Quantum Dissipative Systems
Takeo Kato (ISSP, University of Tokyo)
Collaborators
Takeshi Matsuo (Chiba University)
Naoki Kimura (Osaka City University)

Open Quantum Systems
Langevin Equation

Environment∂V= ξ(t)M q̈ + η q̇ +∂q

SystemCaldeira Leggett modelp2+ V (q) +H=2M

!  p2

1+ mωj2
 2mj
2jj

π ! Cj2
J(ω) =
δ(ω − ωj ) = ηω
2 j mj ω jGeneral form

J(ω) = Aωs%

xj −&2 C qjmωj2

Ohmic dampings > 1 Superohmics < 1 SubrohmicRelevant Systems/Phenomena✓ Macroscopic Quantum Phenomena (MQP)• Josephson devices, quasi-particle tunneling✓ Fabrication of quantum bits (qubits)• Various systems, various decoherence sources✓ Polaron problem, heavy particle motion in metals✓ Impurity problem in Tomonaga-Luttinger Liquids✓ Dynamical mean-ﬁeld study of phonon systems

Outline of the talk
✓ Striking features of Ohmic damping (Analytic)• Localization transition• Relation to the Kondo problem• Experiments of Josephson devices✓ Simulation of quantum dissipation (Numerical)• Path-integral Monte Carlo (PIMC) method• Resistance-shunted Josephson junctions• Dissipative double-well potential systems

Path-integral formalismZ==!

!

Dq(τ )"!

Dq(τ ) exp(−Seff )Eﬀectiveaction"!
Seff =jDxj (τ ) exp(−S)

dτRetardation eﬀect# !
!
1
M q̇ 2 + V (q) + dτ dτ ! k(τ − τ ! )q(τ )q(τ ! )2

Damping kernel

Ak(τ ) ∼(τ − τ ! )s+1

Long-range interaction along imaginary time
Signiﬁcant eﬀects on quantum mechanics of the system

Localization transition
Simple derivation for double-well systems
Adiabatic bases(two-state approx.)|ΨR ! = |σz = 1!

!

|a+!
a!
|ΨL ! = |σz = −1!
|a−!
apωc < ω < ωc|ΨL !

|ΨR !

(pωc ! ∆)Renormalized tunneling matrix " #$ωc2!
a J(ω)!
∆ = ∆ !a + |a−# = ∆ exp −dω
2pωc 2π! ωa= ∆pαForα > 1 ∆ → 0 (p → 0)!

Particle is localized.

ηa2 Dimensionless
α=damping strength
2π!

Kondo problem
Spin-boson model (two-state approximation)∆H = − σx +2!
iωi"1†
ai ai +
2#a !
Ci√+ σz(ai + a†i )22mi ωii

Equivalent! (by bosonization technique)
H=!

J⊥
†
εk ck,σ ck,σ +k,σ2!

(c†k,↑ ck! ,↓ S − + c†k,↓ ck! ,↑ S + )k,k!

J% ! †+(ck,↑ ck,↑ − c†k,↓ ck,↓ )S z
2!
k,kParameter relations !
∆= ρJ⊥
ωc

α=

2δ
1+π

"2πρJ!
δ=−+ O(J!2 )4

Phase shift

Josephson devices
Classical equation of motion!

Φ02π"2!

"21 Φ0∂VCφ̈ +φ̇ +=0
R 2π∂φΦ0V (φ) = −I φ − EJ cos φ2π
EJ : Josephson energy(2e)2
EC =: Charging energy
2C
1/R : Damping factor

Two control parameters
EJ /EC , α = RQ /R(RQ = h/(2e)2 = 6.45 kΩ)

Yagi et.al. 1997

Phase diagram
Renomalization group(Perturbative approach)Experiment

EJ /EC

α = RQ /R

delocalized
localized(insulator) (superconductor)
Fisher-Zwerger 1985

Yagi et. al. 1997

Phase diagram
Renomalization group(Perturbative approach)Experiment

EJ /EC

Beyond perturbation
theoryEJ /EC ∼ 1
α = RQ /R

delocalized
localized(insulator) (superconductor)
Fisher-Zwerger 1985

Yagi et. al. 1997

Temperature dependence
Miyazaki et.al. 2002
EJ /EC ∼ 1

Phase transition at α = 1

Temperature dependence
Miyazaki et.al. 2002
Kimura and TK, 2004
EJ /EC ∼ 1

α<1α>1α=1Numerical simulation by
Path-Integral Monte Carlo
Phase transition at α = 1

Path-Integral Monte Carlo• Imaginary-time path-integral formulation• Discretization of the particle path
• No approximation except for discretization errors- (Exact in the limit of the large Trotter number)• Application to Helium 4, melting problem, etc.
• Local update, Uniform update, Cluster update- (Particle exchange for Bosonic systems)• Improvement on local updates

Discretization of the pathZ=!

Dq(τ ) exp(−Seff )
"# !
!
!
1
Seff = dτM q̇ 2 + V (q) + dτ dτ ! k(τ − τ ! )q(τ )q(τ ! )2

Discretization
Z=! "jq(τ ) → qj

(0 ≤ j ≤ N − 1)

dqj exp(−Seff ({qj }))Seff ({qj }) =!
i<j

K(i − j)qi qj + ∆τ!

V (qj )

jThe problem is mapped to a classical statistics of ‘polymers’Local update (traditional)
V (q) = (q 2 − 1)2+1
Local change0τ

β−1(new)

1. Change the path locally as qj(old)= qj+!

! ∈ [−δ, δ]
for fixed j

2. Generate a random number g(new)(old)−(S[{qj}]−S[{qj}])g<
min[e, 1]
3. Accept this change if
4. Do this procedure for all path fragments

Uniform update
V (q) = (q 2 − 1)2(new)(old)
qj= qj+!

! ∈ [−δ, δ]
for all j

1. Change the path uniformly as
2. Generate a random number g ∈ [0, 1]
(new)
(old)
−(Spot [{qj}]−Spot [{qj}])g<min[e, 1]
3. Accept this change if(This update becomes important for high temperatures.)Cluster update
V (q) = (q 2 − 1)2

Werner and Troyer, 2005
1. Deﬁne a cluster by choosing the path fragments.
(Details of cluster construction is described next.)
2. All the path fragments of the cluster is changed as qj → −qj

Cluster construction
S=!
i<j=!
i<j=!
i<jK(i − j)qi qjK(i − j)|qi ||qj |σi σjJ(i − j)σi σjqi ≡ |qi |σiσi : Ising variable

Potential term is irrelevant
for sign change of the pathV (−q) = V (q)Wolﬀ algorithm (long-range version)
1. Choose j randomly. Current spin = j. Add spin j to a stack.
2. Try to activte the bond between the current spin j andanother spin k with the probability 1 − e−2Jj,k
3. Try this for all possible spin k. If successed, add spin k toa stack. Remove the current spin j from the stack.
4. Repeat 2 and 3 until the stack become empty.

Comparison
Periodic potential V (q) = cos q
Kimura and TK, 2004

Werner and Troyer, 2005

Without cluster updates

With cluster updatesImproved local update
Double-well potential

1V (q) = (q 2 − 1)2

weight0.8

Approximate decomposition

0.60.4−∆τ V (q)e0.2
0-2-10Z=12−λ(q−σm )2µm em=1

(Matsuo and TK, unpublished)!" #{mj }!

M!

dqj exp(−S̃)

j(Step 1) Path {q_j} is changed for ﬁxed {m_j}(Setp 2) Auxiliary ﬁeld {m_j} is changed for ﬁxed {q_j}Improved local update (Step 1)
Consider the update {qj } → {mj }Weightexp(−S[{qj }, {mj }]) = f ({qj })!

w(qj , mj )jw(qj , mj ) = µmj exp(−λ(xj − σmj )2 )1. From the conﬁguration q_j, choose m_j with the probabilityw(qj , mj )!
t(mj |qj ) =mj w(qj , xj )2. This update can be done eﬃciently, because only local
information is needed for it.

Improved local update (Step 2)
Consider the update {mj } → {qj }
S!
!

!"n!
ncn |q̃n | + λN |q̃n − σ̃n |22#$2$$$λNσ̃n$(cn + λN ) $$q̃n −cn + λN $1. Calculate the Fourier components.

16N 2π2 α2 πncn =sin+|n|βg 2N2qj =!

q̃n e−2πijn/Nnσj =!

σ̃n e−2πijn/Nn

2. Generate a random number with the Gaussian distribution
1λN σ̃nwith the averageand width √2 cn + λN
cn + λN

3. All the Fourier components are updated at once.
(No rejection. Very eﬃcient.)

Eﬃciency

Auto-correlation time

100102
100

10-1

10-2

10-210old update

10-1-3

10-4
102

10

3104

new update

310

auto-correlation time [MCS]

effective auto-correlation time [s]Eﬀective CPU time

V (q) = (q 2 − 1)2101

100 3
10

g=2.0, !=0.0
g=2.0, !=0.5
g=2.0, !=1.0
104
number of site

Traditional O(N 2 log N )Local update O(1)Improved O(N log N )Cluster update O(N)

Localization transition!ω0,g=V0

V0 = 1V(x)V0Dq(τ ) exp(−S[q(τ )])%" # $!
24 dq+ (q 2 − 1)2
S[q(τ )] = dτg dτ!
!
− dτ dτ ! k(τ − τ ! )q(τ )q(τ ! )Two-state limit

gω0

Z=!

xg → 0 : αc = 1

Problem : Localization transition for general g
Is there localization transition even when quantum
ﬂuctuation dominates the potential barrier?

Determine the critical point
12000

Susceptability

910000

1 !
1 ! 2
χ≡" 2
qi qj #/"
qj #N i,j
N j

88000

7600064000

αc = 2.0

1.4

1.5

Long-range Ising universality

1.6b1 |α−αc |ν +b2 |α−αc |−νχ∝e

2000
01.4

1.5

1.6

1.7

N = 214 , 215 , 216 , 217 (! 1.2×)105

ν = 1/2

Kosterlitz-Thouless transtion(Luijten 2001)

Phase diagram3
2.5

Localization transition

Localization

2

Incoherent

1.5
1
0.5
0

Coherent-incoherent
crossoverCoherent
01

2

3

4

1. Localization transition occurs even for a small barrier.
cf. Renormalization approach by Fisher et.al.
2. Tendency of saturation in critical dissipation strengthSummary• Brief introduction to ‘dissipative quantum systems’• Brief review on Path-Integral Monte Carlo (PIMC)• Recent improvement of local updates• Future problems: Application to other systems- Impurity problem in Tomonaga-Luttinger Liquid- Field theoretical model, e.g., phi4 model- Phonon: Nonlinearity, electron-phonon