Electronic transport and measurements in mesoscopic systems

S. Gurvitz

Weizmann Institute and NCTS

Outline:

1. Single-electron approach to electron transport.

2. Quantum rate equations.

3. Decoherence and measurement.

1

LECTURE 1

Single-electron approach to electron transport.

2

Nonequilibrium transport in mesoscopic systems

E1E0

E0 E1

2E

2E

Two−dimensional electron gas

Iµµ

L

R

AlGaAs

GaAs

E

Double−dot (qubit)

Emitter

I

Quantum dot (resonant tunneling)

Collector

electrodes

3

Electron motion in the double-dot (double-well).

Electrostatic qubit

E1 2EΩ 0

Schematical representation

Tunneling Hamiltonian: HQ = E1a†1a1 + E2a

†2a2 + Ω0(a

†2a1 + a†1a2)

All parameters in the tunneling Hamiltonian (E1,2 and Ω0) are re-

lated to the double-well potential in the configuration space: J. Bardeen,

PRL 6, 57 (1961), S.G. et al, PRA 69, 042705 (2004).

4

V(x)=U (x)+U (x)1 2

E1

V(x)

xx0

E2

E1

U (x)1

x0

E2

Φ2(x)

U (x)2

xx0

Φ1(x)

x

HQ = E1a†1a1 + E2a

†2a2 + Ω0(a

†2a1 + a†1a2)

Ω0 = −√

2|E|m

Φ1(x0)Φ2(x0)

The coupling Ω0 is real, but it can be of a positive or negative sign.

5

Electrostatic qubit

E1 2EΩ 0

Schematical representation

Electron wave function: |Ψ(t)〉 =[b1(t)a

†1 + b2(t)a

†2

]|0〉.

The Schrodinger equation: i∂t|Ψ(t)〉 = HQ|Ψ(t)〉.ib1(t) = E1b1(t) + Ω0b2(t)

ib2(t) = E2b2(t) + Ω0b1(t)

The initial condition: b1(0) = 1, b2(0) = 0. The probability of finding

the electron in the second dot is |b2(t)|2 = Ω20/ω

2[1 − cos2(ωt)], where

ω =√

Ω20 + ε2 is the Rabi frequency and ε = E1 − E2. Therefore

|b2(t)|2 ¿ 1 for ε À Ω0.

6

1

t

ε=0

ε>>Ω

2 |b (t)| 2

0

N -coupled wells

Ω1 Ω2 ΩN−1

E E EE1 23 N

The probability of finding the electron in the last well for |Ei−Ei+1| À |Ωi|:

|bN(t)|2 ∼ Ω21

ε21

Ω22

ε22

· · · Ω2N−1

ε2N−1

∼(

Ω

ε

)2N

∼ e−N/N0

where N0 is the localization length (Anderson localization).

7

Decay to continuum

0E Ωr

rE

Tunneling Hamiltonian:H = E0a†0a0 +

∑r Era

†rar +

∑r Ωr(a

†ra0 + a†0ar).

The wave function: |Ψ(t)〉 = [b0(t)a†0 +

∑r br(t)a

†r|0〉.

The Schrodinger equation: i∂t|Ψ(t)〉 = H|Ψ(t)〉:ib0(t) = E0b0(t) +

∑r

Ωrbr(t)

ibr(t) = Erbr(t) + Ωrb0(t)

Initial conditions: b0(0) = 1 and br(0) = 0. The Laplace transform:

b(E) =∫∞0 b(t) exp(iEt)dt.

8

0E Ωr

rE

(E − E0)b0(E)−∑r

Ωrbr(E) = i

(E − Er)br(E)− Ωrb0(E) = 0

Substituting br(E) from the second equation to the first one,[E − E0 −

∑r

Ω2r

E − Er

]b0(E) = i

Replace sum over r by an integral:∑

r →∫

ρR(Er)dEr, where

ρR = dr/dEr is the density of states. We assume: Ω(Er)=const and

ρR(Er)=const. Then

∑r

Ω2r

E − Er

→∞∫

−∞

Ω2(Er)ρR

E − Er + iδdEr = −iπΩ2

RρR = −iΓR

2

Therefore b0(E) = iE − E0 + iΓR/2

9

The inverse Laplace transform:

b0(t) =

∞+iδ∫

−∞+iδ

e−iEt

E − E0 + iΓR/2

(dE

2πi

)E

R

2Γ

0E −i

The probability of finding the electron inside the dot |b0(t)|2 = e−ΓRt.

The origin of the irreversibility.

|b (t)|02

t

1Finite reservoir

|b (t)|02

t

1Infinite reservoir

For infinite reservoirs the time of return increases to infinity.

10

Resonant tunneling (Single Electron Transistor).

El

E rl

LµEr0E

Ω ΩµR

l

Tunneling Hamiltonian:

Hset =∑

l

Ela†l al+

∑r

Era†rar+E0a

†0a0+

(∑

l

Ωla†l a0 +

∑r

Ωra†ra0 + H.c.

)

Single-electron approach: The entire system is occupied by a one electron,

which is initially at the level El in the left reservoir. The wave function:

|Ψ(t)〉 =

[∑

l

bl(t)a†l + b0(t)a

†0 +

∑r

br(t)a†r

]|0〉

The initial condition: bl(0) = δll, b0(0) = br(0) = 0.

11

El

E rl

LµEr0E

Ω ΩµR

l

The Laplace transform: |Ψ(E)〉 =∫∞0 |Ψ(t)〉 exp(iEt)dt. Then the

time-dependent Schrodinger equation, i∂t|Ψ(t)〉 = Hset|Ψ(t)〉 reads:

(E − El)bl(E)− Ωlb0(E) = iδll⇐= the initial condition

(E − E0)b0(E)−∑

l

Ωlbl(E)−∑

r

Ωr br(E) = 0

(E − Er)br(E)− Ωr b0(E) = 0

Substituting bl(E) and br(E) from the 1-st and the 3-d equations to the 2-d one:(

E − E0 −∑

l

Ω2l

E − El−

∑r

Ω2r

E − Er

)b0(E) =

iΩl

E − El

Replacing sums by integrals we find: b0(E) = 1

E −E0 + iΓL + ΓR

2

iΩlE − El

12

El

E rl

LµEr0E

Ω ΩµR

l

The amplitude of finding the electron inside the dot (the inverse Laplace transform)

b0(t) =

∞∫

−∞b0(E)

dE

2πi=

Ωl

El − E0 + iΓ/2

(eiElt − e−iE0t−(Γ/2)t

)

where Γ = ΓL + ΓR is the total width of the level E0. The probability of findingthe electron inside the right reservoir (collector):

P(l)R (t) =

∑r

|br(t)|2 =∑

r

∫br(E)b∗r(E

′)ei(E′−E)tdEdE′/(2π)2

The average current:

I(l)R (t) = dP

(l)R (t)/dt =

∑r

∫i(E′ − E)br(E)b∗r(E

′)ei(E′−E)tdEdE′/(2π)2

where br(E) = [Ωr/(E−Er)]b0(E). Finally the average current: I(l)R (t) = ΓR|b0(t)|2.

13

El

E rl

LµEr0E

Ω ΩµR

l

The total current in the collector is a non-coherent sum over all occupied electronstates (El) of the emitter: IR(t) =

∑l I

(l)R (t). Replacing sum over (l) by an integral:

IR(t) =

µL∫

µR

ΓR|b0(t)|2ρLdEl

In the steady-state limit (t → ∞) we find b0(t) → Ωl

(El−E0+iΓ/2)e−iElt.

Then the stationary current, I = I(t →∞)

I =

µL∫

µR

ΓLΓR

(El − E0)2 + Γ2/4

(dEl

2π

)=

µL∫

µR

T (E)dE⇐= The Landauer formula

Here T (E) is the penetration coefficient. For infinite bias (µL−µr À Γ)

the steady-state current I = ΓLΓR

ΓL+ΓR.

14

LECTURE 2

Quantum rate equations.

15

The main drawback of the single-electron approach: non-interacting electrons.

Quantum rate equations (an intuitive approach).

µR

Lµ

LΓ RΓ

LΓ RΓ

µR

(1)

n

(0)

Lµ

n

n is a number of electrons in collector.

σ(n)00 (t) = −ΓLσ

(n)00 (t)︸ ︷︷ ︸

loss term

+ΓRσ(n−1)11 (t)︸ ︷︷ ︸

gain term

σ(n)11 (t) = −ΓRσ

(n)11 (t) + ΓLσ

(n)00 (t)

The accumulated charge: Q(t) =∑

n

n[σ(n)00 (t) + σ

(n)11 (t)].

The current: I(t) = Q(t) =∑

n

n[σ(n)00 (t) + σ

(n)11 (t)] = ΓR

∑n

n[σ(n−1)11 (t)− σ

(n)11 (t)]

= ΓR

∑n

σ(n)11 (t) = ΓRσ11(t) ⇐= the total probability of finding the dot occupied.

In the steady-state limit: σ11(t →∞) = ΓL

ΓL+ΓR. The current: I(t →∞) = ΓLΓR

ΓL+ΓR.

The same result as in the single-electron approach for infinite bias (µL−µR À ΓL,R).

16

Non-interacting electrons

Lµ

µRLΓ LΓ

LΓ LΓ

RΓ

RΓ

RΓ

RΓ

(2)

(1 )(0)

(1 )

n

nn

nσ

(n)00 = −2ΓLσ

(n)00 + ΓR(σ(n−1)

11↑ + σ(n−1)11↓ )

σ(n)11↑ = −(ΓL + ΓR)σ(n)

11↑ + ΓLσ(n)00 + ΓRσ

(n−1)22

σ(n)11↓ = −(ΓL + ΓR)σ(n)

11↓ + ΓLσ(n)00 + ΓRσ

(n−1)22

σ(n)22 = −2ΓRσ

(n)22 + ΓL(σ(n)

11↑ + σ(n)11↓)

The steady-state current: I = 2ΓLΓR/(ΓL + ΓR)

Coulomb blockade (σ22 = 0)

Lµ

µRLΓ

LΓ

LΓRΓ RΓ

RΓ

(1 )

(0) (1 )

n

nn

σ(n)00 = −2ΓLσ

(n)00 + ΓR(σ(n−1)

11↑ + σ(n−1)11↓ )

σ(n)11↑ = −ΓRσ

(n)11↑ + ΓLσ

(n)00

σ(n)11↓ = −ΓRσ

(n)11↓ + ΓLσ

(n)00

The steady-state current: I = 2ΓLΓR/(2ΓL +ΓR)

17

Application of rate equations to qubit measurements

Point-contact as a detector: S.G., Phys. Rev. B 56, 15 215 (1997).J.M. Elzerman, R. Hanson, J.S. Greidanus, L.H.W. van Beveren, S. De Franceschi,L.M.K. Vandersypen, S. Tarucha, L.P. Kouwenhoven, Physica E 25, 135 (2004).

Ω

Ω

Schematic representationPOINT−CONTACT DETECTOR

µR

0

E1

E1

I1 I2

I1µR

0

E2

µL I2E2

E E1 2

µL

18

m

M.H. Devoret and R.J. Schoelkopf, Nature 406, 1039 (2000).T. Hayashi, T. Fujisawa, H.D. Cheong, Y.H. Jeong, Y. Hirayama, Phys. Rev. Lett.91, 226804 (2003).J. Gorman, D.G. Hasko, D.A. Williams, Phys. Rev. Lett., 95, 090502 (2005).S.A. Gurvitz and G.P. Berman, Phys. Rev. B72, 073303 (2005).

19

Qubit measurements with a double−dot detector

QUBIT

DETECTOR DETECTOR

QUBIT

L

ΩE

µ

2

Lµ

µRµ R

E

E

1

2

EΩ

1

0E0E I 0E +U

I=00E

R. Brenner, T.M. Buehler, and D.J. Reily, J. Appl. Phys. 97, 034501 (2005).T.Gilad and S.G., Phys. Rev. Lett., 97, 116806 (2006).HuJun Jiao, Xin-Qi Li, and JunYan Luo, Phys. Rev. B75, 155333 (2007).

20

Qubit measurements with the point-contact detector

2II1n n

Ωlr lr

Ω’E l E r E l E r

ΩE E

1 2

0ΩE E

1 2

0

The density matrix for an entire system, σ(n)ij (t). The Bloch-type rate equations:

σ(n)11 = −D1σ

(n)11 + D1σ

(n−1)11 + iΩ0(σ

(n)12 − σ

(n)21 )

σ(n)22 = −D2σ

(n)22 + D2σ

(n−1)22 + iΩ0(σ

(n)21 − σ

(n)12 )

σ(n)12 = i(E2 − E1)σ

(n)12 + iΩ0(σ

(n)11 − σ

(n)22 )− D1 + D2

2σ

(n)12 +

√D1D2σ

(n−1)12 ,

where D1,2 = I1,2/e. Tracing over the detector states n we find

σ11 = iΩ0(σ12 − σ21)

σ22 = iΩ0(σ21 − σ12)

σ12 = i(E2 − E1)σ12 + iΩ0(σ11 − σ22)− (Γd/2)σ12,

where σij(t) =∑

nσ

(n)ij (t) and Γd = (

√I1/e−

√I2/e)2 is the decoherence rate.

21

Detector current

2II1n n

Ωlr lr

Ω’E l E r E l E r

ΩE E

1 2

0ΩE E

1 2

0

The density matrix for an entire system, σ(n)ij (t). The Bloch-type rate equations:

σ(n)11 = −D1σ

(n)11 + D1σ

(n−1)11 + iΩ0(σ

(n)12 − σ

(n)21 )

σ(n)22 = −D2σ

(n)22 + D2σ

(n−1)22 + iΩ0(σ

(n)21 − σ

(n)12 )

σ(n)12 = i(E2 − E1)σ

(n)12 + iΩ0(σ

(n)11 − σ

(n)22 )− D1 + D2

2σ

(n)12 +

√D1D2σ

(n−1)12 ,

Tracing the qubit states in the density matrix σ(n)ij we obtain the probability of finding n

electrons in the collectror by time t:

Pn(t) = σ(n)11 (t) + σ

(n)22 (t)

The (ensemble) average current is:

< I(t) >= e∑

nPn(t) = (I1 − I2)σ11(t) + I2

22

Quantum-mechanical derivation of rate equations for quantum transport

S.G. and Ya. Prager,

PRB 53, 15 932 (1996);

S.G., PRB 56, 15 215 (1997);

PRB 57, 6602 (1998);

S.G. and Mozyrsky, PRB in press,

arXiv:0704.0194

Resonant tunneling through the SET.

LµEl

Er1EΩ Ω

µRl r

Tunneling Hamiltonian:

Hset =∑

l

Ela†l al +

∑r

Era†rar + E1a

†1a1 +

∑

l,r

(Ωla†l a1 + Ωra

†ra1 + H.c.)

The wave function:

|Ψ(t)〉 =

b0(t) +

∑

l

b1l(t)a†1al +

∑

l,r

brl(t)a†ral +∑

l<l′,r

b1rll′(t)a†1a†ralal′ + · · ·

|0〉.

At t = 0 all the levels in the left (right) reservoirs are occupied up to the Fermienergies µL and µR, i.e. b0(0) = 1 and all other bα(0) = 0. The amplitudes b(t) areobtained from the Schrodinger equation i∂t|Ψ(t)〉 = Hset|Ψ(t)〉.

