Optics and Spectroscopy, Vol. 97, No. 4, 2004, pp. 596–604. Translated from Optika i Spektroskopiya, Vol. 97, No. 4, 2004, pp. 637–645. Original Russian Text Copyright © 2004 by Titov.
PHYSICAL AND QUANTUM OPTICS
Kinetic Equations for the Density Matrix of Two-Level Particles in a Strong Field E. A. Titov Institute of Laser Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail:
[email protected] Received January 22, 2004
Abstract—On the basis of the Green’s function method for nonequilibrium systems, the kinetic equation for two-level particles in a strong field is derived. In the binary approximation, all diagrams are expressed in terms of the reducible vertex part, whose calculation is reduced to the problem of two-channel scattering and finding the corresponding scattering amplitudes. © 2004 MAIK “Nauka/Interperiodica”.
INTRODUCTION By using the Green’s function method for nonequilibrium systems (the Keldysh technique) [1], we obtained [2] the standard set of equations for the density matrix of two-level particles interacting with an electromagnetic wave. In [2], we proceeded from the following statement of the problem. Consider a twolevel weakly imperfect Fermi gas whose atoms undergo collisions with a buffer gas. Collisions of the two-level atoms with one another are not taken into account. At distances exceeding the atomic dimensions, the interaction potential of atoms is described by a power law V
(α)
( α ) –n
( r ) = Cn r ,
where r = |r |, r is the radius vector of the relative motion of atoms; and α = a, b, where the superscript b relates to the upper level of the two-level system and the superscript a, to the lower level. For collisions of a two-level atom with a buffer gas atom, of importance are distances of the order of the Weisskopf radius ρ0 = (Cn /បν)1/n – 1, where ν is the relative velocity of the colliding particles. Thus, the effective distances at which atoms interact substantially exceed atomic dimensions when the velocities v are much smaller than electron velocities ve. Under ordinary conditions, the velocity v lies within the interval n–1 (ប/ µC n )n – 1/n – 2 Ⰶ v Ⰶ ve = e2/ប, where µ is the reduced mass of the colliding atoms. The lower bound on the velocity is caused by the requirement that the motion have quasi-classical character when the de Broglie wavelength is much smaller than the impact parameter ρ0 (ប/µν Ⰶ ρ0). The condition of gas approximation (the binary character of collisions) has the form 3 N B ρ 0 Ⰶ 1, where NB is the buffer gas density. Therefore, in the kinetic equation, we retain only the terms that are bilinear with respect to the densities of the twolevel and buffer gases. In addition, when passing from
the equations for the Green’s functions to the kinetic equation for the density matrix, the quasi-classical condition is used; i.e., we assume that the time and space intervals τ and L within which all the quantities vary satisfy the inequalities εT Ⰷ ប/τ and pT Ⰷ ប/L (here, εT is the atomic thermal energy and pT is the thermal momentum). We also assume that the exact Green’s functions are related to the density matrix by the same formulas as in the case of an ideal gas; i.e., the law of energy conservation is expressed by the δ function and is not spread owing to collisions [1]. Finally, we assume that the Rabi frequency and the detuning of the electromagnetic field frequency relative to the transition frequency are small as compared to the inverse collision time of the particles. This allowed us to discard in [2] –– the scattering amplitudes of the type of Σ ba . Let us compare the scattering amplitude at the level –– –– a( Σ aa ), with the amplitude Σ ba , taking collisions into account according to the perturbation theory. For Σ aa , we have the series (ប = 1) ––
–iΣ–– aa =
+
+
a
,
a
where the dashed line corresponds to the interaction of the level a with a buffer atom; the solid line, to the Green’s function of the two-level particle; and the heavy line, to the Green’s function of the buffer atom. In other words, ( 1 )––
Σ aa = Σ aa ––
(a)
( 2 )––
+ Σ aa
∫
(a)
where V q = drV ( r )e ( 1 )––
(a)
(a)
∫
( a )2
∼ N B V 0 + N B dqq ( V q / v T q ), – iqr
2
, vT is the thermal veloc-
( 2 )––
( 1 )––
ity, Σ aa ~ N B V 0 , and Σ aa ~ Σ aa αξ, where α = 1/mvTρ0 is the quasi-classical parameter, ξ = V (a)(ρ0)/K
0030-400X/04/9704-0596$26.00 © 2004 MAIK “Nauka/Interperiodica”
KINETIC EQUATIONS FOR THE DENSITY MATRIX
597
is the parameter of the perturbation theory, and K = 2 1/m ρ 0 is the average kinetic energy of the particles at
∂ ∆ i ---- + --------- + δ ψ b ( x, t ) = V ψ a ( x, t ) ∂t 2m 1
the distance of action of the potential. For Σ ba , we have
+ dx'V ( x – x' )ϕ ( x', t )ϕ ( x', t )ψ b ( x, t ),
––
–iΣ–ba– (p) =
, b
p
a
where = V/(ω – εp + δ/2 + i0)(ω – εp + i0), p = (p, ω), εp is the kinetic energy of the two-level particle, V is the Rabi frequency, and δ is the detuning of the field frequency relative to the transition frequency. Thus, in the integral over q presented above, an additional factor appears. As a result, we have –– G ba (p)
∫
(a)
(b)
Σ ba ∼ N B dqq V q V q / v T q ( v T q + δ/2 ). ––
2
Therefore, if |δ| Ⰷ 1/τc, then –– Σ ba
~
( 2 )–– (V/δ) Σ aa ;
if |δ| Ⰶ
( 2 )–– (Vτc) Σ aa .
