Photon statistics in resonance fluorescence from laser deflection of an atomic beam

Volume 35, number 3

OPTICS COMMUNICATIONS

December 1980

PHOTON STATISTICS IN RESONANCE FLUORESCENCE

FROM LASER DEFLECTION OF AN ATOMIC BEAM Richard J. COOK University of California, Lawrence Livermore National Laboratory, P.O. Box 5508, Livermore, California 94550, USA Received 30 July 1980

There is a simple relationship between the photon statistics in resonance fluorescence and the statistics of the momentum transferred to an atom by a plane traveling wave. We use this relation and the theory of atomic motion in resonant radiation to derive expressions for the mean (n) and the variance ((An) 2) of the number of photons emitted in a given time by a twolevel atom in a coherent exciting field. We find, in addition to the sub-poissonian statistics [((An) 2) less than (n)] previously reported for the case of resonant excitation, that super-poissonian photon statistics [((An) 2) greater than (n)] occur in resonance fluorescence for certain off-resonance cases. It is suggested that the sub-poissonian and the super-poissonian emission statistics might be demonstrated in a simple photodeflection experiment.

There has been growing interest in recent years in optical phenomena that exhibit purely quantum-mechanical features of the radiation field. Recently Mandel studied the distribution p(n) o f the number of photons emitted in a given time b y a two-level atom in a resonant coherent exciting field [1 ] , and found that the distribution is sub-poissonian in the sense that its variance ((An)2) is less than that o f a Poisson distribution with the same mean (n). The narrowing of the photon number distribution has been attributed to the p h o t o n antibunching effect, and p h o t o n antibunching in time can only occur in a quantized radiation field [2]. In this paper we first point out that there exists a simple relation between the p h o t o n statistics in resonance fluorescence and the statistics of the m o m e n t u m transferred to an atom b y a plane traveling wave. This relation allows us to derive expressions for the mean (n) and the variance ((An) 2) o f the p h o t o n number distribution from the theory of atomic m o t i o n in a traveling wave, for arbitrary detuning of the applied field. The expressions for (n) and ((An) 2) indicate that, in addition to the sub-poissonian statistics mentioned above, super-poissonian p h o t o n statistics [((An) 2) greater than (n)] occur in resonance fluorescence for certain off-resonance cases. This result is of interest because it shows that sub-poissonian p h o t o n statistics

is not a necessary consequence of photon antibunching in time, which is always present in the radiation from a single two-level atom. In addition, the relation between p h o t o n emission statistics and m o m e n t u m transfer in a traveling wave suggests that non-poissonian p h o t o n statistics might be demonstrated in a simple atomicbeam-deflection experiment. Consider the motion of a two-level atom in a travelling wave that propagates in the x-direction. Let co be the frequency of the wave and k = co/c its propagation constant. We focus our attention on m o m e n t u m transfer in the x-direction. In the process of absorption followed by stimulated emission, no net m o m e n t u m is transferred to the atom because the m o m e n t u m hk acquired by the atom in absorption is cancelled by the recoil momentum - h k o f stimulated emission. Therefore only the process of absorption followed b y spontaneous emission need be considered. For n such processes, the atom acquires m o m e n t u m nhk through absorption and experiences n recoils in spontaneous emission. Let h k x be the xcomponent of the recoil tik i in the ith spontaneous event. Then the x-component of atomic m o m e n t u m after n spontaneous events is n

e--e o + .T,k +

i=l

,

(1)

347

Volume 35, number 3

OPTICS COMMUNICATIONS

where P0 is the initial momentum. Since the dipole moment excited by a wave propagating in the x-direction is transverse to this direction, one can readily show (using the dipole distribution for the directions of spontaneous emission and the fact that the directions of different spontaneous recoils are statistically independent) that (k~i) = 0 and (kXky.) = ~k28i/. Moreover, the statistics of the number n otSspontaneous events in a given time and the statistics of k x are independent. Finally, since we shall wish to treat an ensemble of atoms comprising an atomic beam propagating in the z-direction, we allow P0 to be a random variable, independent of n and k~i , with statistics (P0) --0 and (p2) = ((z2tp)2)0 describing the initial divergence of the beam. Using these statistics for n, k x and P0, we obtain from eq. (1) the relation <(ZkP)2) = <(AP)2) 0 + (hk) 2 [((An) 2) + ~-
(2)

for the dispersion of atomic momentum in the x-direction. The above argunient parallels Mandel's more detailed account of momentum transfer in a traveling wave [3]. Now it was shown recently by Cook [4] that the motion of a two-level atom in a monochromatic field, over a time interval that is long compared to the natural atomic lifetime rN, is described by a Fokker-Planck equation with a diffusion coefficient D = D s + D I consisting of a part D s due to recoils in spontaneous emission and a part D I resulting from induced absorption and emission processes. It follows from this theory, for the traveling wave under consideration, that the dispersion of the x-component of atomic momentum satisfies the equation <(AP) 2) = {(a2uP)2)0 + 2(D I + Ds)t,

Decemher 1980

cy between the field frequency co and the atomic resonant frequency coO. Comparing eq. (3) with eq. (2), and noting that the ((An) 2) and the {n) in (2) result from induced and spontaneous processes, respectively, we see that

5D s t /39l 2 t 0~) = = 1 9 } (hk)2 2(A 2 + 132 + ~gg.)

{6)

and ((A/t) 2) = 2Dlt

(hk) 2 =

/3922t

-- /l

2(A 2 +/32 + ½a 2

)[

g22(3/32-- A2)

t

(7)

2(A 2 + 132 + ½U12)2 j

These equations are valid in the limit t >> r N where eq. (3) is valid. A natural measure of the departure of the photon statistics from a Poisson Law is Mandel's 0-parameter Q ~ ((~-An)~2) Z (n) .... (n)

~22(3/32 -- A2) --.

(8)

2(A 2 + 132 + 1o2~2 2 'aa ]

For exact resonance (A = 0, eq. (8) agrees with Mandel's on-resonance expression for Q (eq. (12) of ref. [1 ] ), and in this case the photon statistics are sub-poissonian (((An) 2) < (n)). But for 2x2 > 3/32 , eq. (8) indicates that the photon statistics are super-poissonian (((An) 2) > (n)). The Q-parameter is plotted as a function of A/3 in fig. 1 for f2 = W/23, in which case Q reaches its lower 0.8

(3)

0.6

with

0.4

(hk)2/3~ 2 DS . . . . . . . . . . . i---" 10(A 2 + 132 + gfZ 2)

0.2

(4)

O

0.0 -0.2 0.4

and

-0.6

= DI

(/ik)2/3 a 2 {4(A2 + j32 + ½~22; 1

U12(3/32- A2) /, (5) 2(A2 +/32 + 1~22)2 J

where f2 is the on-resonance Rabi frequency of the twolevel atom in the traveling wave,/3 is half the Einstein A-coefficient, and A = co -- coO is the detuning frequen348

-0.8

-

L 3

-2

-1

i I

I

I

I

0

1

2

3

Fig. l. The variation of Q = (((An)2> (n))/(n) with A/3 for ga = -f23. This case shows the maximum departure of Q from the value Q = 0 of a Poisson process.

Volume 35, number 3 I

I

OPTICS COMMUNICATIONS ]

I

I

I

I

I

0.4 0.3 0.2

December 1980

x-direction is (p) = Iik(n ). If Pz = Moz is the longitudinal m o m e n t u m of the atom, then the mean deflection (assuming the deflection is small) is (0) = (P)/Pz = (n ) h k / M v z .

(9)

0.1

The variance of 0 follows from eq. (2)

0,0

((A0) 2) = ((~°)2}/pz2

-0.1 -0.2 -0.3

= ((a0)2)0 +

-0.4

~1

I

I

1

--8 --6 - 4

-2

0

A/G

I

I

I

I

2

4

6

8

Fig. 2. Q-parameter versus 4//3 for I2 = 10/3. bound of - 3 / 4 at A = 0. It is in this case that the subpoissonian statistics are most pronounced, and the super-poissonian statistics, for A 2 ~ 3/32, enter as a rather weak effect. The super-poissonian statistics enter more strongly as the field strength ~ increases. This is illustrated in fig. 2, where Q is plotted versus A//3 for ~2 = 10/3. Q has maximum values of ~22/4(~22 + 8/32) at dx = -+(7/32 + ~22) 1/2, and the maxima approach Q = 1/4 as ~2 --* ~ . Therefore the maximum super-poissonian excursion of Q is one third as large as the maximum sub-poissonian excursion. It was pointed out to the author by J 2 . Gordon [5] and independently by H.J. Kimble [6], that, for t >> 7"N, Q is proportional to the integral over positive time of the atomic excitation less the steady-state excitation for an atom which starts in the ground state at time zero, and therefore, roughly speaking, super-poissonian statistics result when the average atomic excitation during the natural lifetime following a spontaneous event exceeds the steady-state excitation, and subpoissonian statistics occur when the reverse is true. Mandel has emphasized that the expression for Q derived here is valid only in the limit t >> r N , and that for sufficiently short interaction time (or counting duration) Q is always negative and the photon statistics are subpoissonian for arbitrary detuning of the applied field [71. Next we consider a photodeflection exlSeriment in which an atomic beam of two-level atoms propagates in the z-direction and is transversely illuminated over the interval from z = 0 to z = L by a laser beam propagating in the x-direction. From eq. (1) we find at once that the mean m o m e n t u m delivered to an atom in the

(hk) 2 [((An) 2) + ~(n)] ....... 2 2 -- - - '

(10)

My;

where ((A0)2)0 = ((AP)2)0/P 2 describes the initial beam spreading. Upon solving eqs. (9) and (10) for (n) and ((An)Z), and using the result in (8), we obtain MVz [((A0)2) - ((A0)2)ol Q-

(0>

hk

7 -5'

(11)

which expresses the Q-parameter in terms of the quantities ((A0)2)0, ((A0)2), and (0) that are directly measured in an experiment. It must be emphasized that eq. (I 1) is valid only when vz has a well defihed value, i.e., for a monoenergetic atomic beam. When v z is a random variable, as in the case of a thermal atomic beam, a generalization of the above argument yields, M(1/°2z ) [((A0)2) -- ((A0)2)0 -- $2(0)2]

Q - p&( l/v3>

<0)

7 - - ~ , (12)

where S 2 = [(1/o 4) - ( 1 / o 2 ) 2 ] / ( I / o 2 ) 2. The quantity (0)2S2 is the contribution to the beam spreading ((A0) 2) resulting from the distribution of interaction times t = L / v z . For a thermal atomic beam, (0)2S2 is generally quite large, and the part of ((A0) 2) associated with the photon emission statistics, namely ((A0) 2) - ((A0)2)0 - $2(0} 2, which appears in (12), is easily lost in the uncertainty of the measured value of ((A0)2). To overcome this problem, S must be decreased by velocity selection before the atoms interact with the radiation. When the dispersion of oz is thus decreased to a small value [{(Aoz)2) 1/2 "~ (Oz)] , eq. (12) reads M(Vz)

Q~

hk

[((A0)2) - ((A0)2)0 -- $2(0)2] 7 (0) --~,

(13)

where now S ~ 2((AVz)2)l/2/(Vz ). An accurate measure349

Volume 35, number 3

OPTICS COMMUNICATIONS

ment of Q is then possible if ((A0)2)0, $2(0) 2 ~ ((A0) 2) ((A0)2)n _ $2(0)2, and these conditions are satisfied when ((A0~)2)1/2 <~ hk(n)l/Z/M(u z) and <(Avz)Z)l/2/(Vz ) <~ 1/4(n) 1/2 . In a typical case (k = 105 cm -1 , M = 4 × 10 -23 g,/3 = 3 × 107 s-1, and
350

December 1980

References [1] k. Mandel, Optics Lett. 4 (1979) 205. [2] R.J. Glauber, in: Quantum optics and electronics, eels. C. DeWitt, A. Blandin and C. Cohen-Tannoudji (Gordon and Breach, New York, 1964) p. 65; D. Stoler, Phys. Rev. Lett. 33 (1974) 1397; tl.J. Carmichael and D.F. Walls, J. Phys. B9 L43 (1976) 1199; tl.J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39 (1977) 691. [31 L. Mandel, J. Optics (Paris) 10 (1979) 51. [41 R.J. Cook, Phys. Rev. Lett. 44 (I980) 976. [5] J.P. Gordon (private communication). [6] tt.J. Kimble (private communication). [71 L. Mandel (private communication).