## Abstract

Robust measurement of single quantum bits plays a key role in the realization of quantum computation and communication as well as in quantum metrology and sensing. We have implemented a method for the improved readout of single electronic spin qubits in solid-state systems. The method makes use of quantum logic operations on a system consisting of a single electronic spin and several proximal nuclear spin ancillae in order to repetitively readout the state of the electronic spin. Using coherent manipulation of a single nitrogen vacancy center in room-temperature diamond, full quantum control of an electronic-nuclear system consisting of up to three spins was achieved. We took advantage of a single nuclear-spin memory in order to obtain a 10-fold enhancement in the signal amplitude of the electronic spin readout. We also present a two-level, concatenated procedure to improve the readout by use of a pair of nuclear spin ancillae, an important step toward the realization of robust quantum information processors using electronic- and nuclear-spin qubits. Our technique can be used to improve the sensitivity and speed of spin-based nanoscale diamond magnetometers.

Efforts have recently been directed toward the manipulation of several qubits in quantum systems, ranging from isolated atoms and ions to solid-state quantum bits (*1*, *2*). These small-scale quantum systems have been successfully used for proof-of-concept demonstrations of simple quantum algorithms (*3*–*6*). In addition, they can be used for potentially important practical applications in areas such as quantum metrology (*1*). For example, techniques involving quantum logic operations on several trapped ions have been applied to develop an improved ion state readout scheme, resulting in a new class of atomic clocks (*7*, *8*). We developed a similar technique to enhance the readout of a single electronic spin in the solid state.

Our method makes use of quantum logic between a single electronic spin and nuclear spin qubits in its local environment for repetitive readout. Although such nuclear spins are generally the source of unwanted decoherence in the solid state, recent theoretical (*9*–*11*) and experimental (*12*–*18*) work has demonstrated that when properly controlled, the nuclear environment can become a very useful resource, in particular for long-term quantum memory.

Our experimental demonstration makes use of a single negatively charged nitrogen-vacancy (NV) center in diamond. The electronic ground state of this defect is an electronic spin triplet (with spin *S* = 1) and is a good candidate for a logic qubit on account of its remarkably long coherence times (*19*) and fast spin manipulation by use of microwave fields (*20*). Furthermore, the center can be optically spin-polarized and measured by combining confocal microscopy techniques with spin-selective rates of fluorescence (*12*). In practice, the NV spin readout under ambient room-temperature conditions is far from perfect. This is because laser radiation at 532 nm for readout repolarizes the electronic spin before a sufficient number of photons can be scattered for the state to be reliably determined.

Our approach is to correlate the electronic-spin logic qubit with nearby nuclear spins (*21*), which are relatively unperturbed by the optical readout, before the measurement process (*22*). Specifically, we used one or more ^{13}C nuclei (with nuclear spin *I* = 1/2) nuclear spins in the diamond lattice, coupled to the NV electronic spin via a hyperfine interaction, as memory ancillae qubits. For example, a single ^{13}C nuclear spin has eigenstates |↑〉_{n}_{1} (aligned) or |↓〉_{n}_{1} (anti-aligned) with the local magnetic field. The composite electronic-nuclear system was first prepared in a fiducial state, |0〉* _{e}* |↓〉

_{n}_{1}, by using a sequence of optical, microwave, and radiofrequency (RF) pulses. Next, the electronic spin was prepared in an arbitrary state |Ψ〉

*= α|0〉*

_{e}*+ β|1〉*

_{e}*, where |0,1〉*

_{e}*denote electronic state with projected spin momentum*

_{e}*m*(

_{S}*m*) = 0, 1. Before readout, we performed a sequence of gate operations resulting in the entangled electron-nuclear state |Ψ〉

_{S}*|↓〉*

_{e}

_{n}_{1}→ α|0〉

*|↓〉*

_{e}

_{n}_{1}+ β|1〉

*|↑〉*

_{e}

_{n}_{1}. The optical measurement process projects this state into either |0〉

*|↓〉*

_{e}

_{n}_{1}or |1〉

*|↑〉*

_{e}

_{n}_{1}. When optically excited, these two states fluoresce at different rates dependent on the value of

*m*. Within a typical measurement period, less than one photon was counted before the electron spin was repolarized to |0〉

