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arXiv:cond-mat/9611074 · Physical Review A

Phase-dependent spectrum of scattered light from two Bose condensates

Janne Ruostekoski, Daniel F. Walls

Published May 1, 1997• 37 citations
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Abstract

We calculate the spectrum of the scattered light from quantum degenerate atomic gases obeying Bose-Einstein statistics. The atoms are assumed to occupy two ground states, which are optically coupled through a common excited state by two low-intensity off-resonant light beams. In the presence of a Bose condensate in both ground states, the atoms may exhibit light-induced oscillations between the two condensates analogous to the Josephson effect. The spectrum of the scattered light is calculated in the limit of a low oscillation frequency. In the spectrum we are able to observe qualitative features depending on the phase difference between the macroscopic wave functions of the two condensates. Thus, our optical scheme could possibly be used as an experimental realization of the spontaneous breakdown of the U(1) gauge symmetry in the Bose-Einstein condensation.

DOI
10.1103/physreva.55.3625

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We calculate the spectrum of the scattered light from quantum degenerate atomic gases obeying Bose-Einstein statistics. The atoms are assumed to occupy two ground states, which are optically coupled through a common excited state by two low-intensity off-resonant light beams. In the presence of a Bose condensate in both ground states, the atoms may exhibit light-induced oscillations between the two condensates analogous to the Josephson effect. The spectrum of the scattered light is calculated in the limit of a low oscillation frequency. In the spectrum we are able to observe qualitative features depending on the phase difference between the macroscopic wave functions of the two condensates. Thus, our optical scheme could possibly be used as an experimental realization of the spontaneous breakdown of the U(1) gauge symmetry in the Bose-Einstein condensation.
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 consider two dilute gas Bose-Einstein condensates with opposite velocities from which a monochromatic light field detuned far from the resonance of the optical transition is coherently scattered. In the thermodynamic limit, when the relative fluctuations of the atom number difference between the two condensates vanish, the relative phase between the Bose-Einstein condensates may be established in a superposition state by detections of spontaneously scattered photons, even though the condensates have initially well-defined atom numbers. For a finite system, stochastic simulations show that the measurements of the scattered photons lead to a randomly drifting relative phase and drive the condensates into entangled superpositions of number states. This is because according to Bose-Einstein statistics the scattering to an already occupied state is enhanced.
We study interactions of light with a sample of two-level atoms. Quantum statistical effects of degenerate boson and fermion gases are described by using the quantum field theory of atom-photon interactions, fully including the angular momentum degeneracies. Coupled propagation equations are given for light and matter fields, and plausible general simplifications are enumerated.^ In the limit of large atom-field detuning, multiple scattering of light and QED linewidths and line shifts may be ignored, and the propagation equations of light and matter fields may be decoupled. A general expression for the spectrum of the scattered light is derived, complete with the dependence on atomic level scheme, on various light polarizations, and on the geometry of the experiment. With these simplifications we are able to find distinct qualitative features in the spectrum of the scattered light in the case of both fermions and bosons that depend on the statistics of atoms. This is because for bosons (fermions), those scattering events in which an atom recoils to an already occupied state are enhanced (inhibited).^ As the driving light gets closer to the atomic resonance, multiple scattering and vacuum fluctuations of the electromagnetic fields should also be taken into account. We write the equations of motion for the excited state field operators by eliminating the vacuum electromagnetic field. It is always assumed that the driving external laser field is in the coherent state. We find in the limit of low light intensity, a hierarchy of equations of motion for correlation functions that contain one excited-atom field and one, two, three, etc., ground state atom fields. It is conjectured that the entire linear hierarchy may be solved by solving numerically the classical equations for the coupled system of electromagnetic fields and charged harmonic oscillators. We discuss the emergence of resonant dipole-dipole interactions and collective linewidths, and delineate the limits of validity of the column density approach in terms of non-cooperative atoms by presenting a mathematical example in which this approach is exact. ^
We propose a light-scattering scheme to measure the relative phase of atomic Bose-Einstein condensates and its diffusion rate. The proposal relies on the existence of two independent condensates coupled to a common excited state. To this end, we consider a two-well ground-state potential together with excited-state trap wave functions that extend over the whole region. When the first trap is driven by a weak monochromatic laser field, the light scattered from the second trap has a nonzero mean-electric field amplitude with a phase shift proportional to the difference of the condensate phases. When both condensates are driven, the phases of the two laser fields can be adjusted to cancel the scattering completely by a quantum interference. The particular value of the laser phase difference that gives zero scattering determines the relative phase of the two condensates.
Summary form only given.We consider two overlapping and optically thin dilute gas Bose-Einstein condensates occupying two different Zeeman sublevels. We propose an optical method for increasing the number of condensate atoms. Two laser beams drive Raman transitions between the Zeeman states and induce a coherent quantum tunneling between the Bose-Einstein condensates. The scattering of photons is dominantly coherent and in the forward direction due to the Bose enhancement. This corresponds to the scattering of atoms between the two condensates. Some laser-driven Raman transitions also scatter atoms between the condensate and noncondensate atom fractions. For a range of condensate phase differences there is destructive quantum interference of the amplitudes for scattering atoms out of the condensates. Because the total atom scattering rate into the condensates is unaffected the condensates grow. This mechanism is analogous to that responsible for optical lasing without inversion, and it may find application as a pump for an atom laser.
We study interactions of light with a sample of two-level atoms, with full inclusion of angular momentum degeneracy, at temperatures and densities such that the quantum statistics of the atoms may have an effect. Coupled propagation equations are given for light and matter fields, and plausible general simplifications are enumerated. In particular, the motion of the atoms during the excited-state lifetime may often be ignored. The propagation equations of light and matter fields are decoupled within the assumption that the detuning of the driving light from atomic resonance is large, and the spectrum of scattered light is studied for ideal Bose and Fermi gases. In addition to the expected image of the velocity distribution, the spectra contain qualitatively distinct features that depend on the statistics of the atoms. This is because for bosons (fermions), those scattering events in which an atom recoils to an already occupied state are enhanced (inhibited).