lecture note 1 quantum optics and quantum...
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Lecture Note 1
Quantum Optics and Quantum Information
Jian-Wei Pan
Physikalisches Institut der Universität Heidelberg Philosophenweg 12, 69120 Heidelberg, Germany
Outlines
• A quick review of quantum mechanics• Quantum superposition and noncloning theorem• Quantum Zeno effect• Quantum entanglement • Quantum nonlocality
Basic Principles of Quantum Mechanics
• The state of a quantum mechanical system is completely specified by wave function , which has an important property that is the probability that the particle lies in the volume element at time , satisfying the normalized condition
),( txψ3* ),(),( dxtxtx ψψ
3dx t
1),(),( 3* =∫∞
∞−
dxtxtx ψψ
• To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics (indicates superposition principle ).
Basic Principles of Quantum Mechanics
• In any measurement of the observable associated with operator , the only values that will ever be observed are the eigenvalues , which satisfy the eigenvalue equation
ia
i i iA aψ ψ=)
3* ˆˆ dxAA ψψ∫∞
∞−
>=<
If a system is in a state described by a normalized wave function , then the average value of the observable corresponding to
ψ
A)
Basic Principles of Quantum Mechanics
• The wave function or state function of a system evolves in time according to the time-dependent Schrödinger equation
where is the Hamiltonian.
ψψ Ht
i =∂∂
η
H
• The total wave function must be antisymmetric (or symmetric ) with respect to the interchange of all coordinates of one fermion (boson) with those of another.
Quantum Superposition Principle
.||,||,
,||
222
211
21
212
φφ
φφφφ
==
+==
pp
p
Quantum Superposition Principle
which slit?
| ⟩| ⟩ or
| ⟩| ⟩ +
Classical Physics:“bit”
Quantum Physics:“qubit”
Quantum foundations: Bell’s inequality, quantum nonlocality…Quantum information processing: quantum communication, quantum computation, quantum simulation, and high precision measurement etc …
Quantum Superposition Principle
Qubits: Polarization of Single Photons
One bit of information per photon(encoded in polarization) >>=
>>="1|"|"0|"|
VH
Qubit:⟩+⟩=⟩Φ VH ||| βα
Non-cloning theorem:An unknown quantum state can not be copied precisely!
1|||| 22 =+ βα
2||α
2|| β
Non-Cloning Theorem
,1110,0000
→→
W. Wootters & W. Zurek, Nature 299, 802 (1982) .
)10)(10(11000)10( ++≠+→+
According to linear superposition principle
Zeno Paradox
Origin of Zeno effect
Can the rabbit overtake the turtle?
Quantum Zeno Effect
Quantum Zeno Effect
)1)1(0)1((210 −++=Φ ϕϕ ii eeHH
Quantum Zeno Effect
Interaction-free measurement !
Quantum Zeno Effect
Considering neutron spin evolving in magnetic field, the probability to find it still in spin up state after time T is
where is the Larmor frequency .
)2
(cos2 Tp ω=
ω
up down
Quantum Zeno Effect
T=0 T=π/2 T=π
T=0 T=π/2 T=π
G (cake is good)=G0×
If we cut the bad part of the cake at time T=π/2 , then at T=π we have
G=1/4×G0
)1(,2cos1
=+ ωT
Experiment
P. Kwiat et. al., Phys. Rev. Lett. 74 4763 (1995)
N
NP )]
2([cos2 π
=
In the limit of large N
)(4
1 22
−+−= NON
P π
Bell states – maximally entangled states:( )
( )212112
212112
||||2
1|
||||2
1|
⟩⟩±⟩⟩=⟩Ψ
⟩⟩±⟩⟩=⟩Φ
±
±
HVVH
VVHH
Polarization Entangled Photon Pair
)|||(|2
1
)|||(|2
1|
2121
212112
⟩′⟩′−⟩′⟩′=
⟩⟩−⟩⟩=⟩Ψ−
HVVH
HVVH
)|(|2
1|
)|(|2
1|
⟩⟩−=⟩′
⟩⟩+=⟩′
VHV
VHH
Singlet:
where
45-degree polarization
Polarization Entangled Photons
GHZ states – three-photon maximally entangled states:
)(2
1
)(2
1
)(2
1
)(2
1
321321123
321321123
321321123
321321123
HHVVVH
VHVHVH
HVVVHH
VVVHHH
±=Θ
±=Ξ
±=Ψ
±=Φ
±
±
±
±
Manipulation of Entanglement
0000→0101→1110→1011→
Manipulation of Entanglement
000)10()1000()1100(10000)10(00
→+=+→+→+=+→
Manipulation of Entanglement
00000)10(0)1000(0)1100(110000111000110000
0)1100(0)1000(00)10(000
→+=+→+=+→+→+=
+→+=+→
Einstein-Poldosky-Rosen Elements of Reality
. . .
