Quantum versus classical statistical dynamics of an ultracold Bose gas

We investigate the conditions under which quantum fluctuations are relevant for the quantitative interpretation of experiments with ultracold Bose gases. This requires to go beyond the description in terms of the Gross-Pitaevskii and Hartree-Fock-Bogoliubov mean-field theories, which can be obtained as classical (statistical) field-theory approximations of the quantum many-body problem. We employ functional-integral techniques based on the two-particle irreducible (2PI) effective action. The role of quantum fluctuations is studied within the nonperturbative 2PI $1∕\mathcal{N}$ expansion to next-to-leading order. At this accuracy level memory integrals enter the dynamic equations, which differ for quantum and classical statistical descriptions. This can be used to obtain a classicality condition for the many-body dynamics. We exemplify this condition by studying the nonequilibrium evolution of a one-dimensional Bose gas of sodium atoms, and discuss some distinctive properties of quantum versus classical statistical dynamics.

Figure 1

(Color online) (a) Diagrammatic representation of the correlators in the $\phi \text{−}\tilde{\phi }$ basis. A full line indicates the 1 or $\phi$ component, a broken line the 2 or $\tilde{\phi }$ component. $F_{ab}(x,y)$ is the statistical correlation function, $G_{ab}^{\mathrm{R}}(x,y)={\rho }_{ab}(x,y)\theta (x_0-y_0)$ and $G_{ab}^{\mathrm{A}}(x,y)=-{\rho }_{ab}(x,y)\theta (y_0-x_0)$ the retarded and advanced Green’s functions, respectively. Their representation in terms of the real-valued spectral correlation function ${\rho }_{ab}(x,y)$ exposes the $\theta$ functions which imply the respective time ordering in $x_0$, $y_0$. (b) Diagrammatic expansion of the quantum vertex term entering the action (3). The classical action (53) is lacking the second contribution (red square).

Figure 2

(Color online) Diagrammatic representation of the bubble chains contributing to the functions $I_F^{\prime }$ and $I_{\rho }$ at next-to-leading order in the 2PI $1∕\mathcal{N}$ expansion. The meaning of lines and vertices is explained in Fig. 1. The chains of type (Q1) and (Q2) only appear for a quantum system. Type (C) is present both in quantum and classical systems. These diagrams exhibit that in each term contributing to the functions $I_F^{\prime }$ and $I_{\rho }$, there is at most one loop involving two correlators $F^{\prime }$ or $\rho$. In the classical limit of quantum theory, the loops involving two spectral functions $\rho$ are suppressed compared to those with two statistical functions $F^{\prime }$.

Figure 3

(Color online) Momentum-mode distribution $n(t;p)$ for the initial state (red filled circles, interpolated by red solid line) and six subsequent times $t$ until no change can be observed for $t>0.6\mathrm{s}$. The interpolation of the final distribution is shown as a black solid curve. Note the logarithmic scale. The occupation number is normalized by the total number of atoms in the box, $n_{1}L=853$. The gas is in a far-from-equilibrium state initially, characterized by a Gaussian distribution $n(0;p)$, Eq. (81), with width $\sigma =1.3\times {10}^5{\mathrm{m}}^{-1}$. It is weakly interacting, $\gamma =1.5\times {10}^{-3}$. Since we consider a homogeneous gas and a symmetric initial state, the occupation numbers are invariant under $p\to -p$.

Figure 4

The normalized momentum-mode occupation numbers $n(t;p)∕n_{1}L$, corresponding to those shown in Fig. 3, as functions of time. Shown are the populations of the modes with $p=p_i=2N_s∕L\sin (i\pi ∕N_s)$, $i=0,1,\dots ,N_s∕2$, and one has $n(t;-p)=n(t;p)$. A fast short-time dephasing period is followed by a long quasistationary drift to the final equilibrium distribution. Notice the double-logarithmic scale.

Figure 5

Momentum and time-dependent temperature variable $\Theta (t;p)$ obtained by fitting the distribution $n(t;p)={\mathbf{(}\exp \{[\omega \left(p\right)-\mu ]∕k_B\Theta (t;p)\}-1\mathbf{)}}^{-1}$ to the distribution obtained from the results shown in Fig. 4, for different, equally spaced times between $t=0$ and $t=0.6\mathrm{s}$. One observes that, during the quasistationary period, $0.01\mathrm{s}<t<0.1\mathrm{s}$, no temperature can be associated to the distribution. Only at very large times, $\Theta$ becomes approximately $p$ independent.

Figure 6

(Color online) Ratio of the envelopes of the unequal-time correlation functions, $\xi (t;p)={\{[F_{11}{(t,0;p)}^2+F_{12}{(t,0;p)}^2]∕[{\rho }_{11}{(t,0;p)}^2+{\rho }_{12}{(t,0;p)}^2]\}}^{1∕2}∕n(t;p)$, for four different momentum modes, as a function of time. Due to the normalization with respect to $n(t;p)$ all $\xi (t;p)$ are of the same order of magnitude. $\xi$ is a measure of the interdependence of the statistical and spectral functions, and its settling to a constant value during the quasistationary drift period indicates that these functions become, as in thermal equilibrium, connected through a fluctuation-dissipation relation.

Figure 7

(Color online) Evolution of the kinetic and interaction contributions to the total energy of the gas. During the quasistationary drift these contributions assume the same order of magnitude, calling in mind the virial theorem.

Figure 8

(Color online) The normalized momentum-mode occupation numbers $n(t;p)∕n_{1}L$ for the classical gas (black solid lines) compared to their quantum counterparts from Fig. 4 (red dashed lines), as functions of time. Shown are the populations of the modes with $p=p_i=2N_s∕L\sin (i\pi ∕N_s)$, $i=0,1,\dots ,N_s∕2$, and one has $n(t;-p)=n(t;p)$. In contrast to the quantum statistical evolution there is no quick dephasing in the classical case, such that the initially empty modes become only subsequently filled, as is discussed in the main text. At large times, the classical and quantum gases necessarily evolve to different distributions.

Figure 9

(Color online) The normalized momentum-mode occupation numbers $n(t;p)∕n_{1}L$ for the classical gas (black solid lines) compared to their quantum counterparts from Fig. 4 (red dotted lines), as functions of time. All parameters are chosen as in Fig. 8, except for $\sigma =6.5\times {10}^4{\mathrm{m}}^{-1}$. The dashed (blue) lines show $n(t,p)-1∕2$, i.e., after subtracting the additional flat initial distribution which simulates the quantum “zero point fluctuations.”

Figure 10

(Color online) Ratio of the envelopes of the unequal-time correlation functions, $\xi (t;p)={\{[F_{11}{(t,0;p)}^2+F_{12}{(t,0;p)}^2]∕[{\rho }_{11}{(t,0;p)}^2+{\rho }_{12}{(t,0;p)}^2]\}}^{1∕2}∕n(t;p)$, for four different momentum modes, as a function of time. This figure is analogous to Fig. 6, but shows the difference of the quantum (black lines) and classical (blue lines) evolutions for a gas in the strongly correlated regime $\gamma =15$, for $n_1={10}^5{\mathrm{m}}^{-1}$. The results show that in this regime, where quantum and classical statistical evolution differ considerably, the quantum evolution conserves information about the initial conditions much longer, here roughly by an order of magnitude in time.