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Effects of intense radiation fields on interatomic forces
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
EFFECTS OF INTENSE RADIATION FIELDS ON INTERATOMIC FORCES P. CARBONARO and F. PERSICO Isrituto di Fisjca deli’ Università and Gruppo di Fisica Teorica del Consiglio Nazionale delle Ricerche, Palermo, Italy Received 29 November 1979
The screening, by a strongradiation field, of the interatomic forces in a pair of two-level atoms is studied. The form and the intensity of these forces are obtained as functions of the detuning between the field frequency and the atomic characteristic frequency, and of the field intensity. For a particular “magic” number of radiation photons in the field, some of these interactions vanish effectively, while others are transformed in characteristic ways.
The interaction of an isolated two-level atom with strong electromagnetic fields in the optical range has been a subject of interest in the last few years, especially in connection with the experimental [1] and theoretical [2] aspects ofthe phenomenon of resonance fluorescence. More recently, the attention has shifted towards the manyatom problem [3]. Strictly related to the latter is of course the problem of interatomic forces [41(dipolar, exchange or similar) which tend to modify the dynamics of the interacting atoms immersed in the strong driving field. The effect of this field on the form and on the intensity of the interaction forces, however, has not hitherto been considered. Modifications of the interatomic forces are nonetheless to be expected, as is particularly evident if one looks at the problem from the point of view of dressed atoms [5]. In fact continuous absorption and emission processes in a strong electromagnetic field dress the isolated atom, thereby changing many of its properties. Consequently, it seems likely that also changes in the interatomic forces between dressed atoms are brought about by the strong driving field. On the other hand, the introduction of a simple canonical transformation has provided us with a relatively agile technique for dressing two-level atom operators by the photons of a strong field [61.This opens up the possibility of treating the dressing of interatomic forces in strong fields by the mentioned canonical transformation. It should also be noted that in our approach the radiation field is treated quantum-mechanically, so that our conclusions should be susceptible of generalization to noncoherent states of the field itself. In this paper we wish to report the main results of a theoretical investigation along these lines. Consider a pair of identical two-level atoms 1 and 2, interacting with a monochromatic radiation field of frequency w in the neighbourhood of the energy difference c~(h = 1) between the upper and lower level of each atom, and of wavelength much larger than the interatomic distance. The hamiltonian of the pair in the rotating wave approximation (RWA) is taken as + V, where 1 +S~2), J7 e[(c~~+ atSl) + (aS~+ atS2)J. (1) 9i~= +w 0(5~ Here the a Bose operators pertain to the electromagnetic field mode, while superscript 1 or 2 on the atomic operators S~(1 = +, or z in the pseudospin formalism) refers to a particular atom of the pair. e is the atom—field coupling constant. We add to (1) hamiltonian ~12 which represents interatomic forces of electric or magnetic origin, acting also in the absence of the o-field. We treat these forces in the “secular approximation” (SA), which consists [71of discarding all terms of the form S~S~ and S~S’~ (i #/). These terms connect states of ‘~odiffering in energy by at least ~ so that the assumption beneath SA is that the shifts and splittings caused by 9(12 on the eigenvalues of 9~oare much smaller than con. The total hamiltonian of the pair is thus —
(2) 37
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
