Quantum Imaging
Quantum Imaging
Laboratoire Kastler Brossel Université Pierre et Marie Curie
Paris - France
Nicolas Treps
Ability to separate details XIX th century : Lord Rayleigh
Gemini north dome,
Hawaii
how accurately can I separate two objects in space ?
Resolution is limited by spot size
Diffraction theory : resolution limited by the wavelength
!
XXI st century : image sensor : diode arrays, CCD cameras, …
Object plane
a(y)
Image plane
e(x)
Imaging device with pupil
X
There exist eigenmodes of the system (prolate spheroidal functions ),
f k (x) with eigenvalues t k (transmission coefficient).
The knowledge of these functions, together with e(x), allows the
‘perfect’ reconstruction of the object
CCD camera
Object plane
a(y)
Image plane
e(x) Imaging device
with pupil
X CCD
camera
Quality of the detectors : size, number of pixel, response,…
Classical noise (vibrations, thermal noise,…)Technical limits Fundamental limit
Quantum nature of light (quantum noise)
Limits to resolution
Optical resolution
No a-priori information on the image : smallest details measurable.
In many practical cases : the Rayleigh criteria.
Crossing the standard quantum limits requires very multimode quantum light, i.e. many resources.
Information extraction
A lot of a-priori information : presence and/or modification of a given pattern.
We will show that crossing the standard quantum limit requires a limited amount of resources.
Quantum limit easier to reached : orders of magnitude smaller than the Rayleigh criteria.
Optical read out
Single mode versus multimode light
Quantum limits to resolution
Few modes approach : the quantum laser pointer Many modes approach : multimode cavities
Single mode versus multimode light
Quantum limits to resolution
Few modes approach : the quantum laser pointer Many modes approach : multimode cavities
Paraxial approximation
A beam of light is the result of the excitation of an infinite set of harmonic
oscillators.
The electric field distribution can be expanded over a transverse mode basis : - plane waves basis : very suitable for calculation
However, for the propagation of a beam of light, we make several approximations :
- the light is monochromatic : "(k)!"
- the direction of propagation is well defined : k ! k
z
Where is the slowly varying envelope of the fields that satisfies the propagation equation in the vacuum, projected onto the polarisation axis :
Transverse modes basis
can be expanded on a transverse modes basis such as :
orthonormality completeness
shape of mode i
There is then a unique set of coefficient #i such as :
It contains all the image information
field amplitude in mode i Remark :
As the modes have to satisfy the propagation equation, their knowledge at z=0 is enough.
y
x
z
Examples
- Pixel basis :
Advantages : very natural to describe random images convenient for numerical simulation Drawbacks :
mode diffraction is very important
predicting the field shape under propagation is difficult
light beam
At z=0,
Gaussian modes
Hermite-Gauss modes
Laguerre-Gauss modes
Gaussian modes basis : eigen modes of the propagation These modes have a transverse shape that remain constant under propagation.
They are adapted for light coming out of a cavity (such as laser beams).
Single mode vs. multimode classical light ? Possible to compute the number of modes ?
It depends on the choice of the basis !
For a field coming out of a cavity, one will naturally choose the
Hermite Gauss or Laguerre Gauss basis.
Single mode basis
We have a given image :
We choose the first mode such as :
It is always possible to choose the other modes to satisfy the completeness and orthonormality conditions
In that basis :
No intrinsic definition of multimode at the classical level
Quantum description of the field
Each mode is treated as a single harmonic oscillator
We associate to each mode a set of creation and annihilation operator It allows to define the number of photon in each mode
The electric field operator
classical value quantum fluctuations
Signal given by a detector
light beam
y
x
z
Detector covering a transverse area D
Detector : signal proportional to the number of photons Signal and noise
The signal is given by the mean number of photon
The noise is the variance of the number of photons
Single mode quantum field
Annihilation operators
Single mode field
The field state in all the modes except the first one is a coherent vacuum It then corresponds to the single mode quantum optics studied in the lecture of Hans Bachor.
Electric field operator
Known classical image
It exists a proper definition of single mode at the quantum level
It is based on the quantum fluctuations
The same can be done for a statistical superposition of modes
Outline
Single mode versus multimode light
Quantum limits to resolution
Few modes approach : the quantum laser pointer Many modes approach : multimode cavities
Quantum limits to resolution
Light used in the experiment is single-mode coherent light
modes
0()u r r 1()
u r r ()
n u r r Quantum state Coherent state |#0>
vacuum
vacuum
Classical light !
light beam
Image carried by a Coherent state
i 1(t)i 2(t)i 3(t)i 4(t)
Measurement performed
Photon picture of coherent single mode light
Usual quantum optics description
V
t
Continuous wave regime (1mW ? 1017photons/s)
Photon number : Poisson statistic (also called white noise)
!N =N
Shot noise
Random transverse distribution
Spatial quantum optics description
i 1(t)i 2(t)i 3(t)i 4(t)
Each detectors sees Poissonian noise
i i
N N !=Local Shot noise
!
d
i 1
i 2d
i 1-i 2
d
d lqs
!
<<"=N
d lqs
Example :
?Beam of 1mW ?!=200"m
?Integration time of 10"s =lqs d 5?
6
10.3,41!=="#$%N d Dimensionless quantity
Two pixels case
Smallest displacement detectable Signal scales with Noise scales with
i 1 and i 2 not correlated :Noise of the difference=noise of the sum N N