Theory of Superradiant Scattering of Laser Light from Bose-Einstein Condensates

Abstract

In a recent MIT experiment, a new form of superradiant Rayleigh scattering was observed in Bose-Einstein condensates. We present a detailed theory of this phenomena in which condensate depeletion leads to mode competition, which, together with the directional dependence of the scattering rate, is ultimately responsible for the observed phenomena. The nonlinear response of the system is shown to be highly sensitive to initial quantum fluctuations which cause large run to run variations in the observed superradiant pulses.PACS numbers: 03.75.Fi,42.50.Fx With the recent advent of Bose-Einstein condensation (BEC) in low-density alkali vapors [1,2], a laser-like source of coherent monochromatic atomic matter-waves is now readily available. As the electromagnetic vacuum itself provides a nonlinear medium for atomic fields, an atomic BEC is thus an ideal system to study nonlinear wave-mixing and related phenomena. Indeed, nonlinear atom optics [3-5] is now an experimental reality with the recent observation of atomic four-wave mixing in condensate systems [6,7]. In addition to wave-mixing between atomic matter waves, the ability to generate laser-like atomic fields also raises the possibility to observe direct wave-mixing between atomic and optical fields.In the case where the atoms interact only with far off-resonant optical fields, the dominant atom-photon interaction is two-photon Rayleigh scattering. When the atoms are described as matter waves, Rayleigh scattering is formally equivalent to a cubic nonlinearity, and therefore leads to four-wave mixing between atomic and optical fields. In recent work [8-10] along these lines, the scattering of light by a condensate from a strong pump laser into a weak quantized optical cavity mode was considered. This work was an extension of the collective atomic recoil laser (CARL) [11,12] into the regime of BEC, and focused on exploiting an instability in the lightmatter interaction to parametrically amplify atomic and optical waves as well as to optically manipulate matterwave coherence properties and generate entanglement between atomic and optical fields. Recent experiments by Ketterle and coworkers ar MIT[13], however, have demonstrated that this instability can play an important role also in the case in which laser light is scattered into the vacuum modes of the electromagnetic field. In these experiments, a variation of Dicke superradiance [14] was observed in which the role of electronic coherence, which stores the memory of previous scattering events, is replaced by coherence between center-of-mass momentum states, i.e. interference fringes in the atomic density. In this paper we present a multi-mode theory of condensate superradiance. Beginning with the elimination of the radiated light field as in the Wigner-Weiskopf theory of spontaneous emission, we then derive a linearized model which describes amplification of quantum fluctuations. This is then coupled to a 'classical' nonlinear model in which mode competition quenches scattering in all but the direction(s) of ma...

Robust generation of entanglement in Bose-Einstein condensates by collective atomic recoil

“…However, in view of an experimental observation of entanglement, a detailed analysis of the sources of noise is in order, which in turn may be a serious limitation for entanglement in CARL [11]. Also, it has not yet been proved that BEC superradiance experiments may generate entangled atom-photon states, as suggested in [9]. This issue is investigated for the first time in this paper, where we demonstrate the entangled properties of the atom-atom and atom-photon pairs produced in the linear stage of the superradiant CARL regime in a BEC.…”

“…without the optical cavity. In this case the radiation is emitted along the 'end-fire modes' of the condensate [9] with very large radiation losses (in the mean field model, with κ ≈ c/L, where κ is the cavity decay rate and L is the condensate length). In a recent work [5] it has been shown that atom-atom and atom-photon entanglement can be produced in the linear regime of CARL, in which the ground state of the condensate remains approximately undepleted.…”

Mode competition in superradiant scattering of matter waves

“…As long as depletion of the condensate is not large, i.e. N 0 (0)−N 0 (t) < 0.1N 0 (0), this number is given by N kSR0 (t) = exp(G kSR0 N 0 t) − 1 [15]. At later times, when N kSR0 ∼ 10 >> 1 typically obtained for t > 10µs for V 0 < 5E r ,f † kSR0 can be neglected in Eq.…”

“…The SRB gain is related to G kSR0 the gain for the SR0 process by the relation [15], with L and W the length and width of the Bose gas after its loading inside the optical lattice. In the superfluid phase, the Bose gas conserves its aspect ratio [10] as long as the radial confinement by the lattice laser remains negligible so that this relationship between superradiance gains is assumed to be independent on the lattice depth.…”

“…Interpretation of the data is carried out using the quantum model of references [1,11,15] adjusted to take into account the seeding. The matter wave field is given by Ψ( r, t) = q φ q ( r)e i q rĉ q (t), limited to the first scattering ordersĉ 0 ,ĉ kSR0 ,ĉ 2 kL andĉ −2 kL , the annihilation mode operators of atoms with corresponding momenta and profiles φ q ( r).…”