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arXiv:1907.06974 · Physical Review A

Spinor Bose-Einstein condensate interferometer within the undepleted pump approximation: Role of the initial state

Jianwen Jie, Qingze Guan, D. Blume

Published Oct 11, 2019• 14 citations
arXivHTMLPDF

Abstract

Most interferometers operate with photons or dilute, noncondensed cold atom clouds in which collisions are strongly suppressed. Spinor Bose-Einstein condensates (BECs) provide an alternative route toward realizing three-mode interferometers; in this realization, spin-changing collisions provide a resource that generates mode entanglement. Working in the regime where the pump mode, i.e., the $m=0$ hyperfine state, has a much larger population than the side or probe modes ($m=\ifmmode\pm\else\textpm\fi{}1$ hyperfine states), $f=1$ spinor BECs approximate SU(1,1) interferometers. We derive analytical expressions within the undepleted pump approximation for the phase sensitivity of such an SU(1,1) interferometer for two classes of initial states: pure Fock states and coherent spin states. The interferometer performance is analyzed for initial states without seeding, with single-sided seeding, and with double-sided seeding. The validity regime of the undepleted pump approximation is assessed by performing quantum calculations for the full spin Hamiltonian. Our analytical results and the associated dynamics are expected to guide experiments as well as numerical studies that explore regimes where the undepleted pump approximation makes quantitatively or qualitatively incorrect predictions.