_{S}*, which indicates that the uncertainty of the electronic spin-state measurement is quite large.*

_{e}The nuclear spin can thus reveal the former electronic state because of the correlations established before the electronic spin was reset. To achieve this repetitive readout, we performed a controlled-not operation, which mapped |0〉* _{e}* |↓〉

_{n}_{1}→ |0〉

*|↓〉*

_{e}

_{n}_{1}and |0〉

*|↑〉*

_{e}

_{n}_{1}→ |1〉

*|↑〉*

_{e}

_{n}_{1}, and repeated the optical measurement. Fluorescence counting of these two states can be added to prior measurements so as to decrease the uncertainty for electronic spin-state discrimination. If optical readout does not destroy the orientation of the nuclear spin, the uncertainty in the determination of the electronic spin can be reduced via repetitive measurements. In this way, the overall signal-to-noise of the measurement process of our logic qubit can be increased. After multiple readout cycles and many quantum logic operations, the nuclear spin orientation will finally be destroyed. However, it is possible to further improve the readout scheme by using a pair of ancillary nuclear spins and imprinting the electronic state into a Greenberger-Horne-Zeilinger (GHZ)–like state: |Ψ〉

*|↓〉*

_{e}

_{n}_{1}|↓〉

_{n}_{2}→ α|0〉

*|↓〉*

_{e}

_{n}_{1}|↓〉

_{n}_{2}+ β|1〉

*|↑〉*

_{e}

_{n}_{1}|↑〉

_{n}_{2}. In such a case, the state of the first nuclear spin after repetitive readout sequences can be periodically “refreshed” by using the information stored within the second nuclear spin. These schemes are closely related to a quantum nondemolition (QND) measurement (

*23*,

*24*) because the nuclear spin-population operators

To implement the repetitive readout technique, we used a single NV center in diamond coupled to nearby ^{13}C nuclear spins. These nuclear spins can be polarized and fully controlled and provide a robust quantum memory, even in the presence of optical radiation necessary for electronic spin-state readout (*13*, *22*). This is achieved through a combination of optical, microwave, and RF fields (Fig. 1) and is discussed in (*25*).

To control a single nuclear spin, we chose a NV center with a well-resolved ^{13}C hyperfine coupling near 14 MHz. The degeneracy of the |*m _{S}* = ±1〉

*spin states was lifted by applying a*

_{e}*B*

_{0}= 30 gauss magnetic field along the NV axis. Under these conditions, the transitions of the electronic spin (

**e**) within the subspace of {|0〉

*, |1〉*

_{e}*} can be selectively addressed, conditioned on a certain nuclear state. The model Hamiltonian for this system (Fig. 1A) is*

_{e}*A*is the hyperfine interaction, and γ

*and γ*

_{e}*are the electronic- and nuclear-spin gyromagnetic ratios.*

_{C}*and |1〉*

_{e}*states, conditioned on a single proximal nuclear spin (*

_{e}**n**

_{1}) in |↓〉

_{n}_{1}(or |↑〉

_{n}_{1}), were selectively driven by the microwave field MW1 (or MW2). To control nuclear spin

**n**

_{1}, a resonantly tuned RF field to address the levels |1〉

*|↓〉*

_{e}

_{n}_{1}and |1〉

*|↑〉*

_{e}

_{n}_{1}, which are energetically separated because of the hyperfine interaction (Fig. 1A), was used. After the initialization of spin

**e**, spin

**n**

_{1}was polarized by applying MW1 and RF π pulses, which transfers the polarization from spin

**e**to spin

**n**

_{1}. Rabi oscillations of spin

**n**

_{1}were demonstrated (Fig. 1E) by preparing spin

**e**in the |1〉

*state irrespective of the state of spin*

_{e}**n**

_{1}(by using MW1 and MW2 π-pulses) and increasing the RF pulse length. These data indicate that we can achieve spin

**n**

_{1}preparation (polarization) and readout with combined fidelity

*F*≡ 〈↓| ρ′ |↓〉 ≥ 75%, where ρ′ is the reduced density operator for spin

**n**

_{1}.

We now describe the repetitive readout technique. As illustrated in Fig. 1D, the direct readout of electronic spin is imperfect. We define *n*^{0} and *n*^{1} as the total number of photons detected for the |0〉* _{e}* and |1〉

*states, respectively, during a single measurement interval. The signal is defined as the difference in average counts between the two spin states:*

_{e}*A*

_{0}=

*n*

^{0}−

*n*

^{1}≈ 0.005 (Fig. 1D). Experimentally, photon shot-noise dominated the fluctuations in the counts. Because of this shot noise and the low average count (

*n*

^{0}≈ 0.016), we needed to average over

*N*~ 10

^{5}experimental runs in order to obtain the data in Fig. 1D.

To improve the signal, we used two spins: **e** and **n**_{1}. Both spins were first polarized to the initial state |0〉* _{e}* |↓〉

_{n}_{1}. Next, we performed a unitary operation

*U*(

*t*), which prepares the superposition state |Ψ

_{1}〉 = (α|0〉

*+ β|1〉*

_{e}*) |↓〉*

_{e}

_{n}_{1}that we wanted to measure. Instead of immediately reading out the electronic spin, we use a controlled-not gate (C

*NOT*

_{e}

_{n}_{1}, achieved with an RF π pulse) to correlate spin

**e**with spin

**n**

_{1}(Fig. 2A). We then optically readout/pumped spin

**e**, leaving the spin system in the post-readout state: ρ

_{post}= |0〉 〈0|

*⊗ (|α|*

_{e}^{2}|↓〉 〈↓| + |β|

^{2}|↑〉 〈↑|)

_{n}_{1}. The state of spin

**n**

_{1}via the electronic spin

**e**by performing a controlled-not operation (C

_{n}_{1}NOT

*, achieved with an MW1 or MW2 π pulse) was then readout. This completes a one-step readout of spin*

_{e}**n**

_{1}, which can be repeated.

As a direct illustration of the enhanced readout technique, Fig. 2C shows the accumulated signal for Rabi oscillations of the electronic spin obtained by adding *M* subsequent repetitive readouts for each experimental run. This procedure results in a 10-fold enhancement of spin signal amplitude.

In order to further quantify the performance of this technique, the noise added with each additional repetitive readout must be considered. The repetitive readout spin signal is defined as a weighted sum of difference counts *A _{m}* associated with

*m*th readout:

*A*were determined experimentally by measuring the difference in average counts associated with Rabi oscillations for each

_{m}*m*th repeated readout. The

*w*allowed us to weight the contribution of each repetitive readout to the overall signal. The noise corresponding to the repetitive readout signal is

_{m}*is the uncertainty of the measurement of*

_{m}*A*. Experimentally, this uncertainty was found to be independent of

_{m}*m*.

The signal-to-noise figure of merit is defined as *SNR* (*M*) = *S _{w}*(

*M*)/Δ

*S*(

_{w}*M*). The

*w*weights were chosen by noting that the larger values of

_{m}*A*allow us to extract more information given the fixed uncertainty of each measurement, and we should emphasize these readouts more. As proven in (

_{m}*25*), the optimal choice of weights corresponds to

*A*|, and the SNR would scale with

_{m}In assessing this result, it is noted that various imperfections can affect the repetitive readout, which leads to the imperfect first readout |*A*_{1}|/|*A*_{0}| < 1, the sharp decrease in |*A*_{2}|, and the subsequent exponential reduction |*A _{m}*| = |

*A*

_{2}|η

^{(}

^{m}^{−2)}, with η ≈ 0.95. These behaviors can be attributed to three major imperfections (

*25*): (i) errors from microwave pulses (about 7% error probability for each π pulse), (ii) imperfect optical pumping of the electronic spin after each readout; and most substantially (iii) the depolarization of the nuclear-spin memory under optical illumination.

To quantify the latter process, we studied the decay times for ^{13}C nuclear spins in the presence of optical illumination. For an illumination time *t*_{L} longer than 1 μs, the nuclear spin polarization decays exponentially, with a characteristic time of τ_{n}_{1} = 13 (1) μs (Fig. 3B). Because τ_{n}_{1} is much longer than the time for optical readout and optical spin polarization of the NV electronic spin (350 ns), repetitive readout of **e** is possible. [In the absence of optical illumination, the ^{13}C nuclear spin decay times are >>1 ms (*12*, *13*).] Despite the relatively long τ_{n}_{1}, after many cycles the nuclear spin depolarizes. This degrades the repetitive optical readout for larger *m*, yielding the overall exponential decay in the amplitude |*A _{m}*| with increasing

*m*(

*25*).

