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

2. Properties of an atomic Bose condensate in a double-well potentialThe second-order correction to the ground state is〈 ∣ (2)〉 ∑ ∑(0)∣E ∣mˆV ∣E (0)〉〈 (0)∣E 0 =k E ∣kˆV ∣E (0)〉0m≠0 k≠0whichisanevensuperposition.(E (0)0− E m(0))(E (0)0− E (0)k= −Ω2√ j(j +1) √ j(j +1)− 264κ) ∣ ∣ E(0)m〉(∣ ∣j, 2〉x + ∣ ∣ j, −2〉x), (2.62)2.3.2 Transition temperatureOne characteristic quantity of an ensemble of bosons is the transition temperature T c .This is the temperature at which a macroscopic occupation of the ground state develops,and can be calculated by studying how the condensate fraction N 0 /N varies with temperature.In the thermodynamic limit, the temperature at which the condensed fraction firstappears is sharply defined, but in a system of a finite number of particles, the transitionis smoother[100].The two-mode model provides a good illustration of this.Let us consider a grandcanonical ensemble of weakly interacting atoms, with the populations of the energy levelsE i obeying the Bose-Einstein distribution:N(E i )= ze−β(E i−E 0 )1 − ze −β(E i−E 0 ) , (2.63)where the fugacity z is defined in terms of the chemical potential µ as z =exp(βµ). Thetemperature dependence enters through β =1/(k B T ). The condensate fraction can becalculated from the fugacity: N 0 = z/(1 − z), which in turn can be calculated from thetotal number constraint:N =N∑N(E i )=i=0∞∑z lN ∑l=1 i=0e −lβ(E i−E 0 ) . (2.64)From this equation, the condensate fraction can be calculated numerically using the energyeigenvalues found in the previous section.We consider first the two limiting cases in which the spectrum can be written downin analytic form. When the atom interactions are negligible (i.e. Θ ≪ 1), the energyeigenvalues will be the E i − E 0 = iΩ. This is similar to the one-dimensional harmonicoscillator considered by Ketterle and van Druten[100], except that the spectrum now has46