Optical bistability in a quantum low photon-density regime Synge - - PowerPoint PPT Presentation

Optical bistability in a quantum low photon-density regime

藤堂眞治 Synge Todo

Department of Physics / ISSP, UTokyo Collaborators: Tatsuhiko Shirai (UTokyo → Waseda), Hans de Raedt (Groningen), Seiji Miyashita (UTokyo)

2019/07/29 @ CAQMP2019

[1] T. Shirai, ST, H. de Raedt, S. Miyashita, Phys. Rev. A 98, 043802 (2018). [2] T. Shirai, ST, S. Miyashita, to be submitted

Optical bistability under a driving field

• Dynamical phase transition in dissipative and

nonequilibrium system

• Macroscopic model: Maxwell-Bloch equation
• Microscopic description of optical bistability?
• Analogy and relation to (equilibrium) first-order

phase transition?

• Arecchi, Bonifacio (1965), Lugiato (1984),

Drummond (1981), Rempe et al (1991), Gripp et al (1996)

• System (Dicke model) + external driving field
• Rotating frame approximation
• Dissipation term (Lindblad type)

Quantum master equation

• ωa = ωph = Ω

(transmission) (spontaneous emission)

Mean-field analysis

• Ignoring correlation between spins
• Equation of motion for
• Rescaling variables:
• Cavity cooperativity parameter:
• MF treatment becomes exact in the thermodynamic limit

ρ(t) ≈ ρph(t) ⊗ ρ⊗N

s

(t)

⇒ steady state solution

Bonifacio, Lugiato (1978), Mori (2013)

Numerical methods for finite number of spins

• Quantum master equation
• Vector representation
• Superoperator matrix is huge, but sparse
• Permutation symmetry of spins (uniform interaction)
• Each photon number sector is assigned 


to each CPU core

• Can simulate up to and
• Steady state solution is obtained by 


bi-conjugate gradient (bi-CG) method dimension of :

22N → 1 6(N + 1)(N + 2)(N + 3)

(n1, n2)

N = 100

nmax = 800

Shirai, Todo, de Raedt, Miyashita (2018)

L~ ⇢(t) = 0

Dynamical first order phase transition

• Double peak structure in photon number distribution
• Maxwell's equal area law does not work

Shirai, Todo, de Raedt, Miyashita (2018)

• Relaxation time can be calculated from second eigenvalue of Liouville operator

Spin number dependence of relaxation time

Shirai, Todo, de Raedt, Miyashita (2018)

Detuning effect

• FIG. 10. Dependence of the photon number density n/N on

the detuning frequency ω in the standard case, (κ, γ, C) = (0.01, 0.01, 50), where the single peak appears at ξ = 0.04. The transition of the structures from (a) ξ = 0.01 to (f) ξ = 0.04 is shown in the box: (b) ξ = 0.014, (c) ξ = 0.015, (d) ξ = 0.02, and (e) ξ = 0.03. The stable and unstable MF steady-state solutions are denoted by solid lines (black) and dotted lines (green), respectively.

• FIG. 12. Dependence of the photon number density n/N on

the detuning frequency ω in the present case, (κ, γ, C) = (0.05, 0.002, 50). The transition of the structures from (a) ξ = 0.01 to (f) ξ = 0.04 is shown in the box: (b) ξ = 0.02, (c) ξ = 0.0275, (d) ξ = 0.028, and (e) ξ = 0.03. Between (c) and (d), the topology

• f n(ω) changes. The stable and unstable MF steady-state solu-

tions are denoted by solid lines (black) and dotted lines (green), respectively.

