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This question asks for the experimental result that is theoretically expected or is known already from experience, when one considers the following two scenarios for the micro-mechanical oscillator which can be put in an entangled quantum state with a qubit. See What exactly does Aaron D. O'Connell's experiment show? for a basic theoretical overview.

Experimental Scenario 1

When the qubit and oscillator are in a coherent superposition of direct product states, we make an attempt to view the oscillator, which has dimensions only in the micrometer range. Using a magnifying glass of high resolution, what do we see (or, if a similar operation has already been done, what was seen)?

I know that for viewing the micro mechanical-oscillator using an optical device, we are required to irradiate the oscillator with visible radiation. But, I am not sure if it's always strong enough to break the coherence.

Experimental Scenario 2

This question also assumes the oscillator to be in the same state as that of the previous question. Consider a small massive bob, which is set up such that any tiny changes in the gravitational field in its immediate surrounding could be detected. It is assumed to be placed close to the micro-mechanical oscillator. Let this bob also be kept in an ultra-cold environment as the oscillator, so that one need not worry about environmental decoherence.

In this context, we expect there to be a minuscule variation in the gravitational field experienced by the bob due to the varying distance between the bob and the center of mass of the oscillator. What does our measured value of gravitational field due to the oscillator look like as a function of time? Is it a smooth sine function?

Since these questions are experimental in nature, answers that contain information about similar ideas that have been implemented in experiments of the past are appreciated more than speculations.

Note: In the first question, I am using the word 'see', as it's always in principle possible to observe a micrometer sized object using an appropriate optical lens. To be more precise, let us say that we wish to see a tiny marking on the oscillator.

Emilio Pisanty
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  • Before anyone gives an answer, let me mention a thought I had with regard to the second question. The following possibility appears to be ruled out: The measurement of gravitational field due to oscillator varies as a smooth function while at the same time the oscillator remains in the coherent superposition state, without assigning different physical meanings to 'vibration of oscillator' in the context of gravitational field measurement and to the case in which the qubit-oscillator entangled system is subject to a measurement in the qubit's basis as was done in the original experiment. –  Jan 16 '19 at 19:10

1 Answers1

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When the qubit and oscillator are in a coherent superposition of direct product states,

By this, I imagine that you mean a state of the form $$ |\Psi⟩ = a \left| 1 \right\rangle_q \left| 0 \right\rangle_l + b\left| 0 \right\rangle_q \left| 1 \right\rangle_l $$ as in the answer to the question you've linked.

This state is an entangled state, which means that neither of the components of the system has a (pure, quantum) state of its own. That means, therefore, that if you want to do experiments that deal exclusively with either half of the system, you need to take the density matrix of the system, \begin{align} \rho & = |\Psi⟩⟨\Psi| \\ & = (a \left| 1 \right\rangle_q \left| 0 \right\rangle_l + b\left| 0 \right\rangle_q \left| 1 \right\rangle_l) (a^* \left< 1 \right|_q \left< 0 \right|_l + b^*\left< 0 \right|_q \left< 1 \right|_l) \end{align} and trace out the qubit, leaving you a state \begin{align} \rho_l & = \mathrm{Tr}_q \left(|\Psi⟩⟨\Psi|\right) \\ & = |a|^2 \, |0⟩_l⟨0|_l + |b|^2 \, |1⟩_l⟨1|_l \end{align} where there is no longer any coherence between the different states of the oscillator.

That's a bunch of technical language, the details of which you don't really need to understand, but here's the important part:

For any experiment that only deals with the oscillator and ignores the qubit, the state of the system is completely equivalent to a probabilistic mixture of the two states of the oscillator with respective probabilities $|a|^2$ and $|b|^2$.

Both of your proposed scenarios satisfy this hypothesis, which means that you will be completely unable to measure any quantum effects in either scenario.

Emilio Pisanty
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  • Thanks for the answer. Considering your answer, if we re-consider the second scenario of my question, it seems to me that a hypothetical high sensitive measurement of gravitational field as a function of time gives us a sinusoidal function as we would expect for any ordinary oscillator doing small oscillation. And all the above happens, while at the same time we use qubit thermometry(as was done by O'Connell) to determine the quantum state of the lever. If that were to be the case, the phrase "vibrating and not vibrating" appears to be a sloppy use of language. Is my reasoning right? –  Jan 20 '19 at 19:53
  • Nope, your prediction is incorrect. One useful model of the $|1⟩_l$ state is that it's on a sinusoidal vibration proportional to $\sin(\omega t+\varphi)$ - except that the phase $\varphi$ of that oscillation is completely unknown, and you need to take a probabilistic mixture of all $\varphi\in[0,2\pi]$; this probabilistic mixture then washes out the signal in your second scenario. I appreciate that some of this sounds weird, but I'm deadly serious: there is a point at which you cannot proceed without sitting down and doing a serious read of grown-up QM. You're at that point. – Emilio Pisanty Jan 20 '19 at 20:06
  • As for 'the phrase "vibrating and not vibrating" appears to be a sloppy use of language' $-$ yes and no, but not in the way you'd like it to be. When we say that, we're referring to quantum superpositions, and that concept simply does not fit within classical ontologies - it is a completely new concept. See e.g. this text for some more material on that direction. – Emilio Pisanty Jan 20 '19 at 20:08
  • Thanks for the advice. I am actually done with QM in the level of Sakurai. I didn't write equations as I was not comfortable with latex. By the way, I presume the kind of probabilities associated with density matrices involves both ontic and lack of knowledge probabilities. So, suppose we imagine the second scenario, assuming that we have complete access to all the information that are in principle ex-tractable from the states of the systems, then the second kind of probabilities vanishes. So, under such assumption, what does the function I am looking for look like? –  Jan 20 '19 at 20:21
  • @PVI comments are not for extended back-and-forth. If you can distil that into a precise question (and without skimping on the maths if you want to be taken seriously; here is a MathJax tutorial if you need one) then ask it separately. – Emilio Pisanty Jan 20 '19 at 20:24
  • Also, did you mean in your second comment that our measurement of gravitational field intensity will also have a probabilistic nature, with the probabilities being due to our lack of knowledge? –  Jan 20 '19 at 20:24