23

µE1

lΩµL

R

lrb1lb

ElEr

R

L

µE1

l

µEl

Ω rErΩrΩ

The probability of finding the dot occupied:

σ11(t) =∑

l

|b1l(t)|2

︸ ︷︷ ︸σ(0)11 (t)

+∑

l<l′,r

|b1ll′r(t)|2

︸ ︷︷ ︸σ(1)11 (t)

+∑

l<l′<l′′,r<r′|b1ll′l′′rr′(t)|2

︸ ︷︷ ︸σ(2)11 (t)

+ · · · =∞∑

n=0

σ(n)11 (t) ,

The probability of finding the dot unoccupied:

σ00(t) = |b0(t)|2︸ ︷︷ ︸σ(0)00 (t)

+∑l,r

|blr(t)|2

︸ ︷︷ ︸σ(1)00 (t)

+∑

l<l′,r<r′|bll′rr′(t)|2

︸ ︷︷ ︸σ(2)00 (t)

+ · · · =∞∑

n=0

σ(n)00 (t) .

Rate equations:

σ(n)00 (t) = −ΓLσ

(n)00 (t) + ΓRσ

(n−1)11 (t)

σ(n)11 (t) = ΓLσ

(n)00 (t)− ΓRσ

(n)11 (t) ,

Derivation24

LµEl

Er1EΩ Ω

µRl r

The wave function:

|Ψ(t)〉 =

[b0(t) +

∑l

b1l(t)a†1al +

∑l,r

brl(t)a†ral +

∑l<l′,r

b1rll′(t)a†1a†ralal′ + · · ·

]|0〉.

Substituting |Ψ(t)〉 into the Schrodinger equation i∂t|Ψ(t)〉 = Hset|Ψ(t)〉 we obtain,

ib0(t) =∑

l

Ωlb1l(t)

ib1l(t) = (E1 − El)b1l(t) + Ωlb0(t) +∑

r

Ωrblr(t)

iblr(t) = (Er − El)blr(t) + Ωrb1l(t) +∑

l′Ωl′b1ll′r(t)

· · · · · · · · ·

The Laplace transform: b(E) =∫∞0

eiEtb(t)dt.

25

LµEl

Er1EΩ Ω

µRl r

Eb0(E)−∑

l

Ωlb1l(E) = i

(E + El − E1)b1l(E)− Ωlb0(E)−∑

r

Ωr blr(E) = 0

(E + El − Er)blr(E)− Ωr b1l(E)−∑

l′Ωl′ b1ll′r(E) = 0

· · · · · · · · ·

Replace the amplitude b in each of the sums∑

Ωb by an expression obtained fromthe subsequent equation. For example:

[E −

∑

l

Ω2l

E + El − E1

]b0(E)−

∑

l,r

ΩlΩr

E + El − E1blr(E) = i.

Replace the sums over l and r by integrals:∑

l →∫

ρL(El) dEl , where ρL(El) isthe density of states. Then the first sum becomes an integral

S1 =∫ µL

−Λ

ρL(El)dElΩ2

L(El)E + El − E1

26

S1 =∫ µL

−Λ

ρL(El)dElΩ2

L(El)E + El − E1

In the limit of high bias, µL = Λ →∞:

S1 = −iπΩ2L(Ei − E)ρL(E1 − E) = −i

ΓL

2,

El

E −E−i ε1

We assumed weak energy dependence of ρL and ΩL.The equation for the amplitude b0(E) becomes

[E − i

ΓL

2

]b0(E)−

∑

l,r

ΩlΩr

E + El − E1blr(E) = i.

The second term (“cross” term):∫ Λ

µR

ρR(Er)dEr

∫ µL

−Λ

ρL(El)dElΩL(El)ΩR(Er)blr(E, El, Er)

E + El − E1=⇒ 0 for Λ →∞

All singularities of the amplitude blr(E, El, Er) in the El variable are below the real axis.

27

Indeed, iterating the Scrodinger equation we find

blr(E, El, Er) =Ωr b1l

E + El − Er+

∑l′

Ωl′ b1ll′rE + El − Er

=ΩlΩr b0(E)

(E + El − Er)(E + El − E1)+ · · ·

Therefore all “cross” terms vanish in the large bias

limit, Λ →∞.

