6.3 Spin Qubits in Diamond
6.3.1 Electron Spin Qubits
2
(SxSz+SxSx)+γe(es)B·S + Aes⊥
2 (S+I−+S−I+)+AesqSzIz+QIz2−γnB·I. (6.9) In further discussions, we use the room-temperature values for the hyperfine constantsAesq and Aes⊥. The physical mechanism we exploit in the two-qubit gate scheme derived below only depends on the difference between the ground- and excited-state hyperfine constants; qualitatively, it is therefore rather insensitive to actual parameter values.
6.3 Spin Qubits in Diamond
6.3.1 Electron Spin Qubits
Not only the coherence times have to be long enough for a desired system to be of use in quantum information processing. The ability to initialize, manipulate and readout the state of the qubit with sufficiently high accuracy have to be demonstrated as well (see the criteria on p.2).
Qubit initialization and readout can be achieved via optical spin polarization [173], which is described in more detail in the Sec.6.2.2. When the NV center is optically excited resonant to themS =0 transition, the fluorescence signal crucially depends on the electron spin state. If the
6.3 Spin Qubits in Diamond
system is in themS = 0 state, a finite number of photons will be detected, whereas no signal is expected formS =±1. Hence, the presence or absence of fluorescence will give information about the electron spin state. First studies of single defect centers have been performed in Ref.192 and improved experimental work demonstrated projective single-shot readout with an average fidelity of about 93 % [193]. Even better results for spin readout can be achieved by exploiting coherent hyperfine interactions with nearby nuclear spins, as demonstrated using a13C [194] or a
14N [191] nuclear spin as an ancilla. Initialization of the electron spin to themS =0 state was also implemented using optical spin polarization with fidelities above 99 % at cryogenic temperature [193,195–197].
An external magnetic fieldBsplits spin states with different quantum numbersmS by the Zeeman energy according to the HamiltonianH =γeB·S. Coherent rotations of the NV electron spin state about thex axis on the Bloch sphere can be achieved using microwave radiation (see Sec.6.2.2) [195,198]. Using this method, the implementation of a spin flip or Pauli-Xgate [Eq. (3.3)], also known as aπpulse, could be achieved with a gate time below 1 nanosecond [199] and with gate fidelities reaching 99 % [154]. By a combination of two microwave pulses, a refined version of the Deutsch-Josza quantum algorithm has been experimentally implemented [200]. Rotations about the polar axis have been achieved using various methods, e.g. by utilizing the optical Stark effect with fidelities of 89 % [201]. In a more recent experiment, qubit rotations in the equatorial plane of the Bloch sphere were demonstrated by harnessing the excited state manifold [202]. In a Λconfiguration, coherent population trapping [203] can be exploited to all-optically control the electron spin state. If the two transitions in theΛconfiguration are excited by lasers with Rabi frequenciesΩ0andΩ1, respectively, a ground-state superposition|Diof the form
|Di= 1 q
Ω20+Ω21
(Ω1|mS =0i −Ω0|mS = +1i) (6.10)
cannot be further excited by the optical means due to destructive interference between the two transitions [23,202]. The state|Diis therefore called dark state. If the system decays from the excited state (see Fig.6.2) into the dark state|Di, it will be trapped in this superposition. A proper adjustment of the Rabi frequenciesΩ0andΩ1allows the preparation of any superposition of themS =0 andmS = +1 spin states. Initialization of the spin state to arbitrary positions on the Bloch sphere, readout in an arbitrary basis, and single-spin rotations about arbitrary axes could be demonstrated using this technique [202]. Especially the ultrafast implementation of a Pauli-Z gate [Eq. (3.3)] within 160 ps with a fidelity of 77 % [186]. The devices for optical manipulation have smaller dimensions, which makes this scheme very promising for large-scale application.
Yet another approach enabled high-fidelity single qubits gates with fidelities around 98 % using holonomic transformations [204], but with slower gate times of 160 ns when compared to the all-optical realization mentioned above.
An open issue in the development of diamond-based quantum information processing is
79
Chapter 6: Long-Range Two-Qubit Gate
scalability. As explained in Sec. 3.2, qubits must be coupled to eventually perform universal quantum computation. This means, individual NV centers have to interact with each other in a highly controllable fashion. An ongoing challenge to develop coupling mechanisms between two distant NV center spins, in which the interaction can be arbitrarily turned on and off, and is strong enough to implement quantum gates much faster than the decoherence time. Some of the proposed interaction mechanisms and proof-of-principles experiments are described below.
One of these ideas is to use the of magnetic dipolar coupling between two magnetic moments [see e.g. Eq. (5.37)]. The generation of an entangled state between two electron spins and fidelities of up to 82 % with respect to a maximally entangled state, has been reported for NV centers separated by roughly 10–25 nm and coupled via magnetic dipole-dipole interaction [137,153,154].
A drawback of such a scheme is the random distribution of NV centers in naturally occurring diamond or the lack of positioning accuracy for implanted NV centers, although there has been a lot of progress related to nitrogen-ion implantation in diamond (see e.g. Refs.205and206). The locations of the defects have to be close enough to achieve large enough coupling strengths via the dipolar coupling [cf. Eq. (5.37)]. The maximal displacement is set by the decoherence timeT2 and amounts to roughly 30–50 nm [153,154]. The interaction with proximate nitrogen atoms, that do not form a NV defect, has also been observed [196,207,208]. However, there shouldn’t be any unwanted and disturbing interactions with other defects or nuclear spins present in the vicinity of the system of coupled NV centers. There is thus a piece of randomness left in this scheme. On the other hand, entanglement purification can be directly achieved with magnetic dipolar coupling as developed in Sec.5.4.2, which plays a key role in quantum repeater protocols [8,9,156,157].
Besides the aforementioned works addressing single- and two-qubit controllability, ground-breaking results could also be achieved regarding the feasibility for quantum communication.
It was pointed out in Ref.17that a physical apparatus must have "The ability to interconnect stationary and flying qubits . . . " and ". . . faithfully to transmit flying qubits between specified locations . . . " [cf. criteria (vi.) and (vii.) on p.2]. A first step towards diamond-based quantum communication networks was the demonstration of entanglement between the NV center electron spin and an optical photon [209]. The NV center is excited to a state that decays with equal proba-bility to one of the ground states with spin projectionmS = +1 ormS =−1. This can be achieved for an equal superposition of the spin states, because the optical transition is spin-conserving. Due to the associated orbital angular momentum, the polarization of the emitted photon depends on the spin state to which the system decayed, and is eitherσ+orσ−circular polarization. Therefore, the NV center and the emitted photon reside in the entangled state [209]
ψ= 1
√ 2
( σ−
|mS = +1i+ σ+
|mS =−1i), (6.11)
where|σ±idenote the polarization state of the photon. Since the NV center level structure depends
6.3 Spin Qubits in Diamond
on parameters like strain, magnetic and electric fields, the energy of the emitted photon varies from defect to defect. However, using dc Stark tuning [210], the emission lines between different NV centers can be brought into resonance, rendering the emitted photons indistinguishable. In doing so, two-photon quantum interference between photons that originate from dissimilar NV centers could be observed [211,212]. Going one step further, the combination of spin-photon entanglement and two-photon interference offers the possibility for measurement-based long-distance entanglement generation. In a modified experiment [51], two-photon interference on a beam splitter and a subsequent photon measurement projected the spin qubits onto the maximally entangled states|Ψ±i[Eq. (2.13)], where the sign depends on the specific measurement outcome.
The fidelity of the experimentally realized entangled state was around 70 %, and even more im-pressing, the two entangled NV centers were separated by 3 m [51]. The quantum teleportation protocol, described in Sec.3.4, was implemented in a follow-up experiment [52]. An entangled electron-spin state between distant NV centers was used to unconditionally teleport an arbitrary state of a14N nuclear spin of one NV center onto the electron spin of another NV center, again separated by 3 m. Thus, the demonstration of spin-photon entanglement, the heralded entangle-ment generation between electron spins, and the teleportation of a qubit state make the NV center a highly promising candidate for the realization stationary qubits in quantum communication networks.
The electronic NV spin has also been part of several quantum register architectures that contain one [197,213] or more nuclear spin qubits [47,154,155,214]. Harnessing hyperfine interaction, the electron spin is thereby either utilized to initialize, manipulate and readout the nuclear spin qubits as an ancilla spin, or is part of the quantum register itself. Furthermore, the NV spin was used as ancilla to generate nuclear spin entanglement (see Sec.6.3.2).