Quantum rectifiers from harmonic mixing

– We investigate dissipative quantum transport in extended periodic systems that are subjected to electric harmonic mixing fields Ehm(t) = E1 cos(Ωt)+E2 cos(2Ωt+φ). Although such a drive possesses no net bias on average, the interplay of quantum dissipation and nonlinear response causes a finite directed current. We thus discover the paradigm of a dissipative quantum rectifier. The quantum current exhibits multiple reversals when driven in the nonadiabatic regime. As a function of temperature the quantum current displays a bell-shaped characteristic —constituting the benchmark for quantum stochastic resonance. Moreover, harmonic mixing also serves as a novel tool to selectively control quantum diffusion. The constructive role of time-dependent external driving and inherent dissipation can produce a variety of unexpected phenomena such as fluctuation-induced directed current in periodic structures that lack reflection symmetry (ratchets) [1], or anomalous amplification of weak signals in threshold-like systems (stochastic resonance) [2]. Here, our focus will be on a quantum version of classical ratchet systems; an extension that has been studied only recently for the class of adiabatically rocked ratchets [3]. Such a generalization of classical ratchet work into the world of quantum mechanics is far from being straightforward —due to the necessity of treating dissipation and external driving within a quantum-mechanical setting. As has been demonstrated previously for classical systems, directed current is possible also in symmetric periodic potentials, when driven by unbiased forces with nonvanishing odd numbered cumulant averages of order n ≥ 3 [4]. Such a particular realization appears by considering the harmonic mixing signal of two ac fields of angular frequencies Ω and 2Ω that drive overdamped noise-driven classical transport in a cosine potential [5]. The phenomenon has implicitly been experimentally observed by means of microwave harmonic mixing in one-dimensional organic conductors as early as in 1978 by Seeger and collaborators [6] (see fig. 1 therein). Current, or, equivalently, a finite voltage under open circuit condition [5, 6] (i.e. a nonzero stopping bias) emerges due to a nonlinear response to the unbiased harmonic mixing signal Ehm(t) = E1 cos(Ωt) +E2 cos(2Ωt+ φ) (1) with relative phase φ. However, these prior works [5, 6] did not discuss the rectification phenomenon, or the “ratchet effect” [1, 4]; thus the concept of Brownian machinery has been c © EDP Sciences 504 EUROPHYSICS LETTERS overlooked. As we demonstrate with this work, this harmonic-mixing mechanism is also of profound importance for driving induced dissipative quantum transport in periodic multi-state systems, such as THz-driven superlattices [7]. Driven dissipative quantum transport has recently been studied theoretically for monochromatic periodic driving (i.e. E2 = 0 in (1)) in refs. [8-10], and for symmetric random driving in [11], in the presence of an additional static dc-bias. In both cases the current-voltage characteristics exhibit an interesting behaviour on dissipation strength and —most strikingly— the phenomenon of a zero-bias negative differential conductance (sometimes termed “absolute negative conductance”). This latter effect has been experimentally observed in GaAs-AlGaAssuperlattice structures [7]. Here, we shall demonstrate that similar quantum systems can act as nonadiabatic quantum rectifiers when driven by harmonic mixing signals. To start, let us consider tunneling of a charged particle (electron) among sites in a onedimensional lattice in the presence of an electric field Ehm(t). We restrict the analysis to a single-band tight-binding treatment. The Hamiltonian then reads HTB(t) = − h̄∆ 2 ∞ ∑ n=−∞ (|n〉〈n + 1|+ |n+ 1〉〈n|)− eEhm(t)q̂ , (2) where |n〉 denotes the localized (Wannier) states, h̄∆ is the tunneling coupling energy between neighboring states, and q̂ = a ∑ n n|n〉〈n| is the position operator for the particle on the lattice with period a. In the absence of driving, this Hamiltonian is the archetype for variety of physical quantum transport phenomena [12,14]. For instance, it could be used to describe the current in semiconductor superlattices [14], or in charge-transferring molecular chains [16]. The harmonic mixing signal in (1) constitutes the simplest kind of an asymmetric periodic field possessing a nonvanishing third moment E3 hm(t) = 3 4E 2 1E2 cosφ. Here (...) indicates the time average over the temporal period, T = 2π/Ω. The Hamiltonian in (2) has intensively been studied for the monochromatic case withE2 = 0; it describes, e.g., the remarkable phenomenon of dynamical localization [13-15]. It cannot, however, yield net current for an initially localized particle —because there is no thermal bath that balances the generated heat power— even in the case of additional presence of a dc-field. Thus, to realize finite current one has to invoke a dissipation mechanism. We adopt here the conventional model of quantum dissipation, i.e. we couple the quantum particle bilinearly to a thermal bath of harmonic oscillators [12]: