J. Phys. B: At. Mol. Phys. 17 (1984) 4577-4594. Printed in Great - PDF document

J. Phys. B: At. Mol. Phys. 17 (1984) 4577-4594. Printed in Great Britain of a configuration for S Reynaud Cohen-Tannoudji Spectroscopie Hertzienne I'Ecole Normale Supérieure and Collège de France, 24 rue Lhomond, 7523 1 Received 30 u+ and u- show that such configuration looks than ordinary standing the two counterpropagating waves have same polarisa- tion), and we explain this result as calculation of the various coefficient, diffusion coefficient) mean radiative forces experienced, such laser configuration, with a 1 interest possibility of optical trap (Letokhov et al 1977, Letokhov and Minogin Ashkin 1978): the radiation two counterpropagating dipole add and lead atoms. Unfortunately, such a scheme can photon in the counterpropagating then stimulated photon wave; it thus momentum 2 Since the number such not limited, 4578 J Dalibard, S Reynaud and C orthogonal polarisations, a+ and a- (figure l(a)). For the sake sim- we will (excited level) atomic transition l(b)). It is then obvious that such a scheme does not allow coherent redistribu- processes between after the absorption of a for instance, the atom is in the (e, m = +1) level and, because momentum conservation, emit a a- the other le-) le,) Laser configuration this paper: superposition of two focused a+ and a- (with eventually 'infinite' beam waists: optical 1 transition, a+ and a- waves. In § 2 of this Our notations and Our hypothesis. We then radiative forces (§ 3.1), moving with u ($8 Section 4 implications of work concerning § numerical estimations The total Hamiltonian is sum of HA is atomic Hamiltonian, HF The laser is supposed to be in a coherent state, so that we can. treat it .as a * then two parts (Mollow 1975), the first one (VA-,) the second VA-,) coupling; the quantised taken in its ground state. v atomic Hamiltonian the sum energy of of its introduced the Ib) internal atomic States. In equation (2.2), we have only taken into account le) the ground Ig) (J = O) by the laser light, C Cohen-Tannoudji same the other resulting polarisation E' is linear and rotates along the Oz axis. In order to describe the atom-laser coupling, it is introduce the basis for the atomic excited state equation (2.10)): that the atomic dipole reads: where d is the electric dipole moment of the transition. can finally be written with these the particular case of 'plane o+ and o- waves': this means that the waists of the two o+ and o- very large can dependence on 8(R). As usual, al1 the calculations will be done the rotating-wave keeping only the resonant In this the semiclassical the atomic operator its average is the spatial the atomic is much smaller than the other we will the notion OF atomic-velocity and will, in consider at rest, i.e. with a velocity spread Av such that the kAv the natural r: $ ~AV« r. (2.18) One then immediately notes that the two conditions are compatible Heisenberg inequality: « hr. condition for the semiclassical calculation means recoil energy the natural in other Potentialities of a new laser conjiguration 458 1 atom is still the laser photon absorption that when fulfilled, one can show (Cohen-Tannoudji 1983, Dalibard and Cohen-Tannoudji the forces diffusion coefficient appearing describing the motion, in a fully quantum treatment, are the same the semiclassical ones, which we will calculate here. 3. Radiative forces in the u+-u- usual in semiclassical theory radiative forces (see, example, Cook 1980a, Gordon and 1980), we will calculate these forces in define a force operator time derivative of the atomic momentum: This force can be terms of (2.1), (2.5), (2.12) and (2.15), we get: with Fpo1 = $i d$(~)k(O~..