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Noble-gas broadening of the fine-structure transition in caesium (7P12 - 7P32 ) using tri-level echoes
Volume 58, number 6
OPTICS COMMUNICATIONS
15 July 1986
N O B L E - G A S B R O A D E N I N G OF T H E F I N E - S T R U C T U R E T R A N S I T I O N IN C A E S I U M (7P1/2-7P3/2) U S I N G T R I - L E V E L E C H O E S J. M A N N E R S and A.V. D U R R A N T Department of Physics, Faculty of Science, The Open University, Milton Keynes MK7 6AA, UK Received 23 December 1985; revised manuscript received 19 March 1986
We report tri-level echo measurements of collision cross-sections and broadening constants for the electric-dipole-forbidden fine-structure transition in caesium (7pl/2-7P3/2) perturbed by low pressures ( < 1 torr) of He, Ne, A, Kr, Xe. The hyperfine structures of the levels were not resolved. A brief description of the tri-level echo mechanism is given.
1. Introduction Over the past 10 years a variety of optical echo effects have been developed to study collisions in atomic vapours perturbed by inert foreign gases. The coUision-induced relaxation of two-pulse photon echoes, generated by two collinear laser pulses tuned to an electric-dipole transition, give velocity-averaged collision cross-sections and line broadening constants which complement results obtained by pressurebroadened line prof'fle measurements. The technique has been applied to lithium, sodium and caesium, perturbed by noble gases [ 1 - 4 ] . Several types of threepulse optical echo techniques have also been developed [ 5 - 1 0 ] . Tri-level echoes, generated by three laser pulses and exciting two different transitions having a common level, have been used to study the collisional relaxation of electric-dipole-forbidden transitions. The 3S - nS and 3S - nD transitions in sodium vapour perturbed by noble gases have been studied well into the Rydberg regime using tri-level echoes [9]. The fine-structure transition in sodium (3P1/2 - 3P3/2) has also been studied using a variant of the tri-level echo known as the inverted-differencefrequency tri-level echo [10]. This is the echo effect we have used in the experiments reported here.
2. Outline of theory Details of the theory of the various kinds of triO 0304018/86/$03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
level and other echo phenomena in collisionally perturbed atomic gases can be found elsewhere [11]. This discussion is limited to a brief outline of the inverted-difference-frequency tri-level echo relevant to our experiment. Fig. 1 shows the laser pulse wavelengths and the atomic transitions involved. The first
(a) 7P3/2
7P1/2
459.3 nm
455.5 nm
6Sin (ground state) (b) 455.5 nm
tl
459.3 nm
t2
455.5 nm
t3
459.3 nm
te
¢
Fig. 1. (a) The caesium 6Slr2 ground state and the two finestructure components, 7Plt,,2 and 7P3/2 , of the second resonance level. The levels have hyperfine structure not shown in the figure. (b) The laser pulse excitation sequence and the relative polarisations of the pulses. 389
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and third pulses are tuned to the 6S1/2-7P3/2 (455.5 nm) transition, while the second pulse is tuned to the 6S1/2-7P1/2 (459.3 nm) transition. The echo is generated on the 6S1/2-7P1/2 (459.3 rim) transition. In the experiment the laser pulses excite the atoms from the F = 4 hyperfine component of the 6S1/2 ground state. The hyperfine structures of the excited 7P levels are not resolved by the laser pulses. With the assumptions of ref. [ 11 ], the effect of a resonant laser pulse of wave vector k n and pulse area On arriving at time t n on a two-state atom at position r(tn) is described by
( a/(+)~
(cos(On/2)
isin(On/2)ex p[ik n ' r ( t n ) ]
ai(+)] = \ i sin(On/2 ) exp [ - i k n ' r ( t n ) ] cos(On/2)!
X \ai(_) 1'
(1)
where a i ( - ) and a/.(-) are the lower and upper state amplitudes immediately before the arrival of the pulse, and a/(+) and ai(+) are the amplitudes immediately afterwards. In our experiment, the first pulse of area 01, wave vector k l , and at time t l , excites the 6S1/2-7P3/2 transition. Assuming the initial state of the atom to be the 6S1/2 ground state we can put a i ( - ) = 1, a / ( - ) = 0 in eq. (1), where i labels the 6S1/2 ground state and/" the 7P3/2 state. Thus the amplitudes immediately after the pulse are found to be ai(+ ) = cos (01/2), a/(+) = i sin (01/2) exp [ik 1 • r l ( t l ) ] . These
0 t iS1eikvr(tl) 7P312-- I
1 6Sl/2--
I-01kl
;hI t~
[ I
I
02
/ i~ssl eikl .r(tl )is3e - ik3'r(t3)
1C2
I
I
I I
k~
chI t2
03
k3 i
t3
Fig. 2. The transitions and the pulse-induced amplitudes are shown for the three-pulse sequence. The sine and cosine factors are abbreviated by sn = sin(On~2), cn = cos(On~2), n = 1,2,3. 390
15 July 1986
amplitudes are shown in fig. 2 linked to the initial ground state amplitude, taken to be 1. (The figure uses the abbreviations c n =- cos (On/2), s n = sin (On/2), n = 1,2, 3.) Using eq. (1) again to describe the effect of pulse 2 characterised by 02, k2, t 2 on the 6 S 1 / 2 7P1/2 transition, we obtain the amplitudes shown in the middle part of fig. 2, linked to the pre-pulse ground state amplitude c 1 = cos(½ 01). In a similar way we obtain the final amplitudes after the third pulse characterised by 03, k3, t3 and resonant with the 6S1/2-7P3/2 transition. Of these, figure 2 shows only the final ground state amplitude induced by the action of the third pulse on the previously excited 7P3/2 state, as indicated by the downwards directed arrow. The presence of other amplitude terms induced by the third pulse are indicated by the other arrows but are not shown explicitly because they are not relevant to the tri-level echo mechanism. The final ground state amplitude shown on the right-hand side of fig. 2, and the 7P1/2 state amplitude, also shown in fig. 2, contribute a term of magnitude p to the expectation value of the atomic dipole moment ( t l d l t ) for t > t 3, where p = - i d cos (½01) sin (½01) sin(½02) sin(½03) X exp [i(k 2 .r(t 2) + k 3 . r ( t 3 ) - k 1 " r ( t l ) - cot)] + complex conjugate.
(2)
Here d = I(6S 1/2 [d 17P1/2 )1, and w is the angular frequency of the 6S1/2-7P1/2 transition. The magnitude of the macroscopic polarisation, P ( R , t), at position R at time t > t 3 in a gas of similar atoms can now be found by summing eq. (2) over all atoms per unit volume at R. The Doppler motion of the atoms between pulses can be introduced explicitly by putting r(tn) = R , - o (t - tn) where ~ is the centre of mass velocity of an atom. Thus we obtain
P(R,.t) = - ½nd sin 01 sin(½02) sin (½ 03) X e x p [ i ( k e ' R - cot)] ( e x p ( - i o . A ( t ) ) ) + c c ,
(t>t3)
(3) with k e = k 2 + k 3 - k 1 andA(t) = (t - t2)k 2 + (t - t3)k 3 - (t - t l ) k 1. n is the mean number of atoms per unit volume, and the angular brackets indicate an ensemble average over atomic velocity components. A special case is when the three excitation pulses
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propagate collinearly i.e. k 3 = k I and both are parallel to k 2. We then have k e = k2, and for Ikll > Ik21 there is an instant t e > t 3 given by t e = t 2 + (Ik 1 I/Ik21)(t 3 - t l ) ,
(4)
for which IA(t)l = 0, and the ensemble average in eq. (3) has its maximum possible value of unity. Thus at t = t e (the echo time) the magnitude o f the macroscopic polarisation peaks. We note also that, ignoring dispersion, the phase matching condition is satisfied. Ikel = ~o/c,
(5)
and so the radiation emitted in the direction o f k e from different parts of the sample will be in phase. Thus for an optically thin sample, an echo pulse will be emitted at the echo time t e and of intensity i o ~x ~ n 2 d 2 s i n 2 O l
.~in2[x ~ en2 ) .s m• 2 ( ~l 0 3 ) .
(6)
The optimum pulse area sequence is evidently rr/2, ~r, ft.
Collinear propagation o f the pulses is not always experimentally convenient because o f the need to protect the detector (a photomultiplier tube) from direct exposure to the laser. The angled-beams arrangement indicated in fig. 3 preserves the phase matching condition, eq. (5), and allows a partial rephasing o f the atomic dipoles to give an echo o f reduced magnitude which can be spatially filtered from the excitation pulses. The echo time and the dependence o f the echo intensity on the angle 0 between the directions o f the first and third pulses (see fig. 3) can be found by evaluating the ensemble average in eq. (3) using a gaussian distribution of velocity components characterised by u 0 = ( 2 k T / m ) 1/2 for atoms o f mass m. When this is done it is found that the echo time differs from that of eq. (4) by a term o f order
~
~
ka
J
-kl
Fig. 3. Phase matching with angled beams. The angular frequency to of the macroscopic polarisation is that of the 6Sit 2 -7Pl~ transition, to = c Ik2 I. With the angled beams arrangement shown, the wave number of the macroscopic polarisation is Ikel = Ik2 I, and so the phase matching condition is satisfied.
15 July 1986
0 2 only, and the echo intensity is given by I = I 0 exp {--(2rr2u202/b.2 ) X [t 2 - t 1 + (~./X] -- 1)(t 3 - el)]2} , where I 0 is the echo intensity with collinear excitation pulses and Xi and X! are the transition wavelengths for 6 S l / 2 - 7 P 1 / 2 and 6 S l / 2 - 7 P 3 / 2 . In our experiments t 2 - t I = 2 ns, t 3 - t 1 = 32.5 ns, 0 = 10 mrad, and the caesium vapour temperature was 333 K. This gives I smaller than I 0 by less than 1%. We assume that when the radiating gas is in the presence o f an inert foreign gas the collisional relaxation o f the echo can be described by using phenomenological rate constants describing collision rates between a radiating atom and foreign gas atoms. It is assumed that the pressure of the radiating gas is sufficiently low that collisions between radiating atoms can be neglected. From inspection o f fig. 2 and eq. (2) it can be seen that the relevant factors in the expression for the atomic dipole moment at any time t > t 3 have their origins in the amplitudes o f three different pairs o f levels during the three intervals between the pulses. These are (i) the 6Sl/2 and 7P3/2 levels during the interval t 2 - t l , (i.i) the 7Pl/2 and 7P3/2 levels during the interval t a - t2, and (iii) the 7Pl/2 and 6Sl/2 leyels during the interval t - t 3 . Thus we assume that the collisional relaxation o f the tri-level echo intensity can be described by I o: exp [ - 2 P ' ( t 2 - t l ) ] X e x p [ - 2 P ( t 3 - t2)] ex p [ - 2 P " ( t e - t3) ] , where P ' and F" are the collision rate constants for the relaxation of the electric-dipole-allowed transitions 6S1/2-7P3/2 and 6S1/2-7P1/2, resp. Velocityaveraged collision cross-sections o' and e" for these relaxation processes can be defined by F' = N o ' ~ and F " = N o " O , where ~ = (8kT/rqa) 1/2 is the average relative speed o f colliding atoms of reduced mass/a, and N (>>n) is the number density of perturbing atoms. Similarly I" is the collision rate constant for the relaxation o f the electric-dipole-forbidden time-structure transition, 7P1/2-7P3/2, and is related to a crosssection o by P = Nm3. Echo experiments are carried out at low gas pressures where it can be assumed that only binary encounters occur and the impact approximation is valid. The collision rate constants are then 391
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linear functions of the perturbing gas pressure P. Thus we write Io: e x p ( - ~ P ) .
15 July 1986
6S1/2 ground state. The hyperfme structures of the 7P1/2,3/2 levels were unresolved. The 455.5 nm beam is split at BS1 into two beams with intensity ratio 1:4. The weaker beam is sent directly to the caesium oven, passing through a horizontal polariser on its way. The more intense beam is passed through a 32.5 ns optical delay before it also is horizontally polarised and passed through the caesium oven at an angle of about 10 mrad to the weaker beam. These two pulses constitute pulses 1 and 3 of the sequence. The 459.3 nm beam is passed through a short optical delay (about 4 ns, not shown). This results in pulse 2 arriving at the oven 2 ns later than pulse 1. This beam is then verticaUy polarised and made collinear with pulse 1 at the beam splitter BS2. The echo is then generated at 459.3 nm, vertically polarised and essentially coUinear with pulse 3. This is consistent with the phase matching condition (see fig. 3) bearing in mind that the wavelengths are almost equal, and so ~ -~ 0. The pulses entering the oven had beam diameters of about 3 mm and peak powers of approximately 100 W. Since it is difficult to make a reliable estimate of the laser pulse areas, the beam intensities were adjusted empirically to produce the largest echoes. Typically the maximum echo pulse peak powers were of order 1 mW. The stainless steel oven containing the caesium was 30 cm long with silica ends windows. The central 10 cm length of the oven was heated to a temperature of 60°C. At this temperature the vapour pressure of caesium is approximately 2 X 10 -5 torr.
(7)
with fl = ( 2 ~ / k r ) [a'(t 2 - t 1) * o(t 3 - t2) + o"(t e - t 3)]. The constant/3 can be found experimentally by measuring the echo intensity as a function of perturbing gas pressure with fixed interpulse intervals. If the cross-sections o' and o" are known, from line profile measurements or from two-pulse echo experiments, then the cross-section o for the forbidden 7P1/27P3/2 transition can be determined. The cross-section o can be related to the line broadening constant 3' by o = 3'/-0(see ref. [12]).
3. Experimental arrangement The pulse sequence used to generate the trilevel echo, shown in fig. 1, is achieved by means of the experimental arrangement shown in fig. 4. There are two separate dye laser oscillators: a molectron DLII tuned to 455.5 nm with a linewidth of approximately 1 GHz, and a home-made oscillator (DL) tuned to 459.3 nm with a linewidth of approximately 5 GHz. Both dye lasers are pumped by a molectron UV12 nitrogen laser which produces pulses of duration about 7 ns at a repetition rate of about 20 Hz. The two dye lasers were tuned to select the transitions from the F = 4 hyperfine component of the
Delay p ~ 32.5 ns
DL
459.3 nm
DL II 455.5 nm
Cs
BS1
-=
oven
BS2
Fig. 4. The experimental arrangement.
392
pm
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OPTICS COMMUNICATIONS
Table 1 The cross sections o (in A2) for the fine structur transition 7P1/2 -7Par2 at 60°C are shown in the last column of the table. The line broadening constants FIN are also given (in units of 10 -20 cm-1/cm-a). Shown also are the previously determined [4 ] cross sections and broadening constants of the 6S 1/2 -7PIt2 ,a/2 transitions adjusted to 60°C.
iog~(echo intengit X)
6S1/2-7Pat 2
6Slr2 -7P1/2
7PIt2 -7P3/2
F'/N
a'
r"/N
o"
FIN
o
He Ne Ar Kr Xe
3.07 1.41 2.60 2.37 2A3
431 417 1020 1207 1429
3.78 1.60 2.81 2.47 2.49
529 477 1103 1259 1441
1.79 1.07 1.39 1.57 1A3
250 318 546 798 943
error
-~ 7%
Perturber
0
, 0.2
I I I t I 0.4 O.B O.B neon pressure /torr
15 July 1986
t.
I 1.0
Fig. 5. Microcomputer output showing logel versusP.