As an indication of how this limit can be circumvented, the use of two ancillary nuclear spins was considered. The state of spin **e** may be correlated with a more distant spin **n**_{2} in addition to proximal spin **n**_{1}. As the decay time of spin **n**_{2} is longer than that of spin **n**_{1} because of a weaker interaction with spin **e**, the information stored in spin **n**_{2} persists after spin **n**_{1} has been depolarized under optical illumination. This remaining **n**_{2} polarization can then be transferred to spin **n**_{1} and repetitively readout again.

Control of two nuclear spins is achieved by using the strongly coupled nuclear spin **n**_{1} as a probe for the second nearby ^{13}C nuclear spin **n**_{2}, which cannot be directly observed via the NV center. By placing the NV electronic spin in |0〉* _{e}* state, the hyperfine coupling is removed. This enables proximal

^{13}C nuclear spins with similar Zeeman energy to flip-flop and exchange spin population. Figure 3D shows that the nuclear population,

*p*

_{n}_{1,↑}(τ), oscillates between

*p*

_{n}_{1,↑}(0) ≈ 0.2 and

*p*

_{n}_{1,↑}(

*T*/2) ≈ 0.5 with a period of

*T*= 117 (1) μs (Fig. 3, A and C). The relatively high contrast of these oscillations suggests an interaction with a second nuclear spin (

**n**

_{2}) as the two nuclei flip-flop between the states |↑〉

_{n}_{1}|↓〉

_{n}_{2}and |↓〉

_{n}_{1}|↑〉

_{n}_{2}. Such an excitation exchange requires a similar Zeeman splitting for the two spins, indicating that the second nucleus is also a

^{13}C. The nuclear spin-spin interaction strength determined by our measurements,

*b*= π/

*T*= 4.27 (3) kHz, is several times that of a bare dipolar coupling (2 kHz for two

^{13}C nuclei separated by the nearest neighbor distance, 1.54 Ǻ), signifying that their interaction is mediated by the NV electronic spin (Fig. 3A, inset) (

*25*), which is described by the interaction hamiltonian

*H*

_{int}=

*b*(

*I*

_{1+}

*I*

_{2−}+

*I*

_{1−}

*I*

_{2+}). This interaction can be used to effectively control the state of the second nucleus and of the entire three-spin system. Specifically, a half period of nuclear spin oscillation,

*T*/2, constitutes a SWAP (

*1*) operation between the two nuclear spins. This operation can be used, for example, to polarize the second nuclear spin (Fig. 3, C and D). In addition, by modifying the initial state of spin

**n**

_{1}, we can prepare the initial state of the two nuclei in any of the four possible configurations: ↑↑, ↑↓, ↓↑, or ↓↓ (

*25*). Further control is provided by putting the electronic spin into the |1〉

*state, in which case the flip-flop dynamics between spins*

_{e}**n**

_{1}and

**n**

_{2}disappears (fig. S1). This is because spins

**n**

_{1}and

**n**

_{2}typically have very distinct hyperfine splittings that introduce a large energy difference (Δ

*E*>>

*b*) between |↑〉

_{n}_{1}|↓〉

_{n}_{2}and |↓〉

_{n}_{1}|↑〉

_{n}_{2}and quench the interaction. Therefore, we can implement a controlled-SWAP operation between spins

**n**

_{1}and

**n**

_{2}, enabling full control over spin

**n**

_{2}. We further observed that spin

**n**

_{2}has a decay time of τ

_{n}_{2}= 53 (1) μs (Fig. 3B, inset) under optical illumination. Compared with spin

**n**

_{1}, spin

**n**

_{2}is less perturbed by the optical transitions between different electronic states because it has a weaker hyperfine coupling to the electron (

*22*).

