Paolo Zanardi (USC)

Lorenzo Campos Venuti (ISI)

Obergugl, Austria June 2010

Long time dynamics of a quantum quenchLong time dynamics of a quantum quench

•Unitary Evolution ==>no non-trivial fixed points for t=∞ I.e., no strong (norm) convergence

Hey wait a sec: Equilibration of a finite closed quantumsystem?!? What are you talking about dude???

•Finite size ==>Point spectrum ==>A(t)=measurable quantity is a quasi-periodic function ==> no t=∞ limit (quasi-returns/revivals) ==> not even weak op convergence€

limt →∞ U(t) | Ψ⟩=| Ψ⟩∞ ⇒ | Ψ⟩=| Ψ⟩∞

€

A(t) = D Ap exp(iωp t)⇒ ∀ε > 0∃T(ε,D) /p=1

∑ | A(T) − A(0) |≤ ε

Unitary equilibrations will have to be a different kind of convergence….Unitary equilibrations will have to be a different kind of convergence….

Unitary Equilibration??

€

L(t) =|⟨Ψ | exp(−iHt) | Ψ⟩ |2

€

L(t) =|⟨Ψ | exp(−iHt) | Ψ⟩ |2

Loschmidt Echo:Loschmidt Echo:

€

= pn pm exp[−it(En − Em )]n,m

∑

€

H = En | n⟩⟨n |n

∑

Spectral resolution Probability distribution(s)

Different Time-Scales & Characteristic quantities

•Relaxation Time (to get to a small value by dephasing and oscillate around it)

•Revivals Time (signal strikes back due to re-phasing)

Q1: how all these depend on H, , and system size?Q1: how all these depend on H, , and system size?

€

| Ψ⟩

Q2: how the global statistical features of L(t) depend on H, and system size N?Q2: how the global statistical features of L(t) depend on H, and system size N?

€

| Ψ⟩

€

pn :=|⟨Ψ | n⟩ |2

€

=Tr[| Ψ⟩⟨Ψ | e−iHt | Ψ⟩⟨Ψ | e iHt ] = ⟨ρΨ (t)⟩Ψ

Typical Time Pattern of L(t)

Transverse Ising (N=100)

€

P(y = L(t)) = limT →∞

1

Tδ(y − L(t)) =

0

T

∫ ⟨δ(L − L(t))⟩t

€

P(y = L(t)) = limT →∞

1

Tδ(y − L(t)) =

0

T

∫ ⟨δ(L − L(t))⟩t

For a given initial state L-echo is a RV over the time line [0,∞) with Prob Meas

€

A ⊂[0,∞)

€

μ∞(A) := limT →∞

1

Tχ A (t)dt

0

T

∫ Characteristic function of

Probability distribution of L-echo

Goal: study P(y) to extract global information about theEquilibration process Goal: study P(y) to extract global information about theEquilibration process

€

χA

1 Moments of P(y)

€

κn := y nP(y)dy = limT →∞∫ 1

TLk (t)dt

0

T

∫

Each moment is a RV over the unit sphere (Haar measure) of initial states

Mean:

Long time average of L(t) is the purity of the time-averagedensity matrix (or 1 -Linear Entropy)

Question: How about the other moments e.g., variance andinitial state dependence? Are there “typical” values?Question: How about the other moments e.g., variance andinitial state dependence? Are there “typical” values?

€

κ1 = limT →∞

1

T⟨ρΨ,ρΨ,(t)⟩dt

0

T

∫ = ⟨ρΨ,Π nρΨ,Π m⟩m,n

∑ limT →∞

1

Te−i(En −Em )tdt

0

T

∫

= ⟨ρΨ,Π nρΨ,Π n⟩=n

∑ ⟨ρΨ,D1(ρΨ,)⟩= ⟨D1(ρΨ ),D1(ρΨ,)⟩= TrD1(ρΨ )2

€

limT →∞

1

Te−iHtρΨe iHtdt

0

T

∫ = Π nρΨ,Π n =: D1(n

∑ ρΨ )

€

limT →∞

1

Te−iHtρΨe iHtdt

0

T

∫ = Π nρΨ,Π n =: D1(n

∑ ρΨ )

Remark

€

D1Is a projection on the algebra of the fixed pointsOf the (Heisenberg) time-evolution generated by H

Remark dephased state min purity given the constraints I.e., constant of motion

€

Dn (X) = limT →∞

1

T(e−iHt )⊗n X

0

T

∫ (e iHt )⊗n Dephasing CP-map of the n-copies Hamiltonian, S is a swap in

€

κn (ψ ) = Tr[Dn⊗2(S) |ψ⟩⟨ψ |⊗2n ]

€

κn (ψ ) = Tr[Dn⊗2(S) |ψ⟩⟨ψ |⊗2n ]