As we have shown in a previous paper [6], hamiltonian (I) can be taken to diagonal form, in the limit of intense field (large eigenvalue n of at a), by the canonical transformation (3) T exp {—O(4~)Z)—’I2[(aSk atS1) + (aS~ atS2)]}, —
—
where =
ata + S~1+ S~+ 1 ,
sin 0
=
—2e Xl/2/~
cos 0
=
(w
—
Transformation (3) leads to dressed operators according to the rule
0)/I~, ~
=
[(&
“~‘o)
[~&~+ &t~1)+ (~ + &t~2)]!~)Z—1/2 sin 0 + =~ + + S~)(1 cos 0) + [~& + &t~~1) + + &t~2)]1 c)~—1/2sin0 + ~(C)Z_1) (aSk + atS1) + (aS~+ atS2) = [(& + &t.~)+ (~3 + &t~2)] cos 0+ (~ + ) 2 C)~1/2~m 0 where any dressed operator A is related to its bare counterpart by the relationship S~1+ S~2=
+ ~) cos 0
2+ 4e29t] l/2~
w
(4)
—
—
+ ~(9Z—1)
(5)
(6)
A=TAT-’
and Ô(9Z _1) indicates operators of order 9Z1 with respect to 1. We shall neglect these operators, since we are interested in the limit of large occupation numbers n of the radiation field. Substituting eqs. (4) and (5) into (1) we obtain the dressed atom hamiltonian in diagonal form (7) We also have to transform 9(12 in order to obtain the complete hamiltoman (2) in the dressed atom representation. Because of the SA, 9(12 is a linear combination of the two operators S~1S~2 and (S~.S~ + S~S2÷). We transform these by eqs. (3) and (6), extending the SA to the dressed atom picture, that is neglecting nonsecular dressed atom terms of the form S~S~ and S~S~ (1 r#j). Again, this is justified if the shifts and splittings induced by 9(12 on the eigenvalues of the dressed hamiltonian (7) are much smaller than the energy differences between contiguous levels of (7). Within this approximation, and neglecting terms O( C)z—112) in the limit of intense radiation, we get S~1S~ = ~(3cos20 1) ~ —
(S~S~ + S’S3)
=
+ ~sin20 ~
~(3cos20 l)(~~2 +~ —
+S1S~)]
+~
+ 2sin~0~
+
+
~
(8)
We shall now consider separately different forms of 9(12 corresponding to different types of interactions between the atoms of the pair. (i) When the interatomic forces are of the dipole—dipole type and both atoms have a dipole moment also in the upper and lower state [8] then 9(12 9(~= 2 ~ i-S1S~)]. (9) 7[S~S~ This expression for the dipolar interaction is valid in the ~ is the constant of the dipolar forces. When both atoms are located in the (x, y) plane, 7 ‘S given by —
7.tip
(10)
2/r~2 ,
where jz~and ~2 are the atomic dipole moments and r12 is the interatomic distance. Using eq. (8) in (9) we obtain immediately 9(d =~(3cos2O 1)7L~—~(~S~ +~~)1 (11) We see that transformation (3) preserves the form of 9(d’ only reducing it by a factor —
~(3cos20—1). 38
.
(12)
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
Reduction factor (12) depends on the detuning ~ w~and on the field amplitude as in eqs. (4), and it may reduce to zero for appropriate values of these variables. Taking the average value of 9~on the state of the total system to be about n,and using expression (4) for cos 0, leads us to defining a “magic” number of photons ~M for each detuning, such that (13) oo)2I[(~. “o)2 + ~~2~M] = —
—
(i.’
—
For this magic photon number the effective dipolar interaction between the dressed atoms ofthe pair is seen to vanish. (ii) The Heisenberg exchange in the SA is represented by 9(12 gç~=J[S!S~2 + ~(S~S~+S1S~)]. (14) This form of interaction has the remarkable property ofbeing invariant (within our approximations) under transformation (3), leading to exactly the same hamiltonian (14), only with dressed atomic operators. (iii) In the case of Ising exchange 9(12 = 9(~ JS~1S~. (15) From the first of eqs. (8), Ising exchange can be immediately seen to transform into a form of exchange intermediate between Ising and Heisenberg. For the magic case, interaction (15) is transformed into a pure Heisenberg form. (iv) As for the XY exchange in the SA, 9(l2m 9(xy(S~s~~53). (16) Thus the second of eqs. (8) yields again an exchange form intermediate between XY and Heisenberg, reducing to pure Heisenberg for ~ = (v) Finally, we consider the case in which the atoms of the pair have no permanent electric dipole moment in their ground and excited states, but may develop a “transverse” dipole moment in the (x, y) plane. Then term S~1S~2 is absent, and 9(12 in the SA reduces to [4] g~12 9(~— 7’(S4S~+S~S~), ~‘~‘1 .pj2/r?2 ~ .r12)(j&2 r12)/r~2. (17) —
This form is identical to (16) within our approximations and its reduction, under magic conditions, to pure Heisenberg exchange follows in the same way as in (iv). We now wish to draw some conclusions from the above results and also to emphasize some of the main implications. Diagonalization by dressing transformation (3) of the hamiltonian Of an atomic pair subject to near resonant and strong irradiation, evidences screening effects of the interatomic forces by the radiation field. This leads to the introduction of a magic photon number ~M such that, when the average photon number in the radiation field is ~ important changes take place in the nature of the screened interaction. As for the dipolar interaction in (i), its vanishing under magic conditions confirms the effects found both theoretically and experimentally in paramagnetic systems at NMR [9]and ESR [10]frequencies; the present results extend this effect to the region of noncoherent fields and relate it to the dressed atom concept. Moreover, the marked differences which exist in many respects among magnetic systems coupled by different exchange interactions [11], induce one to think that near resonant irradiation in appropriate experimental conditions might modify their behaviour in more than one observable way [12]. Finally, the behaviour of the dipolar interaction in (17) under the dressing transformation. (3) seems to suggest that appropriate irradiation might transforma set of two-level atoms interacting via nonpermanent dipole— dipole forces into a sort of effectively ferromagnetic-like system. If this is the case, it might be interesting to speculate on the possible meaning of the Curie temperature and of the phase transition associated with it. The authors of this work wish to acknowledge partial support by Comitato Regionale Ricerche Nucleari e di Struttura della Materia. 39
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
References [1] F. Schuda, C.R. Stroud and M. Hercher, J. Phys. B7 (1974) L198; H. Waither, in: Frontiers in laser spectroscopy, eds. R. Balian, S. Haroche and S. Liberman (North-Holland, Amsterdam, 1977) p. 125; R.E. Grove, F.Y. Wu and S. Ezekiel, Phys. Rev. A15 (1977) 227. [2] P.A. Apanasevich, Opt. Spectr. 16 (1964) 387; B.R. Mollow, Phys. Rev. 188 (1969) 1969; H.J. Kimble and L. Mandel, Phys. Rev. A13 (1976) 2123. [3] G.S. Agarwal, A.C. Brown, L.M. Narducci and G. Vetri, Phys. Rev. A15 (1977) 1613; A.S. Jahangir Amin and J.G. Cordes, Phys. Rev. A18 (1978) 1298; Y.C. Lee, C.Y. Chen and Y. Shan, J. Phys. C12 (1979) 1429; G.S. Agarwal, L.M. Narducci, D.H. Feng and R. Gilmore, Phys. Rev. Lett. 42 (1979) 1260. [4] C. Mavroyannis, Phys. Rev. A18 (1978) 185; H.S. Freedhoff, Phys. Rev. A19 (1979) 1132. [5] C. Cohen-Tannoudji and S. Reynaud, J. Phys. BlO (1977) 345. [6] P. Carbonaro, C. Compagno and F. Persico, Phys. Lett. A73 (1979) 97. [7] J.H. Van Vleck, Phys. Rev. 74 (1948) 1168. [8] O.G. Mouritsen and S.J. Knak Jensen, Phys. Rev. B18 (1978) 465. [9] W.I. Goldburg and M. Lee, Phys. Rev. Lett. 11(1963) 255; D. Barnaal and I.J. Lowe, Phys. Rev. Lett. 11(1963) 258; C.S. Yannoni and H.M. Vieth, Phys. Rev. Lett. 37 (1976) 1230; H.T. Stokes and D.C. Ailion, Phys. Rev. Lett. 38 (1977) 1159. [10] K. Tornita, Prog. Theor. Phys. 19(1958)541; M.W.P. Strandberg, Phys. Rev. B6 (1972) 747; R. Boscaino, M. Brai, I. Ciccarello and M.W.P. Strandberg, Phys. Rev. B7 (1973) 50. [11] J. Oitmaa and D.D. Betts, Phys. Lett. 68A (1978) 450. [12] L.P. Regnault, P. Burlet and J. Rossat-Mignod, Physica B86—88 (1977) 661; H.A. Aigra, L.J. De Jongh and WJ. Huiskamp, Physica B86—88 (1977) 737; A.T. Skjeltorp, Physica B86—88 (1977) 1295; D.R. Taylor, Phys. Rev. Lett. 42 (1979) 1302.
40
PHYSICS LETTERS
3 March 1980
EFFECTS OF INTENSE RADIATION FIELDS ON INTERATOMIC FORCES P. CARBONARO and F. PERSICO Isrituto di Fisjca deli’ Università and Gruppo di Fisica Teorica del Consiglio Nazionale delle Ricerche, Palermo, Italy Received 29 November 1979
The screening, by a strongradiation field, of the interatomic forces in a pair of two-level atoms is studied. The form and the intensity of these forces are obtained as functions of the detuning between the field frequency and the atomic characteristic frequency, and of the field intensity. For a particular “magic” number of radiation photons in the field, some of these interactions vanish effectively, while others are transformed in characteristic ways.
The interaction of an isolated two-level atom with strong electromagnetic fields in the optical range has been a subject of interest in the last few years, especially in connection with the experimental [1] and theoretical [2] aspects ofthe phenomenon of resonance fluorescence. More recently, the attention has shifted towards the manyatom problem [3]. Strictly related to the latter is of course the problem of interatomic forces [41(dipolar, exchange or similar) which tend to modify the dynamics of the interacting atoms immersed in the strong driving field. The effect of this field on the form and on the intensity of the interaction forces, however, has not hitherto been considered. Modifications of the interatomic forces are nonetheless to be expected, as is particularly evident if one looks at the problem from the point of view of dressed atoms [5]. In fact continuous absorption and emission processes in a strong electromagnetic field dress the isolated atom, thereby changing many of its properties. Consequently, it seems likely that also changes in the interatomic forces between dressed atoms are brought about by the strong driving field. On the other hand, the introduction of a simple canonical transformation has provided us with a relatively agile technique for dressing two-level atom operators by the photons of a strong field [61.This opens up the possibility of treating the dressing of interatomic forces in strong fields by the mentioned canonical transformation. It should also be noted that in our approach the radiation field is treated quantum-mechanically, so that our conclusions should be susceptible of generalization to noncoherent states of the field itself. In this paper we wish to report the main results of a theoretical investigation along these lines. Consider a pair of identical two-level atoms 1 and 2, interacting with a monochromatic radiation field of frequency w in the neighbourhood of the energy difference c~(h = 1) between the upper and lower level of each atom, and of wavelength much larger than the interatomic distance. The hamiltonian of the pair in the rotating wave approximation (RWA) is taken as + V, where 1 +S~2), J7 e[(c~~+ atSl) + (aS~+ atS2)J. (1) 9i~= +w 0(5~ Here the a Bose operators pertain to the electromagnetic field mode, while superscript 1 or 2 on the atomic operators S~(1 = +, or z in the pseudospin formalism) refers to a particular atom of the pair. e is the atom—field coupling constant. We add to (1) hamiltonian ~12 which represents interatomic forces of electric or magnetic origin, acting also in the absence of the o-field. We treat these forces in the “secular approximation” (SA), which consists [71of discarding all terms of the form S~S~ and S~S’~ (i #/). These terms connect states of ‘~odiffering in energy by at least ~ so that the assumption beneath SA is that the shifts and splittings caused by 9(12 on the eigenvalues of 9~oare much smaller than con. The total hamiltonian of the pair is thus —
(2) 37
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
As we have shown in a previous paper [6], hamiltonian (I) can be taken to diagonal form, in the limit of intense field (large eigenvalue n of at a), by the canonical transformation (3) T exp {—O(4~)Z)—’I2[(aSk atS1) + (aS~ atS2)]}, —
