based strontium atom interferometer with large diffraction orders. This technique can be extended also to small diffraction orders and large atom numbers $N$ by inducing atomic transparency at the frequency of the probe field, reaching an interferometer phase resolution scaling $\mathrm{\Delta }\varphi \sim N^{-3/4}$. We show that for realistic parameters it is possible to obtain a 20 dB gain in interferometer phase estimation compared to the standard quantum limit. Our method is applicable to other atomic species where a narrow transition is available or can be synthesized.

Figure 1

(a) Schematic setup with the probe field ${\widehat{b}}_{\mathrm{in}}$ coupled to the optical cavity and interacting with the atoms (green circle) through the oblique beam with angle $\alpha$ with respect to gravity. The reflected field ${\widehat{b}}_{\mathrm{out}}$ is measured by the detector $D$. The atomic states are manipulated through the Bragg beams $B_1$ and $B_2$. (b) Level diagram for probing on the ${}^1{S}_0–{}^3{P}_1$ transition (red arrow) and for momentum-state manipulation through Bragg diffraction on the ${}^1{S}_0–{}^1{P}_1$ transition (blue arrows). (c) Interferometer trajectories and measurement sequence. The interferometer is formed by a $\pi /2–\pi –\pi /2$ Mach-Zehnder sequence with interrogation time $T$. The cavity-enhanced squeezing measurement with duration $T_m$ is indicated as $M_1$. The Bragg pulse $\theta$ induces a phase-sensitive state for the interferometer, and the final readout measurement is performed ($M_2$). Lower part: Bloch sphere representation of the state evolution.

Figure 2

Expected metrological gain ${\xi }_m$ as a function of the Bragg diffraction order $n$, assuming perfect detection efficiency ${\epsilon }_d=1$, and for $N\eta ={10}^4$. The blue circles refer to the scheme shown in Fig. 1. Values shown as red squares and magenta diamonds refer to the scheme in Fig. 3, where electromagnetically induced transparency (EIT) is present with ${\mathrm{\Omega }}_{\mathrm{eff}}=2\pi \times 400\mathrm{kHz}$ and with perfect transparency ${\mathrm{\Omega }}_{\mathrm{eff}}=\infty$, respectively.

Figure 3

(a) Level diagram for the generation of squeezed momentum states enhanced by induced transparency via the ${}^3{P}_0$ state. The Bragg laser beams are indicated as $B_1$ and $B_2$. The effective control field that couples the states ${}^3{P}_0$ and ${}^3{P}_1$ is obtained by two-photon Raman coupling via the ${}^3{S}_1$ intermediate state. The two Raman beams $R_1$ and $R_2$ operate at the wavelengths 679 nm (${}^3{P}_0–{}^3{S}_1$) and 688 nm (${}^3{P}_1–{}^3{S}_1$), respectively, and are detuned from the transition to the ${}^3{S}_1$ state by ${\mathrm{\Delta }}_R$. (b) Cavity transmission spectrum with (red solid line) and without (blue dashed line) Raman coupling to the ${}^3{P}_0$ state for ${\mathrm{\Omega }}_{\mathrm{eff}}=2\pi \times 400\mathrm{kHz}$, $N\eta =3\times {10}^3$, and $n=1$. The two lateral peaks correspond to the vacuum Rabi splitting for $\kappa =2\pi \times 50\mathrm{kHz}$. The population measurement is performed at the frequency of the transparency region (at probe detuning $\delta =0$), which corresponds to a linewidth of ${\kappa }_{\mathrm{EIT}}=2\pi \times 6\mathrm{kHz}$.

Figure 4

Metrological gain ${\xi }_m$ as a function of the number of photons scattered into free space per atom for different Bragg diffraction orders and for varying Raman coupling strength: $n=5$, ${\mathrm{\Omega }}_{\mathrm{eff}}=0$ (blue solid line); $n=20$, ${\mathrm{\Omega }}_{\mathrm{eff}}=0$ (red dashed line); $n=1$, ${\mathrm{\Omega }}_{\mathrm{eff}}=2\pi \times 400\mathrm{kHz}$ (magenta dot-dashed line). We assume a collective cooperativity $N\eta ={10}^4$ and perfect detection efficiency ${\epsilon }_d=1$.