Simple Quantum Systems

Abstract

In this chapter we study simple quantum systems. A particle in a one-dimensional potential is described by a wave packet which is a solution of the time dependent Schrödinger equation. We discuss two approaches to discretize the second derivative. Finite differences are simple to use but their dispersion deviates largely from the exact relation, except high order differences are used. Pseudo-spectral methods evaluate the kinetic energy part in Fourier space and are much more accurate. The time evolution operator can be approximated by rational expressions like Cauchy’s form which corresponds to the Crank-Nicholson method. These schemes are unitary but involve time consuming matrix inversions. Multistep differencing schemes have comparable accuracy but are explicit methods. Best known is second order differencing. Split operator methods approximate the time evolution operator by a product. In combination with finite differences for the kinetic energy this leads to the method of real-space product formula which can be applied to wavefunctions with more than one component, for instance to study transitions between states. In a computer experiment we simulate a one-dimensional wave packet in a potential with one or two minima. Few-state systems are described with a small set of basis states. Especially the quantum mechanical two-level system is often used as a simple model for the transition between an initial and a final state due to an external perturbation. Its wavefunction has two components which satisfy two coupled ordinary differential equations for the amplitudes of the two states. In several computer experiments we study a two-state system in an oscillating field, a three-state system as a model for superexchange, the semiclassical approximation and the Landau-Zener model for curve-crossing and the ladder model for exponential decay. The density matrix formalism is used to describe a dissipative two-state system in analogy to the Bloch equations for nuclear magnetic resonance. In computer experiments we study the resonance line and the effects of saturation and power broadening. Finally we simulate the generation of a coherent superposition state or a spin flip by applying pulses of suitable duration. This is also discussed in connection with the manipulation of a Qubit represented by a single spin.

Problems

Problem 23.1 Wave Packet Motion

In this computer experiment we solve the Schroedinger equation for a particle in the potential V(x) for an initially localized Gaussian wave packet $\psi (t=0,x)\sim \exp (-a(x-x_{0})^{2})$. The potential is a box, a harmonic parabola or a fourth order double well. Initial width and position of the wave packet can be varied.

• Try to generate the stationary ground state wave function for the harmonic oscillator

• Observe the dispersion of the wave packet for different conditions and try to generate a moving wave packet with little dispersion.

• Try to observe tunneling in the double well potential

Problem 23.2 Two-state System

In this computer experiment a two-state system is simulated. Amplitude and frequency of an external field can be varied as well as the energy gap between the two states (see Fig. 23.9).

• Compare the time evolution at resonance and away from it

Problem 23.3 Three-state System

In this computer experiment a three-state system is simulated.

• Verify that the system behaves like an effective two-state system if the intermediate state is higher in energy than initial and final states (see Fig. 23.13).

Problem 23.4 Ladder Model

In this computer experiment the ladder model is simulated. The coupling strength and the spacing of the final states can be varied.

• Check the validity of the exponential decay approximation (see Fig. 23.15)

Fig. 23.27

(Generation of a coherent mixture by a -pulse) The equations of motion of the Bloch vector (23.268) are solved with the 4th order Runge–Kutta method for an interaction pulse with a Gaussian shape. The pulse is adjusted to obtain a coherent mixture. The influence of dephasing processes is studied. $T_{1}=1000,t_{p}=0.9,V_{0}=0.25$. The occupation difference $\rho _{11}-\rho _{22}=z$ (solid curves) and the coherence $|\rho _{12}|=\frac{1}{2}\sqrt{x^{2}+y^{2}}$ (broken curves) are shown for several values of the dephasing time $T_{2}=5,10,100,1000$

Problem 23.5 Semiclassical Approximation

In this computer experiment we study the crossing between two states along a nuclear coordinate. The time dependent Schrödinger equation for a wave packet approaching the crossing region is solved numerically and compared to the semiclassical approximation.

• Study the accuracy of the semiclassical approximation for different values of coupling and initial velocity

Problem 23.6 Landau–Zener Model

This computer experiment simulates the Landau Zener model . The coupling strength and the nuclear velocity can be varied (see Fig. 23.19).

• Try to find parameters for an efficient crossing of the states.

Problem 23.7 Resonance Line

In this computer experiment a two-state system with damping is simulated. The resonance curve is calculated from the steady state occupation probabilities (see Figs. 23.23, 23.24).

• Study the dependence of the line width on the intensity (power broadening).

Problem 23.8 Spin Flip

The damped two-state system is now subject to an external pulsed field (see Figs. 23.26, 23.27, 23.28).

• Try to produce a coherent superposition state ($\pi /2$ pulse) or a spin flip ($\pi$ pulse).

• Investigate the influence of decoherence.

Fig. 23.28

(Motion of the Bloch vector during and $\pi$ pulses) The trace of the Bloch vector is shown in the laboratory system. Left pulse as in Fig. 23.27 with $T_{2}=1000$. Right $\pi -$pulse as in Fig. 23.26 with $T_{2}=1000$