Quantum enhanced measurement of rotations with a spin-1 Bose-Einstein condensate in a ring trap

We present a model of a spin-squeezed rotation sensor utilizing the Sagnac effect in a spin-1 Bose-Einstein condensate in a ring trap. The two input states for the interferometer are seeded using Raman pulses with Laguerre-Gauss beams and are amplified by the bosonic enhancement of spin-exchange collisions, resulting in spin-squeezing and potential quantum enhancement of the interferometry. The ring geometry has an advantage over separated beam path atomic rotation sensors due to the uniform condensate density. We model the interferometer both analytically and numerically for realistic experimental parameters and find that significant quantum enhancement is possible, but this enhancement is partially degraded when working in a regime with strong atomic interactions.

Figure 1

(a) Energy level diagram illustrating a general Raman transition between two spin states, $|a\rangle$ and $|b\rangle$. The single-photon detuning $\mathrm{\Delta }$ is required to minimize the excited-state population and the two-photon detuning $\delta$ is included as a resonance condition. (b) Energy level diagram in the presence of the quadratic Zeeman shift for a $\pi /2$ pulse (B and C in Fig. 3), which acts as an atomic beam splitter between the $|+1,+\ell \rangle$ and the $|-1,-\ell \rangle$ modes; we have written the atomic states as $|m_{\mathrm{F}},\ell \rangle$, where $m_{\mathrm{F}}$ is the electronic Zeeman sublevel and $\ell \hslash$ is the atomic center-of-mass angular momentum mode.

Figure 2

Schematic of rotation sensing using counter-propagating atoms and Laguerre-Gauss (LG) beams. For illustrative purposes we show the $\ell =5$ case. Color represents the relative phase of the LG beams (the outer ring) and the $m_{\mathrm{F}}=\pm 1$ atomic spin states (inner ring). Beams $B$ and $C$ correspond to the two $\pi /2$ pulses, shown in Fig. 3. The rotation of the LG beams causes a shift in the relative phase, while the atoms provide an inertial reference frame. As the atomic population difference after the second $\pi /2$ pulse (C) depends on the relative phase of the LG beams, (8), the final population difference is sensitive to the rotation.

Figure 3

Schematic of the proposed spin-squeezed atom interferometer. A coherent seeding pulse (beam splitter A) transfers a small number of atoms $N_{\mathrm{seed}}$ from the $|0,0\rangle$ state to each of the $|\pm 1,\pm \ell \rangle$ states. The quadratic Zeeman effect is used to cause resonant spin-exchange collisions by setting an appropriate bias magnetic field. After the desired amount of population transfer, the system is tuned away from resonance and the trap is adiabatically relaxed, perhaps in the $z$ dimension, to reduce the effect of atomic interactions. The modes are mixed with a $\pi /2$ pulse (beam splitter B), which converts the relative number squeezing to phase squeezing. After accumulating a relative phase over interrogation time $T$ in the presence of a rotation, the $|+1,+\ell \rangle ,|-1,-\ell \rangle$ modes are interfered via a final $\pi /2$ pulse (beam splitter C). Finally, the number difference between the $m_{\mathrm{F}}=\pm 1$ Zeeman states is measured, e.g., by destructive imaging using a magnetic-field gradient and Stern-Gerlach separation.

Figure 4

Energy level diagram in the presence of a quadratic Zeeman ${\delta }_Z$ shift for two (simultaneous) seeding pulses, used to transfer a small number of atoms from the original $|0,0\rangle$ ground state to the $|\pm 1,\pm \ell \rangle$ modes. This corresponds to pulses A in Fig. 3. We include the two-photon detuning ${\delta }_T=\hslash {\ell }^2/2m{R}^2$ to ensure that the coupling process conserves kinetic energy.

