A single NV centre in diamond Mark S. Everitt1 Simon Devitt1 Bill Munro2 Kae Nemoto1 1The National Institute of Informatics 2NTT Basic Research Laboratories December 21, 2011 @ NTT BRL 

Introduction • We have a large scale (and we hope realistic) vision of how to build a quantum machine. 

Introduction • We have a large scale (and we hope realistic) vision of how to build a quantum machine. • NV looks great! 

Introduction • We have a large scale (and we hope realistic) vision of how to build a quantum machine. • NV looks great! • Is it too good to be true? 

Introduction Well. . . yes and no. 

Some definitions Select the states of the system to encode qubits on. Subsystem State Computational State electron |+1 |1 e |0 |0 e |−1 × nucleus |↑ |1 n |↓ |0 n With the computational basis set we can move on. 

The Hamiltonian H = H0 + HS + HHF + HD , where H0 = DS2 z + Bγe Sz − Bγn Iz , HS = E 2 (S + S + + S − S −) , HHF = A Sz Iz + A ⊥ 2 (S + I − + S − I +) , HD = Ω0 cos (νt + φ) (γe Sx − γn Ix) , 

Components of the Hamiltonian H = 

Components of the Hamiltonian |0 D |+1 |−1 H = DS2 z 

Components of the Hamiltonian |0 D |+1 γeB |−1 |↓ |↑ 1 2 γnB H = DS2 z + Bγe Sz − Bγn Iz 

Components of the Hamiltonian |0 D |+1 γeB |−1 |↓ |↑ 1 2 γnB E H = DS2 z + Bγe Sz − Bγn Iz + E 2 (S + S + + S − S −) 

Components of the Hamiltonian |0 D |+1 γeB |−1 |↓ |↑ 1 2 γnB E |↓ |↑ A A ⊥ H = DS2 z + Bγe Sz − Bγn Iz + E 2 (S + S + + S − S −) + A Sz Iz + A ⊥ 2 (S + I − + S − I +) 

The interaction picture 

The interaction picture • Choose the interaction picture that removes H0 . 

The interaction picture • Choose the interaction picture that removes H0 . • All gates will be defined in this picture! 

The interaction picture • Choose the interaction picture that removes H0 . • All gates will be defined in this picture! • Provides rotating terms to describe resonances, or lack thereof. 

Strain induced splitting 

Strain induced splitting • We definitely don’t want this, since it takes us out of the computational basis. 

Strain induced splitting • We definitely don’t want this, since it takes us out of the computational basis. ¯ HS = E |1 e −1|e ei2Bγe t + |−1 e 1|e e−i2Bγe t 

Strain induced splitting • We definitely don’t want this, since it takes us out of the computational basis. ¯ HS = E |1 e −1|e ei2Bγe t + |−1 e 1|e e−i2Bγe t • So long as E 2Bγe , we can neglect it. 

Perpendicular hyperfine term −106 −104 −102 −100 −98 0 0.2 0.4 0.6 0.8 1 B [mT] PSWAP At about ∓0.1 T the system hits an “exchange resonance”, when |±1 e |0 n |0 e |1 n . Width of 0.368 mT 

Perpendicular hyperfine term • This has been shown to work as a memory1. i.e. Nucleus always initialised in |0 n . 1G. D. Fuchs et al. (June 26, 2011). “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond”. In: Nature Physics 7.10, pp. 789–793. 

Perpendicular hyperfine term • This has been shown to work as a memory1. i.e. Nucleus always initialised in |0 n . • But this isn’t really a SWAP gate. Taken alone, it’s iSWAP. 1G. D. Fuchs et al. (June 26, 2011). “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond”. In: Nature Physics 7.10, pp. 789–793. 

Perpendicular hyperfine term • This has been shown to work as a memory1. i.e. Nucleus always initialised in |0 n . • But this isn’t really a SWAP gate. Taken alone, it’s iSWAP. • It’s not alone, because there is still the parallel term. 1G. D. Fuchs et al. (June 26, 2011). “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond”. In: Nature Physics 7.10, pp. 789–793. 

Perpendicular hyperfine term • This has been shown to work as a memory1. i.e. Nucleus always initialised in |0 n . • But this isn’t really a SWAP gate. Taken alone, it’s iSWAP. • It’s not alone, because there is still the parallel term. • Errors from iSWAP will screw everything up! 1G. D. Fuchs et al. (June 26, 2011). “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond”. In: Nature Physics 7.10, pp. 789–793. 

Parallel hyperfine term 

Parallel hyperfine term • It commutes with H0 so 

Parallel hyperfine term • It commutes with H0 so ¯ H ≈ A Sz Iz 

Parallel hyperfine term • It commutes with H0 so ¯ H ≈ A Sz Iz • In the computational basis, this is locally equivalent to a controlled-Z gate. 

Parallel hyperfine term • It commutes with H0 so ¯ H ≈ A Sz Iz • In the computational basis, this is locally equivalent to a controlled-Z gate. • Bell state in ≈ 1 µs. 

Parallel hyperfine term • It commutes with H0 so ¯ H ≈ A Sz Iz • In the computational basis, this is locally equivalent to a controlled-Z gate. • Bell state in ≈ 1 µs. |ψ n eiθσz |φ e eiπσz /4 = |ψ n |φ e 

Parallel hyperfine term 0 500 1,000 1,500 2,000 0 0.2 0.4 0.6 0.8 1 τ [ns] Negativity 

The bare Hamiltonian • Without a drive, we have a controlled-Z gate to entangle qubits together. 

The bare Hamiltonian • Without a drive, we have a controlled-Z gate to entangle qubits together. • Must avoid B ≈ ∓0.1 T (iSWAP). 

The bare Hamiltonian • Without a drive, we have a controlled-Z gate to entangle qubits together. • Must avoid B ≈ ∓0.1 T (iSWAP). • That’s big, thin, no problem. 

A big difference • An electron is very light, so it has a large gyromagnetic ratio γe ≈ 28 GHz T−1. 

A big difference • An electron is very light, so it has a large gyromagnetic ratio γe ≈ 28 GHz T−1. • A nucleus is very heavy, and has a correspondingly small gyromagnetic ratio γn ≈ 4.3 MHz T−1. 

A big difference • An electron is very light, so it has a large gyromagnetic ratio γe ≈ 28 GHz T−1. • A nucleus is very heavy, and has a correspondingly small gyromagnetic ratio γn ≈ 4.3 MHz T−1. • It turns out that this makes it very difficult to manipulate the nucleus without affecting the electron. 

A big difference • An electron is very light, so it has a large gyromagnetic ratio γe ≈ 28 GHz T−1. • A nucleus is very heavy, and has a correspondingly small gyromagnetic ratio γn ≈ 4.3 MHz T−1. • It turns out that this makes it very difficult to manipulate the nucleus without affecting the electron. γe γn ≈ 6500 

A crude driving model 10−4 10−3 10−2 10−1 100 101 102 0 0.2 0.4 0.6 0.8 1 Frequency [GHz] Pexcite Ω0 = 0.0010 GHz 

Driving the electron 

Driving the electron • We can neglect the nucleus; weakly driven and far from resonance. 

Driving the electron • We can neglect the nucleus; weakly driven and far from resonance. • Must not neglect the |0 e |−1 e transition! 

Driving the electron • We can neglect the nucleus; weakly driven and far from resonance. • Must not neglect the |0 e |−1 e transition! Ω0γe 2 √ 2 |2Bγe|, 2D 

Driving the electron • We can neglect the nucleus; weakly driven and far from resonance. • Must not neglect the |0 e |−1 e transition! Ω0γe 2 √ 2 |2Bγe|, 2D • i.e. the effective drive must be much less than the detuning between these transitions. 

Driving the electron • We can neglect the nucleus; weakly driven and far from resonance. • Must not neglect the |0 e |−1 e transition! Ω0γe 2 √ 2 |2Bγe|, 2D • i.e. the effective drive must be much less than the detuning between these transitions. • D is an intrinsic limiter for minimum gate time. 

Driving the electron • In the interaction picture, if the aforementioned inequality holds 

Driving the electron • In the interaction picture, if the aforementioned inequality holds H ≈ A Sz Iz − Ω0γe 2 √ 2 |0 e 1|e + |1 e 0|e 

Driving the electron • In the interaction picture, if the aforementioned inequality holds H ≈ A Sz Iz − Ω0γe 2 √ 2 |0 e 1|e + |1 e 0|e • Obvious that we must drive much faster than the hyperfine term. Optimal gate times of ∼ 50 ns which corresponds to a drive amplitude of ∼ 10 MHz. 

Driving the electron • In the interaction picture, if the aforementioned inequality holds H ≈ A Sz Iz − Ω0γe 2 √ 2 |0 e 1|e + |1 e 0|e • Obvious that we must drive much faster than the hyperfine term. Optimal gate times of ∼ 50 ns which corresponds to a drive amplitude of ∼ 10 MHz. • The phase of the drive determines if it is a rotation about x or y. 

Driving the electron2 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 τ [µs] | 1|e +|n Re x |0 e |+ n |2 500 1000 Hz) B A D Simulation Experiment 1 50 100 200 500 Rabi Freq (MH 29 MHz 57 MHz Rabi field calculated from power 20 40 60 80 100 20 40 60 80 100 P 0 P 1 0 0 5 10 15 20 25 30 20 Power increase (dB) 109 MHz C Simulated spin dynamics 440 MHz 20 40 60 80 100 10 20 30 40 20 40 60 80 100 10 20 30 40 0 P 1 0 223 MHz 440 MHz Microwave pulse 2 4 6 8 10 2 4 6 8 10 H 1 1/H 0 P 1 0 1 440 MHz 0 5 10 15 Time (ns) 2 4 6 8 10 Pulse width (ns) 2 4 6 8 10 Pulse width (ns) P 0 on November 6, 2011 www.sciencemag.org Downloaded from B = 200 G, F = 0.998. 2G. D. Fuchs et al. (Dec. 11, 2009). “Gigahertz Dynamics of a Strongly Driven Single Quantum Spin”. In: Science 326.5959, pp. 1520–1522. 

Driving the nucleus Forgetting the electron for the moment • Similar gate times (∼ 50 ns) would require a drive amplitude of ∼ 100 GHz! • Clearly not so realistic. 

Driving the nucleus Forgetting the electron for the moment • Similar gate times (∼ 50 ns) would require a drive amplitude of ∼ 100 GHz! • Clearly not so realistic. Remembering the electron again The electron will see this as a DC field and rotate rapidly. It is possible to place the electron in a state invariant under this rotation, but we must know the state of the electron and it thus cannot be entangled to do so. Not a great solution. 

Driving the nucleus We must drive sloooooowly 

Driving the nucleus We must drive sloooooowly • To avoid perturbing the electron a weak drive is needed. 

Driving the nucleus We must drive sloooooowly • To avoid perturbing the electron a weak drive is needed. • So weak that we can’t neglect hyperfine shift. 

Driving the nucleus We must drive sloooooowly • To avoid perturbing the electron a weak drive is needed. • So weak that we can’t neglect hyperfine shift. • The frequency of the drive depends on the state of the electron. 

Driving the nucleus We must drive sloooooowly • To avoid perturbing the electron a weak drive is needed. • So weak that we can’t neglect hyperfine shift. • The frequency of the drive depends on the state of the electron. • Gate times τ > 100 µs. 

Driving the nucleus We must drive sloooooowly • To avoid perturbing the electron a weak drive is needed. • So weak that we can’t neglect hyperfine shift. • The frequency of the drive depends on the state of the electron. • Gate times τ > 100 µs. That’s a really long gate time! It is, and we have to use it. The only option we have is to minimize the number of uses. 

Driving the nucleus 0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1 τ [µs] Pflip 

What do we actually want to do? Considerations • We want to build a graph state with nuclei. • Electrons and photons are essentially ancillae. • Keep the electron polarised as much as possible to isolate the nucleus from Ising interaction. 

Simplified system A A A A A A 

Entangling the electron and nucleus sin2(θ) |0 0| |0 n eiθσx eiπσz /2 |0 e eiπσx /4 eiπσy /4 z 

Summary • Full set of gates between the spins in a single NV centre. • Manipulation of the nuclear spin is slow. . . • . . . but we only need to do it when initialising or reading out. 

Thank you for your attention. Questions?