User:Wperreault/sandbox

Optical adiabatic passage
Optical adiabatic passage encompasses a broad range of laser-based excitation techniques that make use of the dressed states of the light-matter Hamiltonian to transfer population between quantum states. These techniques cleverly manipulate laser pulses in order to dynamically modify the quantum states of the light-matter system such that a large fraction of the ground state population is smoothly transferred to a single, specific target state. This population transfer is effected by the slow variation of one of the field parameters to create an avoided crossing of the adiabatic eigenstates. Included under this umbrella are the various rapid adiabatic passage techniques, as well as Stimulated Raman adiabatic passage (STIRAP), and Stark-induced Adiabatic Raman Passage (SARP). These techniques differ both in the choice of field parameter to be varied and the molecular systems to which they can be applied, and so the choice of technique must necessarily be based on both the laser sources available as well as the properties of the material to be pumped.

Adiabatic passage
In quantum mechanics, a process is adiabatic if its time variation occurs slowly enough that the system can adapt its configuration and thus remain in an instantaneous eigenstate. Consider the time-dependent Schrödinger equation:

$$i\hbar\frac{d|\Psi\rangle}{dt}=\hat{H}(t)|\Psi\rangle$$.

Assume that at time $$t=t_0$$, the system sits in $$|\Phi_n(t_0)\rangle$$ which is an eigenstate of the Hamiltonian $$\hat{H}(t_0)$$. The Hamiltonian then changes in time in the system, producing $$\hat{H}(t_1)$$. The adiabatic theorem, first proved by Born and Fock, states that if the Hamiltonian changes slowly enough in time, the system will evolve into state $$|\Phi_n(t_1)\rangle$$, an eigenstate of $$\hat{H}(t_1)$$. If it changes too quickly, the system will not have time to adapt, resulting instead in $$|\Psi_n(t_1)\rangle$$, a superposition of eigenstates of $$\hat{H}(t_1)$$. Landau and Zener, working independently, calculated the probability of a non-adiabatic transition for a given linear rate of change of the Hamiltonian, providing some definition of what “slow enough” means in practice (see ). Given this formalism, the fundamental idea behind adiabatic passage is actually rather simple. We want to engineer our optical pulses to meet three conditions:


 * 1) that $$|\Phi_n(t_1)\rangle$$ coincides with the populated ground state, $$|1\rangle$$,
 * 2) that $$|\Psi_n(t_1)\rangle$$ coincides with the target state of choice, $$|2\rangle$$, and
 * 3) that the time variation of the Hamiltonian is slow enough to meet the Landau-Zener condition, meaning that no non-adiabatic transitions take place.

If these three conditions are met, the population will smoothly flow from the ground state to the excited state, giving high state specificity, as well as complete transfer efficiency.

Adiabatic passage in two-level system
Consider the time-dependent Schrödinger equation governing the time dependence of a two-state quantum system exposed to low-intensity, near resonant light:

$$\frac{d}{dt}\begin{bmatrix} C_1(t) \\ C_2(t) \end{bmatrix}=\frac{-i}{2}\begin{pmatrix} -\Delta & \Omega \\ \Omega^* & \Delta \end{pmatrix}\begin{bmatrix} C_1(t)  \\ C_2(t) \end{bmatrix}$$,

where $$\Delta$$ gives the detuning from resonance, and $$\Omega$$ gives the Rabi frequency. This gives the dynamics of the system in terms of the basis states $$|1\rangle$$ and $$|2\rangle$$, which are known as “bare” or “undressed” because they are the eigenstates of the time independent portion of the Hamiltonian, $$\hat{H}^0$$. To understand adiabatic passage, we must reformulate in terms of the dressed states $$|\Phi_n(t)\rangle$$, which are eigenstates of the full Hamiltonian, $$\hat{H}(t)$$.