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arXiv:1406.7757 · Physical Review A

Directed motion of doublons and holes in periodically driven Mott insulators

Maximilian Genske, Achim Rosch

Published Oct 30, 2014• 5 citations
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Abstract

Periodically driven systems can lead to a directed motion of particles. We investigate this ratchet effect for a bosonic Mott insulator where both a staggered hopping and a staggered local potential vary periodically in time. If driving frequencies are smaller than the interaction strength and the density of excitations is small, one obtains effectively a one-particle quantum ratchet describing the motion of doubly occupied sites (doublons) and empty sites (holes). Such a simple quantum machine can be used to manipulate the excitations of the Mott insulator. For suitably chosen parameters, for example, holes and doublons move in opposite directions. To investigate whether the periodic driving can be used to move particles ``uphill,'' i.e., against an external force, we study the influence of a linear potential $\ensuremath{-}gx$. For long times, transport is only possible when the driving frequency $\ensuremath{\omega}$ and the external force $g$ are commensurate, ${n}_{0}g={m}_{0}\ensuremath{\omega}$, with $\frac{{n}_{0}}{2},{m}_{0}\ensuremath{\in}\mathbb{Z}$.

DOI
10.1103/physreva.90.043637

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Periodically driven systems can lead to a directed motion of particles. We investigate this ratchet effect for a bosonic Mott insulator where both a staggered hopping and a staggered local potential vary periodically in time. If driving frequencies are smaller than the interaction strength and the density of excitations is small, one obtains effectively a one-particle quantum ratchet describing the motion of doubly occupied sites (doublons) and empty sites (holes). Such a simple quantum machine can be used to manipulate the excitations of the Mott insulator. For suitably chosen parameters, for example, holes and doublons move in opposite directions. To investigate whether the periodic driving can be used to move particles ``uphill,'' i.e., against an external force, we study the influence of a linear potential \ensuremath−gx\ensuremath{-}gx\ensuremath−gx. For long times, transport is only possible when the driving frequency \ensuremathω\ensuremath{\omega}\ensuremathω and the external force ggg are commensurate, n0g=m0\ensuremathω{n}_{0}g={m}_{0}\ensuremath{\omega}n0​g=m0​\ensuremathω, with n02,m0\ensuremath∈Z\frac{{n}_{0}}{2},{m}_{0}\ensuremath{\in}\mathbb{Z}2n0​​,m0​\ensuremath∈Z.
Controlling and manipulating complex many-body quantum systems will be a key ingredient \nfor the development of next-generation technologies. While the realisation of a \nuniversal quantum machine is still out of reach, in recent years experimental systems of \nultracold atoms have already evolved into a vivid field of research for quantum simulation. \nCrucially, such systems even allow for the successful quantum engineering of targeted \nmany-body systems by means of coherent periodic driving. The essential properties of \nthese Floquet systems encompass two main aspects: fast driving facilitates the simulation \nof effective static systems, and interactions lead to unique heating effects as energy is only \nconserved modulo the driving frequency. Within this thesis we theoretically study both \nof these aspects in respective model systems. \nIn part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott \ninsulator in a designed one-dimensional Floquet system. Here, periodic driving in combination \nwith breaking all mirror symmetries of the system can induce directed motion of \nparticles. In the limit of small excitation densities, the effectively non-interacting quantum \nratchet determines the motion of holes and doublons in the Mott insulator and can in fact \nbe used to manipulate the dynamics of such. This little quantum machine can also be \nused to drive particles against an external force, where transport is possible but requires \nthe fulfilment of a commensurability condition for long times. \nIn part II, we discuss the role of interactions for periodically driven systems by means \nof a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, \nwe develop this semiclassical formalism based on a clear separation of time scales. The \nresult is a description of the dynamics and the scattering of Floquet quasiparticles in \nsuch systems. Here, the property of discrete energy violation is naturally encoded in our \nformalism predicting the heating of interacting Floquet systems to infinite temperatures in \nthe long-time limit. As a first application of this approach, we investigate a cold atom setup \nrealising the Haldane model by means of periodic shaking. While homogeneous systems \nheat up globally, a confining potential evokes thermoelectric transport effects resulting \nfrom spatially dependent heating characteristics. Moreover, we show that the interplay \nof intrinsic heating, macroscopic diffusion and non-trivial topological properties of the \nHaldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous \nparticle transport as the result of developing temperature gradients. \nIn part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing \na theoretical framework for a possible simulation of a topological edge state in a \none-dimensional optical lattice. In this case, the one-dimensional Dirac equation with \nspatially varying mass is important, which captures the topological properties of a corresponding \nsystem of the BDI symmetry class. We analytically discuss such system and \ninvestigate the role of mean-field interaction effects. We also identify the emergence of \ndynamical instabilities in a realisation with bosonic atoms.
