25 May 2000
Optics Communications 179 Ž2000. 537–547 www.elsevier.comrlocateroptcom
Laser control of Mossbauer spectra as a way to gamma-ray lasing 1 Roman Kolesov b
a,b
, Yuri Rostovtsev
a,b
, Olga Kocharovskaya
a,b,)
a Department of Physics, Texas A & M UniÕersity, College Station, TX 77843-4242, USA Institute of Applied Physics Russian Academy of Sciences, 46 UlyanoÕ Street, Nizny NoÕgorod 603120, Russia
Received 16 November 1999; accepted 24 January 2000
Abstract We give a comparative analysis of basic schemes of optical control of Mossbauer spectra providing an elimination of resonant absorption by a gamma-ray nuclear transition and suggest a novel scheme of frequency non-selective optical pumping which turns out to be the most promising for a realization of induced gain in the gamma-ray range. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The possibility of application of lasing without inversion ŽLWI. to novel high-frequency ranges Žspecifically the gamma-ray range, where coherent sources of radiation are absent. was suggested in the first works w1x on the subject Žsee also the review papers w2x.. However, the direct search for possible implementation of gamma-ray LWI schemes at nuclear transitions has been started quite recently w3–8x. The experimental realization of gain w9x and then the first lasers without inversion w10x in optical region have stimulated further activity in this field w11–13x. It should be mentioned that in the recent comprehensive review on the gamma-ray laser Žgraser. problem w14x, lasing without inversion was identified as the most promising direction for further investigation. Indeed, the major idea of LWI is a suppression of resonant absorption Ždue to interference of different absorption channels in an atomic system with a split operating level w1,2x.. Resonant absorption by a nuclear transition is typically 3–5 orders of magnitude larger than off-resonant losses in the active sample w14–16x. Hence if it were possible to suppress resonant absorption up to the level of off-resonant losses, the requirement for incoherent pump would be reduced by the same amount. An additional reduction of the incoherent pump requirement for inversionless schemes comes from the possibility to pump directly at the operating laser transition, avoiding the rather long and inefficient cascade Žstarting from MeV range. which is typically required for realization of population inversion of the nuclear transition. In a sense, the schemes with suppressed resonant absorption allow
) 1
Corresponding author. E-mail:
[email protected] We dedicate this paper to Marlan Scully, quantum cowboy and our hero in science, who has been teaching us physics for many years.
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 5 2 9 - 0
538
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
for the most efficient and economic pumping. In turn, lower power of incoherent pump produces less heat in the sample. This is crucially important in view of the so-called graser dilemma. Namely, in traditional laser schemes, huge heating caused by the incoherent pump required to create population inversion destroys the conditions for the Mossbauer and Bormann effects. In turn, both effects are crucial for obtaining the net gain. The first one excludes Doppler broadening Žwhich otherwise reduces net resonant gain by orders of magnitude. and the second one reduces off-resonant losses by 1–2 orders of magnitude. This is the essence of the graser dilemma. Suppression of resonant absorption apparently suggests a solution of this long-standing contradiction. Namely, the release in the pump requirement and the corresponding reduction of heating allows to pump active nuclei in the host lattice without destroying the conditions of both Mossbauer and Bormann effects. Hence, implementation of LWI schemes Žor any other scheme which could provide efficient suppression of resonant absorption at gamma-ray nuclear transitions. along with the possibility of direct incoherent pumping at the operating transition is very important. 2. Coherent driving of nuclei All experimentally realized optical LWI schemes involve coherent driving of optical transitions in multi-level atomic systems w2x. The first suggestions of similar schemes in nuclei dealt with a direct microwave driving of nuclear hyperfine transition with or without strong magnetic field applied to w3,17x. On one hand, to be efficient, this kind of driving requires low temperature of the sample Žbelow or even far below liquid helium temperature. which is in a very strong contradiction with an essential heating produced by incoherent pump. On the other hand, the inversionless gain condition implies fast decay at the driven microwave transition as compared to nuclear gamma-ray transition which is hardly compatible with a requirement of sufficiently short life time of gamma-ray transition Žof the order of 1 m s – 1 ns w14–17x. which is needed to maximize the ratio of the natural and inhomogeneous linewidths, and, hence, not to decrease a resonant cross-section. Direct resonant laser driving of nuclear transitions is impossible since the energies of photons are typically orders of magnitude away from those corresponding to the nuclear transitions. The possibilities to use multiphoton processes and near field-effects were discussed but turned out to be very inefficient w14x. At the same time laser driving of electronic transitions in the optical range Ži.e. transitions which are due to change of radial distribution of the electron shell. can be made free of both restrictions pointed out for microwave driving. Hence, in principle, a coupling between the electronic and nuclear degrees of freedom could allow for optical manipulation of the response of nucleus at the gamma-ray transition. Coupling via recoil and a possibility of population inversion at the combined nuclear–electron transitions Žinvolving simultaneous transition of nucleus from the excited to ground state and electronic transition from the ground to excited state. was analyzed in w18x. However, the probability of such combined transitions is very small. The possibility to use hyperfine coupling for very efficient manipulation of the nuclear response and realization of nuclear analogues of optical LWI schemes Žsuch as lambda,V and double-lambda schemes w2,12x. had been suggested in w5x and had been studied later in w6–8,11–13x. In the next section we briefly summarize the results of these studies with an emphasis on the difficulties of the realistic implementation of LWI schemes at Mossbauer nuclear transitions. Finally Žin Section 4., we propose a novel scheme of frequency non-selective optical pumping which seems to be the most simple and promising for realization of gain in the gamma-ray range. 3. Nuclear analogues of LWI schemes 3.1. Frequency sensitiÕe schemes The simplest model of the nuclear–electron systems involves at least one optical transition in both ground and excited states of nucleus and a hyperfine ŽHF. coupling between the electronic and nuclear subsystems w11x
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
539
Žsee Fig. 1.. This hyperfine coupling breaks a symmetry of the initially degenerate energy transitions Žsee Fig. 1a. of the compound nuclear–electron system in two ways: Ži. shift of the excited electron state for excited nucleus as compared to the nucleus in the ground state Žisomer or quadruple shift, D ., see Fig. 1b, Žii. hyperfine splitting, D, of at least one of the energy levels, see Fig. 1c. Since the hyperfine coupling is weak Žof the order of 1 m eV. on the scale of both internal electronic Ž1eV. and nuclear transitions Ž10 keV. the selection rules remain practically unaffected by this coupling and hence only transitions with a change of either nuclear or electron state Žbut not both. are allowed. This means that hyperfine interaction cannot provide nuclear excitation or de-excitation under the action of a laser field. Nevertheless, it allows for very efficient laser manipulation of the nuclear response. First, it is due to this interaction that the resonant frequency at which this response occurs depends on whether the electron is excited or not Žsee Fig. 1b.. Second, it obviously allows for a nuclear orientation as result of either optical pumping or coherent population trapping Žsee Fig. 1c.. Both a shift of the resonant frequency of nuclear response and a nuclear orientation lead to a variety of drastic modifications of gamma-ray absorption spectra. The major qualitative modifications of Mossbauer spectra due to optical driving were studied in w11x. They involve vanishing or appearance of additional lines, broadening or narrowing of the lines, splitting and shift of the lines, suppression of absorption, etc. These modifications form the basis of a new kind of laser-Mossbauer spectroscopy w11x. Among them it is a suppression of resonant absorption that is of interest in view of inversionless lasers. There are a few different ways in which optical driving can provide a suppression of resonant absorption on gamma-ray nuclear transition. Consider the scheme shown in Fig. 1b. In the case when the relaxation g of optical transition is much smaller than the isomer shift D Žg < D ., the coherent driving field can be resonant with either 1–2 or 3–4 optical transition. An interaction with nonresonant transition can be neglected until the field is not too strong, namely, the Rabi resonance frequency V Ž V s m opt Eoptr2" . does not exceed HF splitting, V < D. In this case we end up accordingly with the usual V or lambda schemes for LWI and can use the well-known results concerning conditions for LWI in these schemes w2,12x. Namely, in the case when the relaxation G of the nuclear transition is lower than that of the optical transition Ž G - g ., the two-photon Raman gain at corresponding gamma-ray transition exceeds resonant absorption which is suppressed due to the atomic interference. Hence even low intensity of incoherent pump to upper operating state, p ) Grg Žwhere p is the pump parameter showing the rate of the incoherent pump in terms of the relaxation rate at the nuclear transition, p s GpumprG . provides the condition for amplification. Maximum of the gain is achieved when the driving field is of the order of the saturating field for the optical transition: V ; g . A further increase of the intensity of the driving field causes gain decrease and vanish completely in the limit of strong driving field V 4 D. Let us note that the resonant driving of the upper electronic transition Žlambda scheme. may even provide population inversion of the operating 4–2 transition since the lower operating level 2 remains practically empty until driving field is not too strong. In the last case even the lower pump rate Ždefined by off-resonant absorption at 4–2 transition. provides gain.
Fig. 1. Compound nuclear–electron system. In the absence of hyperfine interaction Ža., and in the presence of the hyperfine interactions, leading to the shift Žb. and the splitting Žc. of an energy level.
540
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
It is worthwhile to emphasize that though the possibility to achieve LWI in the scheme considered above does exist in principle, its practical implementation on the Mossbauer transition is questionable. Indeed, the HF structure is typically overlapped by inhomogeneous broadening. Moreover, the requirement of the shorter life time of the optical transition as compared to the gamma-ray transition Žwhich, as has been noted above, should be of the order of 1 m s. means that one has to use electro-dipole allowed optical transitions. The homogeneous broadening of the phonon-free line for such transitions in the compounds containing the elements of the transition group can be made small as compared to the HF splitting at low temperature. However, sufficiently high temperature Žof the order of 100 K. that is unavoidable for a graser operation due to the heating produced by the incoherent pump, phonon broadening would overlap the HF structure. In this case, realization of the LWI schemes described above becomes impossible. It is worthwhile to mention that the condition g ) G can be relaxed while inversionless gain remains when the driving field is detuned Žbut not too far. from one-photon resonance w20x. But the magnitude of the gain in this case essentially decreases, and it becomes progressively more difficult to exceed off-resonant losses. Additional mechanisms of suppression of the resonant absorption appear in the scheme with HF splitting in the ground state. Apparently, in the case when homogeneous broadening of the optical transition does not overlap the HF structure, resonant driving of one of two optical transitions Žlet it be 1–2. can provide the optical pumping of atoms into the HF sublevel uncoupled to the field. As is well-known, the conditions for such optical pumping imply: the low decay rate of populations at the hyperfine transition, w, Ž w - G . and sufficiently strong driving field: < V < 2 ) wg . Similarly, optical pumping into the antisymmetric coherent superposition of the hyperfine sublevels Žso called dark or trapped state. can occur in the case of bichromatic driving field Žso called the double-lambda scheme w19x. when both optical transitions are electric-dipole allowed and < V < 2 ) gg 0 , where g 0 is the relaxation of the coherences between the HF levels. ŽLet us note that at room temperature, w < g 0 and, hence, the threshold for usual optical pumping is lower than that for coherent population trapping.. In both cases, under the condition of fast decay at the optical transition as compared to the gamma-ray transition, the small incoherent pump Ž p ) wrg in the first case and p ) g 0rg in the second case. provides gain. In the first case, this is gain with inversion at the 3–1’ transition. In the second case, gain is developed both at 3–1 and 3–1’ transitions without inversion Žbut with an effective inversion with respect to the bright state.. It is worthwhile to point out that the idea to realize gain in gamma-ray range via frequency-selective optical pumping has been discussed earlier in the literature w14x. Moreover, nuclear orientation via optical pumping was experimentally demonstrated w21x. Though the influence of the driving field on the spatial distribution of gamma-ray emission Žwhich was studied in this experiment. was very small, an in-principle possibility of modification of nuclear response by laser driving has been proven. These experiments were done in gases Žvapors of Rb. where, on one hand, homogeneous line broadening of optical transition was small, but on the other hand Doppler broadening of gamma-ray transition was large as compared to the hyperfine splitting. These conditions are just opposite to those that one has to deal with to realize frequency-selective optical pumping schemes in solids. The absence of Doppler broadening at the Mossbauer transition is very favorable for realization of optical pumping in nuclei. However, the requirement of short lifetime at the optical transition as compared to that of the nuclear transition Žwhich in turn should be of the order of 1 m s. is hardly compatible with the condition of narrow optical linewidth at the room temperature in solids Žg - D .. This problem was already pointed out earlier w14x. A similar problem was also discussed above in connection with the realization of the LWI schemes. 3.2. Degenerate double-lambda scheme In the opposite case Žg ) D . which, as has been mentioned above, is rather typical, the driving field can interact simultaneously with both 1–3 and 1’–3 transitions if they are allowed. As a result, we come to the
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
541
so-called degenerate double-lambda scheme w12,22x. The major conditions of inversionless gain in this case are the same as in the usual double-lambda scheme but the contradiction between the requirement of short decay time and small line broadening is naturally removed. Hence, at the first sight this scheme seems to be quite attractive. However, the problem is that the Zeeman degeneracy of the hyperfine sublevels Žwhich was ignored so far. unavoidably supplies some additional decay channels for the upper optical state Žstate c in Fig. 2.. In turn, these channels lead to the leakage of population from the trapped state. Let us note that the level-crossing technique w13x does not help to avoid these additional decay channels. In order to show that nothing remains in the trapped state in the presence of any additional decay channel we consider the three-level scheme depicted in Fig. 2b. The population of the upper level decays not only to the lower sublevels but can also leave the system. In order to simplify the calculations we restrict ourselves to the completely symmetric case when both transitions have exactly the same dipole moments and, therefore, the same relaxation rates. The only asymmetry is that these two transitions have slightly different frequencies due to the splitting of the ground state. For simplicity we take the frequency of the electromagnetic field to be v s Ž v ac q v b c . r2. The density matrix equations in this case can be written as d ra a dt d rcc dt d sac dt d sb c dt d sa b dt
q i Ž V )sac y Vsc a . s Grcc ,
d rbb dt
q i Ž V )s b c y Vscb . s Grcc ,
y i Ž V )sac y Vsc a . y i Ž V )s b c y Vscb . s y Ž 2 G q gout . rcc , q i Dsac q i V Ž r a a y rc c . q i Vsa b s ygsac , y i Ds b c q i V Ž r b b y rcc . q i Vs b a s ygs b c , q 2i Dsa b q i Ž V )sac y Vsc b . s 0,
where V s mopt Eoptr2" is the Rabi frequency of the optical driving field, gout is the relaxation of population out of the system. Here we neglect relaxation processes between the sublevels. In this case one should expect that the coherence between the sublevels does not decay and, therefore, the population is in the dark state. However, one can easily check, that this set of equations has only one steady-state solution when all the coherences and populations are equal to zero. Thus, the dark state is not populated, and, hence, all the population escapes from the system. The physical reason for it is that, in the case of finite HF splitting the dark state is not a time-independent eigenstate of the total Hamiltonian of the system.
Fig. 2. The dynamic ‘dark’ state is not populated if there are additional decay channels Žgout / 0..
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
542
The question arises if there is any possibility to suppress resonant absorption at the gamma-ray transition by means of laser driving of electronic transitions under the condition that hyperfine structure is completely covered by the optical linewidth. We show below that such possibility does exist.
4. Suppression of resonant gamma-ray absorption via frequency nonselective optical pumping 4.1. Nuclear orientation First, we prove that polarization-selective optical pumping of electronic states under certain conditions can provide nuclear orientation with respect to electron shell orientation Ž² L P I : / 0, where L is the angular momentum of electron, I is the nuclear spin., even when the homogeneous linewidth of the optical transition overlaps the hyperfine splitting. To be concrete, we consider the model system of an ion Žatom. with a nuclear spin, I, equal to 1r2 both in the ground and excited nuclear states, the angular momentum of electron, L, equal to 1 in the ground state and equal to 0 in an excited state and the spin of electron equal to zero both in the ground and excited states. Let us suppose that the ions are implanted into a crystal host which does not affect of the electronic shell. Then the total set of energy levels remains the same as for a free ion. Namely, the hyperfine interaction between the magnetic moments of nucleus and electron, Hh f s l L P I , produces the hyperfine splitting of the states Ž G, g . and Ž E, g . for two sublevels which are accordingly twofold and fourfold degenerate with respect to projection of total moment, m F . The energy diagram is plotted in Fig. 3. The selection rules also remain the same as for a free ion. In the case of broad optical linewidth Žgd ) D ., for
Fig. 3. The simple model of optical pumping
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
543
l l 4, 5 l 6, 5 l 7. One can easily find the
linearly polarized optical field, the allowed transitions are: 2 4, 3 wave functions of all the states involved in the scheme in Fig. 1:
C 1 :s L z s y1, Iz s y1r2 :g , 1
C 3 :s
: ' '3 Ž 0,y 1r2 y 2
C6 :s
' '3 Ž 2
1
C 2 :s
' '3 Ž 2
y 1,1r2 :. g ,
0,1r2 :q 1,y 1r2 :. g ,
1
0,y 1r2 :q y 1,1r2 :. g ,
C4 :s 0,y 1r2 :e , C 7 :s
C5 :s 0,1r2 :e ,
1
: ' '3 Ž 0,1r2 y 2
1,y 1r2 :. g ,
C5 :s 1,1r2 :g . Here indices g and e indicate either ground or excited state of electronic transition. Using these wavefunctions it is easy to calculate the dipole moments of all the transitions:
m 14 s m 85 s m ,
m 24 s m65 s m 35 s m 74 s m'2r3 ,
m 34 s m 75 s m 25 s m64 s m'1r3 .
The density matrix equations describing an interaction of an ion with a linearly polarized field can be written as: d r 11 dt d r 22 dt d r 33 dt d s 24 dt d s 34 dt d s 23 dt
s gr44 q g 0 Ž 2 Ž r 22 q r 33 . y 4 r 11 . , q i V 1 Ž s 24 y s42 . s gr44 q g 0 Ž 2 Ž r 11 q r 33 . y 4 r 22 . , q i V 2 Ž s 34 y s43 . s gr44 q g 0 Ž 2 Ž r 11 q r 22 . y 4 r 33 . ,
r 11 q r 22 q r 33 q r44 s 12 ,
q i V 1 Ž r 22 y r44 . q i V 2 s 23 s ygopt s 24 , q i Ds 34 q i V 2 Ž r 33 y r44 . q i V 2 s 32 s ygopt s 34 , q i Ds 23 q i Ž V 2 s 24 y V 1 s43 . s y5g 0 s 23 .
Here r i i is the population of level i and sm n s rm n P expŽyi v t . is the coherence at the transition m y n, where v s v 24 is the frequency of the driving field. In this set of equations we took into account the obvious symmetry of our system, namely the fact that r 11 s r 88 , r 22 s r66 , r 33 s r 77 , r44 s r 55 , s 23 s s67 , s 34 s s 75 , and s 24 s s65 . The other notations are the following: V 1 s V 2r3 , V 2 s V 1r3 , where V is the Rabi frequency of the driving field, gopt is the total linewidth including broadening of the optical line, and g s g41 is the natural linewidth of the optical transition, g 0 is the decay rate of population between the hyperfine sublevels. As has been already discussed above we assume the following relation between the relaxation rates corresponding to the typical physical situation:
'
'
gopt 4 g , D ,g 0 , g 4 g 0 . We analyze the steady-state populations of the ground levels with an increase of the driving field intensity. Particularly the interesting question is whether it is possible to achieve nuclear orientation with respect to electronic shell, i.e. whether a nonzero expectation value of the product L P I can be induced by the optical driving field. ŽObviously in the absence of the field Ž V s 0. all the populations of the ground state sublevels are
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
544
Fig. 4. The population of levels 1, 2, 3, 4 for the scheme in Fig. 3 vs. V rg . The parameters used for the numerical calculations are g s1, gopt s 50, g 0 s10y3 , D s 2P10y2 .
equal to 1r6 independently of the magnitude of the hyperfine splitting and hence ² L P I :s 0.. The answer to the last question essentially depends on the strength of the hyperfine interaction. Our numerical simulations show that in the case of weak hyperfine interaction it is impossible to set all the population into levels 1 and 8 Žsee Fig. 4. and to induce nuclear orientation at any intensity of the field. In the opposite case Ž D 4 g 0 . with an increase of the field intensity, first, the populations of levels 1 and 8 rise to a maximum value nearly equal to 1r2 Žaccordingly, all the other levels become almost empty. and then go down to some constant value Žsee Fig. 5.. When r 11 s r 88 s 1r2 one gets ² L P I :s 1r2, i.e. nuclei are oriented with respect to the electronic shell. Our numerical analysis shows that the nuclear polarization appears if V 4 g 0 gopt and the maximal polarization develops at V f Dgopt . This means that one can expect amplification at the nuclear transitions which involve the empty levels if incoherent pump is applied from the populated states to the upper states of two operating transitions as shown in Fig. 2 by the one-headed dashed arrows. Apparently, gain should occur at the gamma-ray transitions indicated by dotted two-headed arrows Žsee Fig. 2.. Under the condition g - D, the only populated levels Ž1 and 8. do not contribute to resonant absorption, since the corresponding transitions have a different frequency from the operating transitions due to the HF splitting. In conclusion, we can state that in the case of sufficiently strong HF splitting Ž D 4 g 0 . and relatively small radiative broadening of the gamma-ray transition Ž G < D . the polarized driving field of the appropriate intensity can provide gain at gamma-ray nuclear transition even when the optical linewidth overlaps the hyperfine splitting and the total population of the ground nuclear state is much less than the population of the excited nuclear state. It is worth noting that the radiative linewidth of gamma-ray transition is not required to be smaller than hyperfine splitting if circular polarized driving field is used which provides nuclear orientation with respect to the direction of the field polarization.
(
(
Fig. 5. The population of levels 1, 2, 3, 4 for the scheme in Fig. 3 vs. V rg . The parameters used for the numerical calculations are g s1, gopt s 50, g 0 s10y3 , D s 2.
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
545
4.2. Gain conditions Let us determine now what are the conditions for incoherent pump rate and relaxation constants providing population inversion and gain at gamma-ray transition by means of optical pumping. We introduce a weak unpolarized incoherent pump to populate an excited nuclear state and estimate both threshold field intensity and pump parameter providing gain at the nuclear transitions. In order to estimate the threshold parameters we restrict ourselves to consideration of the simplified four-level system shown in Fig. 6 Žtaking into account that the coherence between hyperfine sublevels vanishes in the steady state, as was shown above.. This system is described by the following set of the density matrix equations: d ra a dt d rbb dt d rd d dt d sb c dt
s G Ž 1 q p . r d d q grcc q wr b b y Ž w q p G . r a a , q i Ž s b c V ) y scb V . s G Ž 1 q p . r d d q grcc q wr a a y Ž w q p G . r b b , s p G Ž r a a q r b b . y 2 G Ž 1 q p . r d d , r a a q r b b q rcc q r d d s 1, q i V Ž r b b y rcc . s ygopt s b c ,
where a s mg Egr2 " is the Rabi frequency of the probe field. Solving the set of equations in the steady-state regime, we obtain the population difference between the upper and the lower states at the probe transition
D n s rd d y r b b s
V 2 Ž p Ž g y G . y 2 w . y ggopt Ž G p q 2 w . V 2 Ž g Ž 3 p q 2 . q Ž 4 p q 3 . Ž G p q 2 w . . q ggopt Ž 3 p q 2 . Ž G p q 2 w .
.
First of all, let us set w s 0 to get the zeroth order approximation. In the case g ) G population inversion of the probe transition can be obtained for any value of the pumping parameter p if the intensity of the driving field exceeds some critical value 2
V s
ggopt G gyG
.
In the opposite case g - G , population inversion cannot be achieved by means of optical driving.
Fig. 6. The simple model of optical pumping.
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
546
Now, we take into account the relaxation processes between the lower sublevels a : and b :. In this case the conditions mentioned above must be satisfied. Moreover, even in the case g ) G there is a critical value of the pumping parameter p ) pcrit s
2
2 V q ggopt
ž
/
2
V Ž g y G . y ggopt G
w
which must be exceeded in order to get population inversion. In the limit of very strong driving field 2 V 4 ggopt , the pumping parameter should obey ) p ) pcrit s 2 wr Ž g y G . . ) Now, we estimate pcrit for the typical parameters of real atoms. Let us consider the Mossbauer nuclear 7 y1 transition with G s 10 s , the atomic transition with g s 3 P 10 7 sy1 , and with the relaxation rate between hyperfine sublevels w s 10 3 sy1 . In this case, as one can see, the population inversion can be achieved for p ) 5 P 10y5 , which corresponds to 0.005% nuclei excited. The corresponding intensity of the driving field can be estimated as
I 4 Icrit s c
2 Eopt
4p
c
2"
s 4p mopt
2
gGgopt gyG
s
4 Ggopt
Ž " vopt .
3p g y G
Ž "c .
2
3
.
We take the energy of the optical transition to be " vopt s 1 eV and the typical inhomogeneous linewidth of the electro-dipole allowed optical transition in solid is gopt s 10 12 sy1 . The critical intensity in this case is Icrit s 160 Wrcm2 . Such intensities can be easily achieved by modern lasers.
5. Conclusion Strong suppression of resonant absorption Žby 3–5 orders of magnitude. allows for an in-principle resolution of the gamma-ray laser dilemma, namely for a realization of incoherent pump Žproviding net resonant gain. without destroying of the conditions of the Mossbauer and Borrmann effects. The last two effects are needed to provide the predominance of the net resonant gain over the off-resonant absorption. This resolution is based on the reduced requirement to the power of incoherent pump which is due to: Ži. the suppression of resonant absorption up to the level of off-resonant absorption in the lattice Žii. the realization of the incoherent pump directly at the operating gamma-ray transition. The suppression of resonant absorption at gamma-ray nuclear transition of the active ions in crystal lattice can be provided by the coherent optical driving of electronic transitions of these ions due to the presence of hyperfine coupling between the nuclear and electronic degrees of freedom. In principle, this can be achieved in two different ways. The first one is based on the frequency selective driving of some of the hyperfine sublevels. It implies a large hyperfine splitting compared to the line broadening of the driven optical transition. Frequency selective driving can provide suppression of resonant absorption Žand hence net resonant gain at very low level of incoherent pump. either due to atomic interference in the inversionless lambda, V and double-lambda schemes, or due to depletion of the ground state in the optical pumping scheme. The realization of all these schemes implies a small life time of the optical transition as compared to the gamma-ray transition. As a result, the necessity to use the short-lived nuclear transitions Žof the order of m s. limits the life-time of the optical transitions, 1rg - 1 m s, implying usage of electro-dipole allowed transitions. At sufficiently high temperature Žwhich is due to the heating of the sample by the incoherent pump. such transitions are typically strongly broadened by means of the optical phonons. This makes it very difficult to satisfy a requirement of narrow optical linewidth.
R. KolesoÕ et al.r Optics Communications 179 (2000) 537–547
547
On the other hand, there is a possibility to suppress the resonant absorption at gamma-ray nuclear transition by means of coherent optical driving even when optical linewidth covers the hyperfine structure. This second way can be achieved either via interference mechanism in inversionless degenerate double lambda scheme or depopulation of ground state by means of polarization selective optical pumping. Unfortunately, additional decay channels which are typically present in the system Žfrom the excited electronic state out of two sublevels forming double-lambda scheme. make it difficult to use interference mechanism. At the same time, a realization of frequency non-selective optical pumping is the most promising for suppression of resonant absorption and realization of the induced gain at gamma-ray transition.
Acknowledgements We thank Alex Belyanin and Chris Bednar for stimulating and helpful discussions, and Chris Bednar for his help in preparation of the paper. This work was supported by the Office of Naval Research, and Texas Advance Technology Program. One of us ŽYu.R.. thanks the North Atlantic Treaty Organization for its support.
References w1x O. Kocharovskaya, Ya.I. Khanin, Pis’ma Zh. Eksp. Teor. Fiz. 48 Ž1988. 581 wJETP Lett. 48, 630, 1988x; S.E. Harris, Phys. Rev. Lett. 62 Ž1989. 1033; M.O. Scully, S.-Y. Zhu, A. Gavrielides, Phys. Rev. Lett. 62 Ž1989. 2813. w2x O. Kocharovskaya, Phys. Rep. 219 Ž1992. 175; M.O. Scully, Phys. Rep. 219 Ž1992. 191; E. Arimondo, in: E. Wolf ŽEd.., Progress in Optics, Elsevier Science, Amsterdam, 1996; S.E. Harris, Phys. Today 50 Ž1997. 36; see also a special issue on LWI Laser Physics 9 Ž1999. 745, O. Kocharovskaya, P. Mandel, M. Scully ŽEds... w3x R. Coussement et al., Phys. Rev. Lett. 71 Ž1993. 1824. w4x O. Kocharovskaya, Laser Physics 5 Ž1995. 284; O. Kocharovskaya, Hyperfine interaction 107 Ž1997. 187. w5x O. Kocharovskaya, in: J.Y. Gao, S.Y. Zhu ŽEds.., Proc. International Symposium on Atomic Coherence and Inversionless Optical Amplification, Changchun, China, 1995. w6x R. Kolesov, Y. Rostovtsev, O. Kocharovskaya, Photon Echo and Coherent Spectroscopy, V.V. Samartsev ŽEd.., Proc. SPIE 3239 Ž1998. 422. w7x R.N. Shakhmuratov, Opt. Spect. 84 Ž1998. 724. w8x Y. Rostovtsev, R. Kolesov, O. Kocharovskaya, in: I.I. Popalescu, C.A. Ur ŽEds.., Proc. First International Induced Gamma Emission Workshop, IGE Foundations, 1999, p. 222. w9x A. Nottelman et al., Phys. Rev. Lett. 70 Ž1993. 1783; E.S. Fry et al., Phys. Rev. Lett. 70 Ž1993. 3235; W.E. Van der Veer et al., Phys. Rev. Lett. 70 Ž1993. 3243; J.A. Kleinfeld, A.D. Streater, Phys. Rev. A 53 Ž1996. 1839; C. Fort et al., Opt. Commun. 139 Ž1997. 31. w10x A.S. Zibrov et al., Phys. Rev. Lett. 75 Ž1995. 1499; G.G. Padmabandu et al., Phys. Rev. Lett. 76 Ž1996. 2053. w11x O. Kocharovskaya, R. Kolesov, Yu. Rostovtsev, Phys. Rev. Lett. 82 Ž1999. 3593. w12x O. Kocharovskaya, R. Kolesov, Yu. Rostovtsev, Laser Phys. 9 Ž1999. 745. w13x R.N. Shakhmuratov, G. Kozyreff, R. Coussement, J. Odeurs, P. Mandel, Opt. Commun. 179 Ž2000. 523. w14x G.C. Baldwin, J.C. Solem, Rev. Mod. Phys. 69 Ž1997. 1085; G.C. Baldwin, J.C. Solem, V.I. Goldanski, Rev. Mod. Phys. 53 Ž1981. 667. w15x V.I. Vysotskii, R.N. Kuzmin, Gamma-ray Lasers. Moscow State University Press, Moscow, 1989. w16x L. Rivlin, Quantum Electronics 29 Ž1999. 467. w17x S. Olariu, J.J. Carrol, C.B. Collins, Europhys. Lett. 37 Ž1997. 177. w18x V. Letokhov, Inst. Phys. Conf. Ser. No 151, Section 13 Ž1996. 483. w19x O. Kocharovskaya, P. Mandel, Phys. Rev. A 42 Ž1990. 523; O. Kocharovskaya, F. Mauri, E. Arimondo, Opt. Commun. 84 Ž1991. 393. w20x S.F. Yelin, M.D. Lukin, M.O. Scully, P. Mandel, Phys. Rev. A 57 Ž1999. 3858. w21x G. Shimkaveg, W.W. Quivers, R.R. Dasari, C.H. Holbrow, P.G. Pappas, M.A. Attlili, J.E. Thomas, D.E. Murnick, M.S. Feld, Phys. Rev. Lett. 53 Ž1984. 2230. w22x C.H. Keitel, O.A. Kocharovskaya, L.M. Narducci, M.O. Scully, S.-Y. Zhu, H.M. Doss, Phys. Rev. A 48 Ž1993. 3196.
