qit.ho.displace

qit.ho.displace(alpha, n=30)

Bosonic displacement operator.

Parameters

• alpha (complex) – displacement parameter

• n (int) – truncation dimension

Returns

displacement operator

Return type

array[complex]

Returns the n-dimensional approximation for the bosonic phase space displacement operator

$$D(\alpha ):=\exp (\alpha a^{†}-{\alpha }^{\ast }a)=\exp \left(i\sqrt{2}(Q\mathrm{Im}\alpha -P\mathrm{Re}\alpha )\right)$$

in the number basis. This yields

$$\begin{array}{rl}{D}^{†}(\alpha )QD(\alpha )& =Q+\sqrt{2}\mathrm{Re}\alpha \mathbb{I},\\ D^{†}(\alpha )PD(\alpha )& =P+\sqrt{2}\mathrm{Im}\alpha \mathbb{I},\end{array}$$

and thus the displacement operator displaces the state of a harmonic oscillator in phase space.