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

Mark Dykman

Department of Physics and Astronomy, Michigan State University

Scaling and Rare Events near Excitation Threshold of a Parametric Oscillator

In collaboration with

• Z. R. Lin, RIKEN Center for Emergent Matter Science
• Y. Nakamura, RIKEN Center for Emergent Matter Science

Example: quasienergy states

quasienergy ≡ Floquet eigenvalue; quantization: 𝜁 → 𝜁𝑜

𝜔𝜁 𝑢 = 𝑓−𝑗𝜁𝑗 ℏ

⁄ 𝑣𝜁 𝑢 ,

𝑣𝜁 𝑢 + 2𝜌 𝜕𝐺 = 𝑣𝜁(𝑢)

Eigenstates of a periodically driven system are not stationary:

𝐼 𝑢 = 𝐼0 𝑟, 𝑞 − 𝑟𝑟cos𝜕𝐺𝑢, 𝑗ℏ𝜔̇ = 𝐼 𝑢 𝜔

Driven mesoscopic vibrational systems of current interest: Josephson junctions, cavity modes in optical and superconducting cavities, nanomechanical systems, cold atoms,…

CNT

Example: quasienergy states

quasienergy ≡ Floquet eigenvalue; quantization: 𝜁 → 𝜁𝑜

𝜔𝜁 𝑢 = 𝑓−𝑗𝜁𝑗 ℏ

⁄ 𝑣𝜁 𝑢 ,

𝑣𝜁 𝑢 + 2𝜌 𝜕𝐺 = 𝑣𝜁(𝑢)

Eigenstates of a periodically driven system are not stationary:

𝐼 𝑢 = 𝐼0 𝑟, 𝑞 − 𝑟𝑟cos𝜕𝐺𝑢, 𝑗ℏ𝜔̇ = 𝐼 𝑢 𝜔

Relaxation, 𝑼 = 𝟏: inter-state transitions with emission of photons, phonons, etc.

E0 E1 E2 E3

) (t F

Fock states Quasienergy states Quasienergy states are linear combinations of Fock

• states. Inter-level transitions down in energy,

𝑂𝐺𝐺𝐺𝐺 → |𝑂𝐺𝐺𝐺𝐺 − 1〉 , correspond to inter-quasi- energy level transitions 𝑜 → 𝑜 ± 𝑛 , “up” and “down” in quasienergy. Even where the energy-level width Γ ≪ Δ𝐹, we can have Γ ≥ Δ𝜁

|𝒐〉 |𝒐 + 𝟑〉 |𝒐 + 𝟐〉 |𝒐 + 𝟒〉

Problems: distribution over the quasienergy states? Effects of the breaking of the discrete-time symmetry? Related features of quantum fluctuations?

Parametric oscillator

Classical phenomenological description, 𝑛 = 1: 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

) cos ( 2

3 2

= + + + Γ + q q t F q q

F

γ ω ω   

Bifurcation diagram

Critical field strength: 𝑟

𝐺 = 2𝛥𝜕𝐺, 𝑟 𝐺 ≪ 𝜕0 2

Relevant dimensionless parameters: Scaled frequency detuning 𝜈𝑞 = 𝜕𝐺 − 2𝜕0

2Γ ⁄

Scaled field amplitude 𝑔

𝑞 = 𝑟/𝑟 𝐺

3 stable states 2 stable states no vibrations critical point

) cos ( 2

3 2

= + + + Γ + q q t F q q

F

γ ω ω   

more complicated than just symmetry-breaking

co-dimension 2 bifurcation point

Quantum mechanics: 𝑞, 𝑟 = −𝑗ℏ → 𝑄, 𝑅 = −𝑗 ℏ

, ℏ = 3|𝛿|ℏ/𝜕𝐺𝑟𝐺 The rotating wave approximation (RWA) 𝑟 𝑢 = 𝐷 𝑅𝑅𝑅𝑅 𝜚 + 𝑄𝑅𝑄𝑄 𝜚 , 𝑞 𝑢 = −

1 2 𝜕𝐺𝐷 𝑅𝑅𝑄𝑄 𝜚 − 𝑄𝑅𝑅𝑅 𝜚 , 𝜚 = 1 2 𝜕𝐺𝑢 + 1 4 𝜌;

dimensionless Planck constant

) cos ( 2

3 2

= + + + Γ + q q t F q q

F

γ ω ω   

Change to variables that slowly vary over the vibration period: Approximations: slow decay, Γ ≪ 𝜕0, + weak quantum noise, ℏ

≪ 1

depends on the nonlinearity!

