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# Emergence of coherence and the dynamics of quantum phase transitions

Edited by William D. Phillips, National Institute of Standards and Technology, Gaithersburg, MD, and approved February 9, 2015 (received for review May 19, 2014)

## Significance

Quantum phase transitions are characterized by a dramatic change of the ground-state behavior; famous examples include the appearance of magnetic order or superconductivity as a function of doping in cuprates. In this work, we explore how a system dynamically crosses such a transition and in particular, investigate in detail how coherence emerges when an initially incoherent Mott insulating system enters the superfluid regime. We present results from an experimental study using ultracold atoms in an optical lattice as well as numerical simulations and find a rich behavior beyond the scope of any existing theory. This quantum simulation of a complex many-body system is an important stepping stone for a deeper understanding of the intricate dynamics of quantum phase transitions.

## Abstract

The dynamics of quantum phase transitions pose one of the most challenging problems in modern many-body physics. Here, we study a prototypical example in a clean and well-controlled ultracold atom setup by observing the emergence of coherence when crossing the Mott insulator to superfluid quantum phase transition. In the 1D Bose–Hubbard model, we find perfect agreement between experimental observations and numerical simulations for the resulting coherence length. We, thereby, perform a largely certified analog quantum simulation of this strongly correlated system reaching beyond the regime of free quasiparticles. Experimentally, we additionally explore the emergence of coherence in higher dimensions, where no classical simulations are available, as well as for negative temperatures. For intermediate quench velocities, we observe a power-law behavior of the coherence length, reminiscent of the Kibble–Zurek mechanism. However, we find nonuniversal exponents that cannot be captured by this mechanism or any other known model.

Phase transitions are ubiquitous but rather intricate phenomena, and it took until the late 19th century until a first theory of classical phase transitions was established. Quantum phase transitions (QPTs) are marked by sudden drastic changes in the nature of the ground state on varying a parameter of the Hamiltonian. They constitute one of the most intriguing frontiers of modern quantum many-body and condensed matter physics (1⇓⇓–4). Although it is typically possible to adiabatically follow the slowly changing ground state in a gapped phase, where the lowest excitation is separated from the ground state by a finite energy, these spectral gaps usually close at a QPT. Because the correlation length simultaneously diverges, adiabaticity is bound to break down, and several important questions emerge. How does a state dynamically evolve across the QPT (i.e., how does the transition literally happen?)? To what extent can the static ground state of a gapless phase be prepared in a realistic finite-time experiment? When entering a critical phase associated with an infinite correlation length, such as superfluid or ferromagnetic order, at what rate and by what mechanism will these correlations build up? Despite the fundamental importance of these questions, satisfactory answers have not been identified so far. Although the intrinsic complexity of the underlying nonintegrable models hinders numerical studies in most cases, the progress in the field of ultracold atoms now enables quantitative experiments in clean, well-isolated, and highly controllable systems.

Here, we study the quantitative dynamics of a transition into a gapless, superfluid phase in the regime of short and intermediate quench times, finding complex behavior outside the scope of any current theoretical model. As a prototypical many-body system with a QPT, we use the transition from a Mott insulator to a superfluid in the Bose–Hubbard model (5⇓⇓–8) by changing (quenching) a parameter of the Hamiltonian. The nonequilibrium settings considered here are relatively well-understood for sudden quenches, where the buildup of superfluid correlations (coherence) can be described by the ballistic spreading of quasiparticles (9, 10). These excitations are generated during the instantaneous parameter change and spread with a group velocity limited by Lieb–Robinson bounds (11). For continuous quenches, the situation is substantially more complex, because the continuous change of the Hamiltonian leads to drastically different elementary excitations throughout the evolution. Moreover, a continuous quench typically starts in the ground state, and the relevant excitations are only created during the ramp. Although there is a large body of literature trying to capture these intricate dynamics of creation and change of quasiparticles in terms of scaling laws (12⇓–14), a comprehensive and fully satisfactory theory is lacking, and many questions are still largely open. These descriptions are built on, for example, adiabatic perturbation theory (3, 15) or scaling collapses (13, 16, 17). Free models allow an exact treatment (18, 19) and can help to build an intuition for more complex physical systems. The Kibble–Zurek framework (12, 20⇓⇓–23) provides a simple guideline for the growth of correlations and predicts the density of defects after asymptotically slow ramps. It is, however, still not satisfactorily understood which correlation length results from crossing a phase transition in a strongly correlated model at a finite rate. The situation is aggravated by the fact that, in 2D and 3D lattice systems, the available numerical techniques do not allow an accurate classical numerical simulation of this setting for long evolution times.

