# The Rabi Frequency

This is a continuation from the previous tutorial - rare earth-doped fibers.

Both the linear susceptibility approach and the rate equation analysis we have developed in the past several tutorials are approximations—though usually very good approximations—to the exact dynamics of an atomic system with an external signal applied.

If a very strong (or very fast) signal is applied to an atomic transition, however, the exact nonlinear behavior of the atomic response becomes more complicated, and the rate-equation approximation is no longer adequate to describe the atomic response.

In this tutorial, therefore, we explore the conditions under which the rate-equation approximation will remain valid, and some of the interesting new effects—particularly the very important Rabi flopping behavior—that a resonant atomic transition will display in response to a strong enough applied signal.

## 1. Validity of the Rate-Equation Model

Let us first do a quick review of the basic equations which lead to rate equations, and of the approximations involved in writing the rate equation for a simple two-level atomic system.

### The Resonant-Dipole Equation

The classical electron oscillator model, with suitable quantum extensions, led us to the "resonant-dipole equation"

$\tag{1}\frac{d^2p(t)}{dt^2}+\Delta\omega_a\frac{dp(t)}{dt}+\omega_a^2p(t)=K\Delta{N}(t)\mathcal{E}(t)$

where the constant $$K$$ is given by

$\tag{2}K\equiv\frac{3^*\omega_a\epsilon\lambda^3\gamma_\text{rad}}{4\pi^2}$

We emphasize once more that this equation is a quantum-mechanically correct equation for the expectation value of the polarization $$\langle{p(t)}\rangle$$ in a two-level quantum system.

To derive the linear susceptibility $$\tilde{\chi}(\omega)$$ for the atomic transition, we solved this equation for sinusoidal steady-state signals, treating the population difference $$\Delta{N(t)}$$ as a constant. This susceptibility turns out to be, of course, directly proportional to the population difference $$\Delta{N}$$.

We then used these linearized sinusoidal results, based on assuming that the level populations are constant, to derive the rate equations that predict a time rate of change for the populations $$N_1(t)$$, $$N_2(t)$$, and $$\Delta{N(t)}$$.

To do this, we made the assumption that both the linear susceptibility description (based on constant $$\Delta{N}$$) and the rate-equation results (which describe a time-varying $$\Delta{N(t)}$$) will remain valid provided that the time rate of change of the population difference $$\Delta{N(t)}$$ is "slow" in some meaningful sense. The conditions for this approximation to be valid are essentially the following.

### Transient Response of the Resonant Dipole Equation

The resonant-dipole equation 5.1 is basically a linear second-order resonant equation, with a linewidth $$\Delta\omega_a$$ or response time $$2/\Delta\omega_a$$ (often written as $$T_2$$).

To examine its transient behavior, let us suppose that the population difference $$\Delta{N(t)}$$ is indeed constant, or nearly so, and that the applied signal $$\mathcal{E}(t)$$ is a cosinusoidal signal at $$\omega=\omega_a$$ that is turned on at $$t=0$$ in the form

$\tag{3}\mathcal{E}(t)=E_1\sin\omega_at,\qquad{t\ge0}$

The induced response $$p(t)$$ will then be given, very nearly, by

$\tag{4}p(t)\approx-K\frac{\Delta{N}E_1}{\omega_a\Delta\omega_a}[1-e^{-\Delta\omega_at/2}]\cos\omega_at$

as illustrated in Figure 5.1.

If the $$2/T_2$$ term dominates in the linewidth expression $$\Delta\omega_a=\gamma+2/T_2$$, as is often the case, or if we simply absorb the $$\gamma$$ contribution into a broadened definition of $$2/T_2$$, this may be written as

$\tag{5}p(t)\approx-K\frac{\Delta{N}E_1}{\omega_a\Delta\omega_a}[1-e^{-t/T_2}]\cos\omega_at$

The transient response is thus a forced cosinusoidal oscillation that builds up to a steady-state value with a time constant $$2/\Delta\omega_a$$, often written as just $$T_2$$ for simplicity.

The important conclusion to be drawn here is the following.

We already know that the forced response of the atomic polarization $$p(t)$$ to a sinusoidal driving signal $$\mathcal{E}(t)$$ will in general be very small unless the driving signal frequency $$\omega$$ is at or close to the resonance frequency $$\omega_a$$ of the system.

We now see that this forced sinusoidal response of $$p(t)$$ will follow any amplitude (or phase) variations in the envelope of the sinusoidal driving term $$\mathcal{E}(t)$$ with a transient time delay that is approximately $$2/\Delta\omega_a\approx{T_2}$$.

It will therefore be a valid approximation to solve the resonant-dipole equation 5.1 and use it to find both the steady-state and transient responses of $$p(t)$$, making the approximation that $$\Delta{N(t)}$$ is a constant, provided that the transient rate of change of $$\Delta{N(t)}$$ itself is slow compared to the time constant $$2/\Delta\omega_a\approx{T_2}$$. It is the slow variation of $$\Delta{N(t)}$$ that is essential—not necessarily the slow variation (or weak amplitude) of $$\mathcal{E}(t)$$.

Note also that within this approximation, variations in either the phase or the amplitude of a signal field $$\mathcal{E}(t)$$ that are rapid compared to $$T_2$$ will simply not be "seen" or responded to by the atomic system.

To put this in another way, the atomic transition has a finite bandwidth $$\Delta\omega_a$$, and rapid variations in phase or amplitude of $$\mathcal{E}(t)$$ represent frequency sidebands that are outside the linewidth $$\Delta\omega_a$$ of the atomic response. Hence these sidebands will induce little or no response (in the small signal limit).

### The Population Difference Equation

Let us now see what determines the time-variation of $$\Delta{N(t)}$$ itself, especially under the influence of stronger applied signals.

From an energy-conservation argument, we showed that the population equation for a simple two-level quantum system may be written in the form

$\tag{6}\frac{d\Delta{N(t)}}{dt}+\frac{\Delta{N(t)}-\Delta{N_0}}{T_1}=-\left(\frac{2}{\hbar\omega}\right)\boldsymbol{\mathcal{E}}(t)\cdot\frac{d\pmb{p}(t)}{dt}$

The population difference $$\Delta{N(t)}$$ is essentially a measure of the energy in the atomic system; and the term on the right-hand side of this equation gives the instantaneous power delivered by the field $$\boldsymbol{\mathcal{E}}(t)$$ to the atomic polarization $$\pmb{p}(t)$$, expressed in photon units.

This equation, written in this form, is also a quantum-mechanically correct equation for the expectation value of the energy, or the population difference $$\Delta{N}$$, in a two-level quantum system.

To make this be generally true even at large signals, however, the right-hand side of this equation must be kept in the more general and fundamental form given here, rather than in the stimulated transition form $$-2W_{12}\Delta{N}$$, because we are now not limiting ourselves to the usual steady-state amplitude and phase relationship between $$\mathcal{E}(t)$$ and $$p(t)$$ that leads to the rate equations.

### Quasi-sinusoidal Applied Signals

Suppose we write the applied signal $$\mathcal{E}(t)$$ in a somewhat more general form, namely

$\tag{7}\mathcal{E}(t)=\text{Re}\tilde{E}(t)e^{j\omega_at}$

That is, we assume that $$\mathcal{E}(t)$$ is basically a sinusoidal signal somewhere near the resonance frequency $$\omega_a$$, but with a possible amplitude or phase modulation that is contained in the time-varying complex amplitude $$\tilde{E}(t)$$.

Similarly, we can write the resulting polarization in the same general form

$\tag{8}p(t)=\text{Re}\tilde{P}(t)e^{j\omega_at}$

Let us put these signals into the right-hand side of the population difference equation 5.6. This right-hand side then becomes

\tag{9}\begin{align}-\frac{2}{\hbar\omega}\mathcal{E}(t)\frac{dp(t)}{dt}&=-\frac{j}{2\hbar}\left(\tilde{E}e^{j\omega{t}}+\tilde{E}^*e^{-j\omega{t}}\right)\times\left(\tilde{P}e^{j\omega{t}}-\tilde{P}^*e^{-j\omega{t}}\right)\\&=\frac{j}{2\hbar}\left(\tilde{E}\tilde{P}^*-\tilde{E}^*\tilde{P}\right)-\frac{j}{2\hbar}\left(\tilde{E}\tilde{P}e^{2j\omega{t}}-\tilde{E}^*\tilde{P}^*e^{-2j\omega{t}}\right)\end{align}

The driving term on the right-hand side of the population equation will thus contain both quasi constant or dc terms proportional to the imaginary part of $$\tilde{E}\tilde{P}^*$$ and second harmonic or $$\pm2\omega$$ terms proportional to the imaginary part of $$\tilde{E}\tilde{P}$$.

### Harmonic-Generation Terms

Let us first consider these $$\pm2\omega$$ terms, which are essentially harmonic-generation terms. The response of the population difference $$\Delta{N}(t)$$ in Equation 5.6 is fundamentally a sluggish response, because of the normally very long relaxation time $$T_1$$ that appears on the left-hand side.

We can expect therefore that the response of this equation to the second harmonic or $$\pm2\omega$$ terms will be very small, compared to the response to the quasi-dc terms on the right-hand side.

To put this in another way, changes in $$\Delta{N}(t)$$ will result from the integration over time of the terms on the right-hand side of Equation 5.6. But those terms with a time-variation of the form $$e^{\pm2j\omega{t}}$$ will tend to integrate to zero within a few optical cycles, whereas quasi-dc terms will tend to integrate into a significant change with time. The $$2\omega$$ terms on the right-hand side of Equation 5.6 can therefore normally be dropped.

At high enough signal levels, the harmonic terms appearing in Equation 5.6, which we are now discarding, will produce some small but nonzero modulation at $$2\omega$$ of the population difference $$\Delta{N}(t)$$.

These small second-harmonic terms in $$\Delta{N}(t)$$ will then carry back into the right-hand side of the resonant-dipole equation 5.1 for $$p(t)$$, where they will mix with the $$\pm\omega$$ terms in $$\mathcal{E}(t)$$ to produce both $$\pm3\omega$$ driving terms, and small additional $$2\omega-\omega=\omega$$ terms.

The $$\pm3\omega$$ driving terms will then produce third harmonic terms in the polarization $$p(t)$$, and these terms may in turn radiate and generate third-harmonic optical signals. This chain of harmonic effects can continue to higher orders as well, though the effects grow rapidly weaker with increasing order.

Large enough driving signals will, therefore, potentially produce even-order harmonic responses in $$\Delta{N}(t)$$, which in turn will feed back to produce odd-order harmonic responses in $$p(t)$$, and vice versa.

These various higher-order harmonic responses can be observed as harmonic generation and intermodulation or mixing effects that occur at large signal intensities in atomic systems. These harmonic-generation effects are one part of the rich repertoire of large-signal nonlinear effects that can be observed in atomic systems, and that form the basis of the very useful field of nonlinear optics.

### Conventional Rate-Equation Approximation

If we ignore these weak harmonic terms, however, the population difference equation 5.6 simplifies to

$\tag{10}\frac{d\Delta{N}(t)}{dt}+\frac{\Delta{N}(t)-\Delta{N_0}}{T_1}=-\frac{j}{2\hbar}\left[\tilde{E}(t)\tilde{P}^*(t)-\tilde{E}^*(t)\tilde{P}(t)\right]$

Now, if the linear-susceptibility or rate-equation condition holds, we can relate $$\tilde{P}$$ and $$\tilde{E}$$ to a good approximation by

$\tag{11}\tilde{P}\approx\tilde{\chi}\epsilon\tilde{E}\approx(\chi'+j\chi^")\epsilon\tilde{E}$

where $$\tilde{\chi}$$ itself is directly proportional to $$\Delta{N}$$.

The right-hand side of this equation then simplifies still further to become

$\tag{12}\frac{d\Delta{N}(t)}{dt}+\frac{\Delta{N}(t)-\Delta{N_0}}{T_1}\approx(\epsilon/\hbar)\chi^"|\tilde{E}|^2\approx-2W_{12}\Delta{N}(t)$

But this is, of course, simply the standard two-level rate equation.

### Conditions for Rate-Equation Validity

Having derived this rate equation for $$\Delta{N}(t)$$ on the assumption that any changes in $$\Delta{N}(t)$$ will be slow, we can then solve it for the predicted variation of $$\Delta{N}(t)$$ and see if the rate of change will in fact be slow.

As we have seen in earlier tutorials, a typical transient solution to the rate equation, assuming a constant-amplitude signal $$W_{12}$$ suddenly turned on at $$t=t_0$$, is

$\tag{13}\Delta{N}(t)=\Delta{N}_{ss}+[\Delta{N}(t_0)-\Delta{N}_{ss}]\times\exp[-(2W_{12}+1/T_1)(t-t_0)]$

where $$\Delta{N}_{ss}$$ is the partially saturated, steady-state value of $$\Delta{N}(t)$$ as $$t\rightarrow\infty$$.

The condition that the time rate of change of the population, or $$(d/dt)\Delta{N}(t)$$, be slow compared to the time constant $$2/\Delta\omega_a$$ in the resonant-dipole equation reduces to the condition that

$\tag{14}[2W_{12}+1/T_1]\ll[\Delta\omega_a\equiv1/T_1+2/T_2]$

One general condition for this to be satisfied, and for a rate equation to be valid, is that $$1/T_1\ll1/T_2$$, or that the energy relaxation time $$T_1$$ be long compared to the dephasing time $$T_2$$.

To put this another way, the atomic transition should have a significant amount of broadening due to dephasing, as compared to the purely lifetime broadening in the system. This condition is generally true for most laser transitions.

Of more significance, once this condition is met, is that the signal strength must be weak enough so that

$\tag{15}W_{12}\ll\Delta\omega_a$

In other words, the stimulated-transition rate $$W_{12}$$ must be small compared to the transition linewidth $$\Delta\omega_a$$.

For electric dipole transitions this can be converted into a condition on the applied signal strength given by

$\tag{16}|\tilde{E}|^2\ll\frac{(\hbar\Delta\omega_a)^2}{\epsilon\hbar\gamma_\text{rad}\lambda^3}$

In a quantum analysis we can show that this condition is equivalent to requiring that the quantum-mechanical perturbation of the energy of the atom caused by the applied field strength $$\tilde{E}$$ must be small compared to the homogeneous linewidth $$\hbar\Delta\omega_a$$ of the atom expressed in energy units. We will express this condition in another and more meaningful way in the following section.

### Rate-Equation Validity in Typical Laser Systems

The great majority of signals present in even high-power laser systems will in fact satisfy the criteria expressed by Equations 5.14 through 5.16, and the rate-equation approximation will thus be valid. Higher-power laser systems, in fact, commonly use materials that have wider atomic linewidths, which helps to preserve this condition.

For example, the atomic linewidths in gas lasers may range from a few hundred Mhz up to a few Ghz, and solid-state linewidths are typically $$10^{11}$$ to $$10^{12}$$ Hz. The corresponding transient response times $$T_2$$ for the polarization equation are in the range from $$10^{-8}$$ to $$10^{-12}$$ sec.

The stimulated-transition rates in these same lasers can be estimated by equating the actual laser power density extracted per unit volume from the laser medium to the inverted population density $$\Delta{N}$$ per unit volume, times $$\hbar\omega$$, times a signal-stimulated-transition rate $$W_{ij}$$. The resulting stimulated-transition rates are typically in the range from $$10^3$$ to $$10^7$$ sec$$^{-1}$$, and thus readily meet the criterion just given.

The level populations $$N_j(t)$$ in an atomic system also change with time as a result of relaxation and laser pumping. In practice, in useful laser materials population changes due to either relaxation or pumping are slow—in fact, most often very slow—compared to the inverse linewidth.

As an elementary example, the relaxation and pumping time constants $$w_{ij}^{-1}$$ and $$W_{ij}^{-1}$$ in solid-state laser materials commonly range from milliseconds (e.g., ruby) to a few hundred microseconds (e.g., Nd:YAG or Nd:glass); whereas inverse linewidths in these materials are in the range $$1/\Delta\omega_a\approx10^{-11}$$ seconds.

In organic dye lasers the relaxation and stimulated transition times can be much faster, e.g., a few nanoseconds ($$10^{-9}$$ sec) down to even a few picoseconds ($$10^{-12}$$ sec). However, the inverse linewidths for these materials are even shorter, e.g., typically $$1/\Delta\omega_a\approx10^{-13}$$ sec.

### Saturation Condition

We might also ask if we can obtain the saturation condition for a population difference $$\Delta{N}$$ while still remaining within the range of validity of the rate equation approach.

The signal intensity required to achieve saturation in, for example, a simple two-level atomic system is given by $$2W_{12}T_1\ge1$$ or $$W_{12}\ge1/2T_1$$, and the condition for remaining within the rate-equation regime is given in Equation 5.14. Combining these two equations then yields the double condition

$\tag{17}[1/2T_1]\le{W_{12}}\ll[\Delta\omega_a\equiv1/T_1+2/T_2]$

This says that a population difference can be saturated without violating the rate-equation limitation only if the time constants $$T_1$$ and $$T_2$$ satisfy the condition, $$1/T_1\ll\Delta\omega_a$$ or $$T_2\ll{T_1}$$.

This condition is met in virtually all useful laser materials: the energy relaxation rates are nearly always slow compared to the atomic linewidth, since the latter is determined primarily by dephasing (or even inhomogeneous) mechanisms that are substantially larger than pure lifetime broadening.

### Large-Signal Effects in Multilevel Systems

The classical oscillator or resonant-dipole model and the associated linear susceptibility, on the one hand, and the multilevel rate equations, on the other hand, provide two complementary sets of equations for analyzing the complete response not merely of a two-level system, but of a multilevel atomic system as well.

In a multienergy-level system, a separate resonant-dipole model must be applied to each individual atomic transition, with the associated populations $$N_j(t)$$ and $$N_i(t)$$ taken as quasi constants.

The resulting polarization and susceptibility on each transition can then be used directly in Maxwell's equations, and can describe accurately the amplitude, phase, polarization, and even tensor characteristics of the atomic response on that particular transition.

Note that the resonant-dipole equation for each transition is a second-order equation, as well as potentially a vector equation. Hence it can give both amplitude and phase, as well as tensor properties, of the response on that transition.

