SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTRY INFORMTION DOI:.38/NPHYS2444 Demonstration of entanglement-by-measurement of solid-state qubits Wolfgang Pfaff, Tim H. Taminiau, Lucio Robledo, Hannes Bernien, Matthew Markham, 2 Daniel J. Twitchen, 2 and Ronald Hanson, Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 546, 26 G Delft, The Netherlands 2 Element Six, Ltd., Kings Ride Park, scot, Berkshire SL5 8BP, United Kingdom EXPERIMENTL SETUP Sample We use a naturally occurring Nitrogen-Vacancy (NV) center in high purity type IIa chemical-vapor deposition grown diamond with a crystal orientation obtained by cleaving a substrate. To improve the photon collection efficiency, a solid immersion lens was deterministically fabricated on top of the center by focussed ion beam milling [ 3]. Microwave and radio-frequency pulses for spin manipulation are applied through a lithographically defined strip line on the sample []. Setup We use a home-built low-temperature confocal microscope that has been described in detail in Robledo et al. [4]. ll experiments are performed at sample temperatures between 8.7 and 8.85 K. small magnetic field ( 5 G) is applied by means of four permanent magnets arranged around the cryostat. The magnetic field is aligned such that no precession of the 3 spin in the = manifold is observed for initial preparation in one of the eigenstates in the = ± manifold. SYSTEM DESRIPTION Level structure Figure S shows the electronic energy levels in the optical ground state for the three-spin system associated to the NV center used: the NV electron spin (S = ), the native Nitrogen ( 4 N) nuclear spin (I = ) and a nearby arbon ( 3 ) nuclear spin (I =/2). We define our qubits as follows. The ancilla consists of the = (labeled a ) and = (labeled a ) electron levels. The 3 and 4 N qubits are labeled and N, respectively. The four two-qubit register eigenstates in the = manifold N =(m I,m IN ) are: (/2, ), (/2, ), ( /2, ), and ( /2, ). Nuclear magnetic Figure S2 shows nuclear magnetic spectra for the 3 and 4 N qubits. From these spectra we determine the transition frequencies = MHz for the 3 and Q + n = MHz for the 4 N. oherence The dephasing times T2 (T 2 N ) for the 3 ( 4 N)-spin is (3.±.2) ms ((.±.7) ms) (Fig. S2B), and for the electron we find T2 e = (. ±.2) µs. The dephasing times of the three spins scale with their gyromagnetic ratios because they share the same environment. Estimation of the coupling between the nuclei From the hyperfine splitting we can determine the lattice site at which the 3 atom is located, relative to the nitrogen atom [5]. We perform a simple estimation for the magnetic dipolar coupling ω N between the two nuclei. The magnetic field of the nitrogen spin at the position of the 3 atom causes a Zeeman-splitting of the carbon-spin, depending on its quantization axis dictated by the hyperfine interaction with the electron. We take the maximum Zeeman-energy (quantization axis aligned with the magnetic field from the nitrogen) as an upper bound for the dipolar coupling and obtain ω N 2 46 Hz. Measurement of the coupling between the nuclei We measure the direct coupling between the nuclei with a Ramseytype experiment as shown in figure F of the main text. The accumulated phases of the 4 N spin, conditional on the state of the 3 spin, for different times are shown in figure S2. From the phase differences we extract a direct coupling strength of (3 ± 3) Hz between the nuclear spins, in agreement with the expected value. NTURE PHYSIS

