Liouvillian analysis of the relaxation
time in open quantum systems
Takashi Mori (Riken)
joint work with Tatsuhiko Shirai (Waseda)
Novel Quantum States in Condensed Matter 2022

Open quantum many-body systems
cold atoms: highly controllable isolated quantum systems
foundation of statistical mechanics (thermalization)
quantum control of many-body systems (Floquet engineering)
important problem: taking into account the effect of dissipation
unavoidable dissipation in condensed matters
controlled dissipation in recent cold-atom experiments→dissipation engineering
environmentsystemor bathJ. T. Barreiro et al., Nature (2011)
G. Barontini et al., Phys. Rev. Lett. (2013)
T. Tomita et al., Sci. Adv. (2017)

Outline
relaxation time τ in open quantum systemsdρS(t)= − ℒρS(t)dt

Liouvillianℒ → τ?

➤ Gap discrepancy problem and its resolution
TM and T. Shirai, Phys. Rev. Lett. 125, 230604 (2020)

g: spectral gap of ℒ “Liouvillian gap”

τ ≲ g −1➤ general upper bound on the auto-correlation function in thesteady state: “instantaneous decay rate”
TM and T. Shirai, to appear soon on arXiv

Lindblad equation (GKLS equation)
dynamics of open quantum systems
Markov approximation: Lindblad (GKLS) equation
dρ(t)
1 †
†
= − i[H, ρ(t)] +Lk ρ(t)Lk − {Lk Lk, ρ(t)}∑()
dt2kLk : Lindblad jump operator
particle loss Lk = bi

particle gain Lk = bi†

two-body loss Lk = (bi)2dephasing Lk = bi†biboundary dissipation

bulk dissipation

dissipation only at the boundaries

dissipation in the bulk

Liouvillian and its eigenmodes
dρ(t)
1 †
†
= − i[H, ρ(t)] +Lk ρ(t)Lk − {Lk Lk, ρ(t)}∑()dt2k= − ℒρ(t)ℒ Liouvillian (superoperator)
eigenvalues and eigenmodesℒρn = λn ρn0 = λ0 < Re λ1 ≤ Re λ2 ≤ …
ρ(t) = e −ℒtρ(0) = ρss +∑n≠0

Cne −λntρnn = 0: unique steady state ρ0 = ρssn ≠ 0: decaying eigenmodes; decay rate = Re λnLiouvillian gap and relaxation time
ρ(t) = e −ℒtρ(0) = ρss +∑n≠0

Cne −λntρn0 = λ0 < Re λ1 ≤ Re λ2 ≤ …

Liouvillian gap g: the smallest decay rateg = Re λ1“asymptotic decay rate”

relaxation time τ: ∥ρ(τ) − ρss∥tr = ϵ∥ρ(t) − ρss∥tr ≲ e −gt (t → ∞)fi

τ ≲ g −1

ϵ: a xed threshold̂ tr = Tr | Â |trace norm ∥A∥boundary dissipated systems
e.g. 1D spin systemenergy ow

Lbulk Hamiltonian: non-integrable (chaotic)
diffusive transport of energy
relaxation time scales as τ ∼ L 2

flLiouvillian gap should be scaled as g ∼ L −2

gap discrepancy problem
numerical evidence of g ∼ L −1 in chaotic spin chains withboundary dissipation
M. Znidaric, Phys. Rev. E (2015)(numerical results)τ ≫ g −1τ ≲ g −1

(naive expectation)

staggered XXZ model
L−1

Ly yx xz zz
H=σσ+σσ+Δσσ+bσi i+1)i i+1
∑ ( i i+1∑ i ii=1
i=1

with boundary dephasingL̂ 1 =γ σ ̂z1

L̂ 2 =γ σ ̂zLg ∼ L −1 when Δ < 1This is not exceptional:
many models show g ≫ L −211bi = −1, − ,0, − 1, − ,0,…
()22

modelbulk Hamiltonian: 1D hard-core Bose-Hubbard11††
H=−hb b + bi+1bi) + Un−n −∑ ( i i+1∑ ( i 2 ) ( i+1 2 )i=1i=1L−1L−111††−h′ (bi bi+2 + bi+2bi) + U′n−n −∑∑ ( i 2 ) ( i+2 2 )i=1i=1L−2L−2h = U = 1, h′ = U′ = 0.24
L

