Dynamic Stark ladders in the mesoscopic metallic rings

The quantum theory for mesoscopic electric circuit with charge discreteness is briefly described. The Schrödinger equation of the mesoscopic electric circuit with external source which is the time function have been proposed. The mesoscopic metallic ring is regarded as a “pure” inductance design, and the dependence of the quasienergy spectrum on a parameter called the electric matching ratio is presented.  2003 Published by Elsevier B.V. PACS: 72.10.-d; 54.30.Bv; 03.65.-w After Büttiker, Imry and Landauer predicted the Josephson-like effects in small and strictly one-dimensional rings of normal metal driven by an external magnetic flux [1], several manifestations of the AB effects and AC effects have been predicted and verified in mesoscopic systems [2]. Three experiments have verified the persistent current in mesoscopic metallic rings and many different theories have been proposed [3]. In previous papers [4], the mesoscopic metallic ring is regarded as a “pure” inductance (L) design electrical circuit. The persistent current formula has been proposed by using the quantum theory for mesoscopic electric circuits in accord with the discreteness of electric charge. Likharev et al. had predicted that there are effects including Bloch oscil* Corresponding author. E-mail address: [email protected] (B. Chen). 0375-9601/03/$ – see front matter  2003 Published by Elsevier B.V. doi:10.1016/S0375-9601(03)00645-5 lations [5], Wannier–Stark ladders [6] and Laudau– Zener tunneling [7] in a mesoscopic Josephson junction [8]. Recently, electrons driven by an external field show a variety of very interesting phenomena, the same effects have been observed in high-quality superlattices, optical ring resonators and more recently on ultra cold atoms in accelerating optical potentials [9]. As Imry pointed out the same interesting things occur in a mesoscopic ring threaded by a flux when the flux changes with time [10]. For the special case, the resulting current will show Bloch oscillation with a Josephson-type frequency [11]. When the change of flux is not slow enough, Zener-type transitions may occur among the bands. The dynamic treatment in the mesoscopic metallic rings is very interesting. In this Letter, we briefly demonstrate a quantum mechanical theory for mesoscopic electric circuit with the discreteness of charge, also consider the external source as a time function. The explicit solutions for the quasi432 B. Chen et al. / Physics Letters A 313 (2003) 431–435 energies and the Floquet states are obtained exactly. The results depend on the number-theoretical property of the matching ratio of the ac frequency and the frequency of the Stark ladders associated with the dc field. The classical equation of motion for an electric circuit of LC design is the same as that for a harmonic oscillator, where the “coordinate” means electric charge [11]. Along with the dramatic achievement in nanotechnology, such as molecular-beam expiatory, atomic-scale fabrication or advanced lithography, mesoscopic physics and nanoelectronics are undergoing a rapid development [12]. The electronic device community has been witnessing a strong and definite trend in the miniaturization of integrated circuits and components toward atomic-scale dimensions [13]. When the transport dimension reaches a characteristic dimension, namely, the charge carrier inelastic coherence length, one must use quantum mechanics to discuss the problems in the mesoscopic systems and also need to consider the charge discreteness. The quantization of the circuit was carried out in the same way as that of a harmonic oscillator [14,15]. In order to take into account the discreteness of electronic charge, we must also impose that the eigenvalues of the selfadjoint operator q̂ (electric charge) take discrete values (1) q̂|n〉 = nqe|n〉, where n ∈ Z (set of integers) and qe = 1.602 × 10−19 C, the elementary electric charge [4]. Since the spectrum of charge is discrete, the inner product in charge representation will be a sum instead of the usual integral and the electric current operator will be defined by the discrete derivatives ∇qe , ∇̄qe . (2) ∇qe = Q̂− 1 qe , ∇̄qe = 1 − Q̂+ qe , where Q̂ = ee is a minimum “shift operator”. It is easy to check that ∇+ qe = −∇̄qe . Thus for the mesoscopic quantum electric circuit one will have finite differential Schrödinger equation: