Tsuneaki Miyahara, Tokyo Metropolitan Univ. 1)Description of light in the phase space First-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, decoherence 5) Pulse compression I . Description of light  in the  (x, x' ,y, y' , ω, t) space Trick : Describe light geometrically and introduce uncertainty principle of light( Fourier limit) ω→ space is treated as same as the position – momentum space
Because of uncertainty principle  the minimum area  of the ellipse ＝λ/4 Downsizing the beam makes the beam divergence  larger aussian beam: Beam with standard deviation of distribution described by an ellipse
Asymmetric Bragg reflection
Intensity
Bending radiation
Figure 5.5: The interference patterns for E = 100 eV at BL-12A. The direction of the double slit is vertical. Undulator radiation bitrary un The interference patterns for E = 100 eV at BL-28A. The Figure 5.15: direction of the double slit is vertical. Undulator radiation  without monochromator Figure 5.21: The interference patterns for the direct beam at BL-28A. The direction of the double slit is vertical. Poor monochromaticity
When  the electron emittance is much smaller than the diffraction limit, Source point
Does it interfere ? Separate sources Interfere! The similar thing happens in ω–t space.
"Dynamical" quantum beats (more degrees of freedom) C₁:probability to come to 1(time-dependent) C¸:probability to come to 2(time-dependent) . . . Cn:probability to come to n(time-dependent) * Special case: coherent motion Cn ∞ (1/n!)¹/²exp(-inΩt)An Experiments by P.Corkum et. al. Phase relation between the two wave packet Second-order coherence =Correlation between intensities (First-order coherence= correlation between amplitudes ) R. Z. Tai et al. Phys. Rev. A 60 (1999) Two-photon correlation  is proportional to wave packet length. Width of the slit D ( γ 12 ) is changed to change Y12 A   : accidental correlation K  : duty ratio of signals T R : response time of detectors τς : wave packet length Y12 : first-order spatial coherence
Tai et. al., Rev. Sci. Instrum. 71 (2000) 1256.
Amplifier Vx = G(D) I1 I2+ Nx G(D) is of the second order spatial coherence on the Fraunhofer slit.
+10.2 V • Output Voltage of Function Generator (0.795Hz) -10.2 V Coincidence Rate (input to the lockin amplifier)
55 eV • Photon Energy (energy resolution E/ΔE ~ 10000) • Coherence in the horizontal direction was measured. •  Accumulation time for the measurement of the two-photon correlation for a slit width was about 4 hours.
•   Tungsten-wire scanner (50 µm thickness) was used. Beamsize Σ = 60.9  μm ● (Gaussian Approximation I(x) = I(0) exp(-x²/(2 Σ²)) )
Characteristic of chaotic radiation R.Z. Tai et. al., Phys. Rev. A60 3262 (1999) Y. Takayama et al. ,J. Synchrotron Rad.10 303 (2003) Two spaces with density matrix

Time evolution

If A does not observe subspace b,
Condition for coherence : For all α, β A: broken symmetry operator space a: electronic system, (creation,  annihilation operators) C α Ca space b:   bosonic system a + A k Interaction Hamiltonian: H ab = c t c βα k 〈α 〉 + c .c .
Then matrix element of A is,
Conclusion of density matrix consideration: Partial observation of the system can reduce the coherence in subspace. Examples: 1) If we observe light coming from one slit in the Young' s double slit experiment, then no interference. 2) If we do not observe the photon field, the expectation value of the dipole moment of the system is zero. Glauber’s coherent satate
represents a classical electromagnetic wave, lasers. Expectation value of the electric field:  sin ω t
Undulator 1 Unduilator 2 Up-chirped pulse Microbuncher t = to+A( ω − ω ) Dispersive system (double grating)
Without taper (non-chirped pulse) Correlation rate 1 : 0.38(±0.20) Compressed to 38%

First-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− t space. 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 of photon 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.