quantum information: from foundations to experiments second lecture luiz davidovich instituto de...
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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