DOI
10.1103/physreva.100.043606

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Most interferometers operate with photons or dilute, noncondensed cold atom clouds in which collisions are strongly suppressed. Spinor Bose-Einstein condensates (BECs) provide an alternative route toward realizing three-mode interferometers; in this realization, spin-changing collisions provide a resource that generates mode entanglement. Working in the regime where the pump mode, i.e., the m=0m=0m=0 hyperfine state, has a much larger population than the side or probe modes (m=\ifmmode±\else\textpm\fi1m=\ifmmode\pm\else\textpm\fi{}1m=\ifmmode±\else\textpm\fi1 hyperfine states), f=1f=1f=1 spinor BECs approximate SU(1,1) interferometers. We derive analytical expressions within the undepleted pump approximation for the phase sensitivity of such an SU(1,1) interferometer for two classes of initial states: pure Fock states and coherent spin states. The interferometer performance is analyzed for initial states without seeding, with single-sided seeding, and with double-sided seeding. The validity regime of the undepleted pump approximation is assessed by performing quantum calculations for the full spin Hamiltonian. Our analytical results and the associated dynamics are expected to guide experiments as well as numerical studies that explore regimes where the undepleted pump approximation makes quantitatively or qualitatively incorrect predictions.
The SU(1,1) interferometer was originally conceived as a Mach-Zehnder interferometer with the beam splitters replaced by parametric amplifiers. The parametric amplifiers produce states with correlations that result in enhanced phase sensitivity. F=1F=1F=1 spinor Bose-Einstein condensates (BECs) can serve as the parametric amplifiers for an atomic version of such an interferometer by collisionally producing entangled pairs of $|F=1,m=\ifmmode\pm\else\textpm\fi{}1\ensuremath{\rangle}$ atoms. We simulate the effect of single- and double-sided seeding of the inputs to the amplifier using the truncated-Wigner approximation. We find that single-sided seeding degrades the performance of the interferometer exactly at the phase the unseeded interferometer should operate the best. Double-sided seeding results in a phase-sensitive amplifier, where the maximal sensitivity is a function of the phase relationship between the input states of the amplifier. In both single- and double-sided seeding we find there exists an optimal phase shift that achieves sensitivity beyond the standard quantum limit. Experimentally, we demonstrate a spinor phase-sensitive amplifier using a BEC of 23Na^{23}\mathrm{Na}23Na in an optical dipole trap. This configuration could be used as an input to such an interferometer. We are able to control the initial phase of the double-seeded amplifier and demonstrate sensitivity to initial population fractions as small as 0.1%.
Abstract We experimentally demonstrate a new type of spin-mixing interferometry in sodium Bose–Einstein condensates (BECs) based on seeded initial states. Seeding is useful because it speeds up the generation of entangled pairs, allowing many collisions to take place quickly, creating large populations in the arms of the interferometer. The entangled probe states of our interferometer are generated via spin-exchange collisions in F = 1 spinor BECs, where pairs of atoms with the magnetic quantum number m F = 0 collide and change into pairs with m F = ± 1 . Our results show that our seeded spin-mixing interferometer beats the standard quantum limit (SQL) with a metrological gain of 3.69 dB with spin-mixing time t = 10 ms in the case of single-sided seeding, and 3.33 dB with spin-mixing time t = 8 ms in the case of double sided seeding. The mechanism for beating the SQL is two-mode spin squeezing generated via spin-exchange collisions. Our results on spin-mixing interferometry with seeded states are useful for future quantum technologies such as quantum-enhanced microwave sensors, and quantum parametric amplifiers based on spin-mixing.
We numerically investigate atomic interferometry based on spin-exchange collisions in F=1F=1F=1 spinor Bose-Einstein condensates in the regime of long evolution times $t\ensuremath{\gg}h/c$, where ccc is the spin-dependent interaction energy. We show that the sensitivity of spin-mixing interferometry can be enhanced by using classically seeded initial states with a small population prepared in the ${m}_{F}=\ifmmode\pm\else\textpm\fi{}1$ states.
Abstract Active interferometers are designed to enhance phase sensitivity beyond the standard quantum limit by generating entanglement inside the interferometer. An atomic version of such a device can be constructed by means of a spinor Bose–Einstein condensate with an F = 1 groundstate manifold in which spin-changing collisions (SCCs) create entangled pairs of m = ±1 atoms. We use Bethe Ansatz techniques to find exact eigenstates and eigenvalues of the Hamiltonian that models such SCCs. Using these results, we express the interferometer’s phase sensitivity, Fisher information, and Hellinger distance in terms of the Bethe rapidities. By evaluating these expressions we study scaling properties and the interferometer’s performance under the full Hamiltonian that models the SCCs, i.e., without the idealising approximations of earlier works that force the model into the framework of SU(1,1) interferometry.
Although SU(1,1) interferometry achieves Heisenberg-limited sensitivities, it suffers from one major drawback: Only those particles outcoupled from the pump mode contribute to the phase measurement. Since the number of particles outcoupled to these "side modes" is typically small, this limits the interferometer's absolute sensitivity. We propose an alternative "pumped-up" approach where all the input particles participate in the phase measurement and show how this can be implemented in spinor Bose-Einstein condensates and hybrid atom-light systems-both of which have experimentally realized SU(1,1) interferometry. We demonstrate that pumped-up schemes are capable of surpassing the shot-noise limit with respect to the total number of input particles and are never worse than conventional SU(1,1) interferometry. Finally, we show that pumped-up schemes continue to excel-both absolutely and in comparison to conventional SU(1,1) interferometry-in the presence of particle losses, poor particle-resolution detection, and noise on the relative phase difference between the two side modes. Pumped-up SU(1,1) interferometry therefore pushes the advantages of conventional SU(1,1) interferometry into the regime of high absolute sensitivity, which is a necessary condition for useful quantum-enhanced devices.
In this paper we discuss Bose-Einstein spinor condensates for F=1F=1F=1 atoms in the context of 87Rb^{87}\text{R}\text{b}87Rb, as studied experimentally by the Stamper-Kurn group [L. E. Sadler et al., Nature (London) 443, 312 (2006)]. The dynamical quantum fluctuations of a sample that starts as a condensate of NNN atoms in a pure F=1F=1F=1, mF=0{m}_{F}=0mF​=0 state are described in analogy to the two-mode squeezing of quantum optics in terms of an su(1,1)\mathfrak{s}\mathfrak{u}(1,1)su(1,1) algebra. In this system the initial mF=0{m}_{F}=0mF​=0 condensate acts as a source (pump) for the creation pairs of ${m}_{F}=1,\ensuremath{-}1$ atoms. We show that even though the system as a whole is described by a pure state with zero entropy, the reduced density matrix for the mF=+1{m}_{F}=+1mF​=+1 degree of freedom, obtained by tracing out the ${m}_{F}=\ensuremath{-}1,0$ degrees of freedom, corresponds to a thermal state. Furthermore, these quantum fluctuations of the initial dynamics of the system provide the seeds for the formation of domains of ferromagnetically aligned spins.