Abstract
Understanding the dynamics of isolated quantum manybody systems is a central open problem at the intersection between statistical physics and quantum physics. Despite important theoretical effort^{1}, no generic framework exists yet to understand when and how an isolated quantum system relaxes to a steady state. Regarding the question of how, it has been conjectured^{2,3} that equilibration must occur on a local scale in systems where correlations between distant points can establish only at a finite speed. Here, we provide the first experimental observation of this local equilibration hypothesis. In our experiment, we quench a onedimensional Bose gas by coherently splitting it into two parts. By monitoring the phase coherence between the two parts we observe that the thermal correlations of a prethermalized state^{4,5} emerge locally in their final form and propagate through the system in a lightconelike evolution. Our results underline the close link between the propagation of correlations ^{2,3,6,7} and relaxation processes in quantum manybody systems.
Main
It has been theoretically suggested that relaxation in generic isolated quantum manybody systems proceeds through the dephasing of the quantum states populated at the onset of the nonequilibrium evolution^{8,9}. It is generally believed that this dynamically leads to relaxed states that can be well described either by the usual thermodynamical ensembles or by generalized Gibbs ensembles that take into account dynamical constraints^{10}. However, it remains an open question how these relaxed states form dynamically, and in particular, whether they emerge gradually on a global scale, or appear locally and then spread in space and time^{3}.
Ultracold atomic gases offer an ideal test bed to explore such quantum dynamics. Their almost perfect isolation from the environment and the many available methods to probe their quantum states make it possible to reveal the dynamical evolution of a manybody system at a very detailed level^{4,7,11,12,13,14,15,16}.
In our experiment, a phasefluctuating ultracold onedimensional (1D) Bose gas^{17} is split coherently^{18}. The splitting creates a nonequilibrium state consisting of two gases with almost identical phase profiles. Interactions in the manybody system drive the relaxation of this highly phasecorrelated state to a prethermalized state, characterized by thermal phase correlations^{4,19}. The dynamics is monitored by timeresolved measurements of the relative phase field using matterwave interferometry^{20}.
The experimental procedure starts with a 1D degenerate gas of 4,000–12,000 ^{87}Rb atoms trapped at temperatures between 30–110 nK in a magnetic trap, formed 100 μm below the trapping wires of an atom chip^{21}. By applying radiofrequency fields through additional wires on the chip, we rapidly transform the initial harmonic trapping potential into a double well, thereby realizing the matterwave analogue of a coherent beamsplitter^{18} (see Methods).
The system is allowed to evolve in the double well for a variable time t, before the gases are released by switching off the trapping potential. They expand and interfere after a timeofflight of 15.7 ms. The resulting interference pattern allows us to extract the relative phase ϕ(z,t) = θ_{1}(z,t)−θ_{2}(z,t) along the length of the system (Fig. 1). Here, θ_{1}(z,t) and θ_{2}(z,t) are the phase profiles of the two individual gases. Repeating this procedure approximately 150 times for each value of t, we determine the twopoint relative phase correlation function
It measures the degree of correlation between the phases at two arbitrary points z and z^{′}, separated by a distance (refs 22, 23). In contrast to the integrated visibility of the interference pattern, which was used in a previous experiment to identify the prethermalized state^{4}, the phase correlation function provides a sensitive probe for the local dynamics, and is therefore ideally suited to study the propagation of correlations.
Typical experimental data are presented in Fig. 2a. Directly after the quench, the phase correlation function is close to unity for any distance . This is a direct manifestation of the longrange phase coherence produced by the splitting process. After a given evolution time t, the phase correlation function decays exponentially up to a characteristic distance and stays nearly constant afterwards: . This means that beyond the distance longrange phase coherence is retained across the system. With longer evolution time, the position of shifts to larger distances and the value of gradually decreases. This evolution continues until the system reaches a quasisteady state, where the correlations decay exponentially throughout the entire system^{19}. This prethermalized state corresponds to the relaxed state of the 1D system and can be described by a generalized Gibbs ensemble^{4,10}. Our observation that the exponentially decreasing parts of the dynamical phase correlation functions match the exponential decay of the relaxed, prethermalized state for allows us to conclude that equilibration occurs locally in our system.
