A universal speed limit for spreading of coherence

TL;DR

The paper investigates a universal speed limit governing the spread of coherence during the formation of a Bose-Einstein condensate in an isolated, homogeneous gas. Using a cylindrical optical box trap, controlled interactions, and dynamic scaling analysis, the authors demonstrate a universal growth law with exponent $\beta=1/2$ for the coherence length $\ell$, and a universal rate $D=\dfrac{d\ell^2}{dt}=3.4\,\hbar/m$ that is independent of the microscopic interaction strength $na$ and of initial conditions. This universal rate can be expressed in terms of $\hbar/m$ (equivalently the quantum of velocity circulation $\kappa=2\pi\hbar/m$), implying a fundamental speed limit for coherence spreading set by quantum mechanics. The findings provide benchmarks for far-from-equilibrium universality, bear significance for quantum technologies relying on large-scale coherence, and invite analogous measurements in other many-body systems including fermionic superfluids and those with long-range interactions.

Abstract

Discoveries of fundamental limits for the rates of physical processes, from the speed of light to the Lieb-Robinson bound for information propagation, often lead to breakthroughs in the our understanding of the underlying physics. Here we observe such a limit for a paradigmatic many-body phenomenon, the spreading of coherence during formation of a weakly interacting Bose-Einstein condensate. We study condensate formation in an isolated homogeneous atomic gas that is initially far from equilibrium, in an incoherent low-energy state, and condenses as it relaxes towards equilibrium. Tuning the inter-atomic interactions that drive condensation, we show that the spreading of coherence through the system is initially slower for weaker interactions, and faster for stronger ones, but always eventually reaches the same limit, where the square of the coherence length grows at a universal rate given by the ratio of Planck's constant and the particle mass, or equivalently by the quantum of velocity circulation associated with a quantum vortex. These observations are robust to changes in the initial state, the gas density, and the system size. Our results provide benchmarks for theories of universality far from equilibrium, are relevant for quantum technologies that rely on large-scale coherence, and invite similar measurements in other systems.

Figures (11)

• Figure 1: Universal coarsening of an isolated Bose gas.a, Real-space cartoon of coarsening. b, Momentum-space relaxation for different far-from-equilibrium initial states. Our initial states P$_{1,2,3}$ (left column) have different momentum distributions $n_k$, but the same energy, so the gas always relaxes towards the same equilibrium state. For P$_{1,2,3}$, the system takes different times, $t_{1,2,3}$, to evolve to the same $n_k$ shown in the middle column, but from this point onwards it always evolves in the same way. The $n_k$ distributions are averages of at least $20$ measurements. The red scale bar (top image) shows $1\,\upmu{\rm m}^{-1}$. c, Growth of the coherence length, $\ell$ (see text). Plotting $\ell^2(t)$ reveals three stages of relaxation: (i) the non-universal initial dynamics, (ii) the scaling regime where $\ell^2$ grows linearly (dashed lines), as expected for the scaling exponent $\beta = 1/2$, and (iii) the breakdown of scaling at long times due to finite-size effects. The curves for P$_{1,2,3}$ are parallel, with the initial-state effects captured by the different time offsets $t^*$ (intercepts of the dashed lines). d, Dynamic scaling. In the scaling regime, the full low-$k$ distributions for all three initial states (left panel) can be collapsed onto the same curve (right panel) according to Eqs. (\ref{['eq:beta']}, \ref{['eq:dynamicScaling']}) with $\beta=1/2$ and $t \rightarrow t_{\textrm{uni}}\xspace \equiv t - t^*$ (see also Extended Data Fig. \ref{['fig:S4']}); $t_0=60\,\textrm{ms}$ is an arbitrary reference time. All error bars show standard errors of the measurements.
• Figure 1: Initial-state preparation. We prepare our initial states using a combination of time-dependent force $F$ and interaction pulsing, as shown in the top two rows. The forcing pulses for P$_1$ and P$_3$ are $8\,\rm{ms}$ long and have strength $F_0=k_{\textrm{B}} \times 1.5\,{\rm nK}/\upmu{\rm m}$. The sinusoidal force for P$_2$ has amplitude $F_0=k_{\textrm{B}} \times 0.3\,{\rm nK}/\upmu{\rm m}$ and angular frequency $\omega = 2\pi \times 10\,$Hz, and is applied for $1\,$s. For P$_3$ the interactions are pulsed to $a=400\,a_0$ for $30\,$ms. After the end of the preparation sequence we wait for $500\,$ms at $a=0$ for $n_k$ to become isotropic. The images (same as in Fig. \ref{['fig:1']}b) show the $k_x - k_y$ distributions for $k_z\approx 0$ just before we turn on interactions to initiate the relaxation; the red scale bar in the left image shows $1\,\upmu{\rm m}^{-1}$. The bottom three rows show different moments of the corresponding momentum distributions. The integrals of $k^2 n_k$ and $k^4 n_k$ are, respectively, proportional to the total atom number and energy, which are the same for all three states; the energy per particle is $\varepsilon = k_{\textrm{B}} \times 20(2)\,\textrm{nK}$.
• Figure 2: Spreading of coherence for different interaction strengths.a, Gas evolution for two different scattering lengths $a$, starting in the same initial state P$_1$. For stronger interactions, the condensate (the peak at $k=0$) emerges sooner. b, However, plotting $\ell^2 (t)$ reveals that the interaction strength affects only the non-universal initial dynamics (open symbols), while the linear growth of $\ell^2$ in the universal coarsening regime (solid symbols) is the same for both $a$ values.
• Figure 2: Equilibrium $n_0$ and $\eta$. We plot the observed $n_0(t)$ for all our $11$ data sets taken with $V=55(12)\times 10^3 \upmu{\rm m}^3$ and $n=5.4(1.2)\,\upmu{\rm m}^{-3}$. The solid line shows the long-time saturation value $\bar{n}_0 = 25(2) \times 10^6 \upmu{\rm m}^3$, obtained by averaging all the measurements for the two longest times ($0.8\,$s and $1.0\,$s for all data sets; the data points for these two times are slightly offset horizontally for visual clarity). The dashed line shows the value of $n_0$ observed for a quasi-pure BEC, $n_0^{\rm BEC}=39(2)\times 10^6 \upmu{\rm m}^3$, which gives $\bar{n}_0/n_0^{\rm BEC} = 0.64(6)$. The inset shows the same data on a linear scale, focusing on long times.
• Figure 3: Universality of the coarsening speed $D$. Our measurements for various interaction strengths show no systematic variation of $D = \textrm{d} \ell^2 / \textrm{d} t$ (in the scaling regime), and give a combined estimate $D=3.4(3)\,\hbar/m$ (solid line and shading). The four data sets shown in Fig. \ref{['fig:1']}c or Fig. \ref{['fig:2']}b are represented here by the corresponding coloured symbols. The purple triangle indicates a measurement where we reduced the gas density $n$ by a factor of $4.2$, which reduces $T_{\textrm{c}} \propto n^{2/3}$ and the equilibrium condensed fraction (from about $60$% to about $30$%), but $D$ is not affected. The green pentagon indicates a measurement where we instead reduced the system volume $V$ by a factor of $3.5$; in this case $\ell^2$ saturates at a correspondingly lower value ($\propto V^{2/3}$), but again $D$ is not affected. For details on the data taken with reduced $n$ or $V$ see Extended Data Fig. \ref{['fig:S5']}. The data points corresponding to the same $na$ values are slightly offset horizontally for visual clarity.