El

E −E−i εrE −E−i ε 1

Finally the Schrodinger equation is transformed in the large bias limit to the following set

of equation:

(E + iΓL/2)b0(E) = i

(E + El − E1 + iΓR/2)b1l(E)− Ωlb0(E) = 0

(E + El − Er + iΓL/2)blr(E)− Ωr b1l(E) = 0

· · · · · · · · ·

Using the inverse Laplace transform we reduce the Schrodinger equation to the rate

equations:

σ(n)00 (t) = −ΓLσ

(n)00 (t) + ΓRσ

(n−1)11 (t)

σ(n)11 (t) = ΓLσ

(n)00 (t)− ΓRσ

(n)11 (t) .

28

LECTURE 3

Decoherence and relaxation processes in open systems

generated by fluctuating environment.

Outline:

1. Decoherence and Relaxation of the two-state system (qubit) due to

interaction with the environment.

2. Quantum mechanical model of random force acting on the qubit.

3. Relation of the decoherence and relaxation rates to the fluctuation

spectrum of the environment.

4. Decoherence and measurement.

1

Phenomenological treatment of the environment.

(Bloch equations).

Dynamic of a two-state (spin 1/2) system.

i∂t|Ψ(t)〉 = H|Ψ(t)〉, where H = εσz+Ωσx, and |Ψ(t)〉 =

b1(t)

b2(t)

Two-state system can be represented by an electrostatic qubit.

ElectrodesSchematic representation

- e(2)

e

(1)(1) (2)

W

Here |b1(2)(t)|2 is a probability of finding the electron in the first (second) dot.

2

ElectrodesSchematic representation

- e(2)

e

(1)(1) (2)

W

For the initial condition: b1(0) = 1, b2(0) = 0.

|b2(t)|2 =Ω2

Ω2 + ε2(1− cos2 ωRt), where ωR =

√Ω2 + ε2.

The density matrix: σ11(t) = |b1(t)|2, σ22(t) = |b2(t)|2, σ12(t) = b1(t)b∗2(t).

The Schrodinger equation i|Ψ(t)〉 = H|Ψ(t)〉 is transformed to

σ11 = iΩ(σ12 − σ21)

σ22 = iΩ(σ21 − σ12)

σ12 = −2i ε σ12 + iΩ(σ11 − σ22)

3

Decoherence due to interaction with the environment

(2)(1)

e- eW

σ11 = iΩ(σ12 − σ21)

σ22 = iΩ(σ21 − σ12)

σ12 = −2i ε σ12 + iΩ(σ11 − σ22)− (Γd/2)σ12,

where Γd is the decoherence rate. The density-matrix becomes the

statistical mixture for any ε. (Qubit loses the energy for ε > 0).

σ(t) =

σ11(t) σ12(t)

σ21(t) σ22(t)

t→∞−→

1/2 0

0 1/2

4

Relaxation (phonon emission)

+ W

E0

EE − =2−

W E+− gE

σ−−(t) = −Γrσ−−(t)

σ+−(t) = i(E− − E+)σ+−(t)− Γr

2σ+−(t) ,

where Γr is the relaxation rate. The qubit appears in a pure (ground) state.

σ(t) =

σ11(t) σ12(t)

σ21(t) σ22(t)

t→∞−→

1/2 1/2

1/2 1/2

5

Combined effect of decoherence and relaxation on the qubit behavior.

The qubit density matrix can be represented as σ(t) = (1 + τ · S(t))/2, whereτ = τx,y,z are the Pauli matrices and S(t) = Sx,y,z(t) is the “polarization” vector.

Sz = −Γr

2Sz − 2ΩSy

Sy = 2Ω Sz − Γd + Γr

2Sy

Sx = −Γd + 2Γr

2(Sx − Sx)

where Sx = Sx(t →∞) = 2Γr/(Γd+2Γr).

⇐= Bloch NMR equations forT−1

1 = (Γd + 2Γr)/2 andT−1

2 = (Γd + Γr)/2.

“Classical” origin of decoherence and relaxation of the qubit:

Random fluctuations of qubit parameters (or magnetic field for spin).

How relaxation and decoherence are related to the fluctuations spectrum?

Quantum mechanical models of the environment.