1/τc, ~ Hence, if Vτc, |δ|τc Ⰶ 1, the terms discarded in [2] are small. This means that the dynamic (the interaction with the electromagnetic field) and collisional terms in the equation for the density matrix can be considered as independent of each other. If Vτc, |δ|τc ≥ 1, the set of equations obtained in [2] becomes nonclosed; i.e., in all the vertex parts, there appear contributions connected with the field and the detuning, which should be calculated. In a great number of studies, different methods were used for solving the problem of behavior of an atom in a strong field in the presence of collisions (see, for example, [3, 4]). In the present paper, we will not analyze these solutions. The aim of the study is to use the approach developed in [2] for deriving a closed set of equations for the density matrix of a two-level particle in an arbitrarily strong field. For this purpose, with the help of the known canonical transformation, we diagonalize the dynamic part of the Hamiltonian and obtain an increment to the collisional part in the form of a nondiagonal exchange potential. Then the calculation of the vertex parts is reduced to the problem of two-channel scattering and they can be expressed in terms of the corresponding scattering amplitudes. Then, solving the Schrödinger equation, we can determine these scattering amplitudes.
+
where ψα(x, t) and ϕ(x, t) are the Heisenberg operators for a two-level particle (α = a, b) and a buffer atom; V (α)(x – x') is the potential of interaction of the buffer atom with the two-level particle at the level α = a, b; V = –dE/ប; and E is the electromagnetic field strength, which is considered to be independent of the coordinate because we neglect the Doppler frequency shift. Further, d is the dipole moment matrix element for the resonant transition, δ = Ω – ω0 is the detuning of the field frequency Ω from the transition frequency, ∆ is the Laplacian, and m1 is the mass of the two-level particle. Consider the stationary states in the strong field ϕa = α– ψ a – α+ ψ b ,
Here, vT q ~ vT /ρ0 = 1/τc, where τc is the collision time. –– Σ ba
(b)
∫
or ψ a = α– ϕa + α+ ϕb ,
ψ b = – α+ ϕa + α– ϕb ,
α ± = ( ( 1 ± δ/δ 0 )/2 ) ,
∂ ∆ i ---- + --------- ψ ( x, t ) = V ψ b ( x, t ) ∂t 2m 1 a
∫
(a)
+ dx'V ( x – x' )ϕ ( x', t )ϕ ( x', t )ψ a ( x, t ), Vol. 97
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2
We will assume that δ = Ω – ω0 < 0; then, for V the following correspondence takes place: ϕa, ϕb ψa, ψb. By making the replacement, we obtain
0,
∂ ∆ i ---- + --------- ϕ ( x, t ) ∂t 2m 1 a = –λ 1 ϕ a ( x, t ) + L 1 ϕ a ( x, t ) + L 0 ϕ b ( x, t ), ∂ ∆ i ---- + --------- ϕ ( x, t ) ∂t 2m 1 b = –λ 2 ϕ b ( x, t ) + L 0 ϕ a ( x, t ) + L 2 ϕ b ( x, t ), δ±δ λ 1, 2 = --------------0 , 2 Li =
∫ dx ϕ ( x , t )V ( x – x )ϕ ( x , t ), +
2
V 1 = α– V 2
2
(a)
i
(b)
+ α+ V , 2
2
2
V 2 = α+ V 2
V 0 = α– α+ ( V
(a)
iλ 1 t
(a)
i = 1, 2, (b)
+ α– V , 2
(b)
– V ).
, we have
∂ ∆ i ---- + --------- ϕ ( x, t ) = L 1 ϕ a ( x, t ) + L 0 ϕ b ( x, t ), ∂t 2m 1 a ∂ ∆ i ---- + --------- – δ 0 ϕ b ( x, t ) = L 0 ϕ a ( x, t ) + L 2 ϕ b ( x, t ). ∂t 2m 1 The dynamic Hamiltonian, i.e., the part of the complete Hamiltonian that describes transitions in the field, is diagonalized; however, the collisional terms have
+
OPTICS AND SPECTROSCOPY
2 1/2
δ 0 = ( δ + 4V ) .
1/2
Assuming that ϕa, b ~ ϕa, b e TRANSFORMATION OF THE OPERATORS The equations of motion for the second quantization operators in the Heisenberg representation have the form [2] (ប = 1)
ϕb = α+ ψ a + α– ψ b
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TITOV
become nondiagonal: an elastic collision results now in the inelastic transition a b. Thus, to write the applicable set of equations for the Green’s functions and, then, for the density matrix, it is necessary to derive expressions for the vertex part Γαβ for scattering by the potential V V Vˆ = 1 0 , V0 V2
Performing the integration over q0 in (1), we obtain
V 0 = V 12 = V 21 ,
∫
– – Γˆ ( p, p'; k ) = Vˆ p – p' + dqVˆ p – q Rˆ ( q, k )Γˆ ( q, p'; k ),
V 1 = V 11 ,
V 2 = V 22 .
