Spectroscopic measurements of single atom emission in an optical resonator

SpectrochimicaActa,Vol.45A,No. 1, pp. 75-79, 1989. Printed in Great Britain.

0584~539/89 $3.00+0.00 © 1989PergamonPresspie

Spectroscopic measurements of single atom emission in an optical resonator D. J. HEINZEN, J. J. CHILDS and M. S. FELD George Harrison Spectroscopy Laboratory and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

(Received 17 August 1988; in final form 29 August 1988; accepted 31 August 1988) Abstract--Experiments are described in which atoms are placed in a concentric optical resonator and probed by cw dye laser radiation. It is found that, depending on the tuning of the resonator, the radiative linewidth of the atomic transition may either increase or decrease and the observed resonance frequency may shift, relative to their respective free space values. A simple classical interpretation of these effects is given. We also present results of a semi-classical calculation which show that under appropriate conditions, stimulated emission may play a significant role in such experiments.

1. INTRODUCTION

classical point of view why the spontaneous emission may be modified, in spite of the large resonator volume. We also present the results of experiments which study the linewidth and center frequency of a resonance line of barium atoms in a concentric resonator. Both changes in linewidth, due to enhanced and suppressed spontaneous emission, and in center frequency, due to radiative level shifts, are observed. Finally, using the results of semi-classical calculations, we discuss the conditions under which stimulated emission may play a role in such experiments.

F o r an atom in free space, the linewidth of an atomic transition is completely determined by the spontaneous radiative decay of the excited state, assuming that other sources of broadening and the decay of the lower state may be neglected. This "natural" linewidth is often assumed to be an unavoidable, unchangeable property of the atom. However, as first pointed out by PURCELL and others [1], the spontaneous emission rate of an atom can be modified by placing it in a resonator. In particular, PURCELL showed that an atom interacting with a single, resonant cavity mode of volume V and quality factor Q has a spontaneous emission rate given by Fre s = rfree [3Q/(47r2)] [23/V],

2. CLASSICALPICTURE OF RADIATION BY AN ATOM IN A RESONATOR Consider a radiating atom placed near the center of a concentric optical resonator, consisting of two identical mirrors of reflectivity R and subtending a half-angle O, placed a distance L = 2a apart, where a is their radius of curvature (Fig. 1). Classically, this atom may be thought of as a dipole radiator of dipole moment d = ex, where e is the electron charge and x the displacement of the electron from the binding center of force. F o r simplicity, we assume that the dipole is polarized perpendicular to the resonator axis. This oscillating classical dipole gives rise to a radiated field according to the usual dipole formula. In free space, this field escapes to infinity and has no

(1)

where Ff,ee is its spontaneous emission rate in free space. F o r low order cavity modes V ~ ).3, so the rate is enhanced by a factor of order Q: Similarly, a cavity whose lowest order mode is tuned far above resonance inhibits the radiation by a factor of order Q. Recently, there has been renewed interest in observing such effects experimentally, and both enhanced I-2, 5-8] and suppressed [3-8"1 spontaneous emission have been observed. In most previous investigations of cavity-modified spontaneous emission, attention has focused on the regime in which the dimension L of the resonator is comparab!e to the emission wavelength 2. This is because the mode density for most resonator configurations becomes identical to that of free space in the limit L/2 >> 1. However, enhanced and inhibited emission by an atom in optical confocal 1-61 and concentric [7"1 resonators, both with L >> 2, have recently been observed. In such resonators a large number of modes are degenerate and the net effect of this mode degeneracy, when the resonator is tuned or detuned from the atomic resonance, can be substantial. In this paper, we discuss the radiation of an atom in a concentric optical resonator, and describe from a

M2

L I-

MI

L =20

Fig. 1 A t o m in a concentric resonator.

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D.J. HEINZEN et al.

