The Sudarshan-Glauber P representation is a suggested way of writing down the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
, this representation, formally equivalent to several other representations, is sometimes preferred over such alternative representations to describe
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
Roy J. Glauber
Roy Jay Glauber (September 1, 1925 – December 26, 2018) was an American theoretical physicist. He was the Mallinckrodt Professor of Physics at Harvard University and Adjunct Professor of Optical Sciences at the University of Arizona. Born ...
, who worked on the topic in 1963.
Despite many useful applications in laser theory and coherence theory, the Glauber–Sudarshan P representation has the peculiarity that it is not always positive, and is not a bona-fide probability function.
Definition
We wish to construct a function with the property that the
density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, usin ...
is
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Gree ...
in the basis of coherent states , i.e.,
:
We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the optical equivalence theorem. This implies that the density matrix should be in ''anti''-normal order so that we can express the density matrix as a power series
:
Inserting the identity operator
:
we see that
:
and thus we formally assign
:
More useful integral formulas for are necessary for any practical calculation. One method is to define the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
:
and then take the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
:
Another useful integral formula for is
:
Note that both of these integral formulas do ''not'' converge in any usual sense for "typical" systems . We may also use the matrix elements of in the Fock basis . The following formula shows that it is ''always'' possible to write the density matrix in this diagonal form without appealing to operator orderings using the inversion (given here for a single mode),
:
where and are the amplitude and phase of . Though this is a full formal solution of this possibility, it requires infinitely many derivatives of
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
If the quantum system has a classical analog, e.g. a coherent state or
thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) is ...
, then is non-negative everywhere like an ordinary probability distribution. If, however, the quantum system has no classical analog, e.g. an incoherent
Fock state
In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an imp ...
or entangled system, then is negative somewhere or more singular than a Dirac delta function. (By a theorem of Schwartz, distributions that are more singular than the Dirac delta function are always negative somewhere.) Such " negative probability" or high degree of singularity is a feature inherent to the representation and does not diminish the meaningfulness of expectation values taken with respect to . Even if does behave like an ordinary probability distribution, however, the matter is not quite so simple. According to Mandel and Wolf: "The different coherent states are not utuallyorthogonal, so that even if behaved like a true probability density
unction
Anointing is the ritual act of pouring aromatic oil over a person's head or entire body.
By extension, the term is also applied to related acts of sprinkling, dousing, or smearing a person or object with any perfumed oil, milk, butter, or ot ...
it would not describe probabilities of mutually exclusive states."
Examples
Thermal radiation
From statistical mechanics arguments in the Fock basis, the mean photon number of a mode with wavevector and polarization state for a
black body
A black body or blackbody is an idealized physical object, physical body that absorption (electromagnetic radiation), absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence (optics), angle of incidence. T ...
at temperature is known to be
:
The representation of the black body is
:
In other words, every mode of the black body is
normally distributed
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu is ...
in the basis of coherent states. Since is positive and bounded, this system is essentially classical. This is actually quite a remarkable result because for thermal equilibrium the density matrix is also diagonal in the Fock basis, but Fock states are non-classical.
Highly singular example
Even very simple-looking states may exhibit highly non-classical behavior. Consider a superposition of two coherent states
:
where are constants subject to the normalizing constraint
:
Note that this is quite different from a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
because and are not orthogonal. As it is straightforward to calculate , we can use the Mehta formula above to compute ,
:
Despite having infinitely many derivatives of delta functions, still obeys the optical equivalence theorem. If the expectation value of the number operator, for example, is taken with respect to the state vector or as a phase space average with respect to , the two expectation values match:
: