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??
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L(t) =|⟨Ψ | exp(−iHt) | Ψ⟩ |2
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L(t) =|⟨Ψ | exp(−iHt) | Ψ⟩ |2
Loschmidt Echo:Loschmidt Echo:
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= pn pm exp[−it(En − Em )]n,m
∑
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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?
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| 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?
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| Ψ⟩
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pn :=|⟨Ψ | n⟩ |2
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=Tr[| Ψ⟩⟨Ψ | e−iHt | Ψ⟩⟨Ψ | e iHt ] = ⟨ρΨ (t)⟩Ψ
Typical Time Pattern of L(t)
Transverse Ising (N=100)
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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
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A ⊂[0,∞)
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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
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χA
1 Moments of P(y)
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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?
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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
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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
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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
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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
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κn (ψ ) = Tr[Dn⊗2(S) |ψ⟩⟨ψ |⊗2n ]
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κn (ψ ) = Tr[Dn⊗2(S) |ψ⟩⟨ψ |⊗2n ]
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κn (ψ )ψ
=Tr[Dn
⊗2(S)P2n+ ]
Tr(P2n+ )
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P2n+ = Projection on the totally symm ss of
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Hilb⊗2n
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(Hilb⊗n )⊗2
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|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
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κn (ψ )ψ
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κ1(ψ )ψ
=Tr[D1
⊗2(S)(1+ S)]
d(d +1)=
2
d +1
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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
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κn (ψ )ψ
2 Moments of P(y)
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κ1(ψ )ψ
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κn (ψ )
Remark
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κn (ψ )ψ
≈d >>n d−2n Tr[Dn⊗2(S)σ ]
σ ∈S2n
∑
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κ1(ψ )ψ
=1+ d j (d j /d∑ )2
d +1
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σ 2(L) := κ 2 −κ12 = pi
2 p j2
i≠ j
∑ = κ12 − tr(ρ eq
4 ) ≤ κ12
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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 …..!
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κ1(ψ ) = ⟨L(t)⟩t€
σ(L)
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∀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)
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∀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 :
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c = c(ε,δ) > 0
Q: Can we do better? E.g., exp in d concentration?
A: yes we can!
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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
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| 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!
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μ∞{t / | LN (t) − ⟨LN (t)⟩t |≥ ε} = Pr{α ∈ T d / | L(α ) − ⟨L⟩α |≥ ε} ≤ exp(−cε 2
pn4∑)
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μ∞{t / | LN (t) − ⟨LN (t)⟩t |≥ ε} = Pr{α ∈ T d / | L(α ) − ⟨L⟩α |≥ ε} ≤ exp(−cε 2
pn4∑)
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c = (128π 2)−1
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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
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H0 | Ψ0⟩= E0 | Ψ0⟩
H = H0 + V
||V ||= o(ε)
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H0 | Ψ0⟩= E0 | Ψ0⟩
H = H0 + V
||V ||= o(ε)
Ground State of an initial HamiltonianQuench-Ham= init-Ham + perturbation
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pn =|⟨ΨQuenchn | Ψ0⟩ |2 Distribution on the eigenbasis of
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p0 ≈1− χ F = o(1)
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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….
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Slin :=1−κ 1
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H = H0 + V
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pn≠0 ≈|⟨Ψ0 |V | Ψn⟩ |2
(En − E0)2= o(ε 2)GS Fidelity: leading term!
Remark:
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σ 2(L) ≈ p0(1− p0) ≈ χ F
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σ 2(L) ≈ p0(1− p0) ≈ χ F
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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)
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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
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V → dd xV (x)∫
Small Quenches: FS Critical scaling
Continuum limit
Scaling transformations
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x →αx;τ →α ζ τ
V →α −ΔV
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χFSing /Ld →| λ − λ c |νΔQ Proximity of the critical point
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χFSing /Ld → L−ΔQ At the critical points
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ΔQ := 2Δ − 2ζ − d
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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
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L(t) = (1− sin2(2α k )sin2(Λk2 t))
k
∏
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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:
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cosθν (λ ) =cos
2πk
L− λ
Λν
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γ=1
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γ=1
Large size limit (TDL)==> spec(H) quasi-continuous ==> Large t limits exist (R-L Lemma) =time averages
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L(t) = e−Ls( t )
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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)
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s(∞) = −1
πln[(1− | cos(α k ) | /2)]dk∫
Inverting limits I.e., 1st t-average, 2nd TDL
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⟨L(t)⟩t = e−Lg(λ1 ,λ2 )
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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
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κ1 = Π k[1−α k /2],
N =100
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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
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α k := sin2(θk (h1) −θk (h1)
2)
P(L=y) Different Regimes
• Large ==>L for (moderately) large (quasi) exponential
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δh
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δh• Small and close to criticality a) Exponentialb) Quasi critical I.e., universal “Batman Hood”
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δh• Small and off critical a) Exponential b) Otherwise Gaussian
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L >>|δh |−2
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L >>|δh |−1
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L <<| h(i) −1 |∝ξ
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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
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P(y = L) ≈ θ(y)exp(−y /⟨L⟩)
⟨L⟩
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P(y = L) ≈ θ(y)exp(−y /⟨L⟩)
⟨L⟩
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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
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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.,
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Y :=H − ⟨H⟩
⟨H 2⟩c
TDL ⏐ → ⏐ P(Y )∝ exp(−Y 2 /2)⇒ L(t) ≈ e−t 2 ⟨H 2 ⟩c
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exp(−TR2⟨H 2⟩c ) = L ⇒ TR :=
−lnL
⟨H 2⟩c
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exp(−TR2⟨H 2⟩c ) = L ⇒ TR :=
−lnL
⟨H 2⟩c
Relaxation time
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TR = O(1)
TR = O(Lζ )∝ξ ζ
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TR = O(1)
TR = O(Lζ )∝ξ ζ
Off critical (or large quench)
Critical (& small quench)
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⟨Y n⟩c =AnN
d + BnNn(d −Δ )
(A2Nd + B2N
2(d −Δ ))n / 2N →∞ ⏐ → ⏐ ⏐
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Δ > d /2•0 for
€
Bn /B2n / 2
for
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Δ < d /2
Gaussian Non Gaussian
(universal)
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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
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L(t) = κ1 + c(Λk )cos(tΛk )k
∑
+ n-body spectrum contributions
Different regimes depend on how manyfrequencies have a non-negligible weight
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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
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L(t) = Acos(ω1t) + Bcos(ω2t)L=20, h(1)=0.1, h(2)=0.11
More generally (Strong non resonance)
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κn ≤ n!(κ1)n ⇒ eλ ⟨L ⟩ ≤ χ (λ ) := ⟨eλL⟩≤
1
1− λ ⟨L⟩
Therefore implies
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κ1 = ⟨L⟩→ 0
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χ(λ ) →1 ⇔ P(y = L) →δ(y)
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⟨L⟩
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κn = Tr S(ρψ⊗n ⊗Dn (ρψ
⊗n ))[ ]
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κn = Tr S(ρψ⊗n ⊗Dn (ρψ
⊗n ))[ ]€
P(y = L) ≈ θ(y)exp(−y /⟨L⟩)
⟨L⟩
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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.,
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ρψ
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ρ1 := ρψ⊗n ⊗ ρψ
⊗n
2) Dephase 2nd n-copies I.e.,
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ρ2 := (Id ⊗D)(ρ in )
3) Measure S
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κn = Tr(Sρ 3)
For small
Higher Moments: direct operational meaning !
€
Dn
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