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further influence on the atom. However, in the concentric resonator the field is reflected back from the cavity mirrors. By simple ray-tracing it may be seen that upon reflection from either mirror, an image dipole is produced directly opposite the optical axis of the resonator and that after one complete round trip the radiation is strongly refocused at the dipole. After each successive round trip the field is again refocused upon the dipole (Fig. 1). Taking into account all of these reflected waves, it is not difficult to show that the total field experienced by the dipole is El) (t) = -- [2o9~/(3c3)1 d ( t ) - i[8o9o/(3c3)]ff(AX¢)

x ER"~l[t - 2nL/c)],

(2)

where Ell (t) is the component of the field parallel to d, o90 is the resonant frequency of the dipole oscillator in free space, and ff(A~2c) = 1 - (3/4)cos O - (1/4)cos30,

(3)

is the fraction of free space spontaneous emission ordinarily emitted into the solid angle AE¢ subtended by both cavity mirrors [91. Note that for small solid anglef(AY.~) ~- 3AY.~/(81t). The first term on the righthand side of Eqn (2) gives the "radiation reaction field" experienced by a dipole in free space, whereas the second term gives the contribution due to all of the waves reflected back onto the dipole that arise from dipole oscillations at discrete past times separated by the cavity round-trip time. Since the atomic electron experiences a field Ell, its equation of motion is simply

il'(t) + o92 d(t) = [e2 /m]Eii .

(4)

An approximate solution is obtained by substituting -o92d(t) for tt"(t) and substituting d = do e*t, with ~ "-" - io90 [10]. This gives = -- ( F b / 2 ) -- i(o9 0 + Aogb),

(5)

with

essential features are that the decay rate of the dipole is modulated by the lineshape of the cavity, and the frequency shift displays a repeating dispersive-shaped behavior as the cavity is tuned, with the resonance shifting to the red when the nearest cavity resonance is tuned to the blue, and vice versa. Both the change in decay rate, and the size of the frequency shift, scale with the fraction of solid angle, f(AXc), intercepted by the cavity mirrors. Note that, for the case (1 - R ) ,~ 1, the maximum radiation rate F~,"h and minimum rate F~,"h are given by F~ "h = r'free{i + [2f(AEc)/(1 - R ) ] } ,

(7)

F~n h = F f r e e [ l - f ( A E c )

(8)

1.

These results show that in order to significantly suppress the total radiation rate, the cavity must intercept a large fraction of the total solid angle. However, the enhancement in the total radiation rate may be large even for a cavity of moderate solid angle, provided that its finesse is sufficiently large. F r o m Eqns (7) and (8) it is apparent that the cavity may also be quite large, with a length L >> 2, since the size of the cavity does not appear explicitly. This may appear somewhat surprising in view of Eqn (1), since for V ~ L 3, 23/V ,~ 1 and Fres '~ Ff .... despite a high Q. However, we should note that Eqn (1) applies only to a single mode cavity, whereas an optical resonator is inherently multi-mode. In our concentric cavity the atom actually excites a superposition of many modes whose frequencies are degenerate. The combined effect of these modes may, in fact, be viewed as being equivalent to an "effective single mode" of volume Veff=SIE(~')I2dV/E2(F=), with Fa the position of the atom and E the electric field. For our concentric resonator it may easily be shown that V~fr = 322L/[4nj~AZc)] . This is considerably less than L 3, and takes into account the strong focusing of the field back onto the atom. Thus, substituting Vcff in Eqn (1) and using Q = 2nL/[2(1 - R)], we find that rr,s = 2f(AE~)rfr~J(1 -- R).

F b = Ffree[l + {[%/1 +

- 1}f(A~)],

F/(1 + Fsin 2 kL)] (6a)

(9)

We see from Eqn (7) that Eqn (9) exactly equals the increased decay rate of the atom when the concentric resonator is tuned to resonance.

and Ao9b = Ff~,=[f(AE~)/4] [Fsin2kL/(1 + Fsin 2 KL)], (6b) with ['free = 2e2o9~/(3mc3) the usual free space decay rate of the dipole and F = 4 R / ( 1 - R ) 2. Note that ( x / F + 1)/(1 + Fsin 2 kL) is the Airy lineshape function of an optical resonator, with k = 2n/2, and that F is related to the cavity finesse, ~, b y / = ~x/~/2. Equations (6) give the classical predictions for the decay rate and frequency shift of a classical dipole oscillator in a concentric cavity. They are in complete agreement with results of a quantized field calculation [71 with Ffr,= taken to be with the quantum mechanical free space spontaneous emission decay rate. The

3. EXPERIMENTAL STUDY OF SINGLE ATOM SPONTANEOUS EMISSION BY ATOMIC BARIUM IN A CONCENTRIC RESONATOR

In order to test these results, experiments have been done using barium in a concentric resonator. An atomic beam of barium atoms is first collimated by an aperture to a diameter of 1 mm and intercepted outside the cavity at right angles by a beam from a cw dye laser. The atomic beam is then recollimated by a second aperture to a diameter of 25 #m, and intercepted inside the cavity at right angles by a second beam from the same laser. The laser is tuned near the 1So-IP 1 transition of 13SBa at 2 = 553.5 nm. The IP t

Spectroscopic measurements of single atom emission state has a free space radiative linewidth of Ff,e,/2rc = 19 MHz. Two regions of excited atoms are thus created, the first "Region 1" outside the cavity and the second "Region 2" inside the cavity. The laser beam in Region 2 is focused to a diameter of 20/am and has a power of 0.02/tW, and the excited atoms are carefully positioned at the center of the cavity. The laser field is polarized perpendicular to the cavity axis, and since the ~3aBa isotope is resolved from other isotopic components, only a single two-level Am = 0 transition is excited. The concentric cavity mirrors have a radius of curvature of 2.50 cm, which gives L = 5.00 cm and a free spectral range (FSR) of AvFsR=C/(2L) = 3000 MHz. The mirror reflectivity is R = 0.93, so that F = 759 (,¢= 43.2), and the theoretical cavity linewidth is AVFsR/,¢=69 MHz. Their clear diameter is 1.88 cm, so that 0 = 2 2 °, and f(AEc)=0.106. The davity is carefully aligned to concentricity with three piezoelectric transducers, and is tuned by means of small linear displacements (~< 1 FSR) of one cavity mirror. Fluorescence emitted by the atoms in Region 1 and out the sides of the cavity in Region 2 is collected by optical fiber bundles and detected by photomultiplier tubes. In the experiment the cavity tuning is held fixed, and the fluorescence intensity from both detectors is simultaneously recorded as a function of laser frequency. A typical scan is shown in Fig. 2. For each

(a)

c D

Cb)

Laser tuning ( 2 0 MHz/Oiv)

Fig. 2 Fluorescence intensity vs laser tuning. (a) Signal from atoms outside the cavity (region 1); (b) signal from atoms inside the cavity (region 2).

40

77

m

32

J

24 4 2 0

"F -2 t~0 -4

Cavity tuning

(164 MHz/Div)

Fig. 3 Linewidth F and frequency shift Ato vs cavity tuning. The cavity length decreases from left to right. scan the width F, and frequency shift Ao~ are measured, with F the full width at half maximum of the emission curve of the atoms in Region 2, and At, the difference in center frequency between the atoms in Region 2 (inside the cavity) and Region 1 (outside the cavity). Note that Region 1 provides a very accurate frequency reference for the free space atomic resonance frequency. The experiment is then repeated at a succession of equally-spaced cavity tunings, yielding the width F and the frequency shift At, as functions of cavity tuning. The results are shown in Fig. 3. The top set of data shows the observed widths, F, and the lower set shows the frequency shifts, Aco. The straight lines show the observed width F~/(2~) = 28.7 MHz and shift Atoo/(2~) - 0 . 0 MHz with the cavity blocked. The width F~ = Ffre, + Fo contains a contribution Fo from a combination of transit-time broadening, laser frequency jitter and Doppler broadening. The curve in the top set of data shows a fit of the function F = Fb + Fo, and that in the lower set by At, = Acob, where Fb and Ao~b are given by Eqns (6), and the measured values of Ffr,e/(2n)= 19MHz, f(AEc)=0.106, and F0/(2n ) = 9.7 MHz are used. The value of F was adjusted to produce a good fit to the data at F = 23. This value is less than the value calculated from R because in Eqn (6), it was assumed that the mirrors have perfectly shaped surfaces, and that effects of finite atomic displacement from the center of the cavity and of Doppler shifts may be neglected. If these conditions are not satisfied, the cavity lineshape will be broadened, and as a first approximation, they may be taken into account by reducing the value of F. We note that the value F = 23 is consistent with the expected size of these broadening mechanisms. Good agreement with the data is then obtained. There are a number of important features to notice about tile data. First, the linewidth increases by about 8 MHz when the cavity is tuned to a resonance, and decreases by about 1.5 MHz when it is fully detuned,

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D.J. HEINZENet al.

exactly as expected, These are the largest such changes in a natural linewidth which have been observed to date. Also, the shift vanishes when the cavity is tuned to resonance or is exactly half way between resonances, and the transition shifts to the blue when the nearest cavity mode is tuned to the red, and vice versa. This behavior may be understood by noting that the system behaves very much like two coupled oscillators, and the atom-cavity coupling pushes their eigenfrequencies apart. We note that the suppression of the natural linewidth may some day play a role in precision spectroscopy, but that radiative shifts may limit this application. Our experiment shows that in a concentric optical resonator the frequency shift vanishes when the suppression is greatest, i.e. when the atomic frequency is laalf way between two cavity modes. We also note that there is no reason, in principle, that we cannot obtain f ( A E ¢ ) = 1, and obtain nearly total suppression of the linewidth. 4. R O L E O F S T I M U L A T E D E M I S S I O N I N A T O M CAVITY EXPERIMENTS

Since the peak absorption (or stimulated emission) cross-section of a two-level atom is ~r = 322/(2n), the single pass gain of a single atom in the cavity is G = a A N L / V e n = 2ANf(AZ,), with AN the atomic inversion ( - 1 ~
-

-

(G/(1 - R))] } Lb,

(10) where

Lb = Nuhv~,~,/(A2 + ~,~,),

(11)

with A the atom-cavity detuning, 7p=Ff,ee/2 the homogeneous atomic linewidth, Yc the cavity linewidth, Nu the excited state occupation probability

and T2 = (yp +':,)2 E1 - ( G / ( 1 - R))].

(12)

The total emission rate is F = ~)end "~- ]/side,

(l 3)

where 7e,d is the emission rate out the ends of the cavity, given by 7e,d = P/(Nuhv).

(14)

Finally, we can write the emission rate out the sides of the cavity as 7side = E1 +f(AZ¢)] r r....

(15)

r = rf,~,(1 + rend/I"free--f(AZc)).

(16)

to give

Note that in the broad cavity limit (7c ~> 7p) for G ~ 0 Eqn (16) agrees with F b in Eqn (6a) when the atom interacts only with the nearest single mode ( ( 1 - R ) , ¢ 1 in Eqn (6a)) 1-12]. (The case 7p~>~¢ is considered further in Ref. [11].) However, if G is not negligible compared to ( 1 - R), then two additional effects occur. First, the peak emitted power, P, will further increase or decrease, .depending on the sign of G. Second, the effective linewidth 7s of the atom-cavity resonance will either increase or decrease, again depending on the sign of G. The semi-classical model also provides an expression for the power spectrum of the light emitted through the mirrors of the atom-cavity system. The calculations show that in the broad cavity limit and assuming A = 0, the linewidth of the emitted radiation is r b - { [(aoc/Veff)(Tp/?¢)] + rfree x [ 1 - f ( A Z c ) ] } { 1 - [ G / ( 1 - R)]}.

