Rabi Formula in the Weak-Drive Limit

The Rabi formula and first-order time-dependent perturbation theory describe the same driven two-level system in an overlap regime, but they organize the physics differently. The Rabi formula resums repeated transitions and preserves bounded population dynamics. First-order theory keeps only the leading transition amplitude and is simpler, more general, and directly connected to the Fourier spectrum of a weak pulse.

Required background. Rabi Oscillations explains the two-level model; First-Order Transition Probability supplies its weak-excitation limit. Resonant Driving treats finite-pulse widths, and Rotating-Wave Approximation states when counter-rotating terms can be neglected.

The Two-Level Rabi Hamiltonian

Assume an isolated two-level subspace with lower state $\lvert g\rangle$, upper state $\lvert e\rangle$, and bare transition angular frequency $\omega_0$. A monochromatic drive at angular frequency $\omega$ is reduced, after a rotating-frame transformation and the rotating-wave approximation, to

$$H_{\mathrm{eff}} = \frac{\hbar}{2} \left( \Delta\sigma_z + \Omega\sigma_x \right),$$

where

$$\Delta = \omega_0-\omega$$

is the detuning and $\Omega$ is the on-resonance Rabi angular frequency in this convention. The factor of $1/2$ in the Hamiltonian matters: other conventions can differ by a factor of two in the symbol called the Rabi frequency.

Choose

$$\sigma_z\lvert e\rangle = \lvert e\rangle, \qquad \sigma_z\lvert g\rangle = -\lvert g\rangle.$$

The generalized Rabi angular frequency is

$$\Omega_R = \sqrt{\Omega^2+\Delta^2}.$$

For a square pulse of duration $T$ and initial state $\lvert g\rangle$, the exact transition probability within this effective model is

$$P_{\mathrm{R}}(T) = \frac{\Omega^2}{\Omega_R^2} \sin^2\left( \frac{\Omega_RT}{2} \right).$$

“Exact” here means exact for the time-independent two-level effective Hamiltonian. It does not undo errors from truncating other levels, making the rotating-wave approximation, neglecting dissipation, or idealizing the pulse as rectangular.

Exact Amplitude and Its Phase

The Pauli identity $(\Delta\sigma_z+\Omega\sigma_x)^2=\Omega_R^2I$ gives

$$U_{\mathrm{eff}}(T)=\cos(\Omega_RT/2)I -i\frac{\Delta\sigma_z+\Omega\sigma_x}{\Omega_R}\sin(\Omega_RT/2),$$

with the continuous value $U=I$ at $\Omega=\Delta=0$.

The rotating-frame excited-state amplitude is therefore

$$b_e(T) = -i\frac{\Omega}{\Omega_R} \sin\left( \frac{\Omega_RT}{2} \right).$$

Transforming back to the interaction-picture coefficient used in transition theory supplies a convention-dependent phase. With the choices above,

$$c_e(T) = -i e^{i\Delta T/2} \frac{\Omega}{\Omega_R} \sin\left( \frac{\Omega_RT}{2} \right).$$

The phase cancels from the population but matters in Ramsey sequences, coherent pathway interference, and state tomography.

First-Order Transition Result

In the interaction picture, the retained rotating term is

$$V_I^{\mathrm{RWA}}(t) = \frac{\hbar\Omega}{2} \left( \sigma_+e^{i\Delta t} + \sigma_-e^{-i\Delta t} \right).$$

Starting from $\lvert g\rangle$, first order gives

$$\begin{aligned} c_e^{(1)}(T) &= -\frac{i\Omega}{2} \int_0^Tdt\,e^{i\Delta t} \\ &= -i e^{i\Delta T/2} \frac{\Omega T}{2} \operatorname{sinc} \left( \frac{\Delta T}{2} \right). \end{aligned}$$

Thus

$$P_{\mathrm{PT}}(T) = \frac{\Omega^2T^2}{4} \operatorname{sinc}^2 \left( \frac{\Delta T}{2} \right).$$

This is the leading probability obtained by squaring the first-order amplitude. It is second order in the drive strength even though it is conventionally called a first-order perturbation-theory result.

Dimensionless Matching Variables

Introduce

$$a \equiv \Omega T, \qquad d \equiv \Delta T.$$

The exact Rabi probability becomes

$$P_{\mathrm{R}}(a,d) = \frac{a^2}{a^2+d^2} \sin^2\left[ \frac{1}{2} \sqrt{a^2+d^2} \right],$$

whereas first-order transition theory gives

$$P_{\mathrm{PT}}(a,d) = \frac{a^2}{4} \operatorname{sinc}^2 \left( \frac{d}{2} \right).$$

These variables expose the two competing accumulated phases:

• $a=\Omega T$ is the pulse area for a constant real coupling;
• $d=\Delta T$ is the detuning phase accumulated during the pulse.

