C 2 2 e t 29 4 ε c 2 2 e i t 29 i j k l e j t 29 e k

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c 2 [ ] 2 E t ( 29 4 = ε 0 c 2 [ ] 2 E i t ( 29 i , j , k , l E j t ( 29 E k t ( 29 E l t ( 29 = ε 0 c 2 [ ] 2 E i t ( 29 4 + E i t ( 29 2 E j t ( 29 2 i j i = ε 0 c 2 [ ] 2 E i t ( 29 4 + E i t ( 29 2 E j t ( 29 2 i j i = ε 0 c 2 [ ] 2 N atom E i t ( 29 4 + 2 N atom N atom - 1 ( 29 E i t ( 29 2 2 [ VIA-46 ] Again, for the collision- and Doppler-broadening models, this average can be expressed as
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 22 R. Victor Jones, April 27, 2000 I t ( 29 [ ] 2 = N atom [ ] 2 ε 0 c 2 [ ] 2 a 4 2 - N atom ( 29 - 1 [ ] 2245 2 I t ( 29 [ ] 2 [ VIA-47 ] and, thus, we find to a very good approximation that I t ( 29 [ ] 2 - I t ( 29 2 [ ] 2245 I [ VIA-48 ] which is consistent with the experimental results presented above. Let us now look at the two-time average of the intensity from a collection of statistically independent oscillators -- viz. I t 1 ( 29 I t 2 ( 29 = ε 0 c 2 [ ] 2 E t 1 ( 29 2 E t 2 ( 29 2 = ε 0 c 2 [ ] 2 E i t 1 ( 29 i , j , k , l E j t 2 ( 29 E k t 1 ( 29 E l t 2 ( 29 [ VIA-49 ] As a result of statistical independence, most of the cross-terms vanish so that I t 1 ( 29 I t 2 ( 29 = ε 0 c 2 [ ] 2 E i t 1 ( 29 2 E i t 2 ( 29 2 i + E i t 1 ( 29 2 E j t 2 ( 29 2 i j + E i t 1 ( 29 E i t 2 ( 29 E j t 1 ( 29 E j t 2 ( 29 i j [ VIA-50 ] Again, for the collision- and Doppler-broadening models
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 23 R. Victor Jones, April 27, 2000 I t 1 ( 29 I t 2 ( 29 = ε 0 c 2 [ ] 2 N text E i t 1 ( 29 2 E i t 2 ( 29 2 + N text N text - 1 ( 29 E i t 1 ( 29 2 2 + E i t 1 ( 29 E i t 2 ( 29 2 [ VIA-51 ] and, hence, γ 2 ( 29 τ ( 29 = 1 + γ 1 ( 29 τ ( 29 2 [ VIA-52 ] which is important result which holds for any chaotic source -- see the figure on next page. Notice that this result predicts a bunching of the intensity variations. D EGREE OF S ECOND -O RDER C OHERENCE T HE H ANBURY B ROWN - T WIST E XPERIMENT
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 24 R. Victor Jones, April 27, 2000
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 25 R. Victor Jones, April 27, 2000 T HE H ANBURY B ROWN - T WIST S PECTROMETER The Hanbury Brown - Twist Spectrometer is designed to measure 12 I 1 r r ref , t 1 ( 29 - I 1 [ ] I 2 r r ref , t 2 ( 29 - I 2 [ ] 1 4 I 1 r r ref , t 1 ( 29 I 2 r r ref , t 2 ( 29 - I 2 { } [ VIA-53 ] This correlation may then be written I 1 r r ref , t 1 ( 29 - I 1 [ ] I 2 r r ref , t 2 ( 29 - I 2 [ ] I 1 I 2 = γ 2 ( 29 τ ( 29 - 1 [ VIA-54 ] Generally, the results of these experiments are consistent with Equation [ VIA-52 ], but some of the more detailed observations require a quantum mechanical interpretation. 12 R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. , A 242 , 300 (1957).
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 26 R. Victor Jones, April 27, 2000
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