The Glauber–Sudarshan P representation is a suggested way of writing down the
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
distribution of a quantum system in the
phase space formulation
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
*Phase space, a mathematica ...
of quantum mechanics. The P representation is the
quasiprobability distribution
A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobability distributions arise naturally in the study of quantum mechanics ...
in which
observable
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
s are expressed in
normal order
Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
. In
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction ...
, this representation, formally equivalent to several other representations, is sometimes preferred over such alternative representations to describe
light
Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...
in
optical phase space
In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an ''optical system''. For any such system, a plot of ...
, because typical optical observables, such as the
particle number operator
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.
The following is in bra–ket notation: The number operator acts on Fock space ...
, are naturally expressed in normal order. It is named after
George Sudarshan
Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian American theoretical physicist and a professor at the University of Texas. Prof.Sudarshan has been credited with numerous co ...
and
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 in N ...
, who worked on the topic in 1963.
Despite many useful applications in laser theory and coherence theory, the Sudarshan–Glauber 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 used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...
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 Greek � ...
in the basis of
coherent states
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
, i.e.,
:
We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the
optical equivalence theorem The optical equivalence theorem in quantum optics asserts an equivalence between the expectation value of an operator in Hilbert space and the expectation value of its associated function in the phase space formulation with respect to a quasiproba ...
. 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 resolution of the identity
:
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 points ...
:
and then take the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
:
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 mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
If the quantum system has a classical analog, e.g. a coherent state or
thermal radiation
Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electro ...
, 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 im ...
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 oth ...
it would not describe probabilities of mutually exclusive states."
Examples
Fock states
Fock states, for integer , correspond to a highly singular ''P'' distribution, which can be written asWhile this is not a function, this expression corresponds to a
tempered distribution
Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, an ...
. In particular for the vacuum state the ''P'' distribution is a
Dirac delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
at the origin, as . Similarly, the Fock state givesWe can also easily verify that the above expression for works more generally observing thattogether with the identityThe same reasoning can be used to show more generally that the ''P'' function of the operators is given by
Another concise formal expression for the ''P'' function of Fock states using the
Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation:
xy'' + (1 - x)y' + ny = 0,\
y = y(x)
which is a second-order linear differential equation. Thi ...
is
Thermal radiation
From
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
arguments in the Fock basis, the mean photon number of a mode with
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
and polarization state for a
black body
A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The radiation emitted by a black body in thermal equilibrium with its environment is ...
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 probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
f(x ...
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 syste ...
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:
:
See also
*
*
Nonclassical light
In optics, nonclassical light is light
Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as h ...