From the experimental data, we extract the crossover points through the level of longrange phase coherence. To this end, we consider for each t the region where the correlation function is constant, extrapolate the constant value to smaller and determine the position where it crosses the prethermalized correlation function (Supplementary Information). The result of this procedure is shown in Fig. 2b. We observe a clear linear scaling of the position , characterizing the local decay of correlations with time. This observation reveals that an arbitrary point in the gas loses its correlations with other points up to a certain separation , whereas longrange phase coherence persists outside this horizon. The experimental data thus show that the prethermalized state locally emerges in a lightconelike evolution, where c plays the role of a characteristic velocity for the propagation of correlations in the quantum manybody system. For the data presented in Fig. 2b a linear fit allows us to extract a velocity of c = 1.2±0.1 mm s^{−1}.
Lightconelike effects in quantum manybody dynamics have been previously predicted using results from conformal field theory^{2}, and for 2D superfluids^{24}. Similarly, it is known that some quantum spin models exhibit an intrinsic maximum velocity^{6} that limits the propagation of correlations and entanglement to an effective light cone ^{7,25,26}. It has been conjectured that this leads to a local establishment of thermal properties^{3}.
The lightconelike emergence of thermal correlations that we observe in this work can be understood using a homogeneous Luttingerliquid model that effectively describes the interacting manybody system in terms of lowenergy excitations^{27}. Within the Luttingerliquid model, these excitations are superpositions of phase and density fluctuations. They are characterized by a linear dispersion relation ω_{k} = c_{0}k, with k being the momentum of the excitation and c_{0} the speed of sound, the latter defining the characteristic velocity in the homogeneous system.
The coherent splitting process equally distributes energy among the excitations, resulting in a 1/k dependence of their occupation numbers^{28}. Each excitation is initialized with small relative phase fluctuations and high relative density fluctuations. Over time, the amplitude of the phase (density) fluctuations increases (decreases), resulting in a progressive randomization of the relative phase field ϕ(z). Eventually, the energy associated with the phase fluctuations equilibrates with the energy associated with the density fluctuations, leading to the thermal phase correlations of the prethermalized state^{28}.
For a given evolution time t, the dephasing of the excitations with different wavelengths (2π/k) randomizes the relative phase field only up to a characteristic distance . This effect can be understood in the following way (see Methods for mathematical details): the degree of randomization of the phase is related to the amplitude of the contributing phase fluctuations. For large distances they are associated with the highly occupied longwavelength excitations that take a long time (∼1/ω_{k}) to be converted from the initial density fluctuations into phase fluctuations. At time t, there exists a characteristic distance beyond which the contribution of these longwavelength fluctuations to the randomization of the phase is compensated by a decrease of the contribution from the faster shortwavelength fluctuations (see Supplementary Fig. S3 for an illustration). Therefore, the phase does not randomize any further and longrange phase coherence remains beyond . The sharpness of the transition at results from the interference of the many excitations with different momenta.
Alternatively, the excitations in the Luttingerliquid model can also be identified as pairs of quasiparticles, which propagate in opposite directions with momenta k and −k, respectively^{2,7}. This interpretation naturally leads to the lightcone condition, as two points separated by can establish thermal correlations if quasiparticles originating from these points meet after a time .
In Fig. 2a we compare the results of the Luttingerliquid calculation to our measured data, taking into account the finite resolution of the imaging system (Supplementary Information). We find good agreement, using independently measured experimental parameters as the input for the theory. This quantitative agreement validates our interpretation of the observations as the local emergence of thermal correlations.
When increasing the number of particles in our quantum manybody system, we expect interaction effects to play a more important role, leading to a faster local relaxation. In the homogeneous limit this is captured by the scaling of the speed of sound with the 1D density ρ of each gas^{17}. To investigate the scaling of the characteristic velocity, we perform the experiment for a varying number of atoms N in the system. We observe the lightconelike emergence of the thermal correlations over the whole range of probed atom numbers (N∼4,000–12,000). In the experimentally realized trapped system, the density varies along the length of the gases, resulting in a spatially dependent speed of sound. Nevertheless, the superposition of many excitations still leads to a single characteristic velocity for the dynamics, which is slightly reduced with respect to the homogeneous case (Supplementary Information). In Fig. 3 we show the measured characteristic velocities. A Luttingerliquid calculation including the trapping potential describes the experimental data within the experimental error, whereas a purely homogeneous calculation clearly overestimates the characteristic velocity.
