Smits, Stoof, and van der Straten [Phys. Rev. A 104, 023318 (2021)] observed spontaneous breaking of a Z2\mathbb{Z}_2 symmetry in a driven-dissipative Bose-Einstein condensate: a parametrically driven high-order axial collective mode settles, run by run with equal probability, into one of two phase lags, ϕ\phi or ϕ+π\phi+\pi, relative to the drive — a discrete time ('space-time') crystal. The effective model rests on a phenomenological, fitted complex fourth-order coupling gg and a semi-classical Langevin treatment of fluctuations. The paper's conclusion poses the follow-up agenda: induce tunneling of the phase lag between the two values, and drive the system from a superposition of the two phase states into a chosen one with tailored pulses; the companion theory paper explicitly leaves open the microscopic origin of the coupling and of the noise strength. Later literature delivered the fully quantum Fokker–Planck formulation and deterministic switching between period-doubled states in adjacent classical and spin-cavity platforms, but neither coherent manipulation of the two phase states nor a first-principles derivation of the effective coupling has been realized for this driven-dissipative condensate system. This task asks for: a predictive microscopic foundation of the effective model, control over the two symmetry-broken phase states, and a quantified quantum/thermal/technical noise budget.

Background

A Bose-Einstein condensate (BEC) is a dilute ultracold gas in which a macroscopic fraction of the atoms occupies a single quantum state, so that the gas behaves as a coherent quantum fluid whose collective excitations can be followed in real time. In the experiment of Smits, Stoof, and van der Straten [Phys. Rev. A 104, 023318 (2021)], about 5×1065\times10^6 sodium atoms were confined in a strongly elongated magnetic trap with radial and axial trapping frequencies 2π×(92,5)2\pi\times(92,5) Hz. A short modulation of the radial confinement excited the radial breathing mode at ωD/2π≃183.7\omega_D/2\pi\simeq183.7 Hz, close to twice the frequency of a high-order axial collective mode (≃92\simeq92 Hz). The breathing oscillation therefore acts as a parametric drive of the axial mode — the many-body analogue of a degenerate parametric amplifier, in which a pump at frequency ωD\omega_D resonantly excites an oscillation at ωD/2\omega_D/2. The axial mode was monitored non-destructively by off-axis holographic imaging (an interferometric technique that measures the cloud's phase shift while destroying almost no atoms), so the mode amplitude and phase could be followed along each experimental run.

The driven mode settles into a steady subharmonic oscillation whose phase lag ϕ\phi with respect to the drive takes one of two values, ϕ\phi and ϕ+π\phi+\pi, that are exactly degenerate because the drive couples to the square of the mode amplitude. Across 140 identical runs, 69 settled near lag 0 and 71 near π\pi (p=0.49±0.04p=0.49\pm0.04), showing that the choice is made spontaneously. A periodically driven many-body system that responds robustly at an integer fraction of the drive period is called a discrete time crystal, because it breaks the discrete time-translation symmetry of the drive; realized in a superfluid, it is also called a space-time crystal.

The mode is described by a single bosonic annihilation operator a^\hat a obeying, in the frame rotating at half the drive frequency, the effective Hamiltonian

H^=−ℏδ a^†a^+ℏωDAD8(a^†2+a^2)+ℏg2 a^†2a^2,\hat H=-\hbar\delta\,\hat a^\dagger\hat a+\frac{\hbar\omega_D A_D}{8}\left(\hat a^{\dagger 2}+\hat a^2\right)+\frac{\hbar g}{2}\,\hat a^{\dagger 2}\hat a^2,

with δ\delta the detuning from parametric resonance, ADA_D the drive amplitude, and g=g′+ig′′g=g'+ig'' a complex effective fourth-order interaction parameter: the imaginary part g′′g'' encodes nonlinear dissipation caused by the coupling of the condensate mode to the surrounding thermal cloud, and the real part g′g' the associated conservative frequency shift. The Hamiltonian is invariant under a^→−a^\hat a\to-\hat a, a Z2\mathbb{Z}_2 symmetry (invariance under a sign flip, with exactly two symmetry-related configurations), so a nonzero steady expectation value ⟨a^⟩\langle\hat a\rangle of either sign — corresponding to the two phase lags — breaks the symmetry. Fluctuations are included semi-classically through a Langevin equation (a stochastic differential equation for the classical amplitude a(t)a(t) containing a random force) with Stratonovich multiplicative noise (noise whose strength scales with ∣a∣|a| itself), whose correlation strength is fixed by g′′g'' through the fluctuation-dissipation theorem, the requirement that dissipation and its accompanying fluctuations share one microscopic origin. The corresponding Fokker–Planck equation — the partial differential equation for the probability distribution P(a∗,a;t)P(a^*,a;t) — keeps the ensemble distribution symmetric: each trajectory picks a phase, but the ensemble average never does. Technical noise (run-to-run preparation fluctuations, modeled as Gaussian spreads of δ\delta and gg) broadens the outcome distribution without breaking the symmetry.

Two limitations of this description motivate the open problem. First, gg is phenomenological: in the precursor experiment [New J. Phys. 22, 105001 (2020)] the model was fitted to the observed growth curves, giving ∣g∣=0.19(25)×10−4 s−1|g|=0.19(25)\times10^{-4}\ \mathrm{s}^{-1} at fixed AD=0.1A_D=0.1, and no microscopic derivation of g′g' and g′′g'' — or of their dependence on temperature, atom number, and mode structure — exists. The fluctuation budget is dominated by thermal occupation (about 45 initial thermal quanta, and Nfluc≃456N_{\mathrm{fluc}}\simeq456 effective quanta in the growth stage, against a quantum zero-point contribution of 1/21/2). Second, the two symmetry-broken states have only ever been observed to be chosen spontaneously: the paper's conclusion proposes to induce tunneling of the phase lag between the two values — building on the instanton-like tunneling excitations between degenerate discrete-time-crystal phases described by Yang and Cai — and to drive the system from a linear superposition of both phase lags to one particular phase with a tailored π/2\pi/2-type pulse, in analogy with the control pulses of spin resonance. The companion theory paper of the same groups [Phys. Rev. A 104, 043324 (2021)] formulated the semi-classical and fully quantum Fokker–Planck descriptions and explicitly listed the first-principles origin of the coupling and the tunneling dynamics as open ends.

