Efficient simulation of quantum dynamics system using GPU

Lipeng Chen

Division of Materials Science

Nanyang Technological University, Singapore

Schrodinger Equation Liouville equation

1. MCTDH

2. propagator

1. QUAPI

2. HEOM

Exciton diffusion in chlorosome antennas

Single molecule spectra

Dissipative effect on the dynamics at a conical intersection

Polaron dynamics of Holstein model

Dynamics of sub-Ohmic spin boson model

Natural Light Harvesting Complexes

Fenna-Matthews-Olson (FMO)Trimer complex in green sulfur bacteria 4

Light harvesting system in purple bacteriaLHCII in green plants

Chlorosomes antennas in green sulfur bacteria

5

Exciton diffusion in chlorosomes antennas

6S. Ganapathy et al, PNAS. 2009, 106 , 8525

Variational Parameters

7

Quantum state of the system can be approximated, by the Davydov D1 Ansatz

SchrÖdinger Equation

Dirac-Frenkel time-dependent variational method

Equations of motion for the variational parameters

( ) ( )i t H tt

( ) ( )2iL t H t

t

�

( ) 0m

m

d L Ldt

SMX: Higher performance and efficiency increase processing cores reduce control logic

Dynamic parallelism: spawns new threads without going back to the CPU

Hyper-Q: increases GPU utilization provide streams access to 32 independent hardware work queues or MPI ranks.

KEPLER - The World‘s fastest, most efficient HPC architecture

30 rings of 18 sites each540 sites in total

Phonon bandwidth: W=0.5Coupling: g=0.4

18 rings with 18 site each (324 sites)

Coherence size: 175 sites

1

( )

,

20

( ) ( , ) 0 0

( )( ) [coth (1 cos ) (sin )]2

0 ( )

S

S

H tg tph ph

n m

B

iH tph D

R t e C n m n e m

Dg t d t i t tk T

e n t

Single molecule spectra

R Hildner, D Brinks, J. B. Nieder, R. J. Cogdell, N F van Hulst, Science 340, 1448 (2013)

Ultrafast phase coherent excitation of individual LH2 complexes

Emission of single LH2 complexes as a function of delay time

Histogram of the oscillation period T

Single molecule spectra

1

2

†

1 2

1

( )2

[ ( ) ( ) ]( ) ( ) ( )

( ) ( )

( ) ( )

F

i t

i t

H t X t Xt t t

t E t e

t E t e

Field-matter interaction Hamiltonian

Liouville Equation

( ) [ , ]Fd t i H Hdt

( ) { ( )}

( ) 2 ( ) ( )

P t Tr X t

S Im dt t P t

2

1

( ) ( )kk

k

S S

2

1, 1

1 1 10

( )

,

20

( ) ( )

2Re ( ) ( ) ( )

( ) ( , ) 0 0

( )( ) [coth (1 cos ) (sin )]2

S

aba b

ab a b

H tg tph ph

n m

B

S

dt dt t t t R t

R t e C n m n e m

Dg t d t i t tk T

1

1

†

0 ( )

( ) ( ) exp [ ( ) . .] 0

SiH tph D

D n nq q phn q

e n t

t t n t b H c

D1 dynamics

1 1( ) ( ) ( )

2

( ) 0

( ) 0

D S D

nn

nqnq

iL t H tt t

d L Ldt

d L Ldt

� � � � � � � � � � � � � �

Pump pulse 1Pump pulse 2

Signal

Room temperature LHCII absorption spectra

Single molecule signal for 3 values of the exciton-phononcoupling strength (decreasing)

same signals but calculated without the heat bath

the effect of the single molecule orientation The effect of disorder

Effect of a dissipative environment on the dynamics at a conical intersection

Polaron dynamics of Holstein model

J: transfer integralN: number of sitesϕ : off-diagonal couplingg: diagonal couplingW: phonon band width

A linear dispersion phonon band is assumed as

0

21 1q

qW

Multitude of Davydov D1 and D2 trial states

1D t 2D t

† †2 ,

0

0 ( )exp 0M

Mn i n iq q iq qex ph

n i q

D a t b b

† †1 ,

0

0 ( )exp 0M

Mn i n inq q inq qex ph

n i q

D a t b b

Multi-D2 and multi-D1 Ansatz with the multiplicity number M

Measurement of the validity for the trial state

1,2( ) ( )

Mtt i H D

t t t

Multi-D1 results in the diagonal coupling case

Case of J=0.5. W=0.5, g=0.1 and ϕ=0

Multi-D2 results in the diagonal coupling case

In the off-diagonal coupling case with g=W=J=0 and ϕ≠0

1,2( ) ( )

( )=max ( )

Mt

ph

t i H D

t t t

tE t

Reduced density matrix

Feynman-Vernon influence functional

HEOM treatment of Holstein model

Auxiliary density matrix

1

N

q trunq

m N

Terminator

Ntrun

N

3 4 5 6 7 8 9 10

6 455 1820 6188 18564 50388 125970 293930

8 969 4845 20349 74613 245157 735471

9 1330 7315 33649 134596

10 1771 10626 53130 230230

12 2925 20475 118750 593770

16 6545 58905 435897

Total number of hierarchy

1 !! 1 !

trun trun

trun

N N NN N N

0.2, 0.5, 0.1, 0J W g

0.5, 0.5, 0.1, 0J W g

0.2, 0.5, 0.2, 0J W g

0.5, 0.5, 0.2, 0J W g

Dynamics of sub-Ohmic spin-boson model

Δ: the tunneling constantε: the spin bias λl: the coupling strength of the spin to the bath

 

Multi-D1 results v.s D1 results Sub-Ohmic bath s=0.25 Δ=0.1 Relative error

Multi-D1 compare with HEOM

Factorized initial condition bath

Multi-D1 compare with HEOM

Factorized initial condition bath

Multi-D1 compare with PIMC

Polarized initial condition bath

Entropy of the system

Factorized initial condition

Some useful libraries