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Repetitive Readout of a Single Electronic Spin via Quantum Logic with Nuclear Spin Ancillae

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Science  09 Oct 2009:
Vol. 326, Issue 5950, pp. 267-272
DOI: 10.1126/science.1176496

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 (36). 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 (911) and experimental (1218) 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 13C 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 13C nuclear spin has eigenstates |↑〉n1 (aligned) or |↓〉n1 (anti-aligned) with the local magnetic field. The composite electronic-nuclear system was first prepared in a fiducial state, |0〉e |↓〉n1, by using a sequence of optical, microwave, and radiofrequency (RF) pulses. Next, the electronic spin was prepared in an arbitrary state |Ψ〉e = α|0〉e + β|1〉e, where |0,1〉e denote electronic state with projected spin momentum mS (mS) = 0, 1. Before readout, we performed a sequence of gate operations resulting in the entangled electron-nuclear state |Ψ〉e |↓〉n1 → α|0〉e |↓〉n1 + β|1〉e |↑〉n1. The optical measurement process projects this state into either |0〉e |↓〉 n1 or |1〉e |↑〉n1. When optically excited, these two states fluoresce at different rates dependent on the value of mS. Within a typical measurement period, less than one photon was counted before the electron spin was repolarized to |0〉e, which indicates that the uncertainty of the electronic spin-state measurement is quite large.

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 |↓〉n1 → |0〉e |↓〉n1 and |0〉e |↑〉n1 → |1〉e |↑〉 n1, 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 |↓〉n1 |↓〉n2 → α|0〉e |↓〉n1 |↓〉n2 + β|1〉e |↑〉n1 |↑〉n2. 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 I^zn1,n2 do not evolve throughout the electronic-spin readout and constitute “good” QND observables. Although imperfect, optical NV electronic spin detection precludes an ideal QND measurement, our scheme nevertheless allows substantial improvement in the spin readout.

To implement the repetitive readout technique, we used a single NV center in diamond coupled to nearby 13C 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).

Fig. 1

Repetitive readout of an electronic spin. (A) Illustration of the NV center and its proximal 13C nuclear spins. (Inset) Energy levels of the coupled spin system formed by the NV electronic spin (e) and the first proximal 13C nuclear spin (n1). With a static magnetic field applied along the NV axis, spin n1 keeps the same quantization axis when spin e is |0〉e or |1〉e (25). When spin n1 is |↓〉n1 (or |↑〉n1), the microwave field MW1 (or MW2) resonantly drives spin e between |0〉e and |1〉e, which can implement the Cn1NOTe gate. When spin e is |1〉e, the RF field resonantly drives spin n1 between |↓〉n1 and |↑〉 n1, which can implement the CeNOTn1 gate. (B) Illustration of repetitive readout. The red down arrow represents the electronic spin state |0〉e, the red up arrow represents the electronic spin state |1〉e, the blue down arrow represents the nuclear spin state |↓〉n1, and the blue up arrow represents the nuclear spin state |↑〉n1. (C) Experimental pulse sequences that polarize spin n1 to |↓〉 n1 and spin e to |0〉e, followed by various probe operations, before fluorescence readout of spin e. (D) Measured electronic spin Rabi oscillations driven by MW1 and MW2 fields for polarized spin n1. The small wiggles for MW2 are due to off-resonant driving of the majority population in the |↓〉n1 state. The data are in agreement for finite detunings and microwave power (solid curves). The right vertical axis shows the average counts for a single readout. The left vertical axis shows the probability in the |0〉e state, obtained from the average counts (25). (E) Measured nuclear spin Rabi oscillation driven by the RF field.

To control a single nuclear spin, we chose a NV center with a well-resolved 13C hyperfine coupling near 14 MHz. The degeneracy of the |mS = ±1〉e spin states was lifted by applying a B0 = 30 gauss magnetic field along the NV axis. Under these conditions, the transitions of the electronic spin (e) within the subspace of {|0〉e, |1〉e} can be selectively addressed, conditioned on a certain nuclear state. The model Hamiltonian for this system (Fig. 1A) isH=(Δ+γeB0)S^z+γCB0I^zn1+AS^zI^zn1(1)where Δ = 2π × 2.87 GHz is the zero field splitting, A is the hyperfine interaction, and γe and γC are the electronic- and nuclear-spin gyromagnetic ratios. S^z=121^+S^z is a pseudo-spin one-half operator for the electronic spin subspace, 1^ is the identity matrix, and I^zn1 and S^z are the spin-1/2 angular momentum operators. Coherent oscillations between the |0〉e and |1〉e states, conditioned on a single proximal nuclear spin (n1) in |↓〉n1 (or |↑〉n1), were selectively driven by the microwave field MW1 (or MW2). To control nuclear spin n1, a resonantly tuned RF field to address the levels |1〉e |↓〉n1 and |1〉e |↑〉n1, which are energetically separated because of the hyperfine interaction (Fig. 1A), was used. After the initialization of spin e, spin n1 was polarized by applying MW1 and RF π pulses, which transfers the polarization from spin e to spin n1. Rabi oscillations of spin n1 were demonstrated (Fig. 1E) by preparing spin e in the |1〉e state irrespective of the state of spin n1 (by using MW1 and MW2 π-pulses) and increasing the RF pulse length. These data indicate that we can achieve spin n1 preparation (polarization) and readout with combined fidelity F ≡ 〈↓| ρ′ |↓〉 ≥ 75%, where ρ′ is the reduced density operator for spin n1.