(17)

For negligible gain, the power spectrum linewidth is determined by enhanced spontaneous emission. Equation (17) then becomes equal to Eqn (16), i.e. the linewidth of the emitted light is equal to the emission (decay) rate. However, as G increases the atom begins to undergo stimulated emission, resulting in a gain narrowing of the emitting system. This points out an important qualitative difference between enhanced spontaneous emission and stimulated emission, and indicates a clear experimental signpost. We note that the case of G comparable to (1 - R) should be possible to achieve and, in fact, experiments are currently in progress in our laboratory to observe such stimulated emission effects. The semi-classical results also give. an expression for the radiative level shift which reduces to the quantized field expression [7] and the classical expression, Eqn (6b), in the broad cavity limit for G--* 0. Also, note that in the semi-classical picture, the radiative shift is manifest as the frequency pulling between two coupled oscillators, an atomic oscillator and an E M field oscillator, whereas in the quantized field picture

Spectroscopic measurements of single atom emission it arises from absorption and re-emission of virtual photons. It is quite remarkable that these different points of view lead to the same result.

5. CONCLUSION In this paper we have pointed out that an atom in an optical resonator can experience changes in natural linewidth and transition frequency. Q u a n t u m mechanically, these changes can be viewed as resulting from a change in mode density, but we have shown how such effects may also be understood from a simple classical point of view, by considering the effect of the reflection of the atom's radiated field back onto itself. We have also presented new results in which resonator effects cause the natural linewidth of an atomic optical transition to increase by 42% and decrease by 8%. These are the largest resonator induced changes which have so far been reported. Finally, we discussed the conditions under which stimulated emission becomes important to the interaction of a single atom with the resonator. For the effective gain, G, not negligible compared to the resonator loss, stimulated emission increases the power emitted into the cavity and narrows the peaks in power output vs cavity detuning. work was performed at the Massachusetts Institute of Technology Laser Research Center, a National Science Foundation Regional Instrumentation Facility, and was supported by the National Science FoUndation under Grant No. PHY-8706753. Acknowledgements--This

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REFERENCES

[1] E. M. PURCELL,Phys. Rev. 69, 681 (1946). Another early discussion can be found in C. H. TOWNESand A. L. SCHALOW, Microwave Spectroscopy, p. 336. McGraw-Hill, New York (1955). [2] P. GOY, J. M. RAIMOND,M. GROSSand S. HAROCHE, Phys. Rev. Lett. 50, 1903 (1983). [3] R. G. HULET, E. S. HILFER and D. KLEPPNER, Phys. Rev. Lett. 55, 2137 (1985). [4] W. JHE, A. ANDERSON,E. A. HINDS, D. MESCHEDE,L. MOi and S. HAROCHE,Phys. Rev. Lett. 58, 666 (1987). [5] D. J. HEINZEN,J. J. CHILDS,C. R. MONROEand M. S. FELD,in Laser Spectroscopy VIII, p. 36 (edited by W. PERSSON and S. SVANBERG).Springer-Verlag, Berlin (1987). [6] D. J. HEINZEN,J'. J. CHILDS,J. E. THOMASand M. S. FELD, Phys. Rev. Lett. 58, 1320 (1987). [7] D. J. HEINZEN and M. S. FELD, Phys. Rev. Lett. 59, 2623 (1987). [8] F. DEMARTIN1,G. INNOCENTI,G. R. JACOaOVITZand P. MATALONI,Phys. Rev. Lett. 59, 2955 (1987). [9] D. J. HEINZEN, Ph.D. Thesis, M.I.T., June (1988), unpublished. [10] Note that this neglects decay in the reflected waves of the atomic oscillator and is equivalent to assuming the broad cavity limit discussed in Section 4 in which the atomic decay rate is much less than the cavity decay rate. [11] J. J. CHILDS,D. J. HEINZEN and M. S. FELD, to be published. [12] The classical results [Eqns (6)] for the decay rate and shift of the atom-cavity system include the interaction of the atom with all cavity modes. One can reduce these expressions to the single cavity mode results [Eqn (16)] by assuming large mirror reflectivity .and small atom-cavity detuning (A ~ c/Lo), and writing K L = k(L o + AL) = tooL(1 - A/coo)/C = nn - LoA/c, with Lo the cavity length for A = 0.