Quantum information: Quantum information: From foundations to experimentsFrom foundations to experiments

Second LectureSecond Lecture

Luiz DavidovichInstituto de Física

Universidade Federal do Rio de Janeiro BRAZIL

Possible level schemes

Measurement of the parity of the fieldMeasurement of the parity of the field

e g

20

int phase shift per photon4

if is oddIf , atom will come out in state

if is even

t

e n

g n

εδ

ε π

Ω= →

⎧⎪= ⎨⎪⎩

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

e ge ge g

e; g;

e; g;

e; g;

Atom in superposition of two states superposition of two refraction indices (two media) superposition of two fields with different phases

Atom in superposition of two states superposition of two refraction indices (two media) superposition of two fields with different phases

i

Feψα αΨ ∝ + − 0g

e ψ πψ=

⇒ =⇒M. Brune, J.M. Raimond,

S. Haroche, L.D. et N. Zagury, PRA 45, 5193 (1992)

M. Brune, J.M. Raimond, S. Haroche, L.D. et N. Zagury, PRA 45, 5193 (1992)

Measuring decoherence in cavity QED

HOW TO DETECT THE COHERENCEHOW TO DETECT THE COHERENCE??HOW TO DETECT THE COHERENCEHOW TO DETECT THE COHERENCE??

Send a second atom! Send a second atom! [L.D., A. Maali, M. Brune, [L.D., A. Maali, M. Brune, J.M. Raimond, and S. Haroche, PRL J.M. Raimond, and S. Haroche, PRL 7171, 2360 , 2360 (1993); L.D., M. Brune, J.M. Raimond, and S. (1993); L.D., M. Brune, J.M. Raimond, and S. Haroche, PRA Haroche, PRA 5353, 1295 (1996)]., 1295 (1996)].

Results for phase difference equal to Results for phase difference equal to ππ::

•Coherent superposition: Coherent superposition: preparation and probing preparation and probing atoms detected in the same stateatoms detected in the same statePPeeee

•Statistical mixture: Statistical mixture: second atom detected in second atom detected in ee or or gg with 50 % chance with 50 % chance P Peeee/2/2

PHYSICAL INTERPRETATION FOR PHYSICAL INTERPRETATION FOR ππ: : DETECTION OF FIELD PARITYDETECTION OF FIELD PARITY

PHYSICAL INTERPRETATION FOR PHYSICAL INTERPRETATION FOR ππ: : DETECTION OF FIELD PARITYDETECTION OF FIELD PARITY

π/2 rotationπ/2 rotation

Even number of photons: 2kπ rotation (dispersive interaction)

Even number of photons: 2kπ rotation (dispersive interaction)

π/2 rotationπ/2 rotation

( )

( )

2

0

2 1

0

22 !

2 12 1 !

k

k

k

k

kk

kk

αα α

αα α

∞

=

+∞

=

+ − ∝

− − ∝ ++

∑

∑

( )

( )

2

0

2 1

0

22 !

2 12 1 !

k

k

k

k

kk

kk

αα α

αα α

∞

=

+∞

=

+ − ∝

− − ∝ ++

∑

∑

Bloch sphere

A VARIANTA VARIANT

Displace field in the cavity by α (by turning on the microwave field):

( ) ( )1 12 0α α α+ − → +

Ν Ν( ) ( )1 1

2 0α α α+ − → +Ν Ν

What about dissipation? Exit of just one photon is enough to destroy superposition! If damping time of field is tcav, then it takes tcav/n tcav/4α2 for one photon to leave the cavity if state is 2α. Since there is only a 50% chance that the field is in this state, the time should be twice as large: tcav/2α2

Superposition of dark and lighted cavity

Superposition of dark and lighted cavity

EFFECT OF DISSIPATIONEFFECT OF DISSIPATIONEFFECT OF DISSIPATIONEFFECT OF DISSIPATION

π

tcavn Decoherence time: tcavD

Decoherence time: tcavD

n average number of photons in cavity

n average number of photons in cavity

L. D., M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A 53, 1295 (1996).

L. D., M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A 53, 1295 (1996).

EXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTS

[Brune et al., PRL 77, 4887 (1996)]

[Brune et al., PRL 77, 4887 (1996)]

Plot of Pee PegPlot of Pee Peg

What about state of What about state of the field in the cavity?the field in the cavity?

• For classical particle with position q and momentum p, state is defined by distribution of points in phase space (just one point if one has precise information on q and p).

• Could this be done for the quantized electromagnetic field? Can one measure this phase-space representation?

PHASE-SPACE REPRESENTATIONPHASE-SPACE REPRESENTATIONPHASE-SPACE REPRESENTATIONPHASE-SPACE REPRESENTATION

Look for representation with following properties:

( ) ( )ˆ ˆ, , , dpW q p q q dqW q p p pρ ρ= =∫ ∫( ) ( )ˆ ˆ, , , dpW q p q q dqW q p p pρ ρ= =∫ ∫Pure state:

( ) ( )2 2ˆ ˆ, q q q p p pρ ψ ρ ψ= = %

Property must remain true if axes are rotated:

( )( ) ( ) ( )†

cos sin , sin cos

ˆ ˆˆ

W q p q p dp

P q q U U q

θ θ θ θ θ

θ

θ θ θ θ

θ ρ θ

− +

= =

∫ ( )( ) ( ) ( )†

cos sin , sin cos

ˆ ˆˆ

W q p q p dp

P q q U U q

θ θ θ θ θ

θ

θ θ θ θ

θ ρ θ

− +

= =

∫

RADON TRANSFORM (1917)RADON TRANSFORM (1917)RADON TRANSFORM (1917)RADON TRANSFORM (1917)

P(q) uniquely determines W(q,p)! Radon inverse transform

tomography

P(q) uniquely determines W(q,p)! Radon inverse transform

tomography

Cormack and Hounsfield: Nobel prize in Medicine (1979)

Cormack and Hounsfield: Nobel prize in Medicine (1979)

Quantum mechanics: P(q) Wigner function (Bertrand and Bertrand, 1987)

THE WIGNER DISTRIBUTIONTHE WIGNER DISTRIBUTIONTHE WIGNER DISTRIBUTIONTHE WIGNER DISTRIBUTION

Wigner, 1932: Quantum corrections to classical statistical mechanics

( ) 2 '/1ˆ, ' ' 'ipxW x p x x x x e dxρ

π−= + −∫ h

h( ) 2 '/1

ˆ, ' ' 'ipxW x p x x x x e dxρπ

−= + −∫ h

hMoyal, 1949: Average of operators in symmetric form: ( ) ( )ˆ ˆˆ ˆ ˆ / 2 ,Tr xp px dxdpW x p xpρ + =⎡ ⎤⎣ ⎦ ∫( ) ( )ˆ ˆˆ ˆ ˆ / 2 ,Tr xp px dxdpW x p xpρ + =⎡ ⎤⎣ ⎦ ∫

Density matrix in terms of W:

( ) 2 '/ˆ' ' , /ipxx x x x W x p e dpρ+ − = ∫ h h( ) 2 '/ˆ' ' , /ipxx x x x W x p e dpρ+ − = ∫ h h

x x x x=x x x x=

PAULI’S QUESTIONPAULI’S QUESTIONPAULI’S QUESTIONPAULI’S QUESTION

Handbuch der Physik, 1933 – “The mathematical problem, as to whether for given functions W(x) and W’(p) [position and momentum probability densities], the wave function ψ, if such a function exists, is always uniquely determined has not been investigated in all its generality.”

Handbuch der Physik, 1933 – “The mathematical problem, as to whether for given functions W(x) and W’(p) [position and momentum probability densities], the wave function ψ, if such a function exists, is always uniquely determined has not been investigated in all its generality.”

( ) ( ) ( ) ( )2 2 and ' '

do not form a tomograp

Ans

hic

w

complet

e

e s t

r

!

:

e

W x x W p pψ ψ= =( ) ( ) ( ) ( )2 2 and ' '

do not form a tomograp

Ans

hic

w

complet

e

e s t

r

!

:

e

W x x W p pψ ψ= =

EXAMPLES OF WIGNER DISTRIBUTIONSEXAMPLES OF WIGNER DISTRIBUTIONSEXAMPLES OF WIGNER DISTRIBUTIONSEXAMPLES OF WIGNER DISTRIBUTIONS

Ground stateGround state Fock state n=3Fock state n=3

Mixture ααααMixture αααα

Superposition αα Superposition αα

Experimentally produced (ions, cavities)

MEASUREMENT OF THE MOTIONAL MEASUREMENT OF THE MOTIONAL QUANTUM STATE OF A TRAPPED IONQUANTUM STATE OF A TRAPPED IONMEASUREMENT OF THE MOTIONAL MEASUREMENT OF THE MOTIONAL

QUANTUM STATE OF A TRAPPED IONQUANTUM STATE OF A TRAPPED ION

Wineland’s group – PRL 77, 4281 (1996)Wineland’s group – PRL 77, 4281 (1996)

Field quadraturesField quadratures

( ) ( )† †

Correspond to position and momentum of harmonic oscil

1ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ, , , 1/ 22 2

lator:

iX a a Y a a X Y i X Y⎡ ⎤= + = − = ⇒ Δ Δ ≥⎣ ⎦( ) ( )† †

Correspond to position and momentum of harmonic oscil

1ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ, , , 1/ 22 2

lator:

iX a a Y a a X Y i X Y⎡ ⎤= + = − = ⇒ Δ Δ ≥⎣ ⎦

( ) ( ) ( ) ( )Electromagnetic field in terms of quadratur

ˆ ˆ, 2 cos n

e :

s

s

iE r t E u r X t Y tω ω φ ω φ ε⎡ ⎤= + − +⎣ ⎦r rr r( ) ( ) ( ) ( )

Electromagnetic field in terms of quadratur

ˆ ˆ, 2 cos n

e :

s

s

iE r t E u r X t Y tω ω φ ω φ ε⎡ ⎤= + − +⎣ ⎦r rr r

( ) ( )ˆ 0ˆ ˆLike ( ) 0 cos sin

px t x t t

mω ω

ω= +

Phase-shift operator and Phase-shift operator and generalized quadraturesgeneralized quadratures

( ) ( ) ( ) ( ) ( )† †ˆ ˆ ˆˆ ˆ ˆ ˆexp expU i a a U aU a iθ θ θ θ θ= − ⇒ = −( ) ( ) ( ) ( ) ( )† †ˆ ˆ ˆˆ ˆ ˆ ˆexp expU i a a U aU a iθ θ θ θ θ= − ⇒ = −

( ) ( )

( ) ( )

††

†

ˆ ˆˆ ˆ ˆ ˆ ˆ ˆcos sin2

ˆ ˆ ˆ ˆ ˆ ˆ ˆsin cos2

i iae a eX U XU X Y

Y X U YU X Y

θ θ

θ

θ

θ θ θ θ

πθ θ θ θ θ

− += = = +

⎛ ⎞= + = = − +⎜ ⎟⎝ ⎠

( ) ( )

( ) ( )

††

†

ˆ ˆˆ ˆ ˆ ˆ ˆ ˆcos sin2

ˆ ˆ ˆ ˆ ˆ ˆ ˆsin cos2

i iae a eX U XU X Y

Y X U YU X Y

θ θ

θ

θ

θ θ θ θ

πθ θ θ θ θ

− += = = +

⎛ ⎞= + = = − +⎜ ⎟⎝ ⎠

Generalized quadratures:

Special cases:

evolution operator

parity operator

t ω π= →= →

evolution operator

parity operator

t ω π= →= →

ˆ ˆ

ˆ ˆ

X X

Y Y

π

π

=−

=−

ˆ ˆ

ˆ ˆ

X X

Y Y

π

π

=−

=−X

YX

Y

MEASUREMENT OF QUADRATURESMEASUREMENT OF QUADRATURESMEASUREMENT OF QUADRATURESMEASUREMENT OF QUADRATURES

( )

0ˆ ˆ field

to be measured

b

local oscillator

i t

i t

a a e

e

ω

ω θβ

−

− +

= →

→ →( )

0ˆ ˆ field

to be measured

b

local oscillator

i t

i t

a a e

e

ω

ω θβ

−

− +

= →

→ →

Risken and Vogel, 1989: homodyne measurements P(q) Wigner function for EM field

Risken and Vogel, 1989: homodyne measurements P(q) Wigner function for EM field

EXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTS

Smithey et al., PRL 70, 1244 (1993)

Smithey et al., PRL 70, 1244 (1993)

Squeezed Vacuum

Breitenbach et al, Nature 387, 471 (1997)

Breitenbach et al, Nature 387, 471 (1997)

MEASUREMENT OF THE WIGNER MEASUREMENT OF THE WIGNER FUNCTION FOR ONE PHOTONFUNCTION FOR ONE PHOTON

MEASUREMENT OF THE WIGNER MEASUREMENT OF THE WIGNER FUNCTION FOR ONE PHOTONFUNCTION FOR ONE PHOTON

Lvovsky et al, PRL 87, 050402 (2001)Lvovsky et al, PRL 87, 050402 (2001)

WIGNER FUNCTION AND THE CLASSICAL WIGNER FUNCTION AND THE CLASSICAL LIMIT OF QUANTUM MECHANICSLIMIT OF QUANTUM MECHANICS

WIGNER FUNCTION AND THE CLASSICAL WIGNER FUNCTION AND THE CLASSICAL LIMIT OF QUANTUM MECHANICSLIMIT OF QUANTUM MECHANICS

Dissipation leads to disappearance of interference fringes plus evolution towards ground state

Dissipation leads to disappearance of interference fringes plus evolution towards ground state

Decay time for fringes =dissipation time/2|α|2Decay time for fringes =dissipation time/2|α|2

Evolution of coherent superposition of coherent states of harmonic oscillator, with dissipation

Evolution of coherent superposition of coherent states of harmonic oscillator, with dissipationψ ααψ αα

Fast decoherence: one needs a snapshot!

Fast decoherence: one needs a snapshot!

Another expression for the Wigner functionAnother expression for the Wigner function

( )

( ) ( )

†

†

2 '

ˆ ˆ 2 '

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ†

1ˆ, ' ' '

1ˆ' ' '

1ˆ' ' '

1ˆ' ' '

1 ˆ ˆˆTr , ,

ipq

iqp iqp ipq

ipq iqp iqp ipq

ipq iqp iqp ipq i a a

i a a

W q p q q q q e dq

q e e q e dq

q e e e e q dq

q e e e e e q dq

D D e

π

π

ρπ

ρπ

ρπ

ρπ

α α ρ α απ

−

− −

− −

− −

∗ ∗

= + −

= −

= −

=

⎡ ⎤= ⎣ ⎦

∫

∫

∫

∫

( )

( ) ( )

†

†

2 '

ˆ ˆ 2 '

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ†

1ˆ, ' ' '

1ˆ' ' '

1ˆ' ' '

1ˆ' ' '

1 ˆ ˆˆTr , ,

ipq

iqp iqp ipq

ipq iqp iqp ipq

ipq iqp iqp ipq i a a

i a a

W q p q q q q e dq

q e e q e dq

q e e e e q dq

q e e e e e q dq

D D e

π

π

ρπ

ρπ

ρπ

ρπ

α α ρ α απ

−

− −

− −

− −

∗ ∗

= + −

= −

= −

=

⎡ ⎤= ⎣ ⎦

∫

∫

∫

∫( )

2

Change of

normalizati

,

:

1

2

on

dW

αα α

π

π

∗

⇒

=

×

∫ ( )2

Change of

normalizati

,

:

1

2

on

dW

αα α

π

π

∗

⇒

=

×

∫

Displacement operatorDisplacement operator

• Translates position and momentum (or quadratures) in phase space

( ) ( )†ˆ ˆ ˆ, expD a aα α α α∗ ∗= −( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ/ 2, / 2a q ip a q ip= + = −

3 †ˆ ˆ*IH j Ad x a aα α= ⋅ ∝ −∫rr 3 †ˆ ˆ*IH j Ad x a aα α= ⋅ ∝ −∫rr

Corresponds to action of external force for harmonic oscillator, or external current for the field