Quantum Langevin equations

In slow time

𝑅̇ = −

𝑗 ℏ [𝑅, 𝑕] − 𝑅 + 𝜊𝑅 𝜐 , 𝑄̇ = − 𝑗 ℏ [𝑄, 𝑕] − 𝑄 + 𝜊𝑄 𝜐

Quantum noise is 𝜀-correlated in slow time:

𝜊𝑅 𝜐 𝜊𝑅 𝜐′ = 𝜊𝑄 𝜐 𝜊𝑄 𝜐′ = 2𝐸𝜀 𝜐 − 𝜐′ 𝐸 = ℏ 𝑜 + 1 2 , 𝑜 = 𝑓ℏ𝜕0 𝐺𝐶𝑈

⁄

− 1

−1,

[𝜊𝑅 𝜐 , 𝜊𝑄 𝜐′ ] = 2𝑗ℏ 𝜀(𝜐 − 𝜐′)

Noise intensity 𝐸 ∝ ℏ for 𝑙𝐶𝑈 < ℏ𝜕0; for 𝑙𝐶𝑈 ≫ ℏ𝜕0, 𝐸 ∝ 𝑈 𝒉 𝑹, 𝑸 = 𝟐 𝟓 𝑹𝟑 + 𝑸𝟑 𝟑 − 𝟐 𝟑 𝝂𝒒 𝑹𝟑 + 𝑸𝟑 + 𝟐 𝟑 𝒈𝒒(𝑹𝑸 + 𝑸𝑹)

Adiabatic approximation near criticality

Linear equations without noise near the critical point, 𝑔

𝑞 = 1, 𝜈 𝑞 = 0:

𝑅̇ ≈ 𝑔

𝑞 − 1 𝑅 − 𝜈𝑞𝑄,

𝑄̇ ≈ − 𝑔

𝑞 + 1 𝑄 + 𝜈𝑞𝑅

Q

P

𝑅̇ = −

𝑗 ℏ [𝑅, 𝑕] − 𝑅 + 𝜊𝑅 𝜐 , 𝑄̇ = − 𝑗 ℏ [𝑄, 𝑕] − 𝑄 + 𝜊𝑄 𝜐

Q is a “soft mode”

𝑄 𝜐 adiabatically follows 𝑅 𝜐 ⇒ on times 𝜐 ≫ 1 Γt ≫ 1 eliminate 𝑄 𝜐 ⇒

an adiabatic classical equation for the soft mode with quantum noise

𝑅̇ = −𝜖𝑅𝑉 𝑟 + 𝜊𝑅 𝜐 ,

𝑉 𝑅 = 1 4 𝜈𝑞

2 − 𝑔 𝑞 2 − 1

𝑅2 − 𝜈𝑞 4 𝑅4 + 1 12 𝑅6

an analog of the 𝜚6 Landau theory reminder: 𝑔

𝑞 = 𝑟 𝑟 𝐺

⁄ , 𝜈𝑞 ∝ 𝜕𝐺 − 2𝜕0

Stationary distribution

Q

P

× × ×

Critical region: the typical scales are 𝛦𝑅 ∼ 𝐸1/6∝ ℏ1/6, 𝛦𝑔

𝑞 ∼ 𝐸2 3 ⁄ , 𝛦𝜈𝑞 ∼ 𝐸1 3 ⁄

The Wigner distribution 𝜍𝑋 𝑅, 𝑄 ∝ exp − 𝑄 − 𝑄

𝑏𝑏 𝑅 2 2 𝑔 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

𝑋

𝑡𝑡 = Ω𝑡𝑡 exp − 𝑆

𝐵 ℏ

• ⁄

, 𝑆 𝐵 = Δ𝑉/(𝑜 + 1 2) 𝑆 𝐵 ∝ 𝑔

𝑞 2 − 1 3/2

simple power-law scaling only for 𝜈𝑞 = 0 (exact resonance, 𝜕𝐺 = 2𝜕0)

Scaling of the interstate switching rates II

× 𝑆 𝐵1 ∝ 𝑔

𝑞 2 − 1 3/2 independent of 𝜈𝑞,

i.e. of the driving frequency detuning 𝑺𝑩𝟏 𝑺𝑩𝟐

𝑋

𝑡𝑡 = Ω𝑡𝑡 exp − 𝑆

𝐵 ℏ

• ⁄

, 𝑆 𝐵 = Δ𝑉/(𝑜 + 1 2)