—
where =
ata + S~1+ S~+ 1 ,
sin 0
=
—2e Xl/2/~
cos 0
=
(w
—
Transformation (3) leads to dressed operators according to the rule
0)/I~, ~
=
[(&
“~‘o)
[~&~+ &t~1)+ (~ + &t~2)]!~)Z—1/2 sin 0 + =~ + + S~)(1 cos 0) + [~& + &t~~1) + + &t~2)]1 c)~—1/2sin0 + ~(C)Z_1) (aSk + atS1) + (aS~+ atS2) = [(& + &t.~)+ (~3 + &t~2)] cos 0+ (~ + ) 2 C)~1/2~m 0 where any dressed operator A is related to its bare counterpart by the relationship S~1+ S~2=
+ ~) cos 0
2+ 4e29t] l/2~
w
(4)
—
—
+ ~(9Z—1)
(5)
(6)
A=TAT-’
and Ô(9Z _1) indicates operators of order 9Z1 with respect to 1. We shall neglect these operators, since we are interested in the limit of large occupation numbers n of the radiation field. Substituting eqs. (4) and (5) into (1) we obtain the dressed atom hamiltonian in diagonal form (7) We also have to transform 9(12 in order to obtain the complete hamiltoman (2) in the dressed atom representation. Because of the SA, 9(12 is a linear combination of the two operators S~1S~2 and (S~.S~ + S~S2÷). We transform these by eqs. (3) and (6), extending the SA to the dressed atom picture, that is neglecting nonsecular dressed atom terms of the form S~S~ and S~S~ (1 r#j). Again, this is justified if the shifts and splittings induced by 9(12 on the eigenvalues of the dressed hamiltonian (7) are much smaller than the energy differences between contiguous levels of (7). Within this approximation, and neglecting terms O( C)z—112) in the limit of intense radiation, we get S~1S~ = ~(3cos20 1) ~ —
(S~S~ + S’S3)
=
+ ~sin20 ~
~(3cos20 l)(~~2 +~ —
+S1S~)]
+~
+ 2sin~0~
+
+
~
(8)
We shall now consider separately different forms of 9(12 corresponding to different types of interactions between the atoms of the pair. (i) When the interatomic forces are of the dipole—dipole type and both atoms have a dipole moment also in the upper and lower state [8] then 9(12 9(~= 2 ~ i-S1S~)]. (9) 7[S~S~ This expression for the dipolar interaction is valid in the ~ is the constant of the dipolar forces. When both atoms are located in the (x, y) plane, 7 ‘S given by —
7.tip
(10)
2/r~2 ,
where jz~and ~2 are the atomic dipole moments and r12 is the interatomic distance. Using eq. (8) in (9) we obtain immediately 9(d =~(3cos2O 1)7L~—~(~S~ +~~)1 (11) We see that transformation (3) preserves the form of 9(d’ only reducing it by a factor —
~(3cos20—1). 38
.
(12)
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
Reduction factor (12) depends on the detuning ~ w~and on the field amplitude as in eqs. (4), and it may reduce to zero for appropriate values of these variables. Taking the average value of 9~on the state of the total system to be about n,and using expression (4) for cos 0, leads us to defining a “magic” number of photons ~M for each detuning, such that (13) oo)2I[(~. “o)2 + ~~2~M] = —
—
(i.’
—
For this magic photon number the effective dipolar interaction between the dressed atoms ofthe pair is seen to vanish. (ii) The Heisenberg exchange in the SA is represented by 9(12 gç~=J[S!S~2 + ~(S~S~+S1S~)]. (14) This form of interaction has the remarkable property ofbeing invariant (within our approximations) under transformation (3), leading to exactly the same hamiltonian (14), only with dressed atomic operators. (iii) In the case of Ising exchange 9(12 = 9(~ JS~1S~. (15) From the first of eqs. (8), Ising exchange can be immediately seen to transform into a form of exchange intermediate between Ising and Heisenberg. For the magic case, interaction (15) is transformed into a pure Heisenberg form. (iv) As for the XY exchange in the SA, 9(l2m 9(xy(S~s~~53). (16) Thus the second of eqs. (8) yields again an exchange form intermediate between XY and Heisenberg, reducing to pure Heisenberg for ~ = (v) Finally, we consider the case in which the atoms of the pair have no permanent electric dipole moment in their ground and excited states, but may develop a “transverse” dipole moment in the (x, y) plane. Then term S~1S~2 is absent, and 9(12 in the SA reduces to [4] g~12 9(~— 7’(S4S~+S~S~), ~‘~‘1 .pj2/r?2 ~ .r12)(j&2 r12)/r~2. (17) —