Completeness
Locality
non-commuting operators, i.e., momentum P and position X of a particle
12 1 2 1 2
1 2 1 2
1 2 1 2
1| (| | | | )2
1 (| | | | )2
1 (| | | | )2
H V V H
H V V H
R L L R
−Ψ ⟩ = ⟩ ⟩ − ⟩ ⟩
′ ′ ′ ′= ⟩ ⟩ − ⟩ ⟩
= ⟩ ⟩ − ⟩ ⟩
zVH σ,10
,01
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
x
VHV
VHHσ,
)(2
1
)(2
1
−=′
+=′
y
iVHL
iVHRσ,
)(2
1
)(2
1
−=
+=
Polarization Entangled Photon Pairs shared by Alice and Bob
Bohm converted the original thought experiment into something closer to being experimentally testable.
Bohm’s Argument
Bohm’s Argument
According to the EPR argument, there exist threeelements of reality corresponding to and !yx σσ ,
zσHowever, quantum mechanically, and are three noncommuting operators !
yx σσ ,
zσ
Einstein-Podolsky-Rosen Elements of Reality
In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality.
Quantum-mechanical description of physical reality cannot be considered complete!
Bell’s Inequality and Violation of Local Realism
The limitation of the EPR !
Both a local realistic (LR) picture and quantum mechanics (QM) can explain the perfect correlations observed.
Non-testable!
[J. S. Bell, Physics 1, 195 (1964)]
Bell’s inequality states that certain statistical correlations predicted by QM for measurements on two-particle ensembles cannot be understood within a realistic picture based on local properties of each individual particle.
Sakurai's Bell Inequlaity
Correlation measurements between Alice’s and Bob’s detection events for different choices for the bases ( indicted by a and b for the orientation of the PBS).
)(2
12121 HVVH −=Ψ−
Pick three arbitrary directions a, b, and c
Sakurai's Bell Inequlaity
43),( PPbaP +=++
42),( PPcaP +=++
73),( PPbcP +=++
724343 PPPPPP +++≤+
It is easy to see
Sakurai's Bell Inequlaity
The quantum-mechanical prediction is
For example , the inequality would require
),(),(),( +++++≤++ bcPcaPbaP
2)2
(sin21),( babaP −
=++
。。。 ,, 04590 === bca
1464.02500.0
5.22sin215.22sin
2145sin
21 222
≤
+≤ 。。。
CHSH Inequality
Considering the imperfections in experiments, generalized Bell equality — CHSH equality,
Quantum Mechanical prediction: 22=MAXS
Local Reality prediction: 2≤MAXS
J. Clauser et al., Phys. Rev. Lett. 23, 880 (1969)
A. Aspect et al., Phys. Rev. Lett. 49, 1804 (1982)
Experimental Test of Bell Inequality
Drawbacks: 1. locality loop hole2. detection loop hole
020.0101.0exp ±=S violates a generalized inequality 0≤Sby 5 standard deviations.
G. Weihs et al., Phys. Rev. Lett. 81, 5039 (1998)
02.073.2exp ±=S
Experimental Test of Bell Inequality
violates CHSH inequality 2≤S
by 30 standard deviations.
Drawback: detection loop hole