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Undamped two-level atoms driven by a coherent electromagnetic wave
UNDAMPED
10 August 1987
PHYSIC8 LETTERS A
Volume 123, number 5
TWO-LEVEL
ATOMS DRIVEN BY A COHERENT
ELECTROMAGNETIC
WAVE
Stanislaw P. PRAJSNAR Institute ofPhysics, University of Szczecin, ui. Wielkopolska IS. 70-451 Szczecin, Poland Received 13 May 1987; accepted for publication 8 June 1987 Communicated by B. Fricke
The cooperative interaction of a small sample of two-level atoms with external coherent electromagnetic field is considered. The incoherent interactions including spontaneous emission are neglected. The P-representation for the atomic density operator has been found for an arbitrary number of atoms. The average values of the atomic variables are calculated.
1. Introduction
Since the famous paper of Dicke [I], a great deal of attention has been given to the subject of interaction of two-level systems with electromagnetic fields. Special initerest is directed towards the models for which exact solutions exist. The simplest possible system is the undamped two-level atom, excited by a one-mode classical electromagnetic field. A characteristic feature of the solution is the Rabi oscillations of the inversion and probability amplitudes
[VI. In a more realistic semiclassical treatment the decay of states is taken into account by adding damping terms. In the literature there are many theoretical papers and experimental works devoted to the transient effects in the sample of two-level atoms (see for example ‘refs. [ 3,4 ] and the literature quoted therein). The cooperative interaction of atoms with a coherent field has been extensively studied in the last several years [ $61. Recently the interaction of atoms with the f&Id in a cavity has been considered and the oscillations with a cooperative Rabi frequency have been observed [ 7,8]. The aim of our work is to study some dynamical and statistical properties of N atoms driven by a monochromatic wave. For such a simple model the density operator and the average values can be found. In order to simplify the calculations we neglect all incoherent interactions.
2. The P-representation for the density operator
We consider the interaction of a small sample of two-level atoms with an external monochromatic wave. The hamiltonian for the system in the dipole and the rotating-wave approximation is given by H=fio,S,
+gS_ exp(io&)
+g*S+ exp( -iWot) , (1)
where g= io,e*dAo. The collective spin operators S,, S+, S_ satisfy the well-known angular momentum commutation relations. The quantum statistical and dynamical properties of atoms are described by the density operator, which satisfies the following equation in the interaction picture (see for example ref. [91), @~'"'ldt(t)=fO[S_ exp(-tit) -S+ exp( i&), pi*‘{t)] ,
(2)
where 8=2w,&*&-’ is the Rabi frequency and the detuning d=o,oo. It can be shown that ( Sz( t) ) is a constant of motion and therefore the cooperation number S is conserved. It allows the introduction of the weight function P(8, q,t) defined by the diagonal representation in the basis of atomic coherent states [ IO] : p’“‘(t)=
J~~mu,t)~e,twe,B,~,(3)
where
037%9601/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
241
dp=sint7 df3 dp,, 0<47<27~,
0<8
If we have the weight function P, we can easily calculate the average values of the atomic variables,
10 August 1987
PHYSICS LETTERS A
Volume 123. number 5
>
where (Y’= 1+p* and B = AlQ. The Cauchy problem for (6) is formulated in the following way: Find a function P(e, 9, t) of class C’ ( t 2 0) satisfying eq. (6) and the initial condition p(e, P, 0) =po(e, V) .
(9)
We assume that P,,(8, q) is known. If P(0, rp, t) is taken in the form
XX,, ewt-iwAS3t) 10,P >
(4)
,
p(6, V, t)=p,(u,(~r,
with i,, .... in=+, -, 3. In order to derive the equation for P we have to know how the spin opeerators S,, S_, act on I8,
where
w4
u2(‘l/, >v/2)
VI wi:
w2), u~(w,, VIM)),
u,(v/,, ~2)=arccos[(~2-B~~)/~21
(10)
,
s- ie,~xe,tPi =exp( -iqr)[Ssin -titg(fe)ala(~]
=i*na(o12-~:-~:)“2+i(W1
e-sin2(~e)alae ie,~)(e,~o
po(6, q) =lEoc,o V(e, P) ,
aPfat=8cos(dt+~)aPlae
c,~=(l,s,o,
ctg(e)apia~ .