Weak instantaneous coupling is not enough. The expansion that connects the formulas is an expansion in $a$ at fixed $d$, or an off-resonant expansion in $\Omega/\lvert\Delta\rvert$ with its own long-time caveats.

Leading Weak-Drive Expansion

For fixed $d\ne0$ and $a\to0$,

$$\sqrt{a^2+d^2} = \lvert d\rvert \left[ 1+ \frac{a^2}{2d^2} +O\!\left( \frac{a^4}{d^4} \right) \right].$$

At leading order,

$$\begin{aligned} P_{\mathrm{R}}(a,d) &= \frac{a^2}{d^2} \sin^2\left( \frac{d}{2} \right) +O(a^4) \\ &= \frac{a^2}{4} \operatorname{sinc}^2\left( \frac{d}{2} \right) +O(a^4). \end{aligned}$$

Therefore

$$P_{\mathrm{R}}(a,d) = P_{\mathrm{PT}}(a,d) +O(a^4)$$

at fixed $d$. At the amplitude level,

$$c_e(a,d) = c_e^{(1)}(a,d) +O(a^3).$$

The next correction to the amplitude is third order because each interaction flips the two-level state: an odd number of flips is required to end in $\lvert e\rangle$. The probability correction begins at fourth order through interference between the first- and third-order amplitudes.

First Correction and Error Structure

Expanding one order further gives

$$P_{\mathrm{R}}(a,d) = P_{\mathrm{PT}}(a,d) +a^4 C(d) +O(a^6),$$

where

$$C(d) = \frac{ \tfrac14 d\sin d - \sin^2(d/2) }{d^4}.$$

Although this expression appears singular at $d=0$, the singularity is removable. Series expansion gives

$$C(0) = -\frac{1}{48}.$$

At a zero of the leading sinc profile, a relative error based on $P_{\mathrm{PT}}$ is not useful because the denominator vanishes. For $d=2\pi k$ with nonzero integer $k$, both the quadratic probability and $C(d)$ vanish. Expanding the exact sine shows instead

$$P_{\mathrm R}(a,2\pi k)=\frac{a^6}{16(2\pi k)^4}+O(a^8).$$

The amplitude starts at third order there; absolute error and the first nonzero order give meaningful diagnostics.

For a requested tolerance, one should compare the actual observable rather than invoke “weak drive” qualitatively. At fixed $d$ away from a leading zero, the relative population error generally scales as $a^2$.

Exact Resonance

Setting $d=0$ gives

$$P_{\mathrm{R}}(a,0) = \sin^2\left( \frac{a}{2} \right),$$

while

$$P_{\mathrm{PT}}(a,0) = \frac{a^2}{4}.$$

Their expansions agree initially:

$$\begin{aligned} P_{\mathrm{R}}(a,0) &= \frac{a^2}{4} - \frac{a^4}{48} +O(a^6), \\ P_{\mathrm{PT}}(a,0) &= \frac{a^2}{4}. \end{aligned}$$

The relative overestimate from first order is

$$\frac{ P_{\mathrm{PT}}-P_{\mathrm{R}} }{P_{\mathrm{R}}} = \frac{a^2}{12} +O(a^4).$$

Some representative values are:

Pulse area $a$	Exact $\sin^2(a/2)$	First order $a^2/4$	Relative overestimate
$0.1$	$0.00249792$	$0.00250000$	$0.083\%$
$0.5$	$0.06120872$	$0.06250000$	$2.110\%$
$1$	$0.22984885$	$0.25000000$	$8.767\%$
$\pi$	$1$	$2.46740110$	$146.740\%$

The $\pi$-pulse row makes the failure unmistakable. A pulse can be weak relative to the carrier frequency and still have order-one area because it acts for a long time.

Far-Off-Resonant Limit

When

$$\lvert\Delta\rvert \gg \lvert\Omega\rvert,$$

the exact maximum population obeys

$$P_{e,\max} = \frac{\Omega^2}{\Omega^2+\Delta^2} \simeq \frac{\Omega^2}{\Delta^2}.$$

Population transfer can therefore remain perturbatively small for times much longer than $1/\Omega$. However, the generalized frequency contains the shift

$$\Omega_R = \lvert\Delta\rvert + \frac{\Omega^2}{2\lvert\Delta\rvert} + O\!\left( \frac{\Omega^4}{\lvert\Delta\rvert^3} \right).$$

The second-order frequency correction accumulates a substantial phase on the scale

$$T_{\mathrm{shift}} \sim \frac{\lvert\Delta\rvert}{\Omega^2}.$$

Thus “the transition probability stays small” and “the full state is accurately described to first order” are different claims. Off-resonant drives can produce important phases and effective energy shifts without transferring much population.

