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Multipartite Entanglement Among Single Spins in Diamond

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Science  06 Jun 2008:
Vol. 320, Issue 5881, pp. 1326-1329
DOI: 10.1126/science.1157233

Abstract

Robust entanglement at room temperature is a necessary requirement for practical applications in quantum technology. We demonstrate the creation of bipartite- and tripartite-entangled quantum states in a small quantum register consisting of individual 13C nuclei in a diamond lattice. Individual nuclear spins are controlled via their hyperfine coupling to a single electron at a nitrogen-vacancy defect center. Quantum correlations are of high quality and persist on a millisecond time scale even at room temperature, which is adequate for sophisticated quantum operations.

Schrödinger coined the term “entanglement” to mean a peculiar mutual quantum interaction in which the properties of two or more physical objects can be correlated, even when separated. Since then, the generation and retrieval of entanglement among several qubits have become of fundamental importance in quantum science and technology. Quantum teleportation error correction, computation, and communication all benefit from (or require) entanglement. One current challenge for the field of quantum information processing has been to engineer a sufficiently large and complex controllable system in which questions related to entanglement can be precisely explored. Hence, proving entanglement among an increasing number of qubits is typically a benchmark for physical systems, in demonstrating their relevance to engineer quantum states. On the level of single quantum systems, entanglement has been proven for photons (1), ions (2, 3), atoms (4), and superconductors (5). All solid-state qubit devices require low temperature to achieve sufficiently long entanglement lifetime. Whereas single electron spins can be accessed by charge transport (6) or optically (7), nuclei are more promising for quantum engineering because of their long coherence times, even under ambient conditions. Because of their weak interaction with the environment, nuclear spins are not directly affected by (for example) lattice phonons, which are a prominent source for dephasing in most solid-state systems. In ensemble studies, (pseudo)entanglement among nuclei has been demonstrated (810), but this has not been confirmed for single nuclear spins so far.

For the present experiments, 13C nuclei coupled to a single nitrogen-vacancy (NV) defect center in diamond (Fig. 1A) were chosen. This system allows for high-fidelity polarization and detection of single electron and nuclear spin states, even under ambient conditions (1116). The NV center's electron spin (S = 1) exhibits extraordinarily slow relaxation, with a longitudinal relaxation time T1 (i.e., the time for spontaneous transition between pure states) on the order of milliseconds (17). The phase memory time T2 is found to be around 0.6 ms (18). Hence, this defect has been identified as a prominent candidate for engineering quantum states and quantum information processing (1922), as well as for high-resolution magnetometry (23). Scalability toward larger-scale quantum registers was proposed, for example, through optical interactions between NV centers. Alternatively, probabilistic entanglement based on photonic channels can provide efficient scaling up (24).

Fig. 1.

(A) Atomic structure of the NV center. The nitrogen atom, vacancy, and three nearest-neighbor carbon atoms are shown. Two of the carbon atoms are 13C isotopes with nuclear spin 1/2. (B) Energy-level scheme of electronic ground state of the NV center. Two of three hyperfine split electron spin sublevels (mS = 0,–1) are depicted. Allowed transitions are shown by solid arrows (blue, electron spins; orange, nuclear spins). Zero and double quantum transitions are shown as dashed arrows. For further information, see (15). (C) ODMR spectrum (red curve) showing the mS = 0↔–1 transition. The simulated spectrum (blue curve) accounts for the hyperfine splitting of a single electron spin with two nearest-neighbor 13C atoms. a.u., arbitrary units.

The NV center is a point defect (Fig. 1A): That is, the electron spin is mostly localized at the defect site. However, about 11% of its electron spin density is distributed over the nearest-neighbor carbon atoms (25), mostly those at the dangling bonds caused by the vacancy. As a result, substantial hyperfine and dipolar coupling are detectable for single nuclei localized close to the defect (26, 27). Here, we use such coupling (Fig. 1B) to effectively control two nuclear spins on an individual basis, and by using this technique, we are able to demonstrate entanglement of two 13C nuclear spins (N1, N2) in the first coordination shell of the vacancy. All four maximally entangled states, namely the Bell states Math(1) Math(2) are generated, where “0” and “1” denote the two possible nuclear spin orientations (Math) (Fig. 1B). These entangled states cannot be achieved by independently bringing the spins into an individual superposition state: Instead, they are global states of the two spins. To demonstrate even tripartite entanglement, we take the NV center's electron spin (E) into account and generate maximally entangled states such as the Greenberger-Horne-Zeilinger (GHZ) states, as well as the so-called W state Math(3) Math(4) where φ and θ are arbitrary phases. Again, “0” and “1” denote the two possible spin states of all three spins (Math).

The measured optically detected magnetic resonance (ODMR) spectrum of the electron spin of a single center is shown in Fig. 1C. Four lines separated by hyperfine splittings are observed in each mS = 0↔–1 and 0↔1 transition (15). The Bell states Φ± and Ψ± are prepared as follows (Fig. 2A). After initialization by optical pumping, the system is set in the mS = 0 state. The nuclear spin states, however, are undetermined. To initialize a specific nuclear spin starting state, we applied a transition-selective microwave (MW) π pulse and transferred|E N1N2〉 = |000〉 into |100〉. If the system was in |000〉 after the laser irradiation, a pure |00〉 (= |100〉) state is created. If the system was in another state after laser irradiation, a subsequent MW excitation to |00〉 does not occur, and no observable signal is visible. To increase the success rate of initialization, one can, in principle, apply additional preparation steps (27).

Fig. 2.

(A) Pulse sequence for Bell state generation (Φ) among two nuclear spins. Spin-selective pulses are represented by squares, operating on a target qubit. Vertical lines represent logical connections. The control qubit state |1〉 and the state |0〉 are displayed as closed and open circles, respectively. For example, an open circle indicates that the pulse is applied to the target qubit if the quantum state of the controlling qubit is |0〉. Bell states evolve during time τ followed by state tomography. During six different transfer steps, the six possible coherences among the four nuclear spin states are unitarily mapped onto nuclear spin N1. Each coherence is completely analyzed by performing two nutations (nut.) with two 90° phase-shifted rf fields. The results are the density matrix elements. (B) Ramsey fringes of Bell states: (i) Φ+, (ii) Φ+ and Φ, (iii) Ψ+, and (iv) Ψ+ and Ψ. The frequency offsets of rf1 (Δω1), rf2 (Δω2), and fitted curve (ωfit) are: (i) Δω1 = 0.5, Δω2 = 0.5, and ωfit = 1.0 MHz ≈ Δω1+Δω2; (ii)Δω1 = 0, Δω2 = 0.5, and ωfit = 0.5 MHz ≈ Δω1+Δω2; (iii) Δω1 = –0.3, Δω2 = 1.0, and ωfit = 1.3 MHz ≈ Δω2–Δω1; and (iv) Δω1 = 0, Δω2 = 0.4, and ωfit = 0.4 MHz ≈ Δω2–Δω1. (C) Density matrix reconstruction (real part) of states Φ and Ψ+. Imaginary parts are found in (15). The high absolute values of the main off-diagonal elements (Φ: |00〉〈11|, |11〉〈00|; Ψ+: |10〉〈01|, |01〉〈10|) prove that these are, in fact, entangled states and not just incoherent mixtures (otherwise, all off-diagonal elements would be zero). (D) Coherence time of Bell states as compared with the relaxation time of the electron T1.

Generation of Bell states proceeds in two steps. At first, a coherent superposition of states |01〉 and |00〉 is generated with a π/2 pulse on one radio frequency (rf) transition, which yields Math. By the application of a frequency-selective π pulse in resonance with the |01〉 and |11〉 transition, eventually Math is formed. The remaining Bell states are prepared in a similar fashion (28). Having arranged the Bell states, we proved their generation via Ramsey fringes: that is, letting the Bell state evolve freely for a certain time τ and analyzing its phase afterwards (15). Phases of different Bell states evolve differently under rotation around the quantization axis (28). Φ and Ψ are characterized by an angular frequency ΔωΦ, Ψ = Δω1+Δω2 and Δω2–Δω1, respectively (Fig. 2B), where Δω1 and Δω2 are detunings from resonance of the two radio frequency fields rf1 and rf2 involved in generation and detection of the respective Bell states. The “+” and “–” states are distinguishable by their phase with respect to the phase of the driving rf field (Fig. 2B).

A standard procedure to quantify the amount of entanglement is density matrix tomography. By measuring the probabilities to find the nuclear spins in a certain eigenstate and revealing coherent phases among them (e.g., |00〉 + |11〉), researchers have performed density matrix tomography for all Bell states (26) (Fig. 2A). Tomography results are plotted in Fig. 2C. For brevity, only the real part of Φ and Ψ+ is shown. Concurrence and the negative eigenvalue of the partial transpose were calculated (Table 1). To compare the measured density matrix ρ with the ideally expected one σ, we estimated the fidelity F = {tr[(σ1/2ρσ1/2)1/2]}2 [F =tr(σρ) for pure states σ], where tr is the trace of a matrix. As demonstrated in Table 1, especially for Φ±, the respective values get close to ideal. It has been shown that a state ρ of two qubits is entangled if and only if its partial transpose ρpt has a negative eigenvalue (29, 30). As shown in Table 1, this is indeed the case for all four states. We also have measured the coherence decay time T2 of Φ and Ψ (Fig. 2D). Both types of Bell states decay on a time scale of 3 to 5 ms, which is similar to the relaxation time of the electron spin (T1 = 6 ms).

Table 1.

Fidelity and entanglement measures for Bell, GHZ, and W states. Dashes indicate that the respective coherence measures were not calculated.

View this table:

After entangling two nuclear spins, the next step is to generate a tripartite entangled state via the electron spin of the NV center itself as the third qubit. The easiest tripartite entangled state to generate is the GHZ state. For its generation, we start off with a nuclear Φ± state and apply a nuclear spin state–selective π pulse on the electron transition (Fig. 3A). For W state generation and further details of state preparation and readout, see (15). Tomography of these tripartite entangled states can be performed in a fashion similar to that of the Bell states. Note that these three spin coherences are directly observable in our spin resonance experiment. In the present work, we restricted our measurements to the main (ideally nonzero) density matrix entries of the respective entangled states (Figs. 3B and 4B). It can be shown that these elements are sufficient to calculate the fidelity F = tr(σρ) (15) (Table 1).

Fig. 3.

(A) Pulse sequence for generation and detection of the GHZ state. The preparation sequence is similar to that of Bell states, followed by a selective MW pulse. For readout, a tomography is performed for the ideally nonzero density matrix elements. (B) Ramsey fringes of the GHZ coherence quantified by showing the sums of the respective off-diagonal elements of the density matrix (ρ1,88,1 = |000〉〈111| +|111〉〈000|). The decay is fitted to exp[–(τ/T2*)2], with T2* = 1.3 ± 0.2 μs. Error bars indicate the accuracy of the measured data points. (C) Tomogram of GHZ state's main density matrix entries. Unmeasured elements are discarded. The top graph shows the density matrix for evolution time τ = 0 μs, whereas the bottom graph shows the decay of off-diagonal elements for τ = 2.4 μs.

Fig. 4.

(A) Ramsey fringes of all three coherences [sum of the corresponding off-diagonal elements: for example, ρ5,22,5 = |011〉〈110| + |110〉〈011|]. The respective off-diagonal elements are marked with a correspondingly colored arrow in (B). Only coherence between the two energy eigenstates containing the same electron spin orientation (ρ3,2 = |101〉〈110|, ρ2,3 = |110〉〈101|) shows no decay during the observation time. The others decay within 2 μs. The decay is fitted to exp[–(τ/T2*)2], with T2* = 1.2 ± 0.2 μs for the circles and T2* = 1.6 ± 0.3 μs for the triangles. The straight line for the squares is a guide to the eye. Error bars indicate the accuracy of the measured data points. (B) Tomograms of the main entries of the W state. The top and bottom graphs display the density matrix right after preparation and after 4.4 μs, respectively. Gray density matrix elements have not been measured.

Because, for both states, an electron spin is now involved in the entangled states, one would expect the coherences to decay much faster than those where only nuclear spins were involved. Indeed, by measuring Ramsey fringes of GHZ coherence after a waiting time of 2 μs, we find that only an incoherent mixture of 111〉 and |000〉 is left (Fig. 3, B and C). The Ramsey fringe decay time is determined by inhomogeneities caused by slow spectral diffusion. Refocusing such inhomogeneities or decoupling these interactions will greatly lengthen the respective coherence time. Interaction with other spin impurities limits T1 and T2 of the NV center's electron spin [highest reported values: T1 = 380 s, (31); T2 = 350 μs, (18)]. Thus, the GHZ state prepared here would be an even more robust resource for quantum state engineering in a purer lattice environment.

The aforementioned GHZ decoherence is only expected to be due to the electron spin. This is reflected in the dephasing behavior of the W state. After the electron spin coherence has decayed, the remaining W state is written as an incoherent mixture of states |0〉|11〉 and |1〉(|10〉 + |01〉). Thus, for the electron spin measured to be in state |1〉, nuclear spin entanglement is expected to be found, namely the Ψ+ nuclear spin Bell state. Observation of Ramsey fringes of all W state coherences indeed shows the expected behavior. The two coherences involving electron spin states decay within 2 μs, whereas the coherence with mainly nuclear spin character (|1〉|Ψ+〉) persists on this time scale (Fig. 4, A and B). Nuclear spin coherence outlasts electron spin decoherence.

Given the long decoherence times found for nuclear spin entangled states, there is ample opportunity to create even higher entangled states: for example, by including the nitrogen nuclear spin that is present in the defect but which was not used here. Large-scale networks may be achieved by entangling distant defects through emitted photons. Proposals similar to this have been published (22). Finally, we note that GHZ-type spin states are ideal candidates for quantum-improved measurements. In magnetometry applications, they outperform mixed states in signal-to-noise ratio by a factor of Math, where N is the number of entangled spins (32). Entangled spin states in a diamond-based magnetometer thus might enhance sensitivity considerably.

Supporting Online Material

www.sciencemag.org/cgi/content/full/320/5881/1326/DC1

Materials and Methods

Figs. S1 to S4

Tables S1 to S3

References

References and Notes

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