2 2 E/ћ =+ a ω B a =- m I,N = m I,N =+ { m I, =-/2 -Q + N m I,N =- -Q - N { m I, =+/2 a = m I,N = m I,N =± -Q electron + 4 N + 3 FIG. S. Level diagram for the NV center used in this work. The zero-field splitting of the electronic spin, = (7) GHz at 8.8 K, separates the = level from the = manifold. The = ± sub-levels are Zeeman-split by an external magnetic field, ω B = (7) MHz. The m I = ± states of the nitrogen spin are lowered with respect to the m I = state due to a nuclear quadrupole splitting, with Q = MHz. Hyperfine interaction between the nitrogen (carbon) nuclear spin splits the levels into three (two) sub levels, with a coupling constant of N = MHz ( = MHz). The computational basis states used in this work are marked blue. The first qubit in the ket-notation always denotes the carbon spin. The arrows on the right mark the pulses that we apply at various points in the experimental sequence: () drives the nitrogen (carbon) spin between its two eigenstates in the computational basis, and xy drives the electron between the states =, selective on the nuclear state xy. We denote the electron spin state = + as. Microwave pulses between = + levels are labeled as xy, they are selective on the state with the same nuclear spin projections as the computational basis state xy. NILL ELETRON SPIN PREPRTION ND REDOUT The electron spin readout and preparation is performed using spin-resolved resonant optical excitation, as explained in detail in Robledo et al. [4]. We use two separate lasers to selectively excite the transition associated to = and the transition associated to = ± (Fig. S3). By applying laser pulses of different length, three different tasks can be performed: preparation, single-shot readout and strictly non-destructive readout. Preparation The ancilla electron spin is prepared in = ± by a long (2 µs) laser pulse. This pulse pumps the spin into = ± due to a finite probability that the spin flips in the excited state. Single-shot readout Single shot readout of the electron is performed by applying a µs pulse. If the number of detected photons during the pulse remains below a certain threshold (typically or 2), the readout result is taken to be ( a ), and ( a ) otherwise. The averaged readout fidelity for the NV center used in this work is 93%. Non-destructive readout non-destructive readout of the electron spin is obtained by applying a short 2 ns pulse. If one or more photons are detected during this short pulse, the readout result is taken as and the electron is projected into a. We measure a post-readout state fidelity of (99 ± )%. If no photons are detected then minimal information about the state is obtained and the measurement is aborted. The measurement succeeds in approximately 6% of the cases for initial preparation in a.

3 3 B τ 2µs 7µs /N /2 /2 9 5 events NMR frq. (MHz) NMR frq. (MHz) P() τ (ms) τ (ms) φ N 25 3 t (ms) FIG. S2. haracterization of the nuclear spin qubits. () NMR frequency scans. We prepare the nuclei by measurement in, sweep the frequency of an RF pulse of fixed length, and finally read out the nuclear state. We show, for 3 on the left and 4 N on the right, the number of readout results for both qubit states, and. Solid lines are fits to the form F 2 /(F ) sin( F /2F ), where F is the Rabi frequency and is the detuning f f between the frequency f and the applied frequency f. (B) Free induction decay for the 3 (left) and 4 N (right) spin qubits. Shown is the probability P to obtain result (after readout calibration). We sweep the phase of the second /2-pulse with the inter pulse delay τ, giving rise to oscillations of the signal. Fitting to a cosine with gaussian decay envelope exp( (t/t2 ) 2 ) (solid lines; envelope only: dashed lines) yields dephasing times T2 of (3. ±.2) ms and (. ±.7) ms for 3 and 4 N, respectively. () Direct coupling of the nuclear spins. We measure, analogously to the experiment presented in figure F of the main text, the accumulated phase of the nitrogen spin for = (filled dots) and (empty dots). Dashed lines are linear fits. For all resonant optical pulses we use a power of nw, corresponding to /3 of the saturation power. The spin flip rate = ± during readout is / µs. B Readout Pump (Init) = Re-pump = ± FIG. S3. () Reduced optical level scheme of the NV center. The electronic spin states = and ± have transitions from the ground to the excited state, and, respectively. Depending on the local strain, they differ in energy (a few GHz in our case) such that the transitions can be accessed with two different lasers (solid straight lines). dditional to decay via the same transition and emission of a photon (solid wavy lines), non-radiative cross-relaxation can occur (dashed wavy lines). These properties can be exploited to initialize and read out the electron via spin-selective optical excitation. (B) Electron readout. By applying a pulse resonant with the transition the electron spin state is measured. Ideally, for = the NV center emits light, and it stays dark for ±. Depending on the length of the pulse, different tasks can be performed in this manner (see text). dditionally, a successive re-pump pulse on the transition can be applied after a readout pulse to ensure the spin state is = after the measurement. EXPERIMENTL PROTOOL schematic of the experimental protocol is shown in figure S4. For the HSH measurements we use a modified protocol as depicted in figure S4B. The details of the individual blocks are discussed in the respective sections below.

4 ondition 2ns x 4 Init /2(φ) /2(φ ) Readout Verify/reset charge & next run B Verify/reset charge & Init /2(φ) /2(φ ) Readout next run FIG. S4. Experimental protocol. () Protocol used for figures 2 and 3 in the main text. We first initialize by measurement ( Init ), create a maximal superposition of the two nuclei by means of /2-rotations around axes determined by phases φ and φ, and then project into an entangled state with a parity measurement ( ). To analyze the created state we read out the qubits. Readout along axes other than ZZ is achieved by single qubit rotations with corresponding amplitude and phase before measuring. Finally, we verify that the NV is in the right charge state and that the lasers are on. The data of a single experiment run is discarded if any of the conditional steps fails. (B) Modification for the HSH measurements. To avoid any post-selection, all steps that are conditional are performed before the entangled state is created. System initialization circuit diagram and pulse scheme for initialization of the qubits and the ancilla is shown in figure S5. Starting from a mixed state for all three spins, we first bring the electron into a mixed state with = ± by spin pumping. We then apply a microwave -pulse selective on. subsequent projective measurement on the electron, conditioned on outcome =, projects the ancilla into a and the qubits into. Initialization fails for outcome = ±, and the measurement is aborted. fter projection, the ancilla is reset to a by a microwave -pulse. Due to microwave power limitations and dephasing of the NV electron the ancilla reset has a limited fidelity, transferring 96()% of the population back to the computational basis. Entanglement by non-destructive projective parity measurement Entanglement by parity measurement is implemented as shown in figure S6. Starting from a pure state, with the electron ancilla in a and the nuclei in an arbitrary two-qubit product state, we first entangle the ancilla with the two-qubit parity by applying Toffoli gates on the ancilla, conditional on and ( and ). Subsequent electron readout, conditioned on outcome =, projects the qubits into the even (odd) subspace. The readout pulse is short enough (2 ns) that we do not observe any dephasing of the nuclear spins due to optical cycling. Finally, we again reset the ancilla to a. Readout of the nuclear spin qubits We measure the two qubit state of the nuclei as schematically shown in figure S7. We probe an eigenstate by mapping its population onto a with a double-controlled NOT and read out the electronic spin. In case this measurement yields outcome =, the qubits are projected onto the probed state. For = ±, the state is projected onto the remaining superposition state. We sequentially probe all four eigenstates. The result is the first state for which a number of photons is obtained that passes a threshold value.

5 5 : continue : abort ondition 5µs : continue 2µs 2ns : abort optional a b ondition 5µs FIG. S5. Initialization of the ancilla and qubits. ircuit diagram (top) and implemented pulse scheme (bottom). First, =- the =+ E ancilla electron is initialized into m x 2µs 2ns s = ± by optical pumping. onditioning on detection of zero photons during the last 5 µs guarantees an initialization fidelity of > 99.9%. We initialize the nuclear state into by measurement, by first applying a microwave -pulse selective on on the electron and reading out the electron spin state for a short time, where we condition E repeat for,, x on outcome =. Pumping on the transition compensates for possible spin flips during this readout. Finally, the electron is reset to the initial state by a -pulse. In the HSH measurements, we confirm the reset by another electron readout. =- = ondition : continue : abort : 2ns continue : abort optional ondition E x 2ns RF 2 /2(φ) Verify/reset Init Readout charge & next run /2(φ ) FIG. S6. Entanglement byb projective parity measurement. Two selective rotations of the electron (conditional on the two even /2(φ) Verify/reset qubit states) are followed by a short, Verify/reset Init conditional readout, re-pumping, RF and finally resetting /2(φ) RF Readout the ancilla. Projection into RF /2(φ ) charge & next arun state of odd parity is analogously done by applying charge & N (θ ) Init Toffoli gates that are conditional N on the two odd qubit Readout states. next run /2(φ ) alibrated readout For all measurements except the HSH measurements, we use a readout calibration as follows. B To avoid any influence of the order in which we probe the states, we take data by averaging over all possible order permutations. The readout result Verify/reset for the presentedrf data /2(φ) is the first state in the sequence for which at least one photon charge & Init Readout next run has been detected. The raw data of a readout characterization /2(φ ) is shown inrffigure N S8. The raw data contains two errors that we correct for. The readout of the electron has a limited fidelity, originating from either (), incorrect photons being detected (i.e., we obtain the wrong result), or (2), no photons are detected due to limited collection efficiency (i.e., we obtain no result) [4]. The readout characterization after calibrating for these effects is shown in figure S8B. The average fidelity is 96(3)%, in agreement with the error in the state initialization, for which we do not ondition a ondition b Tr repeat 3 for, Fa 3