6

Dynamical quantum mechanical model for fluctuations of qubit parameters

with the controllable fluctuations spectrum.

Electrostatic Qubit

Single Electron Transistor (Environment)

(W)Fluctuation of coupling

W

Fluctuation of energy

e

-e

(e)

I I

7

Fluctuation of coupling (Rabi frequency) between qubit states

(2)(2)(1) (1)

E0

- e- e

lE rWµL ErµR

W WQUBIT

SET

e e’

lWE0

H = HQ + HSET + Hint HQ = ε(a†1a1 − a†2a2) + Ω(a†1a2 + a†2a1)

HSET =∑

l Elc†l cl +

∑r Erc

†rcr + E0c

†0c0 +

∑l,r(Ωlc

†l c0 + Ωrc

†rc0 + H.c.)

Hint = δΩ c†0c0(a†1a2 + a†2a1), where δΩ = Ω′ − Ω.

For large bias voltage the Schrodinger equation i∂t|Ψ(t)〉 = H|Ψ(t)〉 is

reduced to rate equations for the density matrix, σjj′(t).

8

Quantum rate equations (S.G. and Ya. Prager, PRB 53, 15 932 (1996),

S.G., PRB 56, 15 215 (1997); PRB 57, 6602 (1998); S.G. and Mozyrsky, arXiv:0704.0194

σaa = −ΓLσaa + ΓRσbb + iΩ(σac − σca),

σbb = −ΓRσbb + ΓLσaa + iΩ′(σbd − σdb),

σcc = −ΓLσcc + ΓRσdd + iΩ(σca − σac),

σdd = −ΓRσdd + ΓLσcc + iΩ′(σdb − σbd),

σac = −2iεσac + iΩ(σaa − σcc)− ΓLσac + ΓRσbd,

σbd = −2iεσbd + iΩ′(σbb − σdd)− ΓRσbd + ΓLσac.

(2)(2)(2) (1)(1) (1)(2)

(d)(c)(b)(a)

QUBIT

SET

The available discrete states of the entire system

(1)

GR GRE0

L

W W W W’ ’

GR GRGLG LGLG

9

(2)(2)(2) (1)(1) (1)(2)

(d)(c)(b)(a)

QUBIT

SET

(1)

GR GRE0

L

W W W W’ ’

GR GRGLG LGLG

Tracing the SET variables we obtain the reduced density matrix of the qubit:

σ11(t) = σaa(t) + σbb(t), σ22(t) = σcc(t) + σdd(t), σ12(t) = σac(t) + σbd(t)

The qubit’s density matrix becomes the mixture for any qubit-SET coupling

(δΩ = Ω′ − Ω)

σ(t)t→∞−→

1/2 0

0 1/2

Therefore there is no relaxation of the qubit, but only decoherence.

10

(2)(2)(2) (1)(1) (1)(2)

(d)(c)(b)(a)

QUBIT

SET

(1)

GR GRE0

L

W W W W’ ’

GR GRGLG LGLG

The fluctuation spectrum of Ω is given by the SET charge correlator:

SQ(ω) =∫ ∞

−∞〈δQ(0)δQ(t)〉eiωtdt

The correlator can be evaluated via P1(t) = σbb(t) + σdd(t), which is a

probability of finding the SET occupied. One find:

SQ(ω) =2ΓLΓR

Γ(ω2 + Γ2)

where Γ = ΓL + ΓR. The fluctuation spectrum is not affected by the qubit.

11

Comparison with Bloch equations.

σ11 = iΩ(σ12 − σ21)

σ22 = iΩ(σ21 − σ12)

σ12 = −2i ε σ12 + iΩ(σ11 − σ22)− (Γd/2)σ12

Damping oscillations with the Rabi frequency: ωR =√

Ω2 + ε2.

