NONUNITARY TRANSFORMATIONS IN QUANTUM OPTICS D.N. KLYSHKO Department ofPhysics, Moscow University, Moscow 119 899, USSR Received 1 March 1989; accepted forpublication 22 March 1989 Communicated by V.M. Agranovich
A general method ofquantum description of linear optical devices — beam splitters, lenses, etc. — with dissipation is discussed. The method is grounded by means of a kinetic equation. As an example the transformation ofthe field’s statistics by an absorbing half-transparent mirror is considered. It is shown that a heated mirror radiates correlated signals into pairs of mirror-conjugated modes.
1. Over the last two years there seems to be a renewed interest in the quantum theory of spectral fluters andbeam splitters [1—5],caused mainly by their possible use in experiments involving squeezed states (see ref. [2]), photon antibunching [1,3,4], quantum nondemolition measurement [61, the EPR-paradox [7] and other essentially nonclassical optical effects. The various methods, used in refs. [1—51, have a common feature: they are applicable only to dissipationless devices, which can be described by unitary transformations. But in real devices there is always some dissipation small in a dielectric mirror or a Nicol prism and deliberately large in a coloured glass or a spacefrequency filter. Dispersion of the refractive index, used for spectral filtration, is necessarily accompanied by absorption. The transformation of the field’s statistics by a photon counter or a heterodyne detector is also nonunitary as the detector’s quanturn yield i~is less than 1. This is also the case if in an ideal beam splitter or other dissipationless device some modes are not taken into account. A quantum amplifier has negative absorption (~>1) and hence should also be described by a nonunitary transformation (an active interferometer with amplification was considered in ref. [81).
dissipation: time- and space-frequency filters, lenses, interferometers, beam splitters, heterodyne detectors, etc. All these “filters” change the input field’s statistics and the task is to define this change through a set of phenomenological parameters of the filters. A convenient method is based on the division of the radiation field, surrounding the filter, into input and output parts and on the notion of a “transformation matrix” (TM) Dk/, connecting the Fourier components of these parts [91: b~=
~ Dk
1a7,
bk=
~ D~1a1,
(1)
—
—
2. In the present communication a general formalism is considered, which is applicable to the description of various optical devices with or without 334
or in vector notation b + = D a b D* a. Here a a (b b) are the creation and annihilation operators of the input (output) field, indices k, 1 include the wave vectors k, I and the polarisation indices Uk, V1. As a,~,bk are supposed to be time independent, so eq. (1) is applicable only to free fields (there is no such limitation in ref. [1]). The linearity of the fluter implies that the TM is diagonal in 1k I = (Ok/C, i.e. Dkl ö (w~ w,) (here continuous mode distribution is implicated), and thus the summation or integration in (1) is actually performed only over various directions of! and over v,. In case of a beam splitter the number of interacting modes n is 2, in case of a lense n=~. The TM of a dissipationless filter is unitary: D- D~= I, so that the “absorption matrix” A = I D• D~ is zero and the transformation (1) ~,
conserves the commutation relations. Note that it is possible to define D and A through the commutators: D= [b~, a], A=I+ [b~, b}. (2) —
It is clear that Dk/ plays the role of propagator from channel k to / (the free space propagators exp (ikr) should be added if necessary). In addition the TM determines the filter’s own thermal radiation via a
29 May 1989
fore averaging (over the input state) and in certain cases the chaotic radiation, connected with the absorption or amplification, should be taken into account. As will be shown below, both these demands are satisfied, if the following characteristic function is used:
x0~1(p)=x,~(b~u)x~(p)
(6)
,
Kirchhoff-type relation [9]. The relation (1) may be presented as the similaritytransformation in the Hubert space: bk= U ‘akU. If n=2, then U has the following structure [3]:
where
U= exp(iaa j~a1
i,,, (p) is the characteristic function of the in-field
+ ifia~a2)
Xexp(iyata2 +iôa~a1).
(3)
Comparison with (1) yields D’~’1=eia cos( W), D’~2=iyeia sinc( ~ D~1= iôe’~sinc ( hJ~), D~2= e’~ cos ( v’), (4) 2= yd and sinc(x) = x —‘ sin (x). The inverse where ~Pare relations
x~=e,~p(_p.~~.p*), A~=N(T~A,
N~T~ =dkIN~(wk)
=ôkI[exp(hwk/kT)—
Dkl=Dlk,
l]~
(the second argument ~ is being omitted for brevity), xTCU) determines the filter’s own radiation (T is the temperature of the absorbing elements, supposed to be homogeneous). This radiation “jams” the input information and nonclassical features (in addition to the destructive influence of the mode mixing and stochastic absorption itself see eq. (15) below) and plays a principal role in active filters with T<0 [8]. —
2ia = in (DD 2ô=
—
11 ID22 )*, 2ifl=in (DD22/D11 )*, C cos~(B), (5)
C where y=
2, D=Det(D),
B*= (D11D22/D)”
C”= (D12 D22/D21 D11 ) 1/2~ If n>2, then U can be expressed as a combination of two-mode operators (3). 4=Ifollows UU~=I(note the differentFrom senseD~D of the symbol “+“ in these equations), so that U has the meaning of the usual evolution operator [2,3]. Now in (4), (5) d11 = d22, d12 = d21 = (1 —d11) 1/2
Quantum optical systems usually are loaded by one or more detectors. Multiplying D by the diagonal matrix of the detector’s efficiency, DJ~”= ôkI?1 k’ , we get the total TM D 10~=D~°~-D, which takes account of the effective absorption in the detectors. Now the function Xtot Cu) =x 0~1( D~°~ -p) defines the statistics of the really observed electric signals. If the in-field is chaotic, i.e. if ~1~(p)=exp(_p.N.p*), Nk,= ~fl, (7) then according to (6) it retains its character (with modified N, which is defined in eq. (10) below). The Fourier transformation of (6) gives the output Glauber—Sudarshan representation P 0~ (ex) as the convolution of P1,, (~)with the Gaussian function: POUl(fl)=[1t”Det(AT)J~
(here dk,= lDk,I, 9~k/=arg(Dk,)),so that a, are real:
fi and
~“
a=—~’11, P~22, y=ô”’=iWexp(i~ 11—i~,12), O~P~it/2
x
J
P~(~) exp[
—
(D*.~_fl) 2ak.
.Afl(D.z*....$*)]
ft d
(8)
(essentially the same result is given in ref. [3fl
The derivatives of (6) at a = 0 determine the normally ordered output moments, for example,
3. IfA,&O then b~,b should be normally ordered be-