Transient Optical Response
4.1 Optical nutation
Transient Optical Response
Although the principal focus of subsequent chapters will be detailed calculations of steady-state response to light under various conditions, it is helpful to have a mental picture of how polarization grows, decays, and mediates transient interactions using the Bloch equations of Chapter 3. Among other things, this helps one develop a sense for what determines when dynamics should be considered “fast” or “slow.” A key part of the successful analysis of dynamics in new systems lies in categorizing processes as
“fast” or “slow” and deciding which quantities should be retained as time-dependent variables once the timescale of the analysis is specified.
This chapter also provides physical insight into the meaning of “dephasing” or
“decoherence,” and provides an easy way to introduce some surprisingly interesting coherent phenomena induced by repeated applications of pulsed applied fields. Inter-esting connections also exist between irreversible transient phenomena and frequency shifts, but these are deferred to a discussion in Appendix F. Finally, some limitations associated with the use of the simple Bloch vector model are noted at the end of the chapter, as a prelude to focusing on the full term-by-term density matrix treatment used in Chapters 5–7, which provides enough degrees of freedom for analytic solutions to all problems of interest.
4.1 Optical nutation
4.1.1 Optical nutation without damping
Consider what happens when atoms are suddenly exposed to light that is resonant with a ground state transition. This gives rise to a coherent transient phenomenon called nutation, which is the driven response of atoms to light illustrated in Fig. 4.1.
Nutation describes the buildup of polarization caused by the application of an optical field.
Simple analytic solutions can be obtained, provided damping is ignored, even for atoms moving with a velocity vzthat modifies their detuning by the Doppler shift. The Bloch equations (Eqs. 3.8.13–3.8.15) describing evolution of the Bloch vector reduce to
R˙1− ∆R2= 0, (4.1.1)
R˙2+ ∆R1+ ΩR3= 0, (4.1.2)
R˙3− ΩR2= 0. (4.1.3)
Optical nutation 77
t
P(t)
P(t)
P(t)
(a)
t
t (b)
(c)
Ω–1
Figure 4.1 Artist’s concept of the onset of polarization due to nutation in (a) over-damped, (b) critically damped, and (c) under-damped conditions.
For a system whose initial population distribution is described by R3(0), these equations are readily solved for t > 0, and give
R1(t) = Ω∆
Ω2RR3(0) [cos ΩRt− 1] , (4.1.4) R2(t) =− Ω
ΩRR3(0) sin ΩRt, (4.1.5)
R3(t) = R3(0)
1 + Ω2
Ω2R(cos ΩRt− 1)
, (4.1.6)
where Ω2R≡ ∆2+ Ω2, ∆≡ ω0− ω − kvz, and Ω≡ µ21E12/.
Exercise: Show that Eqs. (4.1.4)–(4.1.6) are reproduced by applying the evolution matrix in (Eq. (3.8.22)) to initial Bloch vector ¯R(0) = (0, 0, R3(0)).
To calculate the signal field we need to evaluate the Doppler-averaged matrix element in Eq. (3.9.10)
< ˜ρ21> = 1 kv√
π ∞
−∞
1
2(R1+ iR2) exp(−[∆/kv]2)d∆
= ΩR3(0) 2kv√
π ∞
−∞
∆
Ω2R[cos ΩRt− 1] + i
ΩRsin ΩRt '
exp(−[∆/kv]2)d∆
= i√ π
2kvΩR3(0) exp(−[∆0/kv]2)J0(Ωt), (4.1.7) where we have assumed the excitation is tuned near (but not exactly to) the Doppler peak at ∆ = ∆0with the consequence that the R1contribution is approximately zero.
According to Eqs. 3.9.12 and 3.9.16, the signal field is Es(t) = −NΩLR3(0)√
π 8εv0
µ12exp(−[∆0/kv]2)J0(Ωt)e−i(ωt−kz)+ c.c. (4.1.8) The signal at the detector will exhibit a slow oscillation described by the zero-order Bessel function J0at frequency Ω in Eq. (4.1.8), as illustrated in Fig. 4.1c.
Notice that the amplitude of the Bloch vector given by Eqs. (4.1.4)–(4.1.6) is constant.
¯R(t)= (R21(t) + R22(t) + R23(t))1/2= R3(0). (4.1.9) Also, it precesses about the effective field vector ¯β at the frequency
ΩR= (∆2+ Ω2)1/2. (4.1.10)
The motion of the Bloch vector is particularly easy to visualize for the case of exact resonance (∆ = 0). The tip of the Bloch vector sweeps around a circle in the ˆe2, ˆe3plane perpendicular to the effective field ¯β = Ωˆe1. Hence, at one instant of time it points
“up” along +ˆe3and a half period later it points “down” along−ˆe3. This corresponds to “Rabi flopping” behavior in which a collection of atoms alternately occupies the excited state or the ground state under the influence of a resonant driving field.
The frequency of driven population oscillations is the Rabi frequency, and experi-mentally this may be verified by increasing or decreasing the input field as illustrated in Fig. 4.2a. When this is done, the Rabi oscillation frequency tracks the electric field linearly. On resonance, observations of the temporal oscillations (by any means that samples the excited versus ground state populations) provide a convenient way of measuring the transition dipole moment if the effective intensity of the wave is known [4.1]. With pulsed excitation, a precise number of Rabi oscillations may be induced by controlling the pulse area as illustrated in Fig. 4.2b. Measurement of the differential
Optical nutation 79
(a)
(b)
Fluorescence intensity
t E0
E0 2E0
3π 5p
7π 9p
Change in transmission
p
Figure 4.2 Rabi oscillations in (a) transient fluorescence and (b) differential transmission experiments. The oscillation frequency in time-resolved experiments depends on the incident field strength. The oscillations versus power depend on the pulse area.
transmission (described in more detail in Chapter 5) versus input power can also serve to determine the resonant Rabi frequency. This method is commonly applied in the characterization of semiconductor quantum wells and quantum dots [4.2, 4.3].
4.1.2 Optical nutation with damping
Throughout the algebraic treatment of nutation given above, population and polar-ization decay (T1 and T2 processes, respectively) were ignored. For interactions with light that are longer than the characteristic decay times, this omission is obviously unacceptable. However the Bloch equations do not yield to analysis when the decay terms are included, other than for a special case in which T1= T2 [4.4]. This is due to the requirement that the Bloch model be based on the form of a simple gyroscopic equation of motion. Hence we shall await the density matrix methods of Chapters 5–7 which offer more degrees of freedom, and will not extend nutation analysis further here. Despite the limitations of the Bloch model, it works well for picturing the outcome of multiple pulse interactions in which the pulse durations are much shorter than the characteristic times T1 and T2. In Section 4.2 we shall also find that decay
processes that take place in the free precession periods between ultrafast pulses can be incorporated into this simplified analysis.