Abstract
Light-matter coupling involving classical and quantum light offers a wide range of possibilities to tune the electronic properties of correlated quantum materials. Two paradigmatic results are the dynamical localization of electrons and the ultrafast control of spin dynamics, which have been discussed within classical Floquet engineering and in the deep quantum regime where vacuum fluctuations modify the properties of materials. Here we discuss how these two extreme limits are interpolated by a cavity which is driven to the excited states. In particular, this is achieved by formulating a Schrieffer-Wolff transformation for the cavity-coupled system, which is mathematically analogous to its Floquet counterpart. Some of the extraordinary results of Floquet-engineering, such as the sign reversal of the exchange interaction or electronic tunneling, which are not obtained by coupling to a dark cavity, can already be realized with a single-photon state (no coherent states are needed). The analytic results are verified and extended with numerical simulations on a two-site Hubbard model coupled to a driven cavity mode. Our results generalize the well-established Floquet-engineering of correlated electrons to the regime of quantum light. It opens up a new pathway of controlling properties of quantum materials with high tunability and low energy dissipation.
Highlights
Under a time-periodic perturbation, such as the electric field of a laser or a coherently excited phonon, the time evolution and steady states of a quantum system are described by an effective time-independent Floquet Hamiltonian HF, which can be entirely different from the undriven one
One might assume that the classically driven Floquet limit is recovered only when the cavity is put in a coherent state, but, as we will explain in this work, this is not generally true: At strong light-matter coupling, a Hamiltonian similar to the Floquet Hamiltonian can be engineered by putting the cavity in a given photon-number state, while a coherent state will lead to a more complicated dynamics which is not described by a single effective matter Hamiltonian. We address this fundamental question by demonstrating the crossover from cavity coupling to coherent Floquet engineering for two important classes of Floquet problems: (i) the renormalization of tunneling [47,48], which underlies the Floquet bandstructure control, and (ii) effective induced interactions such as kinetic spin exchange emerging from mobile electrons with Coulomb repulsion, which can be obtained from the Schrieffer-Wolff transformation
The Schrieffer-Wolff transformation is mathematically similar for different systems, and we investigate it for the paradigmatic example of the spin exchange interaction
Summary
Under a time-periodic perturbation, such as the electric field of a laser or a coherently excited phonon, the time evolution and steady states of a quantum system are described by an effective time-independent Floquet Hamiltonian HF , which can be entirely different from the undriven one. Cavity quantum-electrodynamical environments provide a new paradigm for using light-matter interactions for the creation of effective Hamiltonians with tunable interactions, with intriguing proposals ranging from light-induced superconducting pairing to magnetic super-exchange or ferroelectricity [38,39,40,41,42,43,44,45,46] While such a control of many-body interactions has been discussed mostly in the deep quantum limit, where vacuum fluctuations alone affect the solid, one can anticipate that driving the cavity state out of equilibrium implies a continuous crossover to the classical limit of Floquet engineering.
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