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

5. Weak force detection using a double Bose-Einstein condensate5.2 Two-mode systemWe consider the case where a condensate has formed in a quartic double-well potential,and use the two-mode Hamiltonian introduced as Eq. (2.49):Ĥ 2 = ΩĴz +2κĴ 2 x . (5.1)The commutation relations for these operators are[Ĵi Ĵj], = iɛ ijk Ĵ k (5.2)where ɛ ijk allows cyclic permutations of i, j, k ∈{x, y, z}.The operator Ĵx gives the condensate particle-number difference between the two wells,Ĵ y corresponds to the momentum induced by tunnelling, and Ĵz is the difference in occupationbetween the upper and lower energy eigenstates of the potential. The splittingbetween these levels is given by Ω, which is the tunnelling frequency, and κ correspondsto the strength of the interparticle hard-sphere interactions.As discussed in Ch. 2, the two-mode approximation is valid in the regime where theoverlap between the single-particle ground-state modes of each well is small and wherethe many-body effects do not affect significantly the properties of these modes. Theseconditions lead to[33, 125]Ω≪ 1, N ≪ r 0ω 0 |a| , (5.3)where a is the scattering length which determines the strength of the two-atom collisionsand where r 0 = √ /2mω 0 characterizes the size of the wells.5.3 Measurement schemeAs Fig. 5.1 shows, the detection of the weak force proceeds in several stages. The firststep is to prepare the initial state of the system in a quantum state which can optimise theprecision of the measurement. With this in mind, we consider the weak tunnelling limitΩ ≪ κN. In this case, the ground state of the Hamiltonian (Eq. (2.49)) for attractiveinteractions (κ