( Rˆ ( q, k ) ) αβ = δ αβ R β ( q, k ),
To distinguish between the transformed quantities and the initial ones, we will number the subscripts as 1 and 2, with the correspondence 1 a and 2 b.
The graphical equations for Γ αβ in the ladder approximation (α, β = 1, 2) –
=
Σ
+
α β
γ = 1, 2
αγ γ β
–1
R 1 = ( ε 1 – ( q /2µ ) + i0 ) , 2
(2)
–1
R 2 = ( ε 2 – ( q /2µ ) + i0 ) , 2
ε1 = εk ,
CALCULATION OF THE VERTEX PARTS
α β
Here, p = µv = (m2p1 – m1p2)/m, p' = (m2p4 – m1p3)/m, v = v1 – v2 = (p1/m1) – (p2/m2), m = m1 + m2, µ = m1m2/m, k = p1 + p2 = p3 + p4, p1 = p + (m1/m)k, p2 = –p + (m2/m)k, p4 = p' + (m1/m)k, p3 = –p' + (m2/m)k, k(k, ω), where m1 is the mass of the two-level particle and m2 is the mass of the buffer gas atom.
ε2 = εk – δ0 ,
ε k = ω – k /2m. 2
Making the replacement – Γˆ ( p, p'; k ) =
∫ dkVˆ
kQ(p
– k, p'; k ),
(3)
ˆ: we obtain the equation for Q
lead to the equations
ˆ ( p, p'; k ) = δ ( p – p' ) Q
(4)
∫
–
Γˆ ( p, p'; k ) = Vˆ p – p'
ˆ ( q, p'; k ). + Rˆ ( p, k ) dqVˆ p – q Q
(1) m m – ( 0 ) –– + i d qVˆ p – q g –q + ------2 k gˆ q + ------1 k Γˆ ( q, p'; k ), We represent Q ˆ as the sum Q ˆ =Q ˆ1 + Q ˆ 2 . The equa m m ˆ i can be written in the form tion for Q where
∫
––
4
– – – Γ Γ Γˆ = 11 12 – – Γ 21 Γ 22
,
gˆ
( 0 )––
( 0 )–– g 0 = 11 ( 0 )–– 0 g 22
ˆ i ( p, p'; k ) = δ ( p – p' )χˆ i χˆ +i Q
∫
ˆ i ( q, p'; k ), + Rˆ ( p, k ) dqVˆ p – q Q
are the free Green’s functions of the two-level particle and the buffer atom, respectively, and g ( q ) = ( q 0 – E q + i0 ) , ––
–1
( 0 )––
g 11
( 0 )––
g 22
( q ) = ( q 0 – ε q + i0 ) , –1
( q ) = ( q 0 – ε q – δ 0 + i0 ) , –1
2
E q = q /2m 2 , d q = dq ( dq 0 /2π ),
ε q = q /2m 1 , 2
dq = ( dq x dq y dq z )/ ( 2π ) .
4
3
In Eq. (1), we passed to the relative momenta and to the momentum of the center of gravity (see [2]): – Γ αβ ( p 1,
=
– Γ αβ ( p,
p 2 ; p 3, p 4 )
(4)
p'; k )δ ( p 1 + p 2 – p 3 – p 4 ).
i = 1, 2;
(5)
0 χˆ 2 = . 1
1 χˆ 1 = , 0
ˆ i is reduced to an inhomogeThe equation for Q neous equation of the problem of two-channel scattering with the ith initial channel in the system of the center of mass in the momentum representation. The soluˆ i can be written in the form of tion of the equation for Q an expansion in the set of functions of the scattering problem. We will consider these functions. Let us start with channel 1. Then p2 – k2 ----------------0 2µ 2 2 p –k 0 + δ0 --------------- 2µ
ψ ˆ (1) ( p ) k
(6)
∫
ˆ (k1 ) ( p – q ) = 0, + dqVˆ q ψ OPTICS AND SPECTROSCOPY
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KINETIC EQUATIONS FOR THE DENSITY MATRIX
ψ ˆ = 1 . The formal solution of this equawhere ψ ψ2 tion is (1) ˆ (k1 ) ( p ) = δ ( p – k )χˆ 1 + Lˆ k ( p ) ˆf ( p, k )χˆ 1 , ψ (1)
ˆ (ki ) (p) equation for scattering, the wave function ψ should satisfy the condition
(1)
2
–1
2
(1) k –p L k1 ( p ) = ----------------- + i0 , 2µ (1) L k2 ( p )
ˆ i to be the Green’s function of the corresponding For Q
∫ dkψˆ
( Lˆ k ( p ) ) αβ = δ αβ L kβ ( p ),
2
(7)
ˆ (k1 ) ( p dqVˆ q ψ
– q ),
(8)
+
∫
ψ ˆ (2) ( p ) k
k ε i – ------ + i0 2µ
(9)
+ –1 (i) (i) k Λ αβ = ε i – ------ + i0 ( Lˆ k ( p' ) ) Rˆ ( p', k ) 2µ (i)
–1
(2) k –p L k1 ( p ) = ----------------- + δ 0 + i0 , 2µ 2
(1) Λ 11
(2) k –p L k2 ( p ) = ----------------- + i0 . 2µ
( i )+
–1
–1
2
(i) k Φ k ( k ) = ε i – ------ + i0 , 2µ
(12)
where i = 1, 2; ε1 = εk, ε2 = εk – δ0, εk = ω – k2/2m, and ( Rˆ ( p', k ) ) αβ = δ αβ R β ( p', k ), –1
–1
2
R 2 = ( ε 2 – ( p' /2µ ) ). –1
OPTICS AND SPECTROSCOPY
(2) Λ 11
k = ε k – ------ – δ 0 + i0 2µ
2
–1
2
2
(16)
–1
2
k – p' + ------------------ – δ 0 – i0 , 2µ
k = ε k – ------ + i0 2µ
(11)
Here, Φ is proportional to the unit matrix and equals
R 1 = ( ε 1 – ( p' /2µ ) ),
–1
2
k – p' + ------------------ – i0 , 2µ
–1
2
–1
2
k – p' + ------------------ + δ 0 – i0 , 2µ –1
2
2
–1
k – p' + ------------------ – i0 . 2µ
Then
ˆ k ( p )ψ ˆ k ( p' )Rˆ ( p', k ). dkΦ k ( k )ψ
–1
2
–1
(1) Λ 22
2
ˆ i ( p, p'; k ) Q
∫
2
k = ε k – ------ + i0 2µ
(2) k Λ 22 = ε k – ------ – δ 0 + i0 2µ
ˆ i in the form We will seek expressions for Q (i)
,
(i)
(10)
–1
2
αβ
Λ αβ = δ αβ Λ β ,
(2) δ αβ L kβ ( p ), 2
–1
2
(2) ˆ (k2 ) ( p ) = δ ( p – k )χˆ 2 + Lˆ k ( p ) ˆf ( p, k )χˆ 2 , ψ
2
+ (i) –1 k ) ( Lˆ k ( p' ) ) ]Rˆ ( p'; k ).
Let us introduce
∫
=
i
ˆ (ki ) ( p ) ψ + - [ χˆ i ˆf +( p', dk --------------------------2
ˆ (k2 ) ( p – q ) = 0, + dqVˆ q ψ
(i)
(i)
+ (i) –1 + + + × [ δ ( p' – k )χˆ i + χˆ i ˆf ( p', k ) ( Lˆ k ( p' ) ) ]Rˆ ( p'; k ) (15) ˆ ( i ) ( p )χˆ + = ψ
initial one, the corresponding equations have the form p2 – k2 ----------------- – δ 0 0 2µ 2 2 p –k ---------------0 2µ
(14)
ˆ k (p) ψ dk --------------------------2 k ε i – ------ + i0 2µ
p'
where ˆf is related to the ordinary amplitude as ˆf = −2π ˆf /µ. If the second channel plays the role of the
=
+
∫
ˆ i ( p, p'; k ) = Q
The scattering amplitude is determined from the equation
(2) ( Lˆ k ( p ) ) αβ
= δ ( p – p' )χˆ i χˆ i .
–1
2
∫
(i) ˆ (ki )+ ( p' ) k ( p )ψ
ˆ ( i )+ its expression in Substituting into (11) instead of ψ terms of the scattering amplitude ˆf , we obtain
k –p = ----------------- – δ 0 + i0 . 2µ
ˆf ( p, k )χˆ = 1
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ˆ i ( p, p'; k ) = ψ ˆ (p'i ) ( p )χˆ +i Q ˆ (ki ) ( p' ). ˆ (ki ) ( p )χˆ +i ˆf +( p', k )Λ + dkψ
∫
ˆ =Q ˆ1 + Q ˆ 2 into formula (3), we obtain Substituting Q – an expression for Γˆ . Below, we present expressions ± for Γˆ , making trivial changes in the derivation (see [2]): ±
Γ 11 ( p, p'; k ) = f 11 ( p, p' )
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TITOV
problem. The irreducible self-energy parts Σˆ αβ can be found in much the same manner as in [2]. One can also obtain, as in [2], relationships between the scattering amplitudes. Since they are cumbersome, we present them in the Appendix.
