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Frequencies of two running circular electromagnetic waves in a rotating frame of reference
Volume 11, number 3
OPTICS COMMUNICATIONS
FREQUENCIES
OF TWO RUNNING IN A ROTATING
CIRCULAR FRAME
July 1974
ELECTROMAGNETIC
WAVES
OF REFERENCE
V.F. CHEL’TZOV Department
of Physics, Moscow Textile institute, M. Kaluzshskaya, I., 117071
Moscow, USSR
Received 29 April 1974
The frequencies of two running electromagnetic waves interacting with a system of identical two-level atoms in a rotating frame of reference have been calculated with the help of quantum electrodynamics and the covariant formalism. The existence of two states of the perturbed electromagnetic field with degenerate and nondegenerate frequencies has been shown.
The results of calculating the eigenfrequencies of two circular electromagnetic waves running in opposite directions and interacting with a system of N identical two-level atoms in a rotating frame of reference are presented below. The field and atoms are confined within a volume of toroidal form and are subject to a weak rotation about the z-axis. This axis passes through the center of the torus circumference and is perpendicular to its plane. The presence of the rotation requires application of the covariant formalism. The obtained formulae are valid under the following assumptions: a) The system “atoms and field” is conservative and its behavior is determined by the hermitian hamiltonian. b) The radiation field is in a coherent state, while the hamiltonian is preserving the coherence. c) The field vector-potential has A,-component only (a simple estimation gives A, <
In eq. (1) A, and An are the covariant and the contravariant components, respectively. The symbols ;az denote covariant differentiation with respect top, An satisfies the Lorentz condition A’;‘*= 0, co is the velocity of light in vacuum, R, in the Ricci tensor. After introduction of the coordinates X1 = I, X2=P,X3=z,X4= co T rotating about the z-axis at the angular velocityR , in the presence of the A,component only we obtain from eq. (1) the following expression in physical coordinates
gkmA n ;m ;k ‘RinAi
cd, = w o-RQK,,
= -_(4’~g)jn
.
(1)
(2) If the circumference radius R is much larger than the torus cross-section radius A, the simplest solution of eq. (2) in the absence of the medium takes the form (3)
In expression (3) Jo is a Bessel function, T$) is its first root, p is a radial coordinate in the cross-section plane,
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Volume
11, number
OPTICS COMMUNICATIONS
3
where KI = t m/R (the + sign is taken in the case of rotation-wise running waves, the - sign is taken for the counter-rotation running waves), m is a positive integer, w0 is an atomic resonance frequency. The solution (3) satisfies the boundary conditions: A, = 0 on the torus surface, A,(p + 2n) = A,(p). Let us expand A, for the free field into the functions U,, considering the expansion coefficients CT and Cl as photon creation and annihilation operators in the second quantization representation. Substituting this expansion into the formula for the electromagnetic energy density [ 1 ] “=(1/47i)(F41F~--~44FlrilF1~z),
(5)
Since mR << w,, , in a good approximation
2m,C0 ,e,
(8) where 07 in its turn is defined by the equality dCf dt = iw;Cf Expanding j, into U, and introducing the spin operators CJ;, with regard to the analogous expansion for A, we arrive at the operator equation
C Ul( j)uy=(w;)2K __ I(lco 1 1 c+.
f;wqc;c, t i).
(6)
The atomic electron hamiltonian contains in the rotating frame the additional term $Zi%ia/a(p which is due to the rotation. This term describes an effective magnetic field of strength x=:
1974
and Ul(i) is the value of U, at the coordinates of the centre-of-mass of thejth atom. The peculiarity of H consists in commuting of Ho with H at exact resonance (wI = wo) so that the unperturbed energy E, coincides with the perturbed one. Besides H preserves the total number of excitations and the coherence of the field [3]. Since X is antiparallel to the z-axis and the active medium is isotropic in the absence of rotation, the atomic susceptibility K~ at the frequency w; is given by the ordinary equation
(4)
and integrating the resulting expression over the torus volume taking into account the boundary conditions, we arrive at the field hamiltonian
H rad =
July
i
(9)
Writing out with the help of (7) the equations of motion for both sides in (9), using (9) once more and averaging the obtained operator expressions over the wave function of the interacting system, we obtain the following expression for the susceptibility KI:
sz.
The field aC removes the atomic levels degeneracy and leads to selection rules for radiative transitions. For electric dipole transitions we get m, = mb and Eb- E, (10)
= fiw,.
