quantum chaos in ultra-strongly coupled nonlinear...

Quantum chaos in ultra-strongly coupled nonlinearresonators

Uta Naether

MARTES CUANTICOUniversidad de Zaragoza, 13.05.14

Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions

Colaborators

⋄ Juan Jose Garcıa-Ripoll, CSIC, Madrid

⋄ Juan Jose Mazo, UNIZAR & ICMA

⋄ David Zueco, UNIZAR, ICMA & ARAID


Outline

1 IntroductionThe selftrapping transitionQuantizationSignatures of ChaosTwo coupled linear oscillatorsUltra-strong coupling

2 Model and resultsNonlinear mean-field DynamicsEigenvalues and -modesPoincare sectionsQuantum simulations

3 The 1D chain of coupled nonlinear oscillatorsEquationsBand structureSelftrapping

4 Conclusions


Nonlinear localized (solitary) modes

Soliton in the laboratory wave channel, Hawaiian coast and in bronze beads

(granular media).

Image sources: Wikipedia/ Robert I. Odom,University of Washington/Scholarpedia


The selftrapping transition in a dimer

nonlinearity and coupling compete in twocoupled nonlinear oscillators

nonlinear integrable Hamiltonians: analyticthresholds of self-trapping (spatiallocalization)

symmetric or anti-symmetric mode becomeunstable due to nonlinearity, whereaslocal(ized) mode stabilizes


Quantization of the dimer

⋄ Perturbation theory is used to compute the (nonlinear) quantum stateswith coupling as the perturbation.

⋄ Only symmetric and anti-symmetric modes are eigenmodes (of the RWAHamiltonian with conserved norm).

⋄ No localized mode, but the tunneling time

τ ∝ (N − 1)!γN−1

2JN

diverges for N ≥ 2 andγN > 2J.

⋄ The divergence of τ with N shows the symmetry breaking.


Signatures of Chaos

Sensibility to initial conditions.

Topological mixing.

Dense periodic orbits.

Fractal attractors or period doublingindicate chaos

weather, dynamics of satellites in the solarsystem, population growth in ecology,economic models,the dynamics of theaction potentials in neurons, molecularvibrations, and synchronization

Hamiltonian chaos: phase spaceconserving (no attractors), nonintegrableHamiltonians

Lorenz equations (’63): hydrody-

namic model to calculate long term

behavior in the atmosphere, de-

scribing circulation in a shallow

layer of fluid, heated uniformly

from below and cooled uniformly

from above


Quantum chaos

First reports of irregular behavior in the70’s, mostly in system, which are chaotic inthe classical limit.

Quantum chaos: How do quantum objectsbehave in a system, which exhibits chaos inthe classical limit? Uncertainty vs.sensibility of initial conditions.

Avoided crossing due to level repulsion leadto changes in the energy level distribution.

Open topic of research: What are theconditions of quantum integrability? Howdoes thermalization happen? Connectionsbetween chaos and decoherence?

Our system is quite special, quantumfluctuations destroy one constant of motionand make our system become chaotic inthe semi-classical limit.


Two coupled linear oscillators

Starting from the Hamiltonian of two coupled oscillators of masses m1 and m2

with frequencies ω0,1, ω0,2 and coupling strength C ,

H0 =p21

2m1+

p22

2m2+

m1ω20,1

2q21 +

m2ω20,2

2q22 + C(q1 − q2)

2

=p21

2m1+

p22

2m2− 2Cq1q2 +

( m1ω20,1

2+ C

︸︷︷︸

≡ 12ω21m1

)

q21 +

( m2ω20,2

2+ C

︸︷︷︸

≡ 12ω22m2

)

q22

we use second quantization qk =√

~

2mkωk(ak + a+k ) and

pk = −i

√~mkωk

2(ak − a+k ) for k = 1, 2.


The resulting Hamiltonian is

H ≡ H0 −~

2(ω1 +ω2) = ~ω1a

+1 a1 + ~ω2a

+2 a2 −

≡~J︷︸︸︷

~C√m1m2ω1ω2

(a+1 + a1)(a+2 + a2).

Since at least the initial frequencies should be real and masses positiv,mk , ω

20,k ≥ 0 is required. We conclude that

2C = 2J√m1m2ω1ω2 ≤ ω2

kmk =⇒ J ≤ 1

2Min

ω1

√ω1m1

ω2m2, ω2

√ω2m2

ω1m1

.

For identical oscillators (symmetric dimer) with ω1 = ω2 = ω and

m1 = m2 = m, we obtain the limit of ”physicality” at J/ω ≤ 1/2.


Ultra-strong coupling

0.0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

5

6

JΩ

Ei

The Rotating wave approximation is used widely inatom and quantum optics, neglecting normconservation violating terms as a+i a

+j or aiaj .

Recently, the first experiments in circuit QEDshowed behavior beyond the Jaynes-Cummingsmodel (RWA).