€

κn (ψ )ψ

=Tr[Dn

⊗2(S)P2n+ ]

Tr(P2n+ )

€

P2n+ = Projection on the totally symm ss of

€

Hilb⊗2n

€

(Hilb⊗n )⊗2

€

|κ n (ψ ) −κ n (φ) |≤|| Dn⊗2(S) ||∞ |||ψ⟩⟨ψ |⊗2n − | φ⟩⟨φ |⊗2n ||1≤ 4n |||ψ −⟩ | φ⟩ ||

All L(t) moments are Lipschitz functions on the unit sphere of Hilb ==> Levi’s Lemma implies exp (in d) concentration around

Remark We assumed NO DEGENERACY, in general bounded above by

€

κn (ψ )ψ

€

κ1(ψ )ψ

=Tr[D1

⊗2(S)(1+ S)]

d(d +1)=

2

d +1

€

κ1(ψ )ψ

=Tr[D1

⊗2(S)(1+ S)]

d(d +1)=

2

d +1

d=dim(Hilb)==> exp (in N) small =(positivity)=> exp (in N) state-space concentration of around

€

κn (ψ )ψ

2 Moments of P(y)

€

κ1(ψ )ψ

€

κn (ψ )

Remark

€

κn (ψ )ψ

≈d >>n d−2n Tr[Dn⊗2(S)σ ]

σ ∈S2n

∑

€

κ1(ψ )ψ

=1+ d j (d j /d∑ )2

d +1

€

σ 2(L) := κ 2 −κ12 = pi

2 p j2

i≠ j

∑ = κ12 − tr(ρ eq

4 ) ≤ κ12

€

σ 2(L) := κ 2 −κ12 = pi

2 p j2

i≠ j

∑ = κ12 − tr(ρ eq

4 ) ≤ κ12

€

μ∞{t / | L(t) − ⟨L(t)⟩t |≥ Mσ } ≤ M−2

Chebyshev’s inequality

goes to zero with N system size ==> M can diverge While ==>Flucts of L(t) are exponentially rare in time …..! goes to zero with N system size ==> M can diverge While ==>Flucts of L(t) are exponentially rare in time …..!

€

κ1(ψ ) = ⟨L(t)⟩t€

σ(L)

€

∀ε,δ > 0,∃N | N ≥ N ⇒ μ∞{t / | Lψ NN (t) −κ1,N (ψ N )

ψ N |≥ ε} ≤ δ

∀ψ N ∈ SN ⊂HilbN & μ H (SN ) ≥1− exp(−cN)

€

∀ε,δ > 0,∃N | N ≥ N ⇒ μ∞{t / | Lψ NN (t) −κ1,N (ψ N )

ψ N |≥ ε} ≤ δ

∀ψ N ∈ SN ⊂HilbN & μ H (SN ) ≥1− exp(−cN)

In the overwhelming majority of time instants L(t) is exponentially close to the “equilibrium value” €

Mσ → 0

Remark: non-resonance assumed

This what we (morally) got :

€

c = c(ε,δ) > 0

Q: Can we do better? E.g., exp in d concentration?

A: yes we can!

€

t →α := (E1t,...,Ed t)∈ T d →| pneiα n∑ |2

HP: energies rationally independent ==> motion on the d-torus ergodic==>Temporal averages=phase-space averages

€

| L(α ) − L(β ) |≤ 2 pn | e iα n − e iβ n |≤∑ pn |α n − β n |=: 4πD(α ,β )∑

L is Lipschitz on the d-torus with metric D ==> known measure concentration phenomenon!

€

μ∞{t / | LN (t) − ⟨LN (t)⟩t |≥ ε} = Pr{α ∈ T d / | L(α ) − ⟨L⟩α |≥ ε} ≤ exp(−cε 2

pn4∑)

€

μ∞{t / | LN (t) − ⟨LN (t)⟩t |≥ ε} = Pr{α ∈ T d / | L(α ) − ⟨L⟩α |≥ ε} ≤ exp(−cε 2

pn4∑)

€

c = (128π 2)−1

€

c = (128π 2)−1

Remark The rate of meas-conc is the inv of purity of the dephasedStates I.e., mean of L==> Typically order d=epx(N), as promised…

Far from typicality: Small Quenches

€

H0 | Ψ0⟩= E0 | Ψ0⟩

H = H0 + V

||V ||= o(ε)

€

H0 | Ψ0⟩= E0 | Ψ0⟩

H = H0 + V

||V ||= o(ε)

Ground State of an initial HamiltonianQuench-Ham= init-Ham + perturbation

€

pn =|⟨ΨQuenchn | Ψ0⟩ |2 Distribution on the eigenbasis of

€

p0 ≈1− χ F = o(1)

€

Slin =1− Tr[D1(ρΨ0)2] =1− ⟨L(t)⟩t =1− pn

2 ≈1− p02 ≈ 2χ F

n

∑

The linear entropy of the dephased state for a small quenchIs given by the fidelity susceptibility: a well-known object!The linear entropy of the dephased state for a small quenchIs given by the fidelity susceptibility: a well-known object!

