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Power broadening and shifts of micromaser lineshapes
Optics Communications85 ( 1991 ) 267-274 North-Holland
OPTICS COMMUNICATIONS
Full length article
Power broadening and shifts of micromaser lineshapes H. Moya-Cessa, V. Bu~ek ~ and P.L. Knight Optics Section, The Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 12 February 1991;revised manuscript 13 May 1991
We study the effects ofac Stark shifts on the dynamicsof atoms interacting with a near-resonant quantized single mode cavity field of relevance to single atom micromasers. We showthat the collapseand revivals of Rabi oscillations are stronglyinfluenced by such Stark shifts, and that transition lineshapes are not only shifted but asymmetricallydistorted by ac Stark shifts which depend on the detailed photon statistics of the cavityfield.
1. Introduction The Jaynes-Cummings model of a two-level atom interacting in rotating-wave approximation with a quantized single mode radiation field is a useful idealization to describe properties of Rydberg atoms in high-Q cavities in micromasers [ 1 ]. However, the non-resonant levels of the atom are expected to play a discernable r61e in micromaser dynamics and lineshapes. Indeed in early experimental studies of micromasers [ 2 ], both Stark shifts and asymmetries in lineshapes were observed and attributed to the effects of nearby but non-resonant levels. Here we analyse the effects of such shifts and nearby levels on the collapse-revival dynamics of the Jaynes-Cummings model [3,4] and on the time-averaged lineshapes. We should stress that our calculations refer to the interaction of a single atom with a suitablyprepared cavity field, whereas the micromaser field is successively established by earlier atoms. Nevertheless, qualitative features are shared by these two problems. We employ two models to describe the effects of nearby and off-resonant levels. In the first, we use a simple two-level model, but incorporate into the Hamiltonian a term describing the intensity-depenPermanent address: Institute of Physics, Slovak Academyof Sciences, Dtibravskd cesta 9, 842 28 Bratislava, Czechoslovakia. Elsevier SciencePublishers B.V.
dent shift of the two-level transition caused by the virtual transitions to off-resonant levels. The frequency shift depends on photon number, and because of the high polarizability of Rydberg atoms is a large quantity at modest photon numbers. The distribution of photon numbers creates a distribution of intensity-dependent detunings from unperturbed atomic resonance and will both broaden and shift the Rydberg resonances by an amount dependent on the photon number distribution. Our second model describes the situation where a single nearby level is responsible for major perturbations of the resonant two-level dynamics, although levels further away from resonance can still lead to intensity-dependent Stark shifts of atomic energies in this three-level model. In passing, we note that Brune et al. [ 5 ] have discussed how intensity-dependent Stark shifts due to off-resonant levels can be employed in quantum non-demolition measurements of cavity photon numbers in micromasers. In section 2 we develop the two-level model and in section 3 describe the effects of a competing nearby transition in the three-level model.
2. Two-level models including ac-Stark shifts The micromaser field cavity is normally chosen to have a resonance frequency very close to the resonance frequency of a convenient Rydberg atom mi267
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crowave transition frequency [2,4 ]. The cavity Q is so high that no other Rydberg transition can be excited within the cavity bandwidth ~Oc/Q, so it may be allowable to use a two-level atomic model. Nevertheless, other off-resonant states Ii) can be virtually accessed (fig. 1) and cause ac Stark shifts to the Rydberg transitions of interest. The Stark shift is proportional to the photon number 6*f inside the cavity, and to the polarizabilities of the two resonant states due to the admixture of far off-resonant states; it is therefore relatively slowly varying with micromaser frequency. We therefore incorporate such shifts as intensity-dependent corrections hz,Gtf to energies 09i of particular Rydberg states IJ ) in the otherwise standard Jaynes-Cummings two-level rotating-wave representation of the micromaser atom-field Hamiltonian:
IT(t=0)>=
j=0,1
~,
n=0
Q. In; 1),
where In) is the photon number state and the amplitudes Q. are determined by the initial photon statistics of the cavity field. We write the state-vector at time t as [~(t))=
~ n=0
x{C~°)(t)
exp{-i[coc(n+3/2)+o~olt} In+l;O)+C~t)(t) In; 1 ) } ,
(2)
where 10) is the lower atomic state. The time-dependent Schrtklinger equation then gives the evolution equations for the probability amplitudes C~°)(t) and C~'~(t)
i(d/dt)C~°)(t) =Xo(n+ 1 )C~°)(t) +2~+
/4=htnc(&tfi + 1/2) + h
1 September 1991
1C~"(t) ,
i(d/dt)C~t)(t) = (AlO+Z, n)C~.~)(t) (1)
+hA(/~o~+~*Ko~).
The field frequency is o9~, the unperturbed atomic transition frequency is tO,o=Og~-Wo, where tni are the frequencies of the upper ( I 1 ) ) and lower ( I0 ) ) atomic states. The atom-field coupling constant is supposed to be real. The atomic operators/~j are defined as usual, i.e./~0 = l i) (Jl. If at t = 0 the atom is in the upper level I 1 ), then the initial state-vector of the atom-field system can be written as
+ ) . x / n + 1C~°)(t) ,
(3)
with the initial conditions C ~ ° ) ( t = 0 ) = 0 and C~,t ) (t = 0) = Qn. The detuning parameter A~o is defined as A,o = o9~o-- o)c. The solutions of eqs. (3) are
• Llw+)fin-Xo(n+ 1) sin6.t/2"~ --
l
On
~
(4)
C~O)(t) =Q. 22x/n + 1 i6.
I'>
× e x p ( _ i z ' n + x ° ( n + l ) t)sin6.t/2 2
I1> 03
03
6]=42=(n+ l )+ [Ato +Ztn-Zo(n+ l ) ]2
(5)
1o
10> Fig. 1. Level scheme indicating the near-resonantly excited pair of atomic states with transition frequency Oa~o, the field frequency o9¢, and a set of off-resonant levels which participate only virtually in the excitation and are responsible for the ac Stark shifts to the transition frequency o)lO.
