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Lie Algebras for quantum Optics

1930s前 lie algebra称为 infinitesimal group 无限小（变换）群；现在区分 李代数 & 李群

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A Lie algebra of $GROUP$ is a vector space $\mathfrak {g}$ over a field $F$ together with a binary operator $[\cdot,\cdot]:g\times g\to g$ (Lie bracket), satisfying the following axioms:

• bilinearity

  两个分量都满足线性性

  $$ [ax+by,z]=a[x,z]+b[y,z]\\ [z,ax+by]=a[z,x]+b[z,y] $$

• alternating

  交错性

  $$ [x,x]=0 $$

  with characteristic 2, alternating property ↔

  $$ ([x+y,x+y]=)\\ [x,y]+[y,x]=0 $$

• Jacobi identity

  $$ [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0 $$

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• 如何构造一个 Lie algebra呢

  e.g. 满足结合律的代数$A$ → with $[x,y]=xy-yx$ 自然形成李代数$A$

  • general linear Lie algebra

    space of n by n matrix over field $F$ with bracket $[X,Y]=XY-YX$

    $$ \mathfrak{gl}_n(F)\\ \mathfrak{gl}_n(\mathbb R) \sim GL(n,\mathbb R) $$

    general linear group 可逆方阵构成的群 group operator is 矩阵乘法

  homomorphism 同态 / representation 表示

  isomorphism 同构

$$ \begin{equation} \begin{aligned} \exp(\hat{A})=\sum_{m=0}^{\infty}\frac{1}{m!}\hat{A}^{m}. \end{aligned} \end{equation}

$$

$$ \begin{equation} \begin{aligned} \left(\frac{\mathrm{d}}{\mathrm{d}t}\hat{A}^n\right) & = \sum_{k = 0}^{n-1}\hat{A}^k\left(\frac{\mathrm{d}}{\mathrm{d}t}\hat{A}\right)\hat{A}^{n-k-1}. \end{aligned} \end{equation}

$$

Parametric Differentiation of the Exponential

beta function

Mathematical Methods of Quantum Optics

Sneddon’s formula: [p37]

$$ \begin{equation}\begin{aligned}\frac{\mathrm{d}}{\mathrm{d}t}\exp[\hat{A}(t)]=\int_0^1\exp[u\hat{A}]\frac{\mathrm{d}\hat{A}}{\mathrm{d}t}\exp[(1-u)\hat{A}]\mathrm{d}u.\end{aligned}\end{equation} $$

If $\hat A(t)=\hat A\ t$, as an example, then this yields the familiar c-number result

$$ \frac{\mathrm{d}}{\mathrm{d}t}\exp(\hat{A}t)=\hat{A}\exp(\hat{A}t)=\exp(\hat{A}t)\hat{A}. $$

its Reduction to a Polynomial when finite-dim

Consider a spin -1/2, any of its component $\hat S_a$ along a direction $\boldsymbol a$ is a two-dimensional operator which can assume the values $\pm 1/2:\hat S_a^2=\frac 1 4+0\cdot \hat S_a.$

$$ \prod_{i=1}^n\left(\hat{X}-\lambda_i\right)=0\to \hat{X}^N=\sum_{m=0}^{N-1}\alpha_m\hat{X}^m. $$

Note that$C_k(0)=\delta {k0}$, make $C{-1}(\theta)=0$

$$ \sum_{m=1}^NC_{m-1}(\theta)\hat{X}^m=\sum_{m=0}^{N-1}\dot{C}_m(\theta)\hat{X}^m. $$

$$ \dot{C}k(\theta)=\alpha_kC{N-1}(\theta)+C_{k-1}(\theta),\quad C_{-1}(\theta)=0, $$

The solution of spin is

$$ \exp(\theta\hat{S}_a)=\cosh\left(\frac{\theta}{2}\right)+2\sinh\left(\frac{\theta}{2}\right)\hat{S}_a. $$

spin-1:

$$ \exp(\theta\hat{J}_a)=1+\sinh(\theta)\hat{J}_a+(\cosh(\theta)-1)\hat{J}_a^2. $$