Mesoscopic quantum measurements

D.V. AverinDepartment of Physics and Astronomy,

SUNY, Stony Brook

A. Di Lorentzo, K. Rabenstein, V.K. Semenov

D. Shepelyanskii, E.V. Sukhorukov

Summary

10 βα +Basic understanding of quantum measurements:

the measurement process is ``decoherence with an output’’;

reduction of the wave-function is a classically trivial act of picking one random outcome out of several possibilities.

Example of the trivial wave-function reduction: ``Quantum measurement

problem’’: wave function is typically reduced in an unusual way, not compatible with the Schrödinger equation.

Outline

1. Basics - superconductor and semiconductor single-charge qubits- QPC as a quantum detector - ideal quantum detectors: trade-off between information gain and back-action dephasing in quantum measurements

2. Dynamics of quantum measurements- linear- quadratic

R.A. Millikan, Phys. Rev. (Ser. I)32, 349 (1911).

• Superconductor can be thought of as a BEC of Cooper pairs: one single-particle state

Quantum dynamics of Josephson junctions

ϕien=Ψ

occupied with macroscopic number of particles. The phase φand the number of particles n are conjugate quantum variables (Anderson, 64; Ivanchenko, Zil’berman, 65):

[n,ϕ] =i.This relation describes dynamics of addition or removal of particles to/from the condensate.

• If quantum fluctuations of phase φ become large, junction behavior can be described as a semiclassical dynamics of charge that leads to controlled transfer of individual Cooper pairs (D.V. Averin, A.B. Zorin, and K.K. Likharev, 1985):

.2)2( 2 CeEC ≡),11(2)(

cos/2

22

nnnnEqnE

EEH

JC

JC

±+±−−=

=−∂∂−= ϕϕ

−0.5 −0.3 −0.1 0.1 0.3 0.5q=VgC/2e

−1

0

1

2

ε 0/E

J

EC/EJ= 10

6

3

1

0.5

(a)

0.0 0.2 0.4 0.6 0.8 1.0q

0.0

0.5

1.0

<n>

(b)

Cooper-pair box as charge qubit

xJzC EqEH σσ )2()21( −−−=

For EJ <<EC and q≈1/2, the charge tunneling dynamics in the junction is directly reduced to the two-state form:

If a small Josephson junction is biased through a capacitor not resistor, one obtains configuration of a ``Cooper-pair box’’(M. Buttiker, 1987, H. Pothier et al., 1989) which avoids the problem of electromagnetic environment.

reservoir 1

probe 1

reservoir 2

probe 2

box 2 box 1

dc gate 2 dc gate 11 mµ

couplingisland

Vg2 Vg1

Vb1Vb2

E , CJ2 J2 E , CJ1 J1

Vp

Cb1Cb2

Cp Cp

Cm

Cg1Cg2

pulse gate

b

a

Yu. A. Pashkin et al., Nature 421, 823 (2003).

Coherent oscillations in two coupled charge qubits

2

3

4

1

2

3

4

5

0

1

2

3

4

5

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

0 10 20 30

d

c

b

I1 (pA)

a

I2 (pA)

I1 (pA)

I2 (pA)

∆t (ns)

13.4 GHz

9.1 GHz

f (GHz)

,)sin()sin(4

4)cos()cos(1

21)1(

221,2

22,1

2,1 ⎥⎦

⎤⎢⎣

⎡Ω

Ω+−

−Ω−= ttEEE

ttp mJJ εε

ε

.2/12

44

224

4

22 ])2sin()2sin(2

)(sin)(sin ttE

tE

tE mmm ε

εε

εδ

ΩΩ

++ΩΩ

∆+

( ) ,2/122mE+∆=Ω

Oscillations of probability pj to have a Cooper on the box j=1,2:

Entanglement between the qubits in the process of oscillations

( ) ,2/122mE+= δε ( ) ,212 JJ EE −=δ ( ) .221 JJ EE +=∆

+ΩΩ

+−ΩΩ

−= )(sin)(sin2

)(sin)(sin121 22

22

42

2

22

2

2[ ttE

tE

tE

r mmm εε

εε

,log211 2 ±

±±∑−= ppE ,11 rp −±=±

0.0 5.0 10.0t∆/2π

0.0

0.5

1.0

E

J.M. Elzerman et al., PRB 767, 161308 (2003).

Quantum-dot qubits

Quantum antidot qubits

• Information encoded in the position of a quasiparticle in the system of two antidots.

• Adiabatic level-crossing dynamics can be used to transfer quasiparticles between the antidots.

• Conditions of operation: T << Ω, ε << ∆E, ∆E – quasiparticle excitation gap

(Averin and Goldman, 2001)

Gj- control gates

Ω- inter-QAD tunnel

amplitude

ε - quasiparticle

"localization"

energy

Fractional Charge of Laughlin Quasiparticles

V.J. Goldman and B. Su, 1995

V.J. Goldman et al., PRB 64, 085319 (01).

Invariance of the quasiparticle charge

Antidot transport of the FQHE quasiparticles

I.J.Maasilta and V.J. Goldman, PRL 84, 1776 (00).

Antidot ``molecule’’ demonstrates coherent quasiparticletransport in the double-antidot system

Quantum point contacts (QPC)

E. Buks et al., Nature 391, 871 (1999).M. Field et al., PRL 70, 1311 (1993).

Mesoscopic quantum measurements II

D.V. AverinDepartment of Physics and Astronomy,

SUNY, Stony Brook

A. Di Lorentzo, K. Rabenstein, V.K. Semenov

D. Shepelyanskii, E.V. Sukhorukov

Summary

10 βα +Basic understanding of quantum measurements:

the measurement process is ``decoherence with an output’’;

reduction of the wave-function is a classically trivial act of picking one random outcome out of several possibilities.

Example of the trivial wave-function reduction: ``Quantum measurement

problem’’: wave function is typically reduced in an unusual way, not compatible with the Schrödinger equation.

Outline

1. Basics - superconductor and semiconductor single-charge qubits- QPC as a quantum detector - ideal quantum detectors: trade-off between information gain and back-action dephasing in quantum measurements

2. Dynamics of quantum measurements- linear- quadratic

Counting statistics in mesoscopic electron transport

Current noise:

∑=k

kDeVI .)2/( hπAverage current:

j

ε+eV

U (x)j(b)(a)

ε IV

I

).1()2/()0( kk

kI DDeVS ∑ −= hπ

Full counting statistics:(one mode)

.)2/(,)1()( teVNDDCP nNnnN

Nn hπ=−= −

L.S. Levitov and G.B. Lesovik, 1993

Counting statistics and detector properties of a QPC

The dephasing rate at finite temperature:

[ ].))()(1det(ln)2( 11

0 SSffdd−+−=Γ ∫ εεπε h

)(1)(0

)10(

1100 LrRtLrRt

in

+⊗++⊗→

→⊗+

βα

βα

outin ρρ →

Detector-induced ``back-action’’dephasing of the qubit

outLRqubit

out Tr ρρ ,)( = βαβαβαρ ∗∗∗∗∗ ≤+→= )( 1010

)(12 rrttqubit

Finally, summing scattering processes of different electrons we obtain back-action dephasing rate by the QPC detector at T=0:

.||ln)2( 0101 rrtteVd∗∗ +−=Γ hπ

The usual counting statistics result for the QPC gives

( ) .ln)2( 1010 dRRDDeVW Γ≤+−= hπ

and

We see that the QPC is an ideal quantum detector with W=Γd’ if no information is contained in the phases of the scattering amplitudes:

)./arg(,10 jjj rt== φφφ

Information acquisition rate is determined by distinguishabilityof the distributions Pj(n) of the transferred charge in different qubit states:

,)]()([ln)/1( 2/110 ⎟

⎟⎠

⎞⎜⎜⎝

⎛−= ∑ nPnPtW

n

( ) ,)]()([ 10102/1

10N

nRRDDnPnP +=∑

D.V.A. and E.V. Sukhorukov, 2005

An electronic Mach-Zehnder interferometer

Y. Ji et al., Nature 422, 415 (2003).

Phase information can be utilized for measurement if the QPC is included in the electronic Mach-Zender interferometer:

jt j

r j ’

χV

tj

r j

’ ⎟⎟⎠

⎞⎜⎜⎝

⎛= −

−

χ

χ

imm

imm

m eRiDeDRi

S,,

The QPC can always be turned into an ideal detector by adjustingthe scattering matrix Sm of the second QPC of the interferometer. If all measurement information is in the phases of the scattering amplitudes:

,,0 1010 DD =≠−= φφφthis happens if

.2/1,2/)( 10 ==−= mm RDφφχ

Conditional qubit evolution

Since transmission and reflection of an electron are classical alternatives, evolution of the qubit density matrix can be conditioned on the particular outcome of scattering (following ``quantum-trajectories’’ approach in quantum optics; or A.N. Korotkov, 1999, in solid-state context). Depending on whether electron is transmitted or reflected,

,]|||[|

1010,

]|||[|

00||00|| 2/1

12

02

**10*

2/11

20

2

202

DD

tt

DD

D

βα

αβαβ

βα

αα

+→

+→

.]|||[|

1010,

]|||[|

00||00|| 2/1

12

02

**10*

2/11

20

2

202

RR

rr

RR

R

βα

αβαβ

βα

αα

+→

+→

or

Important points: • the qubit does not decohere;• probabilities to be in the states 0 and 1 change.

Bell-type relation for mesoscopic tunneling

One can devise a simple sequence of qubit transformations demonstrating that charge is indeed being transferred between the qubit states even if the tunneling amplitude is vanishing. Start with the state σx=1. After transmission of an electron

.][10 2/11010 DDDD ++=ψ

The sequence of two pulses, one creating tunneling for a period of time, and another changing the phase between the states 0,1:

,2//tan2/)(,4//)( 011 πεπ −==∆ −∫∫ DDdttdtt hh

returns the qubit to the initial σx=1 state, i.e. we have a closed cycle which involves tunneling and transfer of charge at ∆=0. For any classical charge state, there is a probability p to find the state σx=-1,

.][,min 2/11010 DDDDp +=

∆T1f =

SFQ pulses Distributed Josephson Junction Voltmeter

Qubit

Ballistic Read-out for Flux Qubits(flux analog of the QPC detector)

)(2

)(2

0 xUmvxUHH z +++= δσ

It is convenient to characterize the qubit-detector interaction by the scattering parameters t0,t1,r0,r1.

A. Wallraff et al., Nature 425, 155 (2003).

Quantum dynamics of a single fluxon

Generic model of a mesoscopic detector

t

detector

|0> |1 >

1∆

qubit 1Detector Hamiltonian

,.).]ˆ[( exHchMtH ++= ξ

.)(,)(

,))((21 )1(*)1(*)(22)(

∫∫ +−

++

+−

−+−+

=Γ=Γ

Γ+Γ++Γ+Γ−=

ξξξξ

ρρρρ

tdttdt

tttttt nkllk

nijlk

nkllk

nkl&

Conditional evolution

where ξ creates excitations in the process of particle transfer between the detector reservoirs.

Ensemble-averaged evolution

.)(21

,],[2

jkkl

klklklkl

tt

Hi

−Γ+Γ=

−−=

−+γ

ρργρ&

.))()(2/1(exp

)2()/)(0()(22

2/)(

kllk

lknn

kln

kl

intt

ttI

ϕ

ρτρ

−+Γ+Γ−

ΓΓΓΓ=

−+

−+−+

Detector dynamics should exhibit macroscopically different trajectories (see, e.g., J.W. Lee et al., 2005) most of the actual mesoscopic detectors are based on tunneling.

Linear quantum measurements

Linear-response theory enables one to develop quantitative description of the quantum measurement process with an arbitrarydetector provided that it satisfies some general conditions:

• the detector/system coupling is weak so that the detector’s response is linear;

• the detector is in the stationary state; • the response is instantaneous.

xfHHH DS ++=

D.V.A., cond-mat/00044364, cond-mat/0301524. S.Pilgram and M. Büttiker, PRL 89, 200401 (2002). A.A. Clerk, S.M. Girvin, and A.D.Stone, PRB, (2003).

Information/back-action trade-off in linear measurements

Dynamics of the measurement process consists of information acquisition by the detector and back-action dephasing of the measured system. The trade-off between them has the simplest form for measurements of the static system with HS=0. If x|j>=xj|j>, we have for the back-action dephasing:

,)0()( ''t

jjjjdet Γ−= ρρ .)( 22

' hfjjd Sxx −=Γ π

Information acquisition by the detector is the process of distinguishing different levels of the output signal <o>=λxj in the presence of output noise Sq. The signal level (and the corresponding eigenstates of x) can be distinguished on the time scale given by the by the measurement time τm:

,)]([8 2'jjqm xxS −= λπτ .21)(8 2 ≥=Γ fqdm SSλπτ h

FDT analog for linear quantum measurements

where λ is the linear response coefficient of the detector, Sf and Sqare the low-frequency spectral densities of the, respectively, back-action and output noise, ReSfq is the classical part of their cross-correlator.