∫
+ dk f 11 ( p, k ) f *11 ( p', k ) 2
k − × ε k – ------ + i0 2µ
–1
2
–1
2
k – p' + ------------------ – i0 2µ
+ f 12 ( p, k ) f *12 ( p', k ) 2
k × ε k – ------ – δ 0 − + i0 2µ
–1
2
KINETIC EQUATIONS
–1
2
k – p' + ------------------ + δ 0 – i0 , 2µ
±
Γ 12 ( p, p'; k ) = f 12 ( p, p' )
∫
* ( p', k ) + dk f 11 ( p, k ) f 21 2
k × ε k – ------ − + i0 2µ
–1
2
–1
2
k – p' + ------------------ – δ 0 – i0 2µ
As was done in [2], we write the equations for the –+ + Green’s functions g αβ (1, 2) = i 〈 ϕ β ( 2 )ϕ α ( 1 )〉 , where ϕα(1) = ϕα(x1, t1), 〈…〉; angle brackets denote averaging over an arbitrary state of the system. Further, passing to the new variables R = (x1 + x2)/2, r = x1 – x2, T = (t1 + t2)/2, and t = t1 – t2; making the Fourier transform in the difference variables; and using the quasiclassical conditions, we obtain the set of equations ∂g 11 ( p 1, ω 1, T ) i -----------------------------------∂T –+
+ f 12 ( p, k ) f *22 ( p', k ) 2
k × ε k – ------ – δ 0 − + i0 2µ
–1
2
–1
2
k – p' + ------------------ – i0 , 2µ
= ( Σ 11 ( p 1, ω 1 ) + Σ 11 ( p 1, ω 1 ) )g 11 ( p 1, ω 1, T ) ––
(17)
±
Γ 21 ( p, p'; k ) = f 21 ( p, p' )
––
∫
2
–1
2
++
–1
2
2
–1
k – p' + ------------------ – i0 2µ
2
–+
+ Σ 21 ( p 1, ω 1 )g 12 ( p 1, ω 1, T ) –+
+ Σ 11 ( p 1, ω 1, T ) ( g 11 ( p 1, ω 1 ) + g 11 ( p 1, ω 1 ) ), –+
––
++
∂g 22 ( p 1, ω 1, T ) i -----------------------------------∂T –+
+ f 22 ( p, k ) f *12 ( p', k ) k × ε k – ------ – δ 0 − + i0 2µ
–+
+ Σ 12 ( p 1, ω 1 )g 21 ( p 1, ω 1, T )
+ dk f 21 ( p, k ) f *11 ( p', k ) k − × ε k – ------ + i0 2µ
++
–1
2
k – p' + ------------------ + δ 0 – i0 , 2µ
= ( Σ 22 ( p 1, ω 1 ) + Σ 22 ( p 1, ω 1 ) )g 22 ( p 1, ω 1, T ) ––
++
–+
±
Γ 22 ( p, p'; k ) = f 22 ( p, p' )
+ Σ 21 ( p 1, ω 1 )g 12 ( p 1, ω 1, T ) ––
∫
* ( p', k ) + dk f 21 ( p, k ) f 21 2
k × ε k – ------ − + i0 2µ
–1
2
++
k – p' + ------------------ – δ 0 – i0 2µ
2
–1
2
+ Σ 22 ( p 1, ω 1, T ) ( g 22 ( p 1, ω 1 ) + g 22 ( p 1, ω 1 ) ), –+
2
–1
k – p' + ------------------ – i0 . 2µ
p'; k ) =
– ( Γ βα ( p',
––
++
∂g 21 ( p 1, ω 1, T ) –+ - – δ 0 g 21 ( p 1, ω 1, T ) i -----------------------------------∂T = ( Σ 22 ( p 1, ω 1 ) + Σ 11 ( p 1, ω 1 ) )g 21 ( p 1, ω 1, T ) ––
By using relationships (A4) (see the Appendix), we can show that + Γ αβ ( p,
–+
–+
+ f 22 ( p, k ) f *22 ( p', k ) k × ε k – ------ – δ 0 − + i0 2µ
(18)
+ Σ 12 ( p 1, ω 1 )g 21 ( p 1, ω 1, T ) –1
2
–+
++
+ Σ 21 ( p 1, ω 1 )g 11 ( p 1, ω 1, T ) ––
p; k ) )*.
–+
+ Σ 21 ( p 1, ω 1 )g 22 ( p 1, ω 1, T ) ++
± Γ αβ
We obtained the expression for the vertex part in terms of the scattering amplitudes of the two-channel
–+
–+
+ Σ 21 ( p 1, ω 1, T ) ( g 22 ( p 1, ω 1 ) + g 11 ( p 1, ω 1 ) ), –+
––
OPTICS AND SPECTROSCOPY
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KINETIC EQUATIONS FOR THE DENSITY MATRIX
where –– g 11 ( p )
Let us substitute (20) and (21) into set (18) and integrate it over ω1 by using definitions (19). As the result, we obtain the set of equations for the elements of the density matrix ραβ(p, T)):
= ( ω – ε p + i0 ) , –1
g 22 ( p ) = ( ω – ε p – δ 0 + i0 ) ,
g αα = – ( g αα )*,
–1
––
++
––
g 11 ( p, ω, T ) = 2πiρ 11 ( p, T )δ ( ω – ε p ), –+
(19)
∂ρ 11 ( p 1, T ) ( 11 ) i --------------------------= –s 11 ρ 11 ( p 1, T ) + s 12 ρ 21 ( p 1, T ) ∂T
g 22 ( p, ω, T ) = 2πiρ 22 ( p, T )δ ( ω – ε p – δ 0 ), –+
–+ g 21 ( p,
( 11 )
– ( s 12 )*ρ 12 ( p, T ) + S 11 ,
ω, T ) = 2πiρ 21 ( p, T )δ ( ω – ε p – δ 0 /2 ),
∂ρ 22 ( p 1, T ) ( 22 ) i --------------------------= –s 22 ρ 22 ( p 1, T ) + s 21 ρ 12 ( p 1, T ) ∂T
and ραβ(p, T) are the density matrix elements [1]; P2 –– – iΣ αβ ( 1,
( 22 )
– ( s 21 )*ρ 21 ( p, T ) + S 22 ,
2) =
