Quantum chaos in ultra-strongly coupled nonlinear resonators

Transcription

1 Quantum chaos in ultra-strongly coupled nonlinear resonators Uta Naether MARTES CUANTICO Universidad de Zaragoza,

2 Colaborators Juan José García-Ripoll, CSIC, Madrid Juan José Mazo, UNIZAR & ICMA David Zueco, UNIZAR, ICMA & ARAID

3 Outline 1 Introduction The selftrapping transition Quantization Signatures of Chaos Two coupled linear oscillators Ultra-strong coupling 2 Model and results Nonlinear mean-field Dynamics Eigenvalues and -modes Poincaré sections Quantum simulations 3 The 1D chain of coupled nonlinear oscillators Equations Band structure Selftrapping 4 Conclusions

4 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

5 The selftrapping transition in a dimer nonlinearity and coupling compete in two coupled nonlinear oscillators nonlinear integrable Hamiltonians: analytic thresholds of self-trapping (spatial localization) symmetric or anti-symmetric mode become unstable due to nonlinearity, whereas local(ized) mode stabilizes

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7 Quantization of the dimer Perturbation theory is used to compute the (nonlinear) quantum states with coupling as the perturbation. Only symmetric and anti-symmetric modes are eigenmodes (of the RWA Hamiltonian with conserved norm). No localized mode, but the tunneling time diverges for N 2 and τ (N 1)!γN 1 2JN γn > 2J. The divergence of τ with N shows the symmetry breaking.

8 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 doubling indicate chaos weather, dynamics of satellites in the solar system, population growth in ecology, economic models,the dynamics of the action potentials in neurons, molecular vibrations, and synchronization Hamiltonian chaos: phase space conserving (no attractors), nonintegrable Hamiltonians Lorenz equations ( 63): hydrodynamic model to calculate long term behavior in the atmosphere, describing circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above

9 Quantum chaos First reports of irregular behavior in the 70 s, mostly in system, which are chaotic in the classical limit. Quantum chaos: How do quantum objects behave in a system, which exhibits chaos in the classical limit? Uncertainty vs. sensibility of initial conditions. Avoided crossing due to level repulsion lead to changes in the energy level distribution. Open topic of research: What are the conditions of quantum integrability? How does thermalization happen? Connections between chaos and decoherence? Our system is quite special, quantum fluctuations destroy one constant of motion and make our system become chaotic in the semi-classical limit.

10 Two coupled linear oscillators Starting from the Hamiltonian of two coupled oscillators of masses m 1 and m 2 with frequencies ω 0,1,ω 0,2 and coupling strength C, H 0 = = ˆp ˆp2 2 + m1ω2 0,1 ˆq m2ω2 0,2 ˆq 2 2 +C(ˆq 1 ˆq 2) 2 2m 1 2m ˆp 1 2 ( + ˆp2 2 m1ω 2 ) ( 0,1 m2ω 2Cˆq 1ˆq 2 + +C ˆq 2 2 ) 0, C ˆq 2 2 2m 1 2m 2 } 2 {{}} 2 {{} 1 2 ω2 1 m 1 we use second quantization ˆq k = 2m k ω k (â k +â + k ) and m ˆp k = i k ω k (â 2 k â + k ) for k = 1, ω2 2 m 2

11 The resulting Hamiltonian is J {}}{ H H 0 2 (ω1+ω2) = ω1â+ 1 â 1+ ω 2â + C 2 â 2 (â + m1m 1 +â 1)(â + 2 +â 2). 2ω 1ω 2 Since at least the initial frequencies should be real and masses positiv, m k,ω 2 0,k 0 is required. We conclude that 2C = 2J m 1m 2ω 1ω 2 ω 2 km k = J 1 2 Min { ω 1 ω1m 1 ω 2m 2,ω 2 For identical oscillators (symmetric dimer) with ω 1 = ω 2 = ω and m 1 = m 2 = m, we obtain the limit of physicality at J/ω 1/2. ω2m 2 ω 1m 1 }.

12 Ultra-strong coupling Ei 3 The Rotating wave approximation is used widely in atom and quantum optics, neglecting norm conservation violating terms as a + i a + j or a i a j. Recently, the first experiments in circuit QED showed behavior beyond the Jaynes-Cummings model (RWA). Price to pay for the new physics: Conservation of norm is lost (and integrability in some cases), stationary modes become quasistationary J Ω

13 Model equations Quantum Hamiltonian H = [ ωâ kâk + γ ] 2 (â k )2 âk 2 J(â 0â1 +λâ 0â 1 +H.c.), (1) k=0,1 â k and â k - (bosonic) annihilation and creation operators of both oscillators with frequency ω ˆn i = â iâi 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 u k = ωu k J(u 1 k +λu 1 k)+ γ u k 2 u k, (2) â k â k := u k field amplitudes at site k = (1,2)

14 Nonlinear mean-field Dynamics u 1, u 2, N u 1, u 2, N Λ 0,Γ t Λ 1,Γ t u 1, u 2, N u 1, u 2, N Λ 0,Γ t Λ 1,Γ t i u k = ωu k J(u 1 k +λu 1 k )+ γ u k 2 u k, J/ω = 1/2

15 Eigenvalues and -modes We search for the symmetric and antisymmetric mode and their nonlinear continuation, so we assume u 1 = u 2 = u and separate u i = a i +ib i. a (ω + γu 2 ) J(1 λ) a 1 β a 2 b 1 = 0 0 J(1 λ) (ω + γu 2 ) (ω + γu 2 a 2 ) J(1 + λ) 0 0 b 1 b 2 J(1 + λ) (ω + γu 2 ) 0 0 b 2 which yields for ν = iβ ν sym = ± [(ω +γu 2 ) J(1+λ)][(ω +γu 2 ) J(1 λ)] ν ants = ± [(ω +γu 2 )+J(1+λ)][(ω +γu 2 )+J(1 λ)]. ( ) ( ) NRWA: ω+2j u 2 < γ < 2J ω u 2 RWA: ν = ±[(ω +γu 2 )±J] J/ω = 2, u 2 = 1

16 Helpful quantities We will use the transformations u i = n i exp(iθ i ) to define the population imbalance ρ = n 1 n 2 and the phase difference φ = θ 1 θ 2. Only for RWA, the norm N = n 1 +n 2 is conserved, thus the system integrable. We use the discrete Fourier transform ũ i (ν), to obtain the normalized spectral density g(ν,γ) = ũ1(ν) 2 + ũ 2(ν) 2 ν ũ1(ν) 2 + ũ 2(ν) 2.

17 (a) (b) (c) ν (d) γ (a) ρ min vs. γ and J/ω, the analytic γ th,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 u 2 = 1/2 are shown in (d), ν for the antisymmetric modes is plotted with a blue/green) line, the symmetric cases in black/gray.

18 Poincaré sections γ =3 γ = 3 (a) (b) (c) (d) 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.

19 γ =7 γ = 7 RWA (a) (b) NRWA (c) (d) Mean-field simulations, Poincaré 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.

20 Quantum simulations (a) Spectral density g(ν,γ) = ñ 1 (ν) + ñ 2 (ν) ν [ ñ 1(ν) + ñ 2 (ν) ] (b) for CR (a) and RWA(b). n 1 (t = 0) = 17, n 2 (t = 0) = 0, ω = 2 (c): first crossing time τ vs. γn, for N(t = 0) = n 1 (t = 0) = 17 (black) and N(t = 0) = n 1 (t = 0) = 2 green (gray), the full (dashed) lines correspond (c) to CR (RWA), respectively.

21 0.8 Ρ t N t (a) Jt n 1(0) = 0 n 1(0) = 1 n 1(0) = 3 n 1(0) = Ρ t N t (b) Jt Quantum dynamics for J/ω = 0.1, N 0 = 17, ρ(t)/n(t) vs. Jt is plotted for the USC model at γ = 20 (top figure) and γ = 20 (bottom). n 1 (0) = 0,1,3,5 (black, blue, red and green, respectively).

22 p Τ Θ 0 Θ Τ Probability p( τ) of the tunneling times τ for J/ω = 0.5, N 0 = 17, γ = 1. The case of θ = 0 is shown in orange, θ = 1 in blue.

23 The 1D chain of coupled nonlinear oscillators The Hamiltonian H = n [ωa na n J(a n +a n)(a n+1 +a n+1 )+ γ 4 a na na na n ] has the equations of motion ta n = i [H,an] = iωan +ij(an+1 +an 1 +a n+1 +a n 1 ) iγa na 2 n, which, in the mean field limit with a n = ψ n, yield tψ n H (iψ n) = iωψn +ij(ψn+1 +ψn 1 +ψ n+1 +ψ n 1) iγ ψ n 2 ψ n. To have a meaningful physical interpretation, ω > 4J. In the case of weak coupling, we can use RWA ( ω J), and get tψ n = iωψ n +ij(ψ n+1 +ψ n 1) iγ ψ n 2 ψ n.

24 Band structure We separate ψ n = a n +ib n and the equation set into real and imaginary part, and obtain the eigenvalue equations βa n = ωb n & βb n = ωa n +2J(a n+1 +a n 1) β 2 a n = ω 2 a n +2Jω(a n+1 +a n 1) (4) which, for an Ansatz of plane waves a n = exp(ikn), gives us the band spectrum λ = iβ = ± ω 2 4Jωcos(k). Thus, we have extended and propagating modes inside the bands λ ±[ ω 2 4Jω, ω 2 +4Jω]. The density of states of such band modes in an 1D chain is defined as g(λ) = d λ N = d λ k 2π λ 1 g(λ) = vs. f(λ) RWA = 2π 4J 2 ω 2 14 (ω2 λ 2 ) 2 2π 4J 2 (ω 2 λ 2 ) 2. (3)

25 7 0.4 Λnrwa,Λrwa 5 3 g Λ, f Λ Π 4 Π 2 k 3Π 4 Π Λ 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.

26 DNLS: dynamical self-trapping transition of an initially localized wave-packet happens at γ/j 3.8 independent of the ratio of J/ω To observe the transition we use 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 = t max. Direct integration with ψ n(t = 0) = δ n,n0 in a chain of length N = 101, t max = N/5 and J = 1.

27 NRWA: R for positive γ; fixed integration time t = t max for growing nonlinearity γ > 0 and ω. Direct integration of (22) with ψ n(t = 0) = δ n,n0 in a chain of length N = 101, t max = N/5, J = 1.

28 NRWA: R for negative γ; fixed integration time t = t max for growing nonlinearity γ > 0 and ω. Direct integration of (22) with ψ n(t = 0) = δ n,n0 in a chain of length N = 101, t max = N/5, J = 1.

29 ω =4 RWA ω =7 RWA ω =4 NRWA ω = 10 NRWA output patterns for fixed integration times t max = N/5 and on the right the corresponding spectral density.

30 ω =4 RWA ω =4 NRWA ω =7 RWA ω =7 NRWA ω =4 NRWA NRWA ω = 10 NRWA output patterns for fixed integration times t max = N/5 and on the right the corresponding spectral density.

31 Conclusions We considered a system of nonlinear dimers with ultra-strong coupling, where quantum fluctuations destroy the integrability of the semi-classical system. For negative nonlinearities, we find chaotic regimes in the mean-field and quantum calculations. Chaos affects the self-trapping transition and makes tunnelling times unpredictable. The irregular behavior of the tunneling time should be verifiable in experiments, e.g in circuit QED. The results are extendible to chains of nonlinear oscillators. see Phys. Rev. Lett. 112,

32 Thank you!