Periodically driven quantum ratchets: Symmetries and resonances

Abstract

We study the quantum version of a tilting and flashing Hamiltonian ratchets, consisting of a periodic potential and a time-periodic driving field. The system dynamics is governed by a Floquet evolution matrix bearing the symmetry of the corresponding Hamiltonian. The dc-current appears due to the desymmetrization of Floquet eigenstates, which become transporting when all the relevant symmetries are violated. Those eigenstates which mostly contribute to a directed transport reside in phase space regions corresponding to classical resonances. Quantum dynamics leads to the dependence of the average velocity on the initial phase of the ac-field. A resonant enhancement (or suppression) of the dc-current, due to avoided crossings between different Floquet states takes place upon tuning some control parameters. Our studies are predominantly aimed at experimental realizations of ac-driven quantum ratchets with cold atoms.

Quantum ratchet transport with minimal dispersion rate

“…The solution of the eigenvalue problem for the corresponding Floquet operator U (t,t 0 ), |ψ(t + t 0 ) = U (t,t 0 )|ψ(t 0 ) provides the set of eigenfunctions {|ψ α (t) }. The eigenfunctions satisfy the Floquet theorem |ψ α,κ (t) = e −iε α [κ]t/h |φ α,κ (t) , |φ α,κ (t) = |φ α,κ (t + T ) , and the Bloch theorem [18]. The Hilbert space of the system is sliced into invariant subspaces, each one of which is spanned by the states bearing the same quasimomentum value κ ∈ [−1/2,1/2].…”

“…We consider a quantum particle loaded into a periodic potential, which is periodically modulated in time [18]. The dynamics of the system is governed by a Schrödinger equation,…”

“…The asymptotic current generated by the system J (t 0 ) = lim t→∞ 1 t−t 0 t t 0 ψ(t )| p|ψ(t ) dt takes on a simple form in the Floquet basis [18]: [7,28]. The Hilbert space of the system can be imagined as a set of "conveyer belt" eigenstates, each one of which is moving with its own velocity, and the overall ratchet current depends on how the initial wave packet |ψ(t = t 0 ) was distributed among the various belts [18,19].…”

“…For a given κ and time t the set of eigenstates |ψ α,κ (t) forms a complete orthonormal basis for the corresponding subspace of the total system's Hilbert space. Any initial state, |ψ(x,t = t 0 ) , can be expanded in the Floquet basis, |ψ(t = t 0 ) = 1/2 −1/2 dκ α C α,κ (t 0 )|φ α,κ (t 0 ) , and the time evolution of the state is governed by [7,18] |ψ(t) =…”

“…An interesting class of ratchet systems are dissipation-free Hamiltonian ratchets, classical [1,6,12] and, even more intriguingly, quantum ones [1,[7][8][9][13][14][15][16][17][18]. For example, a bi-harmonic flashing potential with periodically and uniformly modulated potential height [18], could be turned into a conveyer belt for a Bose-Einstein condensate (BEC) of rubidium atoms [19].…”