User:Ztaghipour/draft article in Ramsey Interferometry

Ramsey Interferometry, the technique for measuring transition frequencies with magnetic resonance, was pioneered by I. I. Rabi in the late 1930's and modified by Norman Ramsey in 1949. The term Ramsey interferometry is used most often for atomic interferometers based on internal states and has found to be an increasingly valuable technique for the definition of the time standard or high precision magnetometry and modern precision measurements, such as atomic clocks.

Ramsey interferometer can be considered as a counterpart of Mach–Zehnder interferometer. Compared to photons, atoms offer the advantage of having an intrinsically more complex structure and therefore allowing a larger range of possible measurements to be undertaken.

Definition
One can show that a Mach-Zehnder-type optical interferometer and the Ramsey rotations on the Bloch sphere are equivalent. Thus, Ramsey interferometry follows SU(2) transformation rules.

A Ramsey sequence consists of at least three building blocks, two π/2 rotations, R1 and R2, and an evolution time, T, in between and is as follows: (1) all spins of the atoms are prepared in the spin up state. (2) A π/2 R1 pulse is applied to rotate each spin into the x direction corresponding to a unitary rotations on the Bloch sphere around x-axis in the equatorial plane, places the atoms in a coherent and equally weighted superposition of i and g, i.e. |g> → (|g>+|i>)/√2 or |i> → (-|g>+|i>)/√2. (3) A fixed time T of free evolution follows the first rotation during which a relative phase ϕ between the two orthogonal states of an atomic system accumulates, and the aim of Ramsey interferometry is to measure this unknown relative phase ϕ. (4) another π/2 pulse is applied, which implements another unitary rotation around an axis in the equatorial plane shifted by 90° with respect to the first one and produces |g> → (|g>+e(iϕ)|i>)/√2 or |i> → (-e(iϕ)|g>+|i>)/√2 transformations.

Simple calculations show that given an atom prepared in level g before R1,the probability Pi for detecting it in level i after R2 is related to the unknown phase ϕ with Pg = 1− Pi = (1 − cosϕ)/2. The atomic superposition and field thus accumulate a phase difference ∆ϕ =2π(νgi− ν)∆t. Therefore, the number of measurements with the outcome being in the specific state, over the total number of readouts deduces the relative phase ϕ. Since there is no possibility to know which states are g and which are i along the path, the result is an interferometric pattern with sinusoidal variation of the population difference versus acquired relative phase ϕ, and is commonly called a “Ramsey fringe”.

In many cases, It is useful to describe the atom undergoing the g → i (or g → e) transition as a pseudospin 1/2 using Bloch sphere representation of the evolution in the rotating frame for the Ramsey-fringe measurement, which is shown in the Fig.1.



Ramsey interferometry and atomic clocks
One of the most well-known applications of Ramsey interferometry is in atomic clocks.

To appreciate how the device works, imagine we have a Ramsey separated field interferometer by applying to an atom two successive π/2 pulses, R1 and R2, at a frequency ν that is nearly resonant with an atomic transition, for instance, the g → i transition at frequency νgi. The resonant field periodically shuffles the atoms between ground and excited states and leads to the Ramsey fringes with a width which is simply the inverse of the time taken by the atoms to cross the intermediate region, as 1/∆t. Locking the microwave frequency on the central fringe at ν = νgi produces a time standard locked to the regular ticking of the atomic electrons. Therefore, in atomic clocks, it is desired to have very slow (ultra cold) atoms with velocities around 10 cm/s. The resulting clock uncertainty is roughly 1 second in 30 million years.

Ramsey interferometer not only can measure time with fantastic accuracy, but also, it provides an experimental format for manipulating quantum states. Also, One can utilize the Ramsey interferometer’s extreme sensitivity to radiation as a tool for counting and manipulating photons without destroying them.

Ramsey interferometry and Schrodinger cats states
A Schrödinger cat (SC) state is a mesoscopic quantum supersposition of field coherent states that have the same mean photon number and nearly opposite phases.

The preparation of SC is essential in quantum physics because of the complementary role played by energy and phase. A high Q factor cavity, sandwiched between two π/2 Ramsey pulses can be used to measure the field’s energy in a quantum non-demolition way. When the atom crosses the cavity in a linear superposition of these levels,(|g>+|e>)/√2, the cavity mode is then in a superposition of states corresponding to two different frequencies at once. This peculiar situation leads to the possibility of studying new kinds of entanglement between a microscopic system (the atom) and a mesoscopic field in the cavity.

As the Ramsey interferometer progressively pins down the photon number (field energy), it randomizes the field phase. The e and g components of atom's state superposition shift the field phase in opposite directions. Thus, after their interaction, the atom and the field are entangled in a global superposition of states. If the difference in the phase shifts of the field due to e and g is π, the two components of the “atom + field” system are associated with two coherent states of opposite phase (The situation is reminiscent of the famous Schrodinger cat entangled with a two-level atom in a superposition of states corresponding to its ‘‘live’’ and ‘‘dead’’ states.) Measuring the state of the atom as it leaves the cavity would reveal the field phase and thereby lift the quantum ambiguity. The role of the second Ramsey pulse is to mix e and g so that the final atomic detection does not yield information about the state of the atom in the cavity. Therefore, the atom’s final detection projects the field into a superposition of the two coherent states.