First-principles quantum simulations of dissociation of molecular condensates: Atom correlations in momentum space

We investigate the quantum many-body dynamics of dissociation of a Bose-Einstein condensate of molecular dimers into pairs of constituent bosonic atoms and analyze the resulting atom-atom correlations. The quantum fields of both the molecules and atoms are simulated from first principles in three dimensions using the positive-$P$ representation method. This allows us to provide an exact treatment of the molecular field depletion and $s$-wave scattering interactions between the particles, as well as to extend the analysis to nonuniform systems. In the simplest uniform case, we find that the major source of atom-atom decorrelation is atom-atom recombination which produces molecules outside the initially occupied condensate mode. The unwanted molecules are formed from dissociated atom pairs with nonopposite momenta. The net effect of this process—which becomes increasingly significant for dissociation durations corresponding to more than about 40% conversion—is to reduce the atom-atom correlations. In addition, for nonuniform systems we find that mode mixing due to inhomogeneity can result in further degradation of the correlation signal. We characterize the correlation strength via the degree of squeezing of particle number-difference fluctuations in a certain momentum-space volume and show that the correlation strength can be increased if the signals are binned into larger counting volumes.

Figure 1

Fractional population in the molecular and atomic fields, $N_0\left(t\right)∕N_0\left(0\right)$ and $N_1\left(t\right)∕2N_0\left(0\right)$, as a function of time $t$. The time is in units of $t_0=1∕g=1∕\left(\chi \sqrt{{\rho }_0}\right)$, which in this example is $t_0=0.2\mathrm{ms}$. The dash-dotted line is the fraction of molecules in the noncondensate modes, $\mathbf{k}\ne \mathbf{0}$. The dashed line is the analytic result for the total atom number in the undepleted molecular field approximation.

Figure 2

(Color online) (a) Slice through the origin of the 3D atomic density in momentum space $n(\mathbf{k},t)$ [in units of $\mu {\mathrm{m}}^3$] at $t=t_0$, where $t_0=1∕\left(\chi \sqrt{{\rho }_0}\right)=0.2\mathrm{ms}$. The momentum components $k_x$ and $k_y$ are in units of $k_0=\sqrt{2m_1\mid {\Delta }_{\mathrm{eff}}\mid ∕\hslash }$, which in this example is $k_0=7.3\times {10}^6{\mathrm{m}}^{-1}$. (b) Angle-averaged density distribution ${⟨n(\mathbf{k},t)⟩}_{\phi ,\theta }$ and mode occupations ${⟨n_{\mathbf{k}}\left(t\right)⟩}_{\phi ,\theta }={⟨n(\mathbf{k},t)⟩}_{\phi ,\theta }{\left(\Delta k\right)}^3$ as a function of the absolute momentum $\mid \mathbf{k}\mid$ at $t=t_0$ (solid line). The lattice spacing is $\Delta k=4.55\times {10}^5{\mathrm{m}}^{-1}$. The dashed line is the result for the analytically soluble case with the undepleted molecular field, Eq. (9).

Figure 3

(Color online) (a) Slice through the origin of the 3D correlation function $g^{\left(2\right)}(\mathbf{k},-\mathbf{k},t)$ at $t=t_0$. The corresponding density profile is shown in Fig. 2a. The sampling errors due to stochastic averaging are small only within the region of $0.65<\mid \mathbf{k}\mid ∕k_0<1.25$, where the correlation function ranges between $\sim 2.8$ and 6. (b) Angle-averaged correlation functions ${⟨g^{\left(2\right)}(\mathbf{k},{\mathbf{k}}^{\prime },t)⟩}_{\phi ,\theta }$ (for ${\mathbf{k}}^{\prime }=-\mathbf{k}$, $\mathbf{k}$, ${\mathbf{k}}_{90}$) at $t=t_0$ as a function of the absolute momentum $\mid \mathbf{k}\mid$ within the spherical shell $0.65<\mid \mathbf{k}\mid ∕k_0<1.25$. The solid line with the dots (right scale) is the respective angle-averaged variance ${⟨V_v(\mathbf{k},-\mathbf{k},t)⟩}_{\phi ,\theta }$.