Periodically driving strongly correlated systems offers new opportunities to control their properties. Heating effects, however, often hinder the actual realization of interesting states. Here, the authors study a model of a Mott insulator with a time-periodic Hubbard on-site potential UUU coupled to a dissipative reservoir. Tuning the frequency above a characteristic value, they unveil a regime where the creation of doublon-holon excitations is greatly enhanced with respect to equilibrium. Below a certain effective temperature, this far-from-equilibrium state would give rise to the exotic $\ensuremath{\eta}$-pairing superconduting state.
In the strongly interacting limit of the Hubbard model localized double occupancies form effective hard-core bosonic excitations, called doublons, which are long lived due to energy conservation. Using time-dependent density-matrix renormalization group methods we investigate numerically the dynamics of doublons arising from the sudden expansion of a spatially confined band-insulating state in one spatial dimension. By analyzing the occupation scaling of the natural orbitals within the many-body state, we show that doublons dynamically quasicondense at the band edges, consistent with the spontaneous emergence of an $\ensuremath{\eta}$ quasicondensate. Building on this, we study the effect of periodically driving the system during the expansion. Floquet analysis reveals that doublon hopping and doublon repulsion are strongly renormalized by the drive, breaking the $\ensuremath{\eta}\text{\ensuremath{-}}\mathrm{SU}(2)$ symmetry of the Hubbard model. Numerical simulation of the driven expansion dynamics demonstrates that the momentum in which doublons quasicondense can be controlled by the driving amplitude. These results point to new pathways for engineering nonequilibrium condensates in fermionic cold-atom experiments and are potentially relevant to driven solid-state systems.
For the fermionic Hubbard model at strong coupling, we demonstrate that directional transport of localized doublons (repulsively bound pairs of two particles occupying the same site of the crystal lattice) can be achieved by applying an unbiased ac field of time-asymmetric (sawtooth-like) shape. The mechanism involves a transition to intermediate states of virtually zero double occupation which are reached by splitting the doublon by fields of the order of the Hubbard interaction. The process is discussed on the basis of numerically exact calculations for small clusters, and we apply it to more complex states to manipulate the charge order pattern of one-dimensional systems.
Intense light can alter the electronic properties of materials through its oscillating electric field. The effects on magnetic excitations in insulators, however, are indirect, since these are neutral and do not see the electric field directly. This study reports on how exchange interactions in single- and multiorbital Mott insulators are modified by electromagnetic radiation, whose frequency is assumed to be on the order of the Mott-Hubbard gap. The latter condition results in a nonvanishing creation rate of doubly occupied sites (doublons), which necessitates taking into account the finite bandwidth of the doublon excitations, a feature which goes beyond the na\"{\i}ve leading order strong correlation expansion. This ultimately leads to a complex-valued exchange parameter, where the real part represents the ``Floquet'' spin-exchange interaction, while the imaginary part reflects the creation rate of doublons.
Classical ratchet potentials, which alternate a driving potential with periodic random dissipative motion, can account for the operation of biological motors. We demonstrate the operation of a quantum ratchet, which differs from classical ratchets in that dissipative processes are absent within the observation time of the system (Hamiltonian regime). An atomic rubidium Bose-Einstein condensate is exposed to a sawtooth-like optical lattice potential, whose amplitude is periodically modulated in time. The ratchet transport arises from broken spatiotemporal symmetries of the driven potential, resulting in a desymmetrization of transporting eigenstates (Floquet states). The full quantum character of the ratchet transport was demonstrated by the measured atomic current oscillating around a nonzero stationary value at longer observation times, resonances occurring at positions determined by the photon recoil, and dependence of the transport current on the initial phase of the driving potential.