Effective Hamiltonians for periodically driven systems

Rahav, Saar; Gilary, Ido; Fishman, Shmuel

doi:10.1103/physreva.68.013820

Cited by 245 publications

(342 citation statements)

References 46 publications

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“…The effective HamiltonianĤ eff and kick operator K(t) can be systematically computed using a perturbative expansion [19,21] in powers of 1/ω, assuming that the Hamiltonian H(t) remains finite in the limit ω → ∞.…”

Section: General Frameworkmentioning

confidence: 99%

“…This approach is rooted in the fact that the dynamics associated with time-dependent Hamiltonians can be well captured by a time-independent effective HamiltonianĤ eff , in the limit of infinitely small driving period T t ch , where t ch denotes a typical time-scale for the dynamical properties under scrutiny, see Refs. [4,5,[18][19][20][21][22]. In this picture, the energy spectrum of the static system is replaced by the (Floquet) spectrum associated withĤ eff , which can potentially present interesting features, such as topological properties.…”

Section: Introductionmentioning

confidence: 99%

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Periodically driven quantum matter: The case of resonant modulations

et al. 2015

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Quantum systems can show qualitatively new forms of behavior when they are driven by fast time-periodic modulations. In the limit of large driving frequency, the long-time dynamics of such systems can often be described by a time-independent effective Hamiltonian, which is generally identified through a perturbative treatment. Here, we present a general formalism that describes time-modulated physical systems, in which the driving frequency is large, but resonant with respect to energy spacings inherent to the system at rest. Such a situation is currently exploited in optical-lattice setups, where superlattice (or Wannier-Stark-ladder) potentials are resonantly modulated so as to control the tunneling matrix elements between lattice sites, offering a powerful method to generate artificial fluxes for cold-atom systems. The formalism developed in this work identifies the basic ingredients needed to generate interesting flux patterns and band structures using resonant modulations. Additionally, our approach allows for a simple description of the micro-motion underlying the dynamics; we illustrate its characteristics based on diverse dynamic-lattice configurations. It is shown that the impact of the micro-motion on physical observables strongly depends on the implemented scheme, suggesting that a theoretical description in terms of the effective Hamiltonian alone is generally not sufficient to capture the full time-evolution of the system.

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Section: General Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Periodically driven quantum matter: The case of resonant modulations

et al. 2015

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“…Away from halffilling this Hamiltonian reduces to the t−J model [2,39]. Using the HFE to perform the SWT offers a few advantages: (i) the SW generator comes naturally out of the calculation, (ii) one can systematically compute higherorder corrections [33][34][35][36][37][38]40], and (iii) the HFE allows for obtaining not only the effective Hamiltonian but also the kick operator, which keeps track of the mixing between orbitals and describes the intra-period dynamics [34,40]. This is important for identifying the fast timescale associated with the large frequency U in dynamical measurements [41], and expressing observables through creation and annihilation operators dressed by orbital mixing [40].…”

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confidence: 99%

Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields

2016

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We generalize the Schrieffer-Wolff transformation to periodically driven systems using Floquet theory. The method is applied to the periodically driven, strongly interacting Fermi-Hubbard model, for which we identify two regimes resulting in different effective low-energy Hamiltonians. In the nonresonant regime, we realize an interacting spin model coupled to a static gauge field with a nonzero flux per plaquette. In the resonant regime, where the Hubbard interaction is a multiple of the driving frequency, we derive an effective Hamiltonian featuring doublon association and dissociation processes. The ground state of this Hamiltonian undergoes a phase transition between an ordered phase and a gapless Luttinger liquid phase. One can tune the system between different phases by changing the amplitude of the periodic drive.

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“…A common approach in the analysis of periodically driven quantum systems is to search for a timeindependent effective Hamiltonian with an energy spectrum approximating the quasienergies of the Floquet Hamiltonian of the system [9][10][11]. The accomplishment of this task typically requires to restrict oneself to specific classes of driving operators.…”

Section: Introductionmentioning

confidence: 99%

Effective time-independent description of optical lattices with periodic driving

Hemmerich

2010

Phys. Rev. A

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For a periodically driven quantum system an effective time-independent Hamiltonian is derived with an eigen-energy spectrum, which in the regime of large driving frequencies approximates the quasi-energies of the corresponding Floquet Hamiltonian. The effective Hamiltonian is evaluated for the case of optical lattice models in the tight-binding regime subjected to strong periodic driving. Three scenarios are considered: a periodically shifted one-dimensional (1D) lattice, a twodimensional (2D) square lattice with inversely phased temporal modulation of the well depths of adjacent lattice sites, and a 2D lattice subjected to an array of microscopic rotors commensurate with its plaquette structure. In case of the 1D scenario the rescaling of the tunneling energy, previously considered by Eckardt et al. in Phys. Rev. Lett. 95, 260404 (2005), is reproduced. The 2D lattice with well depth modulation turns out as a generalization of the 1D case. In the 2D case with staggered rotation, the expression previously found in the case of weak driving by Lim et al. in Phys. Rev. Lett. 100, 130402 (2008) is generalized, such that its interpretation in terms of an artificial staggered magnetic field can be extended into the regime of strong driving.

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Effective Hamiltonians for periodically driven systems

Cited by 245 publications

References 46 publications

Periodically driven quantum matter: The case of resonant modulations

Periodically driven quantum matter: The case of resonant modulations

Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields

Effective time-independent description of optical lattices with periodic driving

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