Switching between period-two states in the range of developed bistability

„First-order“ phase transition

× × × 𝜈𝑞

𝐺𝑑 = 2 𝑔 𝑞 2 − 1 1/2

Critical slowing down

Critical region: the typical scales are ΔQ ∼ 𝐸1/6∝ ℏ1/6,

Δfp ∼ 𝐸2 3

⁄ , Δ𝜈𝑞 ∼ 𝐸1 3 ⁄ , Δ𝜐 ∼ 𝐸−2 3 ⁄ ∝ ℏ−2/3 ,

Reciprocal correlation time as function of the frequency detuning. From top down the scaled field is: (𝑔

𝑞 2−1)/𝐸2 3 ⁄ = −4, −2, 0, 2, 4, 6.

Schematics of the experimental system

Φ M

Pump

𝝏𝑮/2

Pp Pout

𝝏𝑮

𝜕0/2𝜌 = 10.402GHz, Q=340

Temperature: T ~ 10 mK

Vibrational states as a function of driving frequency

Pp = -62.4 dBm ωF /2π/2 = 10.384 GHz ωF/2π/2 = 10.386 GHz ωF 2π/2=10.389GHz ωF /2π/2=10.390GHz ωF /2π/2=10.404GHz ωF /2π/2=10.430GHz

𝜕𝐺/4𝜌 (GHz)

“First order phase transition”

10.384 GHz 10.385 GHz 10.386 GHz 10.387 GHz 10.388 GHz 10.389 GHz 10.390 GHz 10.391 GHz 10.392 GHz ωF/2π/2 = 10.393 GHz

Squeezing?

ωF/2π/2 ~ 10.390 GHz

𝜕𝐺/4𝜌

Nonlinear friction I

Phenomenological nonlinear friction: 𝑔

𝑜𝑜 = −2Γ𝑜𝑜𝑟2𝑒𝑟/𝑒𝑢

A microscopic mechanism for passive quantum vibrational systems:

MD & Krivoglaz, 1975 important for quantum

• ptomechanics (MD, 1978)

nanomechanics: Atalaya & MD, 2015

Nonlinear friction II

Phenomenological nonlinear friction: 𝑔

𝑜𝑜 = −2Γ𝑜𝑜𝑟2𝑒𝑟/𝑒𝑢

𝑅̇ = −

𝑗 ℏ [𝑅, 𝑕𝑞] − 𝑅 + 𝜊𝑅 𝜐 − 1 2 Γ

𝑜𝑜 𝑅, 𝑅2 + 𝑄2 + + 𝜊𝑅

𝑜𝑜 𝑢 ,

𝑄̇ = −

𝑗 ℏ [𝑄, 𝑕𝑞] − 𝑄 + 𝜊𝑄 𝜐 − 1 2 Γ

𝑜𝑜 𝑄, 𝑅2 + 𝑄2 + + 𝜊𝑄

𝑜𝑜 𝑢

Quantum Langevin equations

𝜚6-type theory for the slow variable 𝑟 𝑄ear the 𝑅r𝑄t𝑄𝑅al p𝑅𝑄𝑄t, 𝑉 𝑟 =

1 2 𝐵2𝑟2 + 1 4 𝐵4𝑟4 + 1 6 𝐵6𝑟6

critical point: 𝜈𝑞0 = Γ

𝑜𝑜, 𝑔

𝑞0 = 𝜈𝑞0 2 + 1 1/2

Γ 𝑜𝑜 = 𝐷2Γ𝑜𝑜 4Γ ⁄ , 𝐵2 =

𝜀𝜈𝑞

2

2𝑔

𝑞0 2 − 𝑔

𝑞0 𝜀𝑔 𝑞, 𝐵4 = −𝑔 𝑞0 2 𝜀𝜈𝑞, 𝐵6 = 𝑔 𝑞0 6

2 ⁄ ; 𝜀𝑔

𝑞 = 𝑔 𝑞 − 𝑔 𝑞0 − 𝜈𝑞0𝜀𝜈𝑞/𝑔 𝑞0

• 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 𝝔𝟕-type, for linear and nonlinear friction.

• Along with the time-symmetry breaking transition, the system displays a smeared first-
• rder transition where three stable states are equally populated

Conclusions