Visibility
The classic measurement of the visibility of interference phenomena is of the form
I
I
Vclass  max min
I max  I min
where I max , I min are the maximum and minimum of the corresponding distribution I ( ) .
We show below that this measure of interference is not suitable for general anharmonic
distributions I ( ) , which are becoming usual, specially in the quantum domain. We have
proposed a new measure of the amount of interference which gives the same result than the
classic definition for harmonic fringes but is better behaved when applied to general
anharmonic patterns.
We can naturally define the visibility as the distance of the observed interference pattern I ( )
to the uniform distribution, denoted here as I ,
V2 
1
2 I
2
2
 d I ( )  I  
1
I
2
I
2
 I
2

where
I 
1
dI ( )
2

and
I2 
1
d I ( )2
2

For harmonic patterns I ( )  a  b cos(   ) both definitions provide proportional results
Vclass  2V .
As a suitable example of anharmonic patterns we can consider the output of multiple-beam
interferometers. Specifically, we focus on two particular examples: (1) a bright maximum in a
uniform dark background and (2) a dark minimum in a bright uniform background. For the
sake of simplicity, we can describe these patters by using properly normalized Gaussian
functions
2
2  
 2 
I ( ) 
I ( )  2 e
1  e


2   1 
bright maximum
dark minimum
in a dark background
in a bright background
where  is essentially the inverse of the width of the peaks (bright or dark). We will always
assume narrow enough peaks   1 .
For the bright maximum in a dark background we have Vclass  1 while V 2  2 . This
example demonstrates that V is a visibility measure better behaved than Vclass . It seems clear
that in this case the amount of interference should depend on the width of the peak (i. e. of
 ), as the very same idea of resolving power demonstrates. In this sense we have that, on the
one hand, Vclass does not depend on  appreciably, while V clearly depends on  . Thus V
is better behaved than Vclass .
On the other hand, whenever Vclass  1 the multiple-beam interference would be equivalent
to the harmonic fringes produced by two-beam interferometers. However, it is known that
multiple-beam interferometers provide more efficient examples of interference phenomena
than two-beam arrangements. This fact is clearly reflected by V since it grows without limit
when the width of the peak decreases (    ), while it is bounded from above for harmonic
patterns.
Concerning the case of a dark minimum in a bright background we have that again Vclass  1
while V gives a fully opposite result, V 2  1 /( 2 2 ) , so that V tends to zero as the width
of the minimum decreases. We can show that also in this case V provides the right answer. In
this case, and at difference with the preceding example, as the width of the peak tends to zero
also decreases the relative area enclosed by the minimum, so that the distribution approaches
the uniform distribution. From a practical perspective, as the area enclosed by the peak
decreases, is more difficult to detect the very same existence of the peak distinguishing it
against the bright background. This reasoning is specially clear from a quantum perspective,
since in such a case the area enclosed by the peak represents the probability of detecting the
interference effect.
These examples demonstrate that the new definition is a measure of interference better
behaved than the classic definition, being more sensible to the relevant features of
interference phenomena. It is also interesting to note that, in general, V can be more easily
determined than Vclass , since Vclass involves the determination of the extremes of a highly
nonlinear function depending on a large number of parameters.
Finally we recall that for locally harmonic fringes the classic visibility is proportional to
amplitude correlations. However, this interesting connection is lost when applying Vclass to
general anharmonic fringes. We have demonstrated that these relations are preserved for the
general case when using V .
Visibility for anharmonic fringes
A. Luis, J. Phys. A: Math. Gen. 35, 8805 (2002)
We have extended this idea of visibility as a distance to the case of multi-particle interference,
normally in the form of a probability P ( ) of coincidence of the detection of particles at the
output of an interferometer, where  represent the internal phases of the interferometer. In
order to focus on the multi-particle behaviour it is customary to define the interferometric
distribution of interest as the difference between P ( ) and the single-particle probabilities
Pj ( ) , i. e. P( )  P1 ( ) PN ( ) where N is the number of particles. In this case, the standard
definition of visibility is rather unsuitable. On the one hand usually it is very difficult to
compute the local maximum and minimum of P( )  P1 ( ) PN ( ) since this is an involved
function of a large number of variables. On the other hand P( )  P1 ( ) PN ( ) can be
negative so that the direct application of Vclass is meaningless unless ad-hoc modifications are
performed. In order to avoid these drawbacks we propose to define the multi-particle visibility
as the distance between the joint probability and the product of single-particle probabilities
D  d P( )  P1( ) PN ( )2 .

With this approach we have demonstrated that for two particles the multi-particle visibility
and the individual visibilities are complementary observables (when one increases the other
decreases and vice-versa). Moreover, the visibility of P ( ) and the visibilities of Pj ( ) are
also complementary. Furthermore, we have demonstrated that for three particles there is no
complementarity so that the joint and the single visibilities can increase or decrease
simultaneously.
Visibility for multi-particle interference
A. Luis, Phys. Lett. A 314, 197 (2003)