In our experiment thermal correlations emerge locally in their final prethermalized form. This supports the local relaxation hypothesis^{3} and indicates a general pathway for the emergence of classical properties in isolated quantum manybody systems. In our system, interactions manifest themselves in excitations with a linear dispersion relation (in the homogeneous limit), resulting in a decay of quantum coherence that takes the form of an effective light cone. Whether this scenario holds also for systems with nonlinear dispersion relations, longrange interactions^{29} or systems that are subject to disorder^{30} remains a topic of intense study.
Methods
Splitting process.
The splitting is performed by linearly increasing the amplitude of the radiofrequency current in the chip wires to 24 mA within 12 ms. To minimize longitudinal excitations during the splitting, the initial gas is prepared in a slightly dressed radiofrequency trap that has the same longitudinal confinement as the final doublewell potential (see Supplementary Information for more details). The increase of radiofrequency current results in a rapid decay of the tunnel coupling between the two gases. Simulations of the chip potential and experiments with quasicondensates in thermal equilibrium^{23} indicate that the decoupling of the two gases happens within less than 500 μs. This is faster than the characteristic timescale of the dynamics (∼ 10 ms; ref. 19) and therefore realizes a quench.
Relative phase measurement.
The interference patterns are recorded after a timeofflight expansion of 15.7 ms using absorption imaging. The point spread function of the optical system has a measured r.m.s. width of 3.6 μm. The phase ϕ(z) of the interference patterns is extracted by fitting each pixel line (of size σ_{px} = 2 μm) with a cosinemodulated Gaussian function.
Theoretical model.
Within the Luttinger Liquid theory the phase correlation function can be written as C(z,z′,t) = exp(−(1/2)〈Δϕ_{z z′}(t)^{2}〉), with Δϕ_{z z′}(t) = ϕ(z,t)−ϕ(z′,t). In the homogeneous limit, the local relative phase variance is given by^{28,31}
with L being the length of the system, k = 2πn/L the momentum of the excitations ( integer) and K the Luttinger parameter. The amount of fluctuations is thus determined by the interference of several longitudinal modes of the 1D system.
The first term in the sum (1) represents the growth and subsequent oscillations in the amplitude of the phase fluctuations as they get converted from the initial density fluctuations. The factor 1/k^{2} in the amplitude reflects the 1/k scaling of the excitation occupation numbers associated with the equipartition of energy induced by the fast splitting. The second term in the sum corresponds to the spatial fluctuations. Expression (1) is the Fourier decomposition of a trapezoid with a sliding edge at , which explains the twostep feature of the phase correlation function.
A similar expression can be derived for the trapped system probed in the experiment (Supplementary Information).
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Acknowledgements
We would like to thank D. Adu Smith and M. Gring for contributions in the early stage of the experiment, I. Mazets, V. Kasper and J. Berges for discussions and JF. Schaff and T. Schumm for comments on the manuscript. This work was supported by the Austrian Science Fund (FWF) through the Wittgenstein Prize and the EU through the projects QIBEC and AQUTE. T.L. and M.K. thank the FWF Doctoral Programme CoQuS (W1210); R.G. is supported by the FWF through the Lise Meitner Programme M 1423.
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T.L. and R.G. performed the experiment, analysed the data and carried out the theoretical modelling. J.S. conceived the experiment and the leading scientific questions. All authors contributed to the interpretation of the data and the writing of the manuscript.
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Langen, T., Geiger, R., Kuhnert, M. et al. Local emergence of thermal correlations in an isolated quantum manybody system. Nature Phys 9, 640–643 (2013). https://doi.org/10.1038/nphys2739
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DOI: https://doi.org/10.1038/nphys2739
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