In weak coupling limit (small fluctuations of the Rabi frequency) δωR ¿ Γ

the qubit behavior obtained from the rate equations for the entire system

coincides with that obtained from the Bloch equations with the decoherence

rate

Γd = 2 (δωR)2 SQ(0)

There is no relation between Γd and S(ω) in a strong coupling limit,

δωR À Γ ∼ S−1Q (0)

12

The probability of finding the electron in the first dot if the qubit for

ε = 2Ω, ΓL = Ω, ΓR = 2Ω and δΩ = 0.5Ω. The solid line is the exact

result (rate equations), whereas the dashed line is obtained from the

Bloch-type rate equations.

10 20 30 40 50tW

-0.4

-0.2

0

0.2

0.4

(a) (b)

10 20 30 40 50tW

-0.4-0.2

00.20.40.60.81 t

t

t

Re 12

Im 1211 s ( )

s ( )

s ( )

13

Fluctuation of the qubit’s energy levels

(2) (1)(1) (2)

lE

E0 ErµR

W We e+U

QUBIT

SETµL

ee- -

rWlW

H = HQ + HSET + Hint HQ = ε(a†1a1 − a†2a2) + Ω(a†1a2 + a†2a1)

HSET =∑

l Elc†l cl +

∑r Erc

†rcr + E0c

†0c0 +

∑l,r(Ωlc

†l c0 + Ωrc

†rc0 + H.c.)

Hint = U c†0c0a†1a1, where U is Coulomb energy.

For the large bias voltage, µL − µR, the Schrodinger equation, i∂t|Ψ(t)〉 =

H|Ψ(t)〉, is reduced to rate equations for the density matrix, σjj′(t),

describing the qubit and the SET at once.

14

The available discrete states of the entire system

−−−

’’’’

(b)(a) (d)(c)

−W We ee+U e+U

e e e e

GR

WW

GL GR GL GRRGLGLG

The tunneling widths are energy independent: Γ′L,R = ΓL,R. The charge

correlator is not affected by the qubit, SQ(ω) = 2ΓLΓR/[Γ(ω2 + Γ2)].

There is no relaxation of the qubit, Γr = 0. In a weak coupling limit,

U ¿ (Ω2 + ΓΩ)1/2, the decoherence rate

Γd = U2 SQ(ωR)

There is no relation between Γd and S(ω) in a strong coupling limit,

U À (Ω2 + ΓΩ)1/2. The qubit is localized in the limit U/Ω →∞.

15

The tunneling widths are energy dependent: Γ′L,R 6= ΓL,R (back-action):

Weak coupling, ∆Γ ∝ U .

(2)(2) (1)(1)(1)(1) (2)

−−−−

’’’’

(2)W

e ee+U e+U

e e e e

GR

WW

GL GR GL GRRGLGL

W

G

The qubit’s density matrix does not turn to the mixture in the steady-state:

σ(t)t→∞−→

1/2− αU βU

βU 1/2 + αU

Relaxation takes place, Γr 6= 0. The charge correlator is affected by the

qubit (the back action). From a comparison with Bloch equations we find:

Γd + 2Γr = U2 SQ(ωR)

16

Decoherence and measurement.

(2)(2) (1)(1)(1)(1) (2)

−−−−

(2)W We ee+U e+U

e e e e

GR

WW

GL GR G GRRGLGLGL

The Coulomb interaction U À ΓL,R, but U ¿ Ω (small qubit’s distortion).

20 40 60 80 100tW0

0.2

0.4

0.6

0.8

1

t11s ( )Decoherence rate Γd in the

case of “measurement” ex-

ceeds by order of magnitude

the decoherence from fluctu-

ations of the qubit’s levels.

17

Summary

• Random fluctuations of the qubit parameters due to interaction with

the environment can be described by simple quantum rate equations.

• In the absence of back-action on the fluctuating environment (”pure

environment”) no relaxation of the qubit takes place.

• In the presence of back-action (“measurement regime”) relaxation of

the qubit takes place, in general.

• In weak coupling limit the decorerence (relaxation) rate is given by the

fluctuation spectrum of the environment.

• No relation between the decoherence (relaxation) rate is found in strong

coupling limit. The qubit is localized.

• The method is mostly suitable for multi-qubit systems. How the

decoherence rate scales with number of coupled qubits?

18