Freeing semiclassical field theory from the UV catastrophe and cutoff dependence
Piotr Deuar Joanna Pietraszewicz
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
Inspiration from: discussions at FINESS meetings Nick Proukakis collaborations Mariusz Gajda with Mirek Brewczyk Kazimierz Rzążewski Outline
● Classical wave fields for bosons (background)
● UV and cutoff issues
● Trying to fix it using truncated Wigner
● Fixing it
● Simple examples
● Serious example: breathing mode at high temperature in 3d
16.03.2018 Swinburne University of Technology, VIC Australia 2/24 Simplest semiclassical method = “classical” wave fields
Replacement:
Reasoning:
16.03.2018 Swinburne University of Technology, VIC Australia 3/24 Simplest evolution equations
● Dynamics:
* Gross-Pitaevskii equation (GPE)
● Thermal ensemble:
* run equations until stationary
More fancy method: Truncated Wigner representation
includes quantum fluctuations (for short times)
16.03.2018 Swinburne University of Technology, VIC Australia 4/24 The point of semiclassical field theory
● Nonperturbative
● T > 0
● Single shots
● No special conditions, symmetries, geometries
● Scales very well with system size
1d gas – phase domains after cooling
Liu, Donadello, Lamporesi, Ferrari, Gou, Dalfovo, Proukakis, arXiv:1712.08074 1d gas – thermal equilibrium 16.03.2018 Swinburne University of Technology, VIC Australia 5/24 Classical wave field models
ZNG PGPE + Hartree-Fock SGPE nonperturbative low E modes many-modes at low E high energy description dynamic static reservoir dynamic Low E – high E coupling temperature zero & set extracted set cutoff dependence no some more numerical effort large medium medium
16.03.2018 Swinburne University of Technology, VIC Australia 6/24 Cutoff dependence of observables
e.g. 1d bose gas detailed study Pietraszewicz, PD, PRA 92, 063620 (2015) (comparison to Bethe ansatz) Pietraszewicz, PD, arXiv:1707.01776
quasicondensate quantum turbulent r r o o r r r r e e
e e v v i i t t a a l l e e r r
momentum cutoff momentum cutoff
16.03.2018 Swinburne University of Technology, VIC Australia 7/24 Ultraviolet catastrophe
Quantum Bose-Einstein distribution ½k T energy per PARTICLE B Classical field Rayleigh-Jeans distribution k T energy per MODE B
particles kinetic energy (idgas) ) ) k k ( ( N E
momentum k / √(k T) momentum k / √(k T) B B
Also applies to truncated Wigner at long times
16.03.2018 Swinburne University of Technology, VIC Australia 8/24 1st fix approach: Optimized cutoff Pietraszewicz, PD, arXiv:1707.01776
● Uniform 1d Bose gas: comparison to exact results
Relative error
Global error bound
Global accuracy of observables min RMS Optimal cutoff opt f c
16.03.2018 Swinburne University of Technology, VIC Australia 9/24 SGPE equation
A variety that is amenable to useful amendments
Hamiltonian evolution energy dependent loss to reservoir complex noise
Local frequency extraction
16.03.2018 Swinburne University of Technology, VIC Australia 10/24 2nd fix approach: stable truncated Wigner
● Collaboration: Joan Pietraszewicz, PD (IF PAN) Nick Proukakis (Newcastle) King Ng, Peter Drummond (Swinburne)
● Re-derived SGPE, taking care to preserve Wigner distribution
● Obtain amended stochastic equation:
Explicit particle correction to energy
Wigner correction to expectation values
● Preserves virtual vacuum noise and quantum fluctuations, even as t→ ∞
16.03.2018 Swinburne University of Technology, VIC Australia 11/24 Add quantum treatment of c-field modes?
● Full truncated Wigner representation of the SGPE model
● Wave-particle duality included PD, Pietraszewicz, Proukakis, Świsłocki, in preparation (2018)
Antibunching present But cutoff dependence same as before! → true quantum fluctuations ) ) x , x , 0 ( 0 ) ( 2 ) ( 2 ( g g
distance cutoff
The remaining culprit: classical BATH
16.03.2018 Swinburne University of Technology, VIC Australia 12/24 Three interpretations of the reservoir
● Direct Models the effect of atoms above the energy cutoff but – their number tends to be small
● Added noise Models the effect of extra modes missing from the model e.g. transverse modes in a low-dim system but – their number tends to be small
● Constraint (state engineering) Its role is to force the high energy modes into known and wanted states
16.03.2018 Swinburne University of Technology, VIC Australia 13/24 Classical bath
SGPE master equation
frequency extractor
Consider: 1 mode coupled to a bath
lattice analogy
Substitute classical occupations
Obtain SGPE master equation
16.03.2018 Swinburne University of Technology, VIC Australia 14/24 Use a quantum bath
Substitute quantum occupations
Obtain amended SGPE
“Only” real difference
Bears similarity to the “generalised” SGPE obtained by Duine+Stoof PRA 65, 013603 (2001)
16.03.2018 Swinburne University of Technology, VIC Australia 15/24 Ideal 1d trapped gas (cutoff dependence)
exact plain SGPE modified SGPE cutoff from arXiv:1707.01776
16.03.2018 Swinburne University of Technology, VIC Australia 16/24 Interacting 1d trapped gas
plain SGPE modified SGPE cutoff from arXiv:1707.01776
16.03.2018 Swinburne University of Technology, VIC Australia 17/24 Interacting 3d trapped gas
density in momentum space higher temperature
Kinetic energy density
plain SGPE – to scale
16.03.2018 Swinburne University of Technology, VIC Australia 18/24 Litmus test for thermal cloud: m=0 collective mode
Jin et al, PRL 78, 764 (1997)
adapted from: Jin et al, PRL 78, 764 (1997)
id so nu si g in ay condensate thermal cloud ec d fit
The above figures from: Jackson, Zaremba, PRL 88, 180402 (2002) Bezett, Blakie, PRA 79, 023602 (2009)
16.03.2018 Swinburne University of Technology, VIC Australia 19/24 Results of m=0 mode tests in the past
ZNG PGPE + HF Jackson, Zaremba, PRL 88, 180402 (2002) Bezett, Blakie, PRA 79, 023602 (2009)
PGPE (higher cutoff) Karpiuk, Brewczyk, Gajda, Rzążewski, PRA 81, 013629 (2010) 2nd order Bogoliubov Morgan, Rusch, Hutchinson, Burnett PRL 91, 250403 (2003)
16.03.2018 Swinburne University of Technology, VIC Australia 20/24 m=0 mode frequency
condensate thermal
condensate
thermal
Experiment: Jin, Matthews, Ensher, Wieman, Cornell, PRL 78, 764 (1997) This work (2018) ZNG: Jackson, Zaremba, PRL 88, 180402 (2002) 2nd order Bogoliubov: Morgan, Rusch, Hutchinson, Burnett PRL 91, 250403 (2003) PGPE+HF: Bezettt, Blakie, PRA 79, 023602 (2009) PGPE: Karpiuk, Brewczyk, Gajda, Rzążewski, PRA 81, 013629 (2010)
16.03.2018 Swinburne University of Technology, VIC Australia 21/24 m=0 mode damping
condensate
condensate
thermal
thermal
Experiment: Jin, Matthews, Ensher, Wieman, Cornell, PRL 78, 764 (1997) This work (2018) ZNG: Jackson, Zaremba, PRL 88, 180402 (2002) 2nd order Bogoliubov: Morgan, Rusch, Hutchinson, Burnett PRL 91, 250403 (2003) PGPE+HF: Bezettt, Blakie, PRA 79, 023602 (2009) PGPE: Karpiuk, Brewczyk, Gajda, Rzążewski, PRA 81, 013629 (2010)
16.03.2018 Swinburne University of Technology, VIC Australia 22/24 Cutoff dependence
Experiment Jin et al 1997
open: thermal Bezett + Blakie 2009 Karpiuk, Brewczyk, Gajda, Rzążewski 2010 This work
filled: condensate Experiment
filled: condensate
open: thermal
16.03.2018 Swinburne University of Technology, VIC Australia 23/24 Concluding remarks
• Entire system can be treated this way within a c-field framework coherent + thermal
• Cutoff dependence slain when lattice sufficiently large
• Numerical effort comparable to SGPE though, lattice may need to be larger
• Still lacks wave-particle duality but this is harder to spot than expected
arXiv:1804.XXXXX
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