One body decoherence: fluctuations, recurrences, and the statistics

of the quantum Zeno suppression due to erratic driving

Doron Cohen, Ben-Gurion University

Christine Khripkov (BGU) [6]

Maya Chuchem (BGU) [1,3,4]

Erez Boukobza (BGU) [1,2,5]

Amichay Vardi (BGU) [1,2,3,5,6]

Tsampikos Kottos (Wesleyan) [3,4]

Michael Moore (Michigan) [5]

Katrina Smith-Mannschott (Wesleyan) [3,4]

Moritz Hiller (Freiburg) [3,4] $ISF, $BSF, $DIP

[1] Dynamics & fluctuations (PRL 2009)

[2] Dynamics & fluctuations (PRA 2009)

[3] Dynamics & fluctuations (PRA 2010)

[4] Sweep operation, Landau-Zener (PRL 2009)

[5] Periodic driving, Chaos, Kapitza (PRL 2010)

[6] Erratic vs Noisy driving, Zeno (PRA 2012)

The Bose-Hubbard Hamiltonian (BHH) for a dimer

H =∑i=1,2

[Eini +

U

2ni(ni − 1)

]−K

2(a†2a1 + a†1a2)

N particles in a double well is like spin j = N/2 system

H = −EJz + UJ2z − KJx

Similar to the Josephson Hamiltonian

H(n, ϕ) = U(n− ε)2 −1

2KN cos(ϕ)

n = Jz = occupation difference

ϕ = conjugate phase

Rabi regime: u < 1 (no islands)

Josephson regime: 1 < u < N2 (sea, islands, separatrix)

Fock regime: u > N2 (empty sea)

K = hopping

U = interaction

E = E2 − E1 = bias

u ≡ NUK

, ε ≡ EK

Assuming u>1 and |ε| < εc

Sea, Islands, Separatrix

εc =(u2/3 − 1

)3/2

WKB quantization (Josephson regime)

h = Planck cell area in steradians =4π

N+1

A(Eν) =(1

2+ ν

)h ν = 0, 1, 2, 3, ...

ω(E) ≡ dE

dν=

[1

hA′(E)

]−1

ωK ≈ K = Rabi Frequency

ωJ ≈√NUK = Josephson Frequency

ω+ ≈ NU = Island Frequency

ωx ≈[log

(N2

u

)]−1

ωJ

Eigenstates |Eν〉 are like strips

along contour lines of H.

Wavepacket dynamics

MeanField theory (GPE) = classical evolution of a point in phase spaceSemiClassical theory = classical evolution of a distribution in phase spaceQuantum theory = recurrences, fluctuations (WKB is very good)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

φ / π

n/j

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

φ / π

n/j

Any operator A can be presented by the phase-space function AW(Ω)

⟨A⟩

= trace[ρ A] =

∫dΩ

hρW(Ω)AW(Ω)

Recurrences and fluctuations

~S = 〈 ~J〉/(N/2) = (Sx, Sy , Sz)

OccupationDifference = (N/2) Sz

OneBodyCoherence = S2x + S2

y + S2z

FringeVisibility =[S2x + S2

y

]1/20 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|S|

φ=πφ=4π/5φ=3π/5φ=2π/5φ=π/5φ=0

time [Josephson periods]

Spectral analysis of the fluctuations: dependence on u and on N , various preparations.

The preparations, and their LDOS P (E)

−1.75 −1.25 −0.75 −0.25 0.25−1

−0.5

0

0.5

1

φ / π

n/j

π 0

e

TwinFock preparation Zero preparation Pi preparation Edge preparation

−500 −400 −300 −200 −100 0 1000

0.01

0.02

0.03

E (scaled)

P(E

ν)

−500 −490 −480 −4700

0.25

0.5

0.75

1

E (scaled)

P(E

ν)

−4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

E (scaled)

P(E

ν)

−100 −50 0 50 1000

0.005

0.01

0.015

0.02

0.025

E (scaled)

P(E

ν)

∼

[1 −(

2ENK

)2]−1/2

∼ BesselI

[E−E-NU

]∼ BesselK

[E−ExNU

]∼ exp

[− 1N

(E−ExωJ

)2]

The participation number M

M ≡[∑ν

P(Eν)2

]−1

= number of participating levels in the LDOS

In the semiclassical analysis there is scaling with respect to (u/N)1/2

which is [the width of the wavepacket] / [the width of the separatix]

M [generic] ≈√N

M [Edge] ≈[log

(Nu

)]√N

M [Pi] ≈[log

(Nu

)]√u

M [Zero] ≈ √u

Analysis

* Spectral content: ω ∼ ωosc

* Fluctuations: Var[S(t)]

Var[S] =1

M

∫Ccl(ω)dω

Zero prep Pi prep Edge prep

0 3 6 90.9

0.92

0.94

0.96

0.98

1

1.02

Sx

t (scaled) 0 2 4 6 8

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Sx

t (scaled) 0 2 4 6 8

−0.8

−0.2

0.4

Sx

t (scaled)

The spectral content of Sx

0.01 1 100u/N

1

10

wos

c/wJ

ωosc ≈ 2ωJ [Zero]

ωosc ≈ 1×[log

(Nu

)]−12ωJ [Pi]

ωosc ≈ 2×[log

(Nu

)]−12ωJ [Edge]

ωosc ≈(uN

)1/22ωJ [u N ]

Fluctuations of Sx

0.01 1 100u/N

-1

-0.5

0

0.5

1

Sx

0.01 1 100u/N

0.001

0.01

0.1

1

N1/

4 * R

MS[

Sx]

100 1000N0.01

0.1

RM

S[S x]

~N-0.26

~N-0.09

N = 100 , N = 500 2, N = 1000 .

Naive expectation: phase spreading diminishes coherence.

In the Fock regime 〈Sx〉∞ ≈ 0 [Leggett’s “phase diffusion”]

In the Josephson regime 〈Sx〉∞ is determined by u/N .

Sx ≈ 1/3 [TwinFock]

Sx ≈ exp[−(u/N)] [Zero]

Sx ≈ −1− 4/ log[

132

(u/N)]

[Pi]

RMS[〈A〉t

]=

[1

M

∫Ccl(ω)dω

]1/2

RMS [Sx(t)] ∼ N−1/4 [ Edge]

RMS [Sx(t)] ∼ (log(N))−1/2 [ Pi]

TwinFock: Self induced coherence leading to Sx ≈ 1/3.

Zero: Coherence maintained if u/N < 1 (phase locking).

Pi: Fluctuations are suppressed by u.

Edge: Fluctuations are suppressed by N (classical limit).

Periodic / Erratic / Noisy driving

H = UJ2z − (K + f(t))Jx

Periodic driving: We distinguish between

nearly resonant driving (; Chaos),

and high frequency driving (; Kapitza).

f(t) ∝ sin(Ωt)

Erratic driving: deterministic but fluctuating f(t). An experimentalist

can repeat the experiment with the same realization, or produce other

realizations, as desired.

Noisy driving: arise due to the interaction with a stationary source

(“high temperature bath”). The realizations of f(t) are not under ex-

perimental control. Nature is doing the averaging. (; Zeno).

f(t)f(t′) = 2Dδ(t− t′)

Kapitza pendulum

Kapitza physics in spherical phase-space

dρ

dt= i[H+ f(t)W, ρ]

f(t) = sin(Ωt)

3 iterations and averaging over a cycle ;

dρ

dt= i[H+ V eff, ρ]

V eff = −1

4Ω2[W, [W,H]]

Here:

W ∝ Jx = (N/2) sin θ cosϕ

; V eff ∝ J2y ∼ sin2 ϕ

Erratic vs Noisy driving

H = UJ2z − (K + f(t))Jx

f(t)f(t′) = 2Dδ(t− t′)

Initial state: S = (−1, 0, 0)

Master Equation:dρ

dt= −i[H0, ρ]−D[Jx, [Jx, ρ]]

Quantum Zeno Effect:

[Khodorkovsky Kurizki Vardi 2008]

|S|noise ≈ exp

−

1

N

w2J

Dt

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|S|

time [Josephson periods]

Statistics for erratic driving:

|S|f(t) ≈ exp

−

2

Nsinh2(Λ)

|S|median ≈ exp

−

2

Nsinh2(µ(t))

|S|average ≈ exp

−

1

N

[e2σ(t)2 cosh(2µ(t))− 1

]

Details of analysis (I)

Traditional approach to analyze the Quantum Zeno effect is based on an FGR picture.

Instead we use a semi-classical picture.

J‖ 7→ [j(j + 1)]1/2 cos(r) [Wigner-Weyl], r = the radial coordinate r ≡ θ if J‖ ≡ Jz

For squeezing along orthogonal directions with e±Λ

|S| =

[1 +

2

N

]1/2 ⟨cos(r)

⟩= exp

−

1

2

(〈r2〉 −

2

N

)= exp

−

2

Nsinh2(Λ)

(a) For a pure squeezing scenario

|S| = exp

−

2

Ncot2(2Θ) sinh2(wJ t)

wJ =

√(NU−K)K, Θ = tan−1(wJ/K)

(b) For an erratic squeezing scenario:

one has to figure out the Λ statistics.

; log-wide distribution.

−20 0 20−20

0

20

J z

Jy

(a)

Λ

Dω t

(b)

0 0.5 10

1

2

0

5

10

0 1 2 30

1

2

P(Λ

,t)

Λ

(c)

0.2 0.6 10

2

4

P(S

,t)

S

(d)

Details of analysis (II)

|S| =

[1 +

2

N

]1/2 ⟨cos(r)

⟩= exp

−

1

2

(〈r2〉 −

2

N

)= exp

−

2

Nsinh2(Λ)

(c) For a noisy squeezing scenario there are two options for analysis.

Via Λ statistics:

|S|average ≈ exp

−

1

N

[e2σ(t)2 cosh(2µ(t))− 1

]0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

3

Dω t

Λ2

0 0.4 0.80

2

P(Λ

,t)

Λ0 0.6 1.2

0

1

2

P(Λ

,t)

Λ

0 0.9 1.80

1

2

P(Λ

,t)

Λ

(a) (b)

(b)(a) (c)

(c)

Via convolution of small steps:

Dw = [cot2(2Θ)]w2J

8Dlog-radial diffusion coefficient in multiplicative process

tD =1

2Dtime to randomize direction due to transverse diffusion

|S|average = exp

−

1

N[exp (8Dwt)− 1]

should be contrasted with exp

−

1

N8Dwt

The many body Landau-Zener transition

c c εε0−ε

preparation

diabatic

adiabatic

sudden

E

Dynamical scenarios:

adiabatic/diabatic/sudden

Occupation Statistics

2 1 0 -1 -2

ε1(t)

-101

0

101

102

En

0

2

4

6

8

10

<n>

1e-06 0.0001 0.01 1

1

2

3

4

5

PN;

Var

(n)x

4 PN Var(n)

(a)

(b)

(c)

adiabatic

sudden

sweep rate

Adiabtic-diabatic (quantum) crossover

Diabatic-sudden (semiclassical) crossover

Summary

• Semiclassical analysis (WKB and Wigner-Weyl are beyond MFT)

• The dependence of the participation number M on u and on N .

• Fluctuations and recurrences, study of ωosc and S(t) and Var[S(t)]

• Noise driven dimer: Improved Quantum Zeno effect analysis for S(t)

• Erratic driving: analysis of the statistics of |S(t)| -

challenging the system-bath paradigm

• Occupation statistics in a time dependent Landau-Zener scenario:

identification of the adiabatic / diabatic / sudden crossovers.

c c εε0−ε

preparation

diabatic

adiabatic

sudden

E

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|S|

φ=πφ=4π/5φ=3π/5φ=2π/5φ=π/5φ=0

time [Josephson periods]