( ) ( ) ( ) ( )1 †ˆ ˆˆ ˆ ˆ, * 2 , * , * expW Tr D D i a aα α α α ρ α α π−⎡ ⎤= ⎣ ⎦

DIRECT MEASUREMENT OF THE DIRECT MEASUREMENT OF THE WIGNER DISTRIBUTIONWIGNER DISTRIBUTION

DIRECT MEASUREMENT OF THE DIRECT MEASUREMENT OF THE WIGNER DISTRIBUTIONWIGNER DISTRIBUTION

L.G. Lutterbach and L.D., PRL 78, 2547 (1997)L.G. Lutterbach and L.D., PRL 78, 2547 (1997)

( ), * 2W α α ≤( ), * 2W α α ≤

Based on following expression for Wigner function (Cahill and Glauber, 1969):

( )†ˆ ˆexp *a aα α−Displacement operator

Parity operatorˆ ˆˆ ˆa aΡ Ρ =−

( ) ( )† † 2ˆ ˆ ˆ ˆ ˆ / 2 , * *Tr aa a a d Wρ α α α αα⎡ ⎤+ =⎣ ⎦ ∫( ) ( )† † 2ˆ ˆ ˆ ˆ ˆ / 2 , * *Tr aa a a d Wρ α α α αα⎡ ⎤+ =⎣ ⎦ ∫

(phase shift of the field)

( ) ( ) ( ) ( )( ) ( )

1 †

Photon counting technique:

ˆ ˆˆ ˆ ˆ, * 2 , * , * exp

ˆ2 1 , *n

n

W Tr D D i a a

n n

α α α α ρ α α π

ρ α α

−⎡ ⎤= ⎣ ⎦

= − − −∑Banaszek and Wódkiewicz (1996), Wallentowitz and Vogel (1996), Messina, Manko and Tombesi (1998), Banaszek et al (1999). Also used by Wineland’s group to measure Wigner function for vibrational state of trapped ion.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

EXPERIMENTAL PROPOSALEXPERIMENTAL PROPOSALEXPERIMENTAL PROPOSALEXPERIMENTAL PROPOSAL

( )ˆ , *D α α

1. Displace field to be measured (turn on microwave)

2. Send atom, displace phase of the field by π iff atomic state is e3. Detect atomic state4. Produce field anew, repeat procedure

( ), * / 2e gP P W α α− = − −Problem: must produce shift equal to π

Problem: must produce shift equal to π

Quantum circuit for measuring the Wigner function

MEASUREMENT OF THE QUANTUM MEASUREMENT OF THE QUANTUM STATE OF A PHOTON IN A CAVITY – ENSSTATE OF A PHOTON IN A CAVITY – ENS

MEASUREMENT OF THE QUANTUM MEASUREMENT OF THE QUANTUM STATE OF A PHOTON IN A CAVITY – ENSSTATE OF A PHOTON IN A CAVITY – ENS

Measurement of sub-Planck phase-space structureMeasurement of sub-Planck phase-space structure

A sensitive instrument…

W. Zurek, Nature 412, 712 (2001)W. Zurek, Nature 412, 712 (2001)

( )cat 2α α= + −

( )compass

2 i iα α α α= + − + + −1

P αΔΧ ≈ ≈

h 22area A 1 | |α=: h

XXpPX

Using this instrument for measuring small displacements and rotations

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

System is prepared in a known input state ψ which experiences a small displacement transforming X into X+

We want to infer with minimum error from measurement performed on the displaced state ψ’exp(iP)ψ, where P is the momentum operator

If one measures X, then precision in the determination of the displacement is limited by ΔΧ (width of wavepacket)

For coherent state,

Measuring small displacements and rotations - standard quantum limit

XΔ ≈ h / 1/N N⇒ Δ ≈ =h h

Nh

hΔ

h

Measurement of weak classical forces

• Classical force acting for a fixed time on a simple harmonic oscillator displaces the complex amplitude of the oscillator in phase space

• Action of the force in the interaction picture:

• Must resolve displacement in order to measure the force( ) ( )† *expD a aα α α= −

Interference regions in Wigner function (“sub-Planck” structures)

22area A 1 | |α=: h

In order to have ˆcat cat 0xU ≈ Minimum translation:

: 1 |α =/ N

[ ]2 1cat cat( ) 1 cos(4 | | )

2ε α ε≈ +

ε→

Effect of small displacement

Weak-force detection Much better than standard limit!