To demonstrate concatenated readout experimentally, both nuclear spins were initialized in the state |↓〉_{n}_{1} |↓〉_{n}_{2}, and a single NV electronic spin that we would like to detect was prepared in a superposition state (α|0〉 + β|1〉)* _{e}*. First, the operation (C

*NOT*

_{e}

_{n}_{1}-SWAP-C

*NOT*

_{e}

_{n}_{1}) was used to prepare the GHZ-type state |Ψ〉 = α|0〉

*|↓〉*

_{e}

_{n}_{1}|↓〉

_{n}_{2}+ β|1〉

*|↑〉*

_{e}

_{n}_{1}|↑〉

_{n}_{2}. Next, we optically readout/pumped spin

**e**, leaving the system in state

^{2}|0 ↓↓〉 〈0 ↓↓| + |β|

^{2}|0 ↑↑〉 〈0 ↑↑|.

*M*− 1 repetitive readouts of spin

**n**

_{1}were then performed in the manner described above until spin

**n**

_{1}was depolarized. At that point, spin

**n**

_{2}was still directly correlated with the first measurement of the

**e**spin. This information can be transferred to spin

**n**

_{1}by means of a nuclear SWAP gate. Thus, the parity information can be measured again by performing a second round of

*M*-step repetitive readout. These operations are summarized in the quantum circuit (Fig. 4A) and pulse sequences (Fig. 4B).

Experimentally, the “revival” in the signal amplitude |*A _{m}*| after the SWAP was demonstrated (Fig. 4C), which led to an associated jump in the SNR curve (Fig. 4D) for

*M*′ = 61. This shows that the second nuclear spin can be used to further enhance the readout efficiency. Although ideally the repetitive readout scheme assisted by two nuclear spins should improve the absolute SNR more than a single nuclear spin, in the present experimental realization this is not yet so because more errors are accumulated for the two-nuclear-spin scheme because of initialization and pulse imperfections. These errors reduce the optical signal amplitudes for the readout assisted by two nuclear spins, compromising the overall SNR improvement. Nevertheless, the experiments clearly demonstrate that it is in principle possible to further boost the relative SNR by using additional nuclear spins.

Although we have demonstrated an enhancement for coherent Rabi oscillations, any set of pulses acting on the electronic spin (such as a spin echo sequence) can be implemented. This should have immediate applications to NV-based nanomagnetometry (*26*, *27*). Because the duration of the entire repetitive readout sequence (~150 μs in Fig. 2B) is shorter than the typical echo duration in pure diamond, SNR improvements directly translate into enhanced sensitivity and increased speed of nanoscale diamond magnetometer (*28*). This may have important applications in probing time-varying processes in biophysical systems. The repetitive readout can also be used to achieve single-shot readout of NV centers. At room temperature, with optimized collection efficiency, an improvement in spin signal on the order of a few hundred is needed to achieve single-shot readout. Potentially, this improvement can be obtained by using nuclei more robust to optical depolarization, such as the nitrogen nuclear spin of the NV center in isotopically pure ^{12}C diamond (*19*) and by using advanced control techniques (*29*, *30*) to suppress the imperfections from microwave pulses. Furthermore, resonant optical excitations [wavelength λ ≈ 637 nm] can be used for NV centers at cryogenic temperatures. Here, the resolved spin structure of optical excited states (*20*, *31*, *32*) can be exploited to readout the electronic spin much more efficiently with reduced perturbation to the nuclear spin (*22*). Under these conditions, a 10-fold spin signal improvement may be sufficient to enable single-shot readout of the NV electronic spin. In turn, this can be used to perform robust, adaptive QND measurements of nuclear-spin qubits, which will be of direct use for distributed quantum networks (*13*, *21*). Our experiments demonstrate that manipulation of several nuclear-spin ancillae surrounding a central electronic spin can be used to implement useful quantum algorithms in solid-state systems.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/1176496/DC1

Materials and Methods

Figs. S1 to S4

References

## References and Notes

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- We thank P. Cappellaro, L. Childress, J. Doyle, M. V. G. Dutt, J. MacArthur, A. Sorenson, P. Stanwix, E. Togan, and A. Trifonov for many stimulating discussions and experimental help. This work was supported by the Defense Advanced Research Projects Agency, NSF, the Packard Foundation, and the Pappalardo Fellowship. The content of the information does not necessarily reflect the position or the policy of the U.S. Government, and no official endorsement should be inferred.