268
where the Stark-shifted generalized Rabi frequency 6n is given by
The resonant Rabi frequency is 2 2 x / ~ 1. In what follows we can suppose the constant Xo to be equal to zero (i.e. we neglect the Stark shift of the level 10) ) and then the effective Stark shift from resonance is positive for positive X~ and depends on photon statistics. The time-dependent atomic inversion W(t) is
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W(t)=(~(t)l
OPTICS COMMUNICATIONS
(/~,,-/~oo) I ~ ( t ) ) ,
1 September 1991
1.0 O.B 0.6 0.4 0.2
(6)
and is given by
=: -o.o W(t)=
n=o ~ e'(
( z / ' ° + Z2l n ) 2 a
+
422(n+ 1 ) ) Oz cos ~, t_,
(o)
-0.2 -0.4 -0.6 -0.8 1.0
(7)
-
sealed where P, is the probability of n photons being present initially in the cavity. The time-dependent inversion will exhibit the now familiar collapses and revivals [ 1,3] but these will be modified by the Stark shift term involving Z,. The lineshape, i.e. the variation of the transition probability or inversion with detuning will depend on the choice of the duration of the interaction time t. We here concentrate on the time-average inversion IV(Z/lo) defined by T
1.0 0.8 0.6 0.4 0.2 ~: -0.0 -0.2 -0.4 -0.6 -0.8 1.0
(b)
-
o
Io
20
30 40 so scaled
so
70
so
90
too
Lime
Fig. 2. Time evolution of two state atomic inversion driven by a coherent field with a mean photon number r7=20 as a function of scaled time 2~ot. In (a), we set the shift parameter g~ = 0 to recover the usual Jaynes-Cummings collapses and revivals, and in (b) we set gl=0.5 (in unitsof2~o).
o
~. p , (Alo +X] n) 2 ,=o (Alo+Zln)2+422(n+l) .
time
(8)
An alternative is to average over the spread of interaction times relevant to the micromaser by calculating the average over the transition times through the cavity for a given atomic velocity distribution. In fig. 2 we plot the time-dependent inversion W ( t ) for an atom excited by a resonant (Zt~o=0) cavity field initially in a coherent state l a ) with a Poissonian distribution P,: P, = e - " ~ " / n ! ,
with mean ~ = 2 0 . In fig. 2a we set the shift coefficient g==0 and recover the conventional JaynesCummings collapse and revivals [ 1,3]. In fig. 2b we set y,=0.5: this gives a mean Stark shift Z~r~ of 10 and a mean detuning of 10 compared with a Rabi frequency of 22 (r~ + 1 ) 1/z _~9.17. This intensity-dependent shift from resonance has essentially suppressed the effectiveness of the field to stimulate transitions out of the initial state so that the average inversion is maintained closer to its initial value. The time for the first revival to appear is shortened for increasing values ofz~. When the Stark shift Zln exceeds the resonant Rabi frequency 2 2 ( n + 1 )i/2, the
generalized Rabi frequency d, given in eq. (5) becomes approximately linearly dependent on photon number rather than dependent on the square root of the photon number ~'. The eigenvalues are then commensurate and the collapses and revivals, although greatly diminished, become periodic. The lineshape, defined as the dependence of the time-averaged inversion W on detuning A,o, similarly depends on the Stark shift. In fig. 3 we plot Igp for an atom excited by a cavity field in a coherent state with mean photon numbers r~=4, 16 and 49. From fig. 3a it is seen that the power broadening is inherent in the standard JCM (Zl = 0 ) as the mean photon number is increased, but provided no admixture of other levels is permitted there is no shift of lineshape (in fact this is true for any initial photon number distribution). On the other hand, when such an admixture is permitted, as the mean photon number is increased, we see the lineshape curve ~ This also occurs in other modifications of the Jaynes-Cummings model: for example when thefield frequency is modified by a Kerr interaction within the cavity [ 6 ] or in two-photon Jaynes-Cummings models (see e.g. ref. [7] ).
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1.0
0.8 0.7 0.6 iz:: o.5 0.4 0.3 o.g 0.2~ 0.1
w(o)= (o)
5
o.o ~o
40
3o
0 20
o ...................... detuning
1.0 0.9 0.8 0.7 0.6 ¢:
0.6
(b)
0.4 0.3 0.~, 0.1 0.0
. . . .
-loo
i , , , , i
-v6
. . . .
|
. . . .
i
. . . .
i
-60 - ~ o ~ detuning
. . . .
i
6o
. . . .
i
,6
. . . .
too
Fig. 3. Time averaged inversion if" for a two-level atom driven by a coherent field as a function of detuning A~o =COio-COc. In (a) we set X~--0.0 and thc mean photon n u m b e r ~ - - 4 (line 1 ); ~ = 16 (line 2) and r7=49 (linc 3), illustrating the power-broadening of the lineshapc. In (b) we set Xl = 1.0 (the other parameters arc same as in (a). From this figure the intcnsity-dependent ac Stark shift is seen clearly.
power-broaden and shift under the influence of the ac Stark shift x~n. It is amusing to note that the "selfshift" factor of - Z o is present in the Stark-shifted generalized Rabi frequency (5) independent of n: this represents the effect of the added photon spontaneously emitted by the initially excited atom into the cavity field, time averaged over the Rabi oscillations, but which nevertheless can induce virtual transitions to the far-off resonant levels and so is responsible for the Stark shift of the atomic frequency of state [0 ) . From eq. (8) it is obvious that the time-averaged lineshape ff'(d~o) is not symmetric with respect to 31o=0, i.e.
~z(ZJlO ) ~: W( --z~lO ) , when X, ~: O. Nevertheless, it is instructive to study the symmetry properties with respect to the effective Stark shift Zjri. To do so we write A~o as Aio=
-Zl
so that 270
ri+ J,
1 September 1991
[J- (n-n)xl] / ~ e, ,=o [ j _ (rT_ n)x, ] 2 + 4 2 2 ( n + 1 ) .
(9)
From this equation it is seen that for high enough intensities of the initial coherent field, when the photon number distribution is centered around a, the main contribution to the function ff'(J) is symmetric with respect to the change of the sign of J. It follows that the coherent lineshape remains essentially symmetric with respect to - Z t t i as is seen from fig. 3b. In most micromaser experiments, there are always a small number of thermal photons in the cavity [ 2 ]. In fig. 4 we show the time-averaged lineshape for an atom interacting with a cavity field initially thermally excited, i.e. e.=a"l(a+l)
.+'
,
but otherwise identical to the conditions of fig. 3b, with mean photon numbers ti= 4, 16 and 49 and with X, chosen to unity. We now see a remarkable asymmetry in the lineshape, given by the effects of the thermal Bose-Einstein photon number distribution on the distribution of ac Stark shifts, and qualitatively resembling the shifted, broadened and asymmetric lineshapes observed by Meschede et al. [ 2]. This asymmetry can be explained using eq. (9), in which the Bose-Einstein photon number distribution has its maximum at n = 0, in contrast to the Poissonian distribution of the coherent field, so the terms with n close to a (which are in fact the most symmetric) do not give the dominant contribution to I~(~). This explains why for the thermal field the lineshape is not symmetric with respect to the effective Stark shift g~ti. If the temperature of the cavity is decreased, then ti decreases and the effects of any 1.0
O.B 0.? 0.6 ~: o.e 0,4 0,3 0,2 O,l
o.O,oo.._~.._~o..;~,
. . .o. . . . . .~e . . . . . . .oo . . . . . . ., e
,oo
detuning Fig. 4. As in fig. 3b but for a thermal field. The asymmetries generated by the Bose-Einstein field-induced shifts arc transparent.
residual thermal field on the atom-field interaction are also decreased. We would therefore expect to recover the symmetric lineshape as the temperature decreases, reflecting the photon statistics of the maser field. The Munich micromaser cavity used in ref. [2 ] operated at 2.0 K. A new cavity has now been employed which operates at the lower temperature of 0.5 K, and symmetric lineshapes are indeed observed at such temperatures [8 ], as expected from the above.
3. Three-level models
2 j=O 2 ~ ^ +h ~ )tjo(Rjofi+fitRoj). jml
/4=hCO¢(tit~i+ 1/2) + h ~ (COj+Zjfit~i)/~jj
l .
.
.
.
.
.
IT'(t)) = ~
tt=O
exp{-i[co~(n+3/2)+coo]t}
X {C~°)(t) I n + 1; O) +Ct,~)(t) In; 1 ) + Ct,2)(t)In;2)},
(11)
with the initial condition [ ~ ( t = 0 ) ) = YQ, In; 1 ). From the time-dependent Schr6dinger equation we find the following equations of motion for the probability amplitudes:
.
.
.
.
.
.
I1> .
.
.
.
.
.
+ ) , l o x / ~ 1C.( ' ) ( t ) + 2 2 o x / ~ 1 C(2)(t) ,
i(d/dt)C(.2)(t) = (A~o +Z~ n ) C,t' ) (t) + 2 ~ox/m-+- 1C~°) ( t ) ,
i(d/dt)C~2)(t) = (A2o +Z2 n)C~ 2) (t) + 2 2 o x / ~ lC~°)( t) ,
(12)
where the detunings are Ajo=%o-co¢ and COjo= % -COo. The solutions of eqs. ( 12 ) are straightforward but tedious and are best displayed graphically. Here we again concentrate on the time-averaged lineshapes, defined (in terms of the occupation of the states I 1 ) and I0 ) of interest) as
T (10)
12> .
We write down the atom field state-vector in terms of the unperturbed states as before
i(d/dt)C~°)(t) =Zo(n + 1 )C~°)(t)
In section 2 we studied the effects on the nearlyresonant transitions of virtual excitation of many levels [i) so far off-resonance as to contribute only a slowly-varying (with frequency) Stark shift. It is possible that one state of the set [i) could be sufficiently close to resonance (although of course far less so than the resonant state) that it genuinely participates in the transition dynamics. In this case such a state needs to be included from the outset, and to do so we now adopt a three-level atomic model (fig. 5 ), with an initial state [ l ) nearly-resonant with state 10>, with a detuning A~o; state ]2> is detuned on amount A2o. The atom-field Hamiltonian is
.
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Volume 85, number 2,3
.
I {l)e
10> Fig. 5. Level scheme indicating the near resonantly-excited pair of stales and a nearby level 12) which perturbes the two-level dynamics.
l,[Z(A~o) = lim I f dt (~u(t) I T~Oo T J
0
(J~l 1 --nO0)
I~(t) )
"
In fig. 6 we plot the lineshapes as a function of detuning Ato, but keeping A2o fixed setting all Stark shifts due to the off-resonant states to zero (i.e. Zj=0) for an atom initially in state [ 1 ) excited by a coherent field. We see both transitions centered on their unperturbed frequencies participate; both are powerbroadened. Moreover, we see that with increasing value of coupling constant 22o an intensity-dependent shift can be observed (compare figs. 6a-6b with 22o equal to 1.0 and 2.0, respectively). We can conclude that the level 12 ) acts as a perturber of the [O) ~ [ 1 ) lineshape and as the coupling of level [O) with level [ 2 ) increases, the shift of the lineshape corresponding to the transition [O) ~ [ 1 ) also increases. It should be underlined here that in fig. 6 we imag271
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1 September 1991
1,0 0.8 0.7 O.B
o.6 0.4 0.8 0.2 0.1 0.0
(o)
-6o-4o-ao-zo-lo
o
,,,
0.9 0.8 0.7 0.8
~= o.6 0,4 0.3 0.2 0.! 0.0
,o"*o"so"4o"e,o
detuning
~o
so
40
(b)
0.1
o.o_s~ ,.~ ._~ ,.~ ':'~''6''''~b'''~'6" ~" ~'6" so detuning Fig. 6. Time-averaged inversion ffl as a function of detuning ZllO for a three-level atom driven by coherent field with a mean photon number a = 4 (line l); n = 1 6 (line 2) and r~=49 (line 3) with ZI2o=20 (in units ofk 1o) and 74= 0. Rabi coupling 21o is equal to l, while k2o= 1.0 (a) and 22o=2.0 (b).
ine the detuning moving the relative position of level I 1 ) (i.e. changing the atomic frequency to~ ), and keeping the detuning ,120=t02-oJo-toe fixed. This (admittedly artificial) procedure allows us to display the effect of resonance competition between the two transitions I0 ) ~ I 1 ) and I0 ) ~ 12). If we instead vary the cavity frequency the keeping all atomic frequencies constant (in this case the detuning ,12o is related to the detuning ,1~o: ,12o=,1,o+tO2-tot), there is never a frequency a~ at which the transition I 0 ) ~ 12 ) influences the I 0 ) ~ I 1 ) inversion except as a shift of lineshape. One can readily understand this effect: while the atom is initially prepared in the state I 1 ) then obviously if the field is out of resonance with the transition I 0 ) ~ I 1 ) then the occupation probability of the level 10) is small. Therefore, even if the field is then in resonance with the transition I 0),--, 12 ) the occupation of the level I 1 ) is not influenced significantly. This effect is clearly seen in fig. 7, where all parameters are same as in fig. 6a, but the detuning A2o is now defined as `12o= d ~ o + t O 2 - t o , . We see from this figure that the second "absorption" line due to the transition I0 ) ~ 12 ) 272
~o
Fig. 7. All parameters are same as in fig. 6a, but A2o=Zllo +¢o2 ¢ol, where oJ2-to] =20 (in units of 21o). The atom is initially prepared in state I 1 ).
