Quantum limits to the precision of interferometric measurements in linear
interferometers
Due to the continuous improving of experimental techniques, the effects of quantum noise in
measuring arrangements is currently a subject of practical interest. As a matter of fact, the quantum
noise is the most important source of uncertainty in current interferometric arrangements. Among
them we may include the arrangement schematized in the figure, which is devised to the detection
of small forces.
This is a Michelson interferometer in which one of the mirrors can be displaced by the action of the
force to be detected. This displacement causes the variation of the light intensity leaving the
interferometer.
In this arrangement, the quantum nature of light introduces to sources of noise. On the one hand the
fluctuations of the number of output photons (shot noise) give rise to some uncertainty in the
inferred position of the mirror. On the other hand, the fluctuations of the radiation pressure also
cause fluctuations in the position of the moving mirror. In a first approach both sources of noise
were considered as statistically independent leading to the conclusion that they impose an
unavoidable quantum limit to the precision of the measurement of the position of the mirror: the
standard quantum limit.
Our approach to this problem begins by noting that these sources of noise cannot be statistically
independent since the field states in and out the interferometer must be strongly correlated.
Moreover, in this experiment the only observables are the intensities leaving the interferometer.
These variables must contain all the information about the measurement (theoretically as well as
experimentally) including the fluctuations. Therefore, all the uncertainty of the measurement must
be derived as a shot noise.
To this end we embodied the moving character of the mirror in the input-output transformation by
taking into account explicitly that the length of the corresponding arm depends on the light intensity
impinging on it. The result is a nonlinear transformation. After this inclusion all the uncertainty is
computed as a shot noise. Following this procedure we have been able to demonstrate that, contrary
to what was believed before, there is no quantum limit to the precision of the measurement.
Mode transformation properties and quantum limits for a Fabry-Perot interferometer
A. Luis and L. L. Sánchez-Soto, J. Mod. Opt. 38, 971 (1991)
Breaking the standard quantum limit for interferometric measurements
A. Luis and L. L. Sánchez-Soto, Opt. Commun.. 89, 140 (1992)
Multimode quantum analysis of an interferometer with moving mirrors
A. Luis and L. L. Sánchez-Soto, Phys. Rev. A. 45, 8228 (1992)
Una excursión por los confines del ruido cuántico
L. L. Sánchez-Soto and A. Luis, Rev. Esp. Fís. 7(2), 17 (1993)
A key ingredient of precision measurements is the issue of the practical generation of the field states
that would allow to approach the quantum limits. Concerning phase-shift measurements (with fixed
mirrors) it has been shown that the minimum detectable phase shift  is proportional to the
inverse of the total number of photons involved in the measurement   1 / n . This is known as the
Heisenberg limit. In order to reach this limit it seems necessary to use nonclassical states of light
(for the standard sources of light the phase uncertainty scales as   1 / n ). Different kinds of
nonclassical states have been proposed to reach the quantum limit. The most developed proposals
involve the use of : i) SU(2) intelligent states or ii) maximally entangled states.
i) We have demonstrated that the SU(2) intelligent states can be generated experimentally in the
practical arrangement illustrated in the figure, where there are two nonlinear crystals producing
spontaneous parametric downconversions when pumped (all the input fields represented in the
figure are initially in vacuum and the pump wave has not been represented). The crystals are
coupled by a beam splitter.
We have shown that performing a measurement of the number of photons at the output modes
represented in the figure, the field state in the rest of output modes verify the eigenvalue equation
cosh( ) S x  isenh( ) S y 

This demonstrates the field state generated in this way is a minimum uncertainty state of the Stokes
operators S x , S y and thus it is a SU(2) intelligent state.
SU(2) coherent states in parametric down-conversion
A. Luis and J. Peřina, Phys. Rev. A. 53, 1886 (1996)
ii) Another family of fields state that would serve to reach the Heisenberg limit are the maximally
entangled states of the form
1
 
 1 0 2  0 1 2
2
where 0 represents the vacuum state. In these states the light is distributed at 50 % in two field
modes in such a way that strong nonclassical correlations appear: if a photon is found in one of the
modes (with 50% probability for each mode), the rest of photons appear on the same mode and
none in the other mode.


