User:EverettYou/Second Quantization

Base on the Lecture note.

Second Quantized States

Minimal Uncertainty States

Heisenberg uncertainty principle: for any Hermitian operator ${\hat {A}}$ and ${\hat {B}}$ and any state $|\psi \rangle$, the following inequality holds

${\sqrt {\langle (\delta {\hat {A}})^{2}\rangle \langle (\delta {\hat {B}})^{2}\rangle }}\geq {\frac {1}{2}}|\langle [{\hat {A}},{\hat {B}}]\rangle |$,

where $\delta {\hat {A}}={\hat {A}}-\langle {\hat {A}}\rangle$, $\delta {\hat {B}}={\hat {B}}-\langle {\hat {B}}\rangle$, and $\langle \cdots \rangle \equiv \langle \psi |\cdots |\psi \rangle$.

The equality is achieved if and only if $|\psi \rangle$ is a solution of the minimal uncertainty equation

$(\cos \theta \;{\hat {A}}+i\sin \theta \;{\hat {B}})|\psi \rangle =0$,

for any $\theta \in [0,2\pi )$. There is an one-to-one correspondence between the angle θ and the state $|\psi \rangle$ that minimize the uncertainty between ${\hat {A}}$ and ${\hat {B}}$.

Coherent State

Displacement operator

Definition: for $\alpha \in \mathbb {C}$,

${\hat {D}}(\alpha )=e^{(\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}})}$.

Unitarity: ${\hat {D}}^{\dagger }(\alpha )={\hat {D}}^{-1}(\alpha )={\hat {D}}(-\alpha )$.

Action of displacement operator performs translation in the phase space

${\hat {D}}(\alpha ){\hat {a}}{\hat {D}}^{\dagger }(\alpha )={\hat {a}}-\alpha$,

${\hat {D}}(\alpha ){\hat {a}}^{\dagger }{\hat {D}}^{\dagger }(\alpha )={\hat {a}}^{\dagger }-\alpha ^{*}$.

Applying to the vacuum state leads to the coherent state $|\alpha \rangle ={\hat {D}}(\alpha )|0\rangle$, such that

${\hat {a}}|\alpha \rangle =\alpha |\alpha \rangle$.

Properties of Coherent State

Expansion in particle number representation

$|\alpha \rangle =e^{-|\alpha |^{2}/2}\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle$

Overlap:

$\langle \alpha |\alpha '\rangle =e^{-{\frac {1}{2}}|\alpha |^{2}-{\frac {1}{2}}|\alpha '|^{2}+\alpha ^{*}\alpha '}$.

Completeness:

$\int {\frac {\mathrm {d} [\alpha ]}{\pi }}|\alpha \rangle \langle \alpha |={\hat {1}}$.

Squeezed State

Squeezing operator

Definition: for $\xi \in \mathbb {C}$,

${\hat {U}}(\xi )=e^{\xi ({\hat {a}}^{\dagger }{\hat {a}}^{\dagger }-{\hat {a}}{\hat {a}})/2}$.

Unitarity: ${\hat {U}}^{\dagger }(\alpha )={\hat {U}}^{-1}(\alpha )={\hat {U}}(-\alpha )$.

Action of squeezing operator performs the Bogoliubov transform

${\hat {U}}(\xi ){\hat {a}}{\hat {U}}^{\dagger }(\xi )=\cosh \xi \;{\hat {a}}+\sinh \xi \;{\hat {a}}^{\dagger }$,

${\hat {U}}(\xi ){\hat {a}}^{\dagger }{\hat {U}}^{\dagger }(\xi )=\cosh \xi \;{\hat {a}}^{\dagger }+\sinh \xi \;{\hat {a}}$.

Applying to the vacuum state leads to the squeezed state $|\xi \rangle ={\hat {U}}(\xi )|0\rangle$, such that

$(\cosh \xi \;{\hat {a}}-\sinh \xi \;{\hat {a}}^{\dagger })|\xi \rangle =0$.