Nonclassical Light

Nonclassical Light

Slide presentation accompanying the

Lecture by Roman SchnabelWinter Semester 2004/2005

Universität HannoverInstitut für Atom- und MolekülphysikZentrum für GravitationsphysikCallinstr. 3830167 HannoverGermany

Roman. [email protected]

Nonclassical LightI. Introduction

1. The Measurement Process and Heisenbergs Uncertainty Relation2. The Einstein-Podolski-Rosen–Paradox and Bells Inequality

II. Single Photons and Discrete Variables3. Experimental Tests of Bells Inequality (The Aspect-Experiment)4. Maximally and Non-Maximally Entangled States

(Bell- and Hardy-states, Schrödinger Cat States)5. More Experiments with Nonclassical Photon States (A)6. More Experiments with Nonclassical Photon States (B)

III. Beams of Light and Continuous Variables7. Squeezing from 2nd and 3rd Order Nonlinearities 8. OPO/OPA Squeezing Experiments in the CW Laser Regime (A)9. OPO/OPA Squeezing Experiments in the CW Laser Regime (B)

10. Polarization Squeezing and Spatial Mode Squeezing11. Entangled Laser Beams 12. Kerr Squeezing Experiments in the Pulsed Laser Regime

IV. Applications of Nonclassical Light (Beams and Single Photons)13. Quantum-Non-Demolition, Teleportation and Entanglement Swapping 14. Loop-Hole Free Bell Test and Quantum Cryptography15. Outlook: Nonclassical Interferometry

4

I. Introduction1. The Measurement Process andHeisenbergs Uncertainty Relation

TopicsInterpretations of Quantum TheoryMeasurement process DecoherenceGeneralized Uncertainty Principle

Literature (today)• David J. Griffiths, Introduction to Quantum Mechanics• W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003),

Decoherence, einselection, and the quantum origins of the classicalGeneral Textbook Literature (next weeks)• H.-A. Bachor and T. C. Ralph,

A Guide to experiments in Quantum Optics, 2nd edition• D. F. Walls and G. J. Milburn, Quantum Optics

5

I. 1. The Measurement Process

The Copenhagen Interpretation (proposed by Bohr)Quantum and classical world are separated from each other by a boundary, each world are governed by its own set of laws. The concept of superposition only exists in the quantum world, where we need to distinguish two types of physical processes: the ordinary ones, in which the wave function evolves smoothly under the Schrödinger Equation and measurement processes in which the wave function suddenly collapses.Problem 1: Indeterminacy reveals a lack of physical reality before a collapse Problem 2: What is the compelling reason for the quantum-classical boundary and two sets of laws?Problem 3: Ultimately also the classical world is made from “quantum stuff”. How does the classical evolve from the quantum world?

The Many-Worlds Interpretation (proposed by Everett)The whole universe is represented by a unitarily evolving state vector which is a gigantic superposition to accommodate all the alternatives. It does not suffer from the sudden collapse of the wave function because all alternatives do still exist.Problem 1: The intuitively obvious “conservation law” is violatedProblem 2: Why are the laws of classical physics so stable in this permanently splitting universe we live in?

6

I. 1. The Measurement Process

Some Answers to some of the questions: Decoherence [Zurek03]Since there is no need for a “collapse” in a universe seen from outside, the collapse and the appearance of the classical should be described in a universe made from interacting quantum systems seen from within. Now environment can destroy coherence of quantum states of a system -decoherence. The point is that decoherence does not affect all superpositions equally. There are some states (pointer states) which are robust against interaction with environment. These states define classical states. It turns out that decoherence goes hand in hand with a spreading of information about the system through the environment which is ultimately responsible for the emergence of “objective reality”. The objective reality of a state can be quantified by the redundancy with which it is recorded throughout the universe.

7

I. Introduction2. The EPR – Paradox and Bells Inequality

TopicsThe Einstein-Podolski-Rosen – ParadoxDerivation of Bell´s InequalityNon-Locality

Literature• A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935), Can quantum

mechanical description of physical reality be considered complete? • L. Hardy, Contemporary Physics, 39, 419 (1998), Spooky action at a distance in

quantum mechanics.• N. D. Mermin, Physics Today, 38 (April 1985), Is the moon there when nobody

looks?• R. Penrose, Phil. Trans. R. Soc. Lond. A 356, 1927 (1998), Quantum computation,

entanglement and state reduction.**(as an example for strange interpretations of quantum mechanics).

8

I. 2. The EPR–Paradox and Bells Inequality

9


Weak electro-magnetic signal(detected signal is about one photon)

Questions:What is the wavelength of thesignal?What is the arrival time (position) of the signal (photon)?

What type of experiment give answers?

What is the consequence of the quanta (photons)?

10

σ A2σ B

2 ≥12i

ˆ A , ˆ B [ ]⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 ˆ A = x, ˆ B = p =

h

i∂∂x

Generalized Uncertainty Principle

Regarding the quantum mechanical wave function the wavelength of Ψ is related to the momentum of the particle by the de Broglie formula:

Thus a spread in wavelength corresponds to a spread in momentum:

⇒ ˆ A , ˆ B [ ] = ih

⇒ σ xσ P ≥h

2


Position – momentum uncertainty of a particle

p = 2πh /λ

11

. Introduction2. The EPR – Paradox and Bells Inequality

..


12

March 1947:``I cannot seriously believe in [quantum theory] because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance.´´

March 1948:``That which really exists in B should … not depend on what kind of measurement is carried out in part of space A; it should also be independent of whether or not any measurement at all is carried out in space A.´´

Do you believe that the moon exists only when you look at it?

Albert Einstein 1930 Max Born ~1930


13

John S. Bell

John S. Bell in 1964:

It makes an observable difference if the particle had a precise (though unknown) position prior to the measurement or not.


14

II. Photons 3. Experimental Tests of Bells Inequality

TopicsThe Aspect-ExperimentLoopholes in experimental tests of Bells inequality

(proves of non-locality?)

Literature• A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 49, 91 (1982).• J. F. Clauser and A. Shimony, Rep. Prog. Phys 41, 1881 (1978).• J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt,

Phys. Rev. Lett. 23, 880 (1969).• A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47, 460 (1981).• A. Aspect, J. Dalibard and G. Roger, Phys. Rev. Lett. 49, 1804 (1982).• A. Zeilinger, Phys. Lett A 118, 1 (1986).• E. Santos, Phys. Rev. A 46, 3646 (1992).

15

E θ;φ( )= P++ θ;φ( )+ P−− θ;φ( )− P+− θ;φ( )− P−+ θ;φ( )= cos 2θ − 2φ( )

Snon− local = E 0°;22,5°( )+ E 45°;22,5°( )+ E 45°;67,5°( )− E 0°;67,5°( )= 2 2

Slocal ≤ 2

Ψ =12

VV + HH( )

II. 3. Experimental Tests of Bells Inequality

Source of entangled pairsθ φ

16

II. Photons 4. Maximally and

Non-Maximally Entangled StatesTopicsExperiments with maximally and non-maximally

entangled states (Bell- and Hardy-states)Defining properties of entanglementCharacterization of entangled and separable statesQuantum CakesSchrödinger Cat states

Literature• P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger

Phys. Rev. Lett. 75, 4227 (1995), New high-intensity source of polarization-entangled photon pairs.

• A. G. White, D. F. V. James, P. H. Eberhard and P. G. Kwiat, Phys. Rev. Lett. 83, 3103 (1999), Non-maximally entangled states: Production, Characterization, and Utilization.

• D. Bruß, J. Math. Phys. 43, 4237 (2002),• L. Hardy, Phys. Rev. Lett. 71, 1665 (1993),• P. G. Kwiat and L. Hardy, Am. J. Phys. 68, 33 (2000),• P. Grangier, Nature 419, 577 (2002), Single photons stick together.

17

Ψ± =12

↔b ± b↔( )

φ ± =12

bb ± ↔↔( )

Generation of Maximally Entangled States

*

II. 4. Maximally and Non-Maximally …

18

Generation of Maximally Entangled StatesII. 4. Maximally and Non-Maximally …

19

Generation of Maximally Entangled StatesII. 4. Maximally and Non-Maximally …

Violation of Bells Inequality by up to ~100 standard deviations

P. G. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 75, 4227 (1995)

20

θ φ

II. 3. Experimental Tests of Bells Inequality

Source of entangled pairs

21

Non- Maximally Entangled States and Non-Locality

A. G. White, D. F. V. James, P. H. Eberhard and P. G. Kwiat, Phys. Rev. Lett. 83, 3103 (1999)


ψ = ↔↔ + ε bb( )/ 1+ ε2 , ε = tanχ

αβ

22

Non- Maximally Entangled States and Non-LocalityII. 4. Maximally and Non-Maximally …

ε = 0.47α = 55.6° ⇒β = 72.1°

pexp(α,−α) ≈ 0

pexp(α ⊥ ,−β) ≈ 0

pexp(β,−α ⊥ ) ≈ 0

pexp(β,−β)

23

P. G. Kwiat and L. Hardy, Am. J. Phys. 68, 33 (2000)


24

Ψ cat =

12

b1b 2b 3 ...bN + ↔1↔2↔3 ...↔N( )= " live cat" + "dead cat"


Mesoscopic Schrödinger Cat States

25

Quantum Interference at the Beam-splitter


26

II. Photons 5. More Experiments with Nonclassical

Photon States (A)

TopicsThe intensity correlation functionExperiments with

- photon pairs - Bell test with space like separation- single photons on demand

Literature• H.-A. Bachor and T.C. Ralph, A Guide to Experiments in Quantum Optics,

Wiley, 2nd edition, 2004.• G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger

Phys. Rev. Lett. 81, 5039 (1998),Violation of Bell’s Inequality under strict Einstein locality conditions.

• B. Lounis and W. E. Moerner, Nature 407, 491 (2000),Single photons on demand from a single molecule at room temperature.

27

Photon Pairs - Bell Test with Space like separation

II. 5. More Experiments with nonclassical …

G. Weihs et al., Phys. Rev. Lett. 81, 5039 (1998)

28

Photon Pairs - Bell Test with Space like separation


29

Photon Statistics – A Simple Experiment


1 2 3 t(ms)

Signal

Single Photon Detector

30

Photon Statistics – A Simple Experiment


Poissonian distribution, of expectation value k=4

k

enkknP k

n

=

= −

2

!),(

σ

31

II. Photons 6. More Experiments with Nonclassical

Photon States (B)TopicsSingle photons from single molecules,

quantum dots and single-ion optical-cavity systems

Single photon detectors

Literature• B. Lounis and W. E. Moerner, Nature 407, 491 (2000).

Single photons on demand from a single molecule at room temperature. • C. Santori, D. Fattal, J. Vuckoviv, G. S. Solomon, and Y. Yamamoto, Nature 419,

594 (2002), Indistinguishable photons from a single-photon device.• M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walter, Nature 431, 1075

(2004), Continuous generation of single photons with controlled waveform in a ion-trap cavity system.

• S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, Appl. Phys. Lett. 74, 1063 (1999), Development of a high-quantum-efficiency single-photon counting system.

• J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, Appl. Phys. Lett. 74, 902 (1999), Multiphoton detection using visible light photon counter.

32

Faint laser pulses with Poissonian Distribution


2),0(),1(

!),(

0 kknPknP

enkknP

k

kn

→

−

≈>>

=

Poissonian distribution of expectation value k=0.1, n>0.

1 2

1 2

Poissonian distribution of expectation value k=0.01, n>0.

33

Single photons on demand from a single molecule


B. Lounis and W. E. Moerner, Nature 407, 491 (2000)

34

Single photons on demand from a single molecule


B. Lounis and W. E. Moerner, Nature 407, 491 (2000)

35

Indistinguishable photons from a single-photon source


(ns)

3 ps100 ps - 300 ps

10 ps

36

Indistinguishable photons from a single-photon source


37

Coherent single-photon generation in ion-trap cavity-QED


38

Coherent single-photon generation in ion-trap cavity-QED


39

High-quantum-efficiency single photon counting system

Visible light photon counter (VLPC) with cryostat system


40



41



42

Multiphoton detection using visible light photon counter


43

III. Beams of Light 7. Squeezing from 2nd and 3rd Order Nonlinearities

TopicsSqueezing in time domain Quadratures of the electro-magnetic field Phasor diagramFormal description of coherent and squeezed statesThe beam splitter and vacuum fluctuationsχ(2)-squeezing and χ(3)-Kerr-squeezing

Literature• D. F. Walls and G. J. Milburn, Quantum Optics, Springer-Verlag, Berlin 1995.• G. Breitenbach and S. Schiller, J. Mod. Opt. 44, 2207 (1997),

Homodyne tomography of classical and non-classical light.• R. E. Slusher et al., Phys. Rev. Lett. 55, 2409 (1985) Observation of squeezed

states generated by four-wave mixing in an optical cavity.

44

III. 7. Squeezing from 2nd and 3rd order ...

The electric field operator at a fixed position for a certain polarization may be written in the following form:

ˆ E (t) = E0,k(ωk) ˆ a ke−iωk t + ˆ a k

†eiωkt[ ]k

∑

ˆ a =ˆ X 1 + i ˆ X 2

2

ˆ a † =ˆ X 1 − i ˆ X 2

2

ˆ a , ˆ a †[ ]=1

With the operators of the dimensionless electric field amplitude:

ℜ( ˆ a ) ≡ ˆ X 1 /2 ˆ X 1 = ˆ X 1† = ˆ a + ˆ a †

ℑ( ˆ a ) ≡ ˆ X 2 /2 ˆ X 2 = ˆ X 2† = −i( ˆ a − ˆ a †)

ˆ X 1, ˆ X 2[ ]= 2i ⇒ ∆ ˆ X 1∆ ˆ X 2 ≥1

⇒

Quantization of the Electromagnetic Field

annihilation op.:

creation op.:

45


The electric field vector at a fixed position for a certain polarization may be written in the following form:

ˆ E (t) = E0,k(ωk) ˆ a ke−iωk t + ˆ a k

†eiωkt[ ]k

∑

Quantization of the Electromagnetic Field

⇒ ˆ E (t) = E0,k(ωk) ˆ X 1,k cos(ωkt) + ˆ X 2,k sin(ωkt)[ ]k

∑

QuadraturesQuadrature amplitudes,(Amplitude quadrature amplitude,Phase quadrature amplitude)

46

lα

mαml+α

φ

Phasor Diagram for a Classical FieldIII. 7. Squeezing from 2nd and 3rd order ...

X2/2

X1/2ℜ(α)

ℑ(α)

47

ℜ(α)


lα

mαml+α

Constructive Interference:

X2/2

X1/2

ℑ(α)

48


lα

mα

Destructive Interference:

ℜ(α)

ℑ(α)

49

Coherent state

)ˆ(aℜ

)ˆ(aℑ

α


α = ˆ a = α ˆ a α

∆ ˆ X 1 = ∆ ˆ X 2 =1

∆ ˆ X 1

(Complex amplitude)

(Minimum uncertainty)

Phasor of the quantized field with Gaussian noise distribution(“ball on the stick”)

50

Coherent state

)ˆ(aℜ

)ˆ(aℑ

α

E

t


α = ˆ a = α ˆ a α

∆ ˆ X 1 = ∆ ˆ X 2 =1

∆ ˆ X 1

(Complex amplitude)

(Minimum uncertainty)

51

Phase squeezed state


t

)ˆ(aℜ

)ˆ(aℑ

α

52

Amplitude squeezed state


t

)ˆ(aℜ

)ˆ(aℑ

α

53


Measured quantum noise; a) coherent vacuum, c) amplitude squeezingb) squeezed vacuum d) phase squeezing

[G. Breitenbach and S. Schiller, J. Mod. Opt. 44, 2207 (1997)]

Time [ms] Time [ms]N

oise

cur

rent

[a.u

.]N

oise

cur

rent

[a.u

.]

