A quantum statistical proof of Von Laue's theorem on entropy of partially coherent radiation

Volume 68A, number 3,4 PHYSICS LETTERS 16 October 1978 A QUANTUM STATISTICAL PROOF OF VON LAUE’S THEOREM ON ENTROPY OF PARTiALLY COHERENT RADIATION ~ M. WIEGAND Theoretische Festk~rperphysik,Technische Hochschule Darmstadt, D-6100 Darmstadt, F.R. Germany Received 7 August 1978 Using information theory and Glauber’s definition of (first-order) coherence the statistkal operator of a partially co- herent radiation field is derived. Von Laue’s formula for the entropy is proved. It seems that M. von Laue’s early studies [1 31 — KT (2 on entropy of a partially coherent radiation field have 11 P) r P I1 P~ not been considered so far in the framework of mod- under the constraints erri quantum statistics. Von Laue’s treatment of the Trp = I (3a) entropy of partially coherent light beams was based + — upon the idea that interference phenomena should Repa~a~,~ (3b) be included in thermodynamic considerations. He Eq. (3b) states that we know the average number of first proved the reversibility of reflection and refrac- photons in each mode and the first-order coherence tion at the surface between non-absorbing media, properties ~‘ . With Lagrange parameters ~ for eq. Using this fact and Planck’s expression for the en- (3b) the generalized canonical statistical operator R tropy of thermal radiation he derived a formula for corresponding to the maximum missing information the entropy of two (or three) partially coherent light takes the form (M~~~iw~j32~) beams. In the following we want to show that by means I + expi— M ,aa, of information theory and Glauber’s concept of co- — ~ A X — * herence [4] it is possible to construct a generalized — + ~ Msj..~— M~. (4) canonical statistical operator for the radiation field, Trex~~~ M~a~a~) leading to an expression for the entropy that, in the special case of only two excited modes, corresponds withvon Laue’s formula. For the entropy we obtain (M matrix M~x) We consider the free radiation field in a volume of r M ~1 quantization. The hamiltonian becomes S = K Tr R mR = KTr~I in (1 eM) I 1 J(5) ~= (I) Tr~denotes the trace in the space of all modes. The For the construction of the statistical operator we For the description of interference experiments we only use the principle of maximum missing information need the correlation function of the electric field [5] , G~’~(r,t;r’, t’) = Volume 68A, number 3,4 PHYSICS LETTERS 16 October 1978 logarithm of the partition function is given by It is possible to show that as/o~< 0 for 0 ~f ( 1: as we would expect the entropy decreases with increas-lnZ — Tr~in M (6) ing coherence. 1 C We ieinaik that von Laue’s statement that the en- From tropy of partially coherent light beams is not addi- A — ‘~‘~M lnZ (7\ tive has to be interpreted in the sense of a generalizedA’A / AX’1 “ ‘ thermodynamics [6] , according to which, in the case we obtain the connection between the Lagrange-pa- of an indivisible observation level, there exists only ranieters MAA~and the expectation values AAA in ma- an entropy for the whole system (all modes). The en- trix-form: tropy of a subsystem (one mode) cannot be defined. A = heM l~ We conclude with some remarks about the time- Jependence. With the help of the expectation values The entropy then reads (a~aA,~(t)we construct for each time t a statistical S TIAL(A), (9a) operator [5] L(x)K[(x+ l)ln(x+1) xlnx]. (9b) With real numbers ‘AA’ and phases ~AA’we define the exp( ~ MAA(t)a~aA~) complex degrees of coherence v7~exp (iPAA) by R(t) — ________________ . (15) (a~a > TrexP( ~MAA(t)a~aA~) A.A (10) AX \/(aa )(a~,a,) A A A A The entropy S(t) is then given by From Schwarz’ inequality it follows S(t) = TrAL(A(t)). (16) 0~j ,~l. (11)AX However, for the free field we have With eq. (10) we are able to express the entropy by - A ,(t)-A ,exp{i(w w )t}, (17) means of the degrees of coherence and the number AX AX A A of ~hotons nA — (a~aA>in the modes X. i.e. the matrixA(t) is related to A = A(t — 0) by a Let us consider some examples: unitary transformation. Therefore the entropy of the (~)An incoherent radiation field is characterized free field is time.independent. (However, note that R by ‘AX’ — 6AA’• Then the entropy is the sum of the is stationary only for a monochromatic radiation entropies of each mode, field.) S.~ - ~ L(n~). (12) I would like to thank Professor Dr. E. Fick for A suggesting tIre problem and a critical reading of the (~3)For a first-order coherent field we havef~— 1, manuscript, and Professor Dr. G. Sauermann for help- PXX’ — ~. ~X’ and the entropy becomes ful discussions. S~ 01~=L(~~A)