Figure 5

Analytic estimate of the spin-squeezing in the input states. (a) The spin-squeezing parameter $\xi (r,N_{\mathrm{seed}})$ is plotted as a function of the squeezing parameter $r$ for $N_{\mathrm{seed}}=10$, which shows that $\xi <1$ for $r>0$. (b) Contour plot showing the $N_{\mathrm{seed}}$ dependence of $\xi$. We can see that $\xi$ is approximately independent of $N_{\mathrm{seed}}$ for $N_{\mathrm{seed}}>1$. In practice, however, the $r$ at which the undepleted pump approximation breaks down depends on $N_{\mathrm{seed}}$, so there is effectively a maximum $r$ for each $N_{\mathrm{seed}}$. The dashed black line indicates the contour plotted in (a).

Figure 6

Spin-squeezing parameter normalized to the Heisenberg limit $\left({\xi }_{\mathrm{HL}}\right)$ for $N_t={10}^5$ particles, i.e., $N_0(t=0)=N_t$. Therefore the sensitivity for $N_{\mathrm{seed}}={10}^5/2$ is the SQL. The minimum occurs for a seed size of $N_{\mathrm{seed}}=1/4$ with a sensitivity $\sqrt{2}$ times the Heisenberg limit.

Figure 7

Comparison of the optimal characteristics of the single-mode (SMTWA) and multimode (MMTWA) interferometer as a function of the seed size. (a) Optimum number of atoms transferred into the $m_{\mathrm{F}}=\pm 1$ Zeeman states via spin-exchange collisions for SMTWA, and total number of $m_{\mathrm{F}}=\pm 1$ in the $\ell =\pm 2$ angular momentum modes for the MMTWA points. (b) The overall sensitivity $\mathrm{\Delta }\varphi =\xi /\sqrt{N_t}$, compared to the SQL and HL for a total number of ${10}^5$ atoms. The number in (a) is the value which optimizes $\mathrm{\Delta }\varphi$. As each seed size has a different number of atoms available for measurement [shown in (a)] the SQL and HL are different for each symbol, but for comparison we simply take the best-case scenario $\left(N_t={10}^5\right)$ for each. Each symbol is the minimum sensitivity achieved for that seed size, i.e., symbols are for the optimum value of $\mathrm{\Delta }\varphi$, and the populations in (a) are the corresponding $N_t\left(r_{\mathrm{opt}}\right)$.

Figure 8

Dynamics of the Wineland parameter during the interrogation time. The solid black curve is close to maximally squeezed, with a large preparation time and a small initial seed. The dashed blue curve is only weakly squeezed, with a larger seed and shorter preparation time, and demonstrates only mild dynamics. The dot-dashed red curve indicates an intermediate regime between the two extremes. The parameters used were ${\tau }_{\mathrm{prep}}/\omega =\{125,60,30\}$ ms, with seed sizes of $N_{\mathrm{seed}}=\{100,5000,10000\}$, respectively. The squeezing parameter is calculated from Eq. (30). Nonlinear interactions during the interrogation time further complicate this analysis, and so for illustrative purposes we have considered the case where the transverse area $A$ has been increased such that the nonlinear interactions are negligible, i.e., ${\tilde{c}}_2,{\tilde{c}}_0=0$. We consider the situation wherein the transverse confinement is such that the interaction cannot be ignored in Sec. 7.

Figure 9

Absolute uncertainty $\mathrm{\Delta }\mathrm{\Omega }$ in the rotation measurement as a function of the interrogation time $T$ for effective 1D spin-dependent interaction strengths of (a) $\delta {\tilde{c}}_2=0$, (b) $\delta {\tilde{c}}_2=0.02$, and (c) $\delta {\tilde{c}}_2=1$ during the interrogation time. We have defined the ratio $\delta {\tilde{c}}_2={\tilde{c}}_2\left(T\right)/{\tilde{c}}_2\left(0\right)$ of the interaction strength during the interrogation time ${\tilde{c}}_2\left(T\right)$ to the interaction strength during the preparation time ${\tilde{c}}_2\left(0\right)$. These depend on the transverse area $A$, which can be controlled by relaxing the confinement in the $z$ direction. Even for low trapping frequencies the rotation sensitivity rapidly becomes worse than the SQL, and so (b) and (c) are plotted on a logarithmic time scale.