The equations describing the system dissipative density operator is (Scully the Quantum Optics. \ (\ S5.3 \) )
\ [\ DOT {P} = - \ FRAC {I} {\ hbar} [H, P] - \ frac {1} {2}
\ {\ Gamma, p \} \ quad \ gamma_ {ij} = \ gamma_ {ij} \ delta_ {ij} \] two-level atoms in the dipole approximation, the above equation with there
\ [\ dot {\ rho} _ {aa} = - \ gamma_ {a} \ rho_ {aa} + \ frac {i} {\ hbar} \ left [\ wp_ {ab} E \ rho_ {ba} - \ text {cc} \ right] \\ \ dot {\ rho} _ {bb} = - \ gamma_ {b} \ rho_ {bb} - \ frac {i} {\ hbar} \ left [\ wp_ {ab} E \ rho_ {ba} - \ text {cc} \ right] \\ \ dot {\ rho} _ {ab} = - (i \ omega + \ gamma_ {ab}) \ rho_ {ab} - \ frac {i} {\ hbar} \ wp_ {ab
} E (\ rho_ {aa} - \ rho_ {bb}) \] where \ (\ Omega \) atomic transition frequency (a is an upper level, b is a lower level), \ (\ gamma_ {ab &} = (\ gamma_a + \ gamma_b) / 2 \) , and \ (\ wp_ {ab} \ ) is the dipole moment (in fact, a vector, taken as real), and \ (E ( T) = E_0 \ COS (\ NU T) \) (also a vector). In the rotating wave approximation, and it is equivalent to \ (\ wp_ {ab} \ )Multiplied \ (E (t) \) are changed to \ (E_0 \ exp (-i \ NU T) / 2 \) , and \ (\ wp_ {ba} \ ) multiplied by \ (E (t) \) to \ (E_0 \ exp (I \ NU T) / 2 \) .
Collisions between atoms, can be considered atomic level experiencing random Stark shift, in \ (E_0 = 0 \) collision still exists, and to \ (\ delta \ omega (t ) \) representing the random Stark displacement. Suppose \ (\ delta \ omega (t ) \) a Gaussian random process, the zero mean \ (\ langle \ Delta \ Omega (T) \ rangle_ \ text {em} = 0 \) (ensemble average subscript em Representative lines followed omitted EM), and satisfies \ (\ langle \ Delta \ Omega (T) \ Delta \ Omega (T ') \ rangle = 2 \ gamma_ \ pH {text} \ Delta (T-T') \) . the above equations thereby in \ (E_0 = 0 \) is rewritten when the correction, because the time derivative of the angle element-free \ (\ Omega \) , so only need to rewrite the third equation. The third form of integral equations to obtain
\ [{\ rho} _ { ab} (t) = \ exp \ left [- \ left (i \ omega + \ gamma_ {ab} \ right) ti \ int_ {0} ^ {t} dt ^ {\
prime} \ delta w \ left (t ^ {\ prime} \ right) \ right] \ left. \ rho_ {ab} (0) \ right. \] then both sides take the ensemble average , find it necessary to calculate the ensemble average of only
\ [\ Left \ langle \ exp \ left (-i \ int_ {0} ^ {t} \ delta \ omega (t ') dt ^ {\ prime} \ right) \ right \ rangle = \ sum_i \ frac {( -i) ^ n} {n! } \ left \ langle \ left (\ int_ {0} ^ {t} \ delta \ omega \ left (t ^ {\ prime} \ right) d t '\ right) ^ { n} \ right \ rangle \]
on the right side of the formula \ (n-\) power can be written as desired (switching ensemble average and time integration)
\ [\ left \ langle \ left (\ {0} ^ {the int_ T } \ delta \ omega \ left ( t ^ {\ prime} \ right) d t '\ right) ^ {n} \ right \ rangle = \ int_0 ^ tdt_1 \ cdots \ int_0 ^ tdt_n \ langle \ delta \ omega (t_1 ) \ cdots \ delta \ omega (
t_n) \ rangle \] where \ (\ langle \ Delta \ Omega (T_l) \ cdots \ Delta \ Omega (T_n) \ rangle \) , may be a Gaussian distribution in accordance with the multi-variable with zero mean moment Theorem [Ref.2], in \ (n-\) to expand is written is an even number (odd number equal to 0 the formula)
\ [\ Langle \ delta \ omega (t_1) \ cdots \ delta \ omega (t_n) \ rangle = \ langle \ delta \ omega (t_1) \ delta \ omega (t_2) \ rangle \ cdots \ langle \ delta \ omega ( t_) \ delta \ omega ()
\ {n-1} t_ {n} rangle + \ cdots \] the first term on the right hand side to match the two mean the product, such a product has a contribution, so to traverse all right All such possibilities paired two by two, a total of \ (n! / [2 ^ {n / 2} (n / 2)!] \) of such terms, the contribution of each one are equal.
Thus obtained after finishing
\ [\ left \ langle \ operatorname {exp} \ left (-i \ int_ {0} ^ {t} \ delta \ omega (t ') dt ^ {\ prime} \ right) \ right \ rangle = \ sum_ {n = 0} ^ {\ infty} \ frac {(- 1) ^ {n}} {n!} \ gamma_ \ text {ph} ^ {n} t ^ {n} = e ^ {- \ gamma_ \ text {ph} t
} \] so in \ (E_0 = 0 \) when there
\ [\ rho_ {ab} ( t) = \ exp [- (i \ omega + \ gamma_ {ab} + \ gamma_ \ text {ph}) t] \
] is \ (\ dot {\ rho} _ {ab} (t) = - (i \ omega + \ gamma_ {ab} + \ gamma_ \ text {ph}) \ rho_ {ab } (T) \) , then the \ (E_0 \ neq0 \) time is (are rotary wave approximation, and taking the electric dipole moment is a real number)
\ [\ DOT {\ Rho} _ {ab &} = - ( i \ omega + \ gamma_ {ab } + \ gamma_ \ text {ph}) \ rho_ {ab} - \ frac {i} {2 \ hbar} \ wp E_0 \ text {e} ^ {- i \ nu t} ( \ rho_ {aa} - \ rho_
{bb}) \] under rotation wave approximation, the other two formulas to
\ [\ Dot {\ rho} _ {aa} = - \ gamma_a \ rho_ {aa} + \ frac {iE_0 \ wp} {2 \ hbar} \ left (\ text {e} ^ {- i \ nu t} \ rho_ {ba} - \ text {e} ^ {i \ nu t} \ rho_ {ab} \ right) \\ \ dot {\ rho} _ {bb} = - \ gamma_b \ rho_ {bb} - \ frac {iE_0 \ wp} {2 \ hbar} \ left (\ text {e} ^ {- i \ nu t} \ rho_ {ba} - \ text {e} ^ {i \ nu t} \ rho_ {ab} \ right) \]
this is the equation of motion after considering the density operator of atomic collisions.
In the laser theory, the relaxation effect is not used \ (gamma_i \ \) be expressed using the expression of the relaxation rates to equilibrium (equilibrium under the action of the pump, instead of thermal equilibrium), the above equation can be further simplified.
In the laser theory, when the \ (E_0 = 0 \) , the difference in the number of particles that \ ((\ rho_ {bb} - \ rho_ {aa}) \) exponentially relax to equilibrium value \ ((\ rho_ {BB} - \ rho_ {AA}) _ 0 \) . originally, using the above differential equation can be obtained
\ [(\ dot {\ rho } _ {bb} - \ dot {\ rho} _ {aa}) = - \ gamma_b \ rho_ {bb} + \ gamma_a \ rho_ {aa} + \ frac {iE_0 \ wp} {\ hbar} \ left (\ text {e} ^ {i \ nu t} \ rho_ {ab} - \ text {e} ^ {i \ nu
t} \ rho_ {ba} \ right) \] discarded now front right side of equations two, and to the balance value \ ((\ rho_ {bb} - \ rho_ {aa}) _ 0 \ ) relaxation in place of, or written as
\ [(\ dot {\ rho } _ {bb} - \ dot {\ rho} _ {aa}) = - \ frac {(\ rho_ {bb} - \ rho_ {aa }) - (\ rho_ {bb } - \ rho_ {aa}) _ 0} {\ tau} + \ frac {iE_0 \ wp} {\ hbar} \ left (\ text {e} ^ {i \ nu t} \ rho_ {ab} - \ text {
e} ^ {i \ nu t} \ rho_ {ba} \ right) \] If the density operator for the use of non-diagonal elements of substitution \ (\ sigma_ {ab} = \ text {e} ^ {i \ nu t} \ sigma_ {ab} \) substitution (i.e. slowly varying amplitude conversion), the equation of motion can be rewritten as
\ [(\ Dot {\ rho } _ {bb} - \ dot {\ rho} _ {aa}) = - \ frac {(\ rho_ {bb} - \ rho_ {aa}) - (\ rho_ {bb} - \ rho_ {aa}) _ 0} {\ tau} + \ frac {iE_0 \ wp} {\ hbar} \ left (\ sigma_ {ab} - \ sigma_ {ab} ^ * \ right) \\ \ dot {\ sigma} _ {ab} = i (\ nu- \ omega) \ sigma_ {ab} - \ gamma_ \ text {ph} \ sigma_ {ab} + \ frac {iE_0} {2 \ hbar} \ wp (\ rho_ { bb} - \ rho_ {aa}
) \] for substitution \ (a \ leftrightarrow2, b \ leftrightarrow1, \ wp \ leftrightarrow \ mu, \ gamma_ \ text {ph} \ leftrightarrow1 / T_2 \) is ethylene in the formula is husband "quantum Electronics" \ ((8.1-11 \ SIM 12) \) type.
Reference.
Scully. Quantum Optics.
K. Triantafyllopoulos. On the Central Moments of the Multidimensional Gaussian, The Mathematical Scientist. 28 (2003).
Yali Fu. Quantum Electronics (1983).