The Moyal Equation in Open Quantum Systems

Karl-Peter Marzlin and Stephen Deering

CAP Congress Ottawa, 13 June 2016 Outline

l Quantum and classical dynamics l Phase space descriptions of quantum systems l Open quantum systems l Moyal equation for open quantum systems l Conclusion

Quantum vs classical

Differences between classical mechanics and quantum mechanics have fascinated researchers for a long time

Two aspects:

• Measurement process: deterministic vs contextual

• Dynamics: will be the topic of this talk

Quantum vs classical

Differences in standard formulations:

Quantum Mechanics Classical Mechanics Class. statistical mech.

complex wavefunction deterministic position probability distribution

observables = operators deterministic observables observables = random variables

dynamics: Schrödinger Newton 2 Boltzmann equation for equation for state probability distribution

However, some differences “disappear” when different formulations are used.

Quantum vs classical

First way to make QM look more like CM: Heisenberg picture

Heisenberg equation of motion

dAˆH 1 = [AˆH , Hˆ ] dt i~ is then similar to Hamiltonian mechanics

dA(q, p, t) @A @H @A @H = A, H , A, H = dt { } { } @q @p @p @q

Still have operators as observables

Second way to compare QM and CM: phase space representation of QM

Most popular phase space representation: the Wigner function W(q,p)

1 W (q, p)= S[ˆ⇢](q, p) 2⇡~

iq0p/ Phase space S[ˆ⇢](q, p)= dq0 q + 1 q0 ⇢ˆ q 1 q0 e ~ h 2 | | 2 i Z The Weyl symbol S [ˆ ⇢ ]( q, p ) maps state ⇢ ˆ to a function of position and momentum (= phase space)

Phase space

The Wigner function is similar to the distribution function of classical statistical mechanics

Dynamical equation = Liouville equation. Derived from Schrödinger eq., resembles the Boltzmann eq.

W can take negative values and is therefore not a probability distribution Phase space

Mean value of an observable A ˆ :

Aˆ = dq dp S[Aˆ](q, p) W (q, p) h i Z è in phase space, observables are represented by their Weyl symbols

We can consider Weyl symbols of operators both in Schrödinger and in Heisenberg picture

Phase space

Weyl symbols in Heisenberg picture obey the Moyal equation

dA(q, p, t) 1 = A, H M , A, H M = (A?H H?A) dt { } { } i~ with the Moyal star product

~ ~ A(q, p) ?B(q, p)=A q + i @ ,p i @ B(q, p) 2 p 2 q ✓ ◆ Now QM looks very similar to classical mechanics Phase space

Summary of QM representations:

Hilbert space Phase space

Schrödinger picture Schrödinger Liouville equation equation

Heisenberg picture Heisenberg Moyal equation equation Open quantum systems

Real quantum systems are not isolated

Interaction with an environment modifies dynamics

Environment = another quantum system

Hilbert space: total = S E H H ⌦H If only system observables can be measured, one can “trace over the environment”

Aˆ 1 =Tr Aˆ Tr (ˆ⇢) h S ⌦ Ei S S ⌦ E ⇣ ⌘ Tr E(ˆ⇢ )= reduced density matrix for the system

Open quantum systems

Tracing over the environment modifies dynamical eqns:

Schrödinger eq. è Master eq. for reduced dens. matr.

i @t⇢ˆ = [H,ˆ ⇢ˆ] (Jˆ†Jˆ⇢ˆ +ˆ⇢Jˆ†Jˆ 2Jˆ⇢ˆJˆ†) ~

Heisenberg eq. quantum Langevin equation i i @tAˆ = [A,ˆ Hˆ ] [B,ˆ Aˆ], Fˆ @tBˆ ~ 2~{ } Fˆ = noise operator

Phase space: Liouville eq. è Fokker-Planck eq.

Open quantum systems

The missing piece is what we contributed

Hilbert space Phase space

Schrödinger picture Master equation Quantum Fokker- Planck equation

Heisenberg picture Quantum Langevin Open Moyal equation equation Open Moyal Equation

Strategy to derive the open Moyal equation:

System = Schrödinger particle

Environment = (large) collection of quantum harmonic oscillators (Ford-Kac-Mazur model)

Linear coupling between system and environment

Weyl symbol of Hamiltonian:

H(q, p, qn,pn)=HS + Hint + HE 1 H = p2 + V (q) S 2m N 1 1 H + H = p2 + k (q q)2 E int 2m n 2 n n n=1 n X ✓ ◆ Open Moyal Equation

Evaluate Moyal bracket for system observable A è

p @tA = A, HS M + K(t)A + F (t) @pA { } m • The system observable A = A(q, p, qn,pn,t) depends on the oscillator variables • F ( t )= F ( q n ,p n ,t ) is a force that acts on the system but only depends on the environment • K(t) = phase space environment operator that represents force fluctuations via

K(t)F (t0) F (t)F (t0) /h i Open Moyal Equation

Markovian environment: F ( t ) F ( t 0 ) drops to zero much faster than the dynamicsh of thei system itself

In this limit, the operator K(t) can be approximated by

K(t) 2m ⇡ F(t)

This turns the open Moyal equation into a functional differential equation:

A @ A(q, p, t)= A, H 2p + F (t)@ A t { }M F(t) p

Open Moyal Equation

Hence, phase space observables turn into functionals of a fluctuating random force induced by the environment.

Example: solution for particle with vanishing potential

t t p t 1 t0 Aq(t)=qe + (1 e )+ dt0 (1 e )F (t0 t) m m 0 t Z t t0 Ap(t)=e (p mq)+ dt0 e F (t0 t) 0 Z Initial position/momentum are damped Particle evolves into a kind of Brownian motion Mean values can be evaluated for given F (t)F (t0) h i

Open Moyal Equation

Open (mean) Weyl symbols for variances:

2 2 t 2 Aq2 (q, p, t)=Aq + q 1 e kBT 2t t + 2 (2t 3 e +4e ) m 2 2t 2t 2 2 2 Ap2 (q, p, t)=Ap + mkBT 1 e + e m q Demonstrates how the fluctuating force leads to thermalization

Conclusion

Hilbert Phase We derived the “missing piece” space space for representations of open Schrödinger Master Quantum picture equation Fokker- quantum systems Planck equation Heisenberg Quantum Open picture Langevin Moyal equation equation Observables turn into functionals of a fluctuating force

Particles undergo Brownian motion

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Reference: J. Phys. A 48 (2015) 205301

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