Wigner function of unperturbed states

Product of Wigner functions: integration over blue areas cancels out integration over red areas

Wigner function of perturbed states

2

2

4 | |

2 | |

cat

compass

π α

π α

=

=

Small rotations

| ,e α | Ψ |

| | , | ,e gf e S g SP e P gΨ ≡ Ψ + Ψ

U

†U

perturba on iˆ txU ≡

2 2

2 2

, cat cat( )1

f

e ge eS S

eP P

α ε

α α

Ψ= − ≈ =

Ψ Ψ

General strategy for measurement: couple oscillator to two-level system

LOSCHMIDT ECHO!

2

2

, fe α Ψ

= Ψ

Revisiting collapses and revivals

• J.H. Eberly, N.B. Narozhny, and J.J. Sanchez Mondragon, Phys. Rev. Lett. 44, 1323 (1980)

• J. Gea-Banacloche, Phys. Rev. A 44, 5913 (1991)• Initial state eα, resonant interaction, described

by Jaynes-Cummings model:

• Atom gets disentangled from field at time

• Field is left in a superposition of two coherent states

( ) ( )† †00

0ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 22 2JC zH a a a aσ σω σω

− +

Ω= + + + +

h hh

T =π N Ω0 N ? ( )

ε

Displace Displace it!it!

Echoes

• Echoes arise when through suitable manipulations in a system the dynamics is reversed and a more or less complete recovery of the initial state is achieved (ex: acoustical echoes arising from reflections of sound at walls)

• J. Loschmidt, Sitzungsber. Kais. Akad. Wiss. Wien Math. Naturwiss. Klasse 73 128–142 (1876)

• A. Peres, Phys. Rev. A 30, 1610–15 (1984) - Application to chaos

How to invert motion?

• Apply percussive 2π pulse to state e• Effect on state: e e• Effect on operators:

e

g

iωp

g e g eσ−= → −( )( )†

0 / 2I IH a a Hσ σ− += Ω + → −h

G. Morigi, E. Solano, B.-G. Englert, and H. Walther, Phys. Rev. A 65, 040102(R) (2002)

Atoms and photons as qubitsAtoms and photons as qubits

Two-level atomsTwo-level atoms e

g

atom e gα β= +atom e gα β= +

Cavity with zero or one photon

Cavity with zero or one photon

field 0 1δ γ= +field 0 1δ γ= +

Measurement of atom:

Measurement of atom:

Measurement of fieldMeasurement of field

ionizationionization

Information transfer and entanglement

( ) ( )( )

pulse:

1/ 2 pul

0 1 0

0 0 1

1

se:2

2 pulse 1:

e e ggc e c g g

e e g

c c

g g

π

π

π

− + ⊗ → ⊗

− ⊗ → ⊗ + ⊗

+

− ⊗ → − ⊗

( ) ( )( )

pulse:

1/ 2 pul

0 1 0

0 0 1

1

se:2

2 pulse 1:

e e ggc e c g g

e e g

c c

g g

π

π

π

− + ⊗ → ⊗

− ⊗ → ⊗ + ⊗

+

− ⊗ → − ⊗

How to calibrate interaction time: apply potential between mirrors, taking through Stark effect atoms in and out of resonance with field mode

Cavity mode: quantum databusCavity mode: quantum databus

CNOT with cavity (not quite…)MicrowaveGenerator

Atoms

Counters

Excitation

R R1 2C

0 0

Exe Srcise:

0

how tha

0

1 1 1 1

t

e g g e

e e g g

→ → −

→ − →

0 0

Exe Srcise:

0

how tha

0

1 1 1 1

t

e g g e

e e g g

→ → −

→ − →

TELEPORTATIONTELEPORTATION

Alice wants to transmit to Bob quantum state of system in her possession (example: photon polarization state).

