Phase space, decoherence and the Wigner function

for finite dimensional systems.James Yearsley

Superviser: Prof. JJ Halliwell.

See: Gibbons et. al. quant-ph/0401155

Zurek quant-ph/0306072

J.Yearsley (Unpublished)

Ting Yu gr-qc/9605071

Overview

1) What is decoherence?-What is measurement?

2) An example; the Quantum Optical Master Equation.

3) Discrete phase space and the discrete Wigner function.-What does it tell us about decoherence?

4) Entangled systems and decoherence.

What is Decoherence?

-What is measurement?

-If there is a record of the measurement, we can choose to look at it at any time.-Record is classical information, no need for observer!

System Black Box

System (?)

Result (Classical information) e.g. position of pointer

Record (?)

System Black Box

System (?)

Result (Classical information) e.g. position of pointer

Record (?)

What is Decoherence?

-Consider photon scattering from dust particle in space. -In principle could measure the position of the photon at a later time. -If we are only interested in dust particle must trace out photons.

-End up with a master equation for dust particle (Zurek):

-Interaction ‘localises’ dust particle.

M

The Quantum Optical Master Equation

-A more developed example.-Two level system in a thermal bath of photons.-3 possible processes:

-absorb a photon-stimulated emission-unitary process

-Have Quantum Optical Master Equation:

-Result of evolution; decoherence and thermalisation.

Phase space formalism

-Aim is to represent this model in phase space.-Phase space for continuous systems well understood.-Discrete systems?

-How do we understand concepts such as;

The discrete Wigner function

-Continuous Wigner function is Quasi-probability density.-Can we have discrete version?-Associate a projector with each line in phase space, Unique!-2D plot gives expectation value and probabilities for 3 independent quantities!

-All subtlety of phase space encoded in how you choose the A’s

Wigner function

-How do familiar properties translate?-Cannot be too tightly peaked:

-Must in general be negative somewhere.-Example, spin half particle, or 2-level system.

Wigner transform of QOME

-It’s a mess!

-Can analyse this, but easier to look at;

-For positive A, diffusion in phase spaceDecoherence

-Equivalent to

Why is diffusion the same as decoherence?

Consider a non-classical initial state:

-Ends up as:

-”Classical” state, ignorance interpretable mixture, not superposition.

-Very simple way of interpreting decoherence in phase space.

Entanglement in phase space

-Recall from earlier;

-Negativity is good, no local hidden variables

-Consider two entangled qubits, now separated and interacting with environment.

-Guess;

-Evolves to; Classical mixture.

-Exponential suppression of entanglement.

Conclusion/What have we done?

1) -Decoherence is an important process.-Can be understood in phase space terms.

-Can represent multiple qubits and entangled systems.

2) -Phase space itself is interesting.-Intimate relationship between structure and e.g. uncertainty relations.

3) -Allows an intuitive approach!!