Price to pay for the new physics: Conservation ofnorm is lost (and integrability in some cases),stationary modes become quasistationary.


Model equations

Quantum Hamiltonian

H =∑

k=0,1

[

ωa†k ak +γ

2(a†k)

2a2k

]

− J(a†0a1 + λ a†0a†1 +H.c.), (1)

⋄ ak and a†k - (bosonic) annihilation and creation operators of

both oscillators with frequency ω⋄ ni = a

†i ai particle number operators

⋄ γ - Kerr nonlinearity, J - coupling strength; J/ω ≤ 0.5,⋄ λ = 0 (1) for RWA (NRWA)

Semiclassical Dynamics (DNLS / discrete Gross-Pitaevskii)

i uk = ωuk − J(u1−k + λ u∗1−k) + γ|uk |2uk , (2)

⋄ ak → 〈ak〉 := uk field amplitudes at site k = (1, 2)


Nonlinear mean-field Dynamics

0 2 4 6 8 100.00.20.40.60.81.0

t

Èu1È

,Èu

2È,N

Λ=0, Γ=3.4

0 2 4 6 8 100.00.20.40.60.81.0

t

Èu1È

,Èu

2È,N

Λ=0, Γ=-3.4

0 2 4 6 8 100.00.20.40.60.81.01.2

t

Èu1È

,Èu

2È,N

Λ=1, Γ=3.4

0 2 4 6 8 100.00.51.01.52.02.53.03.5

t

Èu1È

,Èu

2È,N

Λ=1, Γ=-3.4

i uk = ωuk − J(u1−k + λ u∗1−k) + γ|uk |2uk , J/ω = 1/2


Eigenvalues and -modesWe search for the symmetric and antisymmetric mode and their nonlinearcontinuation, so we assume |u1| = |u2| = u and separate ui = ai + ibi .

β

a1a2b1b2

=

0 0 −(ω + γu2) J(1 − λ)

0 0 J(1 − λ) −(ω + γu2)

(ω + γu2) −J(1 + λ) 0 0

−J(1 + λ) (ω + γu2) 0 0

a1a2b1b2

which yields for ν = iβ

νsym = ±

√

[(ω + γu2)− J(1 + λ)][(ω + γu2)− J(1− λ)]

νants = ±

√

[(ω + γu2) + J(1 + λ)][(ω + γu2) + J(1− λ)].

NRWA: −(

ω+2Ju2

)

< γ <(

2J−ωu2

)

RWA: ν = ±[(ω + γu2)± J]

J/ω = 2, u2 = 1


Helpful quantities

We will use the transformations ui =√ni exp(Iθi ) to define the

population imbalanceρ = n1 − n2

and the phase differenceφ = θ1 − θ2.

Only for RWA, the norm N = n1 + n2 is conserved, thus the system integrable.We use the discrete Fourier transform ui (ν), to obtain the normalized spectraldensity

g(ν, γ) =|u1(ν)|2 + |u2(ν)|2

∑

ν |u1(ν)|2 + |u2(ν)|2.

(a) (b)

(c)

(d)

γ

ν

(a) ρmin vs. γ and J/ω, the analytic |γth,rwa| = 4 is shown with dashed white lines (b),(c) spectral densities

g(ν, γ) for J/ω = 0.5 for the RWA (b) and the CR-case (c). The analytic continuations for J/ω = 0.5 and

u2 = 1/2 are shown in (d), ν for the antisymmetric modes is plotted with a blue/green) line, the symmetric cases

in black/gray.


Poincare sections

(a) (b)

(d)(c)

γ = 3 γ = −3

Mean-field simulations, top(bottom): RWA (USC)for J/ω = 0.5, (a),(c): γ = 3 with ρ(0) ∈ (−1, 1),

φ(0) = 0, π and (b),(d): γ = −3, ρ(0) ∈ (−1, 1), φ(0) = 0, π respectively.

NRWA

RWA

(a) (b)

(d)(c)

γ = 7 γ = −7

Mean-field simulations, Poincare sections at top(bottom): RWA (CR)for J/ω = 0.5, (a),(c): γ = 7 with

ρ(0) ∈ (−1, 1), φ(0) = 0 and (b),(d): γ = −7, ρ(0) ∈ (−1, 1), φ(0) = π respectively.


Quantum simulations

Spectral density

g(ν, γ) =|n1(ν)| + |n2(ν)|

∑

ν[|n1(ν)| + |n2(ν)|]

for CR (a) and RWA(b). n1(t = 0) =

17, n2(t = 0) = 0, ω = 2

(c): first crossing time τ vs. γN, forN(t = 0) = n1(t = 0) = 17 (black)and N(t = 0) = n1(t = 0) = 2 green

(gray), the full (dashed) lines correspond

to CR (RWA), respectively.

