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Anticorrelations in Nondegenerate Parametric Three-Wave Interaction.

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A N N A L E N D E R PHYSIK
7.Folge. Band 38.1981. Heft 2, S. 89-168
Anticorrelations in Nondegenerate Parametric
Three-Wave Interaction
By H. PAULand W. BRUNNER
Zentralinstitut fur Optik und Spekt,roskopieder Akademie der IVissenschaften drr DDR, BerlinAdlershof
Dedicated to Prof.Dr. Gustav Richter on the Occasion of the 70th Anniversary of his Birthday
Abstract. It is shown that the anticorrelations in the photon numbers of the signal and the idler
wave, respectively, generated in parametric three-wave interaction under suitable conditions, violate
an inequality well-known in classical communication theory. Hence, it must be concluded that these
anticorrelations reflect a typical quantum mechanical feature of light. In fact, this nonclassical behaviour is responsible for the occurrence of photon antibunching in the superposition field made up
of both the signal and the idler wave, as will be pointed out in some detail.
Antikorrelationon bei der nichtentarteten parametrischen Drei-Wellen-Wt?chselwirkung
Inhaltsubersicht. Es wird gezeigt, d a l die Antikorrelationen zwischen den Wotonenzahlen
in der Signal- und der Idlerwelle, die bci parametrischer Drei-Wellen-Wechselwirkungunter geeigneten Bedingungen erzeugt werden, eine aus der klassischen Kommunikationstheorie bekannte Ungleichung verletzen. Daraus mul3 man schlielen, daB d i m Antikorrelationcn einen typisch quantenmechanischen Wesenszug des Lichtes wiedergeben. I n der Tat ist, wie im rinzelnen ausgefiihrt wird,
dieses nichtklassische Verhalten verantwortlich fur das Auftreten von Photonen-,4ntibunching in dem
durch Superposition von Signal- und Idlerwelle entstehcnden Gesamtfeld.
1. Introduction
During the last years the photon antibunching effect has attracted considerable
interest, since it has no counterpart in classical optics and, hence, provides new evidence
of the wave-particle dualisin. While experiments thus far are restricted t o the study of
resonance fluorescence froin single atoms [ l , 21, various physical nicchanisms that generate antibunching have been analyzed theoretically (see e.g. [3- lo]). Among them is
the process of degenerate parametric three-wave interaction. It has been shown 141
t h a t a weak fundamental wave will acquire antibunching properties in the course of
interaction with a strong coherent harmonic which is simultaneously irradiated into
the nonlinear crystal, provided the phases of the two waves are adjusted such that the
fundamental wave experiences maximum attenuation, diic t o amplification of the harmonic.
Similarly, photon antibunching may be detected idso in nondegenerate parametric
three-wave interaction provided the superposition field made up of both the signal and
the idler wave, is made the object of investigation[ll, 12].The occurrence of antibunching
properties in this case is intimately connected with the presence of anticorrelations
between the signal and the idler wave, which have been predicted in [13, 141.
7 Ann. Physik. 7. Folge, Bd. 38
H.PAULand W. BRUNNER
90
The main purpose of the present paper is t o point out that those anticorrelations are,
in fact, quantum mechanical in nature. We will show in detail that they violate a n in-
equality well-known in classical (statistical) coimiunication theory, i.e. that they are
stronger than can be understood in classical terms. Moreover, the connection of the anticorrelations with the antihnnching properties of the superposition field will be elucidated.
2. Basic Equations
I n contrast to references [ I Y , 141. in which all three modes were quantized, we use
the so-called parametric approximation, i.e. we assume the pump wave t o be intense
enough t o allow for a classical description in forin of a plane wave with a constant
amplitude not altered by the interaction process. In this way the short time approxiination t o which the treatment in [13, 141 had to be restricted, can be avoided. We start
from the well-known equations of motion for the photon annihilation operators q8 and
qi, corresponding t o the signal and the idler wave, respectively, which read in the interaction picture [15]
Here y, taken positive, denotes the effective coupling constant, which is proportional
to both the nonlinear susceptibility of the (nonlinear) medium and the amplitude of
the pump wave. The phase factor E is defined by the phase of the pump wave vpaccording
to the relation
The solutions t o eqs. (1)) (2) are given by [15]
+ &*WPi+
q"t) = c(t) QI
q d t ) = c ( t ) qi
+
E*s(t)
q,+
7
(4)
-
(5)
Here and in what follows we adopt the convention that operators written without any
argument refer t o the initial state, and we use the following abbreviations
c(t) = cosh (yt), s ( t ) = sinh (yt).
(6)
With the solutions (4), (5) at hand, it is a straight-forward matter to evaluate products
of operators whose expectation values are needed in the calculation of photon statistical
properties.
Using the well-known commutation relations between photon creation and annihilation operators we arrive a t the following normally ordered expressions for the relevant
terms
q,C ( t ) q,(t) = c2q:q,
+ s2[qi+qi + 11 + &scq'qi + E*scqsfqi+
7
(7)
+ s"qi'2qf + 4q:qi + 21 + &2s2c2q:q; + &*"%2q,"2qi+2
+ 2&s3c[qi'q: + 2qJ + ?&*S3CQsf[Qi+2Qj+ 2qifI
(8)
+ 2&c3sq,+q;qi + .'&*c3sqsf~q8qi' + 4c2s2q,+q8[qi+qi + 13
q,f2(t) q?(t) = c 4 q y q ;
Q8
Anticorrelations in Parametric Three-Wave Interaction
91
(t,he corresponding expressions for the idler wave are obtained by interchanging the
subscripts i and 8 on the right-hand sides of eqs. (7)) (S)),
q,fq?qfl8
+
c4p;fq8q?qi S4[q$q8 f ' 1 [q?qi f ' 1
&*282C2q:2qi+2 €s"[q,+q,
qi+qi
+
+
+
f E*'3cq$q$[q2q8 f q?qi f 3i
&*8c3q,4qi+[q,+q8
+ qcqi 11
+
+ '
2 2 2 2 2
+ 31 q8qi
q8qi
(9)
+ E'c3[dq8 f qzqi f ' 1 48qi
+ + &"."Czq;q: + ,*2s2c2q&?qifz
+ .2c2[q,+2q: + Pi'": + 2q:q,qicqi + 3q:q's + 3qifqi + 11 -
We assume the signal and the idler wave t o be initially in GLAUBERstates la8>and
lai>, respectively. This implies that the two fields are statistically independent a t the
beginning of the interaction. Hence, the expectation values of the operators on the righthand sides of eqs. (7)-(9) factorize. Moreover, due t o the well-known properties of
the GLAUBERstates [16]
qjx> =
q+ = a*@ 9
(10)
the calculation of the expectation values for the operator conibinations ( 7 ) - (9) simply
reduces t o the replacement
q: -+ a
, q8 -+ ae,qi+ -+ cw
:
,t qi --f u i .
3. Photon Statistics in the Signal and the Idler Wave
I n the wag described before, one easily finds the mean photon number in the signal
wave to be
+
+ +
(11)
. F i , ( t ) z (a,+(t) qs(t)> = c2 la,l2 82(jai12 1 )
2~ l a 8 1 /ail C O ~ @ ,
while the quantity characteristic of the photon statistical properties of the signal wave
reads
q f ( t ) >- (q:(t) q8(t))2
2 8 C I a,I ai I cos 0 + 82
Here, the angle 0 stands for
I
+
- qli
72
-2 '
= 2s2{c2 a, 12
0 = i p p - ip,
I
I m p } + s4.
(12)
(13)
cxr,
where ips and qi are the phases of the complex numbers a: and
respectively.
The corresponding relations for the idler wave are obtained from eqs. (11) and (12)
by interchanging
and mi. We should like to remind the reader that the sign of the
quantity A, (or Aj) which is proportional t o the excess coincidence counting rate t o be
measured in a HANBURY
BROWN
and TWISStype [17] experiment, indicates whether
bunching (+) or antibunching (-) occurs. It is obvious from eq. ( 1 2 ) that d,(t) is always
positive, i.e. the signal (and, similarly, the idler wave), exhibits bunching. This result
is, in fact, not surprising, since the elementary process - the "fusion"of a pump photon
from both a signal and a n idler photon - has the character of one-photon absorption
for the signal and the idler wave, respectively.
4. Cross Correlations
More interesting are the cross correlations between the two waves. Their magnitude
can be characterized by the quantity
Acrow(t)
7'
= <!I,+ ( t ) !d(4 qdt) q,(tD - <a,+( t ) q,(t)> <qif (0 q&))
(14)
92
R. PAUL
and W. BRUNNEB
Utilizing eqs. (9) and (11)we obtain the simple result
We speak of anticorrelations, when A,,,,, becomes negative. Obviously, the specification
0 = z will provide the best opportunity to achieve this. It is evident from eq. (11)that
this choice corresponds to maximum attenuation of both the signal and the idler wave.
Let us assume, for simplicity, that the initial mean photon numbers are the same in
the signal and the idler wave, lasl2 = 1 0 1 ~ 1 ~ . According t o eq. (11)and the corresponding
equation for iii(t), this remains so for all times. Moreover, the fluctuations are also equal
in both beams, A&) = A&). Then the ratio Ac,oss(t)/i,s(t)is practically independent of
Zs(0) for, say G,(O) = I o I , ~ ~2 100, in the early stage of the interaction process. Its
temporal behaviour is represented in Fig. 1for different initial values E,(O). One recogni-
Fig. 1. Evolution of cross correlations between the signal and the idler wave for different values of
the initial mean photon number z,(O) = Zi(0)
zes that AermB(t)/G8(t)is negative for not too large values of t ; this means that indeed
anticorrelations are generated between the signal and the idler wave. After a certain time
has elapsed, the anticorrelations are converted, however, into correlations. This can be
explained by the fact that at this time the contributions due t o spontaneous processes
(amplified parametric fluorescence signals in the signal and the idler wave) become dominant. Physically, it is clear that this will happen the earlier the smaller i,,(O). This feature is displayed in Fig. 1.
We should like to emphasize that the occwrrence of antic*orreliLtionsbetween two
waves is in no way a matter of surprise in cblassical wave theory. For exaniple. iint,icorrelations have recently been observed in light scattering froin rionsphericttl pnrt ic'les
Anticorrelations in Parametric Three-Wave Interaction
93
in dilute solution [HI. The experimental technique consisted in measuring the cross
correlation of signals from two spatially separated
detectors each of which received light
’
from the same small number of scatterers.
Denoting the (fluctuating)deviation of the intensity in the beam 1or 2, respectively,
from its average value by
-
-
i , = I, - I,, i, = I , - I,,
we can write the classical analog of the cross correlation (14) as
A Ccross(t)
l w
((ii(t) iz(t)>>,
where the double bracket indicates ensemble averaging.
As can easily be proved by means of Schwarz’s inequality, the quantity (17) satisfies
the following inequality well-known in classical statistical communication theory (see
e-g. t191)
2
What is really exciting from the classical point of view is the fact that the anticorrelations produced in nondegenerate parametric three-wave interaction are stronger, in
the early stage of the interaction process, than those allowed inthe framework of a classical description, namely, they violate the inequality
IAcrowI
5A,
(19)
I
Fig. 2. Comparison of cros8 and autocorrelations. Curve a represents -Ac,&#
and curve bA,/Z,,
versus t,ime. The initial values of the mean photon numbers are QO) = QO) = 10
that corresponds to (18) in case A, = A P This is seen from Fig. 2, where both the quantities -Across/%8 and A,F8 have been plotted for a typical case. Hence, the anticorrelations in question are, in fact, incompatible with classical wave theory and, therefore,
reflect a specific quantum mechanical feature of the radiation field. This nonclassical
behaviour manifests itself more directly in antibunching properties displayed by the
superposition field made up by the signal and the idler wave, as will be shown in the
following section.
H. PAULand W. BRL-KSER
94
5. Antibunching
It is well known since the pioneering work of GLAUBER [ 2 0 ] that the coincidelice
counting rate in a HANBURY
BROWN
and TWISStype [ 1 5 ] experiment is proportional
t o the quantity
(20)
Here E ( + ) ,E ( - ) denote the positive and negative frequency part, respectively, of t'he
operator for the electric field strength, at a given point r in space, and z is the response
time of the (broad-band) detectors. For simplicity, the field has been assumed t o he
linearly polarized.
Considering now the fieldcomposed of thesignal and the idler wave, which are both
idealized a s single-mode states of the radiation field, we need to take into account in
eq. ( 2 0 ) only the contributions corresponding t o those two modes. (Note that, due to
the normal ordering of the operators in eq. ( 2 0 ) , non-excited niodes do not contribute
t o the quantity R ( t ) ) This
.
means, we may put
E(*)(r, t ) = @*)(r, t ) + @*)(r, t ) ,
111)
where E!+),E i - ) ,in the considered case of running plane waves, in explicit terml;. rend
(22)
(LO,
circular frequency and k, wave vector of the signal wave, V mode volume). Siinilnr
relations hold for the idler wave.
We now make the essential assumption that the frequency difference Ato = I w g - rot
is large enough to satisfy the condition
I
d M Z 1 .
(25)
Physically, this means that the detectors average out the short-time fluctuation* clue
t o the beating of the two modes. I n fact, it is only in such circumstances that the superposition field can be expected to exhibit antibunching. since otherwise the aforementioned
fluctuations will give rise t o strong bunching.
For the sake of simplicity let us further assume that the value of w,/oi is still close
t o unity which is compatible with (23). (Even when z i s chosen, hy order of magnitude.
as small as 10V2 s, as it has been demanded from an analysis of the correlation length of
the antihunching phenomenon in nondegenerate parametric three-wave interaction [21],
both requirements are fulfilled, e.g., for o,= (1015 1013) s-1 and m i = lof5 s-l).
We thus may neglect the difference between o,and mi,which comes into play through
the square root in eqs. ( 2 2 ) , and utilizing the assumption ( 2 3 ) we find from eq. (20, tlliit
the coincidence counting rate, apart from a constant factor, is given by
+
2(t)= <41+2(t)4 : w
+ ( 4 Z 2 ( t )4 w + 2(4i!-(t)
4dt) 4,(t)>*
(24)
From eq. (24) follows the excess coincidence counting rate (with respect to randoni
coincidences) to be proportional to
d(t)
+A i +
ZAcro,,
where the above notation has been used.
(5)
Anticorrelationa in Parametric Three-Wave Iiiteract,ion
95
Specializing, as before, t o the case E,(O) = Ei(0),we have
A (t) = 2 ( 4
+
(26)
~cr0.9,)*
Due t o the fact stated in the previous section, that the cross correlation term is negative
and violates the inequality (19) in the early stage of the interaction process: A ( t ) becoines
indeed negative which is a n indication of antibunching. Hence the latter effect, in case
of nondegenerate parametric three-wave interaction, clearly has its root in the magnitude
of the anticorrelations which is incompatible with classical wave theory.
1
A convenient measure of the antibunching effect is the quantity A/(%, Zi) = -A/Z8,
2
whose explicit value follows froin eqs. (26),.(12), (14) and (11)as (cf. also [12])
+
A _
Piz,
sinh (yt) (2Z,(O) [sinh (3yt) - cosh (3yt)]+
E,(O) [cosh (2yt) - sinh (2yt)l
1
[sinh (3yt)- sinh(yt)])
2
+ sinh2 ( y t )
(27)
Making a quantitative comparison with degenerate parametric three-wave interaction
[4],one observes that eq. (27),at given t o t a l number of photons in the initial state, differs from the corresponding expression applying t o the degenerate case only in that the
noise contributions are larger by a factor of 2. Since the amplified noise is relevant, in
the early stage of the interaction process, for small initial photon nuinbers only, one can
say that the magnitude of the antibunching effect is practically the same in the
10 and not too large times.
degenerate and the nondegenerate case, for 5,
Finally, we should like t o remark that the restriction (23) can be avoided, when the
signal and the idler wave are linearly polarized in mutually orthogonal directions. I n
this case it has recently been shown [22] that the antibunching effect will be displayed
by the interference field produced by passing the two beams (whose frequencies are
assumed t o coincide) through a n appropriately orientated analyzer.
Acknowledgement. The authors are greateful to Dr. A. BANDILLA for stiinulating
discussions.
References
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and L. MAXDEL,
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96
[16] R. J. GLAUBER,Phys. Rev. 131, 2766 (1963).
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Phys. Rev. Lett. 43, 1100 (1979).
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[19] D. MIDDLETOS,An Introduction t o Statistical Communication Theory, p. 188. McGraw-Hill,
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Bei der Redaktion eingegangen am 10.September 1980.
Anschr. d. Verf.: Prof. Dr.H. PAUL
und Prof. Dr. W. BRUNNER
Zentralinstitut fur Optik und Spektroskopie
der Akademie der Wissenschaften der DDR
DDR-1199 Berlin. Rudower Chaussee G
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