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

7. Quantum simulations of evaporatively cooled Bose condensateswhere µ is the chemical potential, which can be written in terms of the coherent amplitudeα = |α|e iθ as µ = E 0 +2κ|α| 2 . Substituting the expansion for â into Eq. (7.26) andretaining terms quadratic in δâ gives()ˆF ≃ κ −|α| 4 + |α| 2 (e 2iθ δâ † δâ † +2δâ † δâ + e −2iθ δâδâ(= κ −|α| 4 + |α| 2 ( ˆX)2 − 1) , (7.27)where the quadrature operators have been defined asˆX = δâe −iθ + δâ † e iθ (7.28a)(Ŷ = −iδ âe −iθ − iδâ † e iθ) .(7.28b)Thus Eq. (7.27) shows that minimising the free energy minimises the fluctuations in the ˆXquadrature, and hence maximises the fluctuations in the Ŷ quadrature. Now the numbervariance is∆n 2 = 〈 (â † â) 2〉 − 〈 â † â 〉 2( 〈X= |α| 2 2 〉 − 〈 X 〉 (2 〈X〉3 ) )+ O , (7.29)|α|which means that with minimised fluctuations in the ˆX quadrature, the ground state isamplitude squeezed.Appendix B overviews the properties of squeezed states and suggests how suitablephase-space representations can be generated from them. As these preliminary investigationsshow, the price to pay for going beyond coherent states is a higher-dimensionalphase space and an increase in complexity. Generalisation of the phase-space representationsinvolves extra parameters (the squeezing parameter ξ in this case), which must beeither chosen apriorior evolved through simulation to obtain a more compact phase-spacerepresentation. There are a myriad of possible choices for the type of representation, eachwith a large parameter space. No comprehensive survey of representations has been undertaken,so the future in this direction is open. The work presented in this thesis is onlya tentative beginning; but perhaps a seed has been sown that will one day reap a harvestof powerful multimode quantum dynamical simulation techniques.168