- O,.,) Fv,, results quantised-field-atom coupling, Knt and Fpol both come from the laser-atom coupling; more precisely, Fin, and Fpol are respectively due to the R dependence of VA-, via $(R) and via Q,,,- Q,,,. As mentioned above, a semiclassical operator average value interested in interna1 forces on r,, it is clear from the expression of the laser-atom coupling (2.15) that only the sublevel le') (at point r,) is state. It O)-(J = 1) two-level system. Potentialities of a new laser conjiguration 458 1 atom is still the laser photon absorption that when fulfilled, one can show (Cohen-Tannoudji 1983, Dalibard and Cohen-Tannoudji the forces diffusion coefficient appearing describing the motion, in a fully quantum treatment, are the same the semiclassical ones, which we will calculate here. 3. Radiative forces in the u+-u- usual in semiclassical theory radiative forces (see, example, Cook 1980a, Gordon and 1980), we will calculate these forces in define a force operator time derivative of the atomic momentum: This force can be terms of (2.1), (2.5), (2.12) and (2.15), we get: with Fpo1 = $i d$(~)k(O~..- O,.,) Fv,, results quantised-field-atom coupling, Knt and Fpol both come from the laser-atom coupling; more precisely, Fin, and Fpol are respectively due to the R dependence of VA-, via $(R) and via Q,,,- Q,,,. As mentioned above, a semiclassical operator average value interested in interna1 forces on r,, it is clear from the expression of the laser-atom coupling (2.15) that only the sublevel le') (at point r,) is state. It O)-(J = 1) two-level system. given by optical Bloch equations: perer + pgg = 1 (3.6d) a have put 6 w, - wO. stationary solution these we get: f: = -'fis- int 2 1 +s where we have introduced the r-dependent to note that this force exactly equal the usual dipole gradient force) found for two-level system. to the fpol and fVac. The result for fpo, is immediate: since the level le'? (at point ro) not coupled the laser the average value of Qge., using (3.36): fpol = o. (3.9) In order to calculate Lac, we first that the quantised can of view, Erre, +EsOurce, where EsOurce is the field radiated by the atom, and Erre, that would the absence a source. Since no gradient the atom position (Tanguy since the quantised field is taken initially in its ground state. It follows that: fvac = O (3.1 1) a so that average force is then An,. using this force atoms, it is useful this force derives fram the potential: U(r) = $fi6 In(1 and to power and waist, i.e. for a given maximal field amplitude GP,, the optimal given by optical Bloch equations: perer + pgg = 1 (3.6d) a have put 6 w, - wO. stationary solution these we get: f: = -'fis- int 2 1 +s where we have introduced the r-dependent to note that this force exactly equal the usual dipole gradient force) found for two-level system. to the fpol and fVac. The result for fpo, is immediate: since the level le'? (at point ro) not coupled the laser the average value of Qge., using (3.36): fpol = o. (3.9) In order to calculate Lac, we first that the quantised can of view, Erre, +EsOurce, where EsOurce is the field radiated by the atom, and Erre, that would the absence a source. Since no gradient the atom position (Tanguy since the quantised field is taken initially in its ground state. It follows that: fvac = O (3.1 1) a so that average force is then An,. using this force atoms, it is useful this force derives fram the potential: U(r) = $fi6 In(1 and to power and waist, i.e. for a given maximal field amplitude GP,, the optimal the optimal slowly the atom is moving with u, in this order radiative forces, u+-u- configuration. It is clear that f,,, still zero in its but the fi',, and f,,, are now changed, since the atomic motion implies a time %(r(t)) and Q,,, - Q,,,. a coupling quite atom case. order to calculate the average values . of Q,,,, Q,.,, . . appearing in the force, general Heisenberg off,,,. to the operator Q,., and taking we get: we have we have used p,.,.: We now want to extract from equations (3.17) and (3.20), values of p,,. and pet,., to order in velocity. these least of order one in u, since they are zero for an atom at rest. Consequently, the time derivatives of these stationary values are at least of order the time derivative of a stationary quantity is indeed (v V)A. The first-order velocity depen- dences the stationary p,,. and p,,,. can then be obtained from: d%(r) (i8+$~)p,e.