Coherence lifetimes of excitations in an atomic condensate due to the thin spectrum

We study the quantum coherence properties of a finite sized atomic condensate using a toy model and the thin spectrum model formalism. The decoherence time for a condensate in the ground state, nominally taken as a variational symmetry breaking state, is investigated for both zero and finite temperatures. We also consider the lifetimes for Bogoliubov quasiparticle excitations, and contrast them to the observable window determined by the ground-state coherence time. The lifetimes are shown to exhibit a general characteristic dependence on the temperature, determined by the thin spectrum accompanying the spontaneous symmetry breaking ground state.

Figure 1

The comparison of the short time decay character for a coherent state condensate with that for a squeezed state at $\zeta =0.5$ and $\zeta =0.9$. The parameters used are $a_s=10\text{nm}$, $a_{\text{ho}}=1\mu \text{m}$, $n={10}^{21}{\text{m}}^{-3}$, but now for $N=100$. In this case, the dimensionless time in units of $\hslash ∕\tilde{u}$ becomes $\hslash ∕\tilde{u}={\omega }_{\text{tr}}^{-1}$. The fastest decay (solid line) denotes the result for a coherent state, while the dashed (dotted) line refers to that of a squeezed state with $\zeta =0.5$ $\left(\zeta =0.9\right)$. As expected, the choice of a squeezed state with real parameters $\alpha$ and $\zeta$ improves the coherence time.

Figure 2

(Color online) The phase space distributions of the initial states used in Fig. 1. Curves (a), (b), and (c) correspond to $\zeta =0.9$, $\zeta =0.5$, and $\zeta =0$, respectively. Although all of them should be centered at $\alpha =10$, they are shifted for convenience.

Figure 3

(Color online) Time evolution of the $Q$ function for a squeezed-coherent state with $\alpha =10$ and $\zeta =0.5$ for different values of $t{\omega }_{\text{tr}}$. (a), (b), (c), and (d) show the $Q$-function distributions for $t{\omega }_{\text{tr}}=0$, $t{\omega }_{\text{tr}}=0.02$, $t{\omega }_{\text{tr}}=0.10$, and $t{\omega }_{\text{tr}}=0.40$. It is seen that as the order parameter decays, the broken phase symmetry is restored, since the $Q$-function distribution becomes rotationally symmetric.

Figure 4

Decay of the order parameter for the coherent state and squeezed states of $\zeta =0.5$, $\zeta =0.5i$, and $\zeta =-0.5$ as a function of $t{\omega }_{\text{tr}}$. The solid line is the coherent state, the dashed line is the squeezed state with $\zeta =0.5$, and the dotted lines are the squeezed states with $\zeta =0.5i$ and $\zeta =-0.5$. Only the state with real squeezing parameter has a longer lifetime than the coherent state.

Figure 5

(Color online) The phase space distributions of the initial states used in Fig. 4. Curves (a), (b), (c), and (d) correspond to $\zeta =0.5$, $\zeta =0$, $\zeta =-0.5$, and $\zeta =0.5i$, respectively. Although all of them should be centered at $\alpha =10$, they are shifted for convenience.

Figure 6

(Color online) The short time decays for thermal coherent states. The lines correspond, respectively, to $T=1000$, 100, 10, 1, and 0.001 nK from left to right. The humps are entirely due to the ground degeneracy $E_0=E_1$. Even as the temperature approaches zero, the state (12) does not approach the ordinary coherent state $D\left(\alpha \right)|0⟩$. Instead, it is a superposition state $D\left(\alpha \right)\left(|0⟩+|1⟩\right)∕\sqrt{2}$.

Figure 7

The decay of $\left|{\rho }_{\text{od}}^{\left(\text{red}\right)}\right|$ as a function of $t∕t_c$.

Figure 8

The relative decay of the off-diagonal element in Eq. (44) as a function of $t∕t_c$ for unit values of parameters.

Figure 9

Decay of the off-diagonal element at $T=10\text{nK}$ as a function of $t{\omega }_{\text{tr}}$. Dashed line shows the decay in the case of the thermal coherent occupation and the solid line shows the decay in the case of the thermal occupation.

Figure 10

Decay of the off-diagonal element for $T=100\text{nK}$ and thermal coherent occupation of the zero mode as a function of $t{\omega }_{\text{tr}}$. Dashed line shows the decay in the case of the thermal coherent occupation and the solid line shows the decay in the case of the thermal occupation.