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Counting statistics in multiple path geometries and quantum stirring
Doron Cohen
Ben-Gurion University
Maya Chuchem (BGU)
Itamar Sela (BGU)
Refs:
Counting statistics for a coherent transition, MC and DC (PRA 2008)
Counting statistics in multiple path geometries, MC and DC (JPA 2008)
Quantum stirring of electrons in a ring, IS and DC (PRBs 2008)
http://www.bgu.ac.il/∼dcohen
I
$DIP, $BSF
Outline
The counting operator:
Q =
∫ t
0
I(t′)dt′
– Single path coherent transition
– Double path coherent transition
– Quantum stirring (full cycle)
– Condensed particles / interactions
〈Q〉 = ???
Var(Q) = ???
P(Q) = ??? [FCS]
I
I
Stirring = inducing DC current by AC driving.
Q is not an observable
The counting statistics can be determined using a continuous measurement scheme:
The current induces a translation of a Von-Neumann pointer. At the final time, the
position of the pointer is measured.
P(Q) =1
2π
∫ ⟨[T e−i(r/2)Q]† [T e+i(r/2)Q]⟩
e−iQrdr
Q = 〈Q〉
Q2 = 〈Q2〉
where Q =
∫ t
0
I(t′)dt′
H. Everett, Rev. Mod. Phys. 29, 454 (1957).
L.S. Levitov and G.B. Lesovik, JETP Letters (1992/3).
Y.V. Nazarov and M. Kindermann, EPJ B (2003).
MC and DC, PRA (2008)
Single path coherent transition
〈N〉 = p = occupation
〈Q〉 = p = counting
Var(Q) = (1− p)p:::::::::
classical: N = 1 [p],0 [1−p]
Q = 1 [p],0 [1−p]
Quantum: N = 1 [p],0 [1−p]
Q = ±√p [(1±√p)/2]
cu
|0>|1>
E1
time
E = u(t)0
p = 1− PLZ
PLZ = e−2π c2
u
Restricted Quantum to Classical Correspondence (QCC)
N = occupation operator (eigenvalues = 0, 1)
I = current operator
Heisenberg equation of motion:d
dtN (t) = I(t)
Counting vs change in Occupation: N (t)−N (0) = Q
Counting statistics = Occupation statistics:
〈Qk〉 =⟨(N (t)−N (0))k
⟩= 〈N k〉t = p for k = 1, 2 only
fi0
˛(N (t)−N (0)) (N (t)−N (0)) (N (t)−N (0))
˛0
fl6=
fi0
˛N (t)3
˛0
fl
Restricted QCC is robust
Detailed QCC is fragileA. Stotland and D. Cohen, J. Phys. A 39, 10703 (2006).
Double path coherent transition
〈N〉 = p
〈Q〉 = λp
Var(Q) = λ2(1− p)p:::::::::
λ =c1
c1 + c2
= splitting ratio
Coherent splitting is not like incoherent partitioning:
Var(Q) 6= (1− λp)λp
λ 6= |c1|2
|c1|2 + |c2|2
c
u
|0> |1>
|2>c
1
2
E+
E−
λ
u(t)
time
p = 1− PLZ
PLZ = e−π(c1+c2)2
u
Splitting vs Partitioning
|Ψ〉 =(|Q=0〉+ |Q=1〉
)⊗ |q=0〉 q = pointer
The Schrodinger-cat paradigm:
The state of the particle becomes mixed;
One measures Q = 0, 1 with 50%-50% probabilities.
|Ψ〉 = |Q=0〉 ⊗ |q=0〉 + |Q=1〉 ⊗ |q=1〉
The Born-Oppenheimer paradigm:
The state of the particle remains pure;
One measures Q = 12 with 100% probability.
|Ψ〉 =(|Q=0〉+ |Q=1〉
)⊗ |q=1
2〉
Splitting and stirring
The scattering point of view:
The particle has two paths to its destination.
The stirring point of view:
A circulating current is induced due to the driving.
60%
40%
60%
40%
400%
−300%
x3
x4
The splitting ratio approach∗ to quantum stirring∗∗
〈N〉 ≈∣∣∣√P
LZ − eiϕ√
P�LZ
∣∣∣2〈Q〉 ≈ λ − λ�
Var(Q) ≈∣∣∣λ
√P
LZ + eiϕλ�
√P�
LZ
∣∣∣2(*) In the classical context a similar approach has been in-
dependently proposed under the name current decomposition
formula. S.Rahav, J.Horowitz, and C.Jarzynski1 (PRL 2008).