0.7 0.6
0.4 0.3
o
detuning
1.0 0.9 0.8
~: o.5
-6o-4o-ao-~o-~o
0.1 o -0.1 -0.2 -0.3
i¢ -o-4
-0.5 -0.6 • -0.7 • -0.8 • -0.9
-I.o-50 ~ ...... -40 -'~b'"~i5 '-"lb" "b...... 1o"~"~'6"i'6"gb detuning Fig. 8. As in fig. 7, but the atom is initially in state 10). The contribution of the transition I0 ),--*12 ) to the time-averaged value of ( [/~jj -/~oo] ) is clearly seen.
is missing. Nevertheless, it should be stressed that if the atom is initially prepared in the state 10) (instead of I 1 ) ) then, of course the transition 1 0 ) ~ 12 ) contributes to the occupation probability of the level I 1 ). This can be clearly seen in fig. 8, where all parameters are the same as in fig. 7, but the atom is supposed to be prepared initially in the state [0). The contribution of the transition 1 0 ) ~ 1 2 ) is transparent. In what follows we will suppose for simplicity that the detuning ,12o is held fixed. If the ac Stark shift X~n is included (for simplicity we can neglect the Stark shifts of levels I 0 ) and 12), i.e. Zo =X2 = 0 ) , we see in fig. 9 for the same parameters as in fig. 6a an additional broadening and shift as we average over the ensemble of shifted lineshapes. Obviously, if we were to take level 12) very far from resonance, it would cease to be involved in the transition dynamics but would merely contribute to the off-resonant Stark shift as an additional m e m ber of the perturbing virtual states. In this case we
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1.0 O.g 0.8 0.7 0.8
il= o.6 0.4 0.3 0.2 0.! 0.0
. . . .
-I00
i
. . . .
-76
;
. . . .
-60
i
. . . .
!
-26
. . . .
0
i
. . . .
~
i
. . . .
60
~ . . . .
76
I00
detuning Fig. 9. time-averaged inversion I,~ as a function of detuning ~ 0 for a three-level atom driven by coherent field with a mean photon n u m b e r t i = 4 (line 1); ~ = 1 6 (line 2) and ri=49 (line 3) with Z12o=20 (in units of,~ ]o) and Rabi couplings 21o =220 = 1. We set Z] = 1.0 and Zo =Z2 = 0.0. 1.0 0.9 0.8 0.? 0.8
~= 0.6 0.4 0.8 0.2 0.1 0.0
(o)
-6o-4o-so-zo-,o
o
~o
eo
80
40
60
detuning 1.0 0.9 0.8 0.7 0.6
3= 0.6 0.4 0.3 0.2 0.1 0.0
(b)
, , , , i . , , , t , , , , i . , , , i . , , . i
-6o-4o-so-eo-~o
....
o
i ....
,o
i . . . .
20
i ....
SO
i . , ,
4O
6O
detuning Fig. 10. As in figs. 6, but the three-level atom si driven by a thermal field.
recover the results for the two-level atom as discussed in previous section. In fig. 10 we show the lineshapes when the threelevel atom, initially in state J l ), is excited by a thermal field and the Stark shifts due to the off-resonant states are set to zero (ya=0). From this figure we can observe the power-broadening as well as the shift due to the transition [0 ) ~ [ 2 ) . As before, the bigger the coupling constant 22o the greater is the shift of the lineshape. Comparing results for the thermal field (fig. 10) with the results for coherent field (fig. 6)
1 September 1991
we see that, for high enough coupling constant 220 in the case of the thermal field, the lineshapes are asymmetric in the vicinity of the resonant transitions. The reason for this effect is the same as previously, i.e. the photon number distribution for the thermal field is not symmetric with respect to ~. Finally in fig. 11 we show the lineshape when the ac Stark shift of the level I1) is taken into account ( Z , = I . 0 , while Zo = Zz = 0); other parameters are same as in fig. 10a. We see that the ac Stark shift leads to an additional power-broadening and shift of the lineshapes.
4. Conclusions
We have shown that ac Stark shifts can strongly affect the dynamics of the Jaynes-Cummings model. These shifts are linearly dependent on the photon number and can detune the atomic transition by an amount which depends on the detailed photon statistics of the radiation field. This directly affects the collapse and revival of the Rabi oscillations by pulling the atom off resonance; more significantly, at large detunings or shifts, the effective Rabi frequency becomes linearly proportional to the photon number and the collapses and revivals become periodic and regular, reflecting the resultant commensurate atom-field eigenvalues. The lineshapes are affected in a more interesting way: for a fixed photon number, the ac Stark shift moves the power-broadened resonance away from its unperturbed value, but with a distribution of numbers, the distribution of shifts contribute to a mean shift and to an effective broadening of the resonance. For thermal fields, in contrast to coherent fields, this results in a notice1.0 o.g 0.8 0.7 0.6
0.5
0.4 0.3 O~ 0.1 0.0
. . . .
-100
i
. . . .
-75
|
. . . .
-5o
i
. . . .
i
. . . .
i
-2~ o 25 detuning
. . . .
i
50
. . . .
i
. . . .
75
~oo
Fig. 11. As in fig. 10a, but the three-level atom is driven by a thermal field.
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Volume 85, number 2,3
OPTICS COMMUNICATIONS
ably asymmetric lineshape qualitatively similar to that observed in early micromaser experiments [ 2 ]. If the micromaser is operated at substantially lower temperature than in ref. [2], the thermal asymmetries become much less significant [ 8 ]. In a real micromaser, the cavity field is established by previous atoms and has both thermal and quasi-coherent maser contributions [9]. We are currently investigating lineshapes in an appropriately-extended micromaser model including atomic pumping and finite temperatures.
Acknowledgements This work was supported in part by the U.K. Science and Engineering Research Council and the Mexican Consejo Nacional de Ciencia y Tecnologia (CONACyT). We are grateful to a referee for drawing our attention to ref. [8].
274
1 September 1991
References [ 1 ] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 ( 1963 ) 89; reviewed in S.M. Barnett, P. Filipowicz, J. Javanainen, P.L. Knight and P. Meystre, in: Frontiers in Quantum Optics, eds. E.R. Pike and S. Sarkar (Adam Hilger, Bristol, 1986) p. 485; H.-I. Yoo and J.H. Eberly, Phys. Rep. 118 (1985) 239. [2] D. Meschede, H. Walther and G. Miiller, Phys. Rev. Lett. 54 (1985) 551. [3] J.H. Eberly, J.J. Sanchez-Mondragon and N.B. Narozhny, Phys. Rev. Lett. 44 (1980) 1323; N.B. Narozhny, J.J. Sanchez-Mondragon and J.H. Eberly, Phys. Rev. A 23 (1981) 236. [4] G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [5] M. Brune, S. Haroche, V. Lefevre, J.M. Raimond and N. Zagury, Phys. Rev. Lett. 65 (1990) 976. [6] V. Bu~ek and I. Jex, Optics Comm. 78 (1990) 425. [ 7 ] S.J.D. Phoenix and P.L. Knight, J. Opt. Soc. Am. B 7 (1990) 116; and references therein. [ 8 ] G. Rempe, W. Schleich, M.O. Scully and H. Walther, in: Proc. 3rd Intern. Symposium on Foundations of quantum mechanics (Physical Society of Japan, Tokyo, 1989 ) p. 294. [9 ] G. Rempe, F. Schmidt-Kaler and H. Walther, Phys. Rev. Len. 64 (1990) 2783; G. Rempe and H. Walther, Phys. Rev. A 42 (1990) 1650.