L

LL−1
†
N=ni =bi bi is conserved: the sector of N = or∑∑22i=1i=1This model is chaotic L. F. Santos and M. Rigol, Phys. Rev. E (2010)
boundary dephasingγb1†b1,￼￼￼￼

L1 =L2 =γbL†bL

γ=4

Liouvillian gap

g1slope -1.6

0.1

0.01
510Lg∼L−1.6≫L−2

time evolution of the trace distance: two timescalesdT (t) = ∥ρ(t) − ρss∥tr

L=4
L=6
L=8
L=10
L=12
L=14100dT

initial plateau Δt ∼ L 2
sudden decay Δt ∼ O(1)asymptotic decay：g−1

∼L

1.6

overall relaxation time ∝ L210-1

010

20

30

40t

dashed lines: slope -gThe crossover time to the asymptotic regime diverges
in the thermodynamic limit50understanding from eigenmode decays
ρ(t) = e −ℒtρ(0) = ρss +∑n≠0Cne −λntρn∥ρ(t) − ρss∥tr ≤∑n≠0| Cn | e −Re λnt

normalization for right eigenmodes: ∥ρn∥tr = 1⟨πn, ρ(0)⟩Cn =⟨πn, ρn⟩bi-orthogonality

πn: left eigenmodes

πn†ℒ = λnπn†inner product ⟨A, B⟩ = Tr [A †B]⟨πn, ρm⟩ = 0 for n ≠ mnormalization for left eigenmodes:∥πn∥op = 1

trace norm and operator norm are dual: | ⟨A, B⟩ | ≤ ∥A∥op∥B∥trupper bounds on expansion coef cients| ⟨A, B⟩ | ≤ ∥A∥op∥B∥tr| ⟨πn, ρ(0)⟩ | ≤ ∥πn∥op∥ρ(0)∥tr = 1⟨πn, ρ(0)⟩Cn =⟨πn, ρn⟩

1| Cn | ≤⟨πn, ρn⟩vanishingly small eigenmode overlap → diverging expansion coeff.
ρ(t) = e −ℒtρ(0) = ρss +Cne −λntρn∑n≠0decay time for nth eigenmode:ln | Cn |
ln Φn
τn ∼≤
=: τ̄nRe λnRe λnΦn = ⟨πn, ρn⟩−1maximum relaxation time is identi ed with τ ≲ τmax := max τ̄nfifin≠0Comparison with the previous formula
previous formulaln(1/ϵ) ln(1/ϵ)τmax = max=gn≠0 Re λnnew formulaln(Φn /ϵ)τmax = max
Re λnn≠0Φn = ⟨πn, ρn⟩−1

Small eigenmode overlap can alter the relaxation time
If Φn depends on the system size L, it even changes the scaling in Levidence of large Cn (= small eigenmode overlap)
Φn = ⟨πn, ρn⟩−1

1D Bose-Hubbard with boundary dephasing

300

L=9
L=8
L=7
L=6
L=5
L=4ln(\n)/L2\n2000.1

130.051000ln(Φn)/L 2

L=9
L=8
L=7
L=6L=5
L=400

12|Re hn|34typically Φn = e-0.05
0O(L 2)

2|Re hn|

4at Re λn = O(1)ln Φnτ̄n == O(L 2)Re λndiffusive relaxation time!

summary of diffusive relaxation time
TM and T. Shirai, Phys. Rev. Lett. (2020)➤ System size dependence of the relaxation time is notnecessarily determined by the Liouvillian gap(gap discrepancy problem)➤ Trace distance from the steady state shows an initial plateau

over a time interval Δt ∼ L 2, and then rapidly decays andenters the asymptotic regime with the decay rate g➤ Explosively large expansion coef

Re λn = O(1) leads to τ̄n ∼ L 2

cients | Cn | ∼ e

O(L 2)➤ Vanishingly small eigenmode overlaps ⟨πn, ρn⟩ = efifi

rise to such large expansion coef cientsat−O(L 2)

givecomment 1: exceptional points
Large expansion coef cients are typically observed near an
exceptional point
Near an exceptional point | λn − λm | = ϵ, the eigenmode overlap
typically vanishes as ⟨πn, ρn⟩ ∼ ϵ−O(L 2)

Vanishingly small eigenmode overlaps ⟨πn, ρn⟩ = efound inthe boundary dissipated 1D Bose-Hubbard model are not due
to an exceptional point
100sminsmedsmin = min | λn − λm | = e

10-1n,m:n≠m

10-2

10-3

4

5

67System size

8fi

10-4

9−O(L)≫ ⟨πn, ρn⟩ = e−O(L 2)comment 2: effect of non-Hermiticity
Hermitian caseπn ∝ ρn∥πn∥op = 1,∥ρn∥tr = 11⟨πn, ρn⟩ >= e −O(L) D dimension of the Hilbert spaceDGap discrepancy can happen even in Hermitian case,−O(L 2)but super-exponentially small overlap ⟨πn, ρn⟩ = eis impossible
Non-Hermiticity of the Liouvillian is essential for diffusive
relaxation time τ ∼ L 2 in boundary dissipated chaotic systems

comment 2: effect of non-Hermiticity
Liouvillian skin effect
T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Phys. Rev. Lett. (2021)

a simple model of single particle with asymmetric hopping
γRdρ L−11 ††=Li,α ρLi,α − {Li,α Li,α, ρ}∑ ∑ ()
dt2i=1 α=left,rightγL†Li,left = γL bi†bi+1Li,right = γR bi+1bi
γR > γLτ = O(L)γL ↔ γRπn⟨πn, ρn⟩ = e −O(L)ρng = O(1)x=0x=Lcomment 3: classical Markov processes
Gap discrepancy problem also happens in classical Markov jumpprocesses
dPn(t) = −RnmPm(t)∑dtmspectral gap g of the transition rate matrix R
relaxation time τIn some models, τ ≫ g −1 (gap discrepancy)
Gap discrepancy problem is not a quantum effectcomment 4: metastability
So far, metastability has been characterized as a vanishingly small
gap of Liouvillian or transition-rate matrix
0 < g = Re λ1 ≪ Re λ2 ≤ Re λ3 ≤ …
A mathematical theory on how to construct metastable states wasdeveloped
B. Gaveau and L. S. Schulman, J. Phys. A (1987)
B. Gaveau and L. S. Schulman, J. Math. Phys. (1998)
G. Biroli and J. Kurchan, Phys. Rev. E (2001)

metastability in open quantum systems:
K. Macieszczak, M. Guta, I. Lesanovsky,and J. P. Garrahan, Phys. Rev. Lett. (2016)
D. C. Rose, K. Macieszczak, I. Lesanovsky, and J. P. Garrahan, Phys .Rev. E (2016)

Metastability associated with a small eigenmode overlap?

comment 4: metastability

TM, Phys. Rev. Res. (2021)

double-well potential with barrier EBand energy difference εgN+ particles

EBg : interaction between particles in the same wellgN− = N − N+ particles

g N+(N+ − 1) N−(N− − 1)
H = N+ε −+]N[22Hamiltoniand ⃗Markov jump with detailed balance
P (t) = − R P ⃗dt

metastability induced by large EB or large gFor suf ciently large N, the gap of the
independent of g

0.350.3

gap 0.2

metastability at strong g is not capturedby the spectral gapfi0.40.25

Λ1transition rate matrix R is

independent of g

0.15
g=2
g=3
g=40.10.05
010203040
N506070

comment 4: metastability

TM, Phys. Rev. Res. (2021)

double-well potential with barrier EB
and energy difference εgEB

g : interaction between particles in the same wellgN+ particles

N− = N − N+ particles

Hamiltoniang N+(N+ − 1) N−(N− − 1)
H = N+ε −+]N[22d ⃗
Markov jump with detailed balance
P (t) = − R P ⃗dt

35
metastability induced by large EB or large g100omax

For suf ciently30
large N, the gap of thetransition rate matrix R is
independent of25 gfi¾o_metastability with large g is explained by
20
vanishing eigenmodeoverlaps: τmax ∼ e βg

100

12g

34

Outline
relaxation time τ in open quantum systemsdρS(t)= − ℒρS(t)dt

Liouvillianℒ → τ?

➤ Gap discrepancy problem and its resolution
TM and T. Shirai, Phys. Rev. Lett. 125, 230604 (2020)

g: spectral gap of ℒ “Liouvillian gap”

τ ≲ g −1➤ general upper bound on the auto-correlation function in thesteady state: “instantaneous decay rate”
TM and T. Shirai, to appear soon on arXivasymptotic and instantaneous decay rates
autocorrelation in the steady statê Aρ̂ ss]
CAA(t) = Tr[A(t)̂ = e −ℒ̃t Â
A(t)Tr[Aρ̂ ss] = 0ℒ̃ is an adjoint of ℒ̂̂
Tr[Aℒρ]= Tr[(ℒ̃A)ρ]1 ̂† ̂†̂̂L ρL − {L L , ρ}∑( k k 2 k k )k1 ̂† ̂ ̂† ̂ ̂̂̂̂̂
− ℒ̃A = i[H, A] +L A L − {L L , A}∑( k k 2 k k)k−ℒρ = − i[H,̂ ρ] +asymptotic decay rate: Liouvillian gap

CAA(t) ∼ e −gt (t → ∞)but crossover time to the asymptotic regime might be too long
instantaneous decay rate in a transient regime?

characterization of the instantaneous decay rate
̂ ss] ↔ ⟨A,̂ B⟩̂ = Tr[A†̂ B]̂
de ne an inner product ⟨A,̂ B⟩̂ ss = Tr[A†̂ Bρ
R. Alicki, Rep. Math. Phys. (1976)

̂ A⟩̂ ss (Â = A†̂ )CAA(t) = ⟨A(t),ℒ̃* is de ned by ⟨A,̂ ℒ̃B⟩̂ ss = ⟨ℒ̃*A,̂ B⟩̂ sssymmetrized Liouvillian
Theorem| CAA(t) | ≤ eℒ̃ + ℒ̃*=
2− ∫0t KA(s)dsCAA(0)Hermitian in the Hilbert spaceassociated with ⟨ ⋅ , ⋅ ⟩ssfifi𝒦𝒦̂̂ ss⟨A(t),A(t)⟩instantaneous decay rate KA(t) =̂ A(t)⟩
̂ ss⟨A(t),̂̂ ss⟨A(t),A(t)⟩instantaneous decay rate KA(t) =̂ A(t)⟩̂ ss⟨A(t),𝒦ℒ̃ + ℒ̃*=2it converges to the Liouvillian gap in the long-time limitlim KA(t) = gt→∞

it is bounded from below by the spectral gap gK of thesymmetrized Liouvillian
KA(t) ≥ gK(“symmetrized Liouvillian gap”)| CAA(t) | ≤ e −gK tCAA(0)We can show 0 ≤ gK ≤ g

The detailed balance condition implies g = gK
R. Alicki, Rep. Math. Phys. (1976) [ℒ̃, ℒ̃*] = 0𝒦𝒦𝒦

properties of the instantaneous decay rate

(eigenvalue of) = (real part of eigenvalue of ℒ̃)Numerics in Born-Markov-secular Lindblad equationinteracting double quantum dots
ε1ε2v
β ,μLβL = βR,CHH(t)/CHH(0)CHH(t) = ⟨Ĥ S(t)Ĥ S⟩ssL

μL = μR

e −gt = e −gK t

UβR, μRβL ≠ βR,

e −gt

μL ≠ μRe −gK tsummary and outlook
TM and T. Shirai, Phys. Rev. Lett. 125, 230604 (2020)➤ The Liouvillian gap does not necessarily give the maximum

relaxation time (gap discrepancy problem)➤ This problem is resolved by taking into account an explosive

growth of expansion coef cients due to vanishingly small
eigenmode overlaps

TM and T. Shirai, to appear soon on arXiv➤ We give a bound on the autocorrelation function in the steadystate by using the instantaneous decay rate KA(t)➤ The instantaneous decay rate is generally bounded from below by

the spectral gap of the symmetrized Liouvillian➤ Our numerical calculations show that the symmetrized Liouvillian

figap gives a tight bound at short times