Simulating many-body physics with quantum phase-space methods

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Gaussian Basis II: Thermal operatorŝΛ = |I ± n| ∓1 : exp[â(I ∓ I − [I ± n] −1) ]â †:✄ now have a phase space of variances: −→ λ = (Ω,n)✄ defined for bosons (upper sign) and fermions (lower sign)✄ moments:〈〉â † i â j= E [n i j ] ,〈â † i â† jâ jâ i〉= E [n ii n j j ± n i j n ji ]✄ suitable for cold atomsSimulating many-body physics with quantum phase-space methods 21
Gaussian Basis III: General form (including squeezing)̂Λ( −→ ∣∣∣ λ ) = Ω√σ∣∓1: exp[δâ ( )† I ∓ I − σ −1]δâ/2:relative displacement: δâ = â − αannihilation and creation operators: â =()â 1 ,...,â M ,â † 1 ,...,â† Mcoherent offset: α = ( α 1 ,...,α M ,α + 1 ,...,α+ M), (α = 0 for fermions)covariance: σ =[ n T ± I mm + I ± n], I =[ ±I 00 I].upper signs: bosons; lower signs:fermionsSimulating many-body physics with quantum phase-space methods 22
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 24 and 25: Extended phase space−→ λ = (Ω
Page 26 and 27: Application I: photons in a fibreĤ
Page 28 and 29: Quantum Langevin Equations✄ Raman
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

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