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

5. Weak force detection using a double Bose-Einstein condensatewhere ∆ FR describes the width of resonance at B = B 0 .include the term:The Hamiltonian must thenĤ FR = κ 2 c † 2 22 c2( ( ))N N= κ 22 2 − 1 +(N − 1) Ĵx + Ĵ x2 , (5.55)where κ 2 =2π∆/(mV eff (B 0 − B)). The terms proportional to N and N 2 only produceconstant energy shifts and may be ignored, to giveĤ FR = κ 2((N − 1) Ĵx + Ĵ 2 x). (5.56)If, instead of allowing the system to interact with a weak force, we induce this Feshbachresonance for time τ on the superposition state ∣ ∣ E〉(Eq. (5.4)), the output phase is shiftedin the manner described in Sec. 5.3, but by the amountφ FR = κ 2N(N − 1)τ. (5.57)25.9 Les adieuxThroughout the first section of this thesis, we have been using the two-mode systemas a vehicle to study the quantum and semiclassical properties of small Bose-Einsteincondensates. We have viewed it as an isolated quantum system and also as a quantumsystem coupled to the environment, to see how certain exquisite quantum phenomenasurvive under measurement. We have even used it to suggest a practical use for Bose-Einstein condensates. But now it is time to farewell our trusted friend, and with it, manyof the analytic methods that the two-mode approximation enabled. In the second sectionof the thesis, we are flung into new territory. We shall harness the engine of multimodequantum simulations to take us into higher dimensions and greatly expanded Hilbertspaces.121