The Weak and Long-Time Limits Do Not Commute

At fixed $T$,

$$\Omega\to0$$

implies $a\to0$, and first-order theory becomes accurate. But consider a family of resonant pulses with

$$T_\Omega = \frac{\pi}{\Omega}.$$

As $\Omega\to0$, the instantaneous drive becomes arbitrarily weak while the pulse area remains

$$\Omega T_\Omega = \pi.$$

For every member of the family,

$$P_{\mathrm{R}} = 1,$$

whereas first order predicts

$$P_{\mathrm{PT}} = \frac{\pi^2}{4} \gt 1.$$

This is a singular long-time scaling limit. Statements about weak drive must specify what is held fixed as the drive strength decreases.

Four Independent Approximations

The standard Rabi formula often sits behind several layers of approximation. Their control parameters are not interchangeable.

Step	What is neglected?	Typical diagnostic	Can it hold while first order fails?
Two-level reduction	Other states and leakage pathways	spectral gaps and unwanted matrix elements	Yes; an isolated resonant pair can undergo full Rabi cycles
Rotating-wave approximation	Counter-rotating terms	$\lvert\Omega\rvert,\lvert\Delta\rvert\ll\omega_0+\omega$ plus envelope slowness	Yes; a long resonant $\pi$ pulse can satisfy the RWA very well
First-order transition theory	Repeated transitions and depletion	small total depletion and higher-order terms; $a\ll1$ near resonance	No; this row is the condition being tested
Closed-system description	Relaxation, dephasing, measurement backaction	$T\ll T_1,T_2$ for coherent pulsed dynamics	Yes; unitary Rabi cycling can be nonperturbative

The converse separations also matter. A very short, broadband pulse can produce only a small transition probability while invalidating a narrowband rotating-wave treatment. A far-detuned weak drive can leave populations perturbative while generating a measurable second-order phase.

General Pulse Envelope

Let the rotating-wave interaction contain a complex, time-dependent coupling $\Omega(t)$:

$$V_I^{\mathrm{RWA}}(t) = \frac{\hbar}{2} \big[ \Omega(t)e^{i\Delta t}\sigma_+ +\Omega^*(t)e^{-i\Delta t}\sigma_- \big].$$

First order gives the general weak-pulse amplitude

$$c_e^{(1)} = -\frac{i}{2} \int_{t_0}^{t_1}dt\, \Omega(t)e^{i\Delta t}.$$

Define

$$\eta \equiv \frac{1}{2} \left\lvert \int_{t_0}^{t_1}dt\, \Omega(t)e^{i\Delta t} \right\rvert.$$

Then the leading population is

$$P_e^{(1)} = \eta^2,$$

and a necessary small-depletion condition for this channel is

$$\eta \ll 1.$$

This condition alone does not control higher orders when the integral cancels. The triangle inequality gives the conservative sufficient condition for a small norm of the full integrated interaction

$$\eta \le \frac{1}{2} \int_{t_0}^{t_1}dt\, \lvert\Omega(t)\rvert \ll 1.$$

This stronger condition ignores detuning cancellations. It is useful for guaranteed bounds but can be far more restrictive than necessary for an off-resonant pulse.

On exact resonance with fixed drive phase, all effective Hamiltonians point along the same Bloch-sphere axis, and the full result depends on pulse area. Once detuning, chirp, or phase modulation changes that axis, time ordering matters and pulse area alone is insufficient.

Amplitude Accuracy Versus Population Accuracy

A small population error does not guarantee a small phase error. Suppose

$$c_e = c_e^{(1)} + c_e^{(3)} + \cdots.$$

Then

$$P_e = \lvert c_e^{(1)}\rvert^2 +2\operatorname{Re} \left[ c_e^{(1)*}c_e^{(3)} \right] +O(\Omega^6).$$

The leading probability can be accurate even when a small higher-order phase matters to an interferometric observable. Conversely, near a zero of $c_e^{(1)}$, a higher-order amplitude can control the entire nonzero signal even though its absolute magnitude is tiny.

A validity statement should therefore name the target observable: population, complex amplitude, phase, line center, or time-resolved trajectory.

Multilevel Extension

For several possible final states, the first-order amplitudes take the form

$$c_f^{(1)} = -\frac{i}{\hbar} \int_{t_0}^{t_1}dt\, e^{i\omega_{fi}t}V_{fi}(t).$$

A basic depletion diagnostic is

$$\sum_{f\ne i} \lvert c_f^{(1)}\rvert^2 \ll 1$$

for discrete states, with the sum replaced by the appropriate continuum integral when needed. If a set of states is mutually near resonant, that subspace should be retained and solved nonperturbatively rather than forcing independent two-level formulas onto overlapping lines.

The two-level Rabi formula also misses leakage, dark and bright superpositions, Raman pathways, and multiphoton processes. Those effects require the actual coupling graph and the relevant order of the Dyson Expansion for Transition Amplitudes.

Exercises

Compute the first correction

At fixed $d\ne0$, expand

$$P_{\mathrm{R}}(a,d) = \frac{a^2}{a^2+d^2} \sin^2\left[ \frac12\sqrt{a^2+d^2} \right]$$

through order $a^4$. Show that $C(d)$ has the finite limit $-1/48$ as $d\to0$.

Solution

Use

$$\frac{a^2}{a^2+d^2} = \frac{a^2}{d^2} \left( 1- \frac{a^2}{d^2} +O(a^4) \right)$$

and

$$\sqrt{a^2+d^2} = d+ \frac{a^2}{2d} +O(a^4)$$

for positive $d$; the final expression is even in $d$. Then

$$\sin^2\left[ \frac12\sqrt{a^2+d^2} \right] = \sin^2\left( \frac d2 \right) +\frac{a^2}{4d}\sin d +O(a^4).$$

Multiplying gives

$$\begin{aligned} P_{\mathrm{R}} &= \frac{a^2}{d^2} \sin^2\left( \frac d2 \right) +a^4C(d) +O(a^6),\\ C(d) &= \frac{\sin d}{4d^3} - \frac{\sin^2(d/2)}{d^4}. \end{aligned}$$

The bracket is $C(d)$. Expanding $d\sin d/4$ and $\sin^2(d/2)$ around zero cancels the apparent $d^{-2}$ terms and leaves

$$C(d) = -\frac{1}{48} +O(d^2).$$

Weak but nonperturbative pulse

Let $\Omega/\omega_0=10^{-4}$ and take exact resonance. Choose $T=\pi/\Omega$. Which approximations can remain valid, and why does first-order transition theory fail?

Solution

The small ratio $\Omega/\omega_0$ can make the rotating-wave approximation excellent if the envelope is also slow and other levels are isolated. The pulse area is nevertheless

$$a = \Omega T = \pi.$$

The exact effective two-level result is

$$P_e = \sin^2\left( \frac\pi2 \right) = 1,$$

whereas first order gives $\pi^2/4\gt1$. Thus two-level reduction and the RWA may remain controlled while the transition-amplitude truncation fails because repeated resonant transfers are order one.

Off-resonant phase time scale

Assume $\lvert\Delta\rvert\gg\lvert\Omega\rvert$. Expand $\Omega_R$ and estimate when the phase generated by the second-order frequency shift becomes order one.

Solution

The binomial expansion gives

$$\Omega_R = \lvert\Delta\rvert + \frac{\Omega^2}{2\lvert\Delta\rvert} + O\!\left( \frac{\Omega^4}{\lvert\Delta\rvert^3} \right).$$

The correction to the oscillation phase is of order

$$\delta\phi \sim \frac{\Omega^2T}{\lvert\Delta\rvert},$$

up to a convention-dependent factor of order one. It becomes important at

$$T \sim \frac{\lvert\Delta\rvert}{\Omega^2}.$$

At that time the excited-state population can still be only order $\Omega^2/\Delta^2$. Small leakage does not imply negligible phase evolution.

General-pulse bound

Show that

$$\eta = \frac12 \left\lvert \int_{t_0}^{t_1}dt\, \Omega(t)e^{i\Delta t} \right\rvert$$

obeys $\eta\le\frac12\int\lvert\Omega(t)\rvert dt$. Explain why the bound can be loose.

Solution

The triangle inequality gives

$$\begin{aligned} \eta &\le \frac12 \int_{t_0}^{t_1}dt\, \left\lvert \Omega(t)e^{i\Delta t} \right\rvert \\ &= \frac12 \int_{t_0}^{t_1}dt\, \lvert\Omega(t)\rvert. \end{aligned}$$

The bound discards all phase cancellation. For large detuning or a shaped pulse with destructive spectral interference, the Fourier component at $\Delta$ can be much smaller than the unsigned pulse area. When the unsigned integral is itself small, the Dyson norm remainder is controlled. A small signed integral alone can hide higher-order evolution, so the bound supplies more information than the leading transition probability.

References

• L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover, 1987.
• F. Bloch and A. Siegert, “Magnetic Resonance for Nonrotating Fields,” Physical Review 57, 522–527, 1940.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley, 1992.
• I. I. Rabi, “Space Quantization in a Gyrating Magnetic Field,” Physical Review 51, 652–654, 1937.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, 1990.