User:EverettYou/Second Quantization

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Base on the Lecture note.^[1]

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

.

Reference

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