Time-translation-symmetry breaking in a driven oscillator: From the quantum coherent to the incoherent regime

Abstract

We study the breaking of the discrete time-translation symmetry in small periodically driven quantum systems. Such systems are intermediate between large closed systems and small dissipative systems, which both display the symmetry breaking, but have qualitatively different dynamics. As a nontrivial example we consider period tripling in a quantum nonlinear oscillator. We show that, for moderately strong driving, the period tripling is robust on an exponentially long time scale, which is further extended by an even weak decoherence.The breaking of translation symmetry in time, first proposed by Wilczek [1], has been attracting much attention recently. Such symmetry breaking can occur only away from thermal equilibrium [2]. It is of particular interest for periodically driven systems, which have a discrete time-translation symmetry imposed by the driving. Here, the time symmetry breaking is manifested in the onset of oscillations with a period that is a multiple of the driving period t F . Oscillations with period 2t F due to simultaneously initialized protected boundary states were studied in photonic quantum walks [3]; period-two oscillations can also be expected from the coexistence of Floquet Majorana fermions with quasienergies 0 and π/t F in a cold-atom system [4]. The onset of periodtwo phases was predicted and analyzed [5][6][7][8][9][10] in Floquet many-body localized systems, and the first observations of oscillations at multiples of the driving period in disordered systems were reported [11,12].In systems coupled to a thermal bath, on the other hand, the effect of period doubling has been well-known. A textbook example is a classical oscillator modulated close to twice its eigenfrequency and displaying vibrations with period 2t F [13]. The oscillator has two states of such vibrations; they have opposite phases, reminiscent of a ferromagnet with two orientations of the magnetization. Several aspects of the dynamics of a parametric oscillator in the quantum regime have been studied theoretically, cf. [14][15][16][17][18][19][20][21], and in experiments, cf. [22][23][24]. For a sufficiently strong driving field, a quantum dissipative oscillator, like a classical oscillator, mostly performs vibrations with period 2t F . The interplay of quantum fluctuations and dissipation leads to transitions between the period-two vibrational states, but the rate of these transitions is exponentially small [18].The goal of this paper is to study time symmetry breaking in isolated or almost isolated driven quantum systems with a few degrees of freedom. They are intermediate between large closed systems and dissipative systems, where the nature of the symmetry breaking is very different. To this end, we analyze a driven nonlinear quantum oscillator. Time symmetry breaking in this system should not be limited to period doubling. As an illustration of a behavior qualitatively different from period doubling, we consider period tripling and find the conditions where it occurs. We also address the role of decoherence and the co...

Period-tripling subharmonic oscillations in a driven superconducting resonator

“…A detailed description of the eigenfunctions and tunneling rates can be found in Ref. 16 for the Hamiltonian (2) in the single-mode case.…”

“…This difference is analogous to the difference between a first and a second order phase transition 15 . Furthermore, a symmetry breaking aspect of this difference has important implications for the quantum dynamics of the period-tripling oscillations 16 . Although the period-multiplying phenomenon is theoretically explained in textbooks, experimental demonstrations are not common.…”

“…In Josephson circuits, the period-multiplying phenomenon has not received much attention; instead, research was focused on transition to chaos 21,22 and lately on bifurcation phenomena 23,24 and parametric oscillations 5,6,[25][26][27] . Only recently, subharmonic oscillations in the quantum regime were theoretically discussed in the context of cQED 16,28,29 .…”

Floquet many-body engineering: topology and many-body physics in phase space lattices

“…Here, T is the chosen stroboscopic time step and T is the time-ordering operator. Exotic Floquet Hamiltonians [43][44][45][46][47][48][49] which are inaccessible in static systems can be engineered from equation (1.1) and a range of novel physical phenomena, such as Floquet topological physics [50][51][52][53], phase space crystals [54,55], Anderson localization in time domain [56][57][58] and spontaneous breaking of discrete time-translation symmetry (Floquet time crystals) [59][60][61][62][63][64][65][66][67], can be created by Floquet engineering [68][69][70]. While most work focus on the singleparticle physics of (dissipative) Floquet systems, the possible new physics by Floquet many-body engineering has become an active research direction in recent years.…”

Preparing quasienergy states on demand: A parametric oscillator

“…Such degeneracy is possible for a parametric oscillator for a finite driving amplitude [29]. A driven oscillator also provides a platform for investigating more complicated cases of RB degeneracy [30]. The understanding of the spectrum of H RWA can be gained by looking at the Hamiltonian function H RWA in the phase space of the oscillator in the rotating frame, i.e., by writing H RWA in terms of the scaled quadratures P and Q defined as Q = i(a − a † ) λ/2, P = (a † + a) λ/2.…”