Using the energy-spin total hamiltonian
formalism
[2] we get for the This expression has been derived under the assumption that N_ = ( Zj a!) is independent of time and
(7) where H int _--LJjkAkdu=-LJ co v
218
jzAzdu co
v
The last uncoupling may be justified by taking into account the preservation of (+ X&t +C~C~~) and the validity of the uncoupling
OPTICS COMMUNICATIONS
Volume 11, number 3
in the coherent state. Besides, if N_ is temporarily independent and the state of the atomic subsystem is defined uniquely by prescribing the cooperative number r 121, then the wave function of the interacting system might be factorized. Expanding A, in eq. (2) into U, and using the analogous expansion for j, together with eq. (IO), one may reduce the inhomogeneous equation (2) to a homogeneous one for CT, C,, whence it follows immediately (w; +Rs2K,)2 4nl~lw(,0+wl)N_ = u; + -___ ~_.~~__~ ri[w~-(of)2-2nl~121~~_/~ol]’
(11)
The lastzquation determines the sought eigenfrequencies wI . Taking into consideration the smallness of the second term in the right side of (11) in comparison with o;, we shall obtain for U; two approximate solutions of eq. (11): (o:)L
=&,+w;+zfl)
f {f(w,-
~J;+&?)~-A}~/~
Both solutions satisfy the condition E x E. (or ,:t,:= 20~) in the presence of small mistuning (ma << wo). If N > 0 and mR < 1A I 1/2, the frequen ties WT are complex conjugate and the electromagnetic oscillators are unstable. When N_ < 0 the frequencies are real by any mistuning. The solution (12) yields the state with nearly degenerate frequencies. Full degeneracy occurs when mS2 >> lAi1/2. In this case solution (13) leads to the usual frequency splitting * ZZw =c,J - RRK,, 1 0 ml so that o _ - o+ = 2mQ. The author is sincerely indebted to Prof. G.V. Skrotzky for helpful discussions on this paper.
References [ 1] L.D. Landau and E.M. Lifshitz,
=+(wg+W@zR)i
{~(W0-w(+rR)2-‘4}1/2, (13)
where
A=
i974
(12)
and co;>,
July
Theory of fields (Fizmatgiz. M., 1960). (21 R.H. Dicke, Phys. Rev. 93 (1954) 99. [3] R.J. Glauber, Phys. Let?. 21 (1966) 650; C.L. Mehta and E.C.G. Sudarshan, Phys. Lett. 22 (1966) 574.
$-A.
219
OPTICS COMMUNICATIONS
FREQUENCIES
OF TWO RUNNING IN A ROTATING
CIRCULAR FRAME
July 1974
ELECTROMAGNETIC
WAVES
OF REFERENCE
V.F. CHEL’TZOV Department
of Physics, Moscow Textile institute, M. Kaluzshskaya, I., 117071
Moscow, USSR
Received 29 April 1974
The frequencies of two running electromagnetic waves interacting with a system of identical two-level atoms in a rotating frame of reference have been calculated with the help of quantum electrodynamics and the covariant formalism. The existence of two states of the perturbed electromagnetic field with degenerate and nondegenerate frequencies has been shown.
The results of calculating the eigenfrequencies of two circular electromagnetic waves running in opposite directions and interacting with a system of N identical two-level atoms in a rotating frame of reference are presented below. The field and atoms are confined within a volume of toroidal form and are subject to a weak rotation about the z-axis. This axis passes through the center of the torus circumference and is perpendicular to its plane. The presence of the rotation requires application of the covariant formalism. The obtained formulae are valid under the following assumptions: a) The system “atoms and field” is conservative and its behavior is determined by the hermitian hamiltonian. b) The radiation field is in a coherent state, while the hamiltonian is preserving the coherence. c) The field vector-potential has A,-component only (a simple estimation gives A, <
In eq. (1) A, and An are the covariant and the contravariant components, respectively. The symbols ;az denote covariant differentiation with respect top, An satisfies the Lorentz condition A’;‘*= 0, co is the velocity of light in vacuum, R, in the Ricci tensor. After introduction of the coordinates X1 = I, X2=P,X3=z,X4= co T rotating about the z-axis at the angular velocityR , in the presence of the A,component only we obtain from eq. (1) the following expression in physical coordinates
gkmA n ;m ;k ‘RinAi
cd, = w o-RQK,,
= -_(4’~g)jn
.
(1)
(2) If the circumference radius R is much larger than the torus cross-section radius A, the simplest solution of eq. (2) in the absence of the medium takes the form (3)
In expression (3) Jo is a Bessel function, T$) is its first root, p is a radial coordinate in the cross-section plane,
217
Volume
11, number
OPTICS COMMUNICATIONS
3
where KI = t m/R (the + sign is taken in the case of rotation-wise running waves, the - sign is taken for the counter-rotation running waves), m is a positive integer, w0 is an atomic resonance frequency. The solution (3) satisfies the boundary conditions: A, = 0 on the torus surface, A,(p + 2n) = A,(p). Let us expand A, for the free field into the functions U,, considering the expansion coefficients CT and Cl as photon creation and annihilation operators in the second quantization representation. Substituting this expansion into the formula for the electromagnetic energy density [ 1 ] “=(1/47i)(F41F~--~44FlrilF1~z),
(5)
Since mR << w,, , in a good approximation
2m,C0 ,e,
(8) where 07 in its turn is defined by the equality dCf dt = iw;Cf Expanding j, into U, and introducing the spin operators CJ;, with regard to the analogous expansion for A, we arrive at the operator equation
C Ul( j)uy=(w;)2K __ I(lco 1 1 c+.
f;wqc;c, t i).
(6)
The atomic electron hamiltonian contains in the rotating frame the additional term $Zi%ia/a(p which is due to the rotation. This term describes an effective magnetic field of strength x=:
1974
and Ul(i) is the value of U, at the coordinates of the centre-of-mass of thejth atom. The peculiarity of H consists in commuting of Ho with H at exact resonance (wI = wo) so that the unperturbed energy E, coincides with the perturbed one. Besides H preserves the total number of excitations and the coherence of the field [3]. Since X is antiparallel to the z-axis and the active medium is isotropic in the absence of rotation, the atomic susceptibility K~ at the frequency w; is given by the ordinary equation
(4)
and integrating the resulting expression over the torus volume taking into account the boundary conditions, we arrive at the field hamiltonian
H rad =
July
i
(9)
Writing out with the help of (7) the equations of motion for both sides in (9), using (9) once more and averaging the obtained operator expressions over the wave function of the interacting system, we obtain the following expression for the susceptibility KI:
sz.
The field aC removes the atomic levels degeneracy and leads to selection rules for radiative transitions. For electric dipole transitions we get m, = mb and Eb- E, (10)
= fiw,.
Using the energy-spin total hamiltonian
formalism
[2] we get for the This expression has been derived under the assumption that N_ = ( Zj a!) is independent of time and
(7) where H int _--LJjkAkdu=-LJ co v
218
jzAzdu co
v
The last uncoupling may be justified by taking into account the preservation of (+ X&t +C~C~~) and the validity of the uncoupling
OPTICS COMMUNICATIONS
Volume 11, number 3
in the coherent state. Besides, if N_ is temporarily independent and the state of the atomic subsystem is defined uniquely by prescribing the cooperative number r 121, then the wave function of the interacting system might be factorized. Expanding A, in eq. (2) into U, and using the analogous expansion for j, together with eq. (IO), one may reduce the inhomogeneous equation (2) to a homogeneous one for CT, C,, whence it follows immediately (w; +Rs2K,)2 4nl~lw(,0+wl)N_ = u; + -___ ~_.~~__~ ri[w~-(of)2-2nl~121~~_/~ol]’
(11)
The lastzquation determines the sought eigenfrequencies wI . Taking into consideration the smallness of the second term in the right side of (11) in comparison with o;, we shall obtain for U; two approximate solutions of eq. (11): (o:)L
=&,+w;+zfl)
f {f(w,-
~J;+&?)~-A}~/~
Both solutions satisfy the condition E x E. (or ,:t,:= 20~) in the presence of small mistuning (ma << wo). If N > 0 and mR < 1A I 1/2, the frequen ties WT are complex conjugate and the electromagnetic oscillators are unstable. When N_ < 0 the frequencies are real by any mistuning. The solution (12) yields the state with nearly degenerate frequencies. Full degeneracy occurs when mS2 >> lAi1/2. In this case solution (13) leads to the usual frequency splitting * ZZw =c,J - RRK,, 1 0 ml so that o _ - o+ = 2mQ. The author is sincerely indebted to Prof. G.V. Skrotzky for helpful discussions on this paper.
References [ 1] L.D. Landau and E.M. Lifshitz,
=+(wg+W@zR)i
{~(W0-w(+rR)2-‘4}1/2, (13)
where
A=
i974
(12)
and co;>,
July
Theory of fields (Fizmatgiz. M., 1960). (21 R.H. Dicke, Phys. Rev. 93 (1954) 99. [3] R.J. Glauber, Phys. Let?. 21 (1966) 650; C.L. Mehta and E.C.G. Sudarshan, Phys. Lett. 22 (1966) 574.
$-A.
219