User:Jemlzs/AC Stark effect

In atomic and optical physics, the AC Stark effect (or light shift) is a shift in the energy spectrum of an atomic system due to an applied alternating current electric field. It is named in analogy with the DC Stark effect, where a similar shift in energy levels is caused by the application of a stationary (time-independent) electric field. The value of this shift increases with the strength of the applied field, which makes the phenomenon a convenient tool for imposing spatially-dependent potentials upon neutral atoms. This has allowed for, among other things, the development of new techniques for trapping and cooling neutral atoms.

Theory
There are several methods for deriving the existence and magnitude of the AC Stark effect. We choose to use a dressed state approach, where both the atom and the light perturbing it are given a fully quantum description. The main feature of this approach is that we handle the coupling between the atom and the oscillating electric field nonperturbatively, by means of diagonalizing the interaction Hamiltonian. The resultant eigenstates are the dressed states of our atom, which are properly seen as collective states of the atom and field. The energy eigenvalues of these dressed states differ from the eigenvalues of the uncoupled atom + light system, and the amount by which they differ is exactly the AC Stark shift. We will first derive the light shift, and then give an interpretation of our results.

Derivation of the Effect
First, we assume our atom is illuminated by a strong monochromatic field (like a laser) which is oscillating at a frequency $$\omega_L$$. If $$\omega_L$$ is near-resonant with some two-level transition in our atom of frequency $$\omega_0$$, then it is useful to focus only on this two-level submanifold of the total atomic state space. In this approximation, and for the present neglecting the other (vacuum) modes of our field, the Hamiltonian we obtain is of a Jaynes-Cummings form, meaning that $$H_{tot} = H_A + H_L + H_{AL}$$, with $$H_A = \frac{\hbar \omega_0}{2} \sigma_z$$ acting on the two levels of our atom and $$H_L = \hbar \omega_L (a^\dagger a + \frac{1}{2})$$ acting on the primary mode of our field. In the absence of the coupling Hamiltonian, $$H_{AL}$$, the atom-light Hamiltonian has eigenvalues of the form $$E_{n,\pm} = \hbar \left(n \omega_L \pm \frac{\Delta}{2}\right)$$, with $$\Delta = \omega_L - \omega_0$$ the detuning between our light and atom transition frequencies. These states correspond to either having $$n$$ photons in our field and our atom in the ground state ($$E_{n,+}$$), or having $$n-1$$ photons in our field and our atom in the excited state ($$E_{n,-}$$).

If our atom is electrically neutral, the dominant interaction between the atom and field is the electric dipole interaction, which gives an interaction Hamiltonian of the form $$H_{AL} = \frac{\hbar \Omega_0}{2} E S$$. Here, $$E = a^\dagger + a$$ is the dimensionless electric field operator for our mode, and $$S = \sigma_+ + \sigma_-$$ is the dimensionless angular momentum operator for the atom. $$\Omega_0$$ is the Rabi frequency of the transition, which is proportional both to the strength of the field at the atom location and the transition dipole moment of the atom. If we make the rotating wave approximation for our system, the coupling Hamiltonian $$H_{AL}$$ becomes $$H_{AL} = \frac{\hbar \Omega_0}{2} (a^\dagger \sigma_- + a \sigma_+)$$. The total system Hamiltonian is thus:


 * $$H_{tot} = \hbar \left[ \frac{\omega_0 \sigma_z}{2} + \omega_L (a^\dagger a + \frac{1}{2}) + \frac{\Omega_0}{2} (a^\dagger \sigma_- + a \sigma_+) \right]$$
 * $$= \hbar \left[ \omega_L \left( \frac{\sigma_z + 1}{2} + a^\dagger a \right) - \frac{\Delta \sigma_z}{2} + \frac{\Omega_0}{2} (a^\dagger \sigma_- + a \sigma_+) \right]$$

The first term is a constant of motion (it commutes with $$H_{tot}$$, and in particular, $$H_{AL}$$), so we can choose our eigenstates by first block diagonalizing it, and then diagonalizing the interaction Hamiltonian within each block. Physically, this says that our total Hamiltonian conserves excitation number (photons + atomic excited states), and we wish to choose a dressed state basis for each n-excitation subspace. With no coupling ($$\Omega_0 = 0$$), this gives us the eigenvalues $$E_{n, \pm}$$ from above. For non-zero coupling, we can treat the two states in the n-excitation block using a Bloch sphere picture, giving us a total Hamiltonian of $$H_{tot} = \hbar \left( n\omega_L I + \frac{1}{2} (-\Delta \sigma_z + \sqrt{n} \Omega_0 \sigma_x) \right)$$. The Pauli operators in this pseudo-spin picture act with respect to the basis vectors $$|0\rangle = |n-1\rangle_L \otimes |e\rangle_A $$ and $$|1\rangle = |n\rangle_L \otimes |g\rangle_A $$. By using standard results about operators on the Bloch sphere, we find the eigenvalues of the above Hamiltonian to be $$E'_{n,\pm} = \hbar \left( n \omega_L \pm \sqrt{\Delta^2 + n\Omega_0^2} \right)$$. Since the AC Stark shift is defined as the difference between these energy eigenvalues and the corresponding uncoupled values, we have:


 * $$\delta E_{n,\pm} = E'_{n,\pm} - E_{n,\pm} = \pm \frac{\hbar}{2} \left( \sqrt{\Delta^2 + n\Omega_0^2} - \Delta \right)$$

This shift depends on which $$n$$-excitation sector we're working in, but if our light source has a well-defined number of photons (which is always the case for high intensity coherent states), we can assume that the shift imposed on the atom by our light source is approximately constant. In this case, which we will assume in the following, we define $$\Omega = \sqrt{n} \Omega_0$$. In a semiclassical derivation of the light shift, it is this $$\Omega$$ that we would have been working with from the outset. In either case, $$\Omega$$ is proportional to the strength of the electric field at the location of our atom.

Discussion
If the applied field is sufficiently weak, we can view the light shift as a small perturbation of the energy levels of our system. More precisely, if $$\Omega \ll \Delta$$, then the value of this shift is $$\delta E_{\pm} \approx \pm \frac{\hbar \Omega^2}{4 \Delta}$$. This is quadratic in the electric field strength, which is the same lowest-order behavior as in the DC Stark effect. However, because it is non-perturbative, the dressed state approach provides us with the energy shift caused by fields of arbitrary strength, within the validity of our above assumptions.

One important point to mention is that we've neglected all of the modes of our electromagnetic field other than the laser mode. The effect of these modes, while small compared to the effect of the laser mode, is still significant, as perturbations by these modes can cause the state of our atom to leak out of its two-level submanifold. This effect can be approximated by including an imaginary term in the atomic Hamiltonian that leads to a decay in the probability amplitude for our atom to be in the excited state. Because both of the dressed states involve superpositions of the atomic ground and excited states, this means that our lower-energy dressed state will also decay at some rate. This is known as the scattering rate, and to lowest order is $$\gamma_S = \frac{\hbar \Omega^2 \Gamma}{4 \Delta^2}$$, where $$\Gamma$$ is the natural linewidth of the two-level transition. Because the relative magnitudes of $$\delta E$$ and $$\gamma$$ are set by the ratio $$\frac{\Delta}{\Gamma}$$, by increasing both the detuning and the intensity of our laser, we can make the scattering rate negligible in comparison to the light shift that a given atom feels, allowing for longer interaction times in an experiment.

Applications
The AC Stark effect has many uses in atomic, molecular, and optical physics, as well as within experimental implementations of quantum information processing.

Neutral Atom Trapping and Cooling
Historically, schemes for trapping neutral atoms developed much later than those for charged atoms. The reason for this is that neutral atoms, lacking an overall charge, don't interact as strongly with electromagnetic forces as do ions. There are several ways to get around this difficulty, one of which is by using the AC Stark shift. The basic mechanism here is that the light shift experienced by an atom increases with the intensity of the electric field at its location. Thus, by creating a field whose intensity varies throughout space, we can create regions that are energetically favorable or unfavorable for an atom to reside in. By clever arrangement of our applied field, it is possible to create a variety of trap geometries in this manner.

The light shift that an atom experiences also depends on the internal state of the atom. Using our two-level picture from above, if we were to suddenly flip the atom's internal state from the ground to the excited dressed state, a light shift that had formerly been repulsive would become attractive, and vice versa. This interplay of spatially varying light shifts and internal degrees of freedom can be used to cool an atom, as in the phenomenon of Sisyphus cooling.

Optical Lattices
Rather than configuring a laser to create a single potential minimum or maximum, we can instead imagine creating a standing wave whose intensity varies periodically in space. Such a configuration is known as an optical lattice, and through the AC Stark effect, subjects an atomic sample to a periodic potential energy distribution. Due to its periodic nature, such an experimental setup can be used to simulate particular systems studied in condensed matter or solid-state physics. More generally, optical lattices populated with neutral atoms have been suggested as a setting for universal quantum computation, where the electrical neutrality of the atoms leads to a substantially decreased rate of decoherence.