6 6 continue abort 2ns F F N optional a b FIG. S7. Register readout. We read out the two-qubit eigenstates (denoted a, b, c, d) in succession by mapping theequentially onto the electron by applying a conditional -pulse, and then reading out the electron. The order in which we probe depends on the type of readout, see text. correct. =- =+ ondition repeat for,, and B 3x Readout for the HSH measurements In order to obtain a single-shot readout of the two-qubit register, we modify the readout protocol as shown in figure S9. We do not use permutations of the readout order, so the exact same readout is used throughout the experiment. Note that we not Verify/reset only probe the computational basis states xy, but also the corresponding states xy in the Readout charge & next run = + manifold. During optical excitation, the electron spin can flip from = to either = or +. Thus, probing only the states can lead to the state being trapped in = +. The result of the readout is obtained as follows. We look for the first probe where the obtained number of photons exceeds a threshold (initially two), which determines the readout result. If no attempt surpasses the threshold, its value is decremented by one, and the procedure is repeated. Only in case where no photon at all has been obtained during the whole sequence, the measurement result remains unknown. This happens in <.2% of all cases for the presented Readout next run data. When determining the HSH parameter S, we always assume that unknown outcomes lower its absolute value (worst case scenario). result c True 3 False state prepared result d state prepared FIG. S8. haracterization of the two-qubit readout. () Raw characteristics for the readout using permutations, and (B), after calibration. We first initialize into an eigenstate and subsequently probe all two-qubit states. harge and condition fraction of events s shown in figure S, we verify that the NV center is in its negative charge state and that the lasers are on with the and transitions. If required, we perform a reset of the electronic state by applying a green laser pulse. In the HSH measurements this is done before creating the entanglement, in all other experiments at the end of the experimental sequence.

7 : continue : abort : continue optional a b c d : abort 7 2ns optional a b c d 2ns E =- =+ x repeat for m,, and s =- =+ 3x repeat for,, and FIG. S9. Modified readout for the HSH measurements. We twice read out, then twice the corresponding state in the tinue m True 3x rt s = + manifold. This is repeated for,, and (in this order). The whole sequence is repeated three times. ondition 3 False True ondition 3 False ns ail /2(φ) /2(φ ) (φ) (φ ) FIG. S. harge and verification. We apply pulses on both optical transitions. If a threshold value of 3 photons is RF not reached during this time, we Verify/reset assume that the NV center is either ionized or not on (due to spectral diffusion). In this case, we apply Readout a short green charge (532& nm) laser next pulse run and repeat until the threshold is surpassed. Verify/reset Readout charge & next run RESULTS State error Readout next run We can fully account for the error in the final entangled state by the error in the ancilla-reset after initialization and parity RF measurement. These resets are accomplished by microwave -pulses. Their fidelity is limited by the dephasing of the electron and the available Readout microwave next run power. The used -pulses transfer 96()% of the electron population from = to. The population that is not transferred dephases and is lost for the further procedure; therefore the fidelity of the final state is limited. During the entanglement protocol, this occurs twice after initialization and after the parity measurement and our final state fidelity thus has an upper bound of F 92(2)%. Tomography We determine the elements of the densiy matrix ρ M, defined in the Z-basis as ρ M = x x 2 + iy 2 x 3 + iy 3 x 4 + iy 4 x 2 iy 2 x 22 x 23 + iy 23 x 24 + iy 24. () x 3 iy 3 x 23 iy 23 x 33 x 34 + iy 34 x 4 iy 4 x 24 iy 24 x 34 iy 34 x 44 fter creation of the desired state, we measure the qubits in different bases by performing single-qubit rotations U U N before readout, where U (U N ) is either the identity I, a /2 rotation around Y or a /2 rotation around

