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.