We have demonstrated that these states allow to reach the Heisenberg limit via an effective
amplification of the applied phase-shift. Moreover, we have found a feasible experimental
procedure within the reach of current technology that would serve to generate these states.
In a related work we have demonstrated that these states are equivalent to the so-called
Schroedinger cat states: quantum superpositions of distinguishable states. The interest of this
relation is that all the methods proposed so far to generate the Schroedinger cat states would serve
to generate the states reaching the ultimate quantum limits in phase-shift measurements. As a matter
of fact there are experimental results confirming this possibility.
Generation of maximally entangled states via dispersive interactions
A. Luis, Phys. Rev. A 65, 034102 (2002)
Equivalence between macroscopic quantum superpositions and maximally entangled states:
Application to phase-shift detection
A. Luis, Phys. Rev. A 64, 054102 (2001)
As we have shown, the problem of the optimal input states maximizing resolution when the mirrors
are fixed has been well studied. In another work we have addressed the optimization of
interferometers with moving mirrors, whose position can be affected by the radiation pressure.
We have found a universal nonlinear relation linking the optimal states of the interferometers with
moving and fixed mirrors that solves the problem once for all. This relation allows to easily
determine the optimum states for interferometry with moving mirrors in terms of the well-known
optimum states for interferometers with fixed mirrors. Moreover, this shows that the quantum limits
to the resolution are the same for both kinds of interferometers (this is the Heisenberg limit).
Optimum quantum states for interferometers with fixed and moving mirrors
A. Luis, Phys. Rev. A 69, 045801 (2004)
We have developed an assessment of metrological resolution that can be achieved with a given
detection arrangement. The detection of a signal  arises because the signal produces an
observable change in a given probe state transforming the input probe state  into the output state
  , normally in the form
   exp iG exp  iG ,
where G is the generator of the transformation induced by the signal.
A natural performance assessment of a detection process is to compute the distance between the
input  and transformed   probe states. This is an intrinsic assessment in the sense that it does
not depend on the final measurement performed on the output probe state. As a simple and
meaningful distance we can consider the Hilbert Schmidt so that for weak signals (the only case on
interest in precise quantum metrology)



tr     2  2  2 2 (  , G)
where


2 (  , G)  tr  2G 2  tr GG  .
Not that  (  , G ) is fully symmetric on  and G, this is (  , G )  (G,  ) . We have that  (  , G )
is a kind of generalization of variance since for pure states
 
we have
2 (  , G )   G 2    G 
2




  G 2 and in general 2 (  , G )    G 2 . However,
 (  , G ) is not an uncertainty measure. Instead this is a measure of how much sensitive is  to the
changes generated by G, a rather simple version of the quantum Fisher information. For example,
for every state  commuting with G, [  , G ]  0 we have (  , G )  0 although   G can be
arbitrarily large. Moreover, there is no lower bound for the product (  , G2 )(  , G1 ) for tw
noncommuting gnerators G1,2 . Moreover, the following formula shows that in general  (  , G )
strongly depends on the coherences g j  g k for k  j with the eigenvectors g j
  G is independent of them ,
2 (  , G ) 

of G, while

2
1
 g j  gk 2 g j  gk
2 j, k
An interesting result we have obtained is that when  (  , G ) is larger than the value reached by
coherent states then  is a nonclassical state.
Intrinsic metrological resolution as a distance measure and nonclassical light
A. Rivas y A. Luis, Phys. Rev. A 77, 063813 (2008)
Up to now, all proposals devised to optimize the interferometric resolution are based on linear
interferometers in which the light amplitudes at the output is proportional to the light amplitudes at
the input. We have found that it is possible to perform much better measurements with nonlinear
interferometers in which the light amplitudes at the output are non linear function of the light
amplitudes at the input. This idea is developed in a different section of this same web page.