memo
A short memo for quantum optics
Table of content
Basics
Continuous variable system. A continuous variable (CV) system is a physical system defined with respect to a continuous degree of freedom, taking values ranging over $\mathbb{R}$.
Position. The position basis is the natural basis of a quantum CV system. With each value $x\in\mathbb{R}$ is associated a position eigenstate $\vert x\rangle$. The position operator is obtained through its spectral decomposition as $\hat{x}=\int x\vert x\rangle\langle x\vert\mathrm{d}x$.
Momentum. The momentum basis is the Fourier dual of the position basis. Momentum eigenstates are defined from the inner product $\langle x\vert p\rangle=\exp(ixp)/\sqrt{2\pi}$. The position operator is then defined as $\hat{p}=\int p\vert p\rangle\langle p\vert\mathrm{d}p$.
Mode operators. The mode operators both encode position and momentum in a single (non-Hermitian) operator. Define the annihilation operator $\hat{a}=(\hat{x}+i\hat{p})/\sqrt{2}$ and the creation operator $\hat{a}^{\dagger}=(\hat{x}-i\hat{p})/\sqrt{2}$.
Vector notation. Define the mode-vector $\hat{\boldsymbol{b}}=(\begin{smallmatrix} \hat{a} \ \hat{a}^{\dagger} \end{smallmatrix})$ and the quadrature-vector $\hat{\boldsymbol{q}}=(\begin{smallmatrix} \hat{x} \ \hat{p}\end{smallmatrix})$. And the vectorial complex number $\boldsymbol{\alpha}=(\begin{smallmatrix} \alpha\ \alpha^{\ast} \end{smallmatrix})$. Define the vectorial commutator $[\boldsymbol{x},\boldsymbol{y}]=\boldsymbol{x}\boldsymbol{y}^{\intercal}-\boldsymbol{y}\boldsymbol{x}^{\intercal}$.
Commutation relations. Position and momentum obey the canonical commutation relation $[\hat{x},\hat{p}]=i$. The mode operators obey the bosonic commutation relation $[\hat{a},\hat{a}^{\dagger}]=1$. Introducing the symplectic form as the matrix $\boldsymbol{\omega}=(\begin{smallmatrix} 0 & 1 \ -1 & 0 \end{smallmatrix})$, we write compactly $[\hat{\boldsymbol{q}},\hat{\boldsymbol{q}}]=i\boldsymbol{\omega}$ and $[\hat{\boldsymbol{b}},\hat{\boldsymbol{b}}]=\boldsymbol{\omega}$.
Photon-number basis. The photon-number operator $\hat{n}=\hat{a}^{\dagger}\hat{a}$ has a discrete spectrum. Its eigenstates form the Fock basis $\ket{n}$ with $n\in\mathbb{N}$, related to the position basis as $\langle x\vert n\rangle=(\sqrt{\pi 2^n n!})^{-\tfrac12}H_n(x)\exp(-x^2/2)$.
Displacement. The displacement operator is $\hat{D}(\alpha)=\exp(\boldsymbol{\alpha}^{\intercal}\boldsymbol{\omega}\boldsymbol{b})$. Equivalently, $\hat{D}(\alpha)=\exp(\alpha\hat{a}^{\dagger}-\alpha^{\ast}\hat{a})=\exp(i(p\hat{x}-x\hat{p}))$ with $\alpha=(x+ip)/\sqrt{2}$, $\hat{D}(\alpha)=\exp(i\boldsymbol{q}^{\intercal}\boldsymbol{\omega}\hat{\boldsymbol{q}})$.
Parity. The parity operator is defined as $(-1)^{\hat{n}}$. Or $\int\vert x\rangle\langle -x\vert\mathrm{d}x$. We have the relation $\hat{\Pi}=\int\hat{D}(\alpha)\tfrac{\mathrm{d}^2\alpha}{\pi}$. Parity is not a trace-class operator, i, some bases its trace converges and gives $\mathrm{Tr}[\hat{\Pi}]=1/2$.
Convolutions
Phase-space convolution. $(f\ast g)(\alpha):=\int f(\beta)g(\alpha-\beta)\tfrac{\mathrm{d}^2\beta}{\pi}$
Quantum convolution. $(\hat{\rho}\ast\hat{\sigma})(\alpha):=\mathrm{Tr}\big[\hat{\rho}\hat{D}(\alpha)\hat{\Pi}\hat{\sigma}\hat{\Pi}\hat{D}^{\dagger}(\alpha)\big]$
Hybrid convolution. $f\ast\hat{\rho}=\hat{\rho}\ast f:=\int f(\alpha)\hat{D}(\alpha)\hat{\rho}\hat{D}^{\dagger}(\alpha)\mathrm{d}^2\alpha$
Kernels
The kernel is the tip of the pencil drawing the phase-space distribution.
Phase-point operator. The phase-point operator is $\hat{\Delta}(\alpha)=2\hat{D}(\alpha)\hat{\Pi}\hat{D}^{\dagger}(\alpha)$.
Thermal state. The thermal state with mean photon-number $\bar{n}$ is $\hat{\tau}(\bar{n})=\frac{1}{\bar{n}+1}\sum_{n=0}^{\infty}\left(\frac{\bar{n}}{\bar{n}+1}\right)^n\ket{n}\bra{n}$.
| Name | Notation | Definition |
|---|---|---|
| Phase-point | $\hat{\Delta}$ | $2\hat{\Pi}$ |
| Thermal | $\hat{\tau}(\bar{n})$ | $\tfrac{1}{\bar{n}+1}\left(\tfrac{\bar{n}}{\bar{n}+1}\right)^{\hat{n}}$ |
Phase space
Wigner transform. The Wigner transform of a quantum operator $\hat{\rho}$ is defined as $W_{\hat{\rho}}=\hat{\rho}\ast\hat{\Delta}$.
Inverse Wigner transform. A quantum operator $\hat{\rho}$ can be reconstructed from its Wigner function via $\hat{\rho}=W_{\hat{\rho}}\ast\hat{\Delta}$.
Define the Hilbert-Schimdt (HS) inner product $\langle\hat{\rho},\hat{\sigma}\rangle=\mathrm{Tr}\big[\hat{\rho}^{\dagger}\hat{\sigma}\big]$, and the $L^{2}$ inner product $\langle f,g\rangle=\int f^{\ast}(\alpha)g(\alpha)\tfrac{\mathrm{d}^2\alpha}{\pi}$.
Traciality (Overlap formula). The Hilbert–Schmidt inner product between two operators equals the $L^2$ inner product of their Wigner functions: $\langle\hat{\rho},\hat{\sigma}\rangle=\langle W_{\hat{\rho}},W_{\hat{\sigma}}\rangle$.
Table of Wigner functions
| Operator $\hat{\rho}$ | Wigner function $W_{\hat{\rho}}(\alpha)$ |
|---|---|
| $\hat{1}$ | 1 |
| $\hat{\Delta}(\beta)$ | $\pi\delta(\alpha-\beta)$ |
| $\hat{D}(\beta)$ | $\exp(\beta\alpha^{\ast}-\beta^{\ast}\alpha)$ |
| $\hat{a}$ | $\alpha$ |
| $\hat{x}$ | $\sqrt{2}\mathrm{Re}[\alpha]$ |
| $\hat{p}$ | $\sqrt{2}\mathrm{Im}[\alpha]$ |
| $\hat{H}$ | $\vert\alpha\vert^2$ |
Inner products
Position, momentum, photon-numer, coherent
| $\vert x\rangle$ | $\vert p\rangle$ | $\vert n\rangle$ | $\vert \alpha \rangle$ | |
|---|---|---|---|---|
| $\langle y\vert$ | $\delta(x-y)$ | $\frac{e^{iyp}}{\sqrt{2\pi}}$ | $\frac{e^{-\tfrac{y^2}{2}}H_n(y)}{\sqrt{\sqrt{\pi}2^n n!}}$ | $\frac{e^{\tfrac{\alpha^2-\vert\alpha\vert^2}{2}-\tfrac{(y-\sqrt{2}\alpha)^2}{2}}}{\sqrt[4]{\pi}}$ |
| $\langle q\vert$ | $\frac{e^{-ixq}}{\sqrt{2\pi}}$ | $\delta(p-q)$ | $\frac{e^{-\tfrac{q^2}{2}}(-1)^n H_n(q)}{\sqrt{\sqrt{\pi}2^n n!}}$ | $\frac{e^{\tfrac{\alpha^2-\vert\alpha\vert^2}{2}-\tfrac{(q+\sqrt{2}i\alpha)^2}{2}}}{\sqrt[4]{\pi}}$ |
| $\langle m\vert$ | $\frac{e^{-\tfrac{x^2}{2}}H_m(x)}{\sqrt{\sqrt{\pi}2^m m!}}$ | $\frac{e^{-\tfrac{p^2}{2}}(-1)^m H_m(p)}{\sqrt{\sqrt{\pi}2^m m!}}$ | $[n=m]$ | $\frac{e^{-\tfrac{\vert\alpha\vert^2}{2}}\alpha^m}{\sqrt{m!}}$ |
| $\langle \beta\vert$ | $\frac{e^{\tfrac{\beta^{\ast 2}-\vert\beta\vert^2}{2}-\tfrac{(x-\sqrt{2}\beta^{\ast})^2}{2}}}{\sqrt[4]{\pi}}$ | $\frac{e^{\tfrac{\beta^{\ast 2}-\vert\beta\vert^2}{2}-\tfrac{(p-\sqrt{2}i\beta^{\ast})^2}{2}}}{\sqrt[4]{\pi}}$ | $\frac{e^{-\tfrac{\vert\beta\vert^2}{2}}\beta^{\ast n}}{\sqrt{n!}}$ | $e^{i\mathrm{Im}[\beta^{\ast}\alpha]}e^{-\tfrac{\vert\alpha-\beta\vert^2}{2}}$ |
Rotated position states
Denote the rotated position state as $\vert x,\theta\rangle\equiv\hat{R}(\theta)\vert x\rangle$.
\[\begin{align*} \langle y,\varphi \vert x, \theta \rangle = \sqrt{\tfrac{1-i\cot(\theta-\varphi)}{2\pi}} \exp\left( i\cot(\theta-\varphi)y^2 -2\pi i\left(\csc(\theta-\varphi)\right)yx-\tfrac{\cot(\theta-\varphi)}{2}x^2 \right) \end{align*}\]See fractional Fourier transform.
Displaced Fock states
Denote the dispalced Fock states as $\vert\alpha,n\rangle\equiv\hat{D}(\alpha)\ket{n}$.
\[\begin{align*} \langle \beta \vert \alpha \rangle &= \exp\left(\alpha\beta^{\ast}-\tfrac12(\alpha\alpha^{\ast}+\beta\beta^{\ast})\right) \\&= \exp\left(-\tfrac12 \vert\alpha-\beta\vert^2+i\mathrm{Im}[\beta^{\ast}\alpha]\right) \end{align*}\] \[\begin{align*} \langle \beta, m \vert \alpha, n \rangle &= \langle \beta \vert \alpha \rangle \sqrt{m!n!} \sum\limits_{j=0}^{\min(m,n)}\frac{(\alpha-\beta)^{m-j}(\beta^{\ast}-\alpha^{\ast})^{n-j}}{j!(m-j)!(n-j)!} \\&= \langle \beta \vert \alpha \rangle (-1)^{\left[m\lt n\right]}\sqrt{\tfrac{\min(m,n)!}{\max(m,n)!}}(\alpha-\beta)^{[m-n]}L^{(\vert m-n\vert)}_{\min(m,n)}\left(\vert\alpha-\beta\vert^2\right) \end{align*}\]In the latter equation we have introducd the notation $\alpha^{[p]}=\alpha^{p}$ if $p\geq 0$ and $\alpha^{[p]}=(\alpha^{\ast})^{-p}$ if $p<0$.