Figure 4

Dynamics of the angle-averaged pair correlation functions ${⟨g^{\left(2\right)}(\mathbf{k},{\mathbf{k}}^{\prime },t)⟩}_{\phi ,\theta }$ at $\mid \mathbf{k}\mid ∕k_0=1$, for ${\mathbf{k}}^{\prime }=-\mathbf{k}$, $\mathbf{k}$, and ${\mathbf{k}}_{90}$. The time is in the same units as in Figs. 1, 2. The solid line with the dots (right scale) is the variance of the number-difference fluctuations for equal but opposite momenta, ${⟨V_v(\mathbf{k},-\mathbf{k},t)⟩}_{\phi ,\theta }$, at $\mid \mathbf{k}\mid ∕k_0=1$.

Figure 5

Comparison between the results of simulations of weakly inhomogeneous $(R_{\mathit{TF}}=11.6\mu \mathrm{m})$ and strongly inhomogeneous $(R_{\mathit{TF}}=4.82\mu \mathrm{m})$ systems, columns 1 and 2, having the same peak density ${\rho }_0\left(0\right)=5\times {10}^{19}{\mathrm{m}}^{-3}$. Column 3 refers to the same physical system as in column 2, except that the length of the simulated spatial domain $L$ is halved. Note that the lines are provided as guides-to-the-eye between calculated data points. (a) Slice through the origin of the initial molecular BEC density profile, ${\rho }_0(x,0,0)$. (b) Angle-averaged momentum density ${⟨n(\mathbf{k},t)⟩}_{\phi ,\theta }$ and distribution of mode occupations ${⟨n_{\mathbf{k}}\left(t\right)⟩}_{\phi ,\theta }$ at $t=t_0$ as a function of the absolute momentum $\mid \mathbf{k}\mid ∕k_0$, where $k_0=7.3\times {10}^6{\mathrm{m}}^{-1}$ as in Fig. 2. (c) Fraction of the total number of dissociated atoms relative to $2N_0\left(0\right)$ vs time. (d) Angle-averaged mode population ${⟨n(\mathbf{k},t)⟩}_{\phi ,\theta }$ vs time. Note the different scales. (e) Second-order pair correlation function ${⟨g^{\left(2\right)}(\mathbf{k},-\mathbf{k},t)⟩}_{\phi ,\theta }$ vs time. The dotted line shows the idealized result ${⟨g_{\max }^{\left(2\right)}(\mathbf{k},-\mathbf{k},t)⟩}_{\phi ,\theta }=2+1∕{⟨n_{\mathbf{k}}\left(t\right)⟩}_{\phi ,\theta }$, for comparison. (f) Number-difference variance ${⟨V_v(\mathbf{k},-\mathbf{k},t)⟩}_{\phi ,\theta }$ vs time. (g) Correlation coefficients $C_{\mathbf{k}}\left(t\right)$ at $\mid \mathbf{k}\mid =k_0$ vs time. The quantities shown in (d)–(g) are at $\mid \mathbf{k}\mid =k_0$ and the time is in units of $t_0=1∕\left(\chi \sqrt{{\rho }_0\left(0\right)}\right)=0.2\mathrm{ms}$ in all cases. The number of stochastic trajectories used for ensemble averaging is 2000.

Figure 6

(a) Binned distribution of the angle-averaged occupation numbers ${⟨n_{\mathbf{K}}\left(t\right)⟩}_{\phi ,\theta }$ at $t=t_0$; and (b) the respective number-difference variance ${⟨V_{\overline{v}}(\mathbf{K},-\mathbf{K},t)⟩}_{\phi ,\theta }$ at $\mid \mathbf{K}\mid =k_0$ as a function of time, for the same physical system as in Fig. 5, column 1. The graphs can be compared to those of Figs. 5b, 5f, column 1. The $\left\{\mathbf{K}\right\}$-sublattice of the central momenta of the bins is setup to correspond to every second lattice point of the original computational grid in each dimension, so that the bin volume is $\overline{v}={\left(2\Delta k\right)}^3$. The number of stochastic trajectories simulated is 4500.