Abstract
Twoqubit operation is an essential part of quantum computation. However, solidstate nuclear magnetic resonance quantum computing has not been able to fully implement this functionality, because it requires a switchable interqubit coupling that controls the time evolutions of entanglements. Nuclear dipolar coupling is beneficial in that it is present whenever nuclear–spin qubits are close to each other, while it complicates twoqubit operation because the qubits must remain decoupled to prevent unwanted couplings. Here we introduce optically controllable internuclear coupling in semiconductors. The coupling strength can be adjusted externally through light power and even allows on/off switching. This feature provides a simple way of switching interqubit couplings in semiconductorbased quantum computers. In addition, its long reach compared with nuclear dipolar couplings allows a variety of options for arranging qubits, as they need not be next to each other to secure couplings.
Introduction
Nuclear magnetic resonance (NMR) quantum computing has attracted broad interest because it is one of the most advanced testbeds for quantum computation. Although the interest began with solution NMR^{1,2,3}, it is now believed that scalable NMR quantum computers in the future will be built on semiconductors based on highly developed semiconductor technology^{4,5,6}. The main challenges include the initialization and the creation of spin entanglement, which are essential features of quantum computation^{7}. Semiconductorbased NMR quantum computers are advantageous as they can be achieved optically; that is, the initialization (nuclear–spin polarization) is provided by optical pumping^{8,9}, and the entanglement is created via internuclear (nuclear spin–spin) couplings between polarized nuclei^{10,11}. In optically pumped semiconductors, the latter manifests itself as dipolar order^{12,13} and doublequantum coherence^{14}.
Switchability is another essential functionality required for internuclear couplings, which should be 'on' during operations and 'off' otherwise. In this respect, nuclear dipolar coupling (Dcoupling, hereafter) is not the best choice for the abovementioned reasons. In addition, the time required for operations increases rapidly with increasing qubit number because of decoupling operations^{15}. Other possible candidates include indirect couplings mediated by donor electrons^{4} and magnons^{15}. Their implementations are fairly challenging, however, given the complicated switching mechanisms. By contrast, the scheme presented in this paper is rather simple; the coupling strength can be controlled externally through light power, and on/off switching can be easily implemented.
In this study, we have preformed crosspolarization (CP) experiments with GaAs under light illumination, and demonstrated that a nuclear spin–spin coupling grows in strength, and extends its reach to farther nuclei as light power is increased. These futures bring about a unique transition of the CP process from oscillatory behaviour in the 'dark' towards exponential relaxation with increasing light power. The experiments provide us with information on the essential features of the optically induced nuclear spin–spin couplings; in particular, we find that the coupling strength is roughly proportional to light power, which is essential for the switching of the couplings.
Results
CP process in GaAs in the dark
The present mechanism is manifested in a CP process from ^{75}As (Ispin) to ^{71}Ga (Sspin) in GaAs under infrared light irradiation. Before detailing this process, we first describe it in the dark (without light irradiation) as a reference. This is an ordinary CP process, for which we expect a contact time (_{cp}) dependence of Smagnetization (M_{S}^{eq}) of the form
where T_{IS} is the crossrelaxation time. A relaxation process in the rotation frame (T_{1ρ}) need not be considered here, as it is sufficiently long because of high crystal symmetry^{16,17}. The reality, however, is more complicated than equation (1). Figure 1 shows M_{S}^{eq}(_{cp}) obtained in the dark, which exhibits a clear transient oscillation.
Transient oscillations have been reported in some molecular crystals, and attributed to discrete S–I coupling spectra of isolated S–I pairs^{18,19,20}. The magnetization is transferred back and forth inside the pair with a frequency corresponding to half the flipflop term of the Dcoupling. The present sample, however, is not a molecular crystal, so isolated pairs are expected to be rare. Here an essential factor is the existence of two Ga isotopes, that is, ^{69}Ga and ^{71}Ga with natural abundances of ^{69}N_{A}=0.604 and ^{71}N_{A}=0.396, respectively. A local ^{75}As ^{71}Ga pair appears when only one of the four nearestneighbour sites of ^{75}As is occupied by ^{71}Ga and the others are occupied by ^{69}Ga. The probability of finding such pairs is _{4}C_{1}·^{71}N_{A}·(^{69}N_{A})^{3}=0.35; that is, about 35% of ^{75}As have a single ^{71}Ga in the vicinity and contribute to the oscillation. The pairs are coupled through indirect scalar coupling J_{IS}, where Dcouplings are absent because the Ga sites are situated at magic angle positions in the (100) crystal orientation^{16,17,21}. The process is described by a damping oscillation^{18},
where Ω is the oscillation frequency (Ω=J_{IS}/2) and R is the damping factor. We fit the data using equation (2) with Ω and R as free parameters. The best result is obtained with Ω=0.74 kHz, which yields J_{IS}=2Ω=1.48 kHz. This value is comparable with that in the InP case (J_{IS}=2.3 kHz)^{17}. The fitting curve is shown by a solid line in Figure 1. The fit is not very good at the beginning of the oscillation; this may be due to the presence of ^{75}As sites with more than one ^{71}Ga nucleus in the