In this work, we use ultracold atoms in an optical lattice to study the Mott to superfluid transition in the Bose–Hubbard model for experimental timescales far away from the adiabatic limit. We extract the coherence length from the width of the interference peaks in time-of-flight (TOF) absorption images and observe that, as expected, the final coherence length depends strongly on the quench rate (Fig. 1*A*). Although the resulting coherence length should diverge in the limit of adiabatic ramps, fast quenches result in short coherence lengths. We are able to probe this phase transition experimentally in 1D, 2D, and 3D systems as well as for negative absolute temperature states (24). We compare our measurements in the 1D case with a numerical analysis, finding excellent agreement, and we place them in context with existing theoretical models, thereby uncovering a rich behavior outside the scope of current analytical understanding.

Our experiments (details in *Materials and Methods* and Fig. 1*B*) started by (Fig. 1*B*, I) loading a large Mott insulator of ^{39}K atoms in a 3D optical lattice of depth *h*, the atomic mass *m*, and the lattice wavelength *U*, and the tunneling matrix element is denoted by *J*. Because of the strong repulsive interaction of *B*, II), the scattering length was then tuned within a wide range of values by a Feshbach resonance at a magnetic field of *SI Appendix*, section G). After this state preparation, the Mott to superfluid phase transition was crossed (Fig. 1*B*, III) by linearly ramping down the lattice depth along the horizontal *x* direction to *z* direction after a TOF of *C*). From the width of the interference peaks, we extracted the coherence length of the system (i.e., the characteristic length scale of an assumed exponential decay of correlations) by calculating the expected TOF profiles for various coherence lengths and fitting them to the experimental data (*Materials and Methods* and Fig. 2*A*). We parameterize the ramp time by the dimensionless quantity

We focus on the short and intermediate ramp time regime, where mass transport is negligible, such that the final density distribution is approximately given by that of the initial Mott insulator, which we take to be one particle per site. Under this assumption, the dynamics are governed by the behavior of the homogeneous system at the multicritical tip of the Mott lobe (5), because a superfluid with unit density connects to the Mott lobe precisely at the tip. Our experiment captures for the first time, to our knowledge, the physics of essentially homogeneous quantum systems entering a gapless, superfluid phase. In contrast, previous work (26) investigated the generic transition through the side of the Mott lobe, which is typical for inhomogeneous systems and dominated by mass transport, and studied the inverse superfluid to Mott insulator transition (27), the vacuum to superfluid transition (28, 29), or the transition of spinor Bose–Einstein condensates to a ferromagnetic state (30).

The experimentally measured coherence length (Fig. 2) displays several distinct dynamical regimes. For very fast ramps, the evolution can be approximated as being sudden, and the measured coherence length-*SI Appendix*, section D). Furthermore, although entropy in an ideal bosonic Mott insulator is located predominantly in the surrounding noninsulating shell, it is distributed more homogeneously in a superfluid. Thus, for short times, the trapped system is indistinguishable from the homogeneous system, whereas for longer times, trapping effects dominate the dynamics (32, 33).

The measured emergence of coherence observed here is, indeed, a generic feature of the homogeneous Bose–Hubbard model (i.e., without trap), which can be seen by directly comparing the experimental data with a classical density matrix renormalization group (34) simulation of the homogeneous system based on matrix-product states (Fig. 2*B*) (35). The sole input parameters for the simulation were *U* and *J*, and no fitting to the experimental data points was performed. By extensive scaling in bond dimension as well as Trotter step size, we have ensured numerical convergence. Additional cross-validation was obtained by an optimized exact diagonalization code performing a Runge–Kutta numerical integration of a homogeneous Bose–Hubbard model on 15 sites (*SI Appendix*, section E). We find excellent agreement between experiment and numerical data for small and intermediate ramp times up to

