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Analog cosmological reheating in an ultracold Bose gas

Aleksandr Chatrchyan, Kevin T. Geier, Markus K. Oberthaler, Jürgen Berges, and Philipp Hauke
Phys. Rev. A 104, 023302 – Published 3 August 2021
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  • INTRODUCTION
  • EXPANDING SPACETIME IN BOSE GASES
  • ANALOG PREHEATING
  • ANALOG REHEATING
  • EXPERIMENTAL PERSPECTIVES
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • Supplemental Material
    References

    Abstract

    Cosmological reheating describes the transition of the postinflationary universe to a hot and thermal state. In order to shed light on the underlying dynamics of this process, we propose to quantum-simulate the reheating-like dynamics of a generic cosmological single-field model in an ultracold Bose gas. In our setup, the excitations on top of an atomic Bose-Einstein condensate play the role of the particles produced by the decaying inflaton field after inflation. Expanding spacetime as well as the background oscillating inflaton field are mimicked in the nonrelativistic limit by a time dependence of the atomic interactions, which can be tuned experimentally via Feshbach resonances. As we illustrate by means of classical-statistical simulations for the case of two spatial dimensions, the dynamics of the atomic system exhibits the characteristic stages of far-from-equilibrium reheating, including the amplification of fluctuations via parametric instabilities and the subsequent turbulent transport of energy towards higher momenta. The transport is governed by a nonthermal fixed point showing universal self-similar time evolution as well as a transient regime of prescaling with time-dependent scaling exponents. While the classical-statistical simulations can capture only the earlier stages of the dynamics for weak couplings, the proposed experiment has the potential of exploring the evolution up to late times even beyond the weak coupling regime.

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    • Received 15 November 2020
    • Revised 25 May 2021
    • Accepted 21 June 2021

    DOI:https://doi.org/10.1103/PhysRevA.104.023302

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Bose-Einstein condensatesEvolution of the UniverseNonequilibrium statistical mechanicsParticle & resonance productionQuantum simulationWeak turbulence
    Gravitation, Cosmology & AstrophysicsAtomic, Molecular & OpticalStatistical Physics & Thermodynamics

    Authors & Affiliations

    Aleksandr Chatrchyan1,*,†, Kevin T. Geier1,2,3,*,‡, Markus K. Oberthaler3, Jürgen Berges1, and Philipp Hauke2,1,3

    • 1Institute for Theoretical Physics, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
    • 2INO-CNR BEC Center and Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy
    • 3Kirchhoff Institute for Physics, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

    • *These authors contributed equally to this work.
    • †chatrchyan@thphys.uni-heidelberg.de
    • ‡geier@thphys.uni-heidelberg.de

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    Issue

    Vol. 104, Iss. 2 — August 2021

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    Images

    • Figure 1
      Figure 1

      Schematic illustration of postinflationary reheating dynamics in the early universe and the simulation of an analogous process in an ultracold Bose gas. We consider a scenario where a single-component homogeneous “inflaton” field oscillates around the minimum of its potential, producing particles via parametric instabilities (“preheating”). Later the system enters a turbulent state where energy is transported towards higher momenta in a self-similar way as the universe approaches thermal equilibrium (“reheating”). In the simulation, expanding spacetime as well as the oscillating inflaton field are mimicked in the nonrelativistic limit by modulating the scattering length of a BEC, whose excitations play the role of particles produced by the decaying inflaton.

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    • Figure 2
      Figure 2

      Radially averaged momentum distribution f(t,p) as a function of the radial momentum p=|p| at different times t, demonstrating preheating dynamics. The coupling is modulated with relative strength r=0.25 at a frequency ω, chosen such that the resonance condition εpres=ℏω/2 for the momentum pres=12×pL with pL=2πℏ/L is fulfilled [see discussion below Eq. (16)]. At early times t≲30×t0, a single narrow resonance can be observed around pres. At later times, a broad resonance band emerges with peaks at higher harmonics of the modulation frequency. These secondary resonances are due to nonlinear interactions, as discussed in Sec. 3e. See the video in the Supplemental Material for a qualitative illustration of the dynamics of a single realization [56].

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    • Figure 3
      Figure 3

      Radially averaged momentum distribution f(t,p) as a function of time t showing the exponential growth of the primary and secondary resonances corresponding to the annotated peaks in Fig. 2. We have extracted the growth rates by fitting a straight line to the quantity lnf(t,p), as shown in the insets. The resulting growth rate of the primary resonance ζnum(1)=0.13×t0−1 agrees well with the analytical prediction 2ζpert(1)=0.15×t0−1 obtained from Eq. (17). The growth of the secondary instability starts later, but its rate ζnum(2)=0.23×t0−1 is approximately twice as large as the one of the primary resonance, as expected from our discussion in Sec. 3e. The exponential growth stops when the number of excited atoms becomes comparable to the number of condensate atoms.

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    • Figure 4
      Figure 4

      Self-similar time evolution of the momentum distribution in form of a direct energy cascade for driven (a) and free turbulence (b). Energy is injected at low momenta by modulating the scattering length according to Eq. (11) with a relative amplitude r=1 at a frequency ω chosen such that ℏω/2=εpres with pres=3×2πℏ/L. In the case of continuous modulation (a), a stationary distribution with a power law close to p−2 develops, whose front is evolving self-similarly. If the driving is switched off once the primary resonance has saturated, corresponding here to t=80×t0 (b), energy is propagated in a self-similar way to higher momenta, but the distribution at a given momentum decreases with time, reflecting energy conservation. A power law proportional to p−2 is shown in form of a dotted line as a guide to the eye. The insets show the distributions rescaled according to Eq. (21) using the numerically extracted scaling exponents displayed below the respective distributions.

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    • Figure 5
      Figure 5

      Prescaling at the transition from driven to free turbulence. The simulation parameters are identical to those in Fig. 4, but the modulation is switched off suddenly at a later time t=256×t0. Before this time (blue curves), turbulence is driven and both the momentum distribution as well as the scaling exponents α and β are the same as in Fig. 4. After switching off the modulation (red curves), the ratio α/β quickly changes to the one expected for free turbulence, reflecting energy conservation. The exponent β gradually changes towards the value obtained for free turbulence in Fig. 4, reducing the speed of energy transport in the cascade. Although the scaling exponents still change in time, the distribution has already attained its universal scaling form. This important hallmark of prescaling is indicated in the inset, where all data points collapse to a single curve after rescaling according to Eq. (21) with the extracted time-dependent scaling exponents α(t) and β(t), as described in Appendix pp7.

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    • Figure 6
      Figure 6

      Freeze-out of the momentum distribution of a system with power-law expansion according to Eq. (38) with H0=0.0045×t0−1 and ν=2/3. The numerical data are the same as in Fig. 5, but the laboratory time t has been transformed back to the cosmic time τ by virtue of Eq. (40). As shown in the inset, the cosmic time diverges at the finite laboratory time t(τ=∞)≈444×t0, which is, in particular, shorter than the reheating time of the associated simulating system. As a result, the evolution of the simulated expanding system slows down and eventually freezes before thermalization is complete.

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    • Figure 7
      Figure 7

      Snapshot of the radially averaged, rescaled momentum distribution for the driven turbulent cascade [cf. Fig. 4] at time t=640×t0 for different values of the coupling g̃. The particle number is chosen according to N(g̃)=Nref×(g̃/g̃ref)−1, such that the product Ng̃=Nrefg̃ref remains constant. The vertical dashed line marks the characteristic momentum (cf. Sec. 4d), calculated for the distribution corresponding to the smallest value of g̃. The horizontal dotted lines mark the location of the “quantum half” after rescaling. Within the range of validity of classical-statistical simulations, the rescaled distributions are expected to collapse to a single curve. Deviations can be observed for g̃/g̃ref≳0.1, indicating a breakdown of the method for larger values of the coupling.

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    • Figure 8
      Figure 8

      Diagrammatic illustration of the local (left) and nonlocal (right) one-loop contributions to the self-energy. The second diagram accounts for secondary instabilities.

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    • Figure 9
      Figure 9

      Scaling functions sα(t) (upper panel) and sβ(t) (lower panel) with respect to the reference time tref=85×t0 extracted from the moments of orders 1≤n1<n2≤4. The vertical dotted line marks the time t≈458.6×t0 when the modulation is switched off instantaneously. Before this point, the oscillatory behavior of the moments is directly reflected in the evolution of the scaling functions. As driven turbulence develops, their time averages approach power laws with exponents close to the predictions from kinetic theory, αdriven=−1 and βdriven=−1/2 (dashed lines). After the modulation is switched off, the scaling functions exhibit a kink and the oscillations vanish. In the subsequent evolution, the scaling functions extracted from different moments evolve asynchronously, until they adopt a power-law behavior again for times t≳1000×t0 with exponents close to the predictions from kinetic theory in the regime of free turbulence, αfree=−1 and βfree=−1/4 (dashed-dotted lines). The system evolves self-similarly where all curves have the same slope, as analyzed in Fig. 10.

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    • Figure 10
      Figure 10

      Time-dependent scaling exponents extracted from the moments of orders 1≤n1<n2≤4. The inset in the upper panel shows the instantaneous exponent β(t), which strongly oscillates due to the modulation. The time-averaged quantities β¯(t) and α¯(t)/β¯(t) are displayed in the main plots in the upper and lower panel, respectively. The data have been smoothed using simple moving means and the shaded regions show the corresponding moving standard deviations. The vertical dotted line represents the time when the modulation is switched off. Before this point, in the regime of driven turbulence, the exponents are approximately constant and close to the analytical predictions from kinetic theory, βdriven=−1/2 and αdriven/βdriven=2 (dashed lines). After the modulation is stopped, the exponents jump discontinuously and the exponents extracted from different combinations of moments exhibit discrepancies. For t≳1000×t0, they converge and continue evolving as a single curve, certifying the existence of a prescaling regime of self-similar time evolution. The exponent β(t) gradually approaches the universal value βfree=−1/4 (dashed-dotted line in the upper panel) predicted from kinetic theory in the regime of free turbulence, while the ratio of the exponents quickly adjusts to the prediction αfree/βfree=4 (dashed-dotted line in the lower panel), reflecting energy conservation.

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