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arXiv:2107.09683 · Physical Review Research

Emergent parametric resonances and time-crystal phases in driven Bardeen-Cooper-Schrieffer systems

H. P. Ojeda Collado, Gonzalo Usaj, C. A. Balseiro, Damián H. Zanette, J. Lorenzana

Published Nov 10, 2021• 38 citations
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Abstract

We study the out-of-equilibrium dynamics of a Bardeen-Cooper-Schrieffer condensate subject to a periodic drive.We demonstrate that the combined effect of drive and interactions results in emerging parametric resonances, analogous to a vertically driving pendulum.In particular, Arnold tongues appear when the driving frequency matches 2 0 /n, with n being a natural number and 0 being the equilibrium gap parameter.Inside the Arnold tongues we find a commensurate time-crystal condensate which retains the U(1) symmetry breaking of the parent superfluid/superconducting phase and shows an additional time-translational symmetry breaking.Outside these tongues, the synchronized collective Higgs mode found in quench protocols is stabilized without the need of a strong perturbation.Our results are directly relevant to cold-atom and condensed-matter systems and do not require very long energy relaxation times to be observed.

DOI
10.1103/physrevresearch.3.l042023

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We study the out-of-equilibrium dynamics of a Bardeen-Cooper-Schrieffer condensate subject to a periodic drive.We demonstrate that the combined effect of drive and interactions results in emerging parametric resonances, analogous to a vertically driving pendulum.In particular, Arnold tongues appear when the driving frequency matches 2 0 /n, with n being a natural number and 0 being the equilibrium gap parameter.Inside the Arnold tongues we find a commensurate time-crystal condensate which retains the U(1) symmetry breaking of the parent superfluid/superconducting phase and shows an additional time-translational symmetry breaking.Outside these tongues, the synchronized collective Higgs mode found in quench protocols is stabilized without the need of a strong perturbation.Our results are directly relevant to cold-atom and condensed-matter systems and do not require very long energy relaxation times to be observed.
We present a systematic study of the dynamical phase diagram of a periodically driven BCS system as a function of drive strength and frequency. Three different driving mechanisms are considered and compared: oscillating density of states, oscillating pairing interaction, and oscillating external paring field. We identify the locus in parameter space of parametric resonances and four dynamical phases: Rabi-Higgs, gapless, synchronized Higgs, and time-crystal phases. We demonstrate that the main features of the phase diagram are quite robust to different driving protocols and discuss the order of the transitions. By mapping the BCS problem to a collection of nonlinear and interacting classical oscillators, we shed light on the origin of time-crystal phases and parametric resonances appearing for subgap excitations.
The search for exotic temporal orders in quantum matter, such as discrete quasi-time crystals (DQTCs), has become an important theme in nonequilibrium physics. However, realizing these phases has so far required complex protocols, such as drives with multiple incommensurate frequencies. Here, we present a significantly simpler mechanism: the emergence of DQTCs in a dissipative collective spin system subjected to only a single periodic drive. Remarkably, the characteristic frequencies of this novel phase are not fixed but can be continuously tuned by varying the strength of the drive. Even more strikingly, this tunability is punctuated by Arnold tongues, within which the response main frequency locks to rational fractions of the drive. Our model further provides a unified framework that also encompasses stationary, discrete time crystals and chaotic phases. This discovery simplifies the requirements for generating complex temporal orders and opens a viable route towards the experimental control and manipulation of quasi-time crystalline matter.
We generalize the dynamical phase diagram of a Bardeen-Cooper-Schrieffer condensate, considering attractive to repulsive, i.e., critical quenches (CQs) and a nonconstant density of states (DOS). We show that different synchronized Higgs dynamical phases can be stabilized, associated with singularities in the DOS and different quench protocols. In particular, the CQ can stabilize an overlooked high-frequency Higgs dynamical phase related to the upper edge of the fermionic band. For a compensated Dirac system we find a Dirac-Higgs mode associated with the cusp singularity at the Fermi level, and we show that synchronized phases become more pervasive across the phase diagram. The relevance of these remarkable phenomena and their realization in ensembles of fermionic cold atoms confined in optical lattices is also discussed.
Discrete time crystals (DTCs) refer to a novel many-body steady state that spontaneously breaks the discrete time-translational symmetry in a periodically driven quantum system. Here, we study DTCs in a Bose-Einstein condensate bouncing resonantly on an oscillating mirror, using a two-mode model derived from a standard quantum field theory. We investigate the validity of this model and apply it to study the long-time behavior of our system. A wide variety of initial states based on two Wannier modes are considered. We find that in previous studies the investigated phenomena in the evolution time window (⪅2000⪅2000⪅2000 driving periods) are actually ``short-time'' transient behavior though DTC formation signaled by the subharmonic responses is still shown if the interboson interaction is strong enough. After a much longer (about 20 times) evolution time, initial states with no ``long-range'' correlations relax to a steady state, where time-symmetry breaking can be unambiguously defined. Quantum revivals also eventually occur. This long-time behavior can be understood via the many-body Floquet quasieigenenergy spectrum of the two-mode model. A symmetry-breaking edge for DTC formation appears in the spectrum for strong enough interaction, where all quasieigenstates below the edge are symmetry breaking while those above the edge are symmetric. The late-time steady state's time-translational symmetry depends solely on whether the initial energy is above or below the symmetry-breaking edge. A phase diagram showing regions of symmetry-broken and symmetric phases for differing initial energies and interaction strengths is presented. We find that, according to this two-mode model, the discrete time crystal survives for times out to at least 250000250\phantom{\rule{0.16em}{0ex}}000250000 driving periods.
Symmetry-breaking phase transitions are central to our understanding of states of matter. When a continuous symmetry is spontaneously broken, new excitations appear that are tied to fluctuations of the order parameter. In superconductors and fermionic superfluids, the phase and amplitude can fluctuate independently, giving rise to two distinct collective branches. However, amplitude fluctuations are difficult to both generate and measure, as they do not couple directly to the density of fermions and have only been observed indirectly to date. Here, we excite amplitude oscillations in an atomic Fermi gas with resonant interactions by an interaction quench. Exploiting the sensitivity of Bragg spectroscopy to the amplitude of the order parameter, we measure the time-resolved response of the atom cloud, directly revealing amplitude oscillations at twice the frequency of the gap. The magnitude of the oscillatory response shows a strong temperature dependence, and the oscillations appear to decay faster than predicted by time-dependent Bardeen-Cooper-Schrieffer theory applied to our experimental setup.
We investigate the dynamical phases that emerge in collective spin models subjected to a spatially non-uniform periodic drive. Taking the paradigmatic Lipkin-Meshkov-Glick (LMG) model as a concrete platform, we establish that a rich landscape of dynamical phases emerges when two regions of the system are driven with different field strengths, h1h_1h1​ and h2h_2h2​. Remarkably, despite the `all-to-all' nature of the interactions, the system can be driven into dynamical phases characterized by distinct kinds of discrete time crystal (DTC) orders in different parts of the system. Apart from these coexisting DTCs, tuning the driving field leads to the emergence of phases where DTCs coexist with Floquet-synchronized or oscillatory phases; the former has been dubbed a chimera DTC. Finally, we demonstrate that a tunable set of global DTC phases emerges when h1h_1h1​ and h2h_2h2​ are proximate. Crucially, these dynamical regimes can be observed both for experimentally relevant finite-size systems and in the thermodynamic limit. Our results establish spatially structured driving as a powerful route to realize non-equilibrium phase coexistence in collective spin systems.