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Nanoscale Research Letters

Displacing, squeezing, and time evolution of quantum states for nanoelectronic circuits

Jeong Ryeol Choi, Byeong Jae Choi, Hyun Deok Kim

Published Jan 15, 2013• 4 citations
Publisher

Abstract

: The time behavior of DSN (displaced squeezed number state) for a two-dimensional electronic circuit composed of nanoscale elements is investigated using unitary transformation approach. The original Hamiltonian of the system is somewhat complicated. However, through unitary transformation, the Hamiltonian became very simple enough that we can easily treat it. By executing inverse transformation for the wave function obtained in the transformed system, we derived the exact wave function associated to the DSN in the original system. The time evolution of the DSN is described in detail, and its corresponding probability density is illustrated. We confirmed that the probability density oscillates with time like that of a classical state. There are two factors that drive the probability density to oscillate: One is the initial amplitude of complementary functions, and the other is the external power source. The oscillation associated with the initial amplitude gradually disappears with time due to the dissipation raised by resistances of the system. These analyses exactly coincide with those obtained from classical state. The characteristics of quantum fluctuations and uncertainty relations for charges and currents are also addressed.

DOI
10.1186/1556-276x-8-30

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•Apr 30, 2021•0 citations
: The time behavior of DSN (displaced squeezed number state) for a two-dimensional electronic circuit composed of nanoscale elements is investigated using unitary transformation approach. The original Hamiltonian of the system is somewhat complicated. However, through unitary transformation, the Hamiltonian became very simple enough that we can easily treat it. By executing inverse transformation for the wave function obtained in the transformed system, we derived the exact wave function associated to the DSN in the original system. The time evolution of the DSN is described in detail, and its corresponding probability density is illustrated. We confirmed that the probability density oscillates with time like that of a classical state. There are two factors that drive the probability density to oscillate: One is the initial amplitude of complementary functions, and the other is the external power source. The oscillation associated with the initial amplitude gradually disappears with time due to the dissipation raised by resistances of the system. These analyses exactly coincide with those obtained from classical state. The characteristics of quantum fluctuations and uncertainty relations for charges and currents are also addressed.
Quantum characteristics of a driven series RLC nanoelectronic circuit whose capacitance varies with time are studied using an invariant operator method together with a unitary transformation approach. In particular, squeezing effects and nonclassical properties of a superposition state composed of two displaced squeezed number states of equal amplitude, but 180° out of phase, are investigated in detail. We applied our developments to a solvable specific case obtained from a suitable choice of time-dependent parameters. The pattern of mechanical oscillation of the amount of charges stored in the capacitor, which are initially displaced, has exhibited more or less distortion due to the influence of the time-varying parameters of the system. We have analyzed squeezing effects of the system from diverse different angles and such effects are illustrated for better understanding. It has been confirmed that the degree of squeezing is not constant, but varies with time depending on specific situations. We have found that quantum interference occurs whenever the two components of the superposition meet together during the time evolution of the probability density. This outcome signifies the appearance of nonclassical features of the system. Nonclassicality of dynamical systems can be a potential resource necessary for realizing quantum information technique. Indeed, such nonclassical features of superposition states are expected to play a key role in upcoming information science which has attracted renewed attention recently.
Quantum properties of a superposition state for a series RLC nanoelectronic circuit are investigated. Two displaced number states of the same amplitude but with opposite phases are considered as components of the superposition state. We have assumed that the capacitance of the system varies with time and a time-dependent power source is exerted on the system. The effects of displacement and a sinusoidal power source on the characteristics of the state are addressed in detail. Depending on the magnitude of the sinusoidal power source, the wave packets that propagate in charge( q )-space are more or less distorted. Provided that the displacement is sufficiently high, distinct interference structures appear in the plot of the time behavior of the probability density whenever the two components of the wave packet meet together. This is strong evidence for the advent of nonclassical properties in the system, that cannot be interpretable by the classical theory. Nonclassicality of a quantum system is not only a beneficial topic for academic interest in itself, but its results can be useful resources for quantum information and computation as well.
Exact quantum solutions of two-dimensional dissipative mesoscopic circuit coupled to a power source are derived using two-step unitary transformation approach. With the choice of appropriate unitary operators, the original Hamiltonian of the system is transformed to a simple form whose quantum behavior is well known. From inverse transformation of the quantum solutions associated to transformed system, we obtained the full wave functions in the original system. Thermofield dynamics is employed in order to investigate thermal states of the system. The vacuum correlations between the so called tilde and non-tilde variables produce the temperature effects. The fluctuations and uncertainty relations for charges and currents are derived in thermal states. Due to the effect of energy dissipation generated in the resistances of circuit, the fluctuation of charges decreases exponentially with time whereas that of currents exhibits exponential increase. Accordingly, their products which gives the uncertainty relation does not depend on such exponential functions and always larger than or equal to ℏ/2\hbar/2ℏ/2 so that the uncertainty principle holds.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.
Time evolution of quantum fluctuations is investigated for an over-damped mesoscopic circuit with the help of the quantum characteristic function. It is found that, for an initial squeezed state, the quantum fluctuations of charge and current evolve with time by hyperbolic functions, and relate not only with the circuit parameters, but also the squeezing parameter. It is also found that with other conditions invariant to reducing the quantum fluctuations of charge and current, we should reduce the squeezing amplitude parameter. The research will be helpful in miniaturizing integrated circuits and electric components. It will also be significant for the further study of quantum information.
Abstract This chapter begins the discussion of the time evolution of an active quantum system. From the earlier chapters, time dependent physics has been observed through the presence of the time evolution of the phase of each eigenstate. The Hamiltonian itself is time independent. This represents the same kind of evolution of a classical system like the vibration of a tuning fork when it has been struck or the oscillation of an LC circuit if the capacitor is charged to some voltage and then the switch is closed. In the quantum case, the Hamiltonian has also been time independent. The time evolution evolves according the full-time dependent Schrödinger equation, depending only on a single initial condition of the state vector or wave function and the corresponding time evolution of the phase factor for each eigenstate. However in this chapter, we consider the case of when there is a time dependent Hamiltonian such as a sinewave generator or laser. As in the case of resonant tunneling, we see the importance in dynamics of resonant coupling. With an oscillating potential energy term, we see the presence of Rabi oscillations in the probability amplitude of a two-state system on resonance, which can be viewed as a quantum flip-flop between two states of a quantum bit (qubit).