--p,.,-= -k. v d%/2h 1 2h i6+$r l+s d%(r) S pgew+ rpege. = k 0- 2h 2(l+ s)' the optimal slowly the atom is moving with u, in this order radiative forces, u+-u- configuration. It is clear that f,,, still zero in its but the fi',, and f,,, are now changed, since the atomic motion implies a time %(r(t)) and Q,,, - Q,,,. a coupling quite atom case. order to calculate the average values . of Q,,,, Q,.,, . . appearing in the force, general Heisenberg off,,,. to the operator Q,., and taking we get: we have we have used p,.,.: We now want to extract from equations (3.17) and (3.20), values of p,,. and pet,., to order in velocity. these least of order one in u, since they are zero for an atom at rest. Consequently, the time derivatives of these stationary values are at least of order the time derivative of a stationary quantity is indeed (v V)A. The first-order velocity depen- dences the stationary p,,. and p,,,. can then be obtained from: d%(r) (i8+$~)p,e.--p,.,-= -k. v d%/2h 1 2h i6+$r l+s d%(r) S pgew+ rpege. = k 0- 2h 2(l+ s)' 4584 J Dalibard, S Reynaud and C finally solve to get fpol, to the order in velocity: Calculation off;,,. Applying (3.16) to Q,,,, Q,.,, Q,,,, and Q,,, one gets the same result as (3.6), to the first order in v. The supplementary terms resulting from the component aA/at in (3.16) are indeed al1 equal to k. o average value operator where e" appearst. These supplementary terms are then at least of order not contribute. It that the v system giving perg, pge,, pefe, and p,, is due to 8(r(t)), and consequently, An, at first order in velocity 4. is the same atom case. found to be (see also Ashkin 1980) R6 Vs(o. Vs) f: =-'fia--- int 2 l)r " ). 1 +S 4r ~(1 +s)~ 6 +:r2 at the can 'crossed' term as (k. o)Vs or k(o. total force. This physical invariance changing k -k which is a+ We will f;,, (3.23) since it is a well known expression, and we will rather focus on fpol. We first fpol can be a or an force depending sign of the detuning example, this agrees following qualitative 'sees', because the Doppler a- wave closer to resonance than a+, and then 'experiences' a total force left which motion. In case where a+ and a- are plane progressive waves, it interesting to no change fpol intensity of waves varies. This has obtained for plane standing sign of light intensity constant detuning (Minogin and Serimaa 1979, Minogin 1981a, In this appears as 'simpler' system than standing-wave configuration. 4 neutral atoms, it may qualities of fpol. the optimal * S-1 6 = -$ damping force: can for the vaiidity terms of this section that the t Equation (3.6d) should be p,.,.+p,.,..+p, = 1, but one can easily show that p,.,.. is at least order two in v. To order one, one can then use p,.,. +pK, = 1. 4584 J Dalibard, S Reynaud and C finally solve to get fpol, to the order in velocity: Calculation off;,,. Applying (3.16) to Q,,,, Q,.,, Q,,,, and Q,,, one gets the same result as (3.6), to the first order in v. The supplementary terms resulting from the component aA/at in (3.16) are indeed al1 equal to k. o average value operator where e" appearst. These supplementary terms are then at least of order not contribute. It that the v system giving perg, pge,, pefe, and p,, is due to 8(r(t)), and consequently, An, at first order in velocity 4. is the same atom case. found to be (see also Ashkin 1980) R6 Vs(o. Vs) f: =-'fia--- int 2 l)r " ). 1 +S 4r ~(1 +s)~ 6 +:r2 at the can 'crossed' term as (k. o)Vs or k(o. total force. This physical invariance changing k -k which is a+ We will f;,, (3.23) since it is a well known expression, and we will rather focus on fpol. We first fpol can be a or an force depending sign of the detuning example, this agrees following qualitative 'sees', because the Doppler a- wave closer to resonance than a+, and then 'experiences' a total force left which motion. In case where a+ and a- are plane progressive waves, it interesting to no change fpol intensity of waves varies. This has obtained for plane standing sign of light intensity constant detuning (Minogin and Serimaa 1979, Minogin 1981a, In this appears as 'simpler' system than standing-wave configuration. 4 neutral atoms, it may qualities of fpol. the optimal * S-1 6 = -$ damping force: can for the vaiidity terms of this section that the t Equation (3.6d) should be p,.,.+p,.,..