We now describe the repetitive readout technique. As illustrated in Fig. 1D, the direct readout of electronic spin is imperfect. We define n0 and n1 as the total number of photons detected for the |0〉e and |1〉e states, respectively, during a single measurement interval. The signal is defined as the difference in average counts between the two spin states: A0 = n0n1 ≈ 0.005 (Fig. 1D). Experimentally, photon shot-noise dominated the fluctuations in the counts. Because of this shot noise and the low average count (n0 ≈ 0.016), we needed to average over N ~ 105 experimental runs in order to obtain the data in Fig. 1D.

To improve the signal, we used two spins: e and n1. Both spins were first polarized to the initial state |0〉e |↓〉n1. Next, we performed a unitary operation U(t), which prepares the superposition state |Ψ1〉 = (α|0〉e + β|1〉e) |↓〉n1 that we wanted to measure. Instead of immediately reading out the electronic spin, we use a controlled-not gate (CeNOTn1, achieved with an RF π pulse) to correlate spin e with spin n1 (Fig. 2A). We then optically readout/pumped spin e, leaving the spin system in the post-readout state: ρpost = |0〉 〈0|e ⊗ (|α|2 |↓〉 〈↓| + |β|2 |↑〉 〈↑|)n1. The state of spin n1 via the electronic spin e by performing a controlled-not operation (Cn1NOTe, achieved with an MW1 or MW2 π pulse) was then readout. This completes a one-step readout of spin n1, which can be repeated.

Fig. 2

Realization of repetitive readout. (A) Quantum circuit for M-step repetitive readout scheme assisted by spin n1. (B) Operations and pulse sequences for M = 60. The initial state |0〉e |↓〉n1 is prepared with a six-step pumping of spins e and n1. The MW1 pulse of duration t induces the Rabi rotation U(t) of spin e, whose parity information is imprinted to spin n1 with an RF π pulse (the CeNOTn1 gate). After fluorescence readout of spin e, (M − 1)–repetitive readouts of spin n1 are performed by means of MW1 or MW2 π pulses (Cn1NOTe gates) followed by fluorescence readout. The m = 1 readout is not preceded by a MW1 pulse. (C) Cumulative signal obtained from repetitive readout measurements, summed from m = 1 to M, for M = 1, 5, 10, 20, and 60 repetitions. Constant background counts are subtracted. (D) Amplitudes |Am| for Rabi oscillation measurements obtained from the mth readout normalized to the signal amplitude without repetitive readout (A0). (E) Improvement in SNR using the repetitive readout scheme. Blue curves in (D) and (E) are simulations with imperfection parameters estimated from independent experiments (25).

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 Am associated with mth readout: Sw(M)=m=1MwmAm. The average values of Am were determined experimentally by measuring the difference in average counts associated with Rabi oscillations for each mth repeated readout. The wm allowed us to weight the contribution of each repetitive readout to the overall signal. The noise corresponding to the repetitive readout signal is ΔSw(M)=m=1Mwm2σm2. Here, σm is the uncertainty of the measurement of Am. Experimentally, this uncertainty was found to be independent of m.

The signal-to-noise figure of merit is defined as SNR (M) = Sw(M)/ΔSw(M). The wm weights were chosen by noting that the larger values of Am allow us to extract more information given the fixed uncertainty of each measurement, and we should emphasize these readouts more. As proven in (25), the optimal choice of weights corresponds to wm=|Am|/σm2, and the optimized SNR is given bySNRopt(M)=m=1M|Amσm|2(2)In the ideal QND case, each repetitive readout would yield the same |Am|, and the SNR would scale with M. For our experiment, the SNR saturates (Fig. 2E) because of the decay of the normalized amplitudes (Fig. 3D). Nevertheless, the experimental data shown in Fig. 2E indicate the enhancement of SNR by more than 220%.

