Defect Centers

12.1 Localized States

A defect is a missing or substituted atom in an otherwise regular crystal, and it can bind electrons in levels of its own. Which of those levels make a qubit, and how is it prepared and rotated? A spin with three levels, split even without a field, lets a microwave pulse pick one pair. Spin-dependent optical cycles then prepare it, and its fluorescence reads it.

12.1.1 Defects and spin levels

Why not use a single electron spin, as in a quantum dot? A spin with two levels has no splitting at zero field, so it needs a magnet to become a qubit. Some defects bind two unpaired electrons whose spins combine to a total spin of one. Such a spin has three states along an axis, labelled \(m_s=0,\pm1\).

We take one example, the nitrogen-vacancy (NV) centre in diamond, and two physical inputs about it. First, the crystal around the defect separates \(m_s=0\) from \(m_s=\pm1\) by an energy \(\hbar D\) even with no applied field; for the NV centre \(D/(2\pi)\approx2.87\) GHz. Second, a field \(B\) along the defect's symmetry axis shifts \(m_s=\pm1\) by \(\pm\hbar\gamma_eB\), where \(\gamma_e=g\mu_B/\hbar\) is the electron's gyromagnetic ratio. The resulting axial model, written with a dimensionless spin-one operator \(S_z\), is

\[H=\hbar DS_z^2+\hbar\gamma_eBS_z.\]

It omits strain, a deformation of the crystal, and any field component across the axis.

Its energies are \(E_m=\hbar Dm^2+\hbar\gamma_eBm\), so the transitions from \(m_s=0\) to \(\pm1\) have angular frequencies \(D\pm\gamma_eB\). A modest field separates the two lines, and a narrow microwave pulse then selects one pair as the qubit. The third level stays present as a possible leakage state.

12.1.2 Initialization and microwave control

How is a defect spin prepared? By optical pumping, because the paths of optical excitation and decay depend on the spin state. To see how a one-way bias builds up, take a toy cycle that moves a fraction \(r\) of the unwanted-state population into the target state and never moves any back. After \(N\) cycles the unwanted population is \(p_N=p_0(1-r)^N\). Real cycles also pump some population the wrong way, which leaves a nonzero limit, so the prepared population must be measured.

Once prepared, the spin is rotated by microwaves resonant with the selected transition, with the Rabi dynamics we derived. The same optical cycle reads it, since the fluorescence is brighter for one spin state; the two photon-count distributions overlap, as in any threshold readout. Strong optical reset of the electron can also disturb nearby nuclear spins, so preparation is tested on the whole node, not only on the electron.

12.2 Spin Registers

The electron is needed for optical attempts and readout, which disturb whatever it stores. Where can a qubit wait meanwhile, and how do the electron and that store interact? A nuclear spin nearby couples to the electron only weakly, which makes it a quiet memory. Keeping only the part of that coupling that survives the large energy mismatch gives a ZZ interaction and a conditional gate.

12.2.1 Nuclear memories

Where can a qubit wait while the electron is busy? In a nearby nuclear spin, whose magnetic moment is far smaller than the electron's, so stray fields disturb it less. The electron and nucleus interact magnetically. Treat each as two-level. The full interaction contains terms that flip one spin while leaving the other. The electron's splitting is much larger than the coupling, so such terms are strongly mismatched, and by the rotating-wave argument they average out. The part that survives, called the secular interaction, is

\[H_{\rm int}=\hbar AZ_eZ_n/4,\]

with \(A\) an angular frequency.

What does it do to the nucleus? With the electron's Z value fixed at \(s=\pm1\), the nuclear Hamiltonian changes by \(s\hbar AZ_n/4\). The nuclear transition frequency therefore shifts in opposite directions for the two values of \(s\), and the two lines are separated by \(|A|\). A pulse narrow enough to resolve that separation acts on the nucleus only for one electron state. The same coupling is a hazard during storage: if the electron flips unexpectedly, the nuclear phase accumulates at the wrong rate. So an idle nuclear coherence time does not predict the memory's quality while the electron is repeatedly reset and excited.

12.2.2 Conditional operations

Can the secular interaction alone make a controlled gate? In the basis 00, 01, 10, 11, it gives the phases \((-At/4,+At/4,+At/4,-At/4)\). Their local-phase-invariant combination is \(-At\). Choosing \(At=\pi\) therefore gives a controlled sign after local Z corrections, exactly as for the ZZ coupling between two qubits.

The phase difference between one even and one odd basis state is only \(At/2\), and it must not be confused with the invariant conditional phase. A selective microwave or radio-frequency pulse gives a conditional operation by a second route, using the resolved transition frequencies instead of accumulated phase. Its selectivity must be large compared with spectator nuclear couplings, since every extra nucleus that helps a gate also couples to the memory.

12.3 Optical Interfaces

A node becomes part of a network only when its qubit can be entangled with a qubit far away. How can photons do that, and how fast? If a detector click cannot reveal which node emitted, the click leaves the two nodes entangled. Counting the photons that must be collected then gives the rate, and the memory must survive every failed attempt.

12.3.1 Readout and heralded entanglement

How can light entangle two distant spins that never interact? Let each node emit a photon whose presence or properties depend on its spin, and send both photons to a beam splitter in the middle. If a detection could have come from either node, it carries no information about which one emitted. The click pattern then announces, or heralds, a joint spin state that is entangled. The accepted pattern and the overlap of the two photons' modes fix that state.

Is a click enough? No. Background light and photons that can be told apart produce the same classical record without the coherence. So the quality of the heralded state depends on how identical the photons are, not only on whether a detector fired.

How often does it succeed? A remote protocol that needs two photons adds collection, transmission, detector efficiency, and an allowed pattern of coincidences. If each required photon is detected with probability \(\eta\) and an accepted pattern has conditional probability \(q\), the success probability per attempt is \(s=q\eta^2\), assuming independence. This is the rate model for that protocol, not a formula for every heralded scheme.

12.3.2 Collection efficiency and scaling

What does a small collection efficiency cost? At \(\eta=0.1\), two required photons contribute only 0.01, before \(q\) and other imperfections. For attempts spaced by \(\tau\), the mean waiting time is \(\tau/s\). Because the photon requirement is squared, raising the collection efficiency speeds up the network quadratically in this model.

What happens to the stored qubit during the wait? It must survive idle evolution and repeated optical disturbance. Suppose each failed attempt keeps a fraction \(c\) of its coherence, and attempts succeed independently with probability \(s\). Success on attempt \(n\) has probability \(s(1-s)^{n-1}\) and follows \(n-1\) failures, so the mean surviving factor at first success is

\[\sum_{n\ge1}s(1-s)^{n-1}c^{n-1}=\frac{s}{1-(1-s)c},\]

before idle decay is included. When success is rare, even a small disturbance per attempt matters. So a fidelity conditioned on success is meaningful only together with the number of attempts and the memory's behaviour during them.

Photons carried entanglement between nodes here. Could photons carry the computation itself, even though they barely interact with one another?

Models and further derivations: Doherty et al.: defect spin and optical physics; Bernien et al.: heralded solid-state entanglement. Numerical examples in this chapter are illustrative, not device specifications.