54


State Representations of the Quantized Field

ˆ H = hωkk

∑ ˆ a k† ˆ a k +

12

⎛ ⎝ ⎜

⎞ ⎠ ⎟

ˆ a † ˆ a n = ˆ n n = n n

ˆ a n = n n −1

ˆ a † n = n +1 n +1 n =ˆ a †( )n

n!0

Fock States (Number States)

Hamiltonian of the electromagnetic field

55


Coherent States

ˆ a α =α α

ˆ a † α =α* α

ˆ n = α ˆ n α =α*α = α 2

α = e− α 2 / 2 αn

n!∑ n ⇒ P(n) = n α

2=

α2n

n!e− α 2

α = ˆ D (α) 0 = exp(αˆ a † −α* ˆ a ) 0

Eigenvalue equation (complex)

Displacement operator

Poissonian distribution

56


Squeezed States

α,ε = ˆ D (α) ˆ S (ε) 0 ε = rSe2iθ S

ˆ S = exp 12

ε* ˆ a 2 −12

εˆ a †2⎛ ⎝ ⎜

⎞ ⎠ ⎟

ˆ n = α,ε ˆ n α,ε = α 2 + sinh2(rS ) ≥ α 2

ˆ S †(ε) ˆ Y 1 + i ˆ Y 2( )ˆ S (ε) = ˆ Y 1e−rS + i ˆ Y 2e

rS

⇒ ∆ ˆ Y 1 = e−rS , ∆ ˆ Y 2 = erS , ˆ Y 1 + i ˆ Y 2 = ˆ X 1 + i ˆ X 2( )e−iθ S

Degree and angle of squeezing

Squeezing operator

57

Optical Parametric Amplification

ˆ X 1

ˆ X 2

Generation of amplitude squeezed light

ϕϕ=90°=90°

ˆ X 1(t) = eχt ˆ X 1(0)ˆ X 2(t) = e−χt ˆ X 2(0)

χ < 0

III. 8. Generation and Detection of Squeezed Light (A)

58


ϕϕ=0°=0°

Generation of phase squeezed light

Generation of amplitude squeezed light

III. 8. Generation and Detection of Squeezed Light (A)


χ > 0

ˆ X 1

ˆ X 2

59

Kerr squeezingIII. 7. Squeezing from 2nd and 3rd order ...

Kerr effect:Intensity dependent phase shift

X1

X2

α2 >> 1

ϕ << 1

60

III. Beams of Light 8. OPO/OPA Squeezing Experiments in the

CW Laser Regime (A)TopicsSqueezing in frequency domain Phasor diagram and modulation sidebandsQuadratures in the 2-Photon-FormalismOptical parametric oscillation and amplification (OPO,OPA)Squeezed light from a degenerate OPO / OPAEquation of motion for the nonlinear cavity

Literature• D. F. Walls and G. J. Milburn, Quantum Optics, Springer-Verlag, Berlin 1995.• C. M. Caves, Phys. Rev. Lett. 31, 3068 (1985),

New Formalism for two-photon quantum optics.• M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984)

Squeezing of intracavity and traveling-wave light fields produced in parametric amplification.

• K. Schneider, M. Lang, J. Mlynek, and S. Schiller, Opt. Exp. 2, 59 (1998), Generation of strongly squeezed light at 1064 nm.

61

Phasors in frequency space (amplitude squeezing)

III. 8. OPA/OPO Squeezing Experiments… (A)

Ω

X1

E0

ω+Ω0

ω−Ω0

X2

62

Quadrature Noise OperatorsNoise operator diagrams for a coherent and phase squeezed beam

( )Ω1Xδ

( )Ω2Xδ

1-1

-1

1

δ ˆ X 1 Ω( )= δˆ a Ω( )+ δˆ a † −Ω( )δ ˆ X 2 Ω( )= −i δˆ a Ω( )−δˆ a † −Ω( )( )

ˆ a Ω( )= α + δˆ a Ω( )δˆ a Ω'( ),δˆ a † Ω( )[ ]= δ Ω− Ω'( )

Linearized annihilation and creation operators


ˆ X 1(Ω) = ˆ X 1†(Ω) = ˆ a (Ω) + ˆ a †(−Ω)

ˆ X 2(Ω) = ˆ X 2†(Ω) = −i( ˆ a (Ω) − ˆ a †(−Ω))

