Definition, properties and applications of the quantum phase
difference
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( a1a2 )  arg( S x  i S y )
where the second equality is expressed in terms of the Stokes operators
S x  a1a2  a2a1
S y  i(a1a2  a2 a1 )
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 
2
r
n 1
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
a1
a2
The modes whose phase difference is to be measured are the modes a1, a2 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
Pob ( )   d P(   )( ) ,
where ( ) 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)