Fig. 3

Coherence and control of two nuclear spins. (A) The coupled spin system formed by the NV electronic spin (e) and two proximal 13C nuclear spins (n1 and n2). (Middle) Energy levels for spins n1 and n2 when spin e is in the |0〉 e state. (Right) Schematic of flip-flop between spins n1 and n2, which is electron-mediated by the second-order hopping via |1〉e |↓〉n1 |↓〉n2. (B) Measured depolarization of spins n1 and n2 under optical illumination. For the duration of optical illumination tL longer than 1 μs, the polarizations for spins n1 and n2 decay exponentially with characteristic times τn1 = 13 (1) μs and τn2 = 53 (5) μs, respectively. For tL less than 1 μs, the decay is slightly faster, which is probably associated with dynamics of the spin-fluctuator model that describe optically induced depolarization of single nuclei (22, 25). These decay times are much longer than the optical readout/pump time of the electronic spin (about 350 ns). (Inset) Log-linear plot. (C) Operations and pulse sequence to probe dynamics between spins n1 and n2. (D) Measured spin flip-flop dynamics between spins n1 and n2. For three different preparations of the initial state [|↓〉n1 |↑〉n2 (blue), |↓〉n1 and n2 in thermal state (black), and |↓〉n1 |↓〉n2 (purple)], the observed population, pn1,↑ (t), oscillates with the same period T = 117 (1) μs. These observations verify the theoretical prediction, with flip-flop coupling strength b = 4.27 (3) kHz.

In assessing this result, it is noted that various imperfections can affect the repetitive readout, which leads to the imperfect first readout |A1|/|A0| < 1, the sharp decrease in |A2|, and the subsequent exponential reduction |Am| = |A2(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 13C nuclear spins in the presence of optical illumination. For an illumination time tL longer than 1 μs, the nuclear spin polarization decays exponentially, with a characteristic time of τn1 = 13 (1) μs (Fig. 3B). Because τn1 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 13C nuclear spin decay times are >>1 ms (12, 13).] Despite the relatively long τn1, after many cycles the nuclear spin depolarizes. This degrades the repetitive optical readout for larger m, yielding the overall exponential decay in the amplitude |Am| 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 n2 in addition to proximal spin n1. As the decay time of spin n2 is longer than that of spin n1 because of a weaker interaction with spin e, the information stored in spin n2 persists after spin n1 has been depolarized under optical illumination. This remaining n2 polarization can then be transferred to spin n1 and repetitively readout again.

Control of two nuclear spins is achieved by using the strongly coupled nuclear spin n1 as a probe for the second nearby 13C nuclear spin n2, 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 13C nuclear spins with similar Zeeman energy to flip-flop and exchange spin population. Figure 3D shows that the nuclear population, pn1,↑ (τ), oscillates between pn1,↑ (0) ≈ 0.2 and pn1,↑ (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 (n2) as the two nuclei flip-flop between the states |↑〉n1 |↓〉n2 and |↓〉n1 |↑〉n2. Such an excitation exchange requires a similar Zeeman splitting for the two spins, indicating that the second nucleus is also a 13C. 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 13C 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 Hint = b(I1+I2− + I1−I2+). 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 n1, 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〉e state, in which case the flip-flop dynamics between spins n1 and n2 disappears (fig. S1). This is because spins n1 and n2 typically have very distinct hyperfine splittings that introduce a large energy difference (ΔE >> b) between |↑〉n1 |↓〉n2 and |↓〉n1 |↑〉n2 and quench the interaction. Therefore, we can implement a controlled-SWAP operation between spins n1 and n2, enabling full control over spin n2. We further observed that spin n2 has a decay time of τn2 = 53 (1) μs (Fig. 3B, inset) under optical illumination. Compared with spin n1, spin n2 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 |↓〉n1 |↓〉n2, 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 (CeNOTn1-SWAP-CeNOTn1) was used to prepare the GHZ-type state |Ψ〉 = α|0〉e |↓〉n1 |↓〉n2 + β|1〉e |↑〉n1 |↑〉n2. Next, we optically readout/pumped spin e, leaving the system in state ρpost' = |α|2 |0 ↓↓〉 〈0 ↓↓| + |β|2 |0 ↑↑〉 〈0 ↑↑|. M − 1 repetitive readouts of spin n1 were then performed in the manner described above until spin n1 was depolarized. At that point, spin n2 was still directly correlated with the first measurement of the e spin. This information can be transferred to spin n1 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).

Fig. 4

Demonstration of the two-level concatenated readout procedure. (A) Quantum circuit for concatenated M-step repetitive readout scheme assisted by both spins n1 and n2. (B) Operations and pulse sequences for M = 60. Ideally, the GHZ-like state α|0〉e |↓〉n1 |↓〉n2 + β|1〉e |↑〉n1 |↑〉n2 with the parity information of spin e imprinted on both spins n1 and n2 is created before the first readout. After the first round of M-step repetitive readout, spin n1 is depolarized, but spin n2 maintains its polarization. The spin state of spin n2 is swapped to spin n1, which is then detected during the second round of M-step repetitive readouts. (C) Normalized amplitude |Am|/|A0| obtained from the mth readout. (D) Measured improvement in the SNR by use of the double repetitive readout scheme. The blue curves in (C) and (D) are simulations with imperfection parameters estimated from independent experiments (25).

Experimentally, the “revival” in the signal amplitude |Am| 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 12C 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

  • * These authors contributed equally to this work.

  • Present address: Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA.

  • Present address: Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA.

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. 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.
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