Relativistic Open Quantum Systems III: Klein's paradox

Klein's paradox is beautifully visualized in the phase space through the relativistic Wigner function.

In introductory quantum mechanics we learn about quantum tunneling and how it allows the transmission of particles through potentials that are otherwise insurmountable in mechanics.

In relativity we have another twist, the transmission is not only present but it can be the dominant effect for strong potentials. In the following animation we illustrate this effect where we clearly see that the transmitted wave-packet is composed of antiparticles (negative momentum but moving to the positive direction).

This type of effect can be seen in effective Dirac materials in terms of an enhanced tunneling. The animation of such process is

These effects are well know in the literature. My contribution in [1] was to show that the presence of quantum decoherence does not affect them in a significant measure.



[1] Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz, Dirac open-quantum-system dynamics: Formulations and simulations , Phys. Rev. A 94, 052111 (2016) https://doi.org/10.1103/PhysRevA.94.052111 

Relativistic Open Quantum Systems II: Majorana particles

Relativistic states are much richer in features than those in non-relativity. One of the most important relativistic hallmarks is the ability to describe of antiparticles. We already know this from books of quantum mechanics, but it is in the phase space where we can visualize them in full glory.

Let us first watch the time-evolution of a free relativistic cat-state:

    This movie might as well describe a non-relativistic cat state, so, no surprise (the dynamics is undergoing quantum coherence).

Now, let us observe the evolution of the corresponding particle-antiparticle coherent superposition:

In this case we observe that the antiparticle at the bottom advances to the positive direction despite of having a negative momentum. This type of dynamics is characteristic of antiparticles!

One critical piece of information is that both states are undergoing the same degree of quantum decoherence due to the contact with an environment. Nevertheless, the particle-antiparticle superposition maintains the interference robust. We just found a state that belongs to a free-decoherence space!

Al should also mention that the relativistic state happens to be a Majorana state.

[1] 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

Relativistic Open Quantum Systems I: Foundations

I recently published a paper about Relativistic Open Quantum Systems

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

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.94.052111

The work on non-relativistic open quantum systems is vast but comparatively, very little can be found involving relativity. This situation is not changing significantly despite of the urgent need to advance in an ever growing number of fields beyond high energy physics, such as solid state, cold atoms, trapped ions, quantum optics and more. This paper reviews the diverse literature and consolidates the fundamental principles to present a unified formalism for Relativistic Open Quantum Systems. As a result, for the first time in the literature, we are able to simulate some relativistic open quantum systems and analyze the effect of quantum decoherence. This allowed us to gain new insights in one of the most fundamental problems of physics such as the quantum to classical transition and to recognize antiparticles as a new potential resource in quantum information.

In this post I will highlight some items treated in the introduction of my paper describing the theoretical foundations of relativistic open quantum systems. More highlights will appear in future posts.

• The formulation of the Manifestly Relativistic Covariant von Neumann Equation for the Dirac equation is absent from the literature in the sense that it is not explicitly stated or treated as it is the case for the non-relativistic analog. Nevertheless, there are many works on the relativistic Wigner function that can be analyzed and traced back to finally obtain
  
  
  where P is the relativistic density operator state  and D is the Dirac generator. Both operators are  4x4 matrices with each entry as an  operator dependent of x and p. This equation is important because it can be considered as the first step to formulate relativistic open quantum systems because P can be a pure or a mixed state in general. Moreover, the only hope to formulate covariant environments must rely on Eq (6) as the underlying generator or coherent dynamics. It must be emphasized that the the study of relativistic covariant environments is completely and utterly absent from the literature. Considering that the vacuum is arguably an example of such environment, this could lead to the discovery of insights on some of the most fundamental problems of modern physics

• An important observation of Eq. (6) is that the time and space are treated on the same footing. The time is itself an operator, that can be parametrized by two degrees of freedom (rows and columns)! In the paper we show how to disentangle those two time parameters and obtain two equations of motion that can be integrated independently. Surprise, surprise, one of those equations is the von Neumann equation that can be written directly using the Dirac Hamiltonian. The second equation of motion is strange and never even suspected in the literature. This additional dynamical equation is not necessary to describe the propagation of an initial state defined at a fixed point in time. So, what is the use of this equation? Answer: It is required to describe the propagation of the state in a different inertial frame of reference.       

• A third important observation of Eq (6) above, is that the mass appears in the generator of motion D. In fact, as we show in the paper, this equation already incorporates the shell mass condition. This is interesting when we compare the corresponding classical Hamiltonian formalism where the mass is not hard-coded and appears later as an integral of motion. Somehow, the mass is diluted in the quantum to classical process.     

•  Final observation in this post: The equation of motion (6) is not made with Hermitian generators of motion. In fact the state P is not Hermitian either !!!!

One of the figures of the paper is now featured in the Kaleidoscope section of PRA: http://journals.aps.org/pra/kaleidoscope/pra/94/5/052111

RUNNING THE CODE:

The CUDA source code for my paper can be found at

https://github.com/cabrer7/PyWignerCUDA

The python notebook files that run specific examples can be found at

https://github.com/cabrer7/PyWignerCUDA/tree/master/instances/WignerDirac2D_4x4

This link also contains many figures and many simulations not part of the paper.

Note that the file "pywignercuda_path.py" must be updated with the information of your own system.

It requires PyCUDA along with other standard python libraries and of course an NVIDIA graphics card.

Please ask me questions if you cannot run it.

Relativistic quantum control

Here is a plot from my upcoming paper on Relativistic Quantum Control. In this particular example I show how to trap relativistic  spin1/2 particles that obey the Dirac equation

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