Quantum Optics on Haifei's Home

Why Does Phase Space Sometimes Look Like a Complex Plane?

hfwang132@gmail.com (hfwang132) — Mon, 01 Jul 2024 23:09:03 +0800

Students who have studied physics all know phase space, which consists of generalized coordinates $q$ and generalized momenta $p$.

Example 1: The angle of a simple pendulum forms a configuration space $S_1$. The angle and angular momentum form a phase space $S_1 \times \mathbb{R}$.

Example 2: Given the Hamiltonian and boundary conditions of a classical electromagnetic field, the electric field amplitude of a certain mode forms a configuration space, while the cosine and sine components of the electric field amplitude in that mode form a phase space.

Derivation of Blackbody Radiation (No-Nonsense Version)

hfwang132@gmail.com (hfwang132) — Sat, 22 Jun 2024 18:12:07 +0800

With nothing better to do, let’s review blackbody radiation~

Many articles on blackbody radiation start by telling you a long history lesson, which can easily get confusing.

This article gets straight to the point: shut up and calculate.

1. What Is the Blackbody Radiation Formula?

The blackbody radiation formula refers to the energy density radiated by a blackbody per unit frequency.

To calculate the blackbody radiation formula, we need to calculate how many quantum states there are between frequencies $\nu$ and $\nu + \mathrm{d}\nu$, as well as how many photons occupy each quantum state, and then multiply them by the photon energy $h\nu$.

Why Does the Intensity of Thermal Light Follow an Exponential Distribution?

hfwang132@gmail.com (hfwang132) — Fri, 21 Jun 2024 00:36:00 +0800

1. An Elegant Argument

Nobel laureate Ketterle gave a very elegant argument for the $g^{(2)}$ of a classical thermal light field in Open Course 8.422:

According to the central limit theorem, the electric field $E$ follows a Gaussian distribution, and thus the light intensity $I$ follows an exponential distribution.

The probability density function (pdf) of the exponential distribution is: $f(x) = \gamma e^{-\gamma x}$ .

The moments of the exponential distribution have the following property:

Farewell to the Rotating-Wave Approximation: An Exact Solution of the Quantum Rabi Model

hfwang132@gmail.com (hfwang132) — Wed, 05 Jun 2024 20:52:29 +0800

1. Introduction

In the early days of quantum mechanics, Rabi proposed the Rabi model to treat the interaction between an electromagnetic field and an atom. Since this model quantized only the atom, rather than the electromagnetic wave, it could explain many phenomena (such as Rabi oscillations), but could not explain phenomena involving quantization of the electromagnetic field, such as spontaneous emission.

To explain spontaneous emission and other phenomena involving quantization of the electromagnetic field, Jaynes and Cummings proposed the Quantum Rabi Model (QRM) in 1962. This model also quantizes the electromagnetic field, meaning that vacuum fluctuations of the electromagnetic field cannot be neglected. Thus, spontaneous emission is simply “stimulated emission” “excited” by vacuum fluctuations of the electromagnetic field.

A Fully Quantum Theory of Lasers

hfwang132@gmail.com (hfwang132) — Fri, 31 May 2024 21:45:30 +0800

Most materials introducing lasers derive them in a classical or semiclassical form. So what does a fully quantum theory of lasers look like? And what is it useful for?

This article briefly introduces the Scully-Lamb theory of lasers. It is a fully quantum theory, and since it is generally introduced only in the final chapters of quantum optics textbooks, while (semi-)classical theories work well in most cases, not many people are familiar with the Scully-Lamb theory.

A Simple Quantum Description of Lasers

hfwang132@gmail.com (hfwang132) — Tue, 21 May 2024 00:41:11 +0800

This article aims to derive, using a simple physical picture, the fact that the state output by a laser is a coherent state.

Consider the interaction between a two-level system and a single-mode optical field, with the field frequency equal to the energy-level spacing.

The two-level system initially occupies the excited state $|e\rangle$, while the optical field is in the vacuum state $|0\rangle$.

According to the Jaynes-Cummings model, spontaneous emission occurs even when the optical field is in the vacuum state. After a short time, the quantum state evolves from $|e,0\rangle$ to:

Does the idler light passing through the optical fiber amplifier still entangle with the signal light?

hfwang132@gmail.com (hfwang132) — Wed, 15 May 2024 01:30:09 +0800

Does the idler light passing through the optical fiber amplifier still entangle with the signal light, which is produced by SPDC?

Let’s consider two extremes:

Extreme 1: The gain of the amplifier is equal to 1, that is, there is no gain at all.

In this case, the amplifier acts as if it has done nothing and is an identity channel. So, of course, the entanglement will still be maintained. (Actually, it’s not necessarily the case. Even if the gain is zero, additional noise may be introduced, but we ignore it here for now).

Orbital Angular Momentum of Photons

hfwang132@gmail.com (hfwang132) — Mon, 13 May 2024 20:24:39 +0800

Preface

A 1992 article published in PRA [1] pointed out that photons also possess orbital angular momentum (OAM). Unlike spin angular momentum (i.e., polarization, SAM, Spin Angular Momentum), which can only take $\pm \hbar$, orbital angular momentum can take any integer multiple of $\hbar$. Such orbital angular momentum can be carried by helically shaped wavefronts.

You may be surprised: people did not discover this until 1992? In fact, helically shaped wavefronts had already been studied before 1992; photons carrying angular momentum greater than $\hbar$ had also long been predicted by atomic physics (except that they arise from higher-order transition processes, which do not satisfy selection rules and therefore have extremely low probabilities, making them essentially impossible to observe experimentally). It was not until 1992 that Allen et al. pointed out that beams with helically shaped wavefronts carry quantized orbital angular momentum.

[Quantum Optics Experiment Notes · III] Quantum State Tomography

hfwang132@gmail.com (hfwang132) — Sat, 20 Apr 2024 02:40:49 +0800

Quantum State Tomography

Quantum state tomography is the process of inferring a quantum state from the measurement results of an ensemble of quantum states. Its formulation is very simple, as follows:

Given a set of measurement operators $\{\Pi_1,\cdots,\Pi_n\}$ and their corresponding measurement probabilities $p_k=\operatorname{Tr}[\rho \Pi_k]$, find the quantum state $\rho$.

In other words, in these n equations $p_k=\operatorname{Tr}[\rho \Pi_k]$, $p_k$ and $\Pi_k$ are known, and $\rho$ is to be found.

What Is the Relationship Between Photons and Electromagnetic Field Wave Packets?

hfwang132@gmail.com (hfwang132) — Wed, 03 Apr 2024 10:47:06 +0800

A Wave Packet Can Correspond to a Photon

Example: A photon can be in a state of coherent superposition of different frequencies: $|\psi\rangle=\sum_{k}c_k|k\rangle,\quad \sum_k|c_k|^2=1$. In this case, the photon can manifest as a wave packet.

You can imagine an atom de-exciting and producing a photon; this photon will of course manifest as a wave packet.

Some may argue: if all nonideal factors are excluded, then the linewidth of this photon depends only on natural broadening (lifetime), and it can be regarded as having a single frequency. It therefore has poor localization and cannot be called a wave packet. This is indeed true.
However, if one considers single photons generated by pulsed pumping and parametric processes in the low-gain regime, their natural linewidth is itself very large. In this case, they are indeed in a coherent superposition of different frequencies and manifest as well-localized wave packets in the time domain.

Wave function of photons

hfwang132@gmail.com (hfwang132) — Sat, 09 Mar 2024 17:33:40 +0800

The wave function of a photon in the spacetime representation is:

$\Psi(\mathbf{r},t)=\langle \mathbf{r},t|\psi\rangle=\langle 0 |E^{+}(\mathbf{r},t)|\psi\rangle$

Where $\begin{aligned} |\mathbf{r},t\rangle = E^{-}(\mathbf{r},t) |0\rangle = \sum_{\mathbf{k},\lambda} \sqrt{\frac{\hbar \omega}{2 \epsilon_0 V}} e^{\mathrm{i}(\mathbf{k}\cdot \mathbf{r}-\omega_{\mathbf{k}} t)} a^\dag_{\mathbf{k},\lambda} |0\rangle \end{aligned}$.

Intuitively, this is to let the field operator $E^{-}(\mathbf{r},t)$ create a state $|\mathbf{r},t\rangle$ at the spacetime point $(\mathbf{r},t)$, and then calculate the overlap between this state and $|\psi\rangle$.

When we talk about the spacetime modes of photons, such as Gaussian pulses, hyperbolic secant pulses, etc., we are actually referring to the wave function in the spacetime representation described above.

Baker-Campbell-Hausdorff Formula

hfwang132@gmail.com (hfwang132) — Thu, 07 Mar 2024 16:13:09 +0800

Chinese version here

Baker-Campbell-Hausdorff Formula can be used to compute operator evolution in the Heisenberg picture:

$e^X Y e^{-X}=Y+[X,Y]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots$

This formula is actually just a younger sibling of the BCH formula.

Because the evolution rule of operators in the Heisenberg picture is $A\rightarrow UAU^{\dag}$, where $U$ is a unitary evolution operator.

If $U$ is generated by $H$, then it becomes $A\rightarrow e^{\frac{t}{i\hbar}H}Ae^{-\frac{t}{i\hbar}H}$.

Example 1: Phase Shifter The Hamiltonian is $H=\varphi n$, and the annihilation operator $a$ evolves as: $\begin{aligned} e^{-i\varphi n} a e^{i\varphi n}&= a + i\varphi [n, a] - \frac{\varphi}{2!} [n,[n,a]] - \cdots \\ &= a (1+i\varphi -\frac{\varphi^2}{2!} - \cdots)\\ &= e^{i\varphi} a \end{aligned}$

What is the significance of complex numbers in describing EM waves?

hfwang132@gmail.com (hfwang132) — Thu, 15 Feb 2024 23:46:31 +0800

In classical mechanics, complex numbers are merely a mathematical tool used to simplify calculations.

In quantum mechanics, complex numbers are not just a mathematical trick, but have a certain physical significance. Consider the classical vector potential:

$\begin{aligned} \mathbf{A}(\mathbf{r},t)=\sum_{\mathbf{k}\lambda} \left( A_{\mathbf{k}\lambda}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega_{\mathbf{k}}t)} + \text{c.c.}\right)\mathbf{e}_{\mathbf{k}\lambda} \end{aligned}$

where $\mathbf{k}$ and $\lambda$ represent the spatial and polarization modes respectively

Quantizing it yields the vector potential operator in the Heisenberg picture:

$\begin{aligned} \mathbf{A}(\mathbf{r},t)=\sum_{\mathbf{k}\lambda} \left( C_{\mathbf{k}\lambda}\hat{a}_{\mathbf{k}\lambda}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega_{\mathbf{k}}t)} + C_{\mathbf{k}\lambda}^{*}\hat{a}^{\dag}_{\mathbf{k}\lambda} e^{i(\mathbf{k}\cdot\mathbf{r}+\omega_{\mathbf{k}}t)}\right)\mathbf{e}_{\mathbf{k}\lambda} \end{aligned}$

[Quantum Optics Experiments Miscellany · II] Measuring Spectral Correlations and Purity of SPDC Multimode Squeezed States with an HBT Experiment

hfwang132@gmail.com (hfwang132) — Mon, 05 Feb 2024 00:10:15 +0800

In the previous article, we discussed the principle of using non-photon-number-resolving single-photon detectors to measure the quantum second-order correlation function in an HBT experiment.

[Quantum Optics Experiments Miscellany · I] The Principle of Measuring the Second-Order Correlation Function (g2) with Single-Photon Detectors

In this article, let us look at the uses of the quantum second-order correlation function. In addition to its well-known use for distinguishing [super-Poissonian statistics/Poissonian statistics/sub-Poissonian statistics] and [photon bunching/antibunching], an HBT experiment can also be used to measure the spectral purity of multimode squeezed states.

[Quantum Optics Experimental Notes I] Principles of Measuring the Second-Order Correlation Function (g2) with Single-Photon Detectors

hfwang132@gmail.com (hfwang132) — Sun, 04 Feb 2024 15:53:47 +0800

HBT Experiment

Anyone doing quantum optics experiments certainly knows that the HBT experiment can be used to measure the second-order correlation function g2. That is, a beam of light is split into two beams using a 50:50 beam splitter, which are then detected by two separate detectors, and the variation of the correlation between the intensities on the two sides with delay is counted, as shown below:

Hanbury Brown and Twiss Experiment