Two different expressions for Rabi Frequency (on resonance)

Question

I've been searching through several articles and books and I found two different expressions for the Rabi Frequency on the semiclassical rotating-wave approximation, and they are different by a factor of two.

Consider a eletric field $$\boldsymbol{E}=E_0\sin(\omega t)$$ and a dipole matrix $$\left(\begin{array}{cc} 0 & \mu\\ \mu & 0 \end{array}\right)$$

In some places (when people derive it using Heisenberg picture), the formula is:

$$E_0=\frac{\pi\hbar}{2\left\langle a\right|\mu\left|b\right\rangle t}$$

(like on Allen,Eberly book called "Optical Ressonance and Two Level Atoms", or these notes by Oliver Benson, eq 293)

In other places (like Wikipedia or the paper arXiv:0707.1883 at eq A.18) the formula is:

$$E_0=\frac{\pi\hbar}{\left\langle a\right|\mu\left|b\right\rangle t}.$$

I've been through both derivations step by step and I could not understand why they are different. Am I misunderstanding the meaning of something on any of the two formulas?

Answer 1

Unfortunately, there are two different definitions of Rabi frequency that differ by a factor of two. Consider a qubit undergoing Rabi oscillations: it has a time-dependent wavefunction given by $|\psi(t)\rangle = c_0(t) |0\rangle + c_1(t)|1 \rangle$. One definition of the Rabi frequency would be to write the time-dependent coefficients as: $$ \begin{equation} c_0(t) = \cos(\Omega t) \\ c_1(t) = i \sin(\Omega t) \end{equation} $$ In this case, $\Omega$ describes the frequency at which the amplitudes of the wavefunction oscillate.

However, commonly one is interested in the oscillation in the populations in the states $|0\rangle$ and $|1\rangle$. The populations are defined by the magnitude squared of the amplitudes: $$ \begin{equation} |c_0(t)|^2 = \cos^2(\Omega t) = \frac{1}{2} + \frac{1}{2}\cos(2\Omega t)\\ |c_1(t)|^2 = \sin^2(\Omega t) = \frac{1}{2} - \frac{1}{2}\cos(2\Omega t) \end{equation} $$ Note that here, the oscillation in the populations actually occurs at twice the frequency; this $2\Omega$ is often used as the definition of Rabi frequency instead.

Answer 2

Yes, two expressions for the Rabi frequency abound which differ by a factor of two.

Overview

The Hamiltonian coupling two states may be written as

$$ H = C \left(\sigma^+ + \sigma^-\right) $$

I'm only showing the bare minimum to help clarify the two conventions.

• I've left out factors of $\hbar$
• If we're thinking about light-matter interactions, I've made a rotating wave approximation and I'm working in an interaction picture
• I'm assuming $C$ is real
• I'm considering a classical drive field. Normally we would use $\Omega$ or $\Omega/2$ as the coupling coefficient in this case, but that's exactly the confusion we're trying to resolve so I've introduced a new coupling symbol $C$ different from any other symbols that might confuse us further.

If the system starts in the ground state, then the time evolution of the excited state (complex) amplitude will be

$$ c_e(t) = \sin\left(Ct\right) $$

The population of the excited state will be $$ P_e(t) = |c_e(t)|^2 = \sin^2(Ct) = \frac{1}{2} - \frac{1}{2}\cos(2Ct) $$

We see that the amplitude of the excited state oscillates at an angular frequency $C$ while the population oscillates at an angular frequency of $2C$. The two conventions are then:

$$ C = \Omega_1 = \frac{\Omega_2}{2} $$

In experimental physics, we introduce a parameter called $T_{\pi}$ called the $\pi$-time which corresponds to the duration it takes, for a given Rabi frequency, the population to flip from the ground state to the excited state. We can see that

$$ T_{\pi} = \frac{\pi}{2C} $$

We summarize key properties of each

First convention - Theory

• $C = \Omega_1$
• $H = \Omega_1 (\sigma^+ + \sigma^-)$
• $c_e(t) = \sin\left(\Omega_1 t\right)$
• $P_e(t) = \frac{1}{2} - \frac{1}{2} \cos\left(2\Omega_1 t \right)$
• $T_{\pi} = \frac{\pi}{2\Omega_1}$

I call this convention the "theory" convention because I think it is more often used by theorists than experimentalists. The main advantage of this convention is the simplicity of the Hamiltonian. One need only remember which states are coupled, and then add a coupling constant coupling them. No need for extraneous factors of 2.

Second convention - Experiment

• $C = \frac{\Omega_2}{2}$
• $H = \frac{\Omega_2}{2}\left(\sigma^+ + \sigma^-\right)$
• $c_e(t) = \sin\left(\frac{\Omega_2}{2}t\right)$
• $P_e(t) = \frac{1}{2} - \frac{1}{2}\cos\left(\Omega_2 t\right)$
• $T_{\pi} = \frac{\pi}{\Omega_2}$

I call this convention the "experiment" convention because I think it is preferred by experimentalists. This is because, in many types of experiments, experimentalists have access to the populations of quantum states, but not necessarily the complex amplitudes. For this reason they are interested in $2C = \Omega_2$, the angular frequency of the population oscillation. Under this convention the $\pi$-time has the expected relationship with the Rabi frequency (i.e. a factor of $\pi$). The downside is that one has to remember the factor of two in the Hamiltonian and some factors of two arise in the equations of motion for the complex amplitudes.

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Obviously both conventions are used by theorists and experimentalists at different times. I've just given them these names to make the distinctions easy to remember and to give us something to call them.