The rate equations, by contrast, treat only the energy flow, or the intensity part of the atomic response, with no phase information being available. However, they do provide a set of simple coupled first-order equations that can tie together the populations of all the energy levels in an atomic system, including relaxation and pumping mechanisms, as well as all the simultaneous signals applied to the system.

As a general approach, then, given a multilevel system with multiple signals applied, we can use the susceptibility and polarization results on each individual transition to find phase shifts, gains, and reactions back on the applied signals, and the rate equations to find the resulting populations on those transitions. Combining these two approaches provides a more or less complete, accurate, and self-consistent description of the atomic response.

There are, of course, situations in which applied signals may violate the rate-equation conditions. We can then expect to find nonlinear effects, including nonlinear mixing, intermodulation, harmonic signals, and the Rabi flopping behavior we will describe in the following section.

Detailed analysis of these effects in a multilevel system requires more general analytical methods, of which the "density matrix" approach of quantum theory is among the most useful and widely employed. The Bloch equations of magnetic-resonance theory are also very useful for analyzing large-signal and nonlinear effects in simple two-level atomic systems.

## 2. Strong-Signal Behavior: the Rabi Frequency

What happens to the atomic behavior when an applied signal is strong enough that the rate-equation approximation is no longer valid?

Much additional insight into the range of validity of the rate equations, and into the quantum behavior of an atomic transition outside this range, can be obtained from a simplified analysis we will present in this section to describe the large-signal response of an elementary two-level electric dipole system.

This analysis will introduce an important new concept, the Rabi frequency for a stimulated atomic transition.

### Simplified Large-Signal Analysis: The Polarization Equation

To carry out a large signal analysis in a simplified and yet meaningful way, we will make three simplifying, though really not very limiting, assumptions.

First, we will assume an on-resonance applied signal, which we will write as $$\mathcal{E}(t)=E_1(t)\exp(j\omega_at)$$, where $$E_1(t)$$ is the slowly varying amplitude of this applied signal. Later on we will assume that this amplitude is constant, although it may be very strong, and may be turned on suddenly at $$t=0$$.

Second, we will allow for possible large-signal and transient effects in the atomic response by writing the polarization $$p(t)$$ in the form

$\tag{18}p(t)=\text{Re}\left[\tilde{P}_1(t)e^{j\omega_at}\right]=\text{Re}\left[-jP_1(t)e^{j\omega_at}\right]$

That is, the polarization amplitude $$\tilde{P}_1(t)$$ is itself assumed to be a time-varying quantity, to account for the transient dynamics of the atomic response. Because we know from experience that in the limiting case of an on-resonance applied signal $$p(t)$$ will turn out to be $$-90^\circ$$ out of time-phase with $$\mathcal{E}(t)$$, we also write this phasor quantity as a real (but time-varying) amplitude $$P_1(t)$$ with a constant factor of $$-j$$ in front, corresponding to a fixed $$90^\circ$$ phase shift.

Substituting Equation 5.18 into the resonant-dipole equation 5.1, and separating the $$e^{+j\omega_at}$$ and $$e^{-j\omega_at}$$ terms leads us to an equation of motion for the phasor amplitude $$P_1(t)$$, namely,

$\tag{19}\frac{d^2P_1(t)}{dt^2}+(2j\omega_a+\Delta\omega_a)\frac{dP_1(t)}{dt}+j\omega_a\Delta\omega_aP_1(t)=jKE_1(t)\Delta{N(t)}$

Now, it is certainly true that $$\Delta\omega_a\ll\omega_a$$; so we can probably drop the $$\Delta\omega_a$$ factor in front of the $$dP_1(t)/dt$$ term. In addition, we can reasonably assume that the time-variation of $$P_1(t)$$ itself, though it may approach in magnitude the quantity $$\Delta\omega_aP_1(t)$$, will surely be slow compared to $$\omega_aP_1(t)$$. In simple terms, the phasor amplitude $$P_1(t)$$ may change significantly within a time of the order of one reciprocal linewidth, or $$1/\Delta\omega_a$$, but not in one optical cycle, or $$1/\omega_a$$.

As a result of this, we can drop the second-derivative term $$d^2P_1(t)/dt^2$$ relative to the $$2\omega_ad{P}_1(t)/dt$$ term, and simplify the transient equation for $$P_1(t)$$ to

$\tag{20}\frac{dP_1(t)}{dt}+\frac{\Delta\omega_a}{2}P_1(t)\approx\frac{K}{2\omega_a}E_1(t)\Delta{N}(t)$

This approximation is commonly referred to as the slowly varying envelope approximation (SVEA). Note that it is a much less restrictive approximation than the rate-equation approximation—that it, it allows much faster time-variations and much stronger signals than in the rate-equation limit.

### The Population Difference Equation

Along with this slowly varying envelope approximation for the resonant-dipole equation, we must also use the population equation of motion (Equation 5.6).

We have already noted that the transient response of that equation will be governed by the generally very slow relaxation time $$T_1$$. Therefore, as a third approximation we will use on the right-hand side of Equation 5.6 the time-averaged value of $$\mathcal{E}\cdot{dp/dt}$$, with the time average being taken over at least a few cycles of the sinusoidal quantities $$\mathcal{E}(t)$$ and $$p(t)$$.

This approximation then takes out the second-harmonic factors, but still allows for relatively rapid envelope variations in either the signal $$\mathcal{E}(t)$$ or the polarization $$p(t)$$.

With this further approximation the population equation becomes

$\tag{21}\frac{d\Delta{N}(t)}{dt}+\frac{\Delta{N}(t)-\Delta{N_0}}{T_1}\approx-\frac{1}{\hbar}E_1(t)P_1(t)$

All complex conjugates have been dropped, since we will find that $$P_1(t)$$ always turns out to be purely real for the on-resonance case, $$\omega=\omega_a$$, which is all we are considering here.

### Large-Signal Solutions: The Rabi Frequency

These last two equations are the basis for our large-signal atomic analysis. Suppose we now assume a constant signal amplitude $$E_1$$ which is turned on suddenly at $$t=0$$. The large-signal polarization and population equations 5.20 and 5.21 with $$E_1$$ constant form a simple pair of linear coupled first-order differential equations for the quantities $$P_1(t)$$ and $$\Delta{N}(t)$$ under the influence of the constant signal field $$E_1$$.

By substituting one of these equations into the other, we can combine the two first-order equations to obtain a single second-order equation for $$\Delta{N}(t)$$, namely,

$\tag{22}\frac{d^2\Delta{N(t)}}{dt^2}+\left(\frac{\Delta\omega_a}{2}+\frac{1}{T_1}\right)\frac{d\Delta{N}(t)}{dt}+\left(\frac{\Delta\omega_a}{2T_1}+\frac{KE_1^2}{2\hbar\omega_a}\right)\Delta{N}(t)=\frac{\Delta\omega_a}{2T_1}\Delta{N_0}$

Now, the quantity $$KE_1^2/2\hbar\omega$$ appearing in the second set of brackets in this equation has the dimensions of a frequency squared. Suppose we define this frequency to be the Rabi frequency $$\omega_R$$, given by

$\tag{23}\frac{KE_1^2}{2\hbar\omega_a}=\frac{3^*}{8\pi^2}\frac{\gamma_\text{rad}\epsilon\lambda^3}{\hbar}\equiv\omega_R^2$

This Rabi frequency $$\omega_R$$ is proportional to the applied signal field strength $$E_1$$, and also depends on the transition strength as measured by $$\gamma_\text{rad}$$. It has a very important physical significance, which we will develop in the following paragraphs.

Using this notation, we can rewrite the population difference equation in the form

$\tag{24}\left[\frac{d^2}{dt^2}+\left(\frac{\Delta\omega_a}{2}+\frac{1}{T_1}\right)\frac{d}{dt}+\left(\frac{\Delta\omega_a}{2T_1}+\omega_R^2\right)\right]\Delta{N(t)}=\frac{\Delta\omega_a}{2T_1}\Delta{N_0}$

We can also write the $$P_1(t)$$ equation in exactly the same form

$\tag{25}\left[\frac{d^2}{dt^2}+\left(\frac{\Delta\omega_a}{2}+\frac{1}{T_1}\right)\frac{d}{dt}+\left(\frac{\Delta\omega_a}{2T_1}+\omega_R^2\right)\right]P_1(t)=\frac{KE_1}{2\omega_aT_1}\Delta{N_0}$

which has exactly the same form as the $$\Delta{N}(t)$$ equation, except for a constant on the right-hand side.

### Large-Signal Limit: Rabi-Frequency Oscillations

Let us consider first the limiting case in which either the applied signal amplitude $$E_1$$ is extremely strong or the relaxation times $$T_1$$ and $$T_2$$ are very long and the linewidth $$\Delta\omega_a$$ is very narrow.

We can then make the large-signal assumption that the Rabi frequency is large compared to all of these other rates, i.e., $$\omega_R\gg\Delta\omega_a$$ and $$\omega_R\gg1/T_1$$. The differential equations 5.24 and 5.25 for the population difference $$\Delta{N}(t)$$ and the polarization amplitude $$P_1(t)$$ then reduce to the very much simplified forms

$\tag{26}\frac{d^2\Delta{N}}{dt^2}+\omega_R^2\Delta{N}\approx0$

and similarly

$\tag{27}\frac{d^2P_1}{dt^2}+\omega_R^2P_1\approx0$

The first of these equations has an elementary solution of the form

$\tag{28}\Delta{N}(t)=\Delta{N_0}\cos\omega_Rt$

and the polarization amplitude $$P_1(t)$$ then has a matching solution of the form

$\tag{29}P_1(t)=\sqrt{K\hbar/2\omega_a}\Delta{N_0}\sin\omega_Rt=P_m\sin\omega_Rt$

where $$P_m$$ is the maximum value of the oscillating polarization.

Figure 5.2 shows the population difference $$\Delta{N(t)}$$ and the envelope of the polarization $$P_1(t)$$ as given by these very-large-signal solutions.

It is apparent that in this very-strong-signal limit, the atomic behavior is very different from the rate-equation limit.

The population difference $$\Delta{N}(t)$$, rather than going exponentially toward a saturated valued $$\Delta{N}_\text{ss}$$ as it does in the rate-equation limit, instead continually oscillates back and forth between its initial value and the opposite of that value, at the Rabi frequency $$\omega_R$$.

At the same time, the induced polarization amplitude $$P_1(t)$$ instead of catching up to the applied signal with a time constant $$\approx{T_2}$$, continually chases but never catches up with the sinusoidal ringing of the population difference $$\Delta{N}(t)$$, so that the magnitude of $$P_1(t)$$ also oscillates sinusoidally, but lags behind $$\Delta{N}(t)$$ by 1/4 of a Rabi cycle.

### Discussion of the Rabi Flopping Behavior

This overall behavior of $$\Delta{N}(t)$$ and $$P_1(t)$$ is generally referred to as Rabi flopping behavior. It is a common result of quantum as well as classical analyses of large-signal atomic response.

The Rabi frequency $$\omega_R$$ in this very-large-signal limit is, by assumption, large compared to either $$\Delta\omega_a\approx2/T_2$$ or to $$1/T_1$$. An essential feature of this regime is therefore that the population difference $$\Delta{N}(t)$$ oscillates through many Rabi cycles in a time interval short compared to either $$T_1$$ or (more important) $$T_2$$.

This Rabi oscillation frequency will still, however, be very small compared to the optical carrier frequency $$\omega_a$$, so that there will still be very many optical cycles within each Rabi cycle. The slowly varying envelope approximation for $$\Delta{N}(t)$$ and $$P_1(t)$$ compared to $$\omega_a$$ is therefore still entirely valid.

It is also important to note that the Rabi frequency $$\omega_R$$ at which these oscillations occur depends directly on the applied signal amplitude $$E_1$$, and on the square root of the transition strength as determined by the $$\gamma_\text{rad}$$ value.

Turning up the applied signal intensity will therefore give an even more rapid oscillation of the atomic population. The two lower plots in Figure 5.2 show two different applied signal strengths, with the stronger applied signal leading to a larger Rabi frequency.

We emphasize again that all this oscillatory behavior occurs during a time interval short compared to either $$T_1$$ or $$T_2$$, and is entirely different from the rate-equation behavior at much lower applied signal levels.

Note also that the polarization amplitude oscillates with a 90° time lag relative to $$\Delta{N}(t)$$, so that the maximum value of $$|P_1|$$ occurs when $$|\Delta{N}|=0$$, in sharp contrast to the usual rate-equation behavior, in which $$|\tilde{P}(\omega)|$$ is directly proportional to $$\Delta{N}(t)$$.

What this means physically is that the oscillating dipoles are all fully aligned in phase in this situation (since the $$T_2$$ dephasing mechanisms are weak compared to the applied signal); in addition, the oscillatory quantum component $$\tilde{a}_1\tilde{a}_2^*$$ that we discussed earlier has its maximum value, subject to the constraint that $$|\tilde{a}_1|^2+|\tilde{a}_2|^2=1$$ at the midway point when $$|\tilde{a}_1|=|\tilde{a}_2|=\sqrt{1/2}$$.

This Rabi flopping behavior is thus, once again, totally different from the usual rate-equation behavior. It represents the most fundamental type of transient large-signal behavior that can be produced in either an isolated atom or a collection of atoms in which the applied signal is strong enough to override all the relaxation and dephasing processes.

### Large-Signal Case: Limiting Behavior at Long Times

To gain further insight into the distinction between small-signal and large-signal atomic behavior, we can write down the exact solutions to the full differential equations for $$\Delta{N}(t)$$ and $$P_1(t)$$, assuming a signal $$E_1$$ of arbitrary (but constant) amplitude that is again turned on at $$t= 0$$.

Let us first look, however, at the long-term steady-state solution to these equations. If we set all time derivatives to zero in the full differential equations 5.24 or 5.25, the eventual steady-state value of $$\Delta{N}(t)$$, when $$d/dt=0$$, is given by

$\tag{30}\lim_{t\rightarrow\infty}\Delta{N}(t)=\Delta{N}_\text{ss}\equiv\frac{\Delta{N_0}}{1+2(\omega_R^2/\Delta\omega_a)T_1}$

But this is exactly the same as the rate-equation saturation result for a two-level system, namely,

$\tag{31}\Delta{N}_\text{ss}=\frac{\Delta{N_0}}{1+2W_{12}T_1}$

Comparing these two equations that the stimulated-transition probability $$W_[{12}$$ which is valid in the small-signal or rate-equation regime can be related to the Rabi frequency $$\omega_R$$ and the transition linewidth $$\Delta\omega_a$$ by the very simple and useful form

$\tag{32}\text{stimulated transition probability},W_{12}\equiv\frac{\omega_R^2}{\Delta\omega_a}$

Even in the very-large-signal or Rabi-flopping regime, so long as there is any finite $$T_1$$ and $$T_2$$, no matter how small, the Rabi flopping behavior will eventually die out. The population difference in our simple two-level mode will eventually saturate (possibly after many Rabi cycles) to a steady-state (and highly saturated) value given by

$\tag{33}\lim_{t\rightarrow\infty}\Delta{N}(t)=\Delta{N_0}\frac{1}{1+2W_{12}T_1}=\Delta{N_0}\frac{1}{1+S}$

where $$S$$ is the "saturation factor" given by

$\tag{34}S\equiv2W_{12}T_1=I/I_\text{sat}=\frac{2T_1}{\Delta\omega_a}\times\omega_R^2$

This factor is, of course, directly proportional to the applied signal power, and will be very much greater than one for any signal falling in the very-large-signal or Rabi-flopping regime.

### Exact Solutions: Transient Response

The differential equations 5.20 and 5.21 (or 5.24 and 5.25) for $$\Delta{N}(t)$$ and $$P_1(t)$$ can, of course, be solved exactly, without approximations, for any level of signal strength.

As a practical hint, the algebra involved in doing this becomes much easier if you convert the equations to a suitable set of normalized variables. A convenient choice is to normalize the time scale to the dephasing time $$T_2$$ by writing $$t'=t/T_2$$; to define a normalized signal amplitude by $$R=\omega_RT_2=2\omega_R/\Delta\omega_a$$, and a time-constant ratio by $$D=T_2/T_1$$ (note that $$D$$ will normally be a small number); and then to use normalized quantities $$\hat{n}=\Delta{N}/\Delta{N_0}$$ and $$\hat{p}=P_1/P_0$$, where $$P_0=\sqrt{\hbar{K}/2\omega_a}\Delta{N_0}$$.

The two coupled equations then become

\begin{align}\frac{d\hat{n}}{dt}+D(\hat{n}-1)&=-R\hat{p},\\\frac{d\hat{p}}{dt}+\hat{p}&=R\hat{n}\end{align}

Since these are linear coupled equations, the exact solutions will have a transient behavior that will take on either overdamped or oscillatory forms, depending on the ratio of $$R$$ to $$(1-D)/2$$, which in real terms corresponds to the ratio of $$\omega_R^2$$ to the quantity $$(\Delta\omega_a/4-1/2T_1)^2$$.

Let us examine each of these limits in turn.

1. The overdamped or weak-signal regime.

In the weak-signal regime the applied signal strength is small enough that $$\omega_R\lt(\Delta\omega_a/4-1/2T_1)$$, which means that the Rabi frequency is small compared to the atomic linewidth $$\Delta\omega_a$$ (and so the stimulated-transition probability $$W_{12}$$ is small compared to $$\Delta\omega_a$$ also).

The exact solution is then overdamped, and has two exponential decay components given by

$\tag{36}-\alpha\pm\beta=-\left(\frac{\Delta\omega_a}{4}+\frac{1}{T_1}\right)\pm\sqrt{\left(\frac{\Delta\omega_a}{4}-\frac{1}{2T_1}\right)^2-\omega_R^2}$

so that $$\beta\lt\alpha$$. This condition corresponds to the usual rate-equation limit, as we will now see.

If we assume for simplicity that the atomic system is initially at rest, so that $$P_1(0)=0$$ and $$\Delta{N}(0)=\Delta{N_0}$$ when the signal is first turned on, then the solution in this limit may written as

$\tag{37}\Delta{N}(t)=\Delta{N}_\text{sat}[1+Se^{-\alpha{t}}(\cosh\beta{t}+(\alpha/\beta)\sinh\beta{t})]$

where $$\Delta{N}_\text{sat}$$ and the saturation factor $$S$$ are as defined earlier. For the limiting case of a very weak signal, $$\omega_R\ll\Delta\omega_a$$, and also slow energy decay, $$1/T_1\ll\Delta\omega_a$$, the two time constants approach the limits

$\tag{38}\alpha+\beta\approx\Delta\omega_a/2\qquad\text{and}\qquad\alpha-\beta\approx(2W_{12}+1/T_1)$

so Equation 5.37 can be approximated by

$\tag{39}\Delta{N}(t)\approx\Delta{N}_\text{sat}\{1+S\exp[-(2W_{12}+1/T_1)t]\}$

But this is exactly the same as the transient two-level behavior developed using rate equations. This result thus verifies that the Rabi-frequency behavior blends smoothly into rate-equation behavior in the appropriate weak-signal limit.

2. The oscillatory or strong-signal regime.

For signals strong enough that $$\omega_R\gt(\Delta\omega_a/4-1/2T_1)$$, the equations become underdamped, and we must use instead a pair of complex conjugate time constants given by

$\tag{40}-\alpha\pm{j}\beta=-\left(\frac{\Delta\omega_a}{4}+\frac{1}{T_1}\right)\pm{j}\sqrt{\omega_R^2-\left(\frac{\Delta\omega_a}{4}-\frac{1}{2T_1}\right)^2}$

The imaginary part $$\beta$$ in particular now corresponds to a kind of modified Rabi frequency $$\omega_R'$$ given by

$\tag{41}\beta=\omega_R'\equiv\sqrt{\omega_R^2-\left(\frac{\Delta\omega_a}{4}-\frac{1}{2T_1}\right)^2}$

when the effects of damping and dephasing are included.

In terms of these quantities, the exact solution for the same initial conditions then becomes

$\tag{42}\Delta{N}(t)=\Delta{N}_\text{sat}\{1+Se^{-\alpha{t}}[\cos\beta{t}+(\alpha/\beta)\sin\beta{t}]\}$

This result is a more exact form of the large-signal Rabi flopping limit given earlier, with the effects of weak relaxation terms $$\Delta\omega_a$$ and $$T_1$$ included.

To illustrate how the transient response of the atomic system changes as the applied signal amplitude increases from the weak-signal or rate-equation regime to the large-signal or Rabi-flopping regime, Figure 5.3 shows the calculated behavior of $$\Delta{N}(t)$$ and $$P_1(t)$$ from these exact solutions plotted versus $$t/T_1$$ in a two-level system, assuming that the dephasing time $$T_2$$ is 1/5 of the energy decay rate $$1/T_1$$ and that the Rabi frequency ranges from 0.15 to 2.2 times the atomic linewidth $$\Delta\omega_a$$.

This obviously covers a range from the weak-signal regime, exhibiting essentially rate-equation behavior, into the lower end of the strong-signal regime, exhibiting a significant amount of Rabi flopping behavior.

In the intermediate regime between weak and very strong applied signals, the population clearly oscillates back and forth at a modified Rabi frequency $$\beta\equiv\omega_R'$$ that is somewhat lower than $$\omega_R$$. This Rabi flopping behavior eventually dies out, however, as the dephasing effects described by $$\Delta\omega_a$$ gradually destroy the coherently driven transient behavior.

### Summary

There are two points concerning the results derived in this chapter that we should especially emphasize here.

• All the results we have developed in this section are quantum-mechanically correct (at least for an ideal two-level quantum system), since the initial polarization and population difference equations from which we started were quantum-mechanically correct. The Rabi flopping behavior is a very general and characteristic quantum phenomenon, readily predicted from Schrodinger's equation for any strongly perturbed two-level system.
• Even in the weak-signal regime where no Rabi flopping behavior is occurring, the Rabi frequency $$\omega_R$$ still provides a natural measure of the strength of the applied signal field, relative to the transition frequency $$\omega_a$$. In quantum-mechanical terms, $$\hbar\omega_R$$ is a measure of the perturbation hamiltonian caused by the applied field acting on the atom, just as $$\hbar\Delta\omega_a$$ is a measure of the random perturbation hamiltonian caused by the relaxation mechanisms and the dephasing or phonon-broadening mechanisms acting on the atoms, and $$\hbar\omega_a$$ is a measure of the static or unperturbed hamiltonian of the atom.

This point is especially illustrated by the fact that the stimulated-transition probability $$W_{12}$$ in any two-level system (electric dipole or any other kind) can always be written in terms of the Rabi frequency $$\omega_R$$ for that transition, in the form

$\tag{43}W_{12}=\frac{\omega_R^2}{\Delta\omega_a}$

where $$\Delta\omega_a$$ is the homogeneous linewidth for that transition.

The condition for rate-equation behavior, which we said earlier was $$W_{12}\ll\Delta\omega_a$$, translates into the condition that

$\tag{44}\omega_R\ll\Delta\omega_a$

In other words, in order to be in the rate-equation regime, the Rabi flopping frequency $$\omega_R$$ itself must be much less than the linewidth $$\Delta\omega_a$$.

In physical terms, rate-equation behavior results when the signal strength and hence the Rabi frequency are small enough that a dephasing event or a relaxation event is sure to occur, and to break up the Rabi flopping behavior, before even a fraction of a Rabi cycle is completed.

So-called coherent or large-signal Rabi-flopping effects occur, on the other hand, when the atoms can be driven through one or several Rabi cycles in a time short compared to either of the relaxation times $$T_1$$ or $$T_2$$.

### Coherent Pulse Effects

Rabi flopping behavior, and other strong-signal effects and departures from elementary rate-equation behavior, are most easily observed by using pulsed signals and transient detection methods.

This is both a practical matter, in that strong applied signals are more easily obtained in pulsed form, and a consequence of the fact that the nonlinear Rabi-frequency kind of behavior shows up most clearly in transient rather than steady-state behavior of the atoms.

Hence a number of different pulsed large-signal experiments have been developed to demonstrate such coherent transient behavior; these are commonly referred to as "coherent pulse" experiments.

As one example, the Rabi flopping behavior predicts that if we apply a sufficiently strong signal pulse with a duration $$T_p$$ such that $$\omega_RT_p\equiv\pi$$ to an initially uninverted and absorbing two-level atomic transition, this pulse can flip the initially absorbing population difference $$\Delta{N_0}$$ over into a completely inverted and hence amplifying condition $$-\Delta{N_0}$$ at the end of the pulse.

We simply turn off the applied signal in the Rabi flopping behavior at the point where the initial population inversion has been completely inverted, and then let this inverted population slowly decay back to equilibrium with time constant $$T_1$$.

This is commonly known as a "$$\pi$$ pulse" or "180° pulse" experiment. It provides one way (though in practice not a very useful way) to obtain pulsed inversion in a two-level system. Note that there is an inverse relationship between the signal amplitude $$E_1$$ and the pulse duration $$T_p$$ needed to produce an exactly 180° pulse.

Similarly, a pulse with an "area" (that is, an $$\omega_RT_p$$ product) such that $$\omega_RT_p=2\pi$$ will first invert the atomic population difference, and then flip it exactly back to its initial condition, as illustrated in Figure 5.4.

As a result such a "$$2\pi$$ pulse" or "360° pulse" will deliver no energy at all to the atoms (at least, not to first order). This means that a sufficiently strong pulse with this area can travel essentially unattenuated through an absorbing atomic medium which is otherwise opaque for lower-intensity signals. This phenomenon is known as "self-induced transparency" and has been demonstrated experimentally.

### Self-Consistent Coherent Pulse Analyses

More rigorous analysis of these coherent pulse experiments requires us to take into consideration not only the effect of the signal on the atoms, but also the reaction of the resulting induced atomic polarization $$p(t)$$ back on the signal.

For instance, in self-induced transparency the first half of the signal pulse delivers energy from the signal to the atoms, but in the second half of the pulse the atoms radiate energy back to the signal.

As a consequence the signal pulse soon distorts from a square pulse, or whatever its initial shape may be, into a unique self-consistent pulse shape. Also, the pulse velocity is reduced much below the free propagation velocity of the electromagnetic wave, in essence because the pulse energy spends a significant fraction of the time stored in the atoms rather than in the wave.

A detailed calculation of pulse propagation through a simple two-level absorbing medium has been carried out by Davis and Lin, using essentially the atomic equations presented in this section, combined with Maxwell's equations for the propagation of the signal pulse itself. Figure 5.5 illustrates some typical results from their calculations.

The left-hand plots show the initial smooth pulse sent into the absorbing medium (plotted as $$E$$ field amplitude, not intensity), and also the resulting modified pulseshapes at two different distances into the absorbing medium.

The time scale for the modified pulses has been delayed in each case by the propagation time from the input plane to the observation plane, so that the two pulses will line up.

The input pulse duration ($$\approx$$ 5 ps) is much shorter than the assumed value of $$T_2$$ for the atomic medium, and the pulse intensity is large enough that the Rabi frequency at the peak is large compared to both the atomic linewidth and the inverse pulse duration. The right-hand plots show the time-variation of the population difference $$\Delta{N}(t)$$ at these same two observation planes, on the same delayed time scale, as the pulse sweeps past each plane.

In the top pair of plots, corresponding to the first observation plane, the early portion of the pulse (up to about 1.5 ps) has been strongly absorbed by the medium; but beyond that time the accumulated pulse energy has been enough to strongly saturate the absorber, so that the trailing edge of the pulse is nearly unattenuated.

The right-hand plot also shows that the pulse intensity near the peak is more than adequate to produce significant Rabi flopping behavior in the atomic system. Note also that the population difference begins to recover toward its unsaturated value (plotted downward) with time constant $$T_1$$ as the pulse intensity dies away.

By the time the pulse reaches the second observation plane, which is five times further into the absorbing medium, the oscillatory Rabi behavior of the atomic polarization has begun to react back on the propagating pulse; and the pulse itself has acquired a strong oscillatory behavior as well.

The first full cycle in the pulse oscillation has become, in fact, almost a full $$2\pi$$ pulse, sufficient to flip the population difference more than 60% of the way to complete inversion to the opposite sign. If this pulse were to propagate further, it would it fact break up into one or several such $$2\pi$$ pulses.

There exist a great many such transient, large-signal, coherent-pulse effects which can be demonstrated on atomic transitions using appropriate pulsed signals.

These transient responses can be described analytically using either an electric-dipole model for the atomic transition, or a magnetic-dipole model, which often provides more insight into the transient behavior even for what are really electric-dipole transitions.

We will therefore consider these transient responses in more extensive detail in later tutorials, after we have introduced the magnetic-dipole model for atomic transitions.

### Multilevel Systems: Mixing and Intermodulation

Strong cw or long-pulse signals can also produce significant harmonic generation and intermodulation effects in a real atomic system, as we mentioned in the previous section.

Suppose, for example, that two cw signals are simultaneously applied to the same atomic transition, with at least one of the signals being strong enough to violate the rate approximation conditions and produce significant Rabi flopping effects.

Alternatively, suppose that this strong signal is applied to one transition, say, the $$i\rightarrow{j}$$ transition, and a weak signal is simultaneously applied to another transition which shares a common energy level, say, the $$j\rightarrow{k}$$ transition.

Then in either case, to give a somewhat simplified description, the strong signal will modulate the populations $$N_i(t)$$ or $$N_j(t)$$ in time according to the modified Rabi frequency. This modulation of the populations will then modulate the net absorption or emission seen by other, weaker signals on the same or on connecting transitions.

In general, we can expect to see intermodulation and distortion products appearing in any strong-signal multiple-frequency experiment, with the weaker signals being modulated at something like the Rabi frequency produced by the stronger signal.

These mixing and intermodulation effects rapidly become very complicated when several energy levels or several applied frequencies are involved. Proper analysis of these effects usually requires carrying out what is called a multilevel quantum-mechanical density-matrix analysis.

Fortunately, intermodulation effects of this type are usually small in most practical laser systems, although they can sometimes be observed. The general criterion for observing them is applying unusually strong signals to an atomic system that has unusually narrow linewidths and strong transitions, so that the Rabi frequencies involved can become larger than the linewidths.