Two-pulse spin echo in two-level systems inside ... - physics.sk

68 L. L. Chotorlishvili et al.
α 1 = cos [Ω 1 (∆ω)t 1 /2] + i cos [θ 1 (∆ω)] sin [Ω 1 (∆ω)t 1 /2] ,
β 1 = sin [θ 1 (∆ω)] sin [Ω 1 (∆ω)t 1 /2] ,
β 2 = sin [θ 2 (∆ω)] sin [Ω 2 (∆ω)t 2 /2] ,
[
Ω 1,2 (∆ω) = γnη 2 2 (∆ω)(h (1,2) ) 2 + ∆ω 2] 1/2
,
tn[θ 1,2 (∆ω)] = γ n h (1,2) η(∆ω)/∆ω .
Here h (1,2) is the radio-frequency field amplitude, t 1 and t 2 denote the duration of radiofrequency
field.
When we have wide angles of rotation (Ω 1,2 t 1,2 >> 2π) for the same radio-frequency
pulses, the integration by ∆ω removes fast oscillating factors sin Ωt and cos Ωt down to zero.
Therefore, the formula (1) takes the simplest form:
|∆M(ω, t)| = 3m δω/2
∫
0
16 ∣ d(∆ω)(ω nd − ω nw ) −1/2 (∆ω d − ∆ω) −1 (∆ω − ∆ω w ) −1/2 ·
−δω/2
(2)
· ∆ωη 4 (∆ω)(γ n h) 3[ ∆ω 2 + (γ n η(∆ω)h) 2] 2 exp [−i∆ω(t − 2τ12 )]
∣ .
The peculiarity of TLS is the dispersion of splitting energy E requiring the obtained expression
to be averaged by distribution function. The further simplification of (2) also depends on
E. When E/¯h ∼ ω n where ω n is the carrier frequency of the receiver, δω >> γ n hη(∆ω) is
valid δω is the TLS line shape width caused by Klauder-Anderson mechanism [17], equals to the
passband width of the receiver). In this case the expression
∆ω exp [−i∆ω(t − 2τ 1,2 )] ÷ [ ∆ω 2 + (γ n hη(∆ω)) 2] 2
is the fast-oscillating factor under the integral (2). Therefore, for ∆ω = 0 other factors can
be removed from under the sign of the integral and thereafter the limits of integration tend to
infinity. As a result, taking into account the averaging by the TLS distribution function, we have
¯p
E∫
max
∫ E
E=0 ∆ 0=0
EdEd∆ 0
∆ 0
√
E2 − ∆ 2 0
,
here ¯p is the TLS state density and ∆ 0 is the TLS tunneling parameter. For the resonant TLS, the
energy of which is within the interval ¯hω n − ¯hδω < E < ¯hω n + ¯hδω, we have
|M(ω, t)| ∼ ¯p
¯hω n+¯hδω ∫
∫ E
¯hω n−¯hδω ∆ 0=0
√ ∣∣∣∣
E/¯h − ω n
E/¯h − ω
∣ |t − 2τ 12| exp
[− (|E/¯h − ω n|) 1/2 ]
EdEd∆ 0
√
|t − 2τ 12 | ∆ 0 E2 − ∆ 2 0
(3)