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Physical Review B

Mesoscopic Aharonov-Bohm loops in a time-dependent potential: Quasistationary electronic states and quantum transitions

I. O. Kulik

Published Sep 18, 2007• 4 citations
Publisher

Abstract

Motivated by the interest to testable, exactly solvable models of quantum behavior in the time-dependent potentials, which may be important in studying (and in proof) basic quantum mechanical laws, as well as in considering the possibility of using the electronic components suggested to hypothetical ``quantum computers,'' we develop a method of solving the Schr\"odinger equation in a certain class of time-periodic and space-dependent one-dimensional potentials. In particular, it is shown that the quasistationary electronic state in the one-dimensional cyclic mesoscopic metallic ring in a rotating-potential field displays periodic variation of quasienergy in the function of magnetic flux threading the ring (the Aharonov-Bohm effect) and oscillation, superposed on the monotonous dependence, in the function of angular velocity of rotating potential (the effect similar to Rabi and/or Bloch oscillation). At large speed of rotation, quasienergy decreases rather than increases with the increase of angular velocity. The dependence of quasienergy on flux in space periodic potential displays standard $hc∕e$ periodicity as well as the periodicity with a larger period, $Nhc∕e$, where $N$ is the number of sites in the loop, corresponding to one flux quantum per lattice site. This is an interference effect similar to one observed in the fractional quantum Hall effect but, unlike in the latter, not requiring the concept of ``fractional electron charge,'' $e∕N$. The physical significance of the quasienergy states is clarified by studying the quantum transitions between the states as well as by investigating the energy flow between the ring and the external source of the potential.

