Susskind–Glogower operator

The Susskind–Glogower operator, first proposed by Leonard Susskind and J. Glogower,^[1] refers to the operator where the phase is introduced as an approximate polar decomposition of the creation and annihilation operators.

It is defined as

$V={\frac {1}{{\sqrt {aa^{{\dagger }}}}}}a$,

and its adjoint

$V^{{\dagger }}=a^{{\dagger }}{\frac {1}{{\sqrt {aa^{{\dagger }}}}}}$.

Their commutation relation is

$[V,V^{{\dagger }}]=|0\rangle \langle 0|$,

where $|0\rangle$ is the vacuum state of the harmonic oscillator.

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

$Va^{{\dagger }}aV^{{\dagger }}=a^{{\dagger }}a+1$,

where $a^{\dagger}a$ is the number operator. So the exponential of the phase operator displaces the number operator in the same fashion as $\exp \left(i{\frac {px_{o}}{\hbar }}\right)x\exp \left(-i{\frac {px_{o}}{\hbar }}\right)=x+x_{0}$.

They may be used to solve problems such as atom-field interactions,^[2] level-crossings ^[3] or to define some class of non-linear coherent states,^[4] among others.

References

1. Susskind, L.; Glogower, J. (1964). "Quantum mechanical phase and time operator". Physica. 1: 49.
2. Rodríguez-Lara, B. M.; Moya-Cessa, H.M. (2013). "Exact solution of generalized Dicke models via Susskind-Glogower operators". Journal of Physics A. 46: 095301. doi:10.1088/1751-8113/46/9/095301.
3. Rodríguez-Lara, B.M.; Rodríguez-Méndez, D.; Moya-Cessa, H. (2011). "Solution to the Landau-Zener problem via Susskind-Glogower operators". Physics Letters A. 375: 3770–3774. doi:10.1016/j.physleta.2011.08.051.
4. León-Montiel, J.; Moya-Cessa, H.; Soto-Eguibar, F. (2011). "Nonlinear coherent states for the Susskind-Glogower operators" (PDF). Revista Mexicana de Física. 57: 133.