The Susskind–Glogower operator, first proposed by

Leonard Susskind Leonard Susskind (; born June 16, 1940)his 60th birth anniversary was celebrated with a special symposium at Stanford University.in Geoffrey West's introduction, he gives Suskind's current age as 74 and says his birthday was recent. is an Americ ...

and Jonathan Glogower, refers to the operator where the phase is introduced as an approximate polar decomposition of the

creation and annihilation operators Creation operators and annihilation operators are Operator (mathematics), mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilatio ...

. It is defined as :

V=\fraca

, and its adjoint :

V^=a^\frac

. Their

commutation relation In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...

is :

, 0\rangle\langle 0,

, where

, 0\rangle

is the vacuum state of the

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive const ...

. They may be regarded as a (exponential of) phase operator because :

Va^a V^=a^a+1

, where

a^a

is the number operator. So the exponential of the phase operator displaces the number operator in the same fashion as the

momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...

acts as the generator of translations in quantum mechanics:

\exp\left(i\frac\right)\hat\exp\left(-i\frac\right)=\hat+x_0

. They may be used to solve problems such as atom-field interactions, level-crossings or to define some class of non-linear coherent states, among others.

References

Quantum optics Leonard Susskind {{quantum-stub