Various techniques must be employed in order to discriminate between the echo pulse and the excitation pulses. First, all beams emerging from the oven are passed through a vertical polariser which discriminates against pulses 1 and 3 with extinction ratios o f approximately 105. A spatial filter, consisting of a 20 cm focal length converging lens and a pinhole discriminates against pulse 2 and a monochromator set to pass 459.3 nm further reduces pulses 1 and 3. The beams then pass through a neutral density filter and then to a photomultiplier tube whose output is processed and averaged, and the echo signal applied to the Y-input of an X Y recorder. The noble gas pressure in the oven is measured by a Baratron capacitance manometer, the output from which is applied to the X-input of the recorder. The relaxation o f the echo is then displayed by allowing the foreign gas to leak slowly into the oven up to a pressure o f approximately 0.8 torr over a period of a few minuts. The resulting traces show an exponential decay of echo intensity with foreign gas pressure• The traces are analysed using a microcomputer to fred the constant/~ in eq. (7). The log plot from a typical trace is shown in fig. 5.
5. Results and discussion The velocity-averaged collision cross-sections, o, for the fine structure transition (7P1/2-7P3/2) were
= 7%
-~ 12%
measured for the five noble gases, He, Ne, Ar, Kr, using the experimentally determined values of/~ found from the XY recorder traces of the echo relaxation, and using the relationship given in eq. (7). The values of the cross-sections o' and o" used in the calculation were taken from the results o f our two-pulse echo experiments on the 6S1/2-7P3/2 and 6 S 1 / 2 7P1/2 transitions reported in a previous paper [4]. The results for the cross-sections o for the five noble gases are displayed in table 1 together with the values of o' and o" used in eq. (7). Shown also are the line broadening constants F I N (= o~). See ref. [12]. The main source o f experimental error was in the determination of/3 from the echo relaxation traces. Between 10 and 20 traces were taken for each noble gas perturber and the average value of/~ determined with a standard deviation o f about 8%. This error was combined with smaller estimated errors originating in the values o f o' and o", the oven temperature and measurements o f the intervals between the pulse peaks displayed on an oscilloscope, to give an overall estimated error in o o f about 12%. Although the positions o f the pulse peaks can be measured on an oscilloscope quite accurately, the relatively long pulse durations (about 7 ns) imply ambiguity in the excitation times o f atoms, and thus introduce some additional uncertainty into our results• Direct measurements of the collisional relaxation of the electric-dipole-forbidden fine-structure transition 7P1/2-7P3/2 (~. = 55.1/am) have not been carried out and our measurements o f 393
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the cross-sections and broadening constants are the first. We assume that the collisional relaxation o f the echo that we have measured occurs as a result o f phase interruption o f the 7P1/2-7P3/2 superposition state due to collisional perturbation o f the relative energy separation o f the levels, and the weakly inelastic collisional transfer among the Zeeman and hyperfine levels. None o f these collisional transfer cross-sections is available for comparison with our results. The fine structure collisional transfer cross-section for the 7P1/2,3/2 levels has been measured [13] and is too small to make a significant contribution to our measured cross-sections. Finally we note that the tri-level echo technique we have used is very much less sensitive to the velocity changing effects o f collisions than ordinary two pulse echoes. It is assumed that these effects can be neglected in any time interval A t provided k u A t ~ 1 where k is the wave number o f the relevant transition for the interval A t and u is the average collisional velocity change (see ref. [ 14]). This criterion is expected to be well satisfied in our experiment where in the longest interval (t 3 - t 2 = 30.5 ns) the relevant wave number is that o f the fine structure transition (about (10 - 4 o f a typical optical wave number) while in the first and last intervals which involve optical wave numbers, the intervals themselves were very short (t 2 - t 1 -~ t e - t 3 ~ 2 ns)~
394
15 July 1986
References [1] A. Flusberg, T. Mossberg and S.R. Hartmann, Optics Comm. 24 (1978) 207. [2] R. Kachru, T.W. Mossberg and S.R. Hartmann, J. Phys. B: Atom. Molec. Phys. 13 (1980) L363. [ 3] R. Kachru, TJ. Chen, T.W. Mossberg and S.R. Hartmann, Phys. Rev. A25 (1982) L546. [4] A.V. Durrant and J. Manners, J. Phys. B: Atom. Molec. Phys. 17 (1984) L701. [5] T. Mossberg, A. Flusberg, R. Kachru and S.R. Hartmann, Phys. Rev. Lett. 42 (1979) 1665. [6] T.W. Mossberg, R. Kachru and S.R. Hartmarm, Phys. Rev. Lett. 44 (1980) 73. [7] R. Kachru, T.J. Chen, S.R. Hartmann, T_W. Mossberg, P.R. Berman, Phys. Rev. Lett. 47 (1981) 902. [ 8] T. Mossberg, A. Flusberg, R. Kachru and S.R. Hartmann, Phys. Rev. Lett. 39 (1977) 1523. [9] A. Flusberg, R. Kachru, T. Mossberg and S.R. Hartmann, Phys. Rev. A19 (1979) 1607. [10] T.W. Mossberg, E. Whittaker, R. Kachru and S.R. Hartmann, Phys. Rev. A22 (1980) 1962. [ 11 ] T.W. Mossberg, R. Kaehru, S.R. Hartman, A.M. Flusberg, Phys. Rev. A20 (1979) 1976. [12] E.L. Lewis, Phys. Reports 58 (1980) 1. [ 13 ] P. Munster and J. Marek, J. Phys. B: Atom. Molec. Phys. 14 (1981) 1009. [14] P.R. Berman, T.W. Mossberg and S.R. Hartmann, Phys. Rev. A25 (1982) 2550.
OPTICS COMMUNICATIONS
15 July 1986
N O B L E - G A S B R O A D E N I N G OF T H E F I N E - S T R U C T U R E T R A N S I T I O N IN C A E S I U M (7P1/2-7P3/2) U S I N G T R I - L E V E L E C H O E S J. M A N N E R S and A.V. D U R R A N T Department of Physics, Faculty of Science, The Open University, Milton Keynes MK7 6AA, UK Received 23 December 1985; revised manuscript received 19 March 1986
We report tri-level echo measurements of collision cross-sections and broadening constants for the electric-dipole-forbidden fine-structure transition in caesium (7pl/2-7P3/2) perturbed by low pressures ( < 1 torr) of He, Ne, A, Kr, Xe. The hyperfine structures of the levels were not resolved. A brief description of the tri-level echo mechanism is given.
1. Introduction Over the past 10 years a variety of optical echo effects have been developed to study collisions in atomic vapours perturbed by inert foreign gases. The coUision-induced relaxation of two-pulse photon echoes, generated by two collinear laser pulses tuned to an electric-dipole transition, give velocity-averaged collision cross-sections and line broadening constants which complement results obtained by pressurebroadened line prof'fle measurements. The technique has been applied to lithium, sodium and caesium, perturbed by noble gases [ 1 - 4 ] . Several types of threepulse optical echo techniques have also been developed [ 5 - 1 0 ] . Tri-level echoes, generated by three laser pulses and exciting two different transitions having a common level, have been used to study the collisional relaxation of electric-dipole-forbidden transitions. The 3S - nS and 3S - nD transitions in sodium vapour perturbed by noble gases have been studied well into the Rydberg regime using tri-level echoes [9]. The fine-structure transition in sodium (3P1/2 - 3P3/2) has also been studied using a variant of the tri-level echo known as the inverted-differencefrequency tri-level echo [10]. This is the echo effect we have used in the experiments reported here.
2. Outline of theory Details of the theory of the various kinds of triO 0304018/86/$03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
level and other echo phenomena in collisionally perturbed atomic gases can be found elsewhere [11]. This discussion is limited to a brief outline of the inverted-difference-frequency tri-level echo relevant to our experiment. Fig. 1 shows the laser pulse wavelengths and the atomic transitions involved. The first
(a) 7P3/2
7P1/2
459.3 nm
455.5 nm
6Sin (ground state) (b) 455.5 nm
tl
459.3 nm
t2
455.5 nm
t3
459.3 nm
te
¢
Fig. 1. (a) The caesium 6Slr2 ground state and the two finestructure components, 7Plt,,2 and 7P3/2 , of the second resonance level. The levels have hyperfine structure not shown in the figure. (b) The laser pulse excitation sequence and the relative polarisations of the pulses. 389
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and third pulses are tuned to the 6S1/2-7P3/2 (455.5 nm) transition, while the second pulse is tuned to the 6S1/2-7P1/2 (459.3 nm) transition. The echo is generated on the 6S1/2-7P1/2 (459.3 rim) transition. In the experiment the laser pulses excite the atoms from the F = 4 hyperfine component of the 6S1/2 ground state. The hyperfine structures of the excited 7P levels are not resolved by the laser pulses. With the assumptions of ref. [ 11 ], the effect of a resonant laser pulse of wave vector k n and pulse area On arriving at time t n on a two-state atom at position r(tn) is described by
( a/(+)~
(cos(On/2)
isin(On/2)ex p[ik n ' r ( t n ) ]
ai(+)] = \ i sin(On/2 ) exp [ - i k n ' r ( t n ) ] cos(On/2)!
X \ai(_) 1'
(1)
where a i ( - ) and a/.(-) are the lower and upper state amplitudes immediately before the arrival of the pulse, and a/(+) and ai(+) are the amplitudes immediately afterwards. In our experiment, the first pulse of area 01, wave vector k l , and at time t l , excites the 6S1/2-7P3/2 transition. Assuming the initial state of the atom to be the 6S1/2 ground state we can put a i ( - ) = 1, a / ( - ) = 0 in eq. (1), where i labels the 6S1/2 ground state and/" the 7P3/2 state. Thus the amplitudes immediately after the pulse are found to be ai(+ ) = cos (01/2), a/(+) = i sin (01/2) exp [ik 1 • r l ( t l ) ] . These
0 t iS1eikvr(tl) 7P312-- I
1 6Sl/2--
I-01kl
;hI t~
[ I
I
02
/ i~ssl eikl .r(tl )is3e - ik3'r(t3)
1C2
I
I
I I
k~
chI t2
03
k3 i
t3
Fig. 2. The transitions and the pulse-induced amplitudes are shown for the three-pulse sequence. The sine and cosine factors are abbreviated by sn = sin(On~2), cn = cos(On~2), n = 1,2,3. 390
15 July 1986
amplitudes are shown in fig. 2 linked to the initial ground state amplitude, taken to be 1. (The figure uses the abbreviations c n =- cos (On/2), s n = sin (On/2), n = 1,2, 3.) Using eq. (1) again to describe the effect of pulse 2 characterised by 02, k2, t 2 on the 6 S 1 / 2 7P1/2 transition, we obtain the amplitudes shown in the middle part of fig. 2, linked to the pre-pulse ground state amplitude c 1 = cos(½ 01). In a similar way we obtain the final amplitudes after the third pulse characterised by 03, k3, t3 and resonant with the 6S1/2-7P3/2 transition. Of these, figure 2 shows only the final ground state amplitude induced by the action of the third pulse on the previously excited 7P3/2 state, as indicated by the downwards directed arrow. The presence of other amplitude terms induced by the third pulse are indicated by the other arrows but are not shown explicitly because they are not relevant to the tri-level echo mechanism. The final ground state amplitude shown on the right-hand side of fig. 2, and the 7P1/2 state amplitude, also shown in fig. 2, contribute a term of magnitude p to the expectation value of the atomic dipole moment ( t l d l t ) for t > t 3, where p = - i d cos (½01) sin (½01) sin(½02) sin(½03) X exp [i(k 2 .r(t 2) + k 3 . r ( t 3 ) - k 1 " r ( t l ) - cot)] + complex conjugate.
