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.

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.

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.

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?

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.

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).

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.

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

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.

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.

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?