Scaling and Rare Events near Excitation Threshold of a Parametric - PowerPoint PPT Presentation

Scaling and Rare Events near Excitation Threshold of a Parametric Oscillator Mark Dykman Department of Physics and Astronomy, Michigan State University In collaboration with Z. R. Lin, RIKEN Center for Emergent Matter Science Y. Nakamura, RIKEN Center for Emergent Matter Science

Example: quasienergy states Eigenstates of a periodically driven system are not stationary : 𝑗ℏ𝜔̇ = 𝐼 𝑢 𝜔 𝐼 𝑢 = 𝐼 0 𝑟 , 𝑞 − 𝑟𝑟 cos 𝜕 𝐺 𝑢 , 𝑢 + 2𝜌 ⁄ 𝑣 𝜁 𝑢 , 𝜔 𝜁 𝑢 = 𝑓 −𝑗𝜁𝑗 ℏ 𝑣 𝜁 = 𝑣 𝜁 ( 𝑢 ) 𝜕 𝐺 quasienergy ≡ Floquet eigenvalue; quantization: 𝜁 → 𝜁 𝑜 Driven mesoscopic vibrational systems of current interest: Josephson junctions, cavity modes in optical and superconducting cavities, nanomechanical systems, cold atoms,… CNT

Example: quasienergy states Eigenstates of a periodically driven system are not stationary: 𝑗ℏ𝜔̇ = 𝐼 𝑢 𝜔 𝐼 𝑢 = 𝐼 0 𝑟 , 𝑞 − 𝑟𝑟 cos 𝜕 𝐺 𝑢 , 𝑢 + 2𝜌 ⁄ 𝑣 𝜁 𝑢 , 𝜔 𝜁 𝑢 = 𝑓 −𝑗𝜁𝑗 ℏ 𝑣 𝜁 = 𝑣 𝜁 ( 𝑢 ) 𝜕 𝐺 quasienergy ≡ Floquet eigenvalue; quantization: 𝜁 → 𝜁 𝑜 Relaxation, 𝑼 = 𝟏 : inter-state transitions with emission of photons, phonons, etc. Quasienergy states Fock states | 𝒐 + 𝟒〉 E 3 Quasienergy states are linear combinations of Fock states. Inter-level transitions down in energy, E 2 | 𝒐 + 𝟑〉 𝑂 𝐺𝐺𝐺𝐺 → | 𝑂 𝐺𝐺𝐺𝐺 − 1 〉 , correspond to inter-quasi- F ( t ) | 𝒐 + 𝟐〉 energy level transitions 𝑜 → 𝑜 ± 𝑛 , “up” and E 1 “down” in quasienergy. Even where the energy-level | 𝒐〉 width Γ ≪ Δ𝐹 , we can have Γ ≥ Δ𝜁 E 0 Problems: distribution over the quasienergy states? Effects of the breaking of the discrete-time symmetry? Related features of quantum fluctuations?

Parametric oscillator + Γ + ω + ω + γ =    2 3 Classical phenomenological description, 𝑛 = 1: q 2 q ( F cos t ) q q 0 0 F Weak damping, resonant modulation 𝜕 𝐺 ≈ 2 𝜕 0 ⇒ excitation for weak field, small nonlinearity. The period-two states differ in phase by 𝜌 - spontaneous breaking of discrete time-translation symmetry

Bifurcation diagram + Γ + ω + ω + γ =    2 3 q 2 q ( F cos t ) q q 0 2 Critical field strength: 𝑟 𝐺 = 2𝛥𝜕 𝐺 , 𝑟 𝐺 ≪ 𝜕 0 0 F Relevant dimensionless parameters: ⁄ Scaled frequency detuning 𝜈 𝑞 = 𝜕 𝐺 − 2 𝜕 0 2 Γ Scaled field amplitude 𝑔 𝑞 = 𝑟 / 𝑟 𝐺 3 stable states no vibrations critical point 2 stable states more complicated than just symmetry-breaking co-dimension 2 bifurcation point

The rotating wave approximation (RWA) + Γ + ω + ω + γ =    2 3 Change to variables that slowly vary over the q 2 q ( F cos t ) q q 0 0 F vibration period: 1 1 1 𝑟 𝑢 = 𝐷 𝑅𝑅𝑅𝑅 𝜚 + 𝑄𝑅𝑄𝑄 𝜚 , 𝑞 𝑢 = − 2 𝜕 𝐺 𝐷 𝑅𝑅𝑄𝑄 𝜚 − 𝑄𝑅𝑅𝑅 𝜚 , 𝜚 = 2 𝜕 𝐺 𝑢 + 4 𝜌 ; � , ℏ � = 3| 𝛿 | ℏ / 𝜕 𝐺 𝑟 𝐺 Quantum mechanics: 𝑞 , 𝑟 = −𝑗ℏ → 𝑄 , 𝑅 = −𝑗 ℏ dimensionless Planck constant � ≪ 1 Approximations: slow decay, Γ ≪ 𝜕 0 , + weak quantum noise, ℏ depends on the nonlinearity!

Quantum Langevin equations In slow time 𝑗 𝑗 𝑅̇ = − � [ 𝑅 , 𝑕 ] − 𝑅 + 𝜊 𝑅 𝜐 , 𝑄̇ = − � [ 𝑄 , 𝑕 ] − 𝑄 + 𝜊 𝑄 𝜐 ℏ ℏ Quantum noise is 𝜀 -correlated in slow time: = 2 𝐸𝜀 𝜐 − 𝜐 ′ 𝜊 𝑅 𝜐 𝜊 𝑅 𝜐 ′ = 𝜊 𝑄 𝜐 𝜊 𝑄 𝜐 ′ � + 1 −1 , [ 𝜊 𝑅 𝜐 , 𝜊 𝑄 𝜐 ′ ] = 2𝑗ℏ � 𝑜 � 𝜀 ( 𝜐 − 𝜐 ′ ) ⁄ � = 𝑓 ℏ𝜕 0 𝐺 𝐶 𝑈 𝐸 = ℏ 2 , 𝑜 − 1 Noise intensity 𝐸 ∝ ℏ for 𝑙 𝐶 𝑈 < ℏ𝜕 0 ; for 𝑙 𝐶 𝑈 ≫ ℏ𝜕 0 , 𝐸 ∝ 𝑈 𝒉 𝑹 , 𝑸 = 𝟐 𝟓 𝑹 𝟑 + 𝑸 𝟑 𝟑 − 𝟐 𝟑 𝝂 𝒒 𝑹 𝟑 + 𝑸 𝟑 + 𝟐 𝟑 𝒈 𝒒 ( 𝑹𝑸 + 𝑸𝑹 )

Adiabatic approximation near criticality 𝑗 𝑗 𝑅̇ = − � [ 𝑅 , 𝑕 ] − 𝑅 + 𝜊 𝑅 𝜐 , 𝑄̇ = − � [ 𝑄 , 𝑕 ] − 𝑄 + 𝜊 𝑄 𝜐 ℏ ℏ Linear equations without noise near the critical point, 𝑔 𝑞 = 1, 𝜈 𝑞 = 0 : 𝑅̇ ≈ 𝑔 𝑄̇ ≈ − 𝑔 𝑞 − 1 𝑅 − 𝜈 𝑞 𝑄 , 𝑞 + 1 𝑄 + 𝜈 𝑞 𝑅 Q is a “soft mode” 𝑄 𝜐 adiabatically follows 𝑅 𝜐 ⇒ on times 𝜐 ≫ 1 Γt ≫ 1 eliminate 𝑄 𝜐 ⇒ P an adiabatic classical equation for the soft mode with quantum noise 𝑅̇ = −𝜖 𝑅 𝑉 𝑟 + 𝜊 𝑅 𝜐 , Q 𝑉 𝑅 = 1 𝑅 2 − 𝜈 𝑞 4 𝑅 4 + 1 2 − 𝑔 2 − 1 12 𝑅 6 4 𝜈 𝑞 𝑞 an analog of the 𝜚 6 Landau theory reminder: 𝑔 𝑞 = 𝑟 𝑟 ⁄ , 𝜈 𝑞 ∝ 𝜕 𝐺 − 2 𝜕 0 𝐺

Stationary distribution × × × P Q ⁄ , 𝛦𝜈 𝑞 ∼ 𝐸 1 3 ⁄ Critical region: the typical scales are 𝛦𝑅 ∼ 𝐸 1 / 6 ∝ ℏ 1 / 6 , 𝛦𝑔 𝑞 ∼ 𝐸 2 3 2 2 𝑔 The Wigner distribution 𝜍 𝑋 𝑅 , 𝑄 ∝ exp − 𝑄 − 𝑄 𝑏𝑏 𝑅 � 𝑞 + 1 𝐸 exp[ − 𝑉 𝑅 ⁄ ] 𝐸

Scaling of the interstate switching rates I Switching between period-two states in the range of developed bistability × � 𝐵 ℏ � 𝑋 𝑡𝑡 = Ω 𝑡𝑡 exp − 𝑆 ⁄ , � + 1 � 𝐵 = Δ𝑉 /( 𝑜 𝑆 2) simple power-law scaling only for 𝜈 𝑞 = 0 (exact resonance, 𝜕 𝐺 = 2 𝜕 0 ) 3 / 2 2 − 1 � 𝐵 ∝ 𝑔 𝑆 𝑞

Scaling of the interstate switching rates II × Switching between period-two states 𝑺 𝑩𝟐 𝑺 𝑩𝟏 in the range of developed bistability � 𝐵 ℏ � 𝑋 𝑡𝑡 = Ω 𝑡𝑡 exp − 𝑆 ⁄ , � + 1 � 𝐵 = Δ𝑉 /( 𝑜 𝑆 2) 3 / 2 independent of 𝜈 𝑞 , 2 − 1 � 𝐵1 ∝ 𝑔 𝑆 𝑞 i.e. of the driving frequency detuning