∂ρ 21 ( p 1, T ) i --------------------------– δ 0 ρ 21 ( p 1, T ) ∂T
α P1 β
∫
= ( – 1 ) d p 2 ig ( p 2 ) ( – iΓ αβ ( p 1, p 2 ; p 2, p 1 ) ). 4
–
––
( 21 )
= –s 21 ρ 21 ( p 1, T ) + s 21 ρ 11 ( p 1, T )
For the buffer atom, we have the hole line; therefore,
∫
dω 2 –– --------- g ( p 2 ) f ( p 2 ) = N p2 f ( p 2, ω 2 = E p2 ), 2π
where N p2 is the momentum distribution function of the buffer gas and f(p) is an arbitrary function. Taking this into account, we have –– Σ αβ ( p 1,
∫
ω1 ) =
– dp 2 N p2 Γ αβ ( p,
p; k ),
(20)
where p = (m2p1 – m1p2)/m is the momentum of the center of gravity of the colliding particles, k = (k, ω), k = ++ –– p1 + p2, ω = ω1 + E p2 ; Σ βα = –( Σ αβ )*. –+ – iΣ αβ ( p 1,
= ( –1 )
ω 1, T ) =
∑ ∫ d qd 4
4
∑
γ
δ β
( 21 )
– s˜ 21 ρ 22 ( p 1, T ) + S 21 , 2π * ( p, p ) ), s αα = ------ dp 2 N p2 ( f αα ( p, p ) – f αα µ
∫
α = 1, 2;
p = ( m 2 p 1 – m 1 p 2 )/m,
2π 2 S 11 = 2πi ------ µ
+
∑ ∫ dq dp N 2
p2 – q
γ , δ = 1, 2
γδ p – (p + q) × A 11 ( p, p + q )ρ γδ ( p 1 + q, T )δ ------------------------------- – l γδ , 2µ 2
2π 2 S 22 = 2πi ------ µ
P2 +
γ , δ = 1, 2 α
601
2
∑ ∫ dq dp N 2
p2 – q
γ , δ = 1, 2
γδ
× A 22 ( p, p + q )ρ γδ ( p 1 + q, T )
p 2 iΓ δβ ( p 1 + q, p 2 – q; p 2, p 1 ) +
γ , δ = 1, 2
× ( – iΓ αγ ( p 1, p 2 ; p 2 – q, p 1 + q ) )ig ( p 2 – q )
p – (p + q) × δ ------------------------------- + δ 0 – l γδ , 2µ 2
2
–+
–
2π 2 S 21 = ------ µ
× ig ( p 2 )ig γδ ( p 1 + q, T ), +–
–+
Σ αβ ( p 1, ω 1, T ) –+
∑ ∫
= ( – 2πi )
(21)
∑ ∫ dq dp N 2
γδ
× A 21 ( p, p + q )ρ γδ ( p 1 + q, T )Z γδ , (p + q) – p = ------------------------------- + l γδ – i0 2µ 2
– dq dp 2 N p2 – q Γ αγ ( p,
p + q; k )
p2 – q
γ , δ = 1, 2
Z γδ
2
–1
γ , δ = 1, 2
2
+
× δ ( ω 1 – ε p1 + q + E p2 – E p2 – q – l γδ ), l 11 = 0,
l 22 = δ 0 ,
( 11 )
s 12
l 12 = l 21 = δ 0 /2.
OPTICS AND SPECTROSCOPY
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(p + q) – p – ------------------------------- + l γδ – δ 0 + i0 , 2µ
× Γ δβ ( p + q, p; k )ρ γδ ( p 1 + q, T )
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2
2π = ------ dp 2 N p2 – f 12 ( p, p ) µ
∫
602
TITOV
2π + ------ dk ( f 11 ( p, k ) f *21 ( p, k )L 1+ ( p ) µ
2π * ( p, k )L 2+ * (p + q) + ------ dk ( f 11 ( p + q, k ) f 11 µ
* ( p, k )L 2– ( p ) ) , - + f 12 ( p, k ) f 22
* ( p, k )L 1– * (p + q) ) , --- + f 12 ( p + q, k ) f 12
∫
( 22 )
s 21
∫
A 11 = ( A 11 )*, 12
2π = ------ dp 2 N p2 – f 21 ( p, p ) --µ
∫
21
22 * ( p, p + q ), A 22 = f 22 ( p, p + q ) f 22
2π * ( p, k )L 2+ ( p ) + ------ dk ( f 21 ( p, k ) f 11 µ
* ( p, p + q ), A 22 = f 21 ( p, p + q ) f 21
* ( p, k )L 1– ( p ) ) , - + f 22 ( p, k ) f 12
A 22 = – f 21 ( p, p + q ) -
∫
11
12
2π * ( p + q, k )L 2+ ( p + q ) + ------ dk ( f 21 ( p, k ) f 11 µ
2π * ( p, p ) --s 21 = ------ dp 2 N p2 f 22 ( p, p ) – f 11 µ
∫
∫
2π * ( p, k )L 2– ( p ) – ------ dk ( f 22 ( p, k ) f 22 µ
∫
* ( p + q, k )L 1– ( p + q ) ) - + f 22 ( p, k ) f 12
* ( p, k )L 1+ ( p ) + f 21 ( p, k ) f 21
* ( p, p + q ) × – f 22
* ( p, k )L 2+ * (p) – f 11 ( p, k ) f 11
2π * ( p, k )L 2– * (p + q) + ------ dk ( f 22 ( p + q, k ) f 22 µ
∫
* ( p, k )L 1– * (p) ) , - – f 12 ( p, k ) f 12 s 21
( 21 )
2π = – ------ dp 2 N p2 f 21 ( p, p ), µ
( 21 ) s˜ 21
2π * ( p, p ), = – ------ dp 2 N p2 f 12 µ
∫
∫
* ( p, k )L 1+ * (p + q) ) , - + f 21 ( p + q, k ) f 21 (22)
–1 –1 2 2 2 2 p –k δ p –k L 1±(p) = ----------------- ± -----0 + i0 – ----------------- ± δ 0 + i0 , 2µ 2µ 2
A 22 = ( A 22 )*, 21
* ( p, p + q ), A 21 = f 21 ( p, p + q ) f 11 11
* ( p, p + q ), A 21 = f 22 ( p, p + q ) f 12 22
–1 –1 2 2 2 2 p – k δ0 p –k L 2±(p) = ----------------- ± ----- + i0 – ----------------- + i0 , 2µ 2µ 2 11 * ( p, p + q ), A 11 = f 11 ( p, p + q ) f 11
* ( p, p + q ), A 11 = f 12 ( p, p + q ) f 12
12
A 21 = – f 22 ( p, p + q ) 21
2π * ( p + q, k )L 2– ( p + q ) + ------ dk ( f 22 ( p, k ) f 22 µ
∫
22
* ( p + q, k )L 1+ ( p + q ) ) - + f 21 ( p, k ) f 21
A 11 = – f 12 ( p, p + q ) 21
* ( p, p + q ) × – f 11 2π * ( p + q, k )L 1+ ( p + q ) + ------ dk ( f 11 ( p, k ) f 21 µ
∫
* ( p + q, k )L 2– ( p + q ) ) --- + f 12 ( p, k ) f 22 * ( p, p + q ) × – f 11
2π * ( p, k )L 2+ * (p + q) + ------ dk ( f 11 ( p + q, k ) f 11 µ
∫
* ( p, k )L 1– * (p + q) ) , - + f 12 ( p + q, k ) f 12 A 21 = ( A 21 )*. 12
OPTICS AND SPECTROSCOPY
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KINETIC EQUATIONS FOR THE DENSITY MATRIX
Since the elements of the initial density matrix are linear combinations of the transformed density matrix, then, finding ραβ, we thus determine the elements of the initial density matrix. An example of application of the set of equations (18) and (22) will be given in a subsequent paper.
603
Summing over i, we obtain the expression for Vˆ in terms of the scattering amplitude Vˆ p – p' = ˆf ( p, p' ) + (1) + + dk [ ˆf ( p, k )χˆ 1 χˆ 1 ˆf +( p', k ) ( Lˆ k ( p' ) )
∫
(A3)
+ (2) + + ˆf ( p, k )χˆ 2 χˆ 2 ˆf +( p', k ) ( Lˆ k ( p' ) ) ].
CONCLUSIONS Here, we obtained the set of equations for the density matrix of two-level particles resonantly interacting with an electromagnetic field for the case of arbitrary values of the ratios of the Rabi frequency and the detuning of the electromagnetic field frequency from the atomic transition frequency to the inverse collision time of the particles. The coefficients in the equations were expressed in terms of the vertex parts (or the scattering operators) taken beyond the mass surface. The calculation of the vertex parts was reduced to the problem of two-channel scattering and they were expressed in terms of the corresponding scattering amplitudes. The latter can be obtained by solving the Schrödinger equation. In the eikonal approximation, the corresponding set of equations is presented in the Appendix. On the basis of this, we can consider the problem solved in the closed form. It is clear that the set of equations obtained is of interest for investigating the absorption of radiation of an arbitrary intensity in the wings of lines. In a subsequent paper, by using the equations obtained, we will present formulas for the absorption coefficient in the statistical wing of an absorption line in the case in which collisions occur with a two-level particle at either of the levels. APPENDIX (1) Here, we will derive important relationships for the scattering amplitudes ˆf . We have the expression ˆf ( p, k )χˆ = i
∫
(i)
ˆ k ( p – q ), dqVˆ q ψ
+ Vˆ p – p' χˆ i χˆ i
ˆ
i
+ i [ δ(p'
2
2
2
k – p' × ------------------ + δ 0 – i0 2µ
Vol. 97
No. 4
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–1
2
–1
2
k –p – ----------------- + δ 0 + i0 , 2µ
2
–1
2
k – p' × ------------------ – δ 0 – i0 2µ
2
–1
2
k –p – ----------------- + i0 2µ
* ( p', k ) + f 12 ( p, k ) f 22 2
2
k – p' × ------------------ – i0 2µ
–1
2
–1
2
k –p – ----------------- + δ 0 + i0 , 2µ
(A4)
∫
* ( p', k ) * ( p', p ) = – dk f 21 ( p, k ) f 11 f 21 ( p, p' ) – f 12 2
2
k – p' × ------------------ – i0 2µ
–1
2
–1
2
k –p – ----------------- – δ 0 + i0 2µ
* ( p', k ) + f 22 ( p, k ) f 12 2
k – p' × ------------------ + δ 0 – i0 2µ
–1
2
2
–1
k –p – ----------------- + i0 , 2µ
∫
* ( p', p ) = – dk f 22 ( p, k ) f 22 * ( p', k ) f 22 ( p, p' ) – f 22 2
k – p' × ------------------ – i0 2µ
–1
2
2
–1
k –p – ----------------- + i0 2µ
* ( p', k ) + f 21 ( p, k ) f 21 2
∫
–1
2
∫
2
k – p' × ------------------ – δ 0 – i0 2µ
+ (i) + + dk ˆf ( p, k )χˆ i χˆ i ˆf +( p', k ) ( Lˆ k ( p' ) ) .