(2)
Here d = I(6S 1/2 [d 17P1/2 )1, and w is the angular frequency of the 6S1/2-7P1/2 transition. The magnitude of the macroscopic polarisation, P ( R , t), at position R at time t > t 3 in a gas of similar atoms can now be found by summing eq. (2) over all atoms per unit volume at R. The Doppler motion of the atoms between pulses can be introduced explicitly by putting r(tn) = R , - o (t - tn) where ~ is the centre of mass velocity of an atom. Thus we obtain
P(R,.t) = - ½nd sin 01 sin(½02) sin (½ 03) X e x p [ i ( k e ' R - cot)] ( e x p ( - i o . A ( t ) ) ) + c c ,
(t>t3)
(3) with k e = k 2 + k 3 - k 1 andA(t) = (t - t2)k 2 + (t - t3)k 3 - (t - t l ) k 1. n is the mean number of atoms per unit volume, and the angular brackets indicate an ensemble average over atomic velocity components. A special case is when the three excitation pulses
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propagate collinearly i.e. k 3 = k I and both are parallel to k 2. We then have k e = k2, and for Ikll > Ik21 there is an instant t e > t 3 given by t e = t 2 + (Ik 1 I/Ik21)(t 3 - t l ) ,
(4)
for which IA(t)l = 0, and the ensemble average in eq. (3) has its maximum possible value of unity. Thus at t = t e (the echo time) the magnitude o f the macroscopic polarisation peaks. We note also that, ignoring dispersion, the phase matching condition is satisfied. Ikel = ~o/c,
(5)
and so the radiation emitted in the direction o f k e from different parts of the sample will be in phase. Thus for an optically thin sample, an echo pulse will be emitted at the echo time t e and of intensity i o ~x ~ n 2 d 2 s i n 2 O l
.~in2[x ~ en2 ) .s m• 2 ( ~l 0 3 ) .
(6)
The optimum pulse area sequence is evidently rr/2, ~r, ft.
Collinear propagation o f the pulses is not always experimentally convenient because o f the need to protect the detector (a photomultiplier tube) from direct exposure to the laser. The angled-beams arrangement indicated in fig. 3 preserves the phase matching condition, eq. (5), and allows a partial rephasing o f the atomic dipoles to give an echo o f reduced magnitude which can be spatially filtered from the excitation pulses. The echo time and the dependence o f the echo intensity on the angle 0 between the directions o f the first and third pulses (see fig. 3) can be found by evaluating the ensemble average in eq. (3) using a gaussian distribution of velocity components characterised by u 0 = ( 2 k T / m ) 1/2 for atoms o f mass m. When this is done it is found that the echo time differs from that of eq. (4) by a term o f order
~
~
ka
J
-kl
Fig. 3. Phase matching with angled beams. The angular frequency to of the macroscopic polarisation is that of the 6Sit 2 -7Pl~ transition, to = c Ik2 I. With the angled beams arrangement shown, the wave number of the macroscopic polarisation is Ikel = Ik2 I, and so the phase matching condition is satisfied.
15 July 1986
0 2 only, and the echo intensity is given by I = I 0 exp {--(2rr2u202/b.2 ) X [t 2 - t 1 + (~./X] -- 1)(t 3 - el)]2} , where I 0 is the echo intensity with collinear excitation pulses and Xi and X! are the transition wavelengths for 6 S l / 2 - 7 P 1 / 2 and 6 S l / 2 - 7 P 3 / 2 . In our experiments t 2 - t I = 2 ns, t 3 - t 1 = 32.5 ns, 0 = 10 mrad, and the caesium vapour temperature was 333 K. This gives I smaller than I 0 by less than 1%. We assume that when the radiating gas is in the presence o f an inert foreign gas the collisional relaxation o f the echo can be described by using phenomenological rate constants describing collision rates between a radiating atom and foreign gas atoms. It is assumed that the pressure of the radiating gas is sufficiently low that collisions between radiating atoms can be neglected. From inspection o f fig. 2 and eq. (2) it can be seen that the relevant factors in the expression for the atomic dipole moment at any time t > t 3 have their origins in the amplitudes o f three different pairs o f levels during the three intervals between the pulses. These are (i) the 6Sl/2 and 7P3/2 levels during the interval t 2 - t l , (i.i) the 7Pl/2 and 7P3/2 levels during the interval t a - t2, and (iii) the 7Pl/2 and 6Sl/2 leyels during the interval t - t 3 . Thus we assume that the collisional relaxation o f the tri-level echo intensity can be described by I o: exp [ - 2 P ' ( t 2 - t l ) ] X e x p [ - 2 P ( t 3 - t2)] ex p [ - 2 P " ( t e - t3) ] , where P ' and F" are the collision rate constants for the relaxation of the electric-dipole-allowed transitions 6S1/2-7P3/2 and 6S1/2-7P1/2, resp. Velocityaveraged collision cross-sections o' and e" for these relaxation processes can be defined by F' = N o ' ~ and F " = N o " O , where ~ = (8kT/rqa) 1/2 is the average relative speed o f colliding atoms of reduced mass/a, and N (>>n) is the number density of perturbing atoms. Similarly I" is the collision rate constant for the relaxation o f the electric-dipole-forbidden time-structure transition, 7P1/2-7P3/2, and is related to a crosssection o by P = Nm3. Echo experiments are carried out at low gas pressures where it can be assumed that only binary encounters occur and the impact approximation is valid. The collision rate constants are then 391
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linear functions of the perturbing gas pressure P. Thus we write Io: e x p ( - ~ P ) .