=measures how initial state fails to be a quenched Hamiltonian Eigenstate. For H(quench) close to H(0) we expect it to be small….

€

Slin :=1−κ 1

€

H = H0 + V

€

pn≠0 ≈|⟨Ψ0 |V | Ψn⟩ |2

(En − E0)2= o(ε 2)GS Fidelity: leading term!

Remark:

€

σ 2(L) ≈ p0(1− p0) ≈ χ F

€

σ 2(L) ≈ p0(1− p0) ≈ χ F

€

χF =|⟨Ψ0 |V | Ψn⟩ |2

(E0 − En )2n≠0

∑ ≤1

Δ2(⟨V 2⟩− ⟨V⟩2) :=

1

Δ2X

€

χF =|⟨Ψ0 |V | Ψn⟩ |2

(E0 − En )2n≠0

∑ ≤1

Δ2(⟨V 2⟩− ⟨V⟩2) :=

1

Δ2X

€

V := V j∑Δ := minn (En − E0)

Local operator (trans inv)

Spectral gap

0)(lim >Δ∞→ LL ∞<−∞→ )(lim LXL d

L (Hastings 06)

€

limL →∞ L−d χ F < ∞Gapped system

For gapped systems 1-LE mean scales at most extensivelyFor gapped systems 1-LE mean scales at most extensively

Superextensive scaling implies gaplessnessSuperextensive scaling implies gaplessness

Small Quenches: The Role of Criticality

€

V → dd xV (x)∫

Small Quenches: FS Critical scaling

Continuum limit

Scaling transformations

€

x →αx;τ →α ζ τ

V →α −ΔV

€

χFSing /Ld →| λ − λ c |νΔQ Proximity of the critical point

€

χFSing /Ld → L−ΔQ At the critical points

€

ΔQ := 2Δ − 2ζ − d

€

ΔQ := 2Δ − 2ζ − d

Scal dim of FS: the smaller the faster the orthogonalization rateSuper-extensivity 2/)( dd −+≤Δ ζ

νλλξ −−= || c

Criticality it is not sufficient, one needs enough relevance….

THE XY Model

)2

1

2

1(),( 11

1

zi

yi

yi

xi

xi

i

LH λσσσγσσγγλ +−

++

= ++=∑ )

2

1

2

1(),( 11

1

zi

yi

yi

xi

xi

i

LH λσσσγσσγγλ +−

++

= ++=∑

=anysotropy parameter, =external magnetic fieldγ λ

QCPs:

0=γ

1±=λ

XX line III-order QPT

Ferro/para-magnetic II order QPT

Jordan-Wigner mapping H Free-Fermion system: EXACTLY SOLVABLE!

L

k

L

kk

πγλπ 2sin)

2(cos 222 +−=Λ

Quasi-particle spectrum: zeroes in the TDL in all the QCPs Gaplessness of the many-body spectrum

€

L(t) = (1− sin2(2α k )sin2(Λk2 t))

k

∏

€

L(t) = (1− sin2(2α k )sin2(Λk2 t))

k

∏

€

α k = θk (λ1) −θk (λ 2)H.T Quan et al, Phys. Rev. Lett. 96, 140604 (2006)

Ising in transverse field:

€

cosθν (λ ) =cos

2πk

L− λ

Λν

€

γ=1

€

γ=1

Large size limit (TDL)==> spec(H) quasi-continuous ==> Large t limits exist (R-L Lemma) =time averages

€

L(t) = e−Ls( t )

€

s(t) =1

2πln[(1− sin2(2α k )sin2(Λk

2 t))]dk∫ t →∞ ⏐ → ⏐ ⏐ s(∞) − Am | t |−3 / 2 cos(Emt + 3π /4) + (m ↔ M)

€

s(∞) = −1

πln[(1− | cos(α k ) | /2)]dk∫

Inverting limits I.e., 1st t-average, 2nd TDL

€

⟨L(t)⟩t = e−Lg(λ1 ,λ2 )

€

g = −1

2πln[(1− sin2(α k ) /2)]dk∫

•g and s are qualitatively the same but when we consider different phases

(m=band min, M=band max

€

κ1 = Π k[1−α k /2],

N =100

€

var(L) = Π k[1−α k +3

8α 2

k ] − Π k[1−α k +1

4α 2

k ]

N =100

First Two Moments

€

α k := sin2(θk (h1) −θk (h1)

2)

P(L=y) Different Regimes

• Large ==>L for (moderately) large (quasi) exponential

€

δh

€

δh• Small and close to criticality a) Exponentialb) Quasi critical I.e., universal “Batman Hood”

€

δh• Small and off critical a) Exponential b) Otherwise Gaussian

€

L >>|δh |−2

€

L >>|δh |−1

€

L <<| h(i) −1 |∝ξ

€

κn ≤ n!(κ1)n ⇒ eλ ⟨L ⟩ ≤ χ (λ ) := ⟨eλL⟩≤

1

1− λ ⟨L⟩

L=20,30,40,60,120

h(1)=0.2, h(2)=0.6

L=10,20,30,40

h(1)=0.9, h(2)=1.2

Approaching exp for large sizes

€

P(y = L) ≈ θ(y)exp(−y /⟨L⟩)

⟨L⟩

€

P(y = L) ≈ θ(y)exp(−y /⟨L⟩)

⟨L⟩

€

κn ≤ n!(κ1)n ⇒ eλ ⟨L ⟩ ≤ χ (λ ) := ⟨eλL⟩≤

1

1− λ ⟨L⟩

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Thanks for the attention!Thanks for the attention!

Summary & ConclusionsSummary & Conclusions

•Unitary equilibrations: measure convergence/concentration

•Moments of Probability distribution of LE (large time)

•Transverse Ising chain: mean, variance, regimes for P(L)

•Universal content of the short-time behaviorof L(t) and criticality

Phys. Rev. A 81,022113 (2010)Phys. Rev. A 81,022113 (2010) Phys. Rev. A 81, 032113 (2010)Phys. Rev. A 81, 032113 (2010)

Short-time behavior

€

L(t) =|⟨e−iHt⟩ |2= exp2(−t)2n

(2n)!⟨H 2n⟩c

n=1

∞

∑

Square of a characteristic function --> cumulant expansion

H sum of N local operator in the TDL N-> ∞ one expects CLT to hold I.e.,

€

Y :=H − ⟨H⟩

⟨H 2⟩c

TDL ⏐ → ⏐ P(Y )∝ exp(−Y 2 /2)⇒ L(t) ≈ e−t 2 ⟨H 2 ⟩c

€

exp(−TR2⟨H 2⟩c ) = L ⇒ TR :=

−lnL

⟨H 2⟩c

€

exp(−TR2⟨H 2⟩c ) = L ⇒ TR :=

−lnL

⟨H 2⟩c

Relaxation time

€

TR = O(1)

TR = O(Lζ )∝ξ ζ

€

TR = O(1)

TR = O(Lζ )∝ξ ζ

Off critical (or large quench)

Critical (& small quench)

€

⟨Y n⟩c =AnN

d + BnNn(d −Δ )

(A2Nd + B2N

2(d −Δ ))n / 2N →∞ ⏐ → ⏐ ⏐

€

Δ > d /2•0 for

€

Bn /B2n / 2

for

€

Δ < d /2

Gaussian Non Gaussian

(universal)

€

Δ =d /2 Non Gaussian & non universal)

L=18, h(1)=0.3, h(2)=1.4 L=20, h(1)=0.1, h(2)=0.11

L=40, h(1)=0.99, h(2)=1.1

€

L(t) = κ1 + c(Λk )cos(tΛk )k

∑

+ n-body spectrum contributions

Different regimes depend on how manyfrequencies have a non-negligible weight

€

c(ω) :=α k

2|ω= Λ k

L=20, h(1)=0.1, h(2)=0.11

L=40, h(1)=0.99, h(2)=1.1

With just two frequencies

€

L(t) = Acos(ω1t) + Bcos(ω2t)L=20, h(1)=0.1, h(2)=0.11

More generally (Strong non resonance)

€

κn ≤ n!(κ1)n ⇒ eλ ⟨L ⟩ ≤ χ (λ ) := ⟨eλL⟩≤

1

1− λ ⟨L⟩

Therefore implies

€

κ1 = ⟨L⟩→ 0

€

χ(λ ) →1 ⇔ P(y = L) →δ(y)

€

⟨L⟩

€

κn = Tr S(ρψ⊗n ⊗Dn (ρψ

⊗n ))[ ]

€

κn = Tr S(ρψ⊗n ⊗Dn (ρψ

⊗n ))[ ]€

P(y = L) ≈ θ(y)exp(−y /⟨L⟩)

⟨L⟩

€

P(y = L) ≈ θ(y)exp(−y /⟨L⟩)

⟨L⟩

is a dephasing super-operator of the n-copies Hamiltonian

S is a “swap” between 1st and 2nd n-copies

Protocol:

1) Prepare 2n copies of I.e.,

€

ρψ

€

ρ1 := ρψ⊗n ⊗ ρψ

⊗n

2) Dephase 2nd n-copies I.e.,

€

ρ2 := (Id ⊗D)(ρ in )

3) Measure S

€

κn = Tr(Sρ 3)

For small

Higher Moments: direct operational meaning !

€

Dn