Coherence
Tsuneaki Miyahara, Japan Women’s Univ.
１) Description of light in the phase spaceFirst-order spatial coherence: Experiments
First –order temporal coherence
Description of light in the 6-dimensional phase space
2) Characteristics of undulator radiation
3) Second-order coherence and photon statistics
Experiments: two-photon correlation
4) Coherence and density matrix
Observation of subspace, decoherence5) Outlook

Ⅰ．Description of light in the (x, x’,y, y’, ω, t) space

Trick：
Describe lightgeometrically andintroduce uncertainty
principle of light( Fourier limit)

ω−t space is treated assame as the position –momentum space⎛⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎝

x1 ⎞⎟ ⎛⎜ 1 l ⎞⎟ ⎛⎜ x0 ⎞⎟⎟⎟=⎜⎟⎜x'1⎟⎟⎟ ⎜⎜⎝ 0 1⎟⎟⎠ ⎜⎜⎜ x '0 ⎟⎟⎟⎠⎝⎠

x1 ⎞⎟ ⎛⎜ 1 0⎞⎟ ⎛⎜ x0 ⎞⎟⎟⎟⎜⎟=⎜x'1⎟⎟⎟ ⎜⎜⎜ ∓ 1 f 1 ⎟⎟⎟⎜⎜⎜ x'0 ⎟⎟⎟
⎠

⎝

⎠⎝

⎠

Changing beamsize

Diffraction limited beam
Because of uncertainty principle the minimum area ofthe ellipse ＝λ/4
Downsizing the beam
makes the beam
divergence larger

Gaussian beam: Beam withstandard deviation ofdistribution described by anellipseConservation of the emittance of diffraction limitted beam

Loss of intensity

Almost losslessFirst- order spatial coherence
Assuming Δω = 0
Young’s double slitexperimentLight source

Imax − Imin
V=Imax + Imin
Contrast: first-order coherenceY. Takayama ( Doctor theses)

Bending radiation

Undulator radiation

Undulator radiation without monochromator

Poor monochromaticity

Where is the source point of undulator radiation?
When the electron emittance is much smaller than thediffraction limit,

Phase
spaceSource point

First -order coherence depends on observation
Source 1slit

Do they interfere？
Source 2
Separate sources

Interfere!
Straight motion

The similar thing happens in ω-t space.

Description in ω−t spaceTemporal “Young’s” interference"Dynamical" quantum beats (more degrees of freedom)

2
1

C1:probability to come to 1(time-dependent)
C2:probability to come to 2(time-dependent)
・・・・・・・・・・
Cn:probability to come to n(time-dependent)＊Special case: coherent motionCn∝(1/n!)1/2exp(-inΩt)AnExperiments by P.Corkum et. al.

Phase relation between the two wave packet

Second-order coherence (Quantum mechanics)

γ (1) =First-orderγ (2) =

Second-order

a2*a1
a1*a1 a2*a2

a2*a1*a1a2
a1*a1 a2*a2

correlation
betweenamplitudes

Correlation
betweenintensities

If 1 and 2 are the same mode,

γ (2) =

a*aa*a − a*aa*a2

Influenced by first-ordercoherenceMeasurement of Second-order coherence
Not simple⎛ ~τc2 ⎞S ∝ I 1 I 2 ⎜⎜ A + κ
γ 12 ⎟⎟TR⎝⎠

How can we eliminate thefalse correlation?
R. Z. Tai et al. Phys. Rev. A 60 (1999)

Two-photon correlation is proportional to wave packet length.
A ：accidental correlation
Width of the slit D is changed
to change γ12κ ：duty ratio of signals

TR ：response time of detectors

τc ：wave packet length
γ12 ：first-order spatial coherenceDesign of the Vacuum Chamber
Side View

Beam Splitter
Fraunhofer SlitSR

MirrorPMT

MirrorTop View
Fraunhofer Slit (width D)
MirrorSR

MirrorBeam Splitter
PMT 1
PMT 2

Wire ScannerTai et. al., Rev. Sci. Instrum. 71 (2000) 1256.

Brief Diagram of the Electric Circuit
I1
PMT1

TACCFDElectrometer
DCstart

RFBias Tee
PMT2Preamplifier(40dB)

Solid State SwitchstopRFDCElectrometer

624 nsec DelayI2Function Generator(0.795 Hz square wave)

VDigital Lockin
AmplifierVx = G(D) I 1 I2 + Nx

Ratemeter

SCA

G(D) is of the second order spatial coherence
on the Fraunhofer slit.

Timing of Delay-Time Modulation
and Control Voltage+10.2 V

Output Voltage of Function Generator (0.795Hz)-10.2 V

Coincidence Rate (input to the lockin amplifier)624 nsec+10.2 VTAC start
signal from the same bunch

Large Coincidence Rate

(0 nsec delay)
TAC stop

624 nsec-10.2 V(624 nsec delay)

TAC start
signal from the same bunch
TAC stop

Small Coincidence Rate

Experimental Condition• Photon Energy
55 eV(energy resolution E/ΔE ∼ 10000)• Coherence in the horizontal direction wasmeasured.
• Accumulation time for the measurement of
the two-photon correlation for a slit width
was about 4 hours.

