1. Theoretical investigation of Sources of Errors in Phase measurement of
Atom Interferometer and its implications on miniaturization
𝐴𝑦𝑢𝑠ℎ𝑖 𝑀𝑎𝑙𝑣𝑖𝑦𝑎1,ϯ
, 𝑆𝑜𝑙𝑜𝑚𝑜𝑛 𝐼𝑣𝑎𝑛1
, 𝑉. 𝑁. 𝑅𝑎𝑑ℎ𝑖𝑘𝑎2
, 𝑈𝑚𝑒𝑠ℎ 𝑅. 𝐾𝑎𝑑ℎ𝑎𝑛𝑒1
1[Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala], 2[NSSG, ISRO Inertial Systems Unit, Trivandrum, Kerala], ϯ[ayushimalviya50@gmail.com]
INTRODUCTION
Atom interferometry is a technique in which coherently split atomic waves are later
recombined. It is widely used for extremely accurate measurements of fundamental
constants like fine structure constant, gravitational constant and properties of
particles by observing quantum effects like electric polarizability, collision cross-
section etc. It is performed using atoms cooled down to a temperature of few micro
Kelvins called ‘cold atoms’.
VELOCITY DISTRIBUTION OF ATOMS
Maxwell Boltzmann particle velocity distribution gives the distribution of kinetic
energy for a large collection of atoms or molecules as a function of temperature.
At room temperature (293 K), velocity of atoms ranges from 100-2000 m/s e.g.
273 m/s for Rubidium, 470 m/s for Nitrogen, 570 m/s for Sodium, 394 m/s for
Argon etc.
𝑓 𝑣 =
𝑚
2𝜋𝑘𝑇
3/2
4𝜋𝑣2
𝑒−
𝑚𝑣2
2𝑘𝑇
LASER COOLING
Laser cooling is a technique used to produce ensemble of cold atoms up to micro
Kelvins of temperature, by interaction of atoms with one or more laser fields [1]. If an
atom is travelling in the direction of laser beam, it absorbs the photon and
experiences a momentum recoil, which slows down the atom.
Light interferometry requires a light source and optical instruments like diffraction
grating, mirrors, beam splitters etc. to manipulate the waves. Lasers are the
preferred light source because they have a larger coherence length, high photon
flux and are well-collimated. In case of atoms, we have a cold atoms cloud as
source, being manipulated by electromagnetic fields in the form of laser beam.
A three level atom interacting with two electromagnetic fields can be used to create
an atom interferometer. The two possibilities on interaction of photon and atom are :
absorption of photon by the atom and stimulated emission of a new photon caused
by the incoming photon. Assuming that the lifetime of excited states is high enough,
spontaneous emission can be prevented. If the atom is initially in ground state with
momentum 𝒑 𝒈, then following figure describes the absorption and stimulated
emission (with counter propagating and co-propagating beams respectively).
REFERENCES
1. Ibnotes, ‘Maxwell Boltzmann Distribution’.
2. Giacomo Lamporesi, UNIVERSITÀ DEGLI STUDI DI FIRENZE, ‘Determination of the
gravitational constant by atom interferometry’, December 2006.
3. Mathias Hauth, Humboldt-Universität zu Berlin, ‘A mobile, high precision atom-
interferometer and its application to gravity observations’, 6 March 2015.
In Doppler cooling, there are six
detuned counter-propagating laser
beams surrounding the atoms, so
that absorption is allowed for only
those atoms moving towards the
laser beam. When accompanied by a
magnetic coil such a setup can be
used to trap cold atoms, called a
Magneto-Optical Trap.
For temperature in micro
Kelvins, thermal velocity of
atoms is in the range of mm/s
and velocity distribution
becomes narrower leading to
cooling of atoms. Lower
velocity allows a larger
interrogation time with atoms.
Atoms have several advantages over photons like higher mass, high sensitivity to
electric and magnetic fields, more degrees of freedom, no unwanted phase shifts
etc. The action of a 𝜋 pulse in an atom interferometer is similar to a mirror i.e.
taking the atoms from one momentum state to another, whereas the action of a
𝜋/2 pulse is similar to a beam splitter i.e. create an equal superposition of the two
momentum states [2].
STIMULATED RAMAN TRANSITIONS
Transition from state |𝑔⟩ to |𝑒⟩ is dipole forbidden but making use of
intermediate state |𝑖1⟩, the above transition can be achieved. There can be an
infinite number of momentum states and three energy Eigenstates for the
system, but by fixing the momentum for one state that for all the other states
can be determined. Thereby reducing the system to an effective number of five
states in the combined Hilbert space. Interference is achieved by inducing a
transition causing fluorescence.
Miniaturization of the system can lead to several errors. Some of them are:
• Uncertainty in velocity distribution of the atoms.
• Laser de-coherence.
• Height of the atomic fountain may not be appropriate.
• Transit time of atoms being shorter than time interval of the pulses.
• Transitions not being sharp, leading to dispersion in energy and momentum.
• Doppler broadening of the Raman Transitions.
• Non-coherent Rabi Oscillations.
Apart from these, there is also a possibility of spontaneous emission occurring.
• |𝒈, 𝒑 𝒈⟩
• 𝒊 𝟏, 𝒑𝒊𝟏 = |𝒊 𝟏, 𝒑 𝒈 + ℏ𝒌 𝟏⟩
• 𝒊 𝟐, 𝒑𝒊𝟐 = 𝒊 𝟐, 𝒑 𝒈 + ℏ𝒌 𝟐
• 𝒆, 𝒑 𝒆 = |𝒆, 𝒑 𝒈 + ℏ 𝒌 𝟏 − 𝒌 𝟐 ⟩
• 𝒊 𝟑, 𝒑𝒊𝟑 = |𝒊 𝟑, 𝒑 𝒈 + ℏ(𝟐𝒌 𝟏 − 𝒌 𝟐)⟩
FUTURE SCOPE
Using rotating wave
approximation (RWA), these
five states can be reduced to
an effective two states system
comprising of only the ground
and excited states, whose
coefficients are given by [3]:
LIGHT VS. ATOM INTERFEROMETRY
𝒄 𝒈 𝒕 = −𝒊
|𝛀 𝒈𝒊 𝟏
| 𝟐
𝟒𝚫
+
|𝜴 𝒈𝒊 𝟐
| 𝟐
𝟒 𝜟−𝝎 𝒆𝒈
𝒄 𝒈 𝒕 −
𝒊𝛀 𝒆𝒊 𝟐
∗
𝛀 𝒈𝒊 𝟏
𝟐
𝒆𝒊 𝜹 𝟏𝟐 𝒕+𝝓 𝟐−𝝓 𝟏 𝒄 𝒆(𝒕)
𝒄 𝒆 𝒕 = −𝒊
|𝛀 𝒆𝒊 𝟏
| 𝟐
𝟒(𝚫+𝝎 𝒆𝒈)
+
|𝜴 𝒆𝒊 𝟐
| 𝟐
𝟒𝚫
𝒄 𝒆 𝒕 −
𝒊𝛀 𝒆𝒊 𝟐
∗
𝛀 𝒈𝒊 𝟏
𝟐
𝒆−𝒊 𝜹 𝟏𝟐 𝒕+𝝓 𝟐−𝝓 𝟏 𝒄 𝒈(𝒕)
SOURCES OF ERRORS
The above system once miniaturized can be used to perform highly accurate
measurements of gravitational acceleration, gravitational constant and gravity
gradient helping in navigation of spacecraft and missiles, Geophysical, Astronomical
and Planetary Studies etc.