teleportation of quantum dot exciton qubits via superradiance

1

Teleportation of Quantum Dot Exciton Qubits

via Superradiance

Aug. 5, 2005

Center for Theoretical Sciences

NCKU

Yueh-Nan Chen ( 陳岳男 ) and Che-Ming Li

Group leader : Prof. Der-San Chuu

Dep. of Electrophysics, NCTU, Taiwan

Collaborator : Prof. Tobias Brandes (Univ. of Manchester)

2

感謝

1. 國科會特約博士後研究計畫： 介觀物理系統在光子晶體中的量子散粒雜訊 NSC 94-2112-M-009-019 計畫主持人：陳岳男 共同主持人：鄭舜仁

2. 國科會奈米計劃： 奈米結構的空腔量子電動力學及量子傳 NSC 94-2120-M-009-002 計畫主持人：褚德三 共同主持人：許世英，林俊源，朱仲夏，趙天生 計畫參與人員：林高進、邱裕煌、李哲明、廖英彥、 簡賸瑞、唐英瓚

3

Outline

1. Brief review of quantum teleportation

2. Brief review of superradiance (collective decay)

3. Teleportation of charge qubits via superradiance

in purely quantum optic system

4. Extension to quantum dot systems

5. Summary

4

Teleportation: Science fiction or science?

From Prof. Beenakker’s web-page

5

In 1993 an international group of six scientists, including IBM fellow Charles H. Bennett, confirmed the intuitions of the majority of science fiction writers by showing that perfect teleportation is indeed possible in principle, but only if the original is destroyed.

Quantum Teleportation

6

QUANTUM TELEPORTATION OF A PERSON (impossible in practice but a good example to aid the imagination) would begin with the person inside a measurement chamber (left) alongside an equal mass of auxiliary material (green).The auxiliary matter has previously been quantum-entangled with its counterpart, which is at the faraway receiving station (right).

PREPARING FOR QUANTUM TELEPORTATION . . .Scientific American, April 2000; by Zeilinger

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... TRANSMISSION OF RANDOM DATA ...

MEASUREMENT DATA must be sent to the distant receiving station by conventional means.This process is limited by the speed of light, making it impossible to teleport the person faster than the speed of light.

8

... RECONSTRUCTION OF THE TRAVELER

RECEIVER RE-CREATES THE TRAVELER, exact down to the quantum state of every atom and molecule, by adjusting the counterpart matter’s state according to the random measurement data sent from the scanning station.

9

R. Ursin et al. describe the high-fidelity teleportation of photons over adistance of 600 metes across the River Danube in Vienna.

Nature 430, 849 (2004)

Quantum teleportation across the Danube

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Teleportation with real atoms:

1. Deterministic quantum teleportation with atoms

M. RIEBE et al., Nature 429, 734 (17 June 2004)

With calcium ions

2. Deterministic quantum teleportation of atomic qubits

M. D. BARRETT et al., Nature 429, 737(17 June 2004)

With atomic (9Be+) ions

11

Proposal for teleportation in solid state system

Phys. Rev. Foucs, 6 February 2004

“Beam Up an Electron!” C. W. J. Beenakker and M. Kindermann, Phys. Rev. Lett. 92, 056801(2004)

12

Creation of an entangled

electron-hole pair. An electron meets a hole.

teleportation

13

Local Unitary Operations

UNOTATION

Qubit is denoted by horizontal lineSingle-qubit unitary transformation U :

H

PATICULAR UNITARY OPERATIONS

Hadamard transform

11

11

2

1H

Unilateral Pauli rotations

01

10x

0

0

i

iy

10

01x

14

Collective Unitary Operations

controlled-NOT(XOR) transformation

a

b

a

baaddition modulo 2

0100

1000

0010

0001

2

1CNOT

TCTC0000 CNOT

TCTC1101 CNOT

15

H

00

0)10(2

1

1)10(2

1

Maximally Entanglement Generation

)1100(2

1 0)10(2

1

01 1)10(2

1

10

11

16

H

H

U

)]1100(2

1)[10(

)]111100011000(2

1 )(

2

100

)(2

111

)(2

101

)(2

110

)]01()01(

)10()10([2

1

M

M

0

0

)]01(11)01(01

)10(10)10(00[2

1

H

Entanglement Source Party I : ALICE

Party II : BOB

y11

z10

x01

00

σ11

σ10

σ01

I00

M U

Party I : ALICE

One qbit Quantum channel

One bit Classical channel

Quantum Network for Teleportation

17

2. Brief review of superradiance

18

Interaction between a two-level atom and the photon reservoir:

In the interaction picture, the state vector :

, where

atomofoperatorcreatingcoperatorphotonb

cHecbDH

q

xqiq

qq

::

..

qq

q tftft

1;)(0;)()( 0

q1;

0;

: an atom initially in the excited state

: a photon of q in the radiation field

• Spontaneous emission of a single two-level atom

19

Results :Results :

,)(0ttietf where is the decay rate

represents the Lamb Shift

, where

0 is the energy spacing

q

q

qq

qc

D

qcD

0

2

0

2),(

20

• Spontaneous emission from two atoms

The The interactioninteraction :

:

:

..2,1

j

j

xqijq

qq

j

c

x

cHecbDH j

position of the j th atomraising operator of the j th atom

One can define the so-called Dicke states :One can define the so-called Dicke states :

1

0

0

1

2

1

2

12

1

2

1

T

S

T

T

21

Decay scheme for two-atom system :Decay scheme for two-atom system :

Limiting Limiting case :case :

<< wavelength of the photon

+=2, - =0

22

• Measurements of superradiance in previous works

Experiment in real atoms:

[R. G. DeVoe and R. G. Brewer, P. R. L. 76, 2049 (1996)]

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3. Teleportation of charge qubits via superradiance

in purely quantum optic system

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1 1 2

Collective decay

Cavity photon

teleportation 2 + 2

entangled

trap

detectordetector

leakage

Teleportation of charge qubit to cavity photon state

25

The scheme:

The interaction between the atom and single-mode cavity:

With the appropriate preparation of the initial state of atom-1 and the control of its passing time through the cavity, the singlet entangled state is created between atom-1 and the cavity photon.

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superradiant detector

subradiant detector

How to distinguish between super- and sub-radiance?

Our proposal:

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The advantages:

It’s a “one-pass” process!

i.e. the Hadamard and CNOT transformations are

omitted and the joint measurements are performed

naturally by collective decay.

The disadvantages:

The maximum successful chance is 50%.

(can be modified to teleportation with insurance by

“redundant encoding”)[S. J. van Enk et al., Phys. Rev. Lett. 78, 4293 (1997)]

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4. Extension to solid-state systems

QD excitons

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Recent experiment on QD excitons (I)

1. The QD exciton states are constructed from electron (e) and heavy hole (h) single-particle basis states with spin projections along the QD growth axis (z) of

2. However, the and eigenstates are often mixed in dots with reduced symmetry, forming two linearly polarized eigenstates separated by the anisotropic e–h exchange splitting of a few times 10 eV.

30

Optically programmable electron spin memory using semiconductor quantum dots

Miro Kroutvar et al., Nature, 432, 81 (2004).

To enable optical selection of pure spin states, magnetic field (B=4T) is applied to the QDs, such that

Zeeman splitting > anisotropic e–h exchange splitting

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Teleportation of QD exciton qubit to photonic qubit

32[Z. Yuan et al., Science 295, 102 (2002).]

It is now possible to generate single-photon electrically!

Recent experiment on QD excitons (II)

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Energy-band diagram of the p-i-n junction:

Typical InAs QD exciton decay time: 1.3ns.

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1. Current through dot-1.

2. Superradiance between dot-1 and dot-2 excitons.

are the super-radiant and sub-radiant decay rate

D (U ): coupling constant between D (U) state and hole (electron) reservoir

Current detection of superradianceCurrent detection of superradiance

35

Double-dot embedded inside a rectangular microcavity with length

z

d

),)(2(

)1(

220

2

,

z

diqqqz

qc

eDdq z

zxy

36

Expectation value of the entangled state <nT> and <nS> in a rectangular microcavity

Solid line

Dashed line

[Y. N. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. Lett. 90, 166802 (2003)]

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p-GaAs

n-GaAs

insulator insulator

Vg1 Vg3

metal contact

InAs QDs1 2 3

collective decay

entangled

Teleportation with semiconductor QD excitons

I. Subradiance-induced singlet entangled state is generated between QD 1 and 2.

II. The bandgap of the exciton in QD 3 (1) is tuned to be (non)-resonant with that in QD 2.

III. A joint measurement is done naturally by collective decay of QD 2 and3.

Steps:

[Y. N. Chen et al., cond-mat/0502412]

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Some remarks about the fidelity of the entangled state:

When one starts to tune the energy band gap of QD 1, the excitons in QD 1 and 2 will no longer decay collectively but are described by the following interactions,

The fidelity of the singlet entangled state after the tuning time t is

( The initial condition is )

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Existing experimental parameters:

1. The lifetime of QD excitons in a microcavity is shown to be greatly inhibited (≥10ns) by tuning the level-spacing 4 meV away from the resonant mode.