15 July 1986
6S1/2 ground state. The hyperfme structures of the 7P1/2,3/2 levels were unresolved. The 455.5 nm beam is split at BS1 into two beams with intensity ratio 1:4. The weaker beam is sent directly to the caesium oven, passing through a horizontal polariser on its way. The more intense beam is passed through a 32.5 ns optical delay before it also is horizontally polarised and passed through the caesium oven at an angle of about 10 mrad to the weaker beam. These two pulses constitute pulses 1 and 3 of the sequence. The 459.3 nm beam is passed through a short optical delay (about 4 ns, not shown). This results in pulse 2 arriving at the oven 2 ns later than pulse 1. This beam is then verticaUy polarised and made collinear with pulse 1 at the beam splitter BS2. The echo is then generated at 459.3 nm, vertically polarised and essentially coUinear with pulse 3. This is consistent with the phase matching condition (see fig. 3) bearing in mind that the wavelengths are almost equal, and so ~ -~ 0. The pulses entering the oven had beam diameters of about 3 mm and peak powers of approximately 100 W. Since it is difficult to make a reliable estimate of the laser pulse areas, the beam intensities were adjusted empirically to produce the largest echoes. Typically the maximum echo pulse peak powers were of order 1 mW. The stainless steel oven containing the caesium was 30 cm long with silica ends windows. The central 10 cm length of the oven was heated to a temperature of 60°C. At this temperature the vapour pressure of caesium is approximately 2 X 10 -5 torr.
(7)
with fl = ( 2 ~ / k r ) [a'(t 2 - t 1) * o(t 3 - t2) + o"(t e - t 3)]. The constant/3 can be found experimentally by measuring the echo intensity as a function of perturbing gas pressure with fixed interpulse intervals. If the cross-sections o' and o" are known, from line profile measurements or from two-pulse echo experiments, then the cross-section o for the forbidden 7P1/27P3/2 transition can be determined. The cross-section o can be related to the line broadening constant 3' by o = 3'/-0(see ref. [12]).
3. Experimental arrangement The pulse sequence used to generate the trilevel echo, shown in fig. 1, is achieved by means of the experimental arrangement shown in fig. 4. There are two separate dye laser oscillators: a molectron DLII tuned to 455.5 nm with a linewidth of approximately 1 GHz, and a home-made oscillator (DL) tuned to 459.3 nm with a linewidth of approximately 5 GHz. Both dye lasers are pumped by a molectron UV12 nitrogen laser which produces pulses of duration about 7 ns at a repetition rate of about 20 Hz. The two dye lasers were tuned to select the transitions from the F = 4 hyperfine component of the
Delay p ~ 32.5 ns
DL
459.3 nm
DL II 455.5 nm
Cs
BS1
-=
oven
BS2
Fig. 4. The experimental arrangement.
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Table 1 The cross sections o (in A2) for the fine structur transition 7P1/2 -7Par2 at 60°C are shown in the last column of the table. The line broadening constants FIN are also given (in units of 10 -20 cm-1/cm-a). Shown also are the previously determined [4 ] cross sections and broadening constants of the 6S 1/2 -7PIt2 ,a/2 transitions adjusted to 60°C.
iog~(echo intengit X)
6S1/2-7Pat 2
6Slr2 -7P1/2
7PIt2 -7P3/2
F'/N
a'
r"/N
o"
FIN
o
He Ne Ar Kr Xe
3.07 1.41 2.60 2.37 2A3
431 417 1020 1207 1429
3.78 1.60 2.81 2.47 2.49
529 477 1103 1259 1441
1.79 1.07 1.39 1.57 1A3
250 318 546 798 943
error
-~ 7%
Perturber
0
, 0.2
I I I t I 0.4 O.B O.B neon pressure /torr
15 July 1986
t.
I 1.0
Fig. 5. Microcomputer output showing logel versusP.
Various techniques must be employed in order to discriminate between the echo pulse and the excitation pulses. First, all beams emerging from the oven are passed through a vertical polariser which discriminates against pulses 1 and 3 with extinction ratios o f approximately 105. A spatial filter, consisting of a 20 cm focal length converging lens and a pinhole discriminates against pulse 2 and a monochromator set to pass 459.3 nm further reduces pulses 1 and 3. The beams then pass through a neutral density filter and then to a photomultiplier tube whose output is processed and averaged, and the echo signal applied to the Y-input of an X Y recorder. The noble gas pressure in the oven is measured by a Baratron capacitance manometer, the output from which is applied to the X-input of the recorder. The relaxation o f the echo is then displayed by allowing the foreign gas to leak slowly into the oven up to a pressure o f approximately 0.8 torr over a period of a few minuts. The resulting traces show an exponential decay of echo intensity with foreign gas pressure• The traces are analysed using a microcomputer to fred the constant/~ in eq. (7). The log plot from a typical trace is shown in fig. 5.
5. Results and discussion The velocity-averaged collision cross-sections, o, for the fine structure transition (7P1/2-7P3/2) were
= 7%
-~ 12%
measured for the five noble gases, He, Ne, Ar, Kr, using the experimentally determined values of/~ found from the XY recorder traces of the echo relaxation, and using the relationship given in eq. (7). The values of the cross-sections o' and o" used in the calculation were taken from the results o f our two-pulse echo experiments on the 6S1/2-7P3/2 and 6 S 1 / 2 7P1/2 transitions reported in a previous paper [4]. The results for the cross-sections o for the five noble gases are displayed in table 1 together with the values of o' and o" used in eq. (7). Shown also are the line broadening constants F I N (= o~). See ref. [12]. The main source o f experimental error was in the determination of/3 from the echo relaxation traces. Between 10 and 20 traces were taken for each noble gas perturber and the average value of/~ determined with a standard deviation o f about 8%. This error was combined with smaller estimated errors originating in the values o f o' and o", the oven temperature and measurements o f the intervals between the pulse peaks displayed on an oscilloscope, to give an overall estimated error in o o f about 12%. Although the positions o f the pulse peaks can be measured on an oscilloscope quite accurately, the relatively long pulse durations (about 7 ns) imply ambiguity in the excitation times o f atoms, and thus introduce some additional uncertainty into our results• Direct measurements of the collisional relaxation of the electric-dipole-forbidden fine-structure transition 7P1/2-7P3/2 (~. = 55.1/am) have not been carried out and our measurements o f 393
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the cross-sections and broadening constants are the first. We assume that the collisional relaxation o f the echo that we have measured occurs as a result o f phase interruption o f the 7P1/2-7P3/2 superposition state due to collisional perturbation o f the relative energy separation o f the levels, and the weakly inelastic collisional transfer among the Zeeman and hyperfine levels. None o f these collisional transfer cross-sections is available for comparison with our results. The fine structure collisional transfer cross-section for the 7P1/2,3/2 levels has been measured [13] and is too small to make a significant contribution to our measured cross-sections. Finally we note that the tri-level echo technique we have used is very much less sensitive to the velocity changing effects o f collisions than ordinary two pulse echoes. It is assumed that these effects can be neglected in any time interval A t provided k u A t ~ 1 where k is the wave number o f the relevant transition for the interval A t and u is the average collisional velocity change (see ref. [ 14]). This criterion is expected to be well satisfied in our experiment where in the longest interval (t 3 - t 2 = 30.5 ns) the relevant wave number is that o f the fine structure transition (about (10 - 4 o f a typical optical wave number) while in the first and last intervals which involve optical wave numbers, the intervals themselves were very short (t 2 - t 1 -~ t e - t 3 ~ 2 ns)~
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References [1] A. Flusberg, T. Mossberg and S.R. Hartmann, Optics Comm. 24 (1978) 207. [2] R. Kachru, T.W. Mossberg and S.R. Hartmann, J. Phys. B: Atom. Molec. Phys. 13 (1980) L363. [ 3] R. Kachru, TJ. Chen, T.W. Mossberg and S.R. Hartmann, Phys. Rev. A25 (1982) L546. [4] A.V. Durrant and J. Manners, J. Phys. B: Atom. Molec. Phys. 17 (1984) L701. [5] T. Mossberg, A. Flusberg, R. Kachru and S.R. Hartmann, Phys. Rev. Lett. 42 (1979) 1665. [6] T.W. Mossberg, R. Kachru and S.R. Hartmarm, Phys. Rev. Lett. 44 (1980) 73. [7] R. Kachru, T.J. Chen, S.R. Hartmann, T_W. Mossberg, P.R. Berman, Phys. Rev. Lett. 47 (1981) 902. [ 8] T. Mossberg, A. Flusberg, R. Kachru and S.R. Hartmann, Phys. Rev. Lett. 39 (1977) 1523. [9] A. Flusberg, R. Kachru, T. Mossberg and S.R. Hartmann, Phys. Rev. A19 (1979) 1607. [10] T.W. Mossberg, E. Whittaker, R. Kachru and S.R. Hartmann, Phys. Rev. A22 (1980) 1962. [ 11 ] T.W. Mossberg, R. Kaehru, S.R. Hartman, A.M. Flusberg, Phys. Rev. A20 (1979) 1976. [12] E.L. Lewis, Phys. Reports 58 (1980) 1. [ 13 ] P. Munster and J. Marek, J. Phys. B: Atom. Molec. Phys. 14 (1981) 1009. [14] P.R. Berman, T.W. Mossberg and S.R. Hartmann, Phys. Rev. A25 (1982) 2550.