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arXiv:0709.3842 · Physical Review A

Bose-Einstein condensate as a nonlinear Ramsey interferometer operating beyond the Heisenberg limit

Seok Jin Choi, Bala Sundaram

Published May 28, 2008• 92 citations
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Abstract

We show that a dynamically evolving two-mode Bose-Einstein condensate (TBEC) with an adiabatic, time-varying Raman coupling maps exactly onto a Ramsey interferometer that includes a nonlinear medium. Assuming a realistic quantum state for the TBEC that has been achieved experimentally, we find that the measurement uncertainty of the ``path-difference'' phase shift scales as the standard quantum limit $(1/\sqrt{N})$, where $N$ is the number of atoms, while that for the interatomic scattering strength scales as $1/{N}^{7/5}$, overcoming the conventional Heisenberg limit of $1/N$.

DOI
10.1103/physreva.77.053613

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We show that a dynamically evolving two-mode Bose-Einstein condensate (TBEC) with an adiabatic, time-varying Raman coupling maps exactly onto a Ramsey interferometer that includes a nonlinear medium. Assuming a realistic quantum state for the TBEC that has been achieved experimentally, we find that the measurement uncertainty of the ``path-difference'' phase shift scales as the standard quantum limit (1/N)(1/\sqrt{N})(1/N​), where NNN is the number of atoms, while that for the interatomic scattering strength scales as 1/N7/51/{N}^{7/5}1/N7/5, overcoming the conventional Heisenberg limit of 1/N1/N1/N.
Dynamical evolution of quantum mechanical squeezing and entanglement in a two-mode Bose-Einstein condensate (TBEC) with an adiabatic, time-varying Raman coupling is studied by finding analytical expressions for these quantities. In particular, we study the entanglement between the atoms in the condensate as well as that between the two modes. The nature of the enhanced quantum correlations in TBEC is clarified by considering squeezing and entanglement both with and without the nonlinear interaction turned on; it is found that entanglement approaching maximal value can be achieved with the nonlinear interactions present. Somewhat counterintuitively, greater squeezing is found in the absence of nonlinear interactions. This is due to the collapses and revivals of the TBEC quantum state induced by the nonlinear interactions. In addition, results involving the self-trapping phase state of TBEC indicate potential for creating a dynamically stable, macroscopic entangled quantum state which is relatively robust with respect to atom number fluctuations.
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We analyze a proposed experiment [Boixo et al., Phys. Rev. Lett. 101, 040403 (2008)] for achieving sensitivity scaling better than 1/N1/N1/N in a nonlinear Ramsey interferometer that uses a two-mode Bose-Einstein condensate (BEC) of NNN atoms. We present numerical simulations that confirm the analytical predictions for the effect of the spreading of the BEC ground-state wave function on the ideal 1/N3/21/{N}^{3/2}1/N3/2 scaling. Numerical integration of the coupled, time-dependent, two-mode Gross-Pitaevskii equations allows us to study the several simplifying assumptions made in the initial analytic study of the proposal and to explore when they can be justified. In particular, we find that the two modes share the same spatial wave function for a length of time that is sufficient to run the metrology scheme.
This chapter describes the use of squeezed states to improve the sensitivity in the estimation of the relative phase accumulated in a Bose--Einstein condensate (BEC) beyond the standard quantum limit. We begin by examining the specific example of a two-component BEC, where atoms in two different hyperfine levels interact with each other. By evolving an initial superposition of states under an effective Hamiltonian, a squeezed state is realized. Next, the squeezing is visualized using the Q-function, giving the mean spin direction and shape of the BEC distribution. We discuss the basic operation of Ramsey interferometry, which is a classic example of two-slit atom diffraction. We then calculate the error in estimating the relative phase shift using the error propagation formula that allows us to define the squeezing parameter. This measures the amount of metrological gain using squeezed states over unsqueezed states. We give an example of the metrological gain in the estimation of the relative phase shift using a squeezed state of the two-component BEC as an input state to an atom Mach--Zehnder interferometer. The Fisher information is used to find which rotation axis gives the optimal information from a given initial state. Finally, we parameterize the nonlinear phase per atom responsible for squeezing and discuss how this parameter is controlled in different experiments.
We propose a feasible scheme to realize nonlinear Ramsey interferometry with a two-component Bose-Einstein condensate, where the nonlinearity arises from the interaction between coherent atoms. In our scheme, two Rosen-Zener pulses are separated by an intermediate holding period of variable duration and through varying the holding period we have observed nice Ramsey interference patterns in the time domain. In contrast to the standard Ramsey fringes our nonlinear Ramsey patterns display diversiform structures ascribed to the interplay of the nonlinearity and asymmetry. In particular, we find that the frequency of the nonlinear Ramsey fringes exactly reflects the strength of nonlinearity as well as the asymmetry of system. Our finding suggests a potential application of the nonlinear Ramsey interferometry in calibrating the atomic parameters such as scattering length and energy spectrum.
Considering matter wave bright solitons from weakly coupled Bose-Einstein condensates trapped in a double-well potential, we study the formation of macroscopic non-classical states, including Schrödinger-cat superposition state and maximally path entangled N00N-state. We examine these macroscopic states by Mach-Zehnder interferometer in the context of parity measurements, which has been done to obtain Heisenberg limit accuracy for linear phase shift measurement. We reveal that the ratio of two-body scattering length to intra-well hopping parameter can be measured with the scaling beyond this limit by using nonlinear phase shift with interacting quantum solitons.