Alice and Bob share an entangled state:

( )1

2↑↓ − ↓↑( )1

2↑↓ − ↓↑

( )α β↑ + ↓( )α β↑ + ↓ Alice has serious problems!

Bennett et al, PRL (1993)Bennett et al, PRL (1993)

Alice implements two binary measurements on her pair of spins and informs Bob, who applies appropriate transformations to his spin so as to reproduce original state

Alice implements two binary measurements on her pair of spins and informs Bob, who applies appropriate transformations to his spin so as to reproduce original state

Just two bits!

Just two bits!

TELEPORTATION IITELEPORTATION II

( ) ( )( ) ( )

1

4

α β α β

α β α β

− +

− +

Ψ⎡− + − −⎣

⎤+ + +

= ↑ ↓ ↑

−

↓

↓ ↑ ↓

Ψ

Φ ↑Φ ⎦

( ) ( )( ) ( )

1

4

α β α β

α β α β

− +

− +

Ψ⎡− + − −⎣

⎤+ + +

= ↑ ↓ ↑

−

↓

↓ ↑ ↓

Ψ

Φ ↑Φ ⎦

( ) ( )1

2α β↑ ⊗↓+ ↑↓ − ↓↑( ) ( )1

2α β↑ ⊗↓+ ↑↓ − ↓↑

Three qubits state:

( )

( ) Bell s

1

2

1a es

2

t t

±

±

⎫= ± ⎪⎪⎬⎪= ±

Φ

Ψ

↑ ↓↑ ↓

→

↑ ↓⎪

↓ ↑⎭

( )

( ) Bell s

1

2

1a es

2

t t

±

±

⎫= ± ⎪⎪⎬⎪= ±

Φ

Ψ

↑ ↓↑ ↓

→

↑ ↓⎪

↓ ↑⎭

Alice measures her Bell states and informs Bob, who applies appropriate transformations to his qubit

Alice measures her Bell states and informs Bob, who applies appropriate transformations to his qubit

Just two bits!Just two bits!

DETECTION OF BELL STATESDETECTION OF BELL STATES

( )

( ) Bell s

1

2

1a es

2

t t

±

±

⎫= ± ⎪⎪⎬⎪= ±

Φ

Ψ

↑ ↓↑ ↓

→

↑ ↓⎪

↓ ↑⎭

( )

( ) Bell s

1

2

1a es

2

t t

±

±

⎫= ± ⎪⎪⎬⎪= ±

Φ

Ψ

↑ ↓↑ ↓

→

↑ ↓⎪

↓ ↑⎭

( )( )( )( )

/ 2

/ 2

/ 2

/ 2

In Out+

+

−

−

↑↑ ↑↑ + ↓↓ ≡ Φ

↑↓ ↑↓ + ↓↑ ≡ Ψ

↓↑ ↑↑ − ↓↓ ≡ Φ

↓↓ ↑↓ − ↓↑ ≡ Ψ

( )( )( )( )

/ 2

/ 2

/ 2

/ 2

In Out+

+

−

−

↑↑ ↑↑ + ↓↓ ≡ Φ

↑↓ ↑↓ + ↓↑ ≡ Ψ

↓↑ ↑↑ − ↓↓ ≡ Φ

↓↓ ↑↓ − ↓↑ ≡ Ψ

Hadamard gate:

0 10

2

0 11

2

+→

−→

H

Reversible: Production and analysis of Bell states

Reversible: Production and analysis of Bell states

Teleportation with cavitiesTeleportation with cavities

( )0 1 21 0 /+( )0 1 21 0 /+

e gα β+

L.D., N. Zagury, et al, PRA 50, R895 (1994)

L.D., N. Zagury, et al, PRA 50, R895 (1994)

“Teleportation machine”“Teleportation machine”

e gα β+

Recent implementationRecent implementation

Zeilinger et al, Nature 430, 849 (2004) Zeilinger et al, Nature 430, 849 (2004)

ConclusionsConclusions

Cavity QED offers the possibility of exploring fundamental phenomena in quantum mechanics• Realization of quantum gates, proposals for

experiments on teleportation of quantum states• Study of the dynamics of the decoherence

process• Direct measurement of the quantum state of the

electromagnetic field, Heisenberg-limited measurements