In the fast and slow quench limits, the physics of the continuous quench in the homogeneous model can be understood from two complementary viewpoints. For fast ramps, *A* and *SI Appendix*, section F). In this picture, fermionic excitations are continuously created during the quench and spread with their corresponding velocity that can be shown to follow a Lieb–Robinson bound (11, 36⇓⇓⇓⇓–41). For intermediate ramps,

We find that, also in our setting, the evolution can be captured in terms of simple power laws (Fig. 3). Within a range of *b* (Fig. 3), finding values that are always substantially lower than *SI Appendix*, section G), the quenches are well-approximated by linear ramps in a large range around the phase transition (*SI Appendix*, section A). More refined studies of this 1D transition, which is of the Kosterlitz–Thouless type, show that, for realistic experimental scales, smaller power laws are expected (44). Our main finding on the dynamics of the 1D phase transition, however, cannot be captured by a simple scaling model. The observed exponent crucially depends on the final point

In the experiment, ramps with different final values *B*), where *SI Appendix*, section A). Because the first part of the evolution is, however, essentially adiabatic, changing the starting point of the ramp does not significantly alter the emerging scaling laws (*SI Appendix*, section G). Because of the rather small resulting coherence lengths, we can also rule out finite-size effects as the origin for the *SI Appendix*, section E). The simulations also show that the inhomogeneous density cannot be the reason for the *SI Appendix*, section D). An inhomogeneous Kibble–Zurek scaling has recently been analyzed for a classical phase transition in ion chains (45, 46) and QPTs (26) as well as thermal (47, 48) phase transitions in ultracold atom systems. In contrast, the agreement between the inhomogeneous experiment and the numerics for the homogeneous system together with the fact that there is no mass transport on these timescales (*SI Appendix*, section B) clearly show that we effectively probe the multicritical QPT of the homogeneous Bose–Hubbard model not influenced by trap effects.

Fig. 4 shows the results of analogous experiments for the 2D and 3D Bose–Hubbard model, which are inaccessible to analytical models as well as current numerical tools. After having verified that the observed quantum dynamics in 1D, indeed, agree with the homogeneous Bose–Hubbard model, the experiments in 2D and 3D can be regarded as analog quantum simulations in a regime out of reach of classical simulation using known methods. Interestingly, the data for higher dimensions show similar power laws as the 1D case, although any critical scaling analysis would strongly depend on dimensionality. Thus, we again find that the dynamics of the Mott to superfluid phase transition on the studied intermediate timescale show complex behavior, which simple approaches based on the critical exponents alone, such as the Kibble-Zurek mechanism, cannot fully capture.

Although the extracted exponents, for the most part, increase for decreasing interaction strength, they start to decrease again for *B*). Furthermore, the full coherence dynamics for

To show that the timescale for the emergence of coherence is not influenced by any possible remnant phase order in the initial state that might seed the dynamics, we additionally studied the emergence of coherence in the attractive Bose–Hubbard model. By crossing the Feshbach resonance and additionally, inverting the external confinement in the deep lattice (II in Fig. 1*B*), we can realize an attractive Mott insulator at negative absolute temperature (24, 49⇓–51) (*SI Appendix*, section G). In Fig. 4*C*, we compare the emergence of coherence between attractive and repulsive interactions in 2D and find essentially identical behavior. Deviations from this symmetric behavior appear only for stronger interactions and can most likely be attributed to multiband effects (52, 53). Because positive and negative temperature superfluids occupy completely different quasimomenta with different correlations (Fig. 4*C*, *Insets*), we conclude that the emergence of coherence observed in the experiment is truly governed by the generic behavior of the continuous quench.

