Quantum Mechanics in the phase space

The representation of quantum mechanics in the phase space (position-momentum) is carried out in terms of the Wigner function. This formulation has the advantage of providing an intuitive visualization of the state and allowing the introduction of interactions with the environment in the context of open quantum dynamics. However, the phase space has the disadvantage of requiring significantly more computational power. Fortunately, new algorithms and modern computational techniques such as GPU computing can be used to overcome this difficulty as I show in a sequence of papers [1,2]. These techniques can be even applied to relativistic systems [3].

The following is a list of some sample simulations. Please click on the links to see the videos.

•  Quantum tunnelling 

• Quartic potential

• Free particle evolution of a coherent superposition of two Gaussian states.

• Free particle evolution of an INCOHERENT superposition  

[1] Efficient method to generate time evolution of the Wigner function for open quantum systems, Renan Cabrera, Denys I. Bondar, Kurt Jacobs, and Herschel A. Rabitz, Phys. Rev. A 92, 042122 – Published 28 October 2015

[2] Efficient computations of quantum canonical Gibbs state in phase space,
Denys I. Bondar, Andre G. Campos, Renan Cabrera, and Herschel A. Rabitz
Phys. Rev. E 93, 063304 – Published 13 June 2016

[3] Dirac open-quantum-system dynamics: Formulations and simulations,
Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz
Phys. Rev. A 94, 052111 – Published 14 November 2016

Simulation of Relativistic quantum systems

The effective and efficient simulation of relativistic quantum systems is now possible to accomplish with a common desktop computer thanks to the development of novel numerical algorithms and the emergence of modern computational techniques such as GPU computing.

I am in the process to post some animations of relativistic quantum systems in two and three dimensions

• The Zitterbewegung in two dimensions 

   

• Klein paradox in two dimensions  

• Klein paradox in three dimensions

Split operator propagator

This is a frameshot from the propagation of a 2D wavefunction

Conservation of Shannon information in classical mechanics

There are many places where to find the proof of the conservation of the quantum Shannon information under unitary evolution. However, I just could not find the equivalent proof that everybody talks in classical mechanics, so here it goes.





The Lioville equation can be expressed in terms of a self-adjoint operator as observed by Koopman and von Neumann, such that



Given that the Liouvillian follows the Leibniz rule, which is true for conservative systems



it is now easy to prove that



This fact is in complete contrast with respect to the thermodynamical entropy, which can be fundamentally irreversible.


-This entry was powered by http://www.codecogs.com/latex/eqneditor.php

DUST

DUST stands for Differential Unitary Space Time coding. This coding method is used in the transmission of information from multiple antennas to multiple antennas.

The transmission of information in a block can be modelled with the following equation



where S is the input matrix, H is the channel matrix, \rho is a scalar coefficient that modulates the channel and W is the noise.

DUST encodes the information in a unitary matrix U_n, which is indirectly encoded through a sequence of two S unitary blocks as



If H is full-rank and the channel does not change significantly in the time the two blocks are transmitted. The unitary matrix U can be recovered from the sequence of two received unitary blocks X without! the need to know the channel H as



where the new noise is defined as


-This entry was powered by http://www.codecogs.com/latex/eqneditor.php

Random Unitary Matrices

The generation of random unitary matrices has many applications in many fields.
The simplest method is based on the QR decomposition of a random complex matrix with independent elements following a Normal distribution.

In Mathematica one can generate a random complex number with Normal distribution for both the real and imaginary parts with the following function

NormalRandomComplex[] := (#[[1]] + #[[2]]*I) &@
  RandomReal[MultinormalDistribution[{0, 0}, {{1, 0}, {0, 1}}]]

where {0, 0} specifies the origin and {{1, 0}, {0, 1}} the covariance matrix (standard deviation 1)

The function that generates the random n x n unitary matrix is

RandomUnitaryMatrix[n_] := Module[{z, q, r},
  z = Table[NormalRandomComplex[], {n}, {n}];
  {q, r} = QRDecomposition[z];
  q.DiagonalMatrix[Sign[Tr[r, List]]]

]

In this code, the QR decomposition of the complex random matrix with normal distribution outputs a unitary matrix "q" and a triangular matrix "r". The sought random unitary matrix is a correction of "q", which must be multiplied by a diagonal matrix filled with 1 and -1 according to the sign of the corresponding diagonal element of "r".

New numerical algorithms

Two recent developments on numerical analysis have called my attention

A new very efficient sparse Fourier transform
http://web.mit.edu/newsoffice/2012/faster-fourier-transforms-0118.html

A significantly faster algorithm for matrix multiplication
http://www.newscientist.com/article/mg21228422.500-mathematical-matrix-multiplier-sees-first-advance-in-24-years.html