This form is identical to (16) within our approximations and its reduction, under magic conditions, to pure Heisenberg exchange follows in the same way as in (iv). We now wish to draw some conclusions from the above results and also to emphasize some of the main implications. Diagonalization by dressing transformation (3) of the hamiltonian Of an atomic pair subject to near resonant and strong irradiation, evidences screening effects of the interatomic forces by the radiation field. This leads to the introduction of a magic photon number ~M such that, when the average photon number in the radiation field is ~ important changes take place in the nature of the screened interaction. As for the dipolar interaction in (i), its vanishing under magic conditions confirms the effects found both theoretically and experimentally in paramagnetic systems at NMR [9]and ESR [10]frequencies; the present results extend this effect to the region of noncoherent fields and relate it to the dressed atom concept. Moreover, the marked differences which exist in many respects among magnetic systems coupled by different exchange interactions [11], induce one to think that near resonant irradiation in appropriate experimental conditions might modify their behaviour in more than one observable way [12]. Finally, the behaviour of the dipolar interaction in (17) under the dressing transformation. (3) seems to suggest that appropriate irradiation might transforma set of two-level atoms interacting via nonpermanent dipole— dipole forces into a sort of effectively ferromagnetic-like system. If this is the case, it might be interesting to speculate on the possible meaning of the Curie temperature and of the phase transition associated with it. The authors of this work wish to acknowledge partial support by Comitato Regionale Ricerche Nucleari e di Struttura della Materia. 39
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
References [1] F. Schuda, C.R. Stroud and M. Hercher, J. Phys. B7 (1974) L198; H. Waither, in: Frontiers in laser spectroscopy, eds. R. Balian, S. Haroche and S. Liberman (North-Holland, Amsterdam, 1977) p. 125; R.E. Grove, F.Y. Wu and S. Ezekiel, Phys. Rev. A15 (1977) 227. [2] P.A. Apanasevich, Opt. Spectr. 16 (1964) 387; B.R. Mollow, Phys. Rev. 188 (1969) 1969; H.J. Kimble and L. Mandel, Phys. Rev. A13 (1976) 2123. [3] G.S. Agarwal, A.C. Brown, L.M. Narducci and G. Vetri, Phys. Rev. A15 (1977) 1613; A.S. Jahangir Amin and J.G. Cordes, Phys. Rev. A18 (1978) 1298; Y.C. Lee, C.Y. Chen and Y. Shan, J. Phys. C12 (1979) 1429; G.S. Agarwal, L.M. Narducci, D.H. Feng and R. Gilmore, Phys. Rev. Lett. 42 (1979) 1260. [4] C. Mavroyannis, Phys. Rev. A18 (1978) 185; H.S. Freedhoff, Phys. Rev. A19 (1979) 1132. [5] C. Cohen-Tannoudji and S. Reynaud, J. Phys. BlO (1977) 345. [6] P. Carbonaro, C. Compagno and F. Persico, Phys. Lett. A73 (1979) 97. [7] J.H. Van Vleck, Phys. Rev. 74 (1948) 1168. [8] O.G. Mouritsen and S.J. Knak Jensen, Phys. Rev. B18 (1978) 465. [9] W.I. Goldburg and M. Lee, Phys. Rev. Lett. 11(1963) 255; D. Barnaal and I.J. Lowe, Phys. Rev. Lett. 11(1963) 258; C.S. Yannoni and H.M. Vieth, Phys. Rev. Lett. 37 (1976) 1230; H.T. Stokes and D.C. Ailion, Phys. Rev. Lett. 38 (1977) 1159. [10] K. Tornita, Prog. Theor. Phys. 19(1958)541; M.W.P. Strandberg, Phys. Rev. B6 (1972) 747; R. Boscaino, M. Brai, I. Ciccarello and M.W.P. Strandberg, Phys. Rev. B7 (1973) 50. [11] J. Oitmaa and D.D. Betts, Phys. Lett. 68A (1978) 450. [12] L.P. Regnault, P. Burlet and J. Rossat-Mignod, Physica B86—88 (1977) 661; H.A. Aigra, L.J. De Jongh and WJ. Huiskamp, Physica B86—88 (1977) 737; A.T. Skjeltorp, Physica B86—88 (1977) 1295; D.R. Taylor, Phys. Rev. Lett. 42 (1979) 1302.
40