Synthesizing lattice structures in phase space

Guo, Lingzhen; Marthaler, Michael

doi:10.1088/1367-2630/18/2/023006

Cited by 42 publications

(48 citation statements)

References 56 publications

(92 reference statements)

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“…The effective interaction coefficients U ij depend on the atomic s-wave scattering length g 0 , shapes of | ( )| w x t , i 2 and how densities of different Wannier wavepackets overlap in the course of time evolution on the classical resonant orbit. Despite the fact that the original interactions between ultra-cold atoms are contact, the effective interactions can be long-range [35,42,93]. Moreover, they can be controlled by changing the s-wave scattering length in space and periodically in time by means of a Feshbach resonance, i.e.…”

Section: A Time Crystal Haldane Phasementioning

confidence: 99%

“…This kind of spontaneous formation of a new crystalline structure in time is dubbed discrete or Floquet time crystals and has been already realized experimentally [30][31][32][33][34]. However, periodically driven systems can also reveal a whole variety of condensed matter phases in the time domain even if no spontaneous process is involved in the emergence of such crystalline structures in time [28,[35][36][37][38][39][40] (see [41][42][43][44] for phase space crystals). Indeed, if a single-or many-body system is driven resonantly, its resonant dynamics can be reduced to solid state-like behavior and importantly such condensed matter physics emerges not in the configuration space but in the time domain-e.g.…”

Section: Introductionmentioning

confidence: 99%

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Topological time crystals

et al. 2019

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By analogy with the formation of space crystals, crystalline structures can also appear in the time domain. While in the case of space crystals we often ask about periodic arrangements of atoms in space at a moment of a detection, in time crystals the role of space and time is exchanged. That is, we fix a space point and ask if the probability density for detection of a system at this point behaves periodically in time. Here, we show that in periodically driven systems it is possible to realize topological insulators, which can be observed in time. The bulk-edge correspondence is related to the edge in time, where edge states localize. We focus on two examples: Su-Schrieffer-Heeger model in time and Bose Haldane insulator which emerges in the dynamics of a periodically driven many-body system. 5 Note that due to the negative effective mass m eff , the first energy band is the highest in energy. 2 New J. Phys. 21 (2019) 052003 where = ¢ J J i 2and J 2i−1 =J, which is identical to the SSH model [86]. The latter describes spinless fermions hopping on a 1D-lattice with staggered hopping amplitudes. Changing the ratio λ 1 /λ in (1), allows one to control the ratio ¢ J J . This effective Hamiltonian belongs to the BDI class of the periodic table of the topological insulators and superconductors [87] and is characterized by a  topological invariant, the winding number ν. For an infinite system with ¢ > J J ( ¢ < J J ), the system is in a topological (trivial) phase with winding number ν=1 (ν=0). For a finite system, the topological phase exhibits zero energy edge states protected by the topology of the bulk. The SSH model has been experimentally realized in quantum simulators and both the presence of edge states and the winding number have been measured [63][64][65]. Let us emphasize that the Wannier states w i (x, t) of equation (2) are localized wavepackets of the effective SSH Hamiltonian of equation (3). We then discuss how such states allow one to detect the topology of the SSH Hamiltonian. corresponding to quasi-energies closest to zero, see top panel. In the topological phase, these eigenstates have zero quasi-energy and are linear combinations of the edge states localized at the edge of time. That is, in the laboratory frame, a detector is placed close to the oscillating mirror (x≈0) and the probability density of clicking of the detector is shown versus time for different values of the ratio ¢ J J of the tunneling amplitudes in (3). This behavior is repeated with the period 2π/ω. At sωt/(2π)=1 and sωt/(2π)=40 there are edges where the eigenstate localizes if ¢ > J J 1. The results correspond to ω=0.067, λ=0.06 in (1) and are obtained within the quantum secular approach [84], see the appendix.

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Section: A Time Crystal Haldane Phasementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Topological time crystals

et al. 2019

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“…where k B is the Boltzmann constant and T eff is the effective temperature in the rotating frame. In [55], we have investigated a similar driven system and showed that the effective temperature T eff equals to the real temperature if the local ground states near the stable points of phase space lattice are the standard coherent states. Actually, the positive and negative sublattices shown in figure 2(a) make no difference in the frame of Floquet theory.…”

Section: T H X P T H X P H X P T H X P T H X P H X Pmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2018

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Hamiltonians which are inaccessible in static systems can be engineered in periodically driven manybody systems, i.e., Floquet many-body systems. We propose to use interacting particles in a onedimensional (1D) harmonic potential with periodic kicking to investigate two-dimensional topological and many-body physics. Depending on the driving parameters, the Floquet Hamiltonian of single kicked harmonic oscillator has various lattice structures in phase space. The noncommutative geometry of phase space gives rise to the topology of the system. We investigate the effective interactions of particles in phase space and find that the point-like contact interaction in quasi-1D real space becomes a long-rang Coulomb-like interaction in phase space, while the hardcore interaction in pure-1D real space becomes a confinement quark-like potential in phase space. We also find that the Floquet exchange interaction does not disappear even in the classical limit, and can be viewed as an effective long-range spin-spin interaction induced by collision. Our proposal may provide platforms to explore new physics and exotic phases by Floquet many-body engineering.

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Phase Space Crystals

Sacha

2020

Time Crystals

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Synthesizing lattice structures in phase space

Cited by 42 publications

References 56 publications

Topological time crystals

Topological time crystals

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

Phase Space Crystals

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