How do they measure quantum observables in general?

Question

In most introductory quantum mechanics books they mention as an experimental example, which demonstrate the quantum nature of measurement, the Stern-Gerlach experiment and how they measure the spin. But is this only quantum experiment in this universe ?

My question is how about other experiments ? where can I find a book that tells what they do in the Lab to prepare states and how they measure observables ? The word observable mathematically means to me a operator (map) on a Hilbert space, but these are for computations. Experimentally it should mean a procedure done on the system forcing it to exhibit a seeable dynamical property. This last sentence is still very abstract to me and I can't find a lot of examples, if any.

For classical mechanics, one can think of any combinations of rigid bodies connected with any joints, ropes, springs or even elastic bodies, and any crazy potential and it makes sense.

For the quantum case the theoretical stuff one learns (at least for me) are abstract and are very difficult to bring down to earth, especially finding examples of how they measure various dynamical observables in practice and/or in theory (not mathematically). Let me state it like this

What are the general procedures to measure the various observables ?

(I have the impression I learned an abstract theoretical dream that has nothing to do with reality, but this must be a wrong impression)

Just two examples,

• in the case of the famous quantum mechanical harmonic oscillator, how can one theoretically define a procedure to measure its energy or momentum ?

• in the case of the Hydrogen atom, how do they measure angular momentum of the electron ?

Are there textbooks for experimental quantum mechanics explaining how they do all that ?

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Added: I found this post which raises quite similar question but yet it hasn't got any useful answer. So I hope nobody closes this post as a duplicate.

Answer 1

One can describe a general quantum measurement as an interaction between a measurement apparatus and a quantum system, followed by a measurement on the apparatus.* This lets you model 'derived' measurements, in which measurement of one systems actually constitutes measurement of another. I'll give some examples below.

First let me list a few of the more basic measurement instruments that experimentalists use:

• Photon counters measure optical power, the number of photons in a field within a given time interval. The associated quantum operator is $\hat{a}^\dagger \hat{a}$.
• A CCD camera also measures power, but one trades photon number resolution for finer position resolution.
• Using beam splitters and a laser, one can measure electric fields oscillating at optical frequencies using a homodyne detection setup, which measures $\hat{a} + \hat{a}^\dagger$ or $(\hat{a}-\hat{a}^\dagger)/i$.
• Analog-to-digital converters measure electrical voltages at lower frequencies (up to 10s of GHz, maybe more). Although they have limited amplitude resolution, it's possible to measure AC electric fields at the quantum limit using Josephson parametric amplifiers.

By combining these and other detectors with additional tools like lasers, particle beams, scanning probe tips etc., it's then a creative process to design an apparatus that can measure a desired quantum operator. There's no general prescription to measure everything. Nevertheless here are a few real-world examples that hopefully cover cases of interest:

• Let's start simple by seeing how experimentalists actually measure spin these days (no one uses the Stern-Gerlach method anymore). Atoms have many energy levels, and one can induce transitions between these energy levels using light. To detect if an atom is in a specific spin state, you can shine laser light that is resonant with the transition from that spin state to some excited state. This excited state will eventually emit a photon, and detection of that photon tells you that the atom was in the given spin state.

• To measure the momentum of a harmonic oscillator, consider the first experiment to detect Bose Einstein condensation. Here they trap a bunch of atoms in a harmonic oscillator potential (using lasers), and then turn off the trapping potential so that they go flying off. Mathematically, their propagation in free space maps their momentum degree of freedom to distance from the center of the trap i.e. position. After a fixed time delay, they measure the atoms' position using a laser and a CCD detector to detect the atomic absorption.

• There has been a big experimental effort to cool a mechanical harmonic oscillator to its ground state, and people needed to figure out ways measure its energy. It's a pretty subtle thing to do. The first paper to do so placed a vibrating membrane inside an optical cavity, which contained a standing wave of infrared light. The influence of the membrane on the cavity is related to the amplitude of the standing wave at the position of the membrane. They used this fact to engineer a measurement of $\hat{x}^2$, which becomes $\hat{x}^2+\hat{p}^2 = $ energy when you time-average over oscillations of the membrane.**

• I'm actually not sure how to measure the angular momentum of an electron in a hydrogen atom. It sounds difficult! If anyone knows how, mention it in the comments.

If you're interested in more experimental examples, try searching on Google Scholar. These measurement techniques aren't really summarized in any textbooks that I know of, but a lot more formalism can be found in 'Quantum Measurement and Control' by Wiseman and Milburne or 'Quantum Measurement Theory and its Applications' by Kurt Jacobs. Also every classical measurement is actually quantum at the end of the day, so think about how more familiar every-day measurements might be given a quantum mechanical description.

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*The precise version of this statement is the POVM formalism, which generalizes standard projective measurements. It isn't usually introduced in quantum classes, but it's pretty straight-forward.

**I'm dropping factors of $\omega$, $\hbar$ and $m$ for simplicity. In these units, $\hat{x}^2+\hat{p}^2$ is just the harmonic oscillator Hamiltonian. Also note that you actually need the response time of the cavity to be longer than oscillation period. This way the time-averaging is performed by the cavity itself, and you never actually measure $\hat{x}^2$ alone. $\hat{x}^2$ and $\hat{x}^2+\hat{p}^2$ are incompatible observables (i.e. they don't commute), so it's not sufficient to simply measure $\hat{x}^2$ and then time-average the resulting signal after the measurement is complete. See page 6 of the paper for the full derivation.

Answer 2

I love this question and would enjoy seeing other answers. You are asking for references, the most comprehensive (=largest number of different experimental setups) one I have come across on the topic so far is the Particle Data Group's PDF review, which has sections listing particle detectors. These detectors measure energy, momentum, position, particle number, and/or arrival time. Maybe I'm missing something besides that as well. It is available online:

https://pdg.lbl.gov/2022/web/viewer.html?file=../download/Prog.Theor.Exp.Phys.2022.083C01.pdf

Partial answers to your question can be found in chapters 35 and 36:

35. Particle Detectors at Accelerators (page 565)

36. Particle Detectors for non-Accelerator Physics (page 618)

But contributions from other users experienced in experimental physics are still very much needed, as this reference is not complete. The PDF above focuses on scattering and does not include even your basic example of a Stern-Gerlach magnet. I know of other techniques as well which I believe not to be in this book, even though it is the most comprehensive such reference I am aware of. So, additional references would be needed before your question is satisfactorily addressed!