In conclusion, by performing an experimental quantum simulation, we have studied the emergence of coherence across a QPT for various interactions, dimensionalities, and positive and negative absolute temperatures. In 1D, we have also performed a detailed theoretical and numerical analysis and found very good agreement between experiment and density matrix renormalization group calculations. The observed dynamics go beyond the regime of free quasiparticles, and despite its complexity, we find that a simple power-law growth emerges in a regime where neither the adiabatic theorem nor Lieb–Robinson bounds can characterize the evolution. Although this power law is reminiscent of a Kibble–Zurek type scaling, we find that, in the studied intermediate ramp time regime, the exponent depends on

Our findings go beyond current theoretical models and raise the question of how well the dynamical features of a QPT in complex models can generally be captured in terms of simple scaling laws by either systematically expanding the free quasiparticle picture or starting from the Kibble–Zurek mechanism. The success of the latter for slow quenches in a variety of specific models suggests that, compared with a full solution of the model, much less knowledge may be sufficient to characterize the evolution. A satisfactory answer to this question will be crucial for a theory of the dynamics of QPTs. Because exact numerical techniques are not available in higher dimensions, this work may inspire a deeper and more systematic analysis of the computational power of analog quantum simulators in general. For example, it seems timely to identify, in the language of complexity theory, the precise way in which quantum simulators are, indeed, more powerful, even in the absence of error correction, than their classical analogs and how accurately experimental quantum simulators can ultimately be certified as functioning quantum devices.

## Materials and Methods

The experiments started with essentially pure condensates of, depending on dataset, ^{39}K atoms in an oblate dipole trap with trap frequencies of *B*). In the 1D case, we then quickly increased the transverse lattice depth to *y* and *z* directions. The scattering length during this loading procedure was

The momentum distribution of the atoms in the optical lattice typically probed using absorption imaging after long TOF is given in refs. 54 and 55:

with the Fourier transform of the onsite Wannier function

where

In the experiment, we probe the momentum distribution using a finite TOF

The second term in the exponential provides a correction to a pure Fourier transform and is equivalent to the quadratic term in the Fresnel approximation of near-field optics. We model the correlators by assuming exponentially decaying correlations between lattice sites:

Here, **4** for individual tubes and then, integrate the result over all tubes. Because the density distribution is approximately constant over the range of the coherence length-**5**, and by approximating them by a Gaussian distribution of width *R*, we finally arrive at (*SI Appendix*, section B)

In the case of negative temperatures, the correlator contains an additional phase term,

where

To extract the coherence length in the system, we integrate the TOF images over a small region of width *y* direction. We fit the resulting interference pattern with calculated patterns of the above model for various-*R* and extract the coherence length-*SI Appendix*, section A). We determine *R* independently by fitting a Gaussian distribution to in situ images (*SI Appendix*, section B). Sample fits are shown in Fig. 2*A*, *Insets* for the 1D case. Although this rather simple ansatz cannot reproduce the numerically calculated correlation functions in detail (*SI Appendix*, section E), it is sufficient to reproduce the experimentally measured interference patterns. Extracted coherence lengths are shown in Fig. 2*A*. In the case of the 2D and 3D sequences, correlations also spread in the transverse directions. To extract the coherence length, we integrate the images in the same range of diameter *y* direction as for 1D and fit the calculated 1D interference patterns to the resulting data.

## Acknowledgments

We thank Marc Cheneau, Masud Haque, Corinna Kollath, Michael Kolodrubetz, Alessandro Silva, Christian Gogolin, Lode Pollet, and Wojciech H. Zurek for discussions; Salvatore Manmana for insights into numerical aspects; and Tim Rom for assistance in setting up the experiment. We acknowledge financial support by Deutsche Forschungsgemeinschaft Grant FOR801 (Deutsch-Israelisches Kooperationsprojekt Quantum Phases of Ultracold Atoms in Optical Lattices), the US Defense Advanced Research Projects Agency (Optical Lattice Emulator Program), Nanosystems Initiative Munich, Spanish Ministerio de Economía y Competitividad Project FIS2011-29287, the Spanish Junta para la Ampliación de Estudios Predoctoral Program, CAM Research Consortium QUITEMAD S2009-ESP-1594, the European Union (Simulations and Interfaces with Quantum Systems), the European Research Council (Taming nonequilibrium quantum systems), Studienstiftung des Deutschen Volkes.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: ulrich.schneider{at}lmu.de.

Author contributions: I.B., J.E., and U.S. designed research; S.B., M.F., S.S.H., M.S., J.P.R., A.R., M.d.R., and J.E. performed research; S.B. and M.F. analyzed data; and S.B., M.F., I.B., J.E., and U.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1408861112/-/DCSupplemental.

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