OPTICS COMMUNICATIONS
Full length article
Power broadening and shifts of micromaser lineshapes H. Moya-Cessa, V. Bu~ek ~ and P.L. Knight Optics Section, The Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 12 February 1991;revised manuscript 13 May 1991
We study the effects ofac Stark shifts on the dynamicsof atoms interacting with a near-resonant quantized single mode cavity field of relevance to single atom micromasers. We showthat the collapseand revivals of Rabi oscillations are stronglyinfluenced by such Stark shifts, and that transition lineshapes are not only shifted but asymmetricallydistorted by ac Stark shifts which depend on the detailed photon statistics of the cavityfield.
1. Introduction The Jaynes-Cummings model of a two-level atom interacting in rotating-wave approximation with a quantized single mode radiation field is a useful idealization to describe properties of Rydberg atoms in high-Q cavities in micromasers [ 1 ]. However, the non-resonant levels of the atom are expected to play a discernable r61e in micromaser dynamics and lineshapes. Indeed in early experimental studies of micromasers [ 2 ], both Stark shifts and asymmetries in lineshapes were observed and attributed to the effects of nearby but non-resonant levels. Here we analyse the effects of such shifts and nearby levels on the collapse-revival dynamics of the Jaynes-Cummings model [3,4] and on the time-averaged lineshapes. We should stress that our calculations refer to the interaction of a single atom with a suitablyprepared cavity field, whereas the micromaser field is successively established by earlier atoms. Nevertheless, qualitative features are shared by these two problems. We employ two models to describe the effects of nearby and off-resonant levels. In the first, we use a simple two-level model, but incorporate into the Hamiltonian a term describing the intensity-depenPermanent address: Institute of Physics, Slovak Academyof Sciences, Dtibravskd cesta 9, 842 28 Bratislava, Czechoslovakia. Elsevier SciencePublishers B.V.
dent shift of the two-level transition caused by the virtual transitions to off-resonant levels. The frequency shift depends on photon number, and because of the high polarizability of Rydberg atoms is a large quantity at modest photon numbers. The distribution of photon numbers creates a distribution of intensity-dependent detunings from unperturbed atomic resonance and will both broaden and shift the Rydberg resonances by an amount dependent on the photon number distribution. Our second model describes the situation where a single nearby level is responsible for major perturbations of the resonant two-level dynamics, although levels further away from resonance can still lead to intensity-dependent Stark shifts of atomic energies in this three-level model. In passing, we note that Brune et al. [ 5 ] have discussed how intensity-dependent Stark shifts due to off-resonant levels can be employed in quantum non-demolition measurements of cavity photon numbers in micromasers. In section 2 we develop the two-level model and in section 3 describe the effects of a competing nearby transition in the three-level model.
2. Two-level models including ac-Stark shifts The micromaser field cavity is normally chosen to have a resonance frequency very close to the resonance frequency of a convenient Rydberg atom mi267
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crowave transition frequency [2,4 ]. The cavity Q is so high that no other Rydberg transition can be excited within the cavity bandwidth ~Oc/Q, so it may be allowable to use a two-level atomic model. Nevertheless, other off-resonant states Ii) can be virtually accessed (fig. 1) and cause ac Stark shifts to the Rydberg transitions of interest. The Stark shift is proportional to the photon number 6*f inside the cavity, and to the polarizabilities of the two resonant states due to the admixture of far off-resonant states; it is therefore relatively slowly varying with micromaser frequency. We therefore incorporate such shifts as intensity-dependent corrections hz,Gtf to energies 09i of particular Rydberg states IJ ) in the otherwise standard Jaynes-Cummings two-level rotating-wave representation of the micromaser atom-field Hamiltonian:
IT(t=0)>=
j=0,1
~,
n=0
Q. In; 1),
where In) is the photon number state and the amplitudes Q. are determined by the initial photon statistics of the cavity field. We write the state-vector at time t as [~(t))=
~ n=0
x{C~°)(t)
exp{-i[coc(n+3/2)+o~olt} In+l;O)+C~t)(t) In; 1 ) } ,
(2)
where 10) is the lower atomic state. The time-dependent Schrtklinger equation then gives the evolution equations for the probability amplitudes C~°)(t) and C~'~(t)
i(d/dt)C~°)(t) =Xo(n+ 1 )C~°)(t) +2~+
/4=htnc(&tfi + 1/2) + h
1 September 1991
1C~"(t) ,
i(d/dt)C~t)(t) = (AlO+Z, n)C~.~)(t) (1)
+hA(/~o~+~*Ko~).
The field frequency is o9~, the unperturbed atomic transition frequency is tO,o=Og~-Wo, where tni are the frequencies of the upper ( I 1 ) ) and lower ( I0 ) ) atomic states. The atom-field coupling constant is supposed to be real. The atomic operators/~j are defined as usual, i.e./~0 = l i) (Jl. If at t = 0 the atom is in the upper level I 1 ), then the initial state-vector of the atom-field system can be written as
+ ) . x / n + 1C~°)(t) ,
(3)
with the initial conditions C ~ ° ) ( t = 0 ) = 0 and C~,t ) (t = 0) = Qn. The detuning parameter A~o is defined as A,o = o9~o-- o)c. The solutions of eqs. (3) are
• Llw+)fin-Xo(n+ 1) sin6.t/2"~ --
l
On
~
(4)
C~O)(t) =Q. 22x/n + 1 i6.
I'>
× e x p ( _ i z ' n + x ° ( n + l ) t)sin6.t/2 2
I1> 03
03
6]=42=(n+ l )+ [Ato +Ztn-Zo(n+ l ) ]2
(5)
1o
10> Fig. 1. Level scheme indicating the near-resonantly excited pair of atomic states with transition frequency Oa~o, the field frequency o9¢, and a set of off-resonant levels which participate only virtually in the excitation and are responsible for the ac Stark shifts to the transition frequency o)lO.
