Simulating many-body physics with quantum phase-space methods

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Quantum Langevin Equations✄ Raman-modified Heisenberg equations for photon-flux field:Z∂∞= − dt∂x̂φ(t,x) ′ g(t −t ′ )̂φ(t ′ ,x) + ̂Γ(t,x) ± i ∂ 2−∞ 2∂t 2̂φ(t,x)[ Z ∞]+ i dt ′ h(t −t ′ )̂φ † (t ′ ,x)̂φ(t ′ ,x) + ̂Γ R (t,x) ̂φ(t,x)−∞✄ correlations of the reservoir fields:〈̂Γ(ω,x)̂Γ † (ω ′ ,x ′ )〉= αAn (ω,x)δ(x − x′ )δ(ω − ω ′ )〈̂Γ † (ω ′ ,x ′ )̂Γ(ω,x)〉= αGn (ω,x)δ(x − x′ )δ(ω − ω ′ )〈̂Γ R† (ω ′ ,x ′ )̂Γ R (ω,x)〉= αRn (|ω|)[n th(|ω|) + Θ(−ω)]δ(x − x ′ )δ(ω − ω ′ )Simulating many-body physics with quantum phase-space methods 27
Phase-Space Equations✄ apply the phase-space recipe, use coherent-state basis✄ two choices:1. +P(a) exact(b) defined on a doubled phase space(c) maps to normally ordered correlations2. Wigner(a) approximation, good for large mode occupations, short times(b) defined on a classical phase space(c) maps to symmetrically ordered correlationsSimulating many-body physics with quantum phase-space methods 28
Page 1: Simulating many-body physics with q
Page 4 and 5: Theoretical Methods✄ deterministi
Page 6 and 7: Overview✄introduction to phase-sp
Page 8: Probability distributions✄ many w
Page 11 and 12: Simulating many-body physics with q
Page 13 and 14: Density operators for quantum evolu
Page 15 and 16: Stochastic Gauges✄ Mapping from H
Page 17 and 18: Overview✄ introduction to phase-s
Page 19 and 20: General Gaussian operatorsa general
Page 22 and 23: Gaussian Basis II: Thermal operator
Page 24 and 25: Extended phase space−→ λ = (Ω
Page 26 and 27: Application I: photons in a fibreĤ
Page 30 and 31: Wigner Equations✄ get stochastic,
Page 32 and 33: +P Equations✄ get two stochastic
Page 34 and 35: Simulations✄ soliton jitter, soli
Page 36 and 37: Solving the Hubbard Model✄ only t
Page 38 and 39: Applying the Gaussian representatio
Page 40 and 41: Stratonovich Equations✄ Itô stoc
Page 42 and 43: Branching✄ averages are weighted,
Page 44 and 45: Application III: Molecules in a wel
Page 46: Summary✄ Generalised phase-space

Simulating many-body physics with quantum phase-space methods

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