Post on 20-Apr-2018
transcript
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]