Spectrum of Light Scattered from a Degenerate Bose Gas

Juha Javanainen
1995 Physical Review Letters  
The spectrum of light scattered from a Bose gas is studied in the limit of large detuning. At low density the spectrum is a direct replica of the velocity distribution of the atoms. In a degenerate gas the spectrum may acquire extra structure, reflecting the Bose-Einstein statistics of the atoms. In the presence of a Bose condensate a characteristic two-peak spectrum develops. PACS numbers: 42.50.Vk, 03.75.Fi, 05.30.Jp An intense effort is currently under way in the trapping and cooling of
more » ... al atoms [1] . With the steady progress toward higher densities and lower temperatures, quantum statistics of the atoms should eventually have measurable consequences. In fact, Bose condensation of a weakly interacting gas is a long-standing goal of cooling. As lasers figure prominently in many experiments, the question of the optical response of the Bose condensate has naturally come up [2] [3] [4] . Another related line of research addresses "nonlinear atom optics" [5] , consequences of the dipoledipole interactions between the atoms in a degenerate gas [5, 6] . A Bose condensate has remarkable optical properties, such as a broad resonance [3, 4] . Unfortunately, Bose condensation is not the only conceivable reason for a broad resonance in a dense gas; the resonance is also broadened in an optically thick sample of noncondensate atoms, or indeed Maxwell-Boltzmann atoms. In the present Letter we search for more unambiguous optical manifestations of atomic degeneracy. We analyze the spectrum of light scattered from a Bose gas, possibly containing a condensate. One expects a replica of the velocity distribution, as in Doppler velocimetry of cold atoms [7] , but it turns out that the spectrum may display additional qualitative features arising directly from the Bose-Einstein statistics. Along the way, we put the combined theories of material response and light propagation in a form conducive to further developments. We begin with slight generalizations and a rewrite of our earlier model [3] . The atoms move in an arbitrary potential, which is included in the center-of-mass (CM) Hamiltonian H CM . The atoms have two internal energy levels, g for "ground" and e for "excited." In addition, we take into account angular momentum degeneracy. The complete internal-state kets therefore read jgm͘ and jem 0 ͘. The frequency of the optical transition is v 0 , and the dipole moment operator of an atom is denoted by d. We introduce the conventional field operators for each internal state, c gm ͑r͒ and c em 0 ͑r͒. Much of our development does not depend on the statistics of the atoms, but for concreteness we assume bosons. We then have the familiar commutators, e.g., ͓c gm ͑r͒, c y gm 0 ͑r 0 ͔͒ d mm 0 d͑r 2 r 0 ͒. As it comes to light, we have found it convenient to retain the plane wave representation for photons, denoting the mode index by q. We occasionally write the electric field operator as a sum of positive-and negative-frequency components, E͑r͒ E 1 ͑r͒ 1 E 2 ͑r͒. The Hamiltonian for the atom-field system finally emerges as the integral over all space of the Hamiltonian density H ͑r͒, (1) We have suppressed the dependence on the position r in our notation; H CM acts on the position argument of the field to the right of it. H gg and H F are the Hamiltonian densities for the interactions between the ground state atoms and for the free electromagnetic field. The rotating-wave approximation (RWA) [8] has not been made in our Hamiltonian. We will apply the theory in the limit of large laser-atom detuning, hence low density of excited atoms. A term H ee analogous to H gg has therefore been ignored. In regard to collisions between ground state and excited state atoms, we only retain the resonant dipole-dipole interactions. They are accounted for by the d ? E terms in (1); see Ref. [9] . We assume from now on that there is a dominant frequency in the light field, V, which is nearly resonant with the atomic transition frequency v 0 . The corresponding wave number is k V͞c. We define field operators that vary "slowly" in the Heisenberg picture, e.g., c em 0 e iVt c em 0 andẼ 1 e iVt E 1 . While considering matter fields, the RWA apparently may be made without adverse consequences. We thus obtain the equations of motion ᠨ c em 0 i µ d 2 H CM h ∂c em 0 1 ih X m c gm ͗em 0 jd ?Ẽ 1 jgm͘ , (2a) ᠨ c gm 2i H CM h c gm 1 ih X m 0 ͗gmjd ?Ẽ 2 jem 0 ͘c em 0 . In Eq. (2a), d V 2 v 0 is the detuning of the light frequency from the atomic resonance. On the other hand, the straightforward RWA would lead to an electromagnetic field that violates causality and 0031-9007͞95͞75(10)͞1927(4)$06.00
doi:10.1103/physrevlett.75.1927 pmid:10059164 fatcat:3ideeafnazc6rfvapkq3xwkfa4