Coherence. Tsuneaki Miyahara, Japan Women s Univ.

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1 Coherence Tsuneaki Miyahara, Japan Women s 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) Outlook

2 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 t space is treated as same as the position momentum space

4 x x' x x' 1 1 l = 0 x x 1 0 x 1 = 0 ' 1 1 ' 1 f x 0

5 Changing beam size

$Diffraction limited beam Because of uncertainty principle$ makes the beam divergence larger Gaussian beam: Beam with

6 Diffraction limited beam Because of uncertainty principle the minimum area of the ellipse =λ/4 Downsizing the beam makes the beam divergence larger Gaussian beam: Beam with standard deviation of distribution described by an ellipse

$diffraction limitted$

7 Conservation of the emittance of diffraction limitted beam Loss of intensity Almost lossless

8 First- order spatial coherence Assuming Δω =0 Young s double slit experiment Light source Imax I V = I + I max min min Contrast: first-order coherence

10 Y. Takayama ( Doctor theses) Undulator radiation Bending radiation

11 Undulator radiation without monochromator Poor monochromaticity

12 Where is the source point of undulator radiation? When the electron emittance is much smaller than the diffraction limit, Phase space Source point

13 First -order coherence depends on observation Source 1 Source 2 Separate sources slit Do they interfere? Interfere! Straight motion The similar thing happens in ω-t space.

14 Description in ω t space

Temporal Young s interference "Dynamical" quantum beats (more degrees of freedom) 2 1 C 1 :probability to come to 1(time-dependent) C 2 :probability to come to 2(time-dependent)

15 Temporal Young s interference "Dynamical" quantum beats (more degrees of freedom) 2 1 C 1 :probability to come to 1(time-dependent) C 2 :probability to come to 2(time-dependent) C n :probability to come to n(time-dependent) *Special case: coherent motion C n (1/n!) 1/2 exp(-inωt)a n Experiments by P.Corkum et. al. Phase relation between the two wave packet

16 Second-order coherence (Quantum mechanics) First-order γ = (1) aa aa * 2 1 aa * * correlation between amplitudes Second-order γ = aaaa * * (2) aa * * 1 1 aa 2 2 Correlation between intensities If 1 and 2 are the same mode, γ = aaaa * * aa * Influenced by first-order aa coherence (2) 2 *

17 Measurement of Second-order coherence Not simple S I 1 I 2 ~ A + τ κ T c R γ 12 2 How can we eliminate the false correlation? R. Z. Tai et al. Phys. Rev. A 60 (1999) Two-photon correlation is proportional to wave packet length. Width of the slit D is changed to change γ 12 A :accidental correlation κ :duty ratio of signals T R :response time of detectors τ c :wave packet length γ 12 :first-order spatial coherence

18 Design of the Vacuum Chamber Side View Fraunhofer Slit Beam Splitter SR Mirror PMT Mirror Top View SR Fraunhofer Slit (width D) Mirror Mirror Beam Splitter PMT 1 PMT 2 Wire Scanner Tai et. al., Rev. Sci. Instrum. 71 (2000) 1256.

19 Brief Diagram of the Electric Circuit Electrometer I1 CFD TAC PMT1 DC RF start PMT2 Bias Tee Preamplifier (40dB) Solid State Switch stop DC RF Electrometer I2 624 nsec Delay Function Generator (0.795 Hz square wave) V Digital Lockin Amplifier Ratemeter SCA Vx = G(D) I1 I2 + Nx G(D) is of the second order spatial coherence on the Fraunhofer slit.

20 Timing of Delay-Time Modulation and Control Voltage Output Voltage of Function Generator (0.795Hz) V V Coincidence Rate (input to the lockin amplifier) 624 nsec V (0 nsec delay) TAC start signal from the same bunch Large Coincidence Rate TAC stop 624 nsec V (624 nsec delay) TAC start signal from the same bunch TAC stop Small Coincidence Rate

21 Experimental Condition Photon Energy 55 ev (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.

22 Beam size Measurement SR A Position (mm) Tungsten-wire scanner (50 μm thickness) was used. Beamsize Σ = 60.9 μm (Gaussian Approximation I(x) = I(0) exp(-x 2 /(2 Σ 2 )) )

23 An example of two-photon correlation Characteristic of chaotic radiation D (μm) R.Z. Tai et. al., Phys. Rev. A (1999) Y. Takayama et al.,j. Synchrotron Rad (2003)

24 Density matrix with two spaces subspace a, b: whole space: a b vectors in a:α, β,γ, δ vectors in b:k, l, m, p, q density matrix ρ: with αβ kl 1)expectation value of operator A k l αβ ρ = α β ρ When A does nothing on b (not observing b) A = ρ β A γ = ρ β A α l βγ ρ ααkk = 1 αk γβll ( ρ ) A = Tr A = ργβ ml β lam γ γm βl k αβ βαkk kl

25 Coherence and density matrix When the space b is not observed, ρa = Trb ρ= α β ραβmm = α β ραβmm m αβ αβ m Here we define, ρ a m ρ αβmm αβ b αβ = ρ b αβ Then we have, = ρ α β and Condition for coherence (pure state) ρ = ρ ρ α β 2 a αγ γβ αβγ ρ a = ρ 2 a ρ = ρ ρ b αβ αγ γβ γ

26 Example of decoherence When then ρ = ψ ψ 1 2 ( R e iθ L ) ψ = α + β R: right polarized L: left polarized 1 = α R R α + β L L 2 β + e i α R L β + ei θ β L R α therefore Tr = 1 2 R R L L + αβ Non-polarized light = Pure state = 0 14 Mixed state

27 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 in the photonmatter interaction, the expectation value of the dipole moment of the matter is zero. (Appendix 2)

28 Glauber s coherent satate α = exp 2 n α α 2 n= 0 n! n a α = αα E a* + a represents a classical electromagnetic wave, lasers. Expectation value of the electric field: sinωt

29 Outlook Producrion of ultrashort pulse < 1 fsec

30 Electronics of the modulation technique to detect correlation Double balanced mixer

31 Results of two- photon correlation Without taper (non-chirped pulse) With taper (up-chirped pulse) Correlation rate 1 : 0.38(±0.20) Compressed to 38%

32 Summary 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.

33 Appendix 1: Time evolution Hamiltonian: H= Ha+ Hb+ Hab Eigen energies of H a in a :E α Eigen energies of H b in b:e k α d dt = exp i Et α = exp If A does not observe subspace b, k i Ekt i A Tr( A) ( ) = ρ = ργβll + Eγ Eβ ργβll β A γ l βγ d dt A b i ( ) b = ργβ + Eγ Eβ ργβ β A γ βγ

34 Appendix 2: broken symmetry operator space a: electronic system, (creation, annihilation operators) space b: bosonic system Interaction Hamiltonian: + Hab = cca α β k α A β + c.c. Assuming correlation (entanglement) ψ = 1 ( 1 ) 2 e n + g n+ Then matrix element of A is, ρ b αβ = ραβll = l α A α = 0 and A = 0 0 ( α β ) + c α a + k c α dipole moment is zero. a k

Linac Based Photon Sources: XFELS. Coherence Properties. J. B. Hastings. Stanford Linear Accelerator Center

Linac Based Photon Sources: XFELS Coherence Properties J. B. Hastings Stanford Linear Accelerator Center Coherent Synchrotron Radiation Coherent Synchrotron Radiation coherent power N 6 10 9 incoherent