ˆ X 1(Ω), ˆ X 2(Ω)[ ]= 2i

⇒ ∆ ˆ X 1(Ω)∆ ˆ X 2(Ω) ≥1Heisenberg Uncertainty Relation

63

Squeezed Light from an OPASqueezed beam

Squeezed beam

Coherent Coherent beambeam


64

A. Franzen


OPA layout

MgO LiNO – hemilithic crystal7.5mm x 5mm x 2.5 mm

Radius of curvature: 10 mm HR=99.97% at 1064 nm

Flat surfaceAR at 1064 nm and 532 nm

Losses0.1 %/cm at 1064 nm4 %/cm at 532 nm

65

A. Franzen


OPA layout




Losses0.1 %/cm at 1064 nm4 %/cm at 532 nm

66

OPA layout




Output coupler: R=94% at 1064 nm

Finesse ~ 100Waist ~ 32 µmFSR ~ 9 GHzγ = 90 MHz

A. Franzen


67


H. Vahlbruch

68


69

Equation of Motion for OPA/OPO (SHG) CavityIII. 8. OPA/OPO Squeezing Experiments… (A)

22211221

221121

ˆ2ˆ2ˆ2ˆ2

ˆ)(ˆ

ˆ2ˆ2ˆ2ˆˆˆ)(ˆ

inlossb

incouplb

incouplbresres

lossb

couplb

couplbres

invac

lossa

incoupla

incouplaresresres

lossa

coupla

couplares

BBBabb

AAAbaaa

δγγγεγγγ

δγγγεγγγ

++++++−=

+++++++−= +

&

&

Equations of motion

)2(χ1ˆ in

frontA

2ˆ inbackA

2ˆ outbackA

1ˆ outfrontA

ˆ a res+ , ˆ a res

ˆ b res+ , ˆ b res

lossγ

1couplγ2couplγ

invacAout

lossA

70


OPA Squeezing

H. Rehbein

Front seeded via R=94%

Back seeded via R=99.97%

71


OPA Squeezing

H. Rehbein





72

III. Beams of Light 9. OPO/OPA Squeezing Experiments in the

CW Laser Regime (B)

TopicsSqueezed light from a degenerate OPO / OPA (revisited)Homodyne detectionQuantum state tomographyFrequency dependent squeezing

Literature• L.-An Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987),

Squeezed states of light from an optical parametric oscillator.• G. Breitenbach and S. Schiller, J. Mod. Opt. 44, 2207 (1997),

Homodyne tomography of classical and non-classical light.• S. Chelkowski, H. Vahlbruch, B. Hage, A. Franzen, N. Lastzka, K. Danzmann, and

R. Schnabel, Phys. Rev A (2005), accepted,Experimental characterization of frequency-dependent squeezed light.

73

III. 9. OPA/OPO Squeezing Experiments… (B)

Electric Polarization

Taylor expansion of electric polarization of media in electric and magnetic fields:

( )...)3()2()3()2(0 +++++= jlkijkljkijklkjijklkjijkjiji EBBhEBhEEEEEEP χχχε

SHG,OPA THG, Kerr effect Faraday effect Cotton-Mouton effect

zyxlkji ,,,,, =

Considering second order terms more explicitly:

( ) ( ) ( ) ( )∑ −=−jk

kkjjkjiijkii EEP ωωωωωχεω ,,)2(0

)2(

0=++− kji ωωω

74


ϕϕ=0°=0°



χ > 0

ˆ X 1

ˆ X 2

75

(Squeezed) signal beam

Intense local oscillator

Phase shift θ

Electrical current

V(ˆ i −) ≅ αLO2 ⋅ δ ˆ X 1

2 cos2 θ + δ ˆ X 22 sin2 θ( )= αLO

2 ⋅ δ ˆ X 2 θ( )

The Balanced Homodyne Detection

Interfering

50/50 beam splitter


76

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

1 11 −1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a Aeiθ

ˆ a B

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ n 1 = ˆ a 1† ˆ a 1 =

12

ˆ a A† e−iθ + ˆ a B

†( ) ˆ a Aeiθ + ˆ a B( ), ˆ n 2 = ˆ a 2† ˆ a 2 =

12

ˆ a A† e−iθ − ˆ a B

†( ) ˆ a Aeiθ − ˆ a B( )ˆ n − = ˆ n 1 − ˆ n 2 = ˆ a A

† ˆ a Be−iθ + ˆ a B† ˆ a Aeiθ

ˆ a =α + δˆ a , α =α*, δˆ a = 0, αA2 >>αB

2 , δˆ a 2 , ˆ X (θ) = ˆ a e−iθ + ˆ a †eiθ

∆2 ˆ n − ≡ ˆ n −2 − ˆ n −

2 ≈ αA2 ∆2 ˆ X B (θ)

The Balanced Homodyne DetectionIII. 9. OPA/OPO Squeezing Experiments… (B)

ˆ a A

ˆ a B

ˆ a 1

ˆ a 2

θ

77

OPA, Noise Power of Coherent and Squeezed Light

δ ˆ X 12

δ ˆ X 22

δ ˆ X coh2

θ=0 θ=π/2

11.2 dB

3.1 dB


78

OPA, Noise Power of Coherent and Squeezed Light


79

Tomography / Noise Histogram


80

Tomography / Wigner-Function Plots


81

Frequency Dependent Squeezing


82

Frequency Dependent Squeezing


83

III. Beams of Light10. Polarization and Spatial Mode Squeezing

TopicsQuantum noise in a polarimeterStokes operators and Quantum Poincaré SphereGeneration of polarization squeezed lightQuantum noise in a pointing measurementThe TEM00 flipped modeGeneration of spatial mode squeezed light

Literature• B. A. Robson, The Theory of Polarization Phenomena (Clarendon, Oxford, 1974)• R. Schnabel, W. P. Bowen, N. Treps, H.-A. Bachor, T. C. Ralph, and P. K. Lam,

Stokes-operator-squeezed continuous-variable polarization states,Phys. Rev. A 67, 012316 (2003), Phys. Rev. Lett. 88, 093601 (2002).

