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arXiv:quant-ph/0602054 · Physical Review A

Bose-Einstein condensates relative phase measurements through atomic homodyne detection

Bruno Requião da Cunha, M. C. de Oliveira

Published Jun 15, 2007• 9 citations
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Abstract

The dynamics of a two-mode Bose-Einstein condensate trapped in a double-well potential results approximately in an effective Rabi oscillation regime of exchange of population between both wells for sufficiently strong overlap between the modes functions and sufficiently large condensates. Facing the effective Rabi oscillation as a temporal atomic beam splitter we propose a nondestructive measurement process allowing an atomic homodyne detection, thus yielding indirect relative phase information about one of the two-mode condensates. This proposal is achieved through a secondary optical homodyne detection, in which the field is allowed to dispersively interact with one of the condensate modes inside an optical cavity.

DOI
10.1103/physreva.75.063615

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The dynamics of a two-mode Bose-Einstein condensate trapped in a double-well potential results approximately in an effective Rabi oscillation regime of exchange of population between both wells for sufficiently strong overlap between the modes functions and sufficiently large condensates. Facing the effective Rabi oscillation as a temporal atomic beam splitter we propose a nondestructive measurement process allowing an atomic homodyne detection, thus yielding indirect relative phase information about one of the two-mode condensates. This proposal is achieved through a secondary optical homodyne detection, in which the field is allowed to dispersively interact with one of the condensate modes inside an optical cavity.
We investigate a nondestructive measurement technique to monitor Josephson-like oscillations between two spatially separated neutral atom Bose-Einstein condensates. One condensate is placed in an optical cavity, which is strongly driven by a coherent optical field. The cavity output field is monitored using a homodyne detection scheme. The cavity field is well detuned from an atomic resonance, and experiences a dispersive phase shift proportional to the number of atoms in the cavity. The detected current is modulated by the coherent tunneling oscillations of the condensate. Even when there is an equal number of atoms in each well initially, a phase is established by the measurement process and Josephson-like oscillations develop due to measurement backaction noise alone.
The properties of an atomic Bose-Einstein condensate in a double-well potential are investigated through a two-mode analysis. An analytic solution for the semiclassical tunnelling and self-trapping dynamics is compared with numerical simulations of the quantum dynamics, which exhibit collapses and revivals for a closed system. A continuous non-destructive measurement technique to monitor the Josephson tunnelling oscillations is presented, in which the condensate in one well dispersively shifts the phase of a coherent probe beam in proportion to atom-number. The evolution of the resulting homodyne photocurrent and Bloch Q distributions shows that oscillations develop even when the initial state possesses phase symmetry. The conditional dynamics of the condensate which result from measurement back-action also appear in certain semiclassical formulations. The homodyne measurement technique is incorporated into a proposed weak-force detector. A maximally entangled initial state, which is the ground state for a double condensate with strong attractive atomic interactions, enables a high-precision measurement. The dynamics of quantum many-body multimode systems of interacting bosons are simulated using phase-space methods. The use of the Wigner technique predicts novel noise effects in fibre solitons. The positive-P representation is used to simulate the formation of mesoscopic Bose-Einstein condensates via evaporative cooling in three dimensional atom traps. The results indicate highly non-classical behaviour near the critical point, and provide evidence for the spontaneous formation of vortices. Comparisons with corresponding mean-field calculations reveal large differences between the semiclassical and fully quantum results. Finally, the possibility of future progress with alternative phase-space methods is considered.
A scheme is proposed, that allows one for performing homodyne detection of the matter-wave field of ultracold bosonic atoms. It is based on a pump-probe lasers setup, that both illuminates a Bose-Einstein condensate, acting as reference system, and a second ultracold gas, composed by the same atoms but in a quantum phase to determine. Photon scattering outcouples atoms from both systems, which then propagate freely. Under appropriate conditions, when the same photon can either be scattered by the Bose-Einstein condensate or by the other quantum gas, both flux of outcoupled atoms and scattered photons exhibit oscillations, whose amplitude is proportional to the condensate fraction of the quantum gas. This setup allows one, for instance, to perform thermometry of a condensate or to monitor the Mott-insulator/superfluid phase transition in optical lattices, and can be extended in order to measure the first-order correlation function of a quantum gas. The dynamics here discussed make use of the entanglement between atoms and photons, which is established by the scattering process, in order to access detailed information on the quantum state of matter.
We study the interaction of light with two Bose-Einstein condensates as an open quantum system. The two overlapping condensates occupy two different Zeeman sublevels and two driving light beams induce a coherent quantum tunneling between the condensates. We derive the master equation for the system. It is shown via stochastic simulations that the measurements of spontaneously scattered photons establish the relative phase between two Bose-Einstein condensates, even though the condensates are initially in pure number states. These measurements are nondestructive for the condensates, because only light is scattered and no atoms are removed from the system. Due to macroscopic quantum interference the detection rate of photons depends strongly on the relative phase between the condensates. This may provide a way to distinguish between initial number states and initial symmetry-breaking states of the condensates.
We theoretically investigate the dispersive interaction of a Bose-Einstein condensate (BEC) trapped inside an optomechanical cavity with a moving end mirror in the presence of the laser phase noise (LPN) as well as the atomic collisions. We assume that the effective frequency of the optical mode is much greater than those of the mechanical and the Bogoliubov modes of the movable mirror and the BEC. In the adiabatic approximation where the damping rate of the cavity is faster than those of the other modes, the system behaves as an effective two-mode model in which the atomic and mechanical modes are coupled to each other through the mediation of the optical field by an effective coupling parameter. We show that in the effective two-mode model, the LPN appears as a classical stochastic pump term which drives the amplitude quadratures of the mechanical and the Bogoliubov modes. It is also shown that a strong stationary mirror-atom entanglement can be established just in the dispersive and Doppler regimes where the two modes come into resonance with each other and the effect of the LPN gets very small.