(6)
The integration of eq. (6) can be done with the help of the method of characteristics [ 121, i.e., we should integrate the equivalent set of ordinary differential equations --=dt 1
de Qcos(At+p)
-dg, = Qsin(At+(p)
ctg6 ’
v, =sin(8) sin(At+p) -p cos(6) ,
+icu sin(e) cos(At+q)] 242
sin(At+p) exp(iaQt) +c.c. ,
(l-2)
where cn, are given by -S(S,
-S)
=(-l)Y-‘(S&M, ~(2S+1)“~(4x)-“~
-MIZ,O) (13)
and (j, ,j,, m,, m2 IJH) denote the Clebsch-Gordan coefficients.
(7)
This set has two independent solutions, which also satisfy eq. (6),
~/,=~[cos(@+Bsin(B)
(11)
(5)
.
On substituting (3) and (5) into (2), and on integrating by parts, we obtain a homogeneous, linear, partial differential equation of the first order,
-Qsin(At+~)
’
then condition (9) will be fulfilled. Thus the function (10) is a solution of the Cauchy problem. In the above consideration we have not yet specified the function Po( 8, cp). In order to simplify further calculations we assume that the atomic system is initially in the Dicke state IS, M). According to ref. [ lo] we have
,
+lictg(te)a/a~]ie,yl)(e,~i
+BW2)
[~4-(W2-PWI)21”2
(8)
3. Results and conchsions We assume now that the atoms are initially in their ground state IS, -S) , and the cooperation number S=N/2. Substituting (11) into (12) we obtain
PHWCS
Volume 123, number 5
P(&
Q,
t)
I’* P/(X) , =,&.(21+ 1> 4n
LETTERS A
10 August 1987
(14)
where P,(x) are the Legendre polynomials and x=(~~-/?~,)la*. Eqs. (14), (4) enable us to find the mean value$ and fluctuations. The average inversion per atom is equal to (&(t))/N=
- [/3’ +cos(cuS2z)]/2a2 .
(15)
This result is timewell-known Rabi oscillation solution [ 31. The inversion reaches the maximum value l/2 in the case ofexact resonance. For inversion flue tuations we obtain
(s+(t)&!_(t))
= sin*(oQf)+2/3*[1-cos(oQt)] 4N04
(16)
One can see that the fluctuations are decreasing if N is increasing, and are equal to zero when the inversion is minimum. In the resonance case they vanish also for the maximum inversion. The dipole moment per atom is equal to (D(t))lN=d(B[l-cos(aQf)]sin(c.~) --(Y sin(aQr)cos(w0t)}la2
.
(17)
The dispersive component (in phase with the incident electric field E(t) N sin( wet)) is equal to zero in the case of exact resonance. The absorption component (out of phase) reaches the maxima values in the resonance. The cooperation number S appears as a scale factor. The dipole moment fluctuations per atom are given by ((D*)
- (0)*)/N*
=d*lN-d*{a
Fig. 1. The time development of (S, ( l)S_ (i) ) (per one atom), for N= 1,2,3 in the case of exact resonance.
= [ 1 -cos(al(at)]/2a2
(S+(t)S_(t))=[5+8f12-(4+8/I*)
)
cos(a52t)
-cos(2a52t)]/4a4,
- 3 cos( 2aQt)]/4a4
.
(19)
Fig. 1 shows the time development of the results (19) when A=O. The curves in fig. 1 illustrate the coop erative behaviour of two-levels atoms in the process of absorption and re-emission of radiation. These results reveal a characteristic feature: there are oscillations with the Rabi frequency a9 when one atom is present, and oscillations with the additional frequency 2aL.J in the case of many atoms. This holds true for arbitrary number N> 1 [ 131. The curve for three atoms has two maxima, which arise when the probability of the system being in the 1312, I/2) state is greatest. In addition we can calculate the fluctuations
sin(aL&)cos(oOt)
-/3[ 1-cos(aGV)]sin(o,,t)}2/N~4.
(18)
This result depends on N like the previous one in eq. (16). The fluctuations are greatest when the dipole moment tends to zero. As another apfilication of the solution (14) let us consider the mean values (S+(t)S_(t)), which describe the collective absorption of radiation by atoms. For one, two, three atoms we have respectively
= [ 1 -cos(aRt)]*/4N04
,
(20)
and it is seen that they depend on N like in (16) and (18). Eqs. (16), (20) can be compared with the results of ref. [ 61 where a similar dependence on N has been obtained for the steady-state fluctuations in the phenomenon of cooperative resonance fluorescence. All our results are in fact oscillatory and do not reach the steady state because of the absence of 243
Volume 123, number 5
PHYSICS LETTERS A
damping terms in the equations of motion. It is connected with the neglecting of the incoherent interactions, in particular the coupling of the atoms with the empty modes of the electromagnetic field. However, this simplified treatment gives a solvable model and informations about the atomic dynamics.
References [ I] R.H. Dicke, Phys. Rev. 93 (1954) 99. [2] M. Sargent, M.O. Scully and W.E. Lamb Jr., Laser physics (Addison-Wesley, Reading, 1974). [ 31 L. Allen and J.H. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975). [4] R.L. Shoemaker, in: Laser and coherence spectroscopy ed. J.I. Steinfeld (Plenum, New York, 1978) p. 197. [ 51L.M. Narducci, M. Orszag and R.A. Tuft, Phys. Rev. A 8 (1973) 1892; G.S. Agarwal, A.C. Brown, L.M. Narducci and G. Vetri, Phys. Rev.A 15 (1977) 1613; G. Compagno and F. Persico, Phys. Rev. A 15 (1977) 2032; C. Mavroyannis, Phys. Rev. A 18 (1978) 185; R. Bonifacio and L.A. Lugiato, Phys. Rev. A 18 ( 1978) 1129;
244
10 August 1987
A.S. Amin and J.G. Cordes, Phys. Rev. A 18 (1978) 1298; H.S. Freedhoff, Phys. Rev. A 19 (1979) 1132; H.J. Carmichael, Phys. Rev. Lett. 43 (1979) 1106; P.D. Drummond and S.S. Hassan, Phys. Rev. A 22 (1980) 662; D.F. Walls, J. Phys. B 13 (1980) 2001; Z. Ficek, R. TanaS and S. Kielich, Phys. Rev. A 29 (1984) 2004. [ 61 G.S. Agarwal, L.M. Narducci, D.H. Feng and R. Gilmore, Phys. Rev. Lett. 42 (1979) 1260. [ 71 Y. Kaluzny, P. Goy, M. Gross, J.M. Raimond and S. Haroche, Phys. Rev. Lett, 51 (1983) 1175. [ 81 A. Crubellier and D. Mayou, Opt. Commun. 50 (1984) 157. [ 91 H. Haken, Handbuch der Physik, Vol. XXV/ZC (Springer, Berlin, 1970). [ lo] F.T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, Phys. Rev.A6(1972)2211. [ 111 L.M. Narducci, in: Cooperative effects, ed. H. Haken (North-Holland, Amsterdam, 1974) p. 119. [ 121 I.N. Snedden, Elements of partial differential equations (McGraw-Hill, New York, 1957). [ 131 S.P. Prajsnar, in preparation.
10 August 1987
PHYSIC8 LETTERS A
Volume 123, number 5
TWO-LEVEL
ATOMS DRIVEN BY A COHERENT
ELECTROMAGNETIC
WAVE
Stanislaw P. PRAJSNAR Institute ofPhysics, University of Szczecin, ui. Wielkopolska IS. 70-451 Szczecin, Poland Received 13 May 1987; accepted for publication 8 June 1987 Communicated by B. Fricke
The cooperative interaction of a small sample of two-level atoms with external coherent electromagnetic field is considered. The incoherent interactions including spontaneous emission are neglected. The P-representation for the atomic density operator has been found for an arbitrary number of atoms. The average values of the atomic variables are calculated.
1. Introduction
Since the famous paper of Dicke [I], a great deal of attention has been given to the subject of interaction of two-level systems with electromagnetic fields. Special initerest is directed towards the models for which exact solutions exist. The simplest possible system is the undamped two-level atom, excited by a one-mode classical electromagnetic field. A characteristic feature of the solution is the Rabi oscillations of the inversion and probability amplitudes
[VI. In a more realistic semiclassical treatment the decay of states is taken into account by adding damping terms. In the literature there are many theoretical papers and experimental works devoted to the transient effects in the sample of two-level atoms (see for example ‘refs. [ 3,4 ] and the literature quoted therein). The cooperative interaction of atoms with a coherent field has been extensively studied in the last several years [ $61. Recently the interaction of atoms with the f&Id in a cavity has been considered and the oscillations with a cooperative Rabi frequency have been observed [ 7,8]. The aim of our work is to study some dynamical and statistical properties of N atoms driven by a monochromatic wave. For such a simple model the density operator and the average values can be found. In order to simplify the calculations we neglect all incoherent interactions.
2. The P-representation for the density operator
We consider the interaction of a small sample of two-level atoms with an external monochromatic wave. The hamiltonian for the system in the dipole and the rotating-wave approximation is given by H=fio,S,
+gS_ exp(io&)
+g*S+ exp( -iWot) , (1)
where g= io,e*dAo. The collective spin operators S,, S+, S_ satisfy the well-known angular momentum commutation relations. The quantum statistical and dynamical properties of atoms are described by the density operator, which satisfies the following equation in the interaction picture (see for example ref. [91), @~'"'ldt(t)=fO[S_ exp(-tit) -S+ exp( i&), pi*‘{t)] ,
(2)
where 8=2w,&*&-’ is the Rabi frequency and the detuning d=o,oo. It can be shown that ( Sz( t) ) is a constant of motion and therefore the cooperation number S is conserved. It allows the introduction of the weight function P(8, q,t) defined by the diagonal representation in the basis of atomic coherent states [ IO] : p’“‘(t)=
J~~mu,t)~e,twe,B,~,(3)
where
037%9601/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
241
dp=sint7 df3 dp,, 0<47<27~,
0<8
If we have the weight function P, we can easily calculate the average values of the atomic variables,
10 August 1987
PHYSICS LETTERS A
Volume 123. number 5
>
where (Y’= 1+p* and B = AlQ. The Cauchy problem for (6) is formulated in the following way: Find a function P(e, 9, t) of class C’ ( t 2 0) satisfying eq. (6) and the initial condition p(e, P, 0) =po(e, V) .
(9)
We assume that P,,(8, q) is known. If P(0, rp, t) is taken in the form
XX,, ewt-iwAS3t) 10,P >
(4)
,
p(6, V, t)=p,(u,(~r,
with i,, .... in=+, -, 3. In order to derive the equation for P we have to know how the spin opeerators S,, S_, act on I8,
where
w4
u2(‘l/, >v/2)
VI wi:
w2), u~(w,, VIM)),
u,(v/,, ~2)=arccos[(~2-B~~)/~21
(10)
,
s- ie,~xe,tPi =exp( -iqr)[Ssin -titg(fe)ala(~]
=i*na(o12-~:-~:)“2+i(W1
e-sin2(~e)alae ie,~)(e,~o
po(6, q) =lEoc,o V(e, P) ,
aPfat=8cos(dt+~)aPlae
c,~=(l,s,o,
ctg(e)apia~ .
(6)
The integration of eq. (6) can be done with the help of the method of characteristics [ 121, i.e., we should integrate the equivalent set of ordinary differential equations --=dt 1
de Qcos(At+p)
-dg, = Qsin(At+(p)
ctg6 ’
v, =sin(8) sin(At+p) -p cos(6) ,
+icu sin(e) cos(At+q)] 242
sin(At+p) exp(iaQt) +c.c. ,
(l-2)
where cn, are given by -S(S,
-S)
=(-l)Y-‘(S&M, ~(2S+1)“~(4x)-“~
-MIZ,O) (13)
and (j, ,j,, m,, m2 IJH) denote the Clebsch-Gordan coefficients.
(7)
This set has two independent solutions, which also satisfy eq. (6),
~/,=~[cos(@+Bsin(B)
(11)
(5)
.
On substituting (3) and (5) into (2), and on integrating by parts, we obtain a homogeneous, linear, partial differential equation of the first order,
-Qsin(At+~)
’
then condition (9) will be fulfilled. Thus the function (10) is a solution of the Cauchy problem. In the above consideration we have not yet specified the function Po( 8, cp). In order to simplify further calculations we assume that the atomic system is initially in the Dicke state IS, M). According to ref. [ lo] we have
,
+lictg(te)a/a~]ie,yl)(e,~i
+BW2)
[~4-(W2-PWI)21”2
(8)
3. Results and conchsions We assume now that the atoms are initially in their ground state IS, -S) , and the cooperation number S=N/2. Substituting (11) into (12) we obtain
PHWCS
Volume 123, number 5
P(&
Q,
t)
I’* P/(X) , =,&.(21+ 1> 4n
LETTERS A
10 August 1987
(14)
where P,(x) are the Legendre polynomials and x=(~~-/?~,)la*. Eqs. (14), (4) enable us to find the mean value$ and fluctuations. The average inversion per atom is equal to (&(t))/N=
- [/3’ +cos(cuS2z)]/2a2 .
(15)
This result is timewell-known Rabi oscillation solution [ 31. The inversion reaches the maximum value l/2 in the case ofexact resonance. For inversion flue tuations we obtain
(s+(t)&!_(t))
= sin*(oQf)+2/3*[1-cos(oQt)] 4N04
(16)
One can see that the fluctuations are decreasing if N is increasing, and are equal to zero when the inversion is minimum. In the resonance case they vanish also for the maximum inversion. The dipole moment per atom is equal to (D(t))lN=d(B[l-cos(aQf)]sin(c.~) --(Y sin(aQr)cos(w0t)}la2
.
