1. What Is a TDC / Time Tagger?

A Time-to-Digital Converter (TDC) is essentially an extremely precise timer (at the picosecond level) that assigns a high-precision timestamp (Timestamp) to every input pulse signal.

Therefore, a TDC is also called a Time Tagger, where “Tag” means assigning a timestamp.

TDCs / Time Taggers are central to experiments such as Hong-Ou-Mandel interference and fluorescence lifetime measurements; it can be said that quantum optics experiments can hardly do without them.

I. Introduction

Recently, I found that many colleagues are not very familiar with impedance matching. They only know that “the impedance of (commonly used SMA/BNC) transmission lines is 50 ohms,” but do not know what this really means. The following situations often occur:

• A 50-ohm signal source connected to a scope with a 1M-ohm input:

• Huh, why did the signal become twice as large?

• A high-frequency square wave connected to a scope with a 1M-ohm input:

• Huh, why is the overshoot so severe? Is there something wrong with this signal source?

• The output signal of an MCU / DAC / FPGA connected directly, without a driver, to a 50-ohm load / 50-ohm oscilloscope input:

• Huh, why did a signal that was supposed to be 3V become only 30 mV?

All of the above result from unfamiliarity with impedance matching and transmission-line theory.

First quantization cannot describe superpositions of particle number, let alone dynamics in which particle number changes.

The terms first quantization and second quantization can easily lead one to believe that they are equivalent, and that one can use whichever is more convenient, much like the Schrödinger and Heisenberg pictures. This is not the case. The descriptive power of second quantization is strictly greater than that of first quantization. First quantization is merely a simplified description of second quantization for situations with a fixed particle number, and is intrinsically deficient.

In a previous article, we said that the electromagnetic field is a connection. This led many readers to think of the connection in general relativity. What are the differences and commonalities between these two kinds of connections?

The electromagnetic connection \(A\) is a connection on a principal bundle, whereas the general-relativistic connection \(\Gamma\) is a connection on a vector bundle. Their definitions appear to be quite different. Is there a way to relate them?

Probability Waves vs. Quantum Fields

The electromagnetic field is not the probability wave of photons, just as the Dirac field is not the probability wave of electrons.

A probability wave refers to the single-particle wavefunction (in the position representation) in nonrelativistic quantum mechanics.

Clearly, probability waves apply only to single-particle states in the nonrelativistic regime. The electromagnetic field and the Dirac field, on the other hand, are quantum fields and must be discussed within the framework of quantum field theory.

Conclusion: integer spin is a classical effect, whereas half-integer spin is a quantum effect and has little to do with relativity.

The reason is simple: spin-1 is the smallest faithful representation of SO(3), while spin-1/2 is the smallest faithful representation of SU(2).

So why can quantum mechanics lift SO(3) to SU(2)? Because quantum states are rays and are equivalent up to a global phase. In other words, quantum mechanics requires projective representations of SO(3). And projective representations of SO(3) are in one-to-one correspondence with representations of SU(2) (Bargmann’s theorem). This is why quantum mechanics considers SU(2) rather than SO(3).

I. Quantum State Tomography

At present, quantum states composed of hundreds or thousands of qubits can already be prepared (twenty references omitted here).

But how do we know that the prepared quantum state \(\rho\) is indeed the one we want \(\rho\)? Or, more broadly: how can we learn (some or all) information about a quantum state \(\rho\)?

To learn a quantum state \(\rho\), we first need to prepare \(\rho\), then measure it in different measurement bases to obtain probability distribution functions, and finally use these probability distribution functions to learn \(\rho\). This process is called quantum state tomography.

My favorite way to understand this is to regard $\Gamma$ as a connection on the frame bundle. Then, the transformation law of the Christoffel symbols

$\boxed{ \begin{aligned} \bar{\Gamma}^i_{\mu j} = \frac{\partial \bar{x}^i}{\partial x^k}\frac{\partial x^l}{\partial \bar{x}^j} \frac{\partial x^\nu}{\partial \bar{x}^\mu} \Gamma^k_{\nu l} \color{red}{ + \frac{\partial \bar{x}^i}{\partial x^k} \frac{\partial^2 x^k}{\partial x^\mu \partial \bar{x}^j}} \end{aligned} }$

is nothing more than a gauge transformation:

$\boxed{ \bar{A} = g^{-1}A g \color{red}{+ g^{-1} \mathrm{d}g} }$

The extra inhomogeneous term $ g^{-1} \mathrm{d}g$ is inherently part of a gauge transformation.

