Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry

We theoretically investigate the generation of atom-light entanglement via Raman superradiance in an optical cavity, and show how this can be used to enhance the sensitivity of atom interferometry. We model a realistic optical cavity, and show that by careful temporal shaping of the optical local oscillator used to measure the light emitted from the cavity, information in the optical mode can be combined with the signal from the atom interferometer to reduce the quantum noise, and thus increase the sensitivity. It was found in Phys. Rev. Lett. 110, 053002 (2013) that an atomic “seed” was required in order to reduce spontaneous emission and allow for single mode behavior of the device. In this paper we find that the optical cavity reduces the need for an atomic seed, which allows for stronger atom-light correlations and a greater level of quantum enhancement.

Figure 1

Scheme for quantum-enhanced atom interferometry. (a),(b) State preparation. State $|1\rangle$ atoms (annihilation operator ${\widehat{a}}_1$) are driven by a classical pump field (Rabi frequency ${\mathrm{\Omega }}_{13}$), leading to the creation of a small number of state $|2\rangle$ atoms (annihilation operator ${\widehat{a}}_2$), and photons in the cavity mode (annihilation operator $\widehat{c}$). (c) Measurement. The classical pump is turned off, and the photons leaking out of the cavity are measured via homodyne detection, and the two atomic modes (${\widehat{a}}_1$ and ${\widehat{a}}_2$) are used as the input to a Mach-Zehnder interferometer.

Figure 2

Energy-level scheme for a three-level Raman transition comprising two nondegenerate hyperfine ground states ($|1\rangle$ and $|2\rangle$). The BEC is initially formed in state $|1\rangle$, and populations are transferred to $|2\rangle$ via the absorption of a photon from the classical pump beam (Rabi frequency ${\mathrm{\Omega }}_{13}$, detuned from the excited state by ${\mathrm{\Delta }}_1$) and the emission of a photon into the cavity mode $\widehat{c}$ (detuned from the excited state by ${\mathrm{\Delta }}_2$).

Figure 3

(a) $N_{a_2}=\langle {\widehat{a}}_2^{†}{\widehat{a}}_2\rangle$ vs $r=\chi \sqrt{N_t}t$ calculated via Eqs. (28) (blue solid line) and Eq. (24) (red dashed line), for an initial state such that $\langle {\widehat{a}}_2^{†}{\widehat{a}}_2\rangle =\langle {\widehat{c}}^{†}\widehat{c}\rangle =0$. (b) ${\xi }_S$ (blue solid line) and ${\xi }_F$ (red dashed line). The black dotted lines are simply to guide the eye, with the lowest line indicating $\mathrm{\Delta }\varphi =\frac{1}{N_t}$. Parameters: ${\alpha }_{10}=\sqrt{{10}^7}$; ${\alpha }_{20}={\mathcal{C}}_0=0$.

Figure 4

(a) $N_{a_2}=\langle {\widehat{a}}_2^{†}{\widehat{a}}_2\rangle$ vs $r=\chi \sqrt{N_t}t$ calculated via Eqs. (28) (blue solid line) and Eq. (24) (red dashed line), for the initial state Eq. (30), with ${\alpha }_{20}=-i\sqrt{N_{\text{seed}}}, {\mathcal{C}}_0=\sqrt{N_{\text{seed}}}$, and ${\alpha }_{10}=\sqrt{N_t-N_{\text{seed}}}$. (b) ${\xi }_S$ (blue solid line) and ${\xi }_F$ (red dashed line). Parameters: $N_t={10}^7$; $N_{\mathrm{seed}}={10}^4$.

Figure 5

(a) $N_{a_2}=\langle {\widehat{a}}_2^{†}{\widehat{a}}_2\rangle$ vs $r=\chi \sqrt{N_t}t$ calculated via Eqs. (28) (blue solid line) and Eq. (24) (red dashed line), for the initial state Eq. (30), with ${\alpha }_{20}=-i\sqrt{N_{\text{seed}}}, {\mathcal{C}}_0=-\sqrt{N_{\text{seed}}}$, and ${\alpha }_{10}=\sqrt{N_t-N_{\text{seed}}}$. (b) ${\xi }_S$ (blue solid line) and ${\xi }_F$ (red dashed line). Parameters: $N_t={10}^7$; $N_{\mathrm{seed}}={10}^4$.

Figure 6

Minimum obtainable ${\xi }_F$ (hollow shapes) and ${\xi }_S$ (solid shapes) vs $N_{\text{seed}}$ for ${\alpha }_{20}=-i\sqrt{N_{\text{seed}}}, \mathcal{C}=\sqrt{N_{\text{seed}}}$ (blue squares), ${\alpha }_{20}=-i\sqrt{N_{\text{seed}}}, \mathcal{C}=-\sqrt{N_{\text{seed}}}$ (black diamonds), and ${\alpha }_{20}=-i\sqrt{N_{\text{seed}}}, \mathcal{C}=0$ (red circles).

Figure 7

$N_{a_2}$ (blue solid line), $N_c$ (red dashed line), and $N_b$ (black dot-dashed line) vs $\tau =\chi \sqrt{N_t}t$. The atom-light coupling, $\chi$, was set to zero at $\tau =9.1$, which was chosen as it roughly corresponds to the time when $N_{a_2}$ is equal to the number of transferred atoms when ${\xi }_S$ is minimum in Fig. 3. Parameters: ${\alpha }_{10}=\sqrt{{10}^7}, {\alpha }_{20}={\mathcal{C}}_0=0, {\beta }_0=0, \kappa =13.0\times {10}^6 \mathrm{rad}{\mathrm{s}}^{-1}, \chi =1.06\times {10}^3 \mathrm{rad}{\mathrm{s}}^{-1}$, and $\gamma =600 \mathrm{rad}{\mathrm{s}}^{-1}$.

Figure 8

Dynamics of a realistic cavity in the absence of a seed. The atom-light coupling was switched off at $\tau =9.1$, as indicated by the vertical black dotted line in both figures. (a) The temporal shaping of $|u_{\mathrm{LO}}{\left(t\right)|}^2$ (blue solid line) used to calculate ${\xi }_S$. For comparison, we have also shown $u_{\mathrm{LO}}\left(t\right)\propto 1$ (red dashed line). (b) ${\xi }_S$ using Eq. (44) (blue line) and $u_{\mathrm{LO}}\left(t\right)\propto 1$ (red dashed line). For each point on the curve, the normalization condition ${\int }_0^t{\left|u_{\mathrm{LO}}\left(t^{\prime }\right)\right|}^{2}d{t}^{\prime }=1$ was enforced. For comparison, we have included ${\xi }_F=\sqrt{N_t}/\sqrt{4V\left(J_y\right)}$ (black dot-dashed line). Parameters: ${\alpha }_{10}=\sqrt{{10}^7}, {\alpha }_{20}={\mathcal{C}}_0=0, {\beta }_0=0, \kappa =13.0\times {10}^6 \mathrm{rad}{\mathrm{s}}^{-1}, \chi =1.06\times {10}^3 \mathrm{rad}{\mathrm{s}}^{-1}$, and $\gamma =600 \mathrm{rad}{\mathrm{s}}^{-1}$.

Figure 9

Dynamics of a realistic cavity with ${\alpha }_{20}=-i\sqrt{{10}^4}$ and ${\mathcal{C}}_0=-\sqrt{{10}^4}$. The atom-light coupling was switched off at $\tau =7$, as indicated by the vertical black dotted line in both figures. (a) The temporal shaping of $|u_{\mathrm{LO}}{\left(t\right)|}^2$ from Eq. (44) (blue solid line) and Eq. (45) (red dashed line) used to calculate ${\xi }_S$. (b) Enlargement to show the details at early times. (c) ${\xi }_S$ using Eq. (44) (blue line) and Eq. (45) (red dashed line), and Eq. (45) with $1\%$ detection loss of light (red circles) and $1\%$ detection loss of atoms (red squares). For each point in time, a different $u_{\mathrm{LO}}\left(t\right)$ was used, such that the normalization ${\int }_0^t{\left|u_{\mathrm{LO}}\left(t^{\prime }\right)\right|}^{2}d{t}^{\prime }=1$ could be enforced. Parameters: ${\alpha }_{10}=\sqrt{N_t-N_{\mathrm{seed}}}, {\alpha }_{20}=-i\sqrt{N_{\mathrm{seed}}}, {\mathcal{C}}_0=-\sqrt{N_{\mathrm{seed}}}, N_{\mathrm{seed}}={10}^4, N_t={10}^7, {\beta }_0=0, \kappa =13.0\times {10}^6 \mathrm{rad}{\mathrm{s}}^{-1}, \chi =1.06\times {10}^3 \mathrm{rad}{\mathrm{s}}^{-1}$, and $\gamma =600 \mathrm{rad}{\mathrm{s}}^{-1}$.