• N. Treps, U. Andersen, B. Buchler, P.K. Lam, A. Maitre, H.-A. Bachor, and C. Fabre, Surpassing the Standard Quantum Limit for Optical Imaging Using Nonclasical Multimode Light, Phys. Rev. Lett. 88, 203601 (2002), Science 301, 940 (2003).

84

Spectrum analyser

Coherent state, H

Polarization rotatingsample

λ/245°

III. 10. Polarization and Spatial Mode Squeezing

Quantum Noise in a Polarimeter

Vacuum, H,V

Vacuum, H,V

By injecting an (amplitude, θ=0) squeezed vacuum at the first beamsplitter the quantum noise in the above measurement is reduced.What observable is squeezed then?

θ

85

Continuous Variable Polarization StatesMeasurement of Stokes Parameters


86


Stokes Parameters in Classical Optics

= α H2

+ αV2,

= α H2

− αV2,

= α HαV eiθ + αVα H e− iθ ,

= −iα HαV eiθ + iαVα H e− iθ .

S0 = αH

2+ αV

2

S1 = αH

2− αV

2

S2 = α+45º

2− α−45º

2

S3 = αRCirc

2− αLCirc

2

Decomposition into two orthogonal fields in H/V-basiswith real amplitudes α of relative phase θ:

87


Stokes Parameters / Poincaré Sphere

Radius ofClassical Poincaré sphere:

For completelypolarized light:

Degree of polarization:

[G.G.Stokes, Trans.Camb.Phil., 9, 399 (1852)]0 ≤

S12 + S2

2 + S32

S0

≤1

Total light intensity:

S12 + S2

2 + S32 = S0

S12 + S2

2 + S32

S0

88

Stokes Operators in Quantum OpticsIII. 10. Polarization Squeezed Light

ˆ S 0 = ˆ a H ˆ a H + ˆ a V

ˆ a V , ˆ S 2 = ˆ a H ˆ a V eiθ + ˆ a V

ˆ a He−iθ

ˆ S 1 = ˆ a H ˆ a H − ˆ a V

ˆ a V , ˆ S 3 = −iˆ a H ˆ a V eiθ + iˆ a V

ˆ a He− iθ ,

ˆ a k, ˆ a l†[ ]= δkl , k, l ∈ H,V , σ A

2σ B2 ≥

12i

ˆ A , ˆ B [ ]⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

ˆ S 1, ˆ S 2[ ]= 2i ˆ S 3 , ˆ S 2, ˆ S 3[ ]= 2i ˆ S 1 , ˆ S 3, ˆ S 1[ ]= 2i ˆ S 2 .

V1V2 ≥ ˆ S 32, V2V3 ≥ ˆ S 1

2, V3V1 ≥ ˆ S 2

2.

Commutation relations of Stokes operators

Uncertainty relations of Stokes operators


89


Stokes Operator VariancesIII. 10. Polarization Squeezed Light

ˆ a H ,V = α H ,V +12

δ ˆ X H ,V+ + iδ ˆ X H ,V

−( )

V0 = V1

V1 = αH2 δ ˆ X H

+( )2+ αV

2 δ ˆ X V+( )2

V2 θ = 0( )= αV2 δ ˆ X H

+( )2+ αH

2 δ ˆ X V+( )2

V3 θ = 0( )= αV2 δ ˆ X H

−( )2+ αH

2 δ ˆ X V−( )2

For a coherent beam:

= αH2 + αV

2 = ˆ n ,

= αH2 + αV

2 = ˆ n ,

= αH2 + αV

2 = ˆ n ,

= αH2 + αV

2 = ˆ n .

90


Quantum Poincaré Sphere

91


Phase difference locked to θ=0.

92

Results from a Coherent Polarization StateIII. 10. Polarization and Spatial Mode Squeezing

93


Polarization Squeezing from two Amplitude Squeezed Beams

94


Polarization Squeezing from two Phase Squeezed Beams

III. 10. Polarization Squeezed Light

95


Uncertainty Volumes of Polarization States

‘Cigar’ State ‘Pancake’ State


(Measured at 8.5 MHz)

96


Polarization Squeezing from two Quadrature Squeezed Beams


97


“Quantum Laser Pointer”

What is the relevant spatial mode that provides the measurement quantum noise?

Spectrum analyser

Coherent state

Laser Pointing

Vacuum, spatial mode

θ

?


98


“Quantum Laser Pointer”

Spectrum analyser

Coherent state

Laser Pointing

Vacuum, spatial mode

θ

?


99


1-Dimensional Spatial Mode Squeezing

N. Treps et al. (2002)


100





101




π 0

0 π


102





103

Coherent State

Coherent State

Coherent State

Laser Beam Input Electrical Noise Power Output

Shot Noise

Shot Noise

Shot Noise-

-

Polarization Squeezed

State

Spatially Squeezed State

Amplitude Squeezed State

Shot NoiseBelow

Shot NoiseBelow

Shot NoiseBelow

[R.E.Slusher et al.,1985]

[N.Treps et al.,2002]

[W.Bowen et al.,2002]

Squeezed States - SummaryIII. 10. Polarization and Spatial Mode SqueezingIII. 10. Polarization and Spatial Mode Squeezing

104

III. Beams of Light11. Entangled Laser Beams

TopicsWhat are the characteristics of an experiment with entangled laser

beams?Entanglement of continuous variablesFirst realization of the EPR Paradox for continuous variablesThe EPR criterion for entanglementThe inseparability criterion for entanglement

Literature• M. D. Reid, Phys. Rev. A 40, 913 (1989),

Demonstration of the EPR-Paradox using nondegenerate parametric amplification.• Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663

(1992), Realization of the EPR Paradox for continuous variables.• W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, Phys. Rev. Lett. 90, 043601

(2003), Phys. Rev. A 69 (2004),Experimental Investigation of Criteria for continuous variable entanglement.

105

Experimental SetupIII. 11. Entangled Laser Beams

106



EPR-entangled pair of laser beams [Einstein, Podolski, Rosen 1935]

Measurement of Amplitude

Measurement of Phase

50/50Beamsplitter

Entanglement

Inferred Quadratures

III. 11. Entangled Laser Beams

‐

+

107

Entanglement Criteria for Continuous Var.III. 11. Entangled Laser Beams

108

CV Entanglement CriteriaIII. 11. Entangled Laser Beams

Conditional variance criterion

Inseparability criterion

[M.D.Reid and P.D.Drummond, Phys.Rev.Lett. 60, 2731 (1988)]

[L-M. Duan, G.Giedke, J.I.Cirac and P.Zoller, Phys.Rev.Lett. 84, 2722 (2000)]

Ε = ming + /− δ ˆ X x

+ − g+δ ˆ X y+( )2

δ ˆ X x− − g−δ ˆ X y

−( )2⎧ ⎨ ⎩

⎫ ⎬ ⎭

≡ ∆2 ˆ X x|y+ ⋅ ∆2 ˆ X x|y

− <1

Ι = δ ˆ X x+ +δ ˆ X y

+( )2δ ˆ X x

− −δ ˆ X y−( )2

≡ ∆2 ˆ X x+y+ ⋅ ∆2 ˆ X x−y

− <1

(g=1, +/- chosen to provide minima)

109

Entangled Quadrature Noise VariancesIII. 11. Entangled Laser Beams

Degree of Inseparability ΙAverage of quadrature noise variances of individual entangled beams

110

Entanglement Criteria for Continuous Var.III. 11. Entangled Laser Beams

Bowen et al. (2003)

Ι=0.44

Ε=0.58

111

III. Beams of Light12. Kerr Squeezing Experiments in the Pulsed

Laser Regime

TopicsOptical Solitons in FibersKerr squeezingEntangled laser pulsesQuadrature noise measurements on laser pulses

Literature• S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs,

Phys. Rev. Lett. 81, 2446-2449 (1998), Photon-number squeezed solitons from an asymmetrical fiber optic Sagnacinterferometer.

• Ch. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs,Phys. Rev. Lett. 86, 4267-4270 (2001),Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fibre.

112

III. 12. Kerr Squeezing Experiments

Electric Polarization

Taylor expansion of electric polarization of media in electric and magnetic fields:

Pi =ε0 χij(1)E j + χijk

(2)E j Ek + χijkl(3)E j EkEl + hijk

(2)BkE j + hijkl(3)BkBl E j + ...( )

SHG,OPA THG, Kerr effect Faraday effect Cotton-Mouton effect

zyxlkji ,,,,, =

Considering third order terms more explicitly:

Pi(3) −ωi( )=ε0d χijkl

(3) −ωi,ω j,ωk,ωl( )E j ω j( )Ek ωk( )jkl∑ El ωl( )

−ωi +ω j +ωk +ωl = 0d=degeneracy

113

Kerr squeezingIII. 7. Squeezing from 2nd and 3rd order ...

Kerr effect:Intensity dependent phase shift;quadrature squeezing is produced for small angles

X1

X2

α2 >> 1

ϕ << 1

114


Squeezed Laser Pulses

115


Squeezed Laser Pulses

116


Entangled Laser Pulses

( )%0.1%5.51 ±(96% visibility)

117


Entangled Laser Pulses

Shot-noise

Shot-noise

p-pol

SQZ4dB

s-pol

SQZ4dB

118

CV Entanglement CriteriaIII. 11. Entangled Laser Beams

Conditional variance criterion

Inseparability criterion

[M.D.Reid and P.D.Drummond, Phys.Rev.Lett. 60, 2731 (1988)]

[L-M. Duan, G.Giedke, J.I.Cirac and P.Zoller, Phys.Rev.Lett. 84, 2722 (2000)]

Ε = ming + /− δ ˆ X x

+ − g+δ ˆ X y+( )2

δ ˆ X x− − g−δ ˆ X y

−( )2⎧ ⎨ ⎩

⎫ ⎬ ⎭

≡ ∆2 ˆ X x|y+ ⋅ ∆2 ˆ X x|y

− <1

Ι = δ ˆ X x+ +δ ˆ X y

+( )2δ ˆ X x

− −δ ˆ X y−( )2

≡ ∆2 ˆ X x+y+ ⋅ ∆2 ˆ X x−y

− <1

(g=1, +/- chosen to provide minima)