Beam size Measurement

SRA

-0.2

-0.1

00.1

0.2

Position (m m)• Tungsten-wire scanner (50 μm thickness) was used.
• Beamsize Σ = 60.9 μm(Gaussian Approximation I(x) = I(0) exp(-x2/(2 Σ2)) )0.3

An example of two-photon correlation
10.5

Characteristicof chaotic
radiation

0-0.5-1-1.5-202040

6080

100

D (μm)
R.Z. Tai et. al., Phys. Rev. A60 3262 (1999)
Y. Takayama et al. ,J. Synchrotron Rad.10 303 (2003)

Density matrix with two spaces
subspace a, b:

whole space： a ⊗ bvectors in a：α, β,γ , δ・・・
vectors in b：k, l, m, p, q・・・・
density matrix ρ:ρ = ∑∑ α k l β ραβ klαβwithkl

ραα = 1
∑αkkk１）expectation value of operator A
A = Tr ( ρ A ) = ∑∑ ργβ ml β l A m γγ m βlWhen A does nothing on b (not observing b)
A = ∑∑ ργβ ll β A γ = ∑∑ ρ βα kk β A αlβγ

kαβCoherence and density matrix
When the space b is not observed,⎛⎞
ρ a = Trb ρ = ∑∑ α β ραβ mm = ∑ α β ⎜ ∑ ραβ mm ⎟m αβαβ⎝ m⎠Here we define,bρ=ρ∑ αβ mm αβmThen we have,bρ a = ∑ ραβα βαβandρ 2a = ∑ ραγ ργβ α βαβγ

Condition for coherence (pure state)ρ a = ρ 2abραβ= ∑ ραγ ργβγ

Example of decoherenceψ

When= 12(αR + eiθ βL)

R: right polarized
L: left polarizedthenρ =ψ ψ= 1 ⎛⎜ α R R α + β L L β + e−iθ α R L β + eiθ β L R α ⎞⎟2⎝⎠
therefore
Non-polarized lightTr ρ = 1 ⎛⎜ R R + L L ⎞⎟αβ2⎝⎠

⎛1 2⎜
⎜1 2⎝

1 2⎞⎟
1 2 ⎟⎠2⎛1 2= ⎜⎜⎝1 2

Pure state1 2⎞⎟
1 2 ⎟⎠

⎛1 2⎜
⎜ 0
⎝0⎞
⎟
1 2 ⎟⎠2⎛1 4= ⎜⎜
⎝ 0

Mixed state0⎞⎟
1 4 ⎟⎠

Conclusion of density matrix consideration:
Partial observation of the system can reduce thecoherence in subspace.
Examples:1) If we observe light coming from one slit in theYoung’s double slit experiment, then no interference.
2) If we do not observe the photon field in the photonmatter interaction, the expectation value of the dipole
moment of the matter is zero. (Appendix 2)

Glauber’s coherent satate⎛ α 2 ⎞ ∞ αnnα = exp ⎜ −⎟∑⎜ 2 ⎟ n =0 n !
⎝⎠a α =α αE ∝ a* + a
represents a classical electromagnetic wave, lasers.
Expectation value of the electric field: sinωtOutlook
Producrion of ultrashort pulse < 1 fsec

Electronics of the modulation technique to detect correlation
Double balanced mixer

Results of two- photon correlation
Without taper (non-chirped pulse)

0.94

Correlationrate

1 : 0.38(±0.20)

With taper (up-chirped pulse)

0.35

Compressed to 38%

SummaryFirst-order coherence
1) First-order coherence depends on how we observe the light.
2) First-order coherence can be improved with sacrifice of intensity.
The loss of intensity is smaller when the source has smaller
emittance.
3) First-order spatial coherence is easily observed in Young’s
experiments.
4) The idea of first-order spatial coherence can be applied to the ω− tspace.
5) Observation of a part of the system could reduce the coherence.,
corresponding to tracing out the density matrix in a sub-space.

Second-order coherence
1) Measurement of two-photon correlation gives information ofphoton statistic and the wave packet length of a photon.
2) Using a tapered undulator and a double grating system, the wave
packet length can be compressed.

Appendix 1: Time evolution
H = H a + H b + H ab
Hamiltonian：
Eigen energies of Ha in a ：EαEigen energies of Hb in b：Ek

⎛ i
⎝

⎞
⎠

α = exp ⎜ − Eα t ⎟⎛ i⎞k = exp ⎜ − Ek t ⎟⎝⎠di⎧⎫
A = Tr ( ρ A ) = ∑∑ ⎨ ργβ ll + ( Eγ − Eβ ) ργβ ll ⎬ β A γdt⎭l βγ ⎩If A does not observe subspace b,d⎧ b ib ⎫A = ∑ ⎨ ργβ + ( Eγ − Eβ ) ργβ ⎬ β A γdt⎭βγ ⎩Appendix 2: broken symmetry operator

space a: electronic system,(creation, annihilation operators)
space b: bosonic system

cα+ak+cαakInteraction Hamiltonian:

H ab = cα+ cβ ak α A β + c.c.
Assuming correlation (entanglement)1ψ =( e n + g n +1 )2Then matrix element of A is,bραβ= ∑ ραβ ll = 0(α ≠ β )lα A α = 0 andA =0“dipole moment” is zero.