8 X applied on the 3 ( 4 N) qubit, yielding 9 possible combinations. From these measurements, we directly determine the entries of the density matrix, without any further estimations: 8 R(ρ init ) = I(ρ init ) = R(ρ superposition ) = I(ρ superposition ) = R(ρ Φ +) = I(ρ Φ +) =.95 ±.6.2 ±.6.2 ±.6.4 ±.5.2 ±.6. ±.3. ±.5. ±.3.2 ±.6. ±.5.2 ±.3. ±..4 ±.5. ±.3. ±..2 ±.3.5 ±.6.7 ±.6.3 ±.5.5 ±.6.4 ±.5. ±.2.7 ±.6.4 ±.5. ±.2.3 ±.5. ±.2. ±.2.24 ±.7.26 ±.4.22 ±.4.26 ±.4.26 ±.4.29 ±.7.2 ±.4.26 ±.4.22 ±.4.2 ±.4.22 ±.6.22 ±.4.26 ±.4.26 ±.4.22 ±.4.25 ±.7. ±.5.3 ±.5. ±.4.2 ±.5.3 ±.4.3 ±.5.3 ±.5.3 ±.5.5 ±.5. ±.4.3 ±.5.5 ±.5.45 ±.8.4 ±.5.7 ±.5.42 ±.4.4 ±.5. ±.2. ±.4. ±.5.7 ±.5. ±.4.3 ±.3.3 ±.5.42 ±.5. ±.5.3 ±.5.52 ±.8. ±.4.3 ±.5.4 ±.5. ±.4. ±.5. ±.5.3 ±.5. ±.5. ±.5.4 ±.5. ±.5. ±.5 The given uncertainties correspond to one statistical standard deviation. ll measurements have been performed with a total of 24 repetitions per measurement basis. Lower bound of the fidelity from correlation measurements We can estimate a lower bound of the fidelity of a created Bell state from correlation measurements in two different bases. Using the diagonal elements of density matrices ρ (Z basis for both qubits) and ρ (basis rotated by /2 for both qubits), we can place a lower bound on the fidelity F [6], F 2 ( ρ, + ρ, 2 ρ, ρ, + ρ, + ρ, ρ, ρ, ). (2) ρ xy,xy and ρ xy,xy are the density matrix elements xy ρ xy and xy ρ xy, respectively.

9 9 Uncertainties The given uncertainties for state characterizations come from photon counting statistics. To calibrate our electron readout, we first initialize the electron by optical spin pumping into either a or a. We then perforingle-shot readout of the electron as described above to obtain a calibration. The standard deviations of the number of events in which zero photons have been collected (outcome a ) and the number of events in which more than photons have been collected (outcome a ) are the basis for all following error analysis. alibration of the readout of the nuclear spin qubits is performed in the same way: We initialize into each of the four two-qubit eigenstates, then read out as described above. The uncertainty in this calibration is again given by the standard deviation of the number of events in which zero/more than zero photons have been collected. Taking into account the uncertainties of the readout calibrations and the statistical uncertainties of any following measurement yields the uncertainties for presented quantities. HSH measurements The measured results for the states Φ + and Ψ are shown in Fig. S. We determine the HSH parameter in the form S = E(, N ) E( 2, N ) + E(, N 2 ) + E( 2, N 2 ). The resulting values for S are 2.43 ±.6 and 2.7 ±.4, respectively.,n 2,N,N 2 2,N 2 Φ +.63 ± ±.3.6 ±.3.52 ±.3 fraction of events Ψ -.48 ±.2 state.62 ±.2 state -.57 ±.2 state -.5 ±.2 state fraction of events FIG. S. HSH measurements for states Φ + (top; 3 repetitions) and Ψ (bottom; 5 repetitions), as described in the main text and Fig. 4. Uncertainties are standard deviation. r.hanson@tudelft.nl [] H. Bernien, L. hildress, L. Robledo, M. Markham, D. Twitchen, and R. Hanson, Phys. Rev. Lett. 8, 4364 (22). [2] L. Marseglia, J. P. Hadden,.. Stanley-larke, J. P. Harrison, B. Patton, Y.-L. D. Ho, B. Naydenov, F. Jelezko, J. Meijer, P. R. Dolan, J. M. Smith, J. G. Rarity, and J. L. O Brien, ppl. Phys. Lett. 98, 37 (2). [3] J. P. Hadden, J. P. Harrison,.. Stanley-larke, L. Marseglia, Y.-L. D. Ho, B. R. Patton, J. L. O Brien, and J. G. Rarity, ppl. Phys. Lett. 97, 9 (2). [4] L. Robledo, L. hildress, H. Bernien, B. Hensen, P. F.. lkemade, and R. Hanson, Nature 477, 574 (2). [5] B. Smeltzer, L. hildress, and. Gali, New Journal of Physics 3, 252 (2).

10 [6] B. B. Blinov, D. L. Moehring, L.-M. Duan, and. Monroe, Nature 428, 53 (24).