In this article, we start from the QED Lagrangian:

This Lagrangian is highly complex: it not only takes into account the electron’s antiparticle—the positron—but the coupling term \(- eA_\mu \bar{\psi} \gamma^\mu \psi\) is also a cubic term, capable of describing various processes such as electron-positron pair creation/annihilation. Due to the presence of the cubic term, this Lagrangian has no analytic solution and can only be solved using perturbation theory in quantum field theory.

Previous article in this series:

Godfly: The electromagnetic field is more than just electric and magnetic fields—The AB effect and Berry connection [Higher and More Wonderful Electrodynamics · 3]

In the previous article, we mentioned that if a manifold is topologically trivial, then integrating the Berry curvature over a closed surface \(\Sigma\) should yield zero. This is because, according to Stokes’ theorem:

However, on a topologically nontrivial manifold, the integral \(\int_\Sigma F\) need not be zero. This is because a globally single-valued connection \(A\) cannot be defined in this case, so Stokes’ theorem no longer applies. What does this mean?

Previous article in this series:

Godfly: Electrodynamics from the Perspective of Gauge Field Theory [Higher and More Elegant Electrodynamics · 2]

The electromagnetic field is not simply the electric field and the magnetic field.

In other words, the electromagnetic field is more than just the electric field and the magnetic field.

What does this mean?

1. Electromagnetic Potentials Have a Higher Status Than Field Strengths

In a previous article, we left a question open:

Previous article in this series:

Godfly: Dirac Magnetic Monopoles—Chern Classes and Chern Numbers [Higher and More Elegant Electrodynamics · 4]

In a previous article, we discussed the Hamiltonians of the electromagnetic field and the Dirac field:

So what are their Hamiltonians? Why does quantum field theory not make much use of Hamiltonians? Read on for the answer.

What Is the Invariance of Differential Forms?

According to textbooks, the invariance of differential forms means that, under a change of variables, the form of a differential equation remains unchanged. That is:

If, when \(\mathrm{d}y\) in \(y\) denotes the function \(y=f(x)\) and \(x\) in \(\mathrm{d}x\) denotes the function \(\phi(x)=x\), \(\mathrm{d}y=a\mathrm{d}x\) holds, that is, \(\mathrm{d}f=a\mathrm{d}\phi\) holds,

then, when \(y\) in \(\mathrm{d}y\) denotes the function \(y=g(t)\) and \(x\) in \(\mathrm{d}x\) denotes the function \(x=\varphi(t)\), \(\mathrm{d}y=a\mathrm{d}x\) likewise holds, that is, \(\mathrm{d}g=a\mathrm{d}\varphi\) holds.

If you have not yet read the previous article, please see:

Electrodynamics from the Perspective of Differential Geometry [Higher and More Elegant Electrodynamics · 1]

This article uses many concepts from the previous article, so please make sure you have read it.

This article continues to use the metric convention of \((-,+,+,+)\).

Why is electrodynamics a gauge field theory?

To understand what this statement means, we must first understand what electrodynamics is and what gauge field theory is.

Preface

You may have heard that Maxwell’s equations have a very simple form:

Or alternatively,

Note: Strictly speaking, \(\mathrm{d}F=0\) (or \(\partial_\mu (\star{F})^{\mu \nu}= 0\)) is not part of the dynamical equations of the electromagnetic field, but rather part of the field’s own structure. This is because \(\mathrm{d}F=0\) follows from the definition of the field strength \(F=\mathrm{d}A\).

Personally, among the interpretations of quantum mechanics, I find Zurek’s Existential Interpretation—also known as Quantum Darwinism [1]—to be the most interesting at present. Zurek himself is one of the founders of decoherence theory.

Quantum Darwinism does not directly provide the dynamics of the measurement process, but it makes several highly illuminating observations:

(1) Measurement outcomes can become macroscopic facts because their information is replicated (cloned) many times and dispersed throughout the environment, allowing different observers to reach a consensus.

Preface

In classical mechanics, a physical quantity is a function on phase space, while the state of a system is a point in phase space (or, for an ensemble, a probability distribution on phase space). Upon quantizing this language, one obtains the phase-space representation of quantum states/operators, namely the Wigner representation. The Weyl representation is its Fourier transform.

Most authors define the Wigner representation as:

and then derive its various properties. However, from a physicist’s perspective, the physical meaning of this expression is unclear, and \((x,p)\) do not have equal status, which is somewhat uncomfortable to look at.

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.

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 \(h\nu\).

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: