tion and detection
A single microwave photon that is created by a certain transition in one qubit carries in- formation about the state of the qubit and can be used as an information bus, e.g., to mediate entanglement between qubits. Also, these single photons could be used for quan- tum cryptography applications in solid-state devices and are eventually just the starting point for interfacing the solid-state and the quantum optics domain.
The special setting of a superconducting resonator as a cavity that is operated with photons in the microwave domain is quite different from the setups in quantum optics experiments. First experiments with microwave photons travelling in conductors usedHan- bury Brown and Twiss(HBT) correlations [265–267] to measure the super-Poissonian and Poissonian photon statistics of a thermal and a coherent photon source [180].
12.2 Deterministic single microwave photon generation and detection 173 optics experiments will be discussed. First, note that the normal conducting part of the setup,e.g., the microwave generators at room temperature, hasnoshot noise but Johnson- Nyquist noise, whereas the superconducting part has no intrinsic electronic shot noise but photon shot noise. A single microwave photon that is generated from a transition in a rf- SQUID or 3jj flux qubit and travels through the superconducting resonator can be viewed as a plane wave inside the waveguide. There are several possible loss mechanisms for this electromagnetic wavepacket, namely
• Static or dynamic charge disorder,
• Magnetic impurities in the superconductor, • Dispersion of the electromagnetic wavepacket.
For the superconducting waveguide, the fabrication in niobium (Nb) technology will be considered in most detail. Static or dynamic charge disorder have been shown to be negligible, see Ref. [268], which follows from the large quality factor Q for the resonances that are associated with the Mooij-Sch¨on modes [268] because the photon itself can also be viewed as a quantum of the Mooij-Sch¨on mode [268, 269]. Magnetic impurities in the Nb can be analyzed via data for the density of states in the gap or subgap tunnel conductance. The data presented in Ref. [270] suggests that the effect of magnetic impurities is also negligible. Thus, it can be shown from an effective circuit model of the superconducting waveguide or transmission line that the dispersion relation for the photon wavepacket is linear. Therefore, the wavepacket will be destroyed only after a significant distance that is estimated to be about one kilometer in the superconductor.
The microwave photon inside the cavity can be generated by an appropriate transition of a rf-SQUID or a 3jj flux qubit, cf. chapter 2.1.2. Here, it is favorable to make use of the rf-SQUID or 3jj flux qubit as a multi-level system. This is due to the fact that if a driving that is resonant with the qubit is applied in order to populate the excited qubit level for the generation of photons, the cavity will also be driven. This happens because the cavity has to be on resonance with the transition that will generate the photon. Thus, a more elaborate scheme is required for the single photon generation. It can be realized, e.g., via stimulated Raman adiabatic passage [271] or a Raman pulse.
For the general rf-SQUID N-level system, the interaction Hamiltonian that describes the coupling between the system and the cavity [260, 272, 273] is given by
HI = − 1 L( ˆΦ−Φx) ˆΦc (12.1) = (a+a†)(X i gii|ii hi|) +g01a|0i h1|+g01a†|0i h1|+g12a|1i h2|+. . .),(12.2)
where the coupling constants are defined as
gij =− 1 L p ~ωc2µ0(hi|Φˆ|ji −δijΦx)Φ0 c, (12.3)
174 12. Circuit-QED – quantum optics in the solid-state where Φ0
c =
R
SB(r)·dSis the time independent part of the flux generated by the resonator field, which is threading the qubit loop. The complete expression for the cavity flux, cf. also appendix F, is Φc = Z S B(r, t)·dS, withB(r, t) = s ~ωc 2µ0 (a(t) +a†(t))B(r, t). (12.4)
For calculation of the coupling strength between the cavity and the qubit, which is deter- mined by thegij, it is necessary to compute the eigenstates |iiof the system Hamiltonian. This can be easily done numerically either in the charge or phase basis. The total flux in the qubit loop is the sum of the externally applied flux and the so-called screening flux, which appears because of the finite self inductance of the qubit loop
Φ = Φx+ Φs. (12.5)
For a 3jj flux qubit with a negligible self-inductance, the so-called screening current is usually disregarded. In this case the expectation value of the total fluxhi|Φ|jiwill vanish at the degeneracy point of the qubit, where the state of the qubit is in an equal superposition state of clockwise and counter-clockwise rotating currents. Due to the small but finite self-inductance of the real qubit, it is most convenient to view the screening current as a perturbation
Φs =LIs =LIcsin(φ1) = LIcsin(φ2), (12.6)
which is induced in the qubit loop. The screening current essential gives the coupling between the cavity and the qubit. In the qubit eigenbasis it can be viewed as completely off-diagonal, mediating the coupling strengthsgij with i6=j. The coupling strengths that are presented in the paper of this chapter were calculated from numerical simulations of the cavity-qubit Hamiltonian and the mutual inductive coupling.
Next, a detection scheme for the detection of photons with different frequencies is required. This can be realized, e.g., by a (non-)linear mixing with another radiation at a certain reference frequency. The basic interferometric setup is the same both for a homodyne and a heterodyne readout scheme. In a homodyne scheme, the reference radiation, i.e., the local oscillator, stems from the same source as the photon, or more generally, the signal that is to be measured. For heterodyne detection the difference is that the reference radiation comes from a different source. For example in a laser scattering experiment with homodyne detection, the laser light is separated into two beams. Then one of the beams is used to perform an experiment and the scattered photons should be read out after the experiment. This radiation is mixed with the other laser beam that acts as a local oscillator. Then the resulting measurement can be made insensitive to fluctuations of the laser frequency.