268
where the Stark-shifted generalized Rabi frequency 6n is given by
The resonant Rabi frequency is 2 2 x / ~ 1. In what follows we can suppose the constant Xo to be equal to zero (i.e. we neglect the Stark shift of the level 10) ) and then the effective Stark shift from resonance is positive for positive X~ and depends on photon statistics. The time-dependent atomic inversion W(t) is
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W(t)=(~(t)l
OPTICS COMMUNICATIONS
(/~,,-/~oo) I ~ ( t ) ) ,
1 September 1991
1.0 O.B 0.6 0.4 0.2
(6)
and is given by
=: -o.o W(t)=
n=o ~ e'(
( z / ' ° + Z2l n ) 2 a
+
422(n+ 1 ) ) Oz cos ~, t_,
(o)
-0.2 -0.4 -0.6 -0.8 1.0
(7)
-
sealed where P, is the probability of n photons being present initially in the cavity. The time-dependent inversion will exhibit the now familiar collapses and revivals [ 1,3] but these will be modified by the Stark shift term involving Z,. The lineshape, i.e. the variation of the transition probability or inversion with detuning will depend on the choice of the duration of the interaction time t. We here concentrate on the time-average inversion IV(Z/lo) defined by T
1.0 0.8 0.6 0.4 0.2 ~: -0.0 -0.2 -0.4 -0.6 -0.8 1.0
(b)
-
o
Io
20
30 40 so scaled
so
70
so
90
too
Lime
Fig. 2. Time evolution of two state atomic inversion driven by a coherent field with a mean photon number r7=20 as a function of scaled time 2~ot. In (a), we set the shift parameter g~ = 0 to recover the usual Jaynes-Cummings collapses and revivals, and in (b) we set gl=0.5 (in unitsof2~o).
o
~. p , (Alo +X] n) 2 ,=o (Alo+Zln)2+422(n+l) .
time
(8)
An alternative is to average over the spread of interaction times relevant to the micromaser by calculating the average over the transition times through the cavity for a given atomic velocity distribution. In fig. 2 we plot the time-dependent inversion W ( t ) for an atom excited by a resonant (Zt~o=0) cavity field initially in a coherent state l a ) with a Poissonian distribution P,: P, = e - " ~ " / n ! ,
with mean ~ = 2 0 . In fig. 2a we set the shift coefficient g==0 and recover the conventional JaynesCummings collapse and revivals [ 1,3]. In fig. 2b we set y,=0.5: this gives a mean Stark shift Z~r~ of 10 and a mean detuning of 10 compared with a Rabi frequency of 22 (r~ + 1 ) 1/z _~9.17. This intensity-dependent shift from resonance has essentially suppressed the effectiveness of the field to stimulate transitions out of the initial state so that the average inversion is maintained closer to its initial value. The time for the first revival to appear is shortened for increasing values ofz~. When the Stark shift Zln exceeds the resonant Rabi frequency 2 2 ( n + 1 )i/2, the
generalized Rabi frequency d, given in eq. (5) becomes approximately linearly dependent on photon number rather than dependent on the square root of the photon number ~'. The eigenvalues are then commensurate and the collapses and revivals, although greatly diminished, become periodic. The lineshape, defined as the dependence of the time-averaged inversion W on detuning A,o, similarly depends on the Stark shift. In fig. 3 we plot Igp for an atom excited by a cavity field in a coherent state with mean photon numbers r~=4, 16 and 49. From fig. 3a it is seen that the power broadening is inherent in the standard JCM (Zl = 0 ) as the mean photon number is increased, but provided no admixture of other levels is permitted there is no shift of lineshape (in fact this is true for any initial photon number distribution). On the other hand, when such an admixture is permitted, as the mean photon number is increased, we see the lineshape curve ~ This also occurs in other modifications of the Jaynes-Cummings model: for example when thefield frequency is modified by a Kerr interaction within the cavity [ 6 ] or in two-photon Jaynes-Cummings models (see e.g. ref. [7] ).
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1.0
0.8 0.7 0.6 iz:: o.5 0.4 0.3 o.g 0.2~ 0.1
w(o)= (o)
5
o.o ~o
40
3o
0 20
o ...................... detuning
1.0 0.9 0.8 0.7 0.6 ¢:
0.6
(b)
0.4 0.3 0.~, 0.1 0.0
. . . .
-loo
i , , , , i
-v6
. . . .
|
. . . .
i
. . . .
i
-60 - ~ o ~ detuning
. . . .
i
6o
. . . .
i
,6
. . . .
too
Fig. 3. Time averaged inversion if" for a two-level atom driven by a coherent field as a function of detuning A~o =COio-COc. In (a) we set X~--0.0 and thc mean photon n u m b e r ~ - - 4 (line 1 ); ~ = 16 (line 2) and r7=49 (linc 3), illustrating the power-broadening of the lineshapc. In (b) we set Xl = 1.0 (the other parameters arc same as in (a). From this figure the intcnsity-dependent ac Stark shift is seen clearly.
power-broaden and shift under the influence of the ac Stark shift x~n. It is amusing to note that the "selfshift" factor of - Z o is present in the Stark-shifted generalized Rabi frequency (5) independent of n: this represents the effect of the added photon spontaneously emitted by the initially excited atom into the cavity field, time averaged over the Rabi oscillations, but which nevertheless can induce virtual transitions to the far-off resonant levels and so is responsible for the Stark shift of the atomic frequency of state [0 ) . From eq. (8) it is obvious that the time-averaged lineshape ff'(d~o) is not symmetric with respect to 31o=0, i.e.
~z(ZJlO ) ~: W( --z~lO ) , when X, ~: O. Nevertheless, it is instructive to study the symmetry properties with respect to the effective Stark shift Zjri. To do so we write A~o as Aio=
-Zl
so that 270
ri+ J,
1 September 1991
[J- (n-n)xl] / ~ e, ,=o [ j _ (rT_ n)x, ] 2 + 4 2 2 ( n + 1 ) .
(9)
From this equation it is seen that for high enough intensities of the initial coherent field, when the photon number distribution is centered around a, the main contribution to the function ff'(J) is symmetric with respect to the change of the sign of J. It follows that the coherent lineshape remains essentially symmetric with respect to - Z t t i as is seen from fig. 3b. In most micromaser experiments, there are always a small number of thermal photons in the cavity [ 2 ]. In fig. 4 we show the time-averaged lineshape for an atom interacting with a cavity field initially thermally excited, i.e. e.=a"l(a+l)
.+'
,
but otherwise identical to the conditions of fig. 3b, with mean photon numbers ti= 4, 16 and 49 and with X, chosen to unity. We now see a remarkable asymmetry in the lineshape, given by the effects of the thermal Bose-Einstein photon number distribution on the distribution of ac Stark shifts, and qualitatively resembling the shifted, broadened and asymmetric lineshapes observed by Meschede et al. [ 2]. This asymmetry can be explained using eq. (9), in which the Bose-Einstein photon number distribution has its maximum at n = 0, in contrast to the Poissonian distribution of the coherent field, so the terms with n close to a (which are in fact the most symmetric) do not give the dominant contribution to I~(~). This explains why for the thermal field the lineshape is not symmetric with respect to the effective Stark shift g~ti. If the temperature of the cavity is decreased, then ti decreases and the effects of any 1.0
O.B 0.? 0.6 ~: o.e 0,4 0,3 0,2 O,l
o.O,oo.._~.._~o..;~,
. . .o. . . . . .~e . . . . . . .oo . . . . . . ., e
,oo
detuning Fig. 4. As in fig. 3b but for a thermal field. The asymmetries generated by the Bose-Einstein field-induced shifts arc transparent.
residual thermal field on the atom-field interaction are also decreased. We would therefore expect to recover the symmetric lineshape as the temperature decreases, reflecting the photon statistics of the maser field. The Munich micromaser cavity used in ref. [2 ] operated at 2.0 K. A new cavity has now been employed which operates at the lower temperature of 0.5 K, and symmetric lineshapes are indeed observed at such temperatures [8 ], as expected from the above.
