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The Moyal Equation in Open Quantum Systems

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The Moyal Equation in Open Quantum Systems 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] γ(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 δ K(t) 2mγ ⇡ δF(t) This turns the open Moyal equation into a functional differential equation: δA @ A(q, p, t)= A, H 2γp + F (t)@ A t { }M − δF(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 γt p γt 1 γt0 A (t)=qe− + (1 e− )+ dt0 (1 e− )F (t0 t) q mγ mγ − 0 − − t Z γt γt0 Ap(t)=e− (p mγq)+ 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 γt 2 Aq2 (q, p, t)=Aq + q 1 e− − kBT 2γt γt + 2 (2γt 3 e− +4e− ) mγ − − 2 2γt 2γt 2 2 2 Ap2 (q, p, t)=Ap + mkBT 1 e− + e− m γ 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 Thanks! Reference: J. Phys. A 48 (2015) 205301 Thanks! .
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