update blog

_members/feiyang-wang.md

@@ -3,6 +3,9 @@ name: Feiyang Wang (王飞洋)
 image: images/yao-lu.jpg
 role: master
 alumni: false
+aliases:
+  - Feiyang Wang
+  - F. Wang
 ---
 
 深圳国际量子研究院硕士一年级

_members/liyun-zhang.md

@@ -2,6 +2,9 @@
 name: Liyun Zhang (张丽昀)
 image: images/yao-lu.jpg
 role: engineer
+aliases:
+  - Liyun Zhang
+  - L. Zhang
 ---
 
 深圳国际量子研究院研究员

_members/qifan-wu.md

@@ -3,6 +3,9 @@ name: Qifan Wu (吴启凡)
 image: images/yao-lu.jpg
 role: phd
 alumni: false
+aliases:
+  - Qifan Wu
+  - Q. Wu
 links:
   home-page: https://www.57f.org/
 ---

_members/qinyang-yu.md

@@ -3,6 +3,9 @@ name: Qinyang Yu (余沁阳)
 image: images/yao-lu.jpg
 role: master
 alumni: false
+aliases:
+  - Qinyang Yu
+  - Q. Yu
 ---
 
 深圳国际量子研究院硕士一年级

_members/xiaotian-feng.md

@@ -3,6 +3,9 @@ name: Xiaotian Feng (丰啸天)
 image: images/yao-lu.jpg
 role: undergrad
 alumni: false
+aliases:
+  - Xiaotian Feng
+  - X. Feng
 links:
   email: [email protected]
   home-page: https://blog.windsky.tech

_members/xueying-mai.md

@@ -3,6 +3,9 @@ name: Xueying Mai (麦雪颖)
 image: images/yao-lu.jpg
 role: phd
 alumni: false
+aliases:
+  - Xueying Mai
+  - X. Mai
 ---
 
 深圳国际量子研究院博士二年级

_members/zhao-wang.md

@@ -2,6 +2,9 @@
 name: Zhao Wang (王钊)
 image: images/yao-lu.jpg
 role: engineer
+aliases:
+  - Zhao Wang
+  - Z. Wang
 ---
 
 深圳国际量子研究院研究员

_posts/2024-04-01-how-to-write-a-post.md

@@ -0,0 +1,77 @@
+---
+title: 博客写作说明
+author: xiaotian-feng
+tags: example
+---
+
+<!-- excerpt start -->
+本篇文章介绍如何写作文章并发布在实验室网站上。
+<!-- excerpt end -->
+
+## 创建文章
+
+{% capture hint %}
+本篇文章的所有内容，以及更详细、更高级的用法都可以在[官方文档](https://greene-lab.gitbook.io/lab-website-template-docs)上找到，如果需要可以仔细阅读。
+{% endcapture %}
+
+{%
+  include alert.html
+  type="tip"
+  content=hint
+%}
+
+文章遵循Markdown语法，可以用Typora, Marktext甚至VS Code等编辑器书写。
+
+要创建文章，直接在`_posts`目录下新建一个`.md`文件，命名格式遵循
+
+```
+YEAR-MONTH-DAY-title.md
+```
+
+<br/>
+
+## 书写文章
+
+### 基本内容
+
+参见[官方说明文档](https://greene-lab.gitbook.io/lab-website-template-docs/basics/write-basic-content)。
+
+插图请放在`asset/img`文件夹下，即
+
+```
+asset/img/yyyy-mm-dd-picname.png
+```
+
+### 数学
+
+数学公式的显示由MathJax引擎驱动，遵循 $\LaTeX$ 语法。书写行内公式可以使用`$...$`或`\(...\)`，例如${}^{171}\mathrm{Yb}^+$与`${}^{171}\mathrm{Yb}^+$`。
+
+书写行间公式可以使用`$$...$$`，例如
+
+```latex
+$$
+E = mc^2
+$$
+```
+
+给出
+$$
+E=mc^2
+$$
+
+### Capture
+
+Capture相当于创建一个变量，其它页面元素的`content`的属性可以调用这个变量。例如本文开头的提示框
+
+```
+{% capture hint %}
+本篇文章的所有内容，以及更详细、更高级的用法都可以在[官方文档](https://greene-lab.gitbook.io/lab-website-template-docs)上找到，如果需要可以仔细阅读。
+{% endcapture %}
+
+{%
+  include alert.html
+  type="tip"
+  content=hint
+%}
+```
+

_posts/2024-04-02-beam-splitter.md

@@ -0,0 +1,214 @@
+---
+title: 分束器的量子理论
+image: assets/img/2024-04-02-beam-splitter.jpg
+author: xiaotian-feng
+tags: quantum optics
+---
+
+<!-- excerpt start -->
+A beam splitter is an optical device that splits a beam of light into a transmitted and a reflected beam. It is a crucial part of many optical experimental and measurement systems, such as interferometers, also finding widespread application in fibre optic telecommunications. 
+<!-- excerpt end -->
+
+## Beam Splitter
+
+### Transfer matrix
+
+The recipe for quantization is that 'replace the classical amplitudes by annihilation operators'. Namely,
+$$
+\begin{bmatrix}
+b_1 \\
+b_2
+\end{bmatrix}
+=
+\begin{bmatrix}
+t & r^\prime\\
+r & t^\prime
+\end{bmatrix}
+\begin{bmatrix}
+a_1 \\
+a_2
+\end{bmatrix}
+$$
+which means
+$$
+\begin{gathered}
+b_1 = t a_1 + r^\prime a_2 \qquad b_2 = r a_1 + t^\prime a_2 \\
+b_1^\dagger =t^* a_1^\dagger+r^{\prime*} a_2^\dagger \qquad b_2^\dagger = r^* a_1^\dagger + t^{\prime *} a_2^\dagger
+\end{gathered}
+$$
+If the outgoing modes are still useful for the quantum theory, they have to satisfy the commutation relations
+$$
+[b_i,b_j^\dagger] = \delta_{ij}
+$$
+which is identical to
+$$
+\begin{cases}
+t r^* + r^\prime t^{\prime*} = 0 \\
+|t|^2+|r^\prime|^2 =1 \\
+|r|^2+|t^\prime|^2=1
+\end{cases}
+$$
+which is also identical to
+$$
+U = \begin{bmatrix}
+t & r^\prime\\
+r & t^\prime
+\end{bmatrix} \text{ is unitary.}
+$$
+Since $U\in\mathrm{SU}(2)$, the transfer matrix could be written as $U = \exp\left[-\frac{\mathrm{i}}{2} \theta\,(\boldsymbol{n} \cdot \boldsymbol{\sigma})\right]$. The most common form of $U$, however, is
+$$
+U= \begin{bmatrix}
+\cos\theta & \sin\theta \mathrm{e}^\mathrm{i \phi} \\
+-\sin\theta \mathrm{e}^\mathrm{-i \phi} & \cos\theta
+\end{bmatrix}
+$$
+Inversely, we have
+$$
+\begin{bmatrix}
+b_1 \\
+b_2
+\end{bmatrix}
+=
+U
+\begin{bmatrix}
+a_1 \\
+a_2
+\end{bmatrix}
+\qquad
+\begin{bmatrix}
+a_1 \\
+a_2
+\end{bmatrix}
+=
+U^\dagger
+\begin{bmatrix}
+b_1 \\
+b_2
+\end{bmatrix}
+$$
+and
+$$
+\begin{gathered}
+a_1^\dagger = b_1^\dagger \cos\theta -b_2^\dagger \sin\theta \mathrm{e}^{-\mathrm{i}\phi} \\
+a_2^\dagger = b_1^\dagger \sin\theta \mathrm{e}^{\mathrm{i}\phi} + b_2^\dagger \cos\theta
+\end{gathered}
+$$
+
+### Beam splitter operator
+
+#### State Space
+
+For fibre optics (i.e. waveguide beam splitter), we can pair two input ports with two output ports.[^1] This correspondence does not imply that a photon on an input port is identical to a photon on an output port (e.g. $\ket{10}_\text{in} \neq \ket{10}_\text{out}$). Rather, the photon number states in the upper and lower fibers, no matter they belong to input or output, can be considered to be in the same state space, and the beam splitter actually acts as a transformation in this state space. 
+
+For example, a 50:50 beam splitter acting on a single-photon incident state should be strictly written as
+$$
+BS(\pi/4) \,(\ket{10}_\text{in} \otimes \ket{00}_\text{out}) = \ket{00}_\text{in} \otimes\frac{1}{\sqrt{2}}(\ket{10}_\text{out}+\ket{01}_\text{out})
+$$
+However, if we focus only on the photon number, we can write
+$$
+BS(\pi/4) \ket{10}= \frac{1}{\sqrt{2}}(\ket{10}-\ket{01})
+$$
+where the first number in the ket represents the photon number in the upper fibre, and the second number represents the photon number in the lower fibre. And, we recognize by default, that the state before BS operators acting on them represents the photon distribution in the input fibers while the state transformed represents the photon distribution in the output fibers.
+
+This can also be glimpsed in the creation and annihilation operators. The linear relationship between $b,b^\dagger$ and $a,a^\dagger$ suggests that they can be considered to act on the same state space.
+
+> Example: Hong–Ou–Mandel interference
+>
+> Answer: Let us consider two single-photon states incident on the same beam splitter $\ket{11}_\text{in}$.
+> $$
+> \begin{aligned}
+> \ket{11}_\text{in} &= a_1^\dagger a_2^\dagger \ket{00}_\text{in}= a_1^\dagger a_2^\dagger \ket{00}_\text{out} = (b_1^\dagger\cos\theta - b_2^\dagger\sin\theta)(b_1^\dagger\sin\theta + b_2^\dagger\cos\theta) \ket{00}_\text{out} \\
+> &= \sqrt{2}\sin\theta\cos\theta(\ket{20}_\text{out}-\ket{02}_\text{out}) +\cos(2\theta)\ket{11}_\text{out}
+> \end{aligned}
+> $$
+> When the beam splitter is 50:50, the output is $\frac{1}{\sqrt{2}} (\ket{20}-\ket{02})$, which means the photons are either in the upper or in the lower fibre!
+
+Overall, beam splitter operator $BS$ has the following properties:
+
+1. Each $BS$ operator associates with a transfer matrix $U$
+2. $BS$ does not change the total photon number of incident beam
+3. $BS \in SU(2)$ 
+4. $BS \ket{n_1,n_2} = \sum_m\ket{lm}$, where $l=n_1+n_2$ and $m=|n_1-n_2|,\cdots,|n_1+n_2|$.
+
+#### Schwinger representation
+
+The angular momentum operator can be expressed by two independent harmonic oscillators.
+$$
+L_x =\frac{1}{2}(a_1^\dagger a_2+a_1 a_2^\dagger) \qquad L_y =\frac{1}{2\mathrm{i}}(a_1^\dagger a_2 - a_1 a_2^\dagger) \qquad L_z=\frac{1}{2}(a_1^\dagger a_1 -a_2^\dagger a_2)
+$$
+
+$$
+L_+ = L_x + \mathrm{i} L_y = a_1^\dagger a_2 \qquad L_-= L_x - \mathrm{i} L_y = a_1 a_2^\dagger
+$$
+
+$$
+L^2= \hat{l}(\hat{l}+1) \qquad \hat{l} =\frac{1}{2}(\hat{n}_1 + \hat{n}_2)=\frac{1}{2} (a_1^\dagger a_1 +a_2^\dagger a_2)
+$$
+
+It is easy to check such angular momentum operators satisfying the commutation rules $[L_i,L_j] = \mathrm{i}\varepsilon_{ijk} L_k$. Therefore, the most general form of $BS$ could be written as
+$$
+BS =\exp\left[\mathrm{i} (\alpha L_x + \beta L_y + \gamma L_z)\right]
+$$
+
+#### Relationship
+
+The relationship between the beam splitter operators $BS$ acting on the states and the transfer matrix $U$ acting on the creation and annihilation operators is[^2]
+$$
+U= \begin{bmatrix}
+\cos\theta & \sin\theta \mathrm{e}^\mathrm{i \phi} \\
+-\sin\theta \mathrm{e}^\mathrm{-i \phi} & \cos\theta
+\end{bmatrix}
+\iff BS= \mathrm{e}^{\xi L_+ -\xi^*L_-} = \exp\left[\theta(\mathrm{e}^\mathrm{i \phi} a_1^\dagger a_2-\mathrm{e}^\mathrm{-i \phi} a_1 a_2^\dagger)\right]
+$$
+
+with
+$$
+\begin{gathered}
+BS^\dagger a_1 BS= a_1 \cos\theta +a_2 \mathrm{e}^{\mathrm{i}\phi}\sin\theta =b_1\\
+BS^\dagger a_2 BS= -a_1 \mathrm{e}^{-\mathrm{i}\phi}\sin\theta +a_2 \cos\theta  =b_2
+\end{gathered}
+$$
+
+Here is a short Python code to check the relations
+
+```python
+import qutip as qt
+import numpy as np
+
+dim1 = 10
+a1 = qt.destroy(dim1)
+a1_dag = qt.create(dim1)
+I1 = qt.qeye(dim1)
+
+dim2 = 10
+a2 = qt.destroy(dim2)
+a2_dag = qt.create(dim2)
+I2 = qt.qeye(dim2)
+
+a1 = qt.tensor(a1,I2)
+a1_dag = qt.tensor(a1_dag,I2)
+a2 = qt.tensor(I1,a2)
+a2_dag = qt.tensor(I1,a2_dag)
+
+# Hong–Ou–Mandel interference
+n1 = 1
+n2 = 1
+theta = np.pi/4
+phi = np.pi/2
+
+input = qt.tensor(qt.basis(dim1,n1),qt.basis(dim2,n2))
+vac = qt.tensor(qt.basis(dim1,0),qt.basis(dim2,0))
+out1_dag = a1_dag*np.cos(theta) - a2_dag*np.sin(theta)*np.exp(-1j*phi)
+out2_dag = a1_dag*np.sin(theta)*np.exp(1j*phi) + a2_dag*np.cos(theta)
+bs = (theta*(a1_dag*a2*np.exp(1j*phi)-a1*a2_dag*np.exp(-1j*phi))).expm()
+output1 = bs*input
+output2 = 1/np.sqrt(np.math.factorial(n1)*np.math.factorial(n2))*out1_dag**n1*out2_dag**n2*vac
+
+print(qt.fidelity(output1,output2))
+```
+
+
+
+[^1]: Makarov D. Theory for the beam  splitter in quantum optics: Quantum entanglement of photons and their  statistics, HOM effect[J]. Mathematics, 2022, 10(24): 4794.
+[^2]: Campos R A, Saleh B E A, Teich M C.  Quantum-mechanical lossless beam splitter: SU (2) symmetry and photon  statistics[J]. Physical Review A, 1989, 40(3): 1371.
+