3. Three-level models
2 j=O 2 ~ ^ +h ~ )tjo(Rjofi+fitRoj). jml
/4=hCO¢(tit~i+ 1/2) + h ~ (COj+Zjfit~i)/~jj
l .
.
.
.
.
.
IT'(t)) = ~
tt=O
exp{-i[co~(n+3/2)+coo]t}
X {C~°)(t) I n + 1; O) +Ct,~)(t) In; 1 ) + Ct,2)(t)In;2)},
(11)
with the initial condition [ ~ ( t = 0 ) ) = YQ, In; 1 ). From the time-dependent Schr6dinger equation we find the following equations of motion for the probability amplitudes:
.
.
.
.
.
.
I1> .
.
.
.
.
.
+ ) , l o x / ~ 1C.( ' ) ( t ) + 2 2 o x / ~ 1 C(2)(t) ,
i(d/dt)C(.2)(t) = (A~o +Z~ n ) C,t' ) (t) + 2 ~ox/m-+- 1C~°) ( t ) ,
i(d/dt)C~2)(t) = (A2o +Z2 n)C~ 2) (t) + 2 2 o x / ~ lC~°)( t) ,
(12)
where the detunings are Ajo=%o-co¢ and COjo= % -COo. The solutions of eqs. ( 12 ) are straightforward but tedious and are best displayed graphically. Here we again concentrate on the time-averaged lineshapes, defined (in terms of the occupation of the states I 1 ) and I0 ) of interest) as
T (10)
12> .
We write down the atom field state-vector in terms of the unperturbed states as before
i(d/dt)C~°)(t) =Zo(n + 1 )C~°)(t)
In section 2 we studied the effects on the nearlyresonant transitions of virtual excitation of many levels [i) so far off-resonance as to contribute only a slowly-varying (with frequency) Stark shift. It is possible that one state of the set [i) could be sufficiently close to resonance (although of course far less so than the resonant state) that it genuinely participates in the transition dynamics. In this case such a state needs to be included from the outset, and to do so we now adopt a three-level atomic model (fig. 5 ), with an initial state [ l ) nearly-resonant with state 10>, with a detuning A~o; state ]2> is detuned on amount A2o. The atom-field Hamiltonian is
.
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OPTICS COMMUNICATIONS
Volume 85, number 2,3
.
I {l)e
10> Fig. 5. Level scheme indicating the near resonantly-excited pair of stales and a nearby level 12) which perturbes the two-level dynamics.
l,[Z(A~o) = lim I f dt (~u(t) I T~Oo T J
0
(J~l 1 --nO0)
I~(t) )
"
In fig. 6 we plot the lineshapes as a function of detuning Ato, but keeping A2o fixed setting all Stark shifts due to the off-resonant states to zero (i.e. Zj=0) for an atom initially in state [ 1 ) excited by a coherent field. We see both transitions centered on their unperturbed frequencies participate; both are powerbroadened. Moreover, we see that with increasing value of coupling constant 22o an intensity-dependent shift can be observed (compare figs. 6a-6b with 22o equal to 1.0 and 2.0, respectively). We can conclude that the level 12 ) acts as a perturber of the [O) ~ [ 1 ) lineshape and as the coupling of level [O) with level [ 2 ) increases, the shift of the lineshape corresponding to the transition [O) ~ [ 1 ) also increases. It should be underlined here that in fig. 6 we imag271
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1 September 1991
1,0 0.8 0.7 O.B
o.6 0.4 0.8 0.2 0.1 0.0
(o)
-6o-4o-ao-zo-lo
o
,,,
0.9 0.8 0.7 0.8
~= o.6 0,4 0.3 0.2 0.! 0.0
,o"*o"so"4o"e,o
detuning
~o
so
40
(b)
0.1
o.o_s~ ,.~ ._~ ,.~ ':'~''6''''~b'''~'6" ~" ~'6" so detuning Fig. 6. Time-averaged inversion ffl as a function of detuning ZllO for a three-level atom driven by coherent field with a mean photon number a = 4 (line l); n = 1 6 (line 2) and r~=49 (line 3) with ZI2o=20 (in units ofk 1o) and 74= 0. Rabi coupling 21o is equal to l, while k2o= 1.0 (a) and 22o=2.0 (b).
ine the detuning moving the relative position of level I 1 ) (i.e. changing the atomic frequency to~ ), and keeping the detuning ,120=t02-oJo-toe fixed. This (admittedly artificial) procedure allows us to display the effect of resonance competition between the two transitions I0 ) ~ I 1 ) and I0 ) ~ 12). If we instead vary the cavity frequency the keeping all atomic frequencies constant (in this case the detuning ,12o is related to the detuning ,1~o: ,12o=,1,o+tO2-tot), there is never a frequency a~ at which the transition I 0 ) ~ 12 ) influences the I 0 ) ~ I 1 ) inversion except as a shift of lineshape. One can readily understand this effect: while the atom is initially prepared in the state I 1 ) then obviously if the field is out of resonance with the transition I 0 ) ~ I 1 ) then the occupation probability of the level 10) is small. Therefore, even if the field is then in resonance with the transition I 0),--, 12 ) the occupation of the level I 1 ) is not influenced significantly. This effect is clearly seen in fig. 7, where all parameters are same as in fig. 6a, but the detuning A2o is now defined as `12o= d ~ o + t O 2 - t o , . We see from this figure that the second "absorption" line due to the transition I0 ) ~ 12 ) 272
~o
Fig. 7. All parameters are same as in fig. 6a, but A2o=Zllo +¢o2 ¢ol, where oJ2-to] =20 (in units of 21o). The atom is initially prepared in state I 1 ).
