quantum chaos in ultra-strongly coupled nonlinear...
TRANSCRIPT
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
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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
Introduction Model and results The 1D chain of coupled nonlinear oscillators 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
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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)
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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.
Introduction Model and results The 1D chain of coupled nonlinear oscillators Conclusions
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!