119


Kerr Squeezing in CW Nonlinear Cavity

21 ˆ2ˆ2ˆˆˆˆ)(ˆ inlossincouplresresresres

losscouplres AAaaaiaa γγµγγ ++++−= +&

)3(χ1ˆ inA

2ˆ invacuumA

2ˆ outlossA

1ˆ outA

resres aa ˆ,ˆ+ lossγcouplγ

Equation of motion

120

IV. Applications of Nonclassical Light 13. QND, Teleportation and Entanglement Swapping

TopicsExperiments with single photons and laser beamsCharacterization of Quantum Non-demolition (QND) measurements:

- Signal transfer and conditional varianceCharacterization of quantum teleportation and entanglement swapping:

- Fidelity- Signal transfer and conditional variance

Literature• P. Grangier. J.A. Levenson, and J.-P. Poizat, Nature 396, 537 (1998),

Quantum non-demolition measurements in optics.• V.B. Braginski and F.Y. Khalili, Rev. Mod. Phys. 68, 1 (1996),

Quantum non-demolition measurements – the route from toys to tools.• D. Bouwmeester et al., Nature 390, 575 (1997),

Experimental quantum teleportation.• W. P. Bowen et al., Phys. Rev. A 67, 032302 (2003),

Experimental investigation of continuous-variable quantum teleportation.

121

Quantum Non-demolition MeasurementsIV. 13. QND, Teleportation ...

T = Tm + Ts =ℜm

out

ℜsin +

ℜsout

ℜsin 0 < T < 2

Signal-Transfer:

Vs|m = Vsout −

δXsoutδXm

out 2

Vmout 0 < V < ∞

Conditional Variance:

T>1 and 0< V < 1QND:

122

Quantum Non-demolition MeasurementsIV. 13. QND, Teleportation ...

Noise-less Amplification

Quantum State Preparation

123

Quantum TeleportationIV. 13. QND, Teleportation ...

• Can quantum information be copied perfectly? No! (No-Cloning-Theorem)

• Can we extract all the information from a system? No!

• Can we send all the information of a system fromone place to another (via classical channels)? Yes, we can!

[Bennett et al. 1993]

124

Quantum Teleportation - CharacterizationIV. 13. QND, Teleportation ...

inoutF ψψ=Fidelity:

,10 ≤≤ FArbitrary amount of copies

Finite amount of copies

No equally good copy can exist

Classical information transfer:

Quantum regime I:

Quantum regime II: ,13/2

,3/25.0

,5.00

≤<

≤<

≤≤

F

F

F

125

Quantum TeleportationIV. 13. QND, Teleportation ...

Bouwmeester et al. (1997)

126

Generation of maximally entangled statesIV. 13. QND, Teleportation ...

Ψ± =12

↔b ± b↔( )

φ ± =12

bb ± ↔↔( )*

127

Quantum Teleportation (Single Photons)III. 13. QND, Teleportation ...


128

IV. 13. QND, Teleportation ...

Teleportation of a single photon state was demonstrated by similar results from measurements on 90° and circular polarized signal states


Quantum Teleportation (Single Photons)

129


Entanglement Swapping (Single Photons)

Entangling photons that never interacted

130


Quantum Teleportation (Laser Beams)

131


Quantum Teleportation (Laser Beams)

k± =α in

± 2⋅ 1− g±( )2

Vin± + Vout

± ,

ˆ X in±

Ω( )= 2α in

±

Ω( )+ δ ˆ X in±

Ω( ) ,

g± = αout± /α in

± .

Vin± = δ ˆ X in

± Ω( )2 ,Variance:

Electronic Gain:

F = e−(k+ +k− )⋅4 ⋅Vin

+Vin−

Vin+ + Vout

+( )⋅ Vin− + Vout

−( )⇒ Fidelity:

Sideband Modulation:

132

SqueezingIII. 11. Entangled Laser Beams

Coherent State Squeezed State

δ ˆ X + Ω( )2 δ ˆ X − Ω( )2 ≥1

δ ˆ X − Ω( )

δ ˆ X + Ω( )

δ ˆ X − Ω( )

δ ˆ X + Ω( )

Heisenberg Uncertainty Relation

133



EPR-entangled pair of laser beams [Einstein, Podolski, Rosen 1935]

Measurement of Amplitude

Measurement of Phase

50/50Beamsplitter

Entanglement

Inferred Quadratures

III. 11. Entangled Laser Beams

‐

+

134


!?

Unknown quantum information

EPR I EPR II

Teleported quantum information

Measurement of phase

Measurement of amplitude

Displacement using g=1

?

50/50

Classical channels

Teleportation of Quadratures

135


Unknown quantum information

“Teleported” classical information

Measurement of phase

Measurement of amplitude

Displacement using g=1

Classical channels

Vacuum Vacuum

!??

50/50

Teleportation of Quadratures

136

W.P. Bowen et al., Phys. Rev. A (2003)

Classical limitNo-cloning limit

Classical limit

No-cloning limit

Perfect teleportation

Teleportation of QuadraturesIV. 13. QND, Teleportation ...

137

Fidelity ResultsIV. 13. QND, Teleportation ...

Nonclassical Light

Documents

Transcript of Nonclassical Light