0.7 0.6
0.4 0.3
o
detuning
1.0 0.9 0.8
~: o.5
-6o-4o-ao-~o-~o
0.1 o -0.1 -0.2 -0.3
i¢ -o-4
-0.5 -0.6 • -0.7 • -0.8 • -0.9
-I.o-50 ~ ...... -40 -'~b'"~i5 '-"lb" "b...... 1o"~"~'6"i'6"gb detuning Fig. 8. As in fig. 7, but the atom is initially in state 10). The contribution of the transition I0 ),--*12 ) to the time-averaged value of ( [/~jj -/~oo] ) is clearly seen.
is missing. Nevertheless, it should be stressed that if the atom is initially prepared in the state 10) (instead of I 1 ) ) then, of course the transition 1 0 ) ~ 12 ) contributes to the occupation probability of the level I 1 ). This can be clearly seen in fig. 8, where all parameters are the same as in fig. 7, but the atom is supposed to be prepared initially in the state [0). The contribution of the transition 1 0 ) ~ 1 2 ) is transparent. In what follows we will suppose for simplicity that the detuning ,12o is held fixed. If the ac Stark shift X~n is included (for simplicity we can neglect the Stark shifts of levels I 0 ) and 12), i.e. Zo =X2 = 0 ) , we see in fig. 9 for the same parameters as in fig. 6a an additional broadening and shift as we average over the ensemble of shifted lineshapes. Obviously, if we were to take level 12) very far from resonance, it would cease to be involved in the transition dynamics but would merely contribute to the off-resonant Stark shift as an additional m e m ber of the perturbing virtual states. In this case we
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1.0 O.g 0.8 0.7 0.8
il= o.6 0.4 0.3 0.2 0.! 0.0
. . . .
-I00
i
. . . .
-76
;
. . . .
-60
i
. . . .
!
-26
. . . .
0
i
. . . .
~
i
. . . .
60
~ . . . .
76
I00
detuning Fig. 9. time-averaged inversion I,~ as a function of detuning ~ 0 for a three-level atom driven by coherent field with a mean photon n u m b e r t i = 4 (line 1); ~ = 1 6 (line 2) and ri=49 (line 3) with Z12o=20 (in units of,~ ]o) and Rabi couplings 21o =220 = 1. We set Z] = 1.0 and Zo =Z2 = 0.0. 1.0 0.9 0.8 0.? 0.8
~= 0.6 0.4 0.8 0.2 0.1 0.0
(o)
-6o-4o-so-zo-,o
o
~o
eo
80
40
60
detuning 1.0 0.9 0.8 0.7 0.6
3= 0.6 0.4 0.3 0.2 0.1 0.0
(b)
, , , , i . , , , t , , , , i . , , , i . , , . i
-6o-4o-so-eo-~o
....
o
i ....
,o
i . . . .
20
i ....
SO
i . , ,
4O
6O
detuning Fig. 10. As in figs. 6, but the three-level atom si driven by a thermal field.
recover the results for the two-level atom as discussed in previous section. In fig. 10 we show the lineshapes when the threelevel atom, initially in state J l ), is excited by a thermal field and the Stark shifts due to the off-resonant states are set to zero (ya=0). From this figure we can observe the power-broadening as well as the shift due to the transition [0 ) ~ [ 2 ) . As before, the bigger the coupling constant 22o the greater is the shift of the lineshape. Comparing results for the thermal field (fig. 10) with the results for coherent field (fig. 6)
1 September 1991
we see that, for high enough coupling constant 220 in the case of the thermal field, the lineshapes are asymmetric in the vicinity of the resonant transitions. The reason for this effect is the same as previously, i.e. the photon number distribution for the thermal field is not symmetric with respect to ~. Finally in fig. 11 we show the lineshape when the ac Stark shift of the level I1) is taken into account ( Z , = I . 0 , while Zo = Zz = 0); other parameters are same as in fig. 10a. We see that the ac Stark shift leads to an additional power-broadening and shift of the lineshapes.
4. Conclusions
We have shown that ac Stark shifts can strongly affect the dynamics of the Jaynes-Cummings model. These shifts are linearly dependent on the photon number and can detune the atomic transition by an amount which depends on the detailed photon statistics of the radiation field. This directly affects the collapse and revival of the Rabi oscillations by pulling the atom off resonance; more significantly, at large detunings or shifts, the effective Rabi frequency becomes linearly proportional to the photon number and the collapses and revivals become periodic and regular, reflecting the resultant commensurate atom-field eigenvalues. The lineshapes are affected in a more interesting way: for a fixed photon number, the ac Stark shift moves the power-broadened resonance away from its unperturbed value, but with a distribution of numbers, the distribution of shifts contribute to a mean shift and to an effective broadening of the resonance. For thermal fields, in contrast to coherent fields, this results in a notice1.0 o.g 0.8 0.7 0.6
0.5
0.4 0.3 O~ 0.1 0.0
. . . .
-100
i
. . . .
-75
|
. . . .
-5o
i
. . . .
i
. . . .
i
-2~ o 25 detuning
. . . .
i
50
. . . .
i
. . . .
75
~oo
Fig. 11. As in fig. 10a, but the three-level atom is driven by a thermal field.
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ably asymmetric lineshape qualitatively similar to that observed in early micromaser experiments [ 2 ]. If the micromaser is operated at substantially lower temperature than in ref. [2], the thermal asymmetries become much less significant [ 8 ]. In a real micromaser, the cavity field is established by previous atoms and has both thermal and quasi-coherent maser contributions [9]. We are currently investigating lineshapes in an appropriately-extended micromaser model including atomic pumping and finite temperatures.
Acknowledgements This work was supported in part by the U.K. Science and Engineering Research Council and the Mexican Consejo Nacional de Ciencia y Tecnologia (CONACyT). We are grateful to a referee for drawing our attention to ref. [8].
274
1 September 1991
References [ 1 ] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 ( 1963 ) 89; reviewed in S.M. Barnett, P. Filipowicz, J. Javanainen, P.L. Knight and P. Meystre, in: Frontiers in Quantum Optics, eds. E.R. Pike and S. Sarkar (Adam Hilger, Bristol, 1986) p. 485; H.-I. Yoo and J.H. Eberly, Phys. Rep. 118 (1985) 239. [2] D. Meschede, H. Walther and G. Miiller, Phys. Rev. Lett. 54 (1985) 551. [3] J.H. Eberly, J.J. Sanchez-Mondragon and N.B. Narozhny, Phys. Rev. Lett. 44 (1980) 1323; N.B. Narozhny, J.J. Sanchez-Mondragon and J.H. Eberly, Phys. Rev. A 23 (1981) 236. [4] G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [5] M. Brune, S. Haroche, V. Lefevre, J.M. Raimond and N. Zagury, Phys. Rev. Lett. 65 (1990) 976. [6] V. Bu~ek and I. Jex, Optics Comm. 78 (1990) 425. [ 7 ] S.J.D. Phoenix and P.L. Knight, J. Opt. Soc. Am. B 7 (1990) 116; and references therein. [ 8 ] G. Rempe, W. Schleich, M.O. Scully and H. Walther, in: Proc. 3rd Intern. Symposium on Foundations of quantum mechanics (Physical Society of Japan, Tokyo, 1989 ) p. 294. [9 ] G. Rempe, F. Schmidt-Kaler and H. Walther, Phys. Rev. Len. 64 (1990) 2783; G. Rempe and H. Walther, Phys. Rev. A 42 (1990) 1650.