DOI
10.1103/physrevb.76.125313

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Motivated by the interest to testable, exactly solvable models of quantum behavior in the time-dependent potentials, which may be important in studying (and in proof) basic quantum mechanical laws, as well as in considering the possibility of using the electronic components suggested to hypothetical quantum computers,'' we develop a method of solving the Schr\"odinger equation in a certain class of time-periodic and space-dependent one-dimensional potentials. In particular, it is shown that the quasistationary electronic state in the one-dimensional cyclic mesoscopic metallic ring in a rotating-potential field displays periodic variation of quasienergy in the function of magnetic flux threading the ring (the Aharonov-Bohm effect) and oscillation, superposed on the monotonous dependence, in the function of angular velocity of rotating potential (the effect similar to Rabi and/or Bloch oscillation). At large speed of rotation, quasienergy decreases rather than increases with the increase of angular velocity. The dependence of quasienergy on flux in space periodic potential displays standard $hc∕e$ periodicity as well as the periodicity with a larger period, $Nhc∕e$, where $N$ is the number of sites in the loop, corresponding to one flux quantum per lattice site. This is an interference effect similar to one observed in the fractional quantum Hall effect but, unlike in the latter, not requiring the concept of fractional electron charge,'' e∕Ne∕Ne∕N. The physical significance of the quasienergy states is clarified by studying the quantum transitions between the states as well as by investigating the energy flow between the ring and the external source of the potential.
The Aharonov-Bohm oscillations in the transmission probability of a one-dimensional mesoscopic ring between two electrodes are considered numerically where a time-periodical flux \ensuremath{\gamma}(t)=\ensuremath{\gamma}+\ensuremath{\lambda} cos\ensuremath{\omega}t is applied. Such a flux simulates a phase-coherent inelastic scattering due to the absorption/emission processes of modulation quanta. An increase of alternating flux amplitude \ensuremath{\lambda} drastically changes the Aharonov-Bohm oscillations of the transmission though the exact period remains ${\mathrm{\ensuremath{\varphi}}}_{0}$=hc/e. A frequency dependence of the transmission has abrupt steps at \ensuremath{\omega}=(E-EF{\mathit{E}}_{\mathit{F}}EF​)/n, caused by the locking of the transmission in the nth channel where EF{\mathit{E}}_{\mathit{F}}EF​ is the Fermi level. Moreover, for transmission of a narrow wave packet the time-periodical flux gives rise to two effects. The first is a fractioning of the input wave packet by the ring in such a way that the output wave function consists of a few separate wave packets. Their number, distance apart, and height essentially depend on the static flux \ensuremath{\gamma}. The second effect is a strong space squeezing of the output wave packet.
Based upon a tight-binding model Hamiltonian and by applying the Jordan-Wigner transformation, the Aharonov-Bohm (AB) effect for charged hard-core bosons in one-dimensional (1D) mesoscopic rings is investigated for the first time. By introducing a specific unitary transformation, the energy spectrum and the persistent charge current are analytically derived for ID perfect lattices. Similar to electron systems, it has been found that the energy and persistent current are periodic in AB flux with the period Φ0=hc/q. More interestingly, we have exactly shown that, in the absence of the external AB flux, the charge current can be self-sustained via the AB effect, which is in fact the ground state of the system regardless of the number of hard-core bosons being even or odd.
We find a fine structure in the Aharonov-Bohm effect, which is characterized by the appearance of a type of periodic oscillation having a smaller fractional period. The effect is illustrated with the aid of a solution of the Bethe ansatz equations for N electrons located on a Hubbard ring in a magnetic field. Specifically, we found that at low density or strong coupling on a Hubbard ring in a magnetic field oscillations with three different periods occur. Along with the conventional Aharonov-Bohm oscillations with the period equal to 1, two additional oscillations with periods equal to 1/N and M/N coexist, where M is the number of down-spin particles and the periods are measured in units of the elementary flux quantum ${\mathrm{\ensuremath{\Phi}}}_{0}$=h/e. The fine structure is due to electron-electron and Zeeman interactions. When M=N/2 there is a coexistence of the fractional 1/N and half-flux quantum periodic oscillations only. With increasing magnetic field, with the spin flips created due to Zeeman interaction, the half-flux quantum period transforms to the integer one. This transformation is not continuous but via the appearance of the fractional M/N oscillations. We discuss the relation of the described fine structure and the M/N effect to existing experiments on single and on arrays of rings, where integer and half-flux quantum periods are, respectively, observed.
We study the effects of correlations on a one-dimensional ring threaded by a uniform magnetic flux. In order to describe the interaction between particles, we work in the framework of the Hubbard model and the t - and t - - models. We focus on the dilute limit. Our results suggest the possibility that the persistent current has an anomalous periodicity , where p is an integer in the range ( is the number of particles in the ring and is the flux quantum). We find that this result depends neither on disorder nor on the detailed form of the interaction, while there is still on-site infinite repulsion.
We consider N strongly interacting electrons in a one-dimensional circular loop that is pierced by a time-periodic magnetic flux a(t)=a0+a1(t){a(t)=a}_{0}{+a}_{1}(t)a(t)=a0​+a1​(t) with the angular frequency $\ensuremath{\omega}.$ Similar to our previous work, where we have considered a static magnetic flux a0,{a}_{0},a0​, the electron positions are expressed in terms of collective and relative coordinates. Strong $e\ensuremath{-}e$ interaction can then be treated in a harmonic approximation for the relative motion. The presently searched solutions of the time-dependent Schr\"odinger equation for a time-periodic flux are given by the Floquet states. The Floquet states for a spatially constant one-particle potential form a complete set of Floquet-basis states, which is used to study the influence of the one-particle potential on the electronic states. While for a spatially constant one-particle potential the time-averaged observables, such as the electronic energy and the electronic current or angular momentum, depend solely on the time-averaged magnetic flux a0,{a}_{0},a0​, a spatially varying one-particle potential leads to pronounced resonances. In the case of moderate electronic relaxation, the stationary properties are determined by the Floquet state with lowest time-averaged energy. For the associated persistent angular momentum we predict jumps with heights proportional to the number of electrons N. We further show that, already by measuring the locations of these jumps in the ${(a}_{0},\ensuremath{\omega})$ plane, one could determine the number of electrons N as well as the effective $e\ensuremath{-}e$ interaction.
We investigate anomalous oscillations due to the Aharonov-Bohm(AB) and Aharonov-Casher(AC) effects of the one-dimensional Hubbard ring with flux in the strong coupling limit. By using the exact diagonalization method and the Shiba transformation, we examine the energies of the ground state and few excited states in the presence of the flux producing the AB or AC effect, where the transformation not only reverses the sign of the interaction U, but also exchanges the role between the AB and AC effects in the model Hamiltonian. We classify the number of oscillation of the ground state as a function of the flux systematically and clarify the close relationship between the AB and AC effects. For example, it is shown that the number of the periods are given by N_L - N_e(N_L - N_↑ + N_↓) for the AB(AC) effect in the very strong attraction, where N_L, N_e, N_↑, and N_↓ are the system size, the number of the total electrons, the number of electrons with up-spin, and the number of electrons with down-spin, respectively, in the condition of N_L > N_e > N_↓ > N_↑. More special case, such as the half-filled band and spin balanced case (N_L = N_e and N_↓ = N_↑ ), we find the periodicity of the system to be 0(2) for the AB(AC) effect in the very strong repulsion. These results show us the nature of interesting phenomena originated by the interplay between the strong correlation and the quantum interference effect in the mesoscopic ring.