Abstract
Quantum phase transitions, transitions between manybody ground states, are of extensive interest in research ranging from condensedmatter physics to cosmology^{1,2,3,4}. Key features of the phase transitions include a stage with rapidly growing new order, called inflation in cosmology^{5}, followed by the formation of topological defects^{6,7,8}. How inflation is initiated and evolves into topological defects remains a hot topic of debate. Ultracold atomic gas offers a pristine and tunable platform to investigate quantum critical dynamics^{9,10,11,12,13,14,15,16,17,18,19,20,21}. We report the observation of coherent inflationary dynamics across a quantum critical point in driven Bose–Einstein condensates. The inflation manifests in the exponential growth of density waves and populations in wellresolved momentum states. After the inflation stage, extended coherent dynamics is evident in both real and momentum space. We present an intuitive description of the quantum critical dynamics in our system and demonstrate the essential role of phase fluctuations in the formation of topological defects.
Main
During a quantum phase transition, a manybody system, originally prepared in the ground state with macroscopic coherence, is suddenly transferred to a metastable state after passing the critical point^{1,6,12}. An example shown in Fig. 1 is a ferromagnetic transition where the Z _{2} inversion symmetry is broken. How does the system evolve toward the new ground states generally with a different symmetry? One can hypothesize two possible scenarios. In the first scenario, fluctuations break the system into locally coherent segments that evolve toward the new ground states independently. After relaxation, the system forms domains with local coherence^{2,12}. In the second scenario, although the macroscopic coherence is maintained, the system undergoes a coherent population transfer of particles toward lower energy states. Here fluctuations determine the domain structure but do not destroy the macroscopic coherence.While both scenarios support rapid evolution toward new ground states, the key differences are the time and length scales of the coherence in the dynamical process.
In this paper, we report the observation of coherent inflationary dynamics in an atomic Bose condensate driven across a quantum critical point. Our experiment is based on caesium Bose–Einstein condensates loaded into a onedimensional phasemodulated optical lattice^{22}. The modulation translates the lattice periodically over time t with displacement \(\Delta x=\frac{s}{2}{\rm{\sin }}\omega t\), where s is the shaking amplitude and ω is the shaking frequency. Shaking hybridizes the ground and excited Bloch bands and results in an effective dispersion ε _{ q } for the condensate^{22}, where the lowest energy state at quasimomentum q = 0 bifurcates into two ground states at \(+{q}^{* }\) and \({q}^{* }\) (named pseudospin up and down), when s exceeds a critical value s _{c}. When the system is driven across the critical point in finite time, domains of pseudospins form in accordance with universal Kibble–Zurek scaling^{21} and excitations within a domain display a roton dispersion^{23}; however, a complete understanding of the processes that underlie the quantum critical dynamics remains evasive.
To reveal the nature of the quantum phase transition, we exploit three schemes to analyse the critical dynamics of the condensate: in situ imaging to record the atomic density profile; timeofflight with a focusing technique^{24} to probe the momentum space distribution n _{ q }; and pseudospin reconstruction to reveal domain structure^{21}. An example is shown in Fig. 2. Here we linearly ramp up the shaking amplitude and interrupt the ramp at time t after passing the critical point to probe the system with the three methods.
We observe two key features indicating coherent evolution. First, from in situ images, a density wave emerges about 20 ms after passing the critical point. Quantified with the density structure factor S _{ q } (ref. ^{25}), the density wave shows an almost fixed wavenumber. Second, from timeofflight images, the atomic population forms sharp side peaks in individual samples; over repeated measurements, the side peaks average to broader features. These observations suggest that atoms occupy a coherent superposition of welldefined momentum states and the density wave emerges from their interference. Although the density wave diminishes after 30 ms, the persistent narrow momentum peaks in atomic population n _{ q } suggest a longlasting coherence. In addition, the period of the density waves approximately matches twice the averaged domain size. Both features will be further discussed in later paragraphs.
A more comprehensive analysis of the density wave and the population distribution in momentum space suggests that the system evolution can be separated into two stages: inflation and relaxation. To see this, we evaluate the density variance \(\delta {n}^{2}=\int {\rm{d}}q{S}_{q}\) from in situ images as well as the total population in finite momentum states \(\Delta N={\sum }_{q > 0}{N}_{q}\) from timeofflight measurements, where N _{ q } is the total atom number in the \(\pm q\) states. For short times after the phase transition, both quantities show a characteristic exponentiallike growth; we name this period the inflation stage (see Fig. 2f). After inflation, all atoms relax toward nonzero momentum states at \(q=\pm {q}^{* }\) while the density wave diminishes. In the following, we investigate the two stages separately.
The exponential growth of excitations can be theoretically understood on the basis of dynamical instability of the condensate^{26}. Shortly after passing the quantum critical point, the \(q=0\) state remains macroscopicly occupied, which justifies the Bogliubov approximation. Owing to the inverted dispersion, the manybody Hamiltonian cannot be diagonalized with bosonic field operators^{27}. Instead, we can write the Hamiltonian as (Supplementary Information)
where \({\hat{\iota }}_{q}^{\dagger }\) and \({\hat{\iota }}_{q}\) are the bosonic creation and annihilation operators of an inflaton with momentum q and growth rate \({\lambda }_{q}=\sqrt{{\varepsilon }_{q}(2\mu +{\varepsilon }_{q})}/\hslash \), μ is the chemical potential and \(\hslash \) is the reduced Planck constant. It is important to emphasize that only modes with negative kinetic energy \({\varepsilon }_{q} < 0\) acquire the inflationary dynamics. According to the Hamiltonian, the excited populations increase exponentially in the inflation phase according to (Supplementary Information)
and the structure factor \({S}_{q}(t)=({\varepsilon }_{q}/\mu )[{N}_{q}(t)+1]\) near the critical point. This result explains the similar exponentiallike growth of both observables in Fig. 2f.
To further test the inflation theory, we perform quench experiments by suddenly driving the system across the critical point, and measure the growth rate of the population in different momentum modes. Right before the quench, we seed a small initial population in the desired momentum states \(\pm {q}^{\prime}\) by imprinting a sinusoidal phase pattern on the condensate \(\delta \phi {\rm{\sin }}({q}^{\prime}x/\hslash )\). Here \(\delta \phi \) is the seed amplitude and the wavenumber \({q}^{^{\prime}}/\hslash \) is externally controlled (see Methods).
After seeding, the condensate quickly grows two side peaks at the seeding momentum \(\pm {q}^{\prime}\) (Fig. 3a). To extract the growth rate, we monitor the population in the momentum states. The population grows exponentially in the beginning but reaches a maximum at a later time when the population in the \(k=0\) state is depleted (Fig. 3b). We fit the fastgrowing interval right after the quench according to equation (2) and compare the growth rate to the prediction (Fig. 3c). Our measured growth rates qualitatively agree with the Bogoliubov result. We find quantitative agreement with our numerical simulation based on the Gross–Piteavskii equation that incorporates the depletion of the condensate (Supplementary Information). In particular, we confirm that only modes \( q \le \) 0.4 q _{L} with kinetic energy \({\varepsilon }_{q} < 0\) exponentially grow and the fastest growth appears at momentum \(q\approx \pm {q}^{* }\). Here \({q}_{{\rm{L}}}=\pi {\rm{}}\hslash /d\) is the lattice momentum with d being the lattice period.
Remarkably, in the absence of seeding, the sample spontaneously grows momentum peaks near \(\pm {q}^{* }\) with a growth rate very close to that seeded at a similar momentum (Fig. 3c and the inset). For unseeded samples, many momentum modes can, in principle, be populated by quantum or thermal fluctuations and then amplified by inflation. The dominance of the modes near \(\pm {q}^{* }\) can be understood since they have the highest growth rate and become dominant during inflation.
Following the inflation stage, the condensates display persistent coherent dynamics in both timeofflight and in situ measurements (Fig. 4). After the rapid growth of the population at seeded momentum \(\pm q^{\prime} \), the system generates higher order harmonics at \(\pm 2q^{\prime} ,\pm 3q^{\prime} ...\), and the atomic populations are coherentlyʹ transferred between these momentum states. The emergence of higher harmonics is due to nonlinear mixing of the matter waves and can be well described based on our numerical model. An example at t = 14 ms shows multiple side peaks that conform to the simulation. Intriguingly, the individual momentum peaks are as narrow as the zero momentum peak of the original condensate; the widths are limited only by the detection resolution. A narrow momentum peak indicates long coherence length based on uncertainty principle. By comparing our measurement with the simulation, we conclude that the lower bound of the coherence length is 15 μm, which is much greater than the average domain size of 4.1(1) μm (Supplementary Information).
Together with the dynamics in momentum space, density waves in seeded samples also display coherent oscillations in quench experiments (Fig. 4c). The density wave appears aligned to the imprinted pattern, and its phase displays multiple alternations (Fig. 4d) that are synchronized with oscillations of the population in momentum space. The contrast of the density wave oscillates and slowly decays with a time constant of τ = 20 ms. Both the alternation and the decay are in good agreement with our simulation. Finally, we find that domains are fully formed as early as \(t=14\) ms, and remain constant afterward. Importantly, the domain structure is deterministic in the seeded experiments, and the domain walls line up with the density wave yielding a domain size half the period of the density wave.
The observed coherent dynamics can be understood on the basis of a simple physical picture. Phase imprinting across the condensate locally breaks the inversion symmetry by inserting a current \(j=\frac{\hslash }{m}{\phi }^{^{\prime} }(x)\), where \(\phi (x)\) is the phase of the condensate wavefunction. Within one period of the imprinted phase pattern, the sign of the local momentum flips twice, resulting in two neighbouring domains with opposite momenta. After the momentum kick, atoms in neighboring domains can flow toward or away from each other determined by the group velocity \({v}_{g}(x)={\rm{d}}{\varepsilon }_{q}/{\rm{d}}q\), leading to the observed density peaks and troughs. Since density waves cost energy in a Bose–Einstein condensate with repulsive interactions, the atom flow reverses after half an oscillation period, yielding the phase alternation of the density wave.
The decisive role of phase imprinting in the real and momentumspace dynamics and domain structure indicates the importance of phase fluctuations in quantum critical dynamics. In addition to the emergence of density waves and atomic occupation in wellresolved momentum states, we present strong evidence supporting the coherent scenario of the quantum phase transition in our system. Furthermore, the phase imprinting technique can find new applications in engineering desired structures of domain walls, which will enable future study on the dynamics and interactions of topological defects.
Methods
Lattice loading
We utilize threedimensional Bose–Einstein condensates of 30,000 caesium atoms confined in an optical dipole trap. The trap is tightly confined in the gravity direction with a trapping frequency of \(2{\rm{}}\pi \times \)226 Hz. Trapping frequencies in the two inplane directions are \(2{\rm{}}\pi \times \)6 and \(2{\rm{}}\pi \times \)9 Hz. The swave scattering length is 2.6 nm. We adiabatically load the Bose–Einstein condensate into a onedimensional optical lattice with a depth of 8.9 E _{R} and period d = 532 nm, where E _{R} = \(h\times \) 1.3 kHz is the recoil energy.
Shaken lattice
We periodically translate the lattice by sinusoidally modulating the phase of one of the lattice beams. The shaking frequency is fixed to ω = \(2{\rm{}}\pi \times \)8 kHz, which is 2π\({\rm{}}\times \)0.87 kHz above the gap at zero momentum between the ground and the first excited Bloch band in the lattice.
Phase imprinting
We imprint the phase pattern across the condensate using a digital micromirror device with a 795 nm laser. To ensure a sinusoidal modulation, we set a grating pattern on the digital micromirror device with twice the desired period \(2{\rm{}}\pi \hslash /q^{\prime} \) and let only the ±1 orders from the diffraction pass in the Fourier plane. The diffracted beams interfere on the atoms, giving a clean sinusoidally varying potential. The imprinting pulse lasts for 20 to 40 µ\({\rm{s}}\), which is very short compared with the condensate and lattice timescale.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank E. Berg, Q. Zhou and B. M. Anderson for helpful discussions. L.W.C. was supported by a Grainger fellowship. A.G. is supported by a Kadanoff–Rice fellowship. This work was supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation under award number DMR1420709, NSF grant PHY1511696 and Army Research OfficeMultidisciplinary Research Initiative grant W911NF1410003.
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The James Franck Institute, Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, IL, USA
 Lei Feng
 , Logan W. Clark
 , Anita Gaj
 & Cheng Chin
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Contributions
L.F. and L.W.C. designed the experiments. L.F. performed the experiments and analysed the data. L.F. and C.C. prepared the manuscript. All authors contributed discussions on experiments and the manuscript. C.C. supervised.
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The authors declare no competing financial interests.
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Correspondence to Cheng Chin.
Supplementary information
Supplementary Information
Supplementary Information, Supplementary Figures 1–3, Supplementary References
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