As one can see, this inequality characterizes the efficiency of the trade-off between the information acquisition by the detector and back-action dephasing of the measured system. The detector that satisfies this inequality as equality is ``ideal’’ or ``quantum-limited’’.

,])(Re[4 2/12fqqf SSS −≤ πλh

Continuous monitoring of the MQC oscillationsThe trade-off between the information acquisition by the detector and back-action dephasing manifests itself in the directly measurable quantity in the case of measurement of coherent quantum oscillations in a qubit.

Spectral density So(ω) of the detector output reflects coherent quantum oscillations of the measured qubit:

Dzx HfH ++−= ∆ σσ2

.)(4

)( 22222

22

ωωπλ

ωd

dqo SS

Γ+∆−∆Γ

+=

.4max ≤qSS

The height of the oscillation peak in the output spectrum is limitedby the link between the information and dephasing:

0 1 2 ω/∆

0

4

8

12

So/

Sq

Ω/∆=1.0 1.5 2.0

Γ=0.1∆

0 1 2 ω/∆

0

4

8

12

So/

Sq

ε=0.7∆

Γ/∆= 1.5

1.0

0.7

0.3

0.1 0.03

A.N.Korotkov and D.V.A., Phys. Rev. B 64, 165310 (2001).

M

L LT CT

Experimental continuous monitoring of the Rabi oscillations

E.Il’ichev et al.,PRL 91,

097906 (2003).

0.96 0.98 1 1.02 1.04

(P/P0)1/2

0.9

1

1.1

ωR/ω

T

(b)

0.96 0.98 1 1.02 1.04

(P/P0)1/2

0

0.5

1

S V,m

ax/S

0

(a) b

ca

bac

Quadratic measurements

,ˆˆ]ˆ[ 2210 McMctMt ++=

)2(2

)1(1

ˆzzM σλσλ +=

.)ˆ( )2()1()2(2

)1(10 zzzztMt σλσσδσδ +++=

Quadratic measurement distinguishes ``parallel’’ from the ``anti-parallel’’ qubit state, i,.e., is the measurement of the product operator of the two qubits σz

(1)σz(2).

The measurement time is

).)((2/)(

),|||)(|(

,/

*0

*0

2200

20

λλ

λ

τ

ttIII

tS

IS

a

am

+Γ−Γ=−=

+Γ+Γ=

=

−+↑↓↑↑

−+

Purely quadratic measurements, δj=0.

00 =H

Example: DC SQUID

),2/1()( 20

22 ΦΦ−=Φ πJEt

.0Φ<<Φ

,)4(

8)( 22222

2

0 ωγωγ

ω+∆−

∆+= a

II

SS

.)(2 2λγ −+ Γ+Γ=

∑∆−=j

jxH )(

0 2 σ

In this case, quadratic measurements can entangle two non-interacting symmetric qubits. Indeed, there are three possible measurement outcomes. In the first two, the qubits are entangled:

).|||)(|(

,,,

,,

,,

2200

0

0

0

λ

ψ

ψ

ψ

λ

λ

+Γ−Γ=

↓↑+↑↓↓↓+↑↑∈=

↓↑−↑↓=−=

↓↓−↑↑=+=

−+ tI

II

III

III

In the third outcome, the qubits perform coherent qubit oscillations at frequency that is twice the oscillation frequency of individual qubits.

W. Mao et al., PRL 93, 056803 (04)

0 1 2 ω/∆

0

4

8

SI/S

0

0

4

8

12

SI/S

0

ν/∆=0.0

0.1

0.21.0

(b)

(a)

Continuously-measured spectroscopy of two qubits

Non-linear measurements, δj≠0, λ≠0.

.)2(2 )2()1()(0 zz

j

jxH σσνσ∑ +∆−=

).12(2 20 −++= zz SStt λδ

.2/)2(,2/)(,2),2/|||)(|(

,)(3

2)(,

21

1222

1

22

2

0

↓↑↓↓↑↑↓↓↑↑

−+

−+=−==+Γ+Γ=

+∆−+=∆≈

IIIIIII

j

ISSjw

j

jjI

γγλδγ

γω

γω

3/)( ↓↑↓↓↑↑ ++= IIII

W. Mao et al., PRB 71, 085320 (05)