Phase space, decoherence and the Wigner function for finite dimensional systems.

1. 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 quant-ph/0401155 Zurek quant-ph/0306072 J.Yearsley (Unpublished) Ting Yu gr-qc/9605071

2. Overview • 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.

3. System • System (?) Black Box • System • System (?) Black Box Result(Classical information) e.g. position of pointer Record (?) Result(Classical information) e.g. position of pointer Record (?) 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!

4. 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

5. 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.

6. 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;

7. 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

8. 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.

9. Wigner transform of QOME • -It’s a mess! • -Can analyse this, but easier to look at; • -For positive A, diffusion in phase space • Decoherence -Equivalent to

10. 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.

11. 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.

12. Conclusion/What have we done? • -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!!