User:Soudachanha/sandbox/LightDressedStates

In the fields of atomic, molecular, and optical science, the term light dressed state refers to the shifted quantum state of an atomic or molecular system resulting from interacting with an external field driven by a laser. This is another effect of the AC Stark shift, first modeled by Autler and Townes in 1995 and termed the Autler-Townes effect. While the Autler-Townes effect refers to any external field, the light dressed states only refer to an external laser field.

Rotating Wave Approximation and Dipole Approximation
In the rotating wave approximation (RWA), the Hamiltonian of the system is taken to be in the interaction frame where rapidly oscillating terms are neglected. The Hamiltonian for the system is the sum of individual Hamiltonians of the atom HA, the laser field HR, and the interaction between atom and laser HAF.

$$H_{tot}=H_A+H_R+H_{AF}$$

With the electric dipole approximation, the Hamiltonian assumes the form where $$i$$ is the bare quantum state of the atom with energy $$E_i-E_0=\hbar\Delta_i$$, $$a_k^\dagger$$ and $$a_k$$ are the creation and annihilation operators, and $$\bf{E}$$ is the electric field. The detuning$$\Delta_i=\omega-\omega_0$$ is defined as the difference laser frequency to the transition frequency between two quantum states.

$$H_{EDA}=\sum_{i}^N \hbar\Delta_i|i\rangle\langle i| + \sum_{k}^N \hbar \nu_k (a^\dagger_ka_k+\frac{1}{2}) + -e\bf{r} \cdot \bf{E} $$

The Hamilton is time evolved by the time-dependent Schröedinger Equation and the Hamiltonian is found in the interaction picture for an N level atom .The coupling strength between quantum levels is given by Rabi frequency $$\Omega_{jk}=\Omega_{kj}$$ where j and k is the quantum levels.

$$H_{RWA}=\sum_{i}^{N-1} \hbar\Delta_{i}|i\rangle\langle i|+ \sum_{j\neq k}^{N-1} \frac{\hbar\Omega_{jk}}{2}e^{i\omega t} |j\rangle\langle k|$$

The eigenvectors and eigenvalues solutions of the approximated Hamiltonian gives the energy levels for the dressed quantum system and is represented by the Jaynes-Cummings model.

Two-Level Systems
The simplest and most studied case of light dressed states are for two-level systems consisting of a ground state $$|0\rangle$$ and an excited state $$|1\rangle$$. The Hamiltonian for the system after applying the EDA and RWA is shown below, assuming the ground state energy level is 0.

$$H_{TLS} = -\hbar\Delta|1\rangle\langle 1| + \frac{\hbar\Omega}{2}(|1\rangle\langle 0| + |0\rangle\langle 1|)$$

The resulting eigenvectors are shown below with corresponding eigenvalues.

$$|+\rangle = cos(\frac{\theta}{2})|e\rangle+sin(\frac{\theta}{2})|g\rangle$$

$$|-\rangle = cos(\frac{\theta}{2})|g\rangle-sin(\frac{\theta}{2})|e\rangle$$

The shift in energy levels is shown and depicted in the figure on the right.

$$E_\plusmn = -\frac{\hbar\Delta}{2} \plusmn \frac{\hbar\sqrt{\Omega^2+\Delta^2}}{2}$$

Applications
Light dressed states have become increasingly impactful the field of quantum optics and quantum information sciences. Implementation of the shifted states has touched quantum qubits, atom trapping and cooling, transport and sensing, and to further studying quantum state shifts and interactions.

Due to the shifted states in a system, the adiabatic transfer from one quantum state to another could be used as qubits. Various publications have laid a theoretical foundation for adiabatic population transfer in variety of atoms and molecules for dressed states. The first demonstration of adiabatic transfer was for helium by Vitanov et al. in 2001. With these, many have created schemes in order to achieve various technologies such as atom interferometer, quantum information transfer. or sensing.

Dressed states have also been used to further research in AMO physics. Groups have used the dark state resulting from adiabatic transfer to study other atom interference effects such as Fano interference in electromagnetically induced transparency of rubidium or simulated lamb shifts the Mollow sidebands in hydrogen. In 2018, Bounds et al. used dressed states of Rydberg atoms in order to create a magneto-optical trap to cool and trap Rydberg state atoms.