+p, = 1, but one can easily show that p,.,.. is at least order two in v. To order one, one can then use p,.,. +pK, = 1. C Cohen-Tannoudji following system, deduced from (3.20) by taking complex conjugates (3.20) in is quite tedious but does not present have put N= 0.5 ïS(k- u)[r2+4(k. ~)~](d8/h)~ 9 = [T2 +4(k. u)~]{Q +$(d81h)~[4~~ +Z(d8/h)' +4(k. u)~]) +k(d8/h)"[A2 +;(dg/ - 3(k. O)'] A2 = S2 +:ï2 +Q(d%/f~)~ Q = [A' - (k- v),]~ + (k- ~)~r,. ratio of polynomials in k. the numerator is of degree the denominator can (3.3 1) is of the previous section interest of formula (3.31) is show that given light radiative force has same sign as S(k.v). Once more, can where a of the (Minogin and Serimaa 1979). We have plotted the variation radiative force optimal case S = -fr and s = 1. One sees that the force is linear until k-u * becomes of the order of ï, reaches a maximum order of 0.2 hkr and then decreases. This means previous section is dominant in the range -$ s k- u s fr, as expected. Note: In order to understand why the atom-laser system studied in this section appears simpler than for the consider dressed-atom picture (Cohen-Tannoudji the case of in a Ig, n,, n,), the atom presence of n, and n, photons in the two counterpropagating waves forming the standing wave, is to the le, n, - 1, n,) and le, n,, n2- 1). These two states Ig, n, - 1, n2+ 1) and Ig, n, + 1, n2- and so 4(a)). analytical expression the force could be obtained the standing-wave C Cohen-Tannoudji following system, deduced from (3.20) by taking complex conjugates (3.20) in is quite tedious but does not present have put N= 0.5 ïS(k- u)[r2+4(k. ~)~](d8/h)~ 9 = [T2 +4(k. u)~]{Q +$(d81h)~[4~~ +Z(d8/h)' +4(k. u)~]) +k(d8/h)"[A2 +;(dg/ - 3(k. O)'] A2 = S2 +:ï2 +Q(d%/f~)~ Q = [A' - (k- v),]~ + (k- ~)~r,. ratio of polynomials in k. the numerator is of degree the denominator can (3.3 1) is of the previous section interest of formula (3.31) is show that given light radiative force has same sign as S(k.v). Once more, can where a of the (Minogin and Serimaa 1979). We have plotted the variation radiative force optimal case S = -fr and s = 1. One sees that the force is linear until k-u * becomes of the order of ï, reaches a maximum order of 0.2 hkr and then decreases. This means previous section is dominant in the range -$ s k- u s fr, as expected. Note: In order to understand why the atom-laser system studied in this section appears simpler than for the consider dressed-atom picture (Cohen-Tannoudji the case of in a Ig, n,, n,), the atom presence of n, and n, photons in the two counterpropagating waves forming the standing wave, is to the le, n, - 1, n,) and le, n,, n2- 1). These two states Ig, n, - 1, n2+ 1) and Ig, n, + 1, n2- and so 4(a)). analytical expression the force could be obtained the standing-wave Potentialities laser configuration the intensity the radiative force (8 = -Lr and s = 1). Figure 4. Dressed-atom picture. (a) Standing-wave case: al1 states corresponding given total number Coherent redistribution can occur. (b) 0,-u- case: manifolds dressed states. coherent redistribu- can occur. In particular, this coupling between al1 the states with a given value of n, +n2 for the redistribution described Consider the case a+ and a- waves. The state Ig, n+, n-) to the le+, n+- 1, n-) and le-, n+, n- - l), and these to the Ig, n+, n-) (figure 4(b)), due to the then contains manifolds involving much compute; one can in particular state, when taken laser frequencies 11 4. Atomic momentum diffusion in the O+-O- the introduction, the is a crucial 7 motion in a light wave, since the corresponding limits in this section a+-a- configuration, one can diffusion coefficient, a standing-wave done at point r interested in the trace (Gordon and 1980a) Potentialities laser configuration the intensity the radiative force (8 = -Lr and s = 1). Figure 4. Dressed-atom picture. (a) Standing-wave case: al1 states corresponding given total number Coherent redistribution can occur. (b) 0,-u- case: manifolds dressed states. coherent redistribu- can occur. In particular, this coupling between al1 the states with a given value of n, +n2 for the redistribution described Consider the case a+ and a- waves. The state Ig, n+, n-) to the le+, n+- 1, n-) and le-, n+, n- - l), and these to the Ig, n+, n-) (figure 4(b)), due to the then contains manifolds involving much compute; one can in particular state, when taken laser frequencies 11 4. Atomic momentum diffusion in the O+-O- the introduction, the is a crucial 7 motion in a light wave, since the corresponding limits in this section a+-a- configuration, one can diffusion coefficient, a standing-wave done at point r interested in the trace (Gordon and 1980a) 4588 J Dalibard, S Reynaud and C Cohen-Tannoudji 4.1. Outline of the calculation In order to compute D, the operators their expression will give nine contributions to D. Actually, al1 terms vanish left with only the various the cancellation Consider terms involving Fvac. One has then take the ground state the single operator VE,(O) or VEi(r). Far VEi(0), that the since only a sum annihilation operators VEi(r), on the other hand is equal to VEi free(~) (see 5 3.1), and one can note that VEi free(r) cornmutes any atomic operator taken at time O, so that terms involving this single-field operator when averaged in vacuum state. then left with crossed terms Fint and Fpol. Physically, it is clear that such terms cannot contribute to D, since they would be proportional k- Vs (see 5 not be a+ and a- Mathematically, they be proportional correlation functions such as and it is to show, using the Qetg(7) and QerSg(r) given in 5 applying the sbch always zero more details, 5 4.3). We have then calculate three terms: Dvac (5 4.2), Dint (5 4.3) and Dpol us remark Dvac and Dint very close the corresponding two-level system. then just briefly outline the explicit algebra for these two terms, referring example Askin (1980) more details. D,,, We first value of (Fvac(0) .Fvac(r) + HC). (Note again that only Ei free(~) contributes to FVac(7); 5 quantised-field part Fvac(0)-Fva,(r) operators. The come the antinormal order (a, on the left, a: on the 4588 J Dalibard, S Reynaud and C Cohen-Tannoudji 4.1. Outline of the calculation In order to compute D, the operators their expression will give nine contributions to D. Actually, al1 terms vanish left with only the various the cancellation Consider terms involving Fvac. One has then take the ground state the single operator VE,(O) or VEi(r). Far VEi(0), that the since only a sum annihilation operators VEi(r), on the other hand is equal to VEi free(~) (see 5 3.1), and one can note that VEi free(r) cornmutes any atomic operator taken at time O, so that terms involving this single-field operator when averaged in vacuum state. then left with crossed terms Fint and Fpol. Physically, it is clear that such terms cannot contribute to D, since they would be proportional k- Vs (see 5 not be a+ and a- Mathematically, they be proportional correlation functions such as and it is to show, using the Qetg(7) and QerSg(r) given in 5 applying the sbch always zero more details, 5 4.3). We have then calculate three terms: Dvac (5 4.2), Dint (5 4.3) and Dpol us remark Dvac and Dint very close the corresponding two-level system. then just briefly outline the explicit algebra for these two terms, referring example Askin (1980) more details. D,,, We first value of (Fvac(0) .Fvac(r) + HC). (Note again that only Ei free(~) contributes to FVac(7); 5 quantised-field part Fvac(0)-Fva,(r) operators. The come the antinormal order (a, on the left, a: on the 4590 J Dalibard, S Starting from optical (3.6) giving evolution of (A(T)), appears that take into account between the, four following This set of coupled equations same case, but since the population lg) can be fed through the decay of levels le") and leo). last correlation of (4.16) (A(0)QerOe.(~)) and (A(O)Q,,(T)). evolution of these two quantities I is very initial values T = 0) are both zero. It follows that these for any T, so that the diffusion coefficient through the (4.16), has the same mathematical expression (Gordon and 1980a, is, however, order of magnitude of Dint is not at al1 the same 1) atom for a two-level system in a same for the Uo s (see standing-wave case (VS)~ is indeed of the order of = 47r2s2/,12, in the (vs)'= (s/ w,)~ where wo diffusion coefficient Dint in the then 47r*(~~/,l)~ smaller than in a corresponding reduction can be considerable much larger than interpretation for such reduction has the introduction paper: the redistribu- \ two beams, responsible the huge diffusion coefficient found in can then only occur inside beam, between various forming this are at the the gradient S. In particular, if there is only one wavevector forming the a+ or a- beam (plane a+ and a- waves, Vs coherent redistribution can occur and Dint vanishes. 4.4. Calculation of D,,, report the Fpo, in the expression Dpo, and we get: D =- PO~ d2k2g2 4 Re Iom dr (B(O)B(r)) 4590 J Dalibard, S Starting from optical (3.6) giving evolution of (A(T)), appears that take into account between the, four following This set of coupled equations same case, but since the population lg) can be fed through the decay of levels le") and leo). last correlation of (4.16) (A(0)QerOe.(~)) and (A(O)Q,,(T)). evolution of these two quantities I is very initial values T = 0) are both zero. It follows that these for any T, so that the diffusion coefficient through the (4.16), has the same mathematical expression (Gordon and 1980a, is, however, order of magnitude of Dint is not at al1 the same 1) atom for a two-level system in a same for the Uo s (see standing-wave case (VS)~ is indeed of the order of = 47r2s2/,12, in the (vs)'= (s/ w,)~ where wo diffusion coefficient Dint in the then 47r*(~~/,l)~ smaller than in a corresponding reduction can be considerable much larger than interpretation for such reduction has the introduction paper: the redistribu- \ two beams, responsible the huge diffusion coefficient found in can then only occur inside beam, between various forming this are at the the gradient S. In particular, if there is only one wavevector forming the a+ or a- beam (plane a+ and a- waves, Vs coherent redistribution can occur and Dint vanishes. 4.4. Calculation of D,,, report the Fpo, in the expression Dpo, and we get: D =- PO~ d2k2g2 4 Re Iom dr (B(O)B(r)) 4592 J Dalibard, S Reynaud and C pure random p space. the atomic during a time AT can be written where N+ (N- respectively) is photons absorbed a+ (a- respec- tively) wave, hki the momentum of the ith fluoresc~e photon. The mean value of Ap is zero, since N+ = fi- and = O. The variance is: where-we have used the fact that the momentum hki given fluorescence not correlated to the other nor to N+ and N-. diffusion coefficient can be identified with Dvac: carried away (N+ + N-/ T is identical to the rate of emission X in (4.1 The second term of (4.28), corresponding to Dpol, describes the fluctuations of the difference N+ - N- photons absorbed is correlation between (a+ after the a+ photon for example, the atom has the same absorbing either a a+ photon or a a- one. It P(N+, N-) finding a realisation (N+, N-) simply related P(N+ + N-) a total N+ + N- emitted photons: then deduces from (4.29): (N+- N-)'= N++ N-. Consequently, Dpol is equal to Dvac (see (4.1 1) and (4.25)).. result (4.30) is same as if N+ and N- were independent Poisson C the case since: I expresses the non-Poisson photons, has the same for the 1979, 1981, 1981b, Reynaud 1983, Stenholm 1983). 5. Implications for optical presented in laser configuration, very the trapping neutral atoms. calc.ulated the optical depth for 4592 J Dalibard, S Reynaud and C pure random p space. the atomic during a time AT can be written where N+ (N- respectively) is photons absorbed a+ (a- respec- tively) wave, hki the momentum of the ith fluoresc~e photon. The mean value of Ap is zero, since N+ = fi- and = O. The variance is: where-we have used the fact that the momentum hki given fluorescence not correlated to the other nor to N+ and N-. diffusion coefficient can be identified with Dvac: carried away (N+ + N-/ T is identical to the rate of emission X in (4.1 The second term of (4.28), corresponding to Dpol, describes the fluctuations of the difference N+ - N- photons absorbed is correlation between (a+ after the a+ photon for example, the atom has the same absorbing either a a+ photon or a a- one. It P(N+, N-) finding a realisation (N+, N-) simply related P(N+ + N-) a total N+ + N- emitted photons: then deduces from (4.29): (N+- N-)'= N++ N-. Consequently, Dpol is equal to Dvac (see (4.1 1) and (4.25)).. result (4.30) is same as if N+ and N- were independent Poisson C the case since: I expresses the non-Poisson photons, has the same for the 1979, 1981, 1981b, Reynaud 1983, Stenholm 1983). 5. Implications for optical presented in laser configuration, very the trapping neutral atoms. calc.ulated the optical depth for