Problem Statement

For the driven-dissipative space-time crystal realized in the parametrically driven Bose-Einstein condensate — where the high-order axial mode spontaneously breaks the Z2\mathbb{Z}_2 symmetry of the drive by settling into phase lag ϕ\phi or ϕ+π\phi+\pi — the unfinished agenda posed by the source experiment and its companion theory is to replace the present phenomenological, passively observed symmetry breaking by a first-principles and actively controlled one. The objective comprises three jointly supporting results:

  1. Microscopic foundation of the effective model. Starting from an explicitly stated microscopic description of the trapped condensate, its thermal cloud, and the drive, derive the effective coupling g=g′+ig′′g=g'+ig'' and the associated noise correlations, including their dependence on temperature, atom number, and mode structure, so that the nonlinear dissipation and fluctuation strengths entering the Langevin/Fokker–Planck description follow from first principles rather than from a fit; state the regime of validity of the semi-classical approximation and confront the derived values with the measured ones.

  2. Controlled manipulation of the two symmetry-broken phase states. Realize and quantify, in this driven-dissipative condensate platform, protocols that (a) switch the phase lag between ϕ\phi and ϕ+π\phi+\pi and (b) prepare a chosen phase from an initially balanced superposition or mixture of the two phase lags, as proposed in the source conclusion. The control operations are quantified within the platform's Langevin/Fokker–Planck or quantum description, validated against the published experimental data on the spontaneous symmetry-breaking dynamics (mode growth and outcome spread); the switching or preparation probabilities are quantified over repeated runs and accounted for quantitatively by the theoretical description.

  3. Noise budget of the symmetry breaking and of the control operations. As part of the results of items 1 and 2, quantify the relative roles of quantum, thermal, and technical fluctuations in the symmetry-breaking dynamics and in the manipulation protocols, including how these roles change with temperature and drive strength as follows from the derived coupling and noise correlations.

These three results jointly deliver the common objective: a quantitative, predictive, and controllable account of spontaneous Z2\mathbb{Z}_2 symmetry breaking in this driven-dissipative system.

References

  1. J. Smits, H. T. C. Stoof, and P. van der Straten, 'Spontaneous breaking of a discrete time-translation symmetry' (preprint title: 'Spontaneous symmetry breaking in a driven-dissipative system'), Phys. Rev. A 104, 023318 (2021). DOI: 10.1103/PhysRevA.104.023318; arXiv:2103.03748, https://arxiv.org/abs/2103.03748
  2. J. Smits, H. T. C. Stoof, and P. van der Straten, 'On the long-term stability of space-time crystals', New J. Phys. 22, 105001 (2020). DOI: 10.1088/1367-2630/abbae9; arXiv:2007.07038, https://arxiv.org/abs/2007.07038
  3. J. N. Stehouwer, H. T. C. Stoof, J. Smits, and P. van der Straten, 'Dynamics of spontaneous symmetry breaking in a space-time crystal', Phys. Rev. A 104, 043324 (2021). DOI: 10.1103/PhysRevA.104.043324; arXiv:2107.14505, https://arxiv.org/abs/2107.14505
  4. Xiaoqin Yang and Zi Cai, 'Dynamical Transitions and Critical Behavior between Discrete Time Crystal Phases', Phys. Rev. Lett. 126, 020602 (2021). DOI: 10.1103/PhysRevLett.126.020602; arXiv:2007.08348, https://arxiv.org/abs/2007.08348
  5. Benjamin Apffel and Romain Fleury, 'Experimental observation of topological transition in linear and nonlinear parametric oscillators', Phys. Rev. E 109, 054204 (2024). DOI: 10.1103/PhysRevE.109.054204, https://doi.org/10.1103/PhysRevE.109.054204
  6. Roy D. Jara, Phatthamon Kongkhambut, Hans Keßler, Andreas Hemmerich, and Jayson G. Cosme, 'Controlled bit flip of period-doubling and discrete time crystalline states in open systems', Phys. Rev. B 112, 024303 (2025). DOI: 10.1103/3l49-xdt1; arXiv:2504.04900, https://arxiv.org/abs/2504.04900
  7. Emmanuel D. G. U, Roy D. Jara Jr., and Jayson G. Cosme, 'Universal logic gates for coupled period-doubling systems', arXiv:2605.19477 (2026). https://arxiv.org/abs/2605.19477
  8. Richelle Jade L. Tuquero and Jayson G. Cosme, 'Impact of quantum noise on phase transitions in an atom-cavity system with limit cycles', Phys. Rev. A 110, 063314 (2024). DOI: 10.1103/PhysRevA.110.063314; arXiv:2407.21390, https://arxiv.org/abs/2407.21390
  9. Hans Keßler, Phatthamon Kongkhambut, Christoph Georges, Ludwig Mathey, Jayson G. Cosme, and Andreas Hemmerich, 'Observation of a Dissipative Time Crystal', Phys. Rev. Lett. 127, 043602 (2021). DOI: 10.1103/PhysRevLett.127.043602; arXiv:2012.08885, https://arxiv.org/abs/2012.08885