Quantum Mechanics in Phase Space

Introduction

The Wigner function (WF) bears the phase space representation of quantum mechanics in terms of position and momentum (x,p) phase space operators, in contrast to conventional approaches based on the Schrödinger picture that expresses a quantum state in terms of either position or momentum operators, but not both at the same time. This particular feature makes the computation time of pure/mixed states more intense, for instance the wavefunction of a quantum state initially stored in a 1D array has to be stored in a 2D array within the Wigner picture, and the situation gets worse if the number of particles grows. In this endeavor, c-field methods like the truncated Wigner Approximation (TWA), provide a tractable approach to describe Bose gases that behave similarly to a classical field, for example, in the case of modes with low occupation in the c-field region, quantum fluctuations are included in the initial conditions by stochastic sampling of the Wigner distribution.

 

1.    1. Wigner Function - Phase space Representation

The Wigner function is a real quasiprobability distribution introduced for the first time by Eugene Wigner in 1932 to study the quantum-classical transition [1-3]

    

 

where B(x,theta;t) is the Blokhintsev function

The Wigner function’s dynamic equation is known as the Moyal equation and is given by

 

with

 

Let’s check the classical limit i.e. hbar->0, which yields the the classical Poisson Bracket

The Bopp operators are defined as,

 

where the bold letters represent Schrodinger operators while (x,p,theta,lambda) are the extended four operators living in the so-called Hilbert phase space [1], which follow the below commuting relations 

 

where Lambda is the conjugate variable of x, and theta is the conjugate variable of p

 

Note that F denotes the Fourier transform. The below table shows all representations hold by the Hilbert phase space, with the explicit form of the extended four operator algebra

 

The below movie shows the Phase space evolution of a Gaussian Wigner function through the doubly double-well potential

 

 

 

The next movie shows the Wigner funtion Vs Denstiy Matrix for the harmonic oscillator eigenstates

 

 

The below movie shows the time evolution of a gaussian Wigner function on a double barrier well potential + Decoherence as an example of an open system

 

 

 

1.    2. Wigner Function – Coherent Representation

fj

In the same footing within the phase space representation, we can define linear combinations of the phase space operators x and p that resembles the standard creation and annihilation operators, as follows

 

with

in the x-p representation the above equations read

 

with the conjugate variables

 

and to formally define the Wigner function in the coherent representation it is useful define the complex variable

 

which in the lambda-theta representation takes the form

so that the Wigner function in the coherent representation is defined as

with

 

Therefore, in the coherent picture the Wigner function is

The coherent and phase space representations are related through

 

2.1 Wigner function for representative states

The overlap of the Fock state and coherent state is [2]

 

Therefore, the Wigner Function is given by 

With the Laguerre polynomial

 

 

Gaussian approximation to the Wigner function for the Fock state is

The density operator for a thermal state is

 

so that the Wigner Function is

 

2.2 Moyal Equation - Coherent Representation

It is easy to show that after performing chain rule

 

 we obtain

 

with

 

Such that the Moyal equation is written as follows

 

Let's check the classical limit

 

2.3         Weyl Symbol for the Bose Hubbard Hamiltonian

For low energies the M-mode Bose-Hubbard Hamiltonian under periodic boundary conditions read

 

so that the Weyl symbol is given by

 

1.    3. Truncated Wigner Approximation (TWA)

The TWA is a semi-classical approximation, and it is very important because it gives the first step in going from classical to quantum description of the dynamics being asymptotically exact at short times

 

where omega is a quantum operator.

References

[1] Cabrera, Renan, et al. "Efficient method to generate time evolution of the Wigner function for open quantum systems." Physical Review A 92.4 (2015): 042122.

[2] Polkovnikov, Anatoli. "Phase space representation of quantum dynamics." Annals of Physics 325.8 (2010): 1790-1852.

[3] Blakie, P. Blair, et al. "Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques." Advances in Physics 57.5 (2008): 363-455.