(**) The splitting ratio approach allows to bypass the Kubo
formula approach to quantum stirring, D.Cohen (PRB 2003),
which is based on the adiabatic transport formalism of
Thouless (1983), Avron (1988), Berry and Robbins (1993).
c
u
|0> |1>
|2>c
1
2
E+
E−
λ λ
u(t)
time
λ = λ − 2λ�
Derivation, using the adiabatic approximation
U(t) ≈Xn
˛n(t)
flexp
»−i
Z t
t0
En(t′)dt′–fi
n(t0)
˛
I(t)nm = 〈n|U(t)†IU(t)|m〉 ≈ 〈n(t)|I|m(t)〉exp
»i
Z t
t0
Enm(t′)dt′–
Q ≡
0@ +Q‖ iQ⊥
−iQ⊥∗ −Q‖
1AVar(Q) = |Q⊥|2 ≈
˛ Z ∞
−∞c eiΦ(t)dt
˛2Φ(t) ≡
Z t
0
qu(t′)2 + (2c)2 dt′ [for a single LZ transition]
−2 −1 0 1 2
−1
0
1
2
3
ℜ (z)
ℑ(z
)
PC
+C
− C0
Var(Q) ≈˛Z ∞
−∞c eiΦ(t)dt
˛2=
˛2c2
u
Z ∞
−∞cosh(z) eiΦ(z)dz
˛2∼
„2c2
u
«2/3
exp
»−π
c2
u
–
PLZ ≈˛Z ∞
−∞
cu
u2+(2c)2eiΦ(t)dt
˛2=
˛1
2
Z ∞
−∞
1
cosh(z)eiΦ(z)dz
˛2∼
“ π
3
”2exp
»−π
c2
u
–
Derivation, using the adiabatic approximation (cont.)
A sequence of two Landau Zener crossings:
〈Q〉 ≈ λ − λ�
[assume for simplicity that only the splitting ratio is different]
E+
E−
λ λ
u(t)
time
Var(Q) =
∣∣∣∣∫ ∞
−∞λc eiΦ(t)dt
∣∣∣∣2 ≈∣∣∣λ eiϕ1 + λ� eiϕ2
∣∣∣2 PLZ
PLZ+LZ =
∣∣∣∣∫ ∞
−∞
cu
u2+(2c)2eiΦ(t)dt
∣∣∣∣2 ≈∣∣∣eiϕ1 − eiϕ2
∣∣∣2 PLZ
Derivation, using the splitting ratio approach
c
u
|0> |1>
|2>c
1
2
H =
0BB@u(t) c1 c2
c1 0 1
c2 1 0
1CCA , I =
0BB@0 ic1 0
−ic1 0 0
0 0 0
1CCAH =
0@ u(t)(c1+c2)√
2(c1+c2)√
21
1A , I =c1√2
0@ 0 i
−i 0
1AH =
0@ u(t) c
c 0
1A , I = λ
0@ 0 ic
−ic 0
1AULZ ≈
„ √PLZ −
√1−PLZ√
1−PLZ√
PLZ
«
U(cycle) =
»T U�
LZ T
–e−iϕ
»U
LZ
–
Q =
ZI(t)dt ≈ λQ
LZ − [T e−iϕULZ]† λ�Q�
LZ [T e−iϕULZ]
Long time Counting Statistics
Naive expectation:
Probabilistic point of view implies
δQ ∝√
t
Quantum result:
The eigenvalues Q± of the Q operator grow linearly with the number of cycles
δQ ∝ t
If we have good control over the preparation we can select it to be a
Floque state of the quantum evolution operator. For such preparation the
linear growth of δQ is avoided, and it oscillates around a residual value.
References and further work
Refs:
Counting statistics for a coherent transition, MC and DC (PRA 2008)
Counting statistics in multiple path geometries, MC and DC (JPA 2008, FQMT proc. 2009)
Quantum stirring of electrons in a ring, IS and DC (PRBs 2008)
Further work:
BEC in 2-sites - Bloch-Josephson oscillations, E.Boukobza, MC, DC and A.Vardi (PRL 2009)
BEC in 2-sites - Landau-Zener transitions, K.Smith-Mannschott, MC, M.Hiller, T.Kottos and DC (PRL 2009)
BEC in 3-sites - Quantum stirring, M.Hiller, T.Kottos and DC (EPL 2008 & PRA 2008)
URL:
http://www.bgu.ac.il/∼dcohen
k2k1
ε ε1 2
ε
kc
The Bose-Hubbard Hamiltonian (BHH) for a dimer
H =X
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 + const
Classical phase space
H(θ, ϕ) =NK
2
»1
2u(cos θ)2 − ε cos θ − sin θ cos ϕ
–H(n, ϕ) = (similar to Josephson/pendulum Hamiltonian)
Jz = (N/2) cos(θ) = n = occupation difference
Jx ≈ (N/2) sin(θ) cos(ϕ), ϕ = relative 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
Ac ≈ 4π`1− u−2/3
´3/2
Wavepacket dynamics
Coherent state |θϕ〉 is like a minimal Gaussian wavepacket
Fock state |n〉 is like equi-latitude annulus
Fock n=0 preparation - exactly half of the particles in each site
Fock coherent θ=0 preparation - all particles occupy the left site
Coherent ϕ=0 preparation - all particles occupy the symmetric orbital
Coherent ϕ=π preparation - all particles occupy the antisymmetric orbital
MeanField theory (GPE) = classical evolution of a point in phase space
SemiClassical theory = classical evolution of a distribution in phase space
Quantum theory = recurrences, fluctuations (WKB is very good!)
