Observation of Scaling in the Dynamics of a Strongly Quenched Quantum Gas

We report on the experimental observation of scaling in the time evolution following a sudden quench into the vicinity of a quantum critical point. The experimental system, a two-component Bose gas with coherent exchange between the constituents, allows for the necessary high level of control of parameters as well as the access to time-resolved spatial correlation functions. The theoretical analysis reveals that when quenching the system close to the critical point, the energy introduced by the quench leads to a short-time evolution exhibiting crossover reminiscent of the finite-temperature critical properties in the system’s universality class. Observing the time evolution after a quench represents a paradigm shift in accessing and probing experimentally universal properties close to a quantum critical point and allows in a new way benchmarking of quantum many-body theory with experiments.

Figure 1

Details of the experimental system. (a) Phase diagram, distinguishing miscible and immiscible phases of the elongated degenerate Bose gas of rubidium atoms in $F=2$ (blue) and $F=1$ (red) hyperfine states. The state of the system is controlled by linear coupling of the levels, with Rabi frequency $\mathrm{\Omega }$, and by tuning the collisional interaction between atoms in the hyperfine states, quantified by the relative strength $\alpha =a_{\uparrow \downarrow }/\sqrt{a_{\uparrow \uparrow }{a}_{\downarrow \downarrow }}$ of inter- and intraspecies scattering lengths (experimentally fixed to $\alpha \approx 1.23$). A quantum phase transition occurs at ${\mathrm{\Omega }}_c=\rho g(\alpha -1)$, with 1D atom density $\rho$ and intraspecies coupling constant $g$. (b) The system is initially prepared far in the miscible regime, and then $\mathrm{\Omega }$ quenched close to ${\mathrm{\Omega }}_c$. After different evolution times the two species are absorption imaged. Snapshots of the patterns emerging on either side of the transition are shown, with corresponding normalized density imbalance $[n_{\uparrow }(y)-n_{\downarrow }(y)]/\rho$ and density correlation functions between spatially separated points $y$ and $y^{\prime }$. The correlations on the miscible side exhibit decay on a characteristic length scale, while oscillations on the immiscible side reflect domain formation as seen in the density.

Figure 2

Time evolution of correlations after quench to the miscible side of the quantum phase transition. (a) Two-component gas as a coupled collective spin ensemble. The spatially resolved density difference between $F=1$ and $F=2$ allows the extraction of the local $z$ component of the collective spin vector $\mathbf{J}(y)$. (b) Measured spin correlation function $G_{zz}(y,y^{\prime },t)=⟨J_z(y)J_z(y^{\prime }){⟩}_t$ at three different times after a quench to $ϵ=\mathrm{\Omega }/{\mathrm{\Omega }}_c-1=0.17$, showing buildup of spatial correlations. Solid black lines show Bogoliubov–de Gennes mean-field predictions. (c) Time evolution of the correlation length $\xi (t;ϵ)$, for three different $ϵ$, deduced by fitting an exponential to the short-distance falloff of the extracted correlation data. The Bogoliubov evolution (solid lines) recovers the initial near-linear rise of $\xi$, with slope given by the speed of (spin-wave) sound, and captures the maximum correlation length at a characteristic time depending on $ϵ$. The predicted oscillatory behavior is experimentally observed for larger $ϵ$, while the maximum correlation length reached at short times is robustly detected in all cases. Dashed lines serve as a guide to the eye, marking the predicted first maxima. For the scaling analysis of the data, the correlation lengths are compared at a fixed time; see Fig. 3.

Figure 3

Spatial and temporal scaling of the spin-spin correlations. (a) Spatial correlations after quenches to different proximities $ϵ$ from the critical point (color coded), on the immiscible (left-hand panels, at $t=39\mathrm{ms}$) and miscible side (right-hand side, $t=12\mathrm{ms}$). Top panels: The correlation functions, under a rescaling $y\to y{ϵ}^{\nu }$ of the distance dependence with mean-field exponent $\nu =1/2$, fall on a universal curve. Bottom panels: $ϵ$ dependence (double-log scale) of the characteristic length scales deduced from the correlation functions. The straight lines reveal values for the critical exponent $\nu =0.51(4)$ on the immiscible and $\nu =0.51(6)$ on the miscible side of the transition. (b) Temporal scaling of the spin-spin correlations. The characteristic time $\tau$ for different $ϵ$ is obtained as the intersection point of the linear fits to the initial rise of $\xi (t)$ (gray symbols for $ϵ=0.1$) and to the behavior after $\xi$ deviates from this rise. The procedure of determining the intersection is exemplarily shown in the upper panel for $ϵ=0.23$. In the lower panel we compare the extracted $\tau (ϵ)$ to a mean-field scaling with $\nu z=1/2$ (dashed line) and to the Bogoliubov prediction $\tau \sim 1/\mathrm{\Delta }$, with gap $\mathrm{\Delta }(ϵ)={\mathrm{\Omega }}_c\sqrt{ϵ(ϵ+1)}$, also applicable at larger $ϵ\gtrsim 1$ (solid line).

Figure 4

Scaling analysis at short times after the quench. Correlation length ${\xi }_0(t;ϵ)$ on the miscible side of the transition, at the time of the first maximum of ${\xi }_0$ after the quench. The solid black line marks the Bogoliubov prediction ${\xi }_{\mathrm{Bog}}=\sqrt{\hslash /(2m{\mathrm{\Omega }}_cϵ)}$ also shown in Fig. 3. Closer to the critical point, the open colored symbols show results of semiclassical simulations of the quench dynamics. In the experimental range ($ϵ\gtrsim 0.1$), simulation data and Bogoliubov mean-field prediction agree. For $ϵ\lesssim 0.1$, the simulations demonstrate a deviation from the mean-field power law and saturation for $ϵ\to 0$.