(a)

(b)

(c)

0 50 100 150 200

0

0.4

0.8

Jt

Ρ!t"#N!t"

0 50 100 150 200

0

0.4

0.8

Jt

Ρ!t"#N!t"

n1(0) = 0 n1(0) = 1 n1(0) = 3 n1(0) = 5

(a)

(b)

Quantum dynamics for J/ω = 0.1, N0 = 17, ρ(t)/N(t) vs. Jt is plotted for the USC model at γ = 20 (top

figure) and γ = −20 (bottom). n1(0) = 0, 1, 3, 5 (black, blue, red and green, respectively).

Θ=1

Θ=0

0 2 4 6 80.0

0.1

0.2

0.3

0.4

DΤ

pHDΤL

Probability p(∆τ) of the tunneling times ∆τ for J/ω = 0.5, N0 = 17, γ = −1. The case of θ = 0 is shown in

orange, θ = 1 in blue.


The 1D chain of coupled nonlinear oscillators

The Hamiltonian

H = ~

∑

n

[

ωa†nan − J(an + a†n)(an+1 + a

†n+1) +

γ

4a†na

†nanan

]

has the equations of motion

∂tan =i

~[H, an] = −iωan + iJ(an+1 + an−1 + a

†n+1 + a

†n−1)− iγa†na

2n,

which, in the mean field limit with 〈an〉 = ψn, yield

∂tψn ≡ ∂H

∂(iψ∗n )

= −iωψn + iJ(ψn+1 + ψn−1 + ψ∗n+1 + ψ∗

n−1)− iγ|ψn|2ψn.

To have a meaningful physical interpretation, ω > 4J. In the case of weakcoupling, we can use RWA ( ω ≫ J), and get

∂tψn = −iωψn + iJ(ψn+1 + ψn−1)− iγ|ψn|2ψn.


Band structure

We separate ψn = an + ibn and the equation set into real and imaginary part,and obtain the eigenvalue equations

βan = ωbn & βbn = −ωan + 2J(an+1 + an−1)

(3)

⇒ β2an = −ω2

an + 2Jω(an+1 + an−1) (4)

which, for an Ansatz of plane waves an = exp(ikn), gives us the band spectrum

λ = iβ = ±√

ω2 − 4Jωcos(k).

Thus, we have extended and propagating modes inside the bandsλ ∈ ±[

√ω2 − 4Jω,

√ω2 + 4Jω]. The density of states of such band modes in

an 1D chain is defined as g(λ) = dλN = dλk2π

g(λ) =λ

2π√

4J2ω2 − 14(ω2 − λ2)2

vs. f (λ)RWA =1

2π√

4J2 − (ω2 − λ2)2.

!6 !4 !2 0 2 4 60.0

0.1

0.2

0.3

0.4

Λ

!g"Λ#,f"Λ#$

0Π

4

Π

2

3 Π

4Π

3

5

7

k

!Λnrwa,Λrwa"

Eigenvalue spectrum λ(k) USC(black) and RWA (blue), densities of

states g(λ) (black) and f (λ) (blue) for ω = 5 and J = 1.

DNLS: dynamicalself-trapping transition of aninitially localized wave-packethappens at γ/J ≃ 3.8

independent of the ratio ofJ/ω

To observe the transition weuse the participation number

R ≡ (∑

n|ψn|2)2

∑

n|ψn|4

→

N ext. modes

1 loc. modes.

RWA: Participation number R for fixed

integration time t = tmax . Direct in-

tegration with ψn(t = 0) = δn,n0 in a

chain of length N = 101, tmax = N/5

and J = 1.

NRWA: R for positive γ; fixed integration time t = tmax for growing

nonlinearity γ > 0 and ω. Direct integration of (22) with ψn(t = 0) = δn,n0 in a

chain of length N = 101, tmax = N/5, J = 1.

NRWA: R for negative γ; fixed integration time t = tmax for growing

nonlinearity γ > 0 and ω. Direct integration of (22) with ψn(t = 0) = δn,n0 in a

chain of length N = 101, tmax = N/5, J = 1.

ω = 4

ω = 7

RWA

RWA

ω = 4

NRWA

ω = 10

NRWA

output patterns for fixed integration times tmax = N/5 and on the right the

corresponding spectral density.

ω = 4

ω = 7

RWA

RWA

ω = 4

NRWA

ω = 10

NRWA

NRWA

NRWA

NRWA

ω = 7

ω = 4

output patterns for fixed integration times tmax = N/5 and on the right the

corresponding spectral density.


Conclusions

We considered a system of nonlinear dimers with ultra-strongcoupling, where quantum fluctuations destroy the integrability ofthe semi-classical system.

For negative nonlinearities, we find chaotic regimes in the mean-fieldand quantum calculations.

Chaos affects the self-trapping transition and makes tunnellingtimes unpredictable.

The irregular behavior of the tunneling time should be verifiable inexperiments, e.g in circuit QED.

The results are extendible to chains of nonlinear oscillators.

see Phys. Rev. Lett. 112, 074101

Thank you!

quantum chaos in ultra-strongly coupled nonlinear...

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