There are many reasons impelling the study of the phase variable in quantum optics.
This a basic variable in classical optics with an outstanding capability to explain very different phenomena, and it seems that the quantum phase difference should play a similar role. Moreover, from a practical perspective, the detection of phase changes is the basis of the most sensitive measuring techniques currently available.
In classical optics the definition of the phase

is straightforward: this is the argument of the complex amplitude of a mode of the electromagnetic field

E
 a e i

( k r
  t ) 

  arg( a ) where a denotes the complex amplitude. However, the quantum phase encounters a basic difficulty: there is no phase operator inheriting all the desirable properties for this variable. There is no simple quantum translation of the relation
  arg( a ) .
Our approach to this problem begins by noting that the phase difference is a more basic and meaningful variable than absolute phase. From a practical perspective, the experimental arrangements always detect relative phases, or difference of phases, but never absolute phases. Following this reasoning we investigated the tentative existence of an operator representing the phase difference directly, without any previous assumption concerning absolute phases. Maybe surprisingly we obtained a positive conclusion. We have demonstrated that there are suitable operator solutions for the equation
  arg( a
1 a

2
)
 arg( S x
 i S y
) where the second equality is expressed in terms of the Stokes operators
S x
 a
1 a
2

 a
2 a
1

S y
 i ( a
1 a
2

 a
2 a
1

)
As it might be expected, this operator cannot be expressed as the difference of phase operators
  
1
 
2
.
It is worth stressing that the phase difference takes always discrete values. The number of allowed values and the spacing depends on the total number of photons n on the form
 n , r
 n
2


1 r r

0 , 1 ,  , n sen

 n

2 n cos


This discreteness of the phase difference explains very easily the existence of an ultimate quantum limit to the precision of any phase measurement (Heisenberg limit) which scales as the inverse of the total number of photons
 n

1 / n .
Phase difference operator
A.
Luis and L. L. Sánchez-Soto, Phys. Rev. A 48 , 4702 (1993)
We have thoroughly examined this operator developing it and applying it to the very diverse problems in different contexts. For example invoking very general arguments we have demonstrated that the measurement of this observable is the optimum strategy for the detection of phase shifts.
Optimum phase-shift estimation and the quantum description of the phase difference
A.
Luis and J. Peřina, Phys. Rev. A 54, 4564 (1996)
We have demonstrated that this operator can be embodied in a phase-space formulation of the quantum theory in terms of Wigner functions.
Discrete Wigner function for finite-dimensional systems
A.
Luis and J. Peřina, J. Phys. A 31, 1423 (1998)
We have applied this operator to the study of the propagation of light in nonlinear media, where phase relations between modes play a relevant role.
Phase properties of light propagating in a Kerr medium: Stokes parameters versus
Pegg-Barnett predictions
A. Luis, L. L. Sánchez-Soto and R. Tanaś, Phys. Rev. A
51, 1634 (1995)
Quantum dynamics of the relative phase in second harmonic generation
J. Delgado, A. Luis, L. L. Sánchez-Soto and A. B. Klimov, J. Opt. B: Quantum
Semiclass. Opt. 2 , 33 (2000)
Concerning the practical measurement we have found that it is possible to measure this operator in an interferometric arrangement (eight-port homodyne detector) made of four beam splitters and a quarter wave plate as illustrated in the figure

a
1 a
2
The modes whose phase difference is to be measured are the modes a
1, a
2 while the other input ports are in the vacuum state. On the one hand, we have demonstrated that the measurement of the number of photons at the four outputs of the interferometer can be interpreted as a noisy simultaneous measurement of the Stokes parameters. We have analyzed all the noise characteristics of the measurement. On the other hand, we have demonstrated that it is also possible to obtain exactly the probability distribution of the phase different operator (and also of many other observables defined as functions of the
Stokes operators).
Generalized measurements in eight-port homodyne detection
A. Luis and J. Peřina, Quantum Semiclass. Opt. 8 , 873 (1996)
Noisy simultaneous measurement of noncommuting observables in eight- and twelveport homodyne detection
A.
Luis and J. Peřina, Quantum Semiclass. Opt.
8 , 887 (1996)
A continuous objective of our work has been to examine whether the quantum phase difference can inherit the properties of its classical counterpart. In a recent work we have applied this phase difference to the study of the origin of complementarity in double beam interferometers.
It is known that quantum systems have mutually excluding properties: the precise knowledge of one of them precludes the precise knowledge of the other. A classic example is the wave-particle duality: the knowledge of the trajectory followed by a particle within an interferometer is incompatible with the existence of interference. doble rendija pantalla
D2
D1
The classic examples of complementarity were explained as consequences of positionmomentum uncertainty relations associated to the measuring apparatus: the destruction of the interference would be caused by the random perturbation of the trajectory caused by the detection scheme. However, some subtle examples of complementarity have been proposed and carried our experimentally recently where the detection scheme does not modify at all the trajectories within the interferometer. These experiences have lead

to many authors to suggest the idea that complementarity is beyond uncertainty relations in the sense that it would not be possible to explain its origin in terms of the alteration of the observed system caused by the observing mechanism. The complementarity would be a consequence of quantum correlations (entanglement) between the system and the apparatus, without involving concepts such as fluctuations and uncertainty relations.
Nevertheless, we think it is still clear that quantum observation disturbs the observed system, and the system-apparatus correlations is precisely the effective mechanism leading to such disturbance. According to this reasoning we have demonstrated that also in those subtle examples of complementarity there is a variable which is clearly disturbed by the measurement: this is the phase difference. As a matter of fact, the alteration of the probability distribution of phase difference P (

) can be easily expressed as where
P ob
(

)
  d

P (
  
)

(

) ,

(

) is a probability distribution of random phase changes. This demonstrates that the observation of the trajectory increases the phase fluctuations, which in turn wipe out the interference.
Complementarity enforced by random classical phase kicks
A. Luis and L. L. Sánchez-Soto, Phys. Rev. Lett.
81, 4031 (1998)
Randomization of quantum relative phase in welcher Weg measurements
A.
Luis and L. L. Sánchez-Soto, J. Opt. B: Quantum Semiclass. Opt. 1 , 668 (1999)
In the above works we have proven the idea that the phase difference is a variable more meaningful and better behaved than absolute phase. We have translated this idea to the framework of the interaction between radiation and matter. We have discovered an operator representing the relative phase between an electromagnetic field mode and the atomic dipole associated to a two-level atom. This subject is interesting given the importance of field-matter phase relations in this context. Also in this case we have demonstrated the good properties of this operator. We have also found some simple and feasible methods to directly measure experimentally this operator.
Quantum atom-field relative phase in the Jaynes-Cummings model
A. Luis and L. L. Sánchez-Soto, Opt. Commun.
133, 159 (1997)
Relative phase for a quantum field interacting with a two-level system
A. Luis and L. L. Sánchez-Soto, Phys. Rev. A
56, 994 (1997)
Determination of atom-field observables via resonant interaction
A.
Luis and L. L. Sánchez-Soto, Phys. Rev. A
57, 3105 (1998)

Recently we have carried out a thoroughly examination of the subject of the quantum phase difference in a context as wide as possible. The main result of this work is to ascertain that most approaches to the problem (theoretical as well as experimental) shear a common structure and arrive at similar conclusions. Most of them are based (directly or indirectly) on the Stokes operators. All of them (explicitly or implicitly) conclude that the quantum phase difference cannot be expressed as difference of individual phases. All of them coincide in the discrete character for this variable. These conclusions confirm the good properties of the phase difference operators we have introduced and developed.
Quantum phase difference, phase measurements and Stokes operators
A.
Luis and L. L. Sánchez-Soto, Progress in Optics,
41 , 421 (2000)
A relevant difficulty of quantum phase is that phase sates are extremely sophisticated from a practical perspective and of very difficult experimental generation. Because of this we have studied their approximation by quadrature coherent squeezed states that may have good phase properties and can be generated experimentally. To this end we have analyzed different approximation criteria such as minimum phase fluctuations, maximum overlap with phase states or maximum variation resolution in phase-shift detection.
Squeezed coherent states as feasible approximations to phase-optimized states
A. Luis, Phys. Lett. A 354, 71 (2006)
In the analysis of coherence and visibility in the interference of an arbitrary number of waves we have found that the quantum phase variable in a finite-dimensional Hilbert space takes part in interesting relations with coherence and interferometric visibility as shown in detail in another notes in this same web page.
Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems
A. Luis, Phys. Rev. A 78 , 025802 (2008)
When studying coherence problems in the classical domain we have found that the statistics of the phase difference provides a simple and useful tool to analyze coherence problems in the quantum and classical domains, as shown in detail in other notes in this same web page. In particular, the phase difference provides a suitable estimator of the interferometric usefulness of quantum light states, including those with vanishing second-order degree of coherence.
Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference
A. Luis, Phys. Rev. A 79 , 053855 (2009)