# Does the dipole moment of an atom modelled as a two level system depends on its frequency?

Consider an atom modelled as a two level system :

$$H=\frac{\hbar \omega}{2} \sigma_z$$

$$|0\rangle$$ and $$|1\rangle$$ are the ground and excited states that span the Hilbert space.

In the Rabi oscillations study, we turn on an interaction Hamiltonian using a laser, and we define the Rabi Frequency as $$\Omega=\frac{\vec{d}.\vec{E_0}}{\hbar}$$ where $$\vec{E_0}$$ is the amplitude of the electric field of the laser, and $$\vec{d}$$ is the quantity :

$$\vec{d}=q\langle 0 | {\widehat{r}} | 1 \rangle$$

$$\widehat{r}$$ being the position operator (which is a vector operator) of the two level system.

My question is :

Do the components of the vector $$\langle 0 | {\widehat{r}} | 1 \rangle$$ actually depend on the frequency $$\omega$$ ?

If so, is there a resource where I can find this law ?

I guess the answer is true, because if I turn on a laser, it will polarize my atom. Thus $$\langle 0 | {\widehat{r}} | 1 \rangle$$ should depend on the strength of the laser. Then it may also depend on the frequency of the atom.

But how can I find the law (if I'm not wrong).

• does the resonance of a harmonic oscillator change when you drive it at different frequencies? No. But it will oscillate at the drive frequency.
– wcc
Feb 1, 2019 at 4:37

Your hamiltonian hides essential physics. Still, the question in the title can be answered. The frequency depends on the dipole moment, not the other way around.

• Doesn't frequency depend on the energies of the two states, whilst the dipole moment (matrix element) simply affect the transition probability (amplitude)?
– Cryo
Jan 31, 2019 at 22:42
• As I said, your hamiltonian does not show the underlying physics. It is likely an electron in a magnetic field, in which case the strength of the field determines the frequency. Jan 31, 2019 at 23:26
• Good point ....
– Cryo
Jan 31, 2019 at 23:57

Usually $$\langle\psi_1 |\mathbf{r}|\psi_2\rangle$$ does depend on time. The reason being is that you usually start in Schrodinger picture (wavefunctions depend on time, operators do not):

$$i\hbar\partial_t|\psi\rangle =\left(H_0+H_1\right)|\psi\rangle$$,

where $$H_0$$ is your default Hamiltonian, and $$H_1$$ is the interaction Hamiltonian. You then go into interaction picture (both wavefunction and operators depend on time):

$$i\hbar\partial_t |\psi_i\rangle = V|\psi_i\rangle$$

$$V=\exp\left(\frac{t H_0}{i\hbar}\right)H_1\exp\left(-\frac{t H_0}{i\hbar}\right)$$

Now, your operator $$\mathbf{r}$$ will arise as a result of decomposition of that interaction operator $$V$$. Lets say that $$\mathbf{r}_s$$ is the Schrodinger picture position operator (time-independent). Then the other operator, the one you will be working with, will be: $$\mathbf{r}=\exp\left(\frac{t H_0}{i\hbar}\right)\mathbf{r}_s\exp\left(-\frac{t H_0}{i\hbar}\right)$$.

Then your matrix element will be:

$$\langle\psi_1 |\mathbf{r}|\psi_2\rangle=\langle\psi_1 |\exp\left(\frac{t H_0}{i\hbar}\right)\mathbf{r}_s\exp\left(-\frac{t H_0}{i\hbar}\right)|\psi_2\rangle=\langle\psi_1 |\exp\left(-i\omega_2 t\right)\mathbf{r}_s\exp\left(i\omega_1 t\right)|\psi_2\rangle$$

The last step is due to the fact that one usually uses eigen-basis of the default Hamiltonian for the decomposition ($$H_0|\psi_{1,2}\rangle=\hbar\omega_{1,2}|\psi_{1,2}\rangle$$). Thus

$$\langle\psi_1 |\mathbf{r}|\psi_2\rangle=\exp\left(-i\left(\omega_1-\omega_2\right) t\right)\langle\psi_1 |\mathbf{r}_s|\psi_2\rangle$$

where $$\langle\psi_1 |\mathbf{r}_s|\psi_2\rangle$$ is time-independent.