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Premise: I work on quantum computer science mainly at a circuit level abstraction. During my studies I spent some time to get an understanding of how abstract gates relate to physical settings. My background is computer science, so it gets hard for me when it comes to QED. I could cover some basics during my free time (e.g. faynman diagrams, B-E condensate, photon cavities, ions, superconductivity).

I am trying to understand how "distant" qubits and, specifically to the case of this question, superconducting-based qubits can interact by means of flying particles.

With some browsing on google.scholar, some papers got my attention as possible starting point. I will not list them all. Rather, there are a few sentences and pictures that I think are fundamental to understand the basic ingredients to achieve such interaction.

1. Superconducting-based can be coupled by means of a (1D) waveguide. I'd like to receive a qualitative explanation of what happens in the diagram below, taken from here

enter image description here

2. In this big survey authors refer to three states systems as a possible configuration to perform photon emission and/or routing. See Figure 4d, also reported below.

enter image description here

I have the feeling that these two pictures are related somehow, in the sense that the coupling between the stationary qubits happens by means of state transitions of the type in figure 4d.

My main doubt come from the fact that in both the papers they refer to photon-photon correlation. I am missing how such an interaction can couple the (distant) qubits.

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    $\begingroup$ It is not clear what your background is an therefore what kind of explanation you need. Are you a physicist? How much do you know about waveguides or photons? $\endgroup$
    – Roger Vadim
    Dec 2, 2022 at 13:48
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    $\begingroup$ I added a bit of my story, should I say something more? $\endgroup$
    – Daniele Cuomo
    Dec 2, 2022 at 13:57

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A common way to couple qubits is via coupling them to a photon mode - in this sense multi-qubit systems can be often viewed as extended versions of Jaynes-cummings model, where each quibit is a two-level system that can be in its ground or excited states, and the interaction is mediated by photons - which can be either photons propagating in space between cavity mirrors (typical for optical cubits) or light in a waveguide or lower frequency electromagnetic oscillations in metallic waveguides (typical for superconducting system.) In either case these are bosonic modes (unlike the two-level systems, which can be seen as fermions with occupation numbers 0 and 1 corresponding to the ground and the excited states.)

The qubits thus interact via physically exchanging photons: a cubit in an excited state emits a photon, which propagates through space/waveguide and then is absorbed by another qubit. Since (ideally) we have no dephasing/decoherence, this absorption/emission picture does not reflect the true state of things - rather the whole system is in a superposition of various energetically allowed states. E.g., for qubit-photon-qubit system with only one qubit exited, we have superposition os states (1,n,0), (0,n+1,0) and (0, n, 1), where the middle number is the number of photons in the mode.

For practical purposes one can perform a canonical transformation that would eliminate the photon modes from the Hamiltonian, replacing them by "effective coupling" constant, as if the qubits are coupled directly. (Sometimes one speaks of integrating out the photon mode - a matter of background/method.)

Remarks

  • More complex coupling schemes are possible, which is probably the case in the second example given in the paper, where additional levels are involved and need to be integrated out to have a set of coupled two-level systems representing qubits only.
  • Other bosonic excitations can be used to carry information - e.g., vibrations of molecules, photons (lattice vibrations), and plasmons (vibrations of electron gas.)
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    $\begingroup$ Great answer, thank you. Just a comment about your sentence: "unlike the two-level systems, which can be seen as fermions". In the case of a transom qubit, I thought this was driven by a condensate (which is a kind bosonic mode?). Am I correct? $\endgroup$
    – Daniele Cuomo
    Dec 2, 2022 at 17:05
  • $\begingroup$ @DanieleCuomo physically qubit could be realised by bosons - e.g., two photon polarizations, or two directions of flux in a Josephson junction, or simply the states with no boson and one boson present. But in the end one needs to somehow isolate these states, to have a two-level system - corresponding to 0 and 1 states. $\endgroup$
    – Roger Vadim
    Dec 2, 2022 at 19:31
  • $\begingroup$ I agree. And in the case of transmons, I would add: I often read of "macroscopic quantum state" given by the generation of a B-E condensate were all the bosons behave coherently. The direction of the flux you refer to, may this be indeed, a flux of such a condensate? $\endgroup$
    – Daniele Cuomo
    Dec 2, 2022 at 21:59
  • $\begingroup$ By reading at the answer to this question: quora.com/… I have the feeling that bosonic modes and fermionic modes should be switched in your answer. Or am I missing something? $\endgroup$
    – Daniele Cuomo
    Dec 5, 2022 at 14:40
  • $\begingroup$ Why would the (effective!) two level system be a fermion? (What does that even mean?) $\endgroup$
    – Norbert Schuch
    Dec 5, 2022 at 15:46
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In this case it's much better to think in terms of waves in the EM field rather than a photon particle. The wave properties are most import in QM... photons like to travel distances that are multiples of their wavelength for example ... a laser cavity only works when the mirrors are n x lambda apart (also said as there are acceptable modes).

The photon is just a result of an electron creating a wave in the field ... but electrons can also interact with each other without a photon! ..... they can exert forces which we attribute to the EM field as well, and the terminology is "virtual" photons for forces, virtual photons can never be observed for certain but the forces are real.

In your 1D waveguide paper the observations are dependent on qbit spacing (because the emitting q-bit must have a unique frequency) and thus wavelength determines the guides optical length between the qbits. The bunching/antibunching effects are observations of the timing of the output of the "circuit" using a photo detector (or microwave detector) .... the antibunching indicates the eventual domination of a single source (qbit) in the circuit after the initial start ... i.e. the whole circuit seems to resonant to one of the qbits ....

Your second paper refers to microwave qbits, microwaves are produced by a small antenna (wavelengths ~1mm) and single photons are possible ... i.e one 1 electron is accelerated ... on a circuit board short paths waveguides are possible and they have good modes because they are designed to be multiples of lambda.

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