Simulating the complete quantum dynamics of very large systems with phase space sampling
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Simulating the complete quantum dynamics
of very large systems with phase space sampling
Piotr Deuar
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
Bose-Hubbard work:
Michał Matuszewski
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
Alex Ferrier, Marzena Szymańska
University College London, UK
Giuliano Orso
LMPQ, Université de Paris, France
Outline
●
Phase-space representations of full quantum mechanics
* can we beat exponential scaling?
* specifically, using the positive-P representation
* example: collisions of Bose-Einstein condensates
●
The case of the driven-dissipative Bose-Hubbard model
* physics of the model
* performance of full quantum simulations
20.10.2021 tfai seminar, Vilnius University, Lithuania 2/25
Difficulties of simulating quantum mechanics
Suppose we have a system with M sites, each site can have 0,1,..., (d-1) particles
orbitals (2,3,...,d-level systems)
modes
How many variables do we need to describe the state?
●
Classical physics: configuration real variables
●
Closed quantum system: state vector complex variables
●
Open quantum system: density matrix complex variables
How can there be hope?
20.10.2021 tfai seminar, Vilnius University, Lithuania 3/25
Quantum complexity
(Type 1) Universal quantum computer, Shor’s, Grover’s algorithms, etc...
* needs precise knowledge of microscopic subsystem observables
(Type 2) Quantum behaviour of (most?) experimental systems
* knowledge of bulk or locally averaged quantities suffices
* statistical uncertainty mirrors experimental reality
Lopes, Imanaliev, Aspect, Cheneau, Boiron,
Westbrook, Nature 520, 66 (2015)
Cabrera, Tanzi, Sanz, Naylor, Thomas,
Cheiney, Tarruell, Science 359, 301 (2018)
20.10.2021 tfai seminar, Vilnius University, Lithuania 4/25
Complete quantum mechanics with limited precision
Approach full quantum predictions as some technical parameter is changed
●
“Asymptotic approach”
* Reduce basis set but treat what happens inside exactly
* matrix product states, “DMRG”
* multi-configuration methods, MCDHF...
* ....
●
“Stochastic approach”
* Use full basis but randomly choose a few configurations
* path integral Monte Carlo
* phase space methods (P, Q, Wigner...)
* ...
●
Bit of both:
* Diagrammatic Monte Carlo
* Quantum trajectories
* ...
●
Each approach has its own quirky pros and cons
20.10.2021 tfai seminar, Vilnius University, Lithuania 5/25
Positive-P representation 1/3: configurations
M subsystems (modes, sites, volumes) labeled by j
Coherent state basis, complex, local
“ket” amplitude Local operator kernel
“bra” amplitude
full system configuration
Full density matrix
Correlations between subsystems are all in the distribution of configurations
The distribution is positive, real let’s SAMPLE IT !
20.10.2021 tfai seminar, Vilnius University, Lithuania 6/25
Positive-P representation 2/3 : Identities and observables
Crucial element: differential identities
Follow from the operator kernel:
Observable: occupation
we have S samples of
the configurations
distributed according to
20.10.2021 tfai seminar, Vilnius University, Lithuania 7/25
Positive-P representation 3/3 : Dynamics. One site example
Density matrix ↔ distribution P+ for the fields ↔ random samples of the fields
Master equation: Bose-Hubbard site
dissipation
apply
identities
Fokker Planck equation
deterministic (ket) deterministic (bra) diffusion (quantum noise)
stochastic
Stochastic (Langevin) equations: correspondence
different noises
White noise
mean field part quantum noise part
20.10.2021 tfai seminar, Vilnius University, Lithuania 8/25
Overall properties for a many-body system
Density matrix ↔ distribution P+ for the fields ↔ random samples of the fields
●
From variables we go to S samples x 2M
●
Therefore ---- Scales extremely well (linear with the number of modes M )
●
Requires no particular symmetries, explicit time-dependence trivially added
●
BUT can be unstable, usually no good for long-time equilibrium.
[ more on this later ! ]
●
Still..... particularly suited to large and “dirty” problems.
Dirty problems done dirt cheap!
20.10.2021 tfai seminar, Vilnius University, Lithuania 9/25
Comparison to path integral Monte Carlo
Path integral Monte Carlo Phase-space representation
t
t
Configuration size: Configuration size:
N particles x T time slices M modes
Typical algorithm: Metropolis Typical algorithm: Langevin equations
weights no weights for dynamics
--> phase problem in dynamics --> no sign problem there
no noise amplification problem noise amplification if dissipation insufficient
20.10.2021 tfai seminar, Vilnius University, Lithuania 10/25Example: ultracold gases. A closed system
local two-body
Boson field trap potential kinetic energy interaction energy
positive-P equations
Mean field GP equation Rest of quantum mechanics
relies on the noise
PD, Drummond, PRL 98, 120402 (2007)
20.10.2021 tfai seminar, Vilnius University, Lithuania 11/25BEC collision expt with He* – makes pairs, detects single atoms
Multi channel plate
short time momentum distribution
=
long time position distribution
Single-atom tomography
E ~ 20eV
4
He in metastable state
Single atom detection efficiency ~ 12%
Supersonic collision
momentum distribution
Perrin, Chang, Krachmalnicoff, Schellekens, Boiron,
Aspect, Westbrook, PRL 99, 150405 (2007)
t=0 Bragg pulse t=0.3 s expansion
20.10.2021 tfai seminar, Vilnius University, Lithuania 12/25positive-P simulations of the halo, 105 atoms 106 modes
●
Above the speed of sound a condensate no longer behaves as a superfluid
k-space density
BEC k correlations in the halo
BEC
-k
PD, Ziń, Chwedeńczuk, Trippenbach, EPJD 65, 19 (2011) PD, Drummond, PRL 98, 120402 (2007)
20.10.2021 tfai seminar, Vilnius University, Lithuania 13/25Achilles heel – noise amplification limits simulation time
single site
here be some
stupid location
in phase space
BEC collission
long time dynamics excluded
Typical simulation time limitation in
Bose-Hubbard system
PD, Drummond, J Phys A 39, 1163 (2006)
20.10.2021 tfai seminar, Vilnius University, Lithuania 14/25Dealing with noise amplification
●
Various ways have been developed to improve this performance:
PD, Drummond, PRA 66, 033812 (2002), J Phys A 39, 2723 (2006);
* stochastic Gauges PD et al, PRA 79, 043619 (2009); Wuster, Corney, Rost, PD, PRE 96, 013309 (2017)
* quantum interpolation PD, PRL 103, 130402 (2009);
Ng, Sorensen, PD, PRB 88, 144304 (2013)
●
Or it can be optimal to just use approximate representations:
Sinatra, Lobo, Castin, J Phys B 35, 3599 (2002)
* truncated Wigner Norrie, Ballagh, Gardiner, PRA 73, 043617 (200
͑(2006͒), PRL 6͒), PRL ), PRL 94, 040401 (2005)
* STAB (Stochastic adaptive Bogoliubov) PD, Chwedeńczuk, Trippenbach, Zin, PRA 83, 063625 (2011)
Kheruntsyan et al, PRL 108, 260401 (2012)
●
It was also found that simulation time grows
with dissipation to an external bath:
PD, Drummond, J Phys A 39, 1163 (2006)
●
But not really tested at the time .......
20.10.2021 tfai seminar, Vilnius University, Lithuania 15/25Driven dissipative Bose-Hubbard model
Arrays of optical cavities
Vincentini, Minganti, Rota, Orso,
Ciuti, PRA 97, 013853 (2018)
Polaritons in micropillars
Also structured lattices – e.g. Lieb lattice
Ultracold atoms in optical lattice with forced losses
Baboux, Ge, Jacqmin, Biondi, Galopin,
Labouvie, Santra, Heun, Ott, Lemaitre, Le Gratiet, Sagnes, Schmidt,
PRL 116, 235302 (2016) Tureci, Amo, Bloch, PRL 116, 066402 (2016)
20.10.2021 tfai seminar, Vilnius University, Lithuania 16/25“Liquid” and “gas” phases
single cavity
lattice,
Biondi, Blatter, Tureci, Schmidt, PRA 96, 043809 (2017) mean-field decoupling approx
20.10.2021 tfai seminar, Vilnius University, Lithuania 17/251st order dissipative phase transition
steady state fluctuating behaviour truncated Wigner approx calculations
Vincentini, Minganti, Rota, Orso, Ciuti, PRA 97, 013853 (2018)
time trace in steady state:
one site
lattice mean
distribution of
steady state densities
20.10.2021 tfai seminar, Vilnius University, Lithuania 18/25Critical slowing down: 2d not 1d
transient dynamics
difference between
n(t) starting from
no saturation
vacuum
for large system
and
nss steady state
critical
slowing
down
saturates
for large system
truncated Wigner approx calculations
Vincentini, Minganti, Rota, Orso, Ciuti, PRA 97, 013853 (2018)
Issues though
●
all calculations in large systems use quite strong approximations
●
mean-field decoupling: crude local density approx.
●
truncated Wigner: throws away part of dynamics, UV divergence
known to be inaccurate if occupations are low
20.10.2021 tfai seminar, Vilnius University, Lithuania 19/25positive-P representation of the driven dissipative BH model
PD, Ferrier, Matuszewski, Orso, Szymańska, PRX Quantum 2, 010319 (2021).
●
Evolution equations for samples
White Gaussian noise deals with interparticle collisions
The rest of the equations is basically mean field
20.10.2021 tfai seminar, Vilnius University, Lithuania 20/25Positive-P simulations stabilised by the dissipation
stationary state can be reached
and studied
in a full quantum description
instability
seems stable over here triggered
γ too low
increasing
dissipation single Bose-Hubard site
20.10.2021 tfai seminar, Vilnius University, Lithuania 21/25Regime of stability for the positive-P approach
●
Remarkably, stability is determined by single-site parameters
( UNSTABLE )
( STABLE )
Regime of stability:
PD, Ferrier, Matuszewski, Orso, Szymańska, PRX Quantum 2, 010319 (2021).
20.10.2021 tfai seminar, Vilnius University, Lithuania 22/25Phase space methods - regimes of applicability
High N
●
Stability is largely
independent of coupling J
High N ●
computational effort scales
linearly with the size of the
system.
●
Computational effort is
independent of
dimensionality.
●
Time-dependent system
Low N parameters and non-
uniformity are trivial to
Here, MPS etc
implement.
can be
accurate Low N
pos-P Stability condition:
PD, Ferrier, Matuszewski, Orso, Szymańska, PRX Quantum 2, 010319 (2021).
20.10.2021 tfai seminar, Vilnius University, Lithuania 23/25Large systems – example simulation 256 x 256 lattice
non-uniform
driving F
(illumination) density matrix elements
slice
GPE mean field
full quantum (positive-P)
local density quantum model
LeBoite, Orso, Ciuti, PRL 110, 233601 (2013).
2 body correlations
site occupation
PD, Ferrier, Matuszewski, Orso, Szymańska, PRX Quantum 2, 010319 (2021).
20.10.2021 tfai seminar, Vilnius University, Lithuania 24/25Summary and outlook
• Quantum dynamics of huge systems can be done
under the right conditions, though stability issues
• Full quantum calculations of large Bose-Hubbard models
positive-P method found to be stable with sufficient dissipation
scalable. e.g. 105 sites is easy
truncated Wigner accurate in complementary regimes
• Other dissipative models may also be possible.
e.g. spins, Jaynes-Cummings-Hubbard
Schwinger bosons Ng, Sorensen, J Phys A 44, 065305 (2011)
Huber, Kirton, Rabl, SciPost Phys 10, 045 (2021) (truncated Wigner)
Ng, Sorensen, PD, PRB 88, 144304 (2013)
SU(n) positive-P-like representations Begg, Green, Bhaseen arXiv:2011.07924 (stochastic gauges)
• Phase space approach also very good for multi-time correlations
similar form to Heisenberg equations
References:
●
PD, Ferrier, Matuszewski, Orso, Szymańska, Fully Quantum Scalable Description of Driven-Dissipative Lattice Models,
PRX Quantum 2, 010319 (2021).
●
PD, Multi-time correlations in the positive-P, Q, and doubled phase-space representations, Quantum 5, 455 (2021).
20.10.2021 tfai seminar, Vilnius University, Lithuania 25/25You can also read
