Cavity Enhanced Spin Measurement of an NV Centre in Diamond

We propose a high efficiency high fidelity measurement of the ground state spin of a single NV center in diamond, using the effects of cavity quantum electrodynamics. The scheme we propose is based in the one dimensional atom or Purcell regime, removing the need for high Q cavities that are challenging to fabricate. The ground state of the NV center consists of three spin levels $^{3}A_{(m=0)}$ and $^{3}A_{(m=\pm1)}$ (the $\pm1$ states are near degenerate in zero field). These two states can undergo transitions to the excited ($^{3}E$) state, with an energy difference of $\approx7-10$ $\mu$eV between the two. By choosing the correct Q factor, this small detuning between the two transitions results in a dramatic change in the intensity of reflected light. We show the change in reflected intensity can allow us to read out the ground state spin using a low intensity laser with an error rate of $\approx5.5\times10^{-3}$, when realistic cavity and experimental parameters are considered. Since very low levels of light are used to probe the state of the spin we limit the number of florescence cycles, thereby limiting the non spin preserving transitions through the intermediate singlet state $^{1}A$.

Addressing single spins is an important route to quantum computation [1].The long decoherence times of spins such as trapped atoms [2,3], ions [4], or charged quantum dots [5], make them ideal candidates for storing and processing quantum information.There are many schemes for using internal spin states in all these architectures [4,6,7], resulting in the demonstration of fundamental quantum logic gates [8,9,10].NV centers in diamond have long decoherence times even at room temperature [11] making them another promising candidate for performing quantum information tasks.Several experiments have shown the manipulation of the ground state spin of a diamond NV center using optically detected magnetic resonance techniques (ODMR) [11,12].This has further led to the coherent control of single 13 C nuclear spins and quantum logic operations [13,14].The main problem in using ODMR is that the detection step involves observing fluorescence cycles from the NV center, which has a probability of destroying the spin memory.Since the energy level transitions of the NV center are not polarization sensitive, we cannot use Faraday rotations to perform quantum non demolition measurements of the spin state as was shown for charged quantum dots [15,16].The scheme we propose here is similar to the ODMR scheme, however by the introduction of a low Q cavity we vastly reduce the number of photons required to probe the spin state, therefore keeping the disturbance of the ground state spin to a minimum and not destroying the spin memory.
If we consider the energy level structure of the NV center in figure 1, the ground state is a spin triplet split by 2.88 GHz due to spin-spin interactions [18].The excited state is a triplet split by spin-spin interactions, but with the further addition of spin-orbit coupling [20].
Recent experimental evidence [17] has uncovered this excited state structure (figure 1).The net effect of spin-spin and spin-orbit interactions is to create a detuning ≈ 1.4 GHz (6 µeV) between the transition from the 3 A (m=0) and the 3 A (m=+1) to 3 E state or ≈ 2.5 GHz(10 µeV) for the 3 A (m=−1) It is exactly this detuning that we plan to exploit to measure the ground state spin of the defect.The energy level structure is not simply a ground and excited triplet state, there also exists an intermediate singlet state 1 A. There is a probability that the 3 E state can decay to this state, with different rates depending on the spin state.For the 3 E m=±1 states (transitions 6,7) both theoretical predictions and experimental results suggest the decay rate is around 0.4 × 1/τ [20,21,22], where τ is the spontaneous emission lifetime (≈ 13 ns).For the 3 E m=0 state (transition 5) theoretically the rate of decay to the singlet should be zero [20], however experimental observations have shown the rate to be 2 ground and excited state splitting [17,18].The defect has zero phonon line at 637 nm, with width of order MHz at low temperatures [19] ≈ 10 −4 × 1/τ [21].Since the 1 A singlet state decays preferentially to the 3 E m=0 state [20,23] (transition 8), then it is clear from the rates above that broadband excitation leads to spin polarization in the spin zero ground state [24].Since transition 8 is non radiative then there will be a dark period in the fluorescence when it becomes populated, and as the decay rate from 3 E m=±1 to the singlet state is much larger than from 3 E m=0 , the change in intensity measures the spin state [21].Clearly using fluorescence intensity to detect the spin state has a probability to flip the spin, therefore it would seem necessary for a scheme to suppress this.However spin flip transitions are essential to initialize the system.Thus a compromise is required between the perfectly cyclic spin preserving transitions required for readout, and the Λ type spin flip transition required for initialization.
We consider the structure in figure 2, which can be modeled as a single sided cavity, where κ is the cavity decay rate (side leakage), η is the coupling of the cavity to external modes, g is the NV center cavity coupling rate and γ is the NV dipole decay rate.We can write down the Heisenberg equations of motion for this structure as [25]: where ω a and ω c are the atomic transition (σ − ) and the intracavity photon (annihilation operator â) frequencies respectively.σz represents a Pauli Z operator on the atomic state and measures the population inversion.If we now combine this with the input output relation for this cavity: then we can find the reflection coefficient for light input into the cavity via b in : where we have set σ z = −1 as is appropriate for the weak excitation limit.At low temperature the zero phonon linewidth is 0.1 µeV [19], we set g = 0.03 meV as appropriate for a cavity mode volume of 0.02 µm  13 ns lifetime.It is desirable for the cavity to be critically coupled to the input output so we will set η to be 50 times faster than κ.
Figure 3 shows the effect of varying κ and η on the reflection coefficient.When κ and η are very low we are in the strong coupling regime with g >> κ, η, γ, here we can clearly see the two Rabi split dressed atom-cavity states in figure 3(d), and all of the light resonant with the cavity mode is reflected.As we increase κ and η we are no longer able to resolve the two states and cross over into the one dimensional atom regime η + κ > g > γ, where in figure 3(b) there is a small peak in reflectivity on resonance, a result of quantum interference [26].
In this one dimensional atom or Purcell regime the damping of the atomic transition (zero phonon line) plays an increasingly dissipative role as a larger proportion of the radiative decay is into non cavity modes.The result of this is a dip in reflectance clearly visible in figure 3(a).
In figure 4 we have plotted the reflectance at the cavity-atom resonance against η as it is the dominant decay channel for the cavity mode.The amount of reflected light drops to zero at a value of η = 4g 2 /γ which corresponds to the transition from the one dimensional atom to the weak coupling regime.Thus at this point all of the light absorbed by the NV center is emitted into non cavity modes.After this turn over point we are in the weak coupling regime, where the NV center has progressively less effect on the dynamics of the system as it so weakly coupled.It is near this transition region that we wish to operate where the narrow feature in the reflection spectrum caused by the zero phonon linewidth dominates (figure 5).
If we consider figure 5, then if we set the cavity to be resonant with the 3 A (m=0) to excited state transition then the reflected intensity for resonant exitation becomes: Almost none of the input light will be reflected when the NV center is in the spin m = 0 ground state.However if it is in the spin m = +1(m = −1) state then the 7 µeV(10 µeV) detuning means that the NV center is effectively uncoupled giving a reflected intensity of: Nearly all of the input light will be reflected.figure 5 corresponds to a total Q factor Q tot = ω/(κ + η) ≈ 55.Since we have set η to be 50 times greater than κ, this means the photonic crystal cavity before coupling needs to have Q = ω/κ ≈ 3000.This is much lower than the cavity Q factor of 300000 that would be required to have the cavity linewidth narrow enough to resolve the two transitions, which when coupled to a waveguide in the same way as here would need to exceed 10 7 .It is possible to further reduce the requirements on the cavity Q factor by reducing the the ratio of η to κ. However the result of this is a reduction in the intensity contrast between the two spin states as a larger proportion of the light confined in the cavity leaks out of the side.The intensity contrast which measures the spin is not influenced by total Q factor, the optimal value being Q tot ≈ 55, the contrast is only influenced by the ratio of η to κ.It is desirable to have this contrast at a maximum in order to minimize errors in state identification.
There are several benefits to this scheme.The first is the obvious increase in collection efficiency of the photons, making low intensity measurements possible.Since less photons are required to probe the spin state there are less fluorescence cycles therefore a reduced probability of a spin flip transition.Additionally as the cavity is resonant with the m = 0 transition by probing with narrow band light then we never excite the spin ±1 transitions which have a higher probability to spin-flip, hence the system is optimized for spin preserving transitions.However if we pump with a broad band laser source we can easily spin polarize the ground state to initialize the system.Since we are in the low Q regime then the Purcell factor is small, F p ≈ 4 for a system with the parameters listed above, thus the rate of spontaneous emission(SE) into the cavity is not significantly modified.If we were operating in the high Q or strong coupling regime then the SE rate into the cavity would be much larger, and the decay rate to the 1 A singlet state would remain unmodified.Therefore with the probability of a spin flip transition greatly reduced, the system would not be simultaneously optimized for readout and initialization.
In order to make the scheme experimentally relevant the limitations of current detector technology must be included.If we consider an overall efficiency of 33% then if we input 60 photons we can expect to detect 20 with unit reflectivity.If the spin is in the ) it is reasonable to expect 18 photons to be detected.If the spin is in the m = 0 state (|r(ω)| 2 < 1%) we may expect 1 photon to be detected.If we set a detection threshold of 6 photons the error in the measurement can then be found from the probability of detecting > 6 photons when we expect 1 and the probability of detecting < 6 photons when we expect 18, giving an error rate of ≈ 1.5 × 10 −3 (assuming poissonian distribution).Standard silicon avalanche photo diodes have a dead time of 50 ns which means that it will take 3 µs to carry out a measurement with 18 detected photons(running at one third of the detector saturation count rate).Since the longest observed spin coherence time of an NV center is 600 µs [27] this introduces a further error rate of 5.5×10 −3 .There are also errors associated with saturation of the NV center.However as the detector dead time is much larger than the modified spontaneous emission lifetime then these are negligible.
Finally there is also an error associated with decay from the 3 E m=0 to the 3 E m=±1 state via the 1 A singlet state which for 60 photons is < 10 −3 .Thus the total error rate is ≈ 7 × 10 −3 .
Simulations of photonic crystals in diamond have shown Q factors larger than 10 6 for mode volumes around 0.02 µm 3 , or larger than 10 5 for mode volumes around 0.008 µm 3 [28].These values are significantly more demanding than those required for this scheme, particularly the Q factor.Experimental evidence suggests that the actual Q factors will be much lower than those simulated.Cavities fabricated by Wang et.al.[29], showed more than a factor of 10 shortfall in the experimental Q factor compared to the simulated, attributed to defects in the nanocrystalline structure.Nevertheless their measured Q factor of 585 would allow a ratio of η ≈ 10κ, in order to have an overall Q factor of 55.This would result in a 65% contrast between the two spin states, increasing the error rate to ≈ 2 × 10 −2 .Theoretical considerations of the absorption in nanocrystalline diamond have predicted a reduction in Q factor from a value of 66300 to a value of around 1350 for a cavity of mode volume 0.02 µm 3 [30].For our purposes this would result in a contrast of 85%, where the error rate would be ≈ 1×10 −2 .So the scheme is clearly robust and can cope with experimental imperfections.
The main difficulty with this scheme, which is true for all schemes, is the positioning of the NV center at the field maximum.If the precision is poor then this can have a detrimental effect on the coupling rate g.This in turn reduces the intensity contrast between the two spin states, which is sensitive to the value of g compared to the zero phonon linewidth γ.There is promise that ion implantation in single crystal diamond could hold the key to fabricating suitable devices [31], the precision of implantation is currently on the nanometer scale [31,32].
The use of single crystal diamond would dramatically reduce the absorption losses caused by defects, so experimental Q factors should be closer to the theoretical predictions.
In conclusion we have proposed an efficient low error measurement of the ground state spin of an NV center.For the realistic parameters proposed here we can achieve error rates of around 7 × 10 −3 .The setup can easily switch between initialization and readout by switching from a broad to narrow band laser source.Low error readout requires modest Q factors, and even with current limitations of photonic crystal cavities the error rate could be as low as 2 × 10 −2 .Work needs to be done on the design and fabrication of photonic crystal cavities coupled to waveguides, particularly in single crystal diamond to minimize absorption losses.We also note that we can measure the spin with a single photon with 92% fidelity (assuming ideal detection), where fidelity is simply the contrast between the two spin states.Hence with some modifications the ideas here could be used to remotely entangle two spatially separated NV centers embedded in cavities, which is a subject for further study.
We acknowledge support from the UK EPSRC (QIP IRC), and the European Commission under projects IST-015848-QAP and IST-034368 EQUIND and Nanoscience ERA project NEDQIT.J.G.R. is supported by a Royal society Wolfson Merit award.
[1] M A Nielsen and I L Chuang.Quantum Computation and Quantum Information.Cambridge University Press, Cambridge, 2000.p 91.

FIG. 1 :
FIG. 1: Energy level diagram of the NV center in diamond showing the experimentally determined FIG. 2: Schematic diagram of an NV center embedded in a photonic crystal cavity with cavity decay rate κ coupled to a photonic crystal waveguide at a rate η.

FIG. 4 :
FIG.4: Calculated reflectance against cavity damping at ω c = ω a and ω c − ω = 0, where we have ignored κ as η = 50κ.The relevant atom-cavity coupling regimes are labeled, the strong coupling regime occurs when η is very low and |r(ω)| ≈ 1.The crossover from the Purcell regime to the weak coupling regime occurs at a value of 36meV as predicted.