Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

3. Homodyne measurements on a Bose-Einstein condensate3.2 Homodyne detection schemeFigure 3.1 illustrates the system under investigation here. One of the wells of the doublewellpotential sits in an optical cavity.A coherent field strongly drives the cavity atthe cavity frequency. We will assume that on the timescale of tunnelling oscillations, thecavity is heavily damped. The cavity field thus relaxes to the steady state on a much fastertimescale than the condensate dynamics. This enables us to make an adiabatic eliminationof the cavity dynamics. We can assume that the cavity field is far off resonance from anydipole transitions in the atomic species, so that the effect of the atoms is then entirelydispersive. The presence of the condensate shifts the phase of the cavity field by an amountproportional to the number of atoms in the cavity at any particular time. If the numberof atoms in the cavity oscillates, so will the phase shift. Thus tunnelling of the condensatewill be manifest as a modulated phase shift in the optical field exiting the cavity.To detect this phase shift we consider a homodyne detection scheme.The opticalcavity resides in one arm of an interferometer that combines the exiting light with areference beam of the same frequency. The result falls on a photodetector, which recordsthe photocurrent. If there is a difference in atom number between the two condensates,then coherent tunnelling can occur and the homodyne current will be modulated at thetunnelling frequency.Assuming that the incoming light is detuned from any atomic resonance, the interactionHamiltonian is∫Ĥ I = −d 3 r ˆψ † (r)µg(r)â † â ˆψ(r), (3.2)where â, â † are the cavity field operators, g(r) is the intensity mode function and µ =g 2 d /4∆ is the coupling strength, with dipole coupling constant g d and optical detuning ∆.Averaging over the optical mode function gives the interaction energy in terms of thecondensate field operators c, c † :where χ is the interaction strength.Ĥ I = −χâ † âĉ † 1ĉ1= − N 2 χ↠â − χâ † âĴx, (3.3)If the optical mode has a beam waist w, and aGaussian profile g(r) =cos 2 (kz)exp(−(x 2 + y 2 )/w 2 ) then the interaction strength isχ =µ2 √ 2(r 0 /w) 2 +1 . (3.4)62