Ramsey interferometry, also known as Ramsey–Bordé interferometry or the separated oscillating fields method, is a form of atom interferometry that uses the phenomenon of magnetic resonance to measure transition frequencies of atoms. It was developed in 1949 by Norman Ramsey, who built upon the ideas of his mentor, Isidor Isaac Rabi, who initially developed a technique for measuring atomic transition frequencies. Ramsey's method is used today in atomic clocks and in the S.I. definition of the second. Most precision atomic measurements, such as modern atom interferometers and quantum logic gates, have a Ramsey-type configuration. A modern interferometer using a Ramsey configuration was developed by French physicist Christian Bordé and is known as the Ramsey–Bordé interferometer. Bordé's main idea was to use atomic recoil to create a beam splitter of different geometries for an atom-wave. The Ramsey–Bordé interferometer specifically uses two pairs of counter-propagating interaction waves, and another method named the "photon-echo" uses two co-propagating pairs of interaction waves.
Contents
Introduction
A main goal of precision spectroscopy of a two-level atom is to measure the absorption frequency
The Rabi Method
A simplified version of the Rabi method consists of a beam of atoms, all having the same speed
The Hamiltonian in the rotating frame (including the rotating wave approximation) is:
The probability of transition from |↓⟩ and |↑⟩ can be found from this Hamiltonian and is:
This probability will be at its maximum when
In reality, however, inhomogeneities such as the atoms having a distribution of velocities or there being an inhomogeneous
The Ramsey Method
Ramsey improved upon Rabi's method by splitting the one interaction zone into two very short interaction zones, each applying a
The Hamiltonian in the rotating frame for the two interaction zones is the same for that of the Rabi method, and in the non-interaction zone the Hamiltonian is only the
This probability function describes the well-known Ramsey Fringes.
If there is a distribution of velocities and a "hard pulse"
By increasing the time of flight in the non-interaction zone,
Because Ramsey's model allows for a longer observation time, one can more precisely differentiate between
Early Ramsey interferometers used two interaction zones separated in space, but it is also possible to use two pulses separated in time, as long as the pulses are coherent. In the case of time-separated pulses, the longer the time between pulses, the more precise the measurement.
Atomic Clocks and the SI Definition of the Second
An atomic clock is fundamentally an oscillator whose frequency
Experiments of Serge Haroche
Serge Haroche won the 2012 Nobel Prize in physics with David J. Wineland for their experiments in cavity quantum electrodynamics (QED) in which they used microwave-frequency photons to verify the quantum description of electromagnetic fields. Essential to their experiments was the Ramsey interferometer, which they used to demonstrate the transfer of quantum coherence from one atom to another through interaction with a quantum mode in a cavity. The setup is similar to a regular Ramsey interferometer, with key differences being there is a quantum cavity in the non-interaction zone and the second interaction zone has its field phase shifted by some constant relative to the first interaction zone.
If one atom is sent into the setup in its ground state
The Ramsey–Bordé Interferometer
Early interpretations of atom interferometers, including those of Ramsey, used a classical description of the motion of the atoms, but Bordé introduced an interpretation that used a quantum description of the motion of the atoms. Strictly speaking, the Ramsey interferometer is not an interferometer in real space because the fringe patterns develop due to changes of the pseudo-spin of the atom in the internal atomic space. However, an argument could be made for the Ramsey interferometer to be an interferometer in real space by thinking about the atomic movement quantumly—the fringes can be thought of as the result of the momentum kick imparted to the atoms by the detuning
The Four Traveling Wave Interaction Geometry
The problem that Bordé et al. were trying to solve in 1984 was the averaging-out of Ramsey fringes of atoms whose transition frequencies were in the optical range. When this was the case, first-order Doppler shifts caused the Ramsey fringes to vanish because of the introduced spread in frequencies. Their solution was to have four Ramsey interaction zones instead of two, each zone consisting of a traveling wave but still applying a
The interaction geometry of two pairs of counter-propagating waves that Bordé et al. introduced allows for improved resolution of spectroscopy of frequencies in the optical range, such as those of Ca and I2.
The Interferometer
Specifically, however, the Ramsey–Bordé interferometer is an atom interferometer that uses this four traveling wave geometry and the phenomenon of atomic recoil. In Bordé's notation, |a⟩ is the ground state and |b⟩ is the excited state. When an atom enters any of the four interaction zones, the wavefunction of the atom is divided into a superposition of two states, where each state is described by a specific energy and a specific momentum: |α,mα⟩, where 'α' is either 'a' or 'b' (see Figure 5). The quantum number mα is the number of light momentum quanta
Looking at the probability to transition to |b⟩ after the atom has passed through the fourth interaction zone, one would find dependence on the detuning in the form of Ramsey fringes, but due to the difference in two quantum mechanical paths. After integrating over all velocities, there are only two closed circuit quantum mechanical paths that do not integrate to zero, and those are the |a, 0⟩ and |b, –1⟩ path and the |a, 2⟩ and |b, 1⟩ path, which are the two paths that lead to intersections of the diagram at the fourth interaction zone in Figure 5. The atom-wave interferometer formed by either of these two paths leads to a phase difference that is dependent on both internal and external parameters, i.e. it is dependent on the physical distances by which the interaction zones are separated and on the internal state of the atom, as well as external applied fields. Another way to think about these interferometers in the traditional sense is that for each path there are two arms, each of which is denoted by the atomic state.
If an external field is applied to either rotate or accelerate the atoms, there will be a phase shift due to the induced de Broglie phase in each arm of the interferometer, and this will translate to a shift in the Ramsey fringes. In other words, the external field will change the momentum states, which will lead to a shift in the fringe pattern, which can be detected. As an example, apply the following Hamiltonian of an external field to rotate the atoms in the interferometer:
This Hamiltonian leads to a time evolution operator to first order in
If
Where
For a calcium atom on the Earth's surface that rotates at
A similar effect can be calculated for the shift in the Ramsey fringes caused by the acceleration of gravity. The shifts in the fringes will reverse direction if the directions of the lasers in the interaction zones are reversed, and the shift will cancel if standing waves are used.
The Ramsey–Bordé interferometer provides the potential for improved frequency measurements in the presence of external fields or rotations.