Coherent seeding of the dynamics of a spinor Bose-Einstein condensate: From quantum to classical behavior

We present experiments revealing the competing effect of quantum fluctuations and of a coherent seed in the dynamics of a spin-1 Bose-Einstein condensate and discuss the relevance of a mean-field description of our system. We first explore a near-equilibrium situation, where the mean-field equations can be linearized around a fixed point corresponding to all atoms in the same Zeeman state $m=0$. Preparing the system at this classical fixed point, we observe a reversible dynamics triggered by quantum fluctuations, which cannot be understood within a classical framework. We demonstrate that the classical description becomes accurate provided a coherent seed of a few atoms only is present in the other Zeeman states $m=\pm 1$. In a second regime characterized by a strong nonlinearity of the mean-field equations, we observe a collapse dynamics driven by quantum fluctuations. This behavior cannot be accounted for by a classical description and persists for a large range of initial states. We show that all our experimental results can be explained with a semiclassical description (truncated Wigner approximation), using stochastic classical variables to model the quantum noise.

Figure 1

Evolution of the number of $(+1,-1)$ pairs $N_{\mathrm{p}}$ (circles) for $q/h\approx 0.22\pm 0.03$ Hz, $N\approx 1880\pm 190$ atoms and various seed sizes: $N_{\mathrm{seed}}\approx 0;0.25;1.8$ from panels (a) to (c). The initial phase is always set to ${\theta }_{\mathrm{ini}}\approx 0$. The solid lines are numerical solutions of the Schrödinger equation with the many-body Hamiltonian in Eq. (2) using $U_s/h=9.9$ Hz. The red dashed lines correspond to the classical prediction (6). Here and in the following, error bars show the statistical error corresponding to two standard errors.

Figure 2

(a) Number of pairs produced after half a period of evolution versus $N_{\mathrm{seed}}$ for $q/h\approx 0.33\pm 0.03$ Hz and $N\approx 2920\pm 280$. The blue diamonds, green circles, and red squares correspond to initial phases ${\theta }_{\mathrm{ini}}\approx 0$; 2.2; and 3.3 rad, respectively. For the three smallest seeds, $N_{\mathrm{seed}}$ is inferred from the calibration of the rf power. The solid lines are the semiclassical predictions given by Eq. (7) with $U_s/h\approx 12$ Hz, assuming $N_{\mathrm{p}}\ll N$. For large $N_{\mathrm{seed}}$, this approximation breaks down, but a numerical solution of the nonlinear classical mean-field (3) and (4) with fixed initial conditions becomes relevant. This fully classical treatment is shown as dashed lines. (b) Scan of the initial phase ${\theta }_{\mathrm{ini}}$ after half a period of evolution for $N_{\mathrm{seed}}\approx 6.0$.

Figure 3

Evolution of the fraction of $(+1,-1)$ pairs $n_{\mathrm{p}}=N_{\mathrm{p}}/N$ in a negligible magnetic field, for $N\approx 124\pm 12$ atoms and various seedings: $N_{\mathrm{seed}}=0;0.54;2.1;4.9;12.8;$ from panel (a) to (e). The initial phase is always set to ${\theta }_{\mathrm{ini}}\approx 0$. The solid lines are numerical solutions of the Schrödinger equation for $U_s/h=24.5$ Hz. In panel (e), the red dashed line is the classical prediction from Eqs. (3), and (4).

Figure 4

(a) Frequency and (b) relaxation rate of the spin-mixing dynamics in a negligible magnetic field. The circles are obtained from a fit to the data of Fig. 3, with the error bars indicating the $95\%$ confidence interval. In panel (a), the red dashed line corresponds to the frequency $\omega$ predicted by the mean-field treatment. In panel (b), the dash-dotted blue line corresponds to the rate ${\gamma }_{\mathrm{c}}$ of the collapse driven by quantum fluctuations, the red dashed line is the damping rate ${\gamma }_{\mathrm{t}}$ due to technical fluctuations, and the solid purple line corresponds to the total damping rate $\mathrm{\Gamma }={[{\gamma }_{\mathrm{c}}^2+{\gamma }_{\mathrm{t}}^2]}^{1/2}$. We use the value $\delta U_s/U_s=0.13\pm 0.04$, obtained from a fit to the data.