WKB quantization (Josephson regime)
h = Planck cell area in steradians =4π
N+1
A(En) =
(1
2+ n
)h
ω(E) ≡ dE
dn=
[1
hA′(E)
]−1
ωK ≈ K = Rabi Frequency
ωJ ≈√
NUK =√
u ωK
ω+ ≈ NU = u ωK
ωx ≈[1
2log
(N2
u
)]−1
ωJ
The many body Landau-Zener transition
c c εε0−ε
preparation
diabatic
adiabatic
sudden
E
Dynamical scenarios:
adiabatic/diabatic/sudden
-20 -10 0 10 20ε
-400
-200
0
200
400
600
800
Ene
rgy
N=30, k=1, U=0.27
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
Sub-binomial scaling of Var(n) versus 〈n〉
0 0.2 0.4 0.6 0.8 1X
0
0.1
0.2
Y
u=0.90, N=18u=2.50, N=10u=4.05, N=30 u=4.05, N=100; SCu=4.05, N=100u=4.05, N=300
K2eff
RateSweep
(NK)2
NK 2
|NUK| NUK
U−K−NK −K/N K/N K NK
2
K / N
D I A B A T I C
Q u a n t u m A D I A B A T I C
S U D D E N P R O C E S S
Quantum Stirring in a 3 site system
k2k1
a bk2k1
ε ε1 2
εε
kc kc
Control parameters:
X1 =
„1
k2−
1
k1
«X2 = E0 (E1=E2=0)
X = (X1, X2)
U = the inter-atomic interaction
H =
2Xi=0
Eini +U
2
2Xi=0
ni(ni − 1)− kc(b†1b2 + b†2b1)− k1(b†0b1 + b†1b0)− k2(b†0b2 + b†2b0)
The induced current: I = −GE (G = G2)
The pumped particles: Q =∮
Idt =∮
G · dX (per cycle)
Stirring of BEC
k2k1
a bk2k1
ε ε1 2
εε
kc kc
strong attractive interaction: classical ball dynamics
negligible interaction (|U| � κ/N): mega-crossing
weak repulsive interaction: gradual crossing
strong repulsive interaction (U � Nκ): sequential crossing
U=−10 −3
U=10−3 −2
U=10
-16.1
-16.05
-16
En
-1 -0.995 -0.99ε
0
5e+07
1e+08
-G
0
5
10
15
n i
-16.2
-16
-15.8
-1.02 -1.01 -1 -0.99ε
0
50
100
0
5
10
15
-17
-16
-15
-14
-1.1 -1 -0.9ε
0
20
40
60
GGanal
0
5
10
15 n0n1n2
-16.05
-16
-15.95
-1.004 -1.002 -1 -0.998 -0.996ε
0
1000
2000
3000
0
5
10
15
U=0 dcba
Results for the geometric conductance
G(R) = −N(k2
1−k22)/2
[(ε−ε−)2+2(k1+k2)2]3/2 mega crossing
G(J) ≈ −[
k1−k2
k1+k2
]1
3Ugradual crossing
G(F ) = −(
k1−k2
k1+k2
) ∑Nn=1
(δεn)2
[(ε−εn)2+(2δεn)2]3/2 , sequential crossing
where:
R = Rabi regime (U � κ/N)
J = Josephson regime (κ/N � U � Nκ)
F = Fock regime (U � Nκ)
Observation:
It is possible to pump Q � N per cycle.
n=0
n=3 n=2 n=1
X2
off−plane section along X1=0
En
R
X1X1
X2 X2
B
ds
Summary
• Semiclassical and WKB analysis of the dynamics
• Occupation statistics in a time dependent scenario
• The adiabatic / diabatic / sudden crossovers
• Quantum stirring: mega / gradual / sequential crossings
c c εε0−ε
preparation
diabatic
adiabatic
sudden
E
Main messages
• FCS for a 2-site coherent transition - the simplest solvable model.
• Counting statistics for a 3-site system - multiple path geometry.
• Restricted QCC fails for multiple path transitions.
• Coherent splitting is not like incoherent partioning.
• Splitting ratio approach to quantum stirring (vs Kubo).
• Interference in the calculation of Var(Q).
• Exact vs adiabatic results for Var(Q).
• Long time counting statistics for multiple cycle stirring process.
• Stirring of BEC - the effect of interactions.
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