2
k –p – ----------------- + i0 2µ
* ( p', k ) * ( p', p ) = – dk f 11 ( p, k ) f 21 f 12 ( p, p' ) – f 21
(A2)
+ = ˆf ( p, p' )χˆ i χˆ i
–1
* ( p', k ) + f 12 ( p, k ) f 12
2
+ (i) – k) + ˆf +(p, k) ( Lˆ k (p') ) ]
OPTICS AND SPECTROSCOPY
2
k – p' × ------------------ – i0 2µ
2
(i)
∫ dk f (p, k)χ χ
∫
* ( p', k ) * ( p', p ) = – dk f 11 ( p, k ) f 11 f 11 ( p, p' ) – f 11
(A1)
ˆ q (p) are the wave functions of the scattering where ψ problem with the ith initial channel. By using condition (14), we obtain
=
Further (see [2]), by using the Hermiticity of the potential Vˆ , we obtain the following relationships for the scattering amplitudes:
–1
2
2
–1
k –p – ----------------- – δ 0 + i0 . 2µ
604
TITOV
(2) Let us find equations for the scattering amplitudes in the eikonal approximation. We will return to definitions and Eqs. (6)–(10): ˆf ( p, p' )χˆ = i
∫
ˆ (p'i ) ( p dkVˆ k ψ
– k)
∫ dre
– ipr
f ( p, p' ) =
∫ dre
– i ( p – p' )r
(1) ψ 1p' ( r ) (1) ψ 2p' ( r )
(2)
Sˆ (r, p') χˆ i .
Vˆ ( r )Sˆ ( r, p' ).
(A5)
If we begin with channel 1, then 1 p') or 0
= e ip'r S 11 . S 21
=e
ip'r
Sˆ (r,
(A6)
Substituting this expression into (6), we obtain, in the coordinate representation, ∆ (1) p' ( 1 ) (1) (1) – ------ψ 1p (r) + V 1 ψ 1p' (r) + V 0 ψ 2p' (r) – ------ψ 1p' (r) = 0, 2µ 2µ 2
∆ (1) (1) (1) – ------ψ 2p' ( r ) + δ 0 ψ 2p' + V 2 ψ 2p' ( r ) 2µ
v = p'/µ,
For ψ p' (r), we have
ip'r
ˆ (p'1 ) (r) ψ
(A7)
where the x axis is directed along the vector p'.
ˆ (p'i ) ( r ). Vˆ ( r )ψ
ˆ (p'i ) (r) = e Let us introduce the S matrix: ψ Then
∂S 11 - = V 1 S 11 + V 0 S 21 , i v --------∂x ∂S 21 - – δ 0 S 21 = V 0 S 11 + V 2 S 21 , i v --------∂x
or, returning to the spatial variables, ˆf ( p, p' )χˆ = i
Substituting here (A6) and neglecting the second derivatives, we obtain
(2) ψ 1p' ( r ) (2) ψ 2p' ( r )
= e ip'r S 12 , S 22
∂S 12 - + δ 0 S 12 = V 1 S 12 + V 0 S 22 , i v --------∂x ∂S 22 - = V 0 S 12 + V 2 S 22 . i v --------∂x
(A8)
After finding the elements of the S matrix from these equations, we can, according to (A5), determine the scattering amplitudes. REFERENCES 1. E. M. Lifshitz and L. P. Pitaevskiœ, Physical Kinetics (Nauka, Moscow, 1979; Pergamon, Oxford, 1981). 2. E. A. Titov, Opt. Spektrosk. 96, 945 (2004) [Opt. Spectrosc. 96, 869 (2004)]. 3. S. I. Yakovlenko, Usp. Fiz. Nauk 136, 593 (1982) [Sov. Phys. Usp. 25, 216 (1982)]. 4. É. G. Pestov, Tr. Fiz. Inst. im. P. N. Lebedeva, Akad. Nauk SSSR 187, 60 (1988).
2
p' ( 1 ) (1) + V 0 ψ 1p' ( r ) – ------ψ 2p' ( r ) = 0. 2µ
Translated by N. Reutova
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No. 4
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