Pairing mean-field theory for the dynamics of dissociation of molecular Bose-Einstein condensates

We develop a pairing mean-field theory to describe the quantum dynamics of the dissociation of molecular Bose-Einstein condensates into their constituent bosonic or fermionic atoms. We apply the theory to one-, two-, and three-dimensional geometries and analyze the role of dimensionality on the atom production rate as a function of the dissociation energy. As well as determining the populations and coherences of the atoms, we calculate the correlations that exist between atoms of opposite momenta, including the column density correlations in three-dimensional systems. We compare the results with those of the undepleted molecular field approximation and argue that the latter is most reliable in fermionic systems and in lower dimensions. In the bosonic case we compare the pairing mean-field results with exact calculations using the positive-$P$ stochastic method and estimate the range of validity of the pairing mean-field theory. Comparisons with similar first-principle simulations in the fermionic case are currently not available, however, we argue that the range of validity of the present approach should be broader for fermions than for bosons in the regime where Pauli blocking prevents complete depletion of the molecular condensate.

Figure 1

Dissociation of a molecular condensate into fermionic atoms. We plot the total number of atoms in each spin component $N_{1,2}\left(\tau \right)$ $\left[N_1\left(\tau \right)=N_2\left(\tau \right)\right]$ relative to the total initial number of molecules $N_0\left(0\right)$ as a function of the dimensionless time $\tau =t/t_0$, where $t_0=1/\kappa \sqrt{N_0\left(0\right)}$ is the time scale. The graphs (a) and (b) are for $\left|\delta \right|=16$ and $\left|\delta \right|=3174$, respectively. The different curves are for 1D, 2D, and 3D cases as shown. In all cases, the initial dimensionless molecular density is $N_0\left(0\right)/l^D=3.1$ $\left(D=1,2,3\right)$, where $l=L/d_0$ is the dimensionless length. The dotted lines next to each solid line are the results from Fermi’s golden rule, Eqs. (11, 12). See text for further details.

Figure 2

Dissociation of a molecular condensate into distinguishable bosons. Apart from bosonic statistics, all parameters are the same as for Fig. 1.

Figure 3

Same as in Figs. 1a, 1b in the 3D case, except that the results of the pairing mean-field theory (solid lines) are compared with those obtained within the undepleted molecular field approximation (dashed line).

Figure 4

Comparison of the results obtained without (solid line) and with (dashed-dotted line) the flat density-of-states approximation in 3D. In both cases we set $\left|\delta \right|=1$ and $N_0/l^3=3.1$, corresponding to ${\rm Y}=3\times {10}^4$. For the dashed-dotted line, only the value of ${\rm Y}$ matters, and the same curve can be obtained with different individual values of $\left|\delta \right|$ and $N_0/l^3$ as long as they result in the same ${\rm Y}$, Eq. (28).

Figure 5

Fractional atomic population $N_1\left(\tau \right)/2N_0\left(0\right)$ in 3D obtained using the present pairing mean-field theory and first-principles (exact) simulations of the same system using the positive-$P$ representation method $\left(+P\right)$ [24]. The dashed line corresponds to the undepleted molecular field approximation. The parameter values are $\left|\delta \right|=4$ and $N_0\left(0\right)/l^3=0.988$.

Figure 6

(Color online) Atomic momentum distribution in each spin state for dissociation into fermionic atoms for a dimensionless absolute detuning $\left|\delta \right|=\left|\Delta \right|t_0=16$ and $N_0\left(0\right)/l^D=3.1$. Only the range of momenta whose population is nonvanishing is shown. (a) $n_{\mathbf{k}}\left(\tau \right)$ as a function of scaled time $\tau =t/t_0$ and scaled absolute momentum $k/k_0$ in 3D. (b) Time slices of the momentum distribution at $\tau =1$ and $\tau =3$. (c) Temporal population of the resonant momentum mode with $k=k_0$. The dashed line represents the respective analytic solution given by ${\text{sin}}^2\left(\tau \right)$ in the undepleted molecular field approximation [50].

Figure 7

(Color online) Same as in Fig. 6 except for $\left|\delta \right|=3174$. In (c) we show the temporal population of the resonant mode $k_0$ in 1D, 2D, and 3D. The analytic result of ${\text{sin}}^2\left(\tau \right)$ corresponding to the undepleted molecular field approximation is practically identical to the 1D curve shown.

Figure 8

(Color online) Same as in Fig. 6 $\left(\left|\delta \right|=16\right)$ except for bosonic statistics of the atoms. The time slices of the momentum distribution in (b) are for $\tau =1$ and $\tau =2$. In (c), we show the temporal population of the resonant mode $k_0$ in 1D, 2D, and 3D (solid line), together with the respective analytic result, ${\text{sinh}}^2\left(\tau \right)$ (dashed line), obtained using the undepleted molecular field approximation [50].

Figure 9

(Color online) Same as in Fig. 7 $\left(\left|\delta \right|=3174\right)$ except for bosonic atom statistics.

Figure 10

(a) Momentum-space column density ${\overline{n}}_{\mathbf{p}}\left(\tau \right)/l$ of each spin component at $\tau =0.36$ for bosonic and fermionic cases in 3D for $\left|\delta \right|=3174$ and $N_0\left(0\right)/l^D=3.1$. The horizontal axis is the absolute value of the 2D reduced momentum $p=\left|\mathbf{p}\right|$, where $\mathbf{p}\equiv \left(k_x,k_y\right)$. (b) Correlation function between momentum column density fluctuations for the atoms with opposite spins and conjugate momenta, ${\overline{G}}_{12}\left(\mathbf{p},-\mathbf{p},\tau \right)$, as a function of $p=\left|\mathbf{p}\right|$, for the same parameter values as in (a).