Simulating many-body physics with quantum phase-space methods

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Extended phase space−→ λ = (Ω,α,α + ,n,m,m + )=⇒ Hilbert-space dimension: 2 M for fermions, N M for bosons=⇒ phase-space dimension: 2(1−M +2M 2 ) for fermions, 2(1+3M +2M 2 )for bosons✄ Moments:〈〉〈â i 〉 = E [α i ]â † i â j = E [ α + i α ]j + n i j〈â i â j 〉 = E [α i α j + m i j ]Simulating many-body physics with quantum phase-space methods 23
Overview✄ introduction to phase-space representations✄ density operator description of quantum evolution (3 classes)➜ static, unitary and open✄ Gaussian operator bases (3 types)➜ coherent, thermal and squeezed✄applications (3 examples)➜ pulse propagation in optical fibres (photons)➜ Hubbard model (atoms)➜ simple atomic-molecular dynamics (molecules)Simulating many-body physics with quantum phase-space methods 24
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 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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