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

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

and its adjoint

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

Their commutation relation is

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

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

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

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

where ${\displaystyle a^{\dagger }a}$ is the number operator. So the exponential of the phase operator displaces the number operator in the same fashion as the momentum operator acts as the generator of translations in quantum mechanics: ${\displaystyle \exp \left(i{\frac {{\hat {p}}x_{o}}{\hbar }}\right){\hat {x}}\exp \left(-i{\frac {{\hat {p}}x_{o}}{\hbar }}\right)={\hat {x}}+x_{0}}$.

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 (9): 095301. arXiv:1207.6551. Bibcode:2013JPhA...46i5301R. doi:10.1088/1751-8113/46/9/095301. S2CID 118671292.
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 (43): 3770–3774. arXiv:1105.4013. Bibcode:2011PhLA..375.3770R. doi:10.1016/j.physleta.2011.08.051. S2CID 118486579.
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. arXiv:1303.2516.