We study the transmission spectra of a Bose Einstein condensate confined in an optical lattice interacting with two modes of a cavity via nonlinear two-photon transition. In particular we show that a nonlinear two-photon interaction between the superfluid (SF) phase and the Mott insulating (MI) phase of a Bose-Einstein condensate (BEC) and the cavity field show qualitatively different transmission spectra compared to the one-photon interaction. We found that when the BEC is in the Mott state, the usual normal mode splitting present in the one-photon transition is missing in the two-photon interaction. When the BEC is in the superfluid state, the transmission spectra shows the usual multiple lorentzian structure. However the separation between the lorentzians for the two-photon case is much larger than that for the one-photon case. This study could form the basis for non-destructive high resolution Rydberg spectroscopy of ultracold atoms or two-photon spectroscopy of a gas of ultracold atomic hydrogen.
Cold atoms in optical lattices exhibit phenomena typical of solid state physics like the formation of energy bands, Josephson effects Bloch oscillations and strongly correlated phases. Many of these phenomena have been already the object of experimental investigations. For a recent review see [1]. Standard methods to observe quantum properties of ultracold atoms are based on destructive matter-wave interference between atoms released from traps [2]. Recently, a new approach was proposed which is based on all optical measurements that conserve the number of atoms. It was shown that atomic quantum statistics can be mapped on transmission spectra of high-Q cavities, where atoms create a quantum refractive index. This was shown to be useful for studying phase transitions between Mott insulator and superfluid states since various phases show qualitatively distinct spectra [3].
Experimental implementation of a combination of cold atoms and cavity QED (quantum electrodynamics) has made significant progress [4,5,6]. Theoretically there have been some interesting work on the correlated atomfield dynamics in a cavity. It has been shown that the strong coupling of the condensed atoms to the cavity mode changes the resonance frequency of the cavity [7]. Finite cavity response times lead to damping of the coupled atom-field excitations [8]. The driving field in the cavity can significantly enhance the localization and the cooling properties of the system [9,10]. It has been shown that in a cavity the atomic back action on the field introduces atom-field entanglement which modifies the associated quantum phase transition [3,11]. The light field and the atoms become strongly entangled if the latter are in a superfluid state, in which case the photon statistics typically exhibits complicated multimodal structures [12]. A coherent control over the superfluid properties of the BEC can also be achieved with the cavity and pump [13] In this work, we show that a nonlinear two-photon interaction between the superfluid (SF) phase and the Mott insulating (MI) phase of a Bose-Einstein condensate (BEC) (confined in an optical lattice) and the cavity field show qualitatively different transmission spectra compared to the one-photon interaction. Two-photon spectroscopy played a very important role in the studies of BEC of atomic hydrogen [14]. Two-photon excitation of 87 Rb atoms to a Rydberg state was also achieved recently [15].
The system we consider here is an ensemble of N two-level atoms with upper and lower states denoted by |1 > and |0 > respectively in an optical lattice with M sites formed by far off resonance standing wave laser beams inside a cavity. A region of k M sites is coupled to two light modes as shown in Fig. 1. Fig. 1 shows two cavities containing the two modes a 1 and a 2 crossed by a one-dimensional optical lattice confining the BEC. In the two-photon process, an intermediate level |i > is involved, which is assumed to be coupled to |1 > and |0 > by dipole allowed transitions. The manybody Hamiltonian in the second quantized form is given by FIG. 1: Schematic diagram of the setup. The BEC atoms are periodically confined in an optical lattice and are made to interact with two laser modes a1 and a2 which are confined in two intersecting cavities. Mode a2 is transmitted and measured by a detector.
In the field part of the Hamiltonian H f , a l are the annihilation operators of light modes with the frequencies ω l , wave vectors k l , and mode functions u l (r), which can be pumped by coherent fields with amplitudes η l . In the atom part, H a , Ψ(r) is the atomic matter-field operator, a s is the s-wave scattering length characterizing the direct interatomic interaction, and H a1 is the atomic part of the single-particle Hamiltonian H 1 . The detuning between the atomic transition frequency and any one of the two modes is nonzero. Under these circumstances, the intermediate state can be adiabatically eliminated and the effective Hamiltonian of the two-level atom can be written in the rotating-wave and dipole approximation as
Here, p and r are the momentum and position operators of an atom of mass m a and resonance frequency ω a , σ + , σ -, and σ z are the raising, lowering, and population difference operators, g 0 is the atom-light coupling constant assumed to be same for both the modes. We will consider nonresonant interaction where the light-atom detunings ∆ = ω 1 + ω 2 -ω a are much larger than the spontaneous emission rate and Rabi frequencies g 0 a 1 a 2 . Thus, in the Heisenberg equations obtained from the single-atom Hamiltonian H 1 (2), σ z can be set to -1 (approximation of linear dipoles). Moreover, the polarization σ -can be adiabatically eliminated and expressed via the fields a 1 and a 2 . An effective single-particle Hamiltonian that gives the corresponding Heisenberg equations for a 1 and a 2 can be written as
Here, we have also added a classical trapping potential of the lattice, V cl (r), co
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