[M. Bayer et al., Phys. Rev. Lett. 86, 3168 (2001)]

2. The tuning pulse from the gate voltage is a step-like function with raising time 40ps.

[Y. Nakamura et al., Nature 398, 786 (1999)]

The fidelity of the entangled state can be as high as 0.98.

40

Detection scheme in QDs:

Angle resolved measurement. (x)Time resolved measurement is required!

But, it’s a statistical average, there must be errors!

The steps:

1. Setting the border line of time to distinguish between

super- and subradiance.

2. Estimating the success probability P.

- For the ration (super/sub) of (1+0.7)/(1-0.7),

P is about 0.47.

41

Another proposal for QD excitons (II)

GaAs 45°

Au

CdTe quantum dots

n+-ZnSe

V

45°

Vi

ZnTe

Experimental setup for entanglement generation Experimental setup for entanglement generation

42

1. We have proposed a teleportation

scheme based on superradiance.

2. This scheme can be applied to both

purely quantum optic and solid state

QD systems.

Summary

Y. N. Chen et al. cond-mat/0502412 (2005).

To appear in “New Journal of Physics” (2004 impact factor: 3.1)

43

superradiant detector

subradiant detector

44

45

Current detection of superradianceCurrent detection of superradiance1. Current through dot-1.

2. Superradiance between dot-1 and dot-2 excitons.

are the super-radiant and sub-radiant decay rate

D (U ): coupling constant between D (U) state and hole (electron) reservoir

46

Some remarks about the fidelity of the entangled state:

When one starts to tune the energy band gap of QD 1, the excitons in QD 1 and 2 will no longer decay collectively but are described by the following interactions,

The fidelity of the singlet entangled state after the tuning time t is

( The initial condition is )

47

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Existing experimental parameters:

1. The lifetime of QD excitons in a microcavity is shown to be greatly inhibited (≥10ns) by tuning the level-spacing 4 meV away from the resonant mode.

[M. Bayer et al., Phys. Rev. Lett. 86, 3168 (2001)]

2. The tuning pulse from the gate voltage is a step-like function with raising time 40ps.

[Y. Nakamura et al., Nature 398, 786 (1999)]

The fidelity can be as high as 0.98.

49

In plotting the figure we have assumed :

D =1 , U =0.2 , and =1/(1.3[ns]) (in free space).

1. As the inter-dot distance is close enough, the current is inhibited.

2. The current shows oscillatory behavior as a function of inter-dot distance — superradiant effect!

• Current through the double-dot

, where is the decay

rate of the quantum

dot exciton

50

t

Decay rate

1/1.3(ns)

1/10(ns)

t

Constant speed

51

Double-dot embedded inside a rectangular microcavity with length

z

d

),)(2(

)1(

220

2

,

z

diqqqz

qc

eDdq z

zxy

52

• Entanglemant of double quantum dot excitons

[Y. N. Chen, D. S. Chuu, and T. Brandes, Phys. Rev. Lett. 90, 166802 (2003)]

, where

By calculating the expectation value of the entangled state

<nT> and <nS>, we can know

the degrees of the entanglement.

Solid line

Dashed line

53

Double-dot embedded inside a rectangular microcavity with length

z

d

),)(2(

)1(

220

2

,

z

diqqqz

qc

eDdq z

zxy

54

In plotting the figure we have assumed :

D =1 , U =0.2 , and =1/(1.3[ns]) (in free space).

1. As the inter-dot distance is close enough, the current is inhibited.

2. The current shows oscillatory behavior as a function of inter-dot distance — superradiant effect!

• Current through the double-dot

, where is the decay

rate of the quantum

dot exciton

55

Energy-band diagram of the p-i-n junction:

Typical InAs QD exciton decay time: 1.3ns.