• FIG. 11. Dependence of the photon number density n/N on the

detuning frequency ω in the low-photon-density case, (κ, γ, C) = (0.1, 0.001, 50), where the two clear peaks at finite ω remain even at ξ = 0.04. The transition of the structures from (a) ξ = 0.01 to (f) ξ = 0.04 is shown in the box: (b) ξ = 0.015, (c) ξ = 0.02, (d) ξ = 0.03, and (e) ξ = 0.036. The stable and unstable MF steady- state solutions are denoted by solid lines (black) and dotted lines (green), respectively.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

• 0.2
• 0.1

0.1 0.2

Photon number density n/N Detuning frequency ∆ω

N=5 N=10 N=20 MF

Shirai, Todo, de Raedt, Miyashita (2018)

Hysteresis under periodic modulation of

• Mean-field analysis
• 0.5

0.01 0.02 0.03 0.04 0.05

Sz

• 0.5

0.01 0.02 0.03 0.04 0.05

Sz

• 0.5

0.01 0.02 0.03 0.04 0.05

Sz

• 0.5

0.01 0.02 0.03 0.04 0.05

Sz

short period P intermediate period P large period P infinite period P analogy to 
 spinodal point

ξ

ξ

P (MF)

sp

≈ 0.1317

Quantum dynamics under periodic field

• Floquet-Liouville operator
• defines stroboscopic dynamics
• Relaxation time to limiting hysteresis cycle is calculated from 2nd eigenvalue of

F = T e

R P

0 L(t) dt

Hartmann et al (2017)

F

ρ(0)

F

→ ρ(P)

F

→ ρ(2P)

F

→ · · ·

F

→ ρlimit

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.01 0.02 0.03 0.04 0.05

Sz Driving amplitude ξ

T = nP

N = 10

limiting cycle convergence to limiting cycle

F

τ = −P/Re log Λ2

Shirai, Todo, Miyashita (unpublished)

Spectrum of Floquet-Liouville operator

• Eigenvalues of FL operator
• Behavior of relaxation time

10-3 10-2 10-1 100 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

∆ ε P

N=40 N=50 N=60 N=70 1 2 3 0.05 0.1 0.15

εi P

N=50

• ℱ "# = Λ# "# , '# = −Re(log Λ#)/1

1

23 ≃ 0.1317

:;<

Fρi = Λiρi

✏i = −Re(log Λi)/P

τ ∼      exp O(N) P < Psp N 1/3 P = Psp O(1) P > Psp

P < Psp

Shirai, Todo, Miyashita (unpublished)

P (MF)

sp

≈ 0.1317

Scaling of relaxation time near

• Relaxation time obeys scaling law:
• Same scaling law observed in (classical) Glauber dynamics at spinodal point of fully

connected Ising model (mean-field model)

Psp

0.5 1 1.5

• 0.5

0.5

∆ ε N1/3 (P-Psp)N2/3

N=50 N=60 N=70 N=80 0.1 0.2 0.3 0.4 0.1 0.11 0.12 0.13 0.14 0.15 0.16

∆ ε P

N=50 N=60 N=70 N=80

• !"/$Δ& = ( !)/$ * − *

,-

*

,-

• N 1/3∆✏ = g(N 2/3(P − Psp))

0.5 1 1.5

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

V1/3∆ ε V2/3(H-Hsp)

V20 V40 V60 V80 V100 V1000 0.05 0.1 0.15 0.2 0.35 0.4 0.45

∆ ε H

V20 V40 V60 V80 V100

• !

Δ& = ( ! * − *,-

PRE 81 011135 (2010)

(* = *,-) 0 = Δ&1" ∼ !1"/$

Paul, Hermann, Binder (2009), Mori, Miyashita, Rikvold (2010)

• !ℋ($) = − (

),+ 2. / /

− (

• ,

034(,)

Summary

• Microscopic model of the optical bistability in terms of quantum master equation
• Efficient parallel simulation method has been developed
• Dynamical first-order-like phase transition between low-photon-density state and

high-photon-density state

• Relaxation times grows exponentially as N increases
• How to define effective free energy?
• Qualitative difference in detuning effect from the classical optical bistability
• T. Shirai, ST, H. de Raedt, S. Miyashita, Phys. Rev. A 98, 043802 (2018)
• Period dependence of hysteresis loop
• Floquet-Liouville operator formalism
• Relaxation time is discussed from eigenvalue spectrum of FL operator
• Dynamical spinodal point has the same scaling form and critical exponents with

classical spinodal point

• T. Shirai, ST, S. Miyashita, in preparation