Atom–Field Interactions
3.8 Bloch equations
exp
−i t
0
δω (t) dt
= exp (−γdepht) . (3.7.10) Using Eq. (3.7.10) in Eq. (3.7.6), we find the result
ρ21(t) = ρ21(0) exp [− (iω0+ Γ21+ γdeph) t] . (3.7.11) By defining a total decay rate for dephasing according to
Γ≡ Γ21+ γdeph= 1
2(γ2+ γ1) + γdeph, (3.7.12) the equation of motion for the coherence becomes
˙
ρ21=− (iω0+ Γ) ρ21+ i
V21(ρ22− ρ11) . (3.7.13) Exercise: Verify that a random frequency shift δω << ω has no effect on the level populations determined by ρ22 and ρ11.
The phenomenological decay terms treated above include two types, differing by whether they affect the diagonal or off-diagonal elements of the density matrix.
The two types are associated with population decay and polarization dephasing, respectively, and are important not only because they extend the applicability of our analysis, but together they provide overall conservation of occupation probabilities, unlike the squared amplitudes |C2|2 and |C1|2 alone. In Section 3.8 we develop a pictorial analogy, called the vector model, between the evolution of the populations and polarizations of a two-level atom and gyroscopic motion. It is useful not only for understanding all the aspects of dynamics covered so far – off-resonant excitation, Rabi flopping, population decay, and dephasing – but will later be used to picture the results of coherent interactions involving multiple pulses. The importance of even weak dephasing is illustrated by Problem 3.3.
3.8 Bloch equations
Having discussed a couple of important categories of relaxation process, and antic-ipating how to handle them, we now modify Eq. (3.6.19) using a phenomenological procedure that preserves the Hermitian character of H while adding decay terms to the equation of motion. The assumed form of radiative decay terms will be fully justified later in Chapter 6.
i ˙ρ = [H, ρ] + relaxation terms
= [H, ρ]± iΛ ± iγρ − iΓρ (3.8.1) In Eq. (3.8.1) we have divided relaxation terms into separate categories for incoherent pumping (Λ), radiative and non-radiative population relaxation (γ), and dephasing (Γ). Population changes due to incoherent pumping processes are particularly chal-lenging to write down at this stage, even if energy pathways of the system are thought
Bloch equations 59 to be fairly well understood. They may contribute population relaxation terms of the form ˙ρii∝ ±iΛ in open systems where external mechanisms can add or remove population without preserving the optically excited number density. On the other hand, incoherent pumping may contribute terms of the form ˙ρii ∝ ±iΛijρjj in closed systems subject to thermal excitation from one level to another. Although relaxation has been introduced in a semiempirical way in Eq. (3.8.1), we shall make extensive use of this equation of motion in Chapters 4 and 5 to get started with the analysis of optical dynamics. Its form will be justified more rigorously later.
Under certain conditions, the density matrix equations of motion can be cast into a convenient form known as the optical Bloch equations, resembling equations for gyroscopic precession. The simplest version of these equations is called the vector model and illustrates that two-level atoms are analogous to spin 1/2 particles. This model provides a geometrical picture of the development of optical polarization in time, and simplifies calculations of coherent aspects of multiple pulse interactions. However, the Bloch equations originated from studies of spin magnetism and assume that diagonal and off-diagonal elements decay at similar rates. Hence, despite the visual
“appeal” of the vector model, as described below, solutions of the Bloch equations should not be confused with full solutions of the density matrix.
Using the RWA, the atom–field interaction is
V21=−12µ21E0e−iωt, (3.8.2) and we introduce the slowly varying envelope approximation (SVEA) by writing
ρ21= ˜ρ21e−iωt, (3.8.3)
where ˜ρ21denotes the slowly varying amplitude of the matrix element ρ21proportional to the amplitude of charge oscillation at the optical frequency. With the use of Eqs. (3.8.2) and (3.8.3), the equation of motion Eq. (3.7.3) becomes
˙˜
ρ21=− [i (ω0− ω) + Γ] ˜ρ21− i 2
µ21E0
(ρ22− ρ11) .
By defining ∆≡ ω0− ω as the detuning of the optical frequency ω from the resonance at ω0, one obtains
d
dt+ i∆ + Γ
˜
ρ21=−i
2Ω (ρ22− ρ11) . (3.8.4) A simple, geometrical picture of system evolution emerges if we define components of a three-space vector by the real quantities
R1= ˜ρ21+ ˜ρ12, (3.8.5)
R2=−i (˜ρ21− ˜ρ12) , (3.8.6)
R3= ρ22− ρ11. (3.8.7)
These quantities vary slowly in an optical period, and because they are mutually independent can be viewed as components of the vector
R = R¯ 1ˆe1+ R2ˆe2+ R3ˆe3 (3.8.8) in a fictitious space spanned by the orthogonal basis set (ˆe1, ˆe2, ˆe3). The time devel-opment of each component is given by
R˙1= ∆R2− ΓR1, (3.8.9)
R˙2=−∆R1− ΓR2− ΩR3, (3.8.10) R˙3=− [γ2ρ22− γ1ρ11] + ΩR2. (3.8.11) The rate Γ is the reciprocal of the dephasing time T2:
Γ = T2−1. (3.8.12)
Equations (3.8.13)–(3.8.15) are the optical Bloch equations. They can be written more compactly as
Because Eq. (3.8.16) is the equation of motion of a gyroscope, it shows that elements of the density matrix can be chosen (in the limiting case above) as components of a Bloch vector ¯R which executes simple precession about an effective field ¯β (Fig. 3.8).
The length of ¯R shrinks with time as exp[−t/T ] due to decay processes.
Off-resonance, the effective field or torque vector ¯β is a constant vector in the ˆ
e1− ˆe3 plane in the absence of decay. In this case, a solution to Eq. (3.8.16) without the relaxation term is obtained by finding a unitary transformation which turns the Bloch vector into a constant vector. To freeze the action of the Bloch vector, we first rotate about ˆe2by angle α. Coordinates transform according to
Bloch equations 61
R β
ê2
ê1
ê3
Figure 3.8 Precessional motion of the Bloch vector ¯R around the effective field ¯β. For the orientation of β that is shown, the detuning is negative.
so the Bloch vector transforms according to
R1
R2 R3
=
cos α 0 sin α
0 1 0
− sin α 0 cos α
R1 R2 R3
. (3.8.18)
Exercise: Draw the direction of the effective field ¯β (see Fig. 3.9) and sketch the motion of the Bloch vector ¯R as a function of time, ignoring decay (T =∞). What is the significance of ¯R passing periodically through the±ˆe3 directions when ∆ = 0? What is the atomic state when ¯R points along ˆe2?
In this process, since the effective field ¯β now lies along the new (primed) ˆe1-axis, the Bloch vector merely executes a precession counter-clockwise about ˆe1. A second rotation about the ˆe1-axis therefore leads to a frame in which ¯R is stationary. The
ê3
ê2
ê1 a
b
Figure 3.9 The effective field ¯β in the Bloch vector model. ¯β makes an angle α with respect to the ˆe1 -axis in the plane of basis vectors ˆe1, ˆe2 such that tan α = ∆/Ω for positive detuning.
rotation angle must be ΩR(∆)t. same as the initial Bloch vector ¯R(0) apart from a single coordinate rotation through angle α which we must “undo.”
R¯=
where the evolution operator U is given by
U =
In the case of nuclear magnetic resonance, the spins precess in real space (with an ˆ
e3-axis determined by the direction of the static field) about an effective magnetic field ¯β consisting of the vector sum of applied static and oscillating magnetic fields.
In the electric dipole or optical case considered here, there is no static field present, so the precession is not in real space. There is no fixed direction associated with the population difference R3= ρ22− ρ11. Nevertheless, there is a polarization axis, and in keeping with the spin analogy T2 is called the “transverse” decay time since it
Inhomogeneous broadening, polarization, and signal fields 63 describes relaxation of transverse components of the Bloch vector. T1 is called the
“longitudinal” relaxation time since it applies to the ˆe3 or ˆz component.