„First-order“ phase transition × × × 1 / 2 𝐺𝑑 = 2 𝑔 2 − 1 𝜈 𝑞 𝑞

Critical slowing down Critical region: the typical scales are ΔQ ∼ 𝐸 1 / 6 ∝ ℏ 1 / 6 , ⁄ , Δ𝜈 𝑞 ∼ 𝐸 1 3 ⁄ , Δ𝜐 ∼ 𝐸 −2 3 ⁄ ∝ ℏ −2 / 3 , Δ f p ∼ 𝐸 2 3 Reciprocal correlation time as function of the frequency detuning. From top down the scaled field ⁄ = − 4, − 2, 0, 2, 4, 6 . 2 − 1)/ 𝐸 2 3 is: ( 𝑔 𝑞

Schematics of the experimental system P out 𝝏 𝑮 /2 Pump 𝝏 𝑮 Φ P p M Temperature: T ~ 10 mK 𝜕 0 /2 𝜌 = 10.402GHz, Q=340

Vibrational states as a function of driving frequency P p = -62.4 dBm ω F /2 π /2 = 10.386 GHz ω F /2 π /2 = 10.384 GHz 𝜕 𝐺 / 4𝜌 (GHz) ω F /2 π /2=10.404GHz ω F /2 π /2=10.430GHz ω F 2 π /2=10.389GHz ω F /2 π /2=10.390GHz

“First order phase transition” ω F /2 π /2 ~ 10.390 GHz ω F /2 π /2 = 10.393 GHz 10.392 GHz 10.391 GHz 10.390 GHz 10.389 GHz 10.388 GHz 10.387 GHz 10.386 GHz 10.385 GHz Squeezing? 10.384 GHz 𝜕 𝐺 /4 𝜌

Nonlinear friction I 𝑜𝑜 = − 2 Γ 𝑜𝑜 𝑟 2 𝑒𝑟 / 𝑒𝑢 Phenomenological nonlinear friction: 𝑔 A microscopic mechanism for passive quantum vibrational systems: MD & Krivoglaz, 1975 nanomechanics: Atalaya & MD, 2015 important for quantum optomechanics (MD, 1978)

Nonlinear friction II 𝑜𝑜 = − 2 Γ 𝑜𝑜 𝑟 2 𝑒𝑟 / 𝑒𝑢 Phenomenological nonlinear friction: 𝑔 𝑜𝑜 𝑢 , 𝑗 1 � 𝑜𝑜 𝑅 , 𝑅 2 + 𝑄 2 + + 𝜊 𝑅 𝑅̇ = − � [ 𝑅 , 𝑕 𝑞 ] − 𝑅 + 𝜊 𝑅 𝜐 − 2 Γ ℏ Quantum Langevin equations � 𝑜𝑜 𝑄 , 𝑅 2 + 𝑄 2 + + 𝜊 𝑄 𝑜𝑜 𝑢 𝑗 1 𝑄̇ = − � [ 𝑄 , 𝑕 𝑞 ] − 𝑄 + 𝜊 𝑄 𝜐 − 2 Γ ℏ 1 / 2 2 + 1 � 𝑜𝑜 , 𝑔 critical point: 𝜈 𝑞0 = Γ 𝑞0 = 𝜈 𝑞0 1 2 𝐵 2 𝑟 2 + 1 4 𝐵 4 𝑟 4 + 1 𝜚 6 -type theory for the slow variable 𝑟 𝑄ear the 𝑅r𝑄t𝑄𝑅al p𝑅𝑄𝑄t , 𝑉 𝑟 = 6 𝐵 6 𝑟 6 2 𝜀𝜈 𝑞 2 𝜀𝜈 𝑞 , 𝐵 6 = 𝑔 � 𝑜𝑜 = 𝐷 2 Γ 𝑜𝑜 4 Γ 6 ⁄ ⁄ ; 𝜀𝑔 Γ , 𝐵 2 = 2 − 𝑔 𝑞0 𝜀𝑔 𝑞 , 𝐵 4 = −𝑔 2 𝑞 = 𝑔 𝑞 − 𝑔 𝑞0 − 𝜈 𝑞0 𝜀𝜈 𝑞 / 𝑔 𝑞0 𝑞0 𝑞0 2𝑔 𝑞0

Conclusions  Near the critical point, parametric oscillators display critical slowing down and anomalously strong quantum fluctuations. The time scale, the fluctuation strength, and the width of the critical region are determined by fractional powers of ℏ .  Quantum dynamics near the critical point is described by a slow variable driven by quantum noise, with a potential of the 𝝔 𝟕 -type, for linear and nonlinear friction.  Along with the time-symmetry breaking transition, the system displays a smeared first- order transition where three stable states are equally populated