(17)
The dispersive component (in phase with the incident electric field E(t) N sin( wet)) is equal to zero in the case of exact resonance. The absorption component (out of phase) reaches the maxima values in the resonance. The cooperation number S appears as a scale factor. The dipole moment fluctuations per atom are given by ((D*)
- (0)*)/N*
=d*lN-d*{a
Fig. 1. The time development of (S, ( l)S_ (i) ) (per one atom), for N= 1,2,3 in the case of exact resonance.
= [ 1 -cos(al(at)]/2a2
(S+(t)S_(t))=[5+8f12-(4+8/I*)
)
cos(a52t)
-cos(2a52t)]/4a4,
- 3 cos( 2aQt)]/4a4
.
(19)
Fig. 1 shows the time development of the results (19) when A=O. The curves in fig. 1 illustrate the coop erative behaviour of two-levels atoms in the process of absorption and re-emission of radiation. These results reveal a characteristic feature: there are oscillations with the Rabi frequency a9 when one atom is present, and oscillations with the additional frequency 2aL.J in the case of many atoms. This holds true for arbitrary number N> 1 [ 131. The curve for three atoms has two maxima, which arise when the probability of the system being in the 1312, I/2) state is greatest. In addition we can calculate the fluctuations
sin(aL&)cos(oOt)
-/3[ 1-cos(aGV)]sin(o,,t)}2/N~4.
(18)
This result depends on N like the previous one in eq. (16). The fluctuations are greatest when the dipole moment tends to zero. As another apfilication of the solution (14) let us consider the mean values (S+(t)S_(t)), which describe the collective absorption of radiation by atoms. For one, two, three atoms we have respectively
= [ 1 -cos(aRt)]*/4N04
,
(20)
and it is seen that they depend on N like in (16) and (18). Eqs. (16), (20) can be compared with the results of ref. [ 61 where a similar dependence on N has been obtained for the steady-state fluctuations in the phenomenon of cooperative resonance fluorescence. All our results are in fact oscillatory and do not reach the steady state because of the absence of 243
Volume 123, number 5
PHYSICS LETTERS A
damping terms in the equations of motion. It is connected with the neglecting of the incoherent interactions, in particular the coupling of the atoms with the empty modes of the electromagnetic field. However, this simplified treatment gives a solvable model and informations about the atomic dynamics.
References [ I] R.H. Dicke, Phys. Rev. 93 (1954) 99. [2] M. Sargent, M.O. Scully and W.E. Lamb Jr., Laser physics (Addison-Wesley, Reading, 1974). [ 31 L. Allen and J.H. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975). [4] R.L. Shoemaker, in: Laser and coherence spectroscopy ed. J.I. Steinfeld (Plenum, New York, 1978) p. 197. [ 51L.M. Narducci, M. Orszag and R.A. Tuft, Phys. Rev. A 8 (1973) 1892; G.S. Agarwal, A.C. Brown, L.M. Narducci and G. Vetri, Phys. Rev.A 15 (1977) 1613; G. Compagno and F. Persico, Phys. Rev. A 15 (1977) 2032; C. Mavroyannis, Phys. Rev. A 18 (1978) 185; R. Bonifacio and L.A. Lugiato, Phys. Rev. A 18 ( 1978) 1129;
244
10 August 1987
A.S. Amin and J.G. Cordes, Phys. Rev. A 18 (1978) 1298; H.S. Freedhoff, Phys. Rev. A 19 (1979) 1132; H.J. Carmichael, Phys. Rev. Lett. 43 (1979) 1106; P.D. Drummond and S.S. Hassan, Phys. Rev. A 22 (1980) 662; D.F. Walls, J. Phys. B 13 (1980) 2001; Z. Ficek, R. TanaS and S. Kielich, Phys. Rev. A 29 (1984) 2004. [ 61 G.S. Agarwal, L.M. Narducci, D.H. Feng and R. Gilmore, Phys. Rev. Lett. 42 (1979) 1260. [ 71 Y. Kaluzny, P. Goy, M. Gross, J.M. Raimond and S. Haroche, Phys. Rev. Lett, 51 (1983) 1175. [ 81 A. Crubellier and D. Mayou, Opt. Commun. 50 (1984) 157. [ 91 H. Haken, Handbuch der Physik, Vol. XXV/ZC (Springer, Berlin, 1970). [ lo] F.T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, Phys. Rev.A6(1972)2211. [ 111 L.M. Narducci, in: Cooperative effects, ed. H. Haken (North-Holland, Amsterdam, 1974) p. 119. [ 121 I.N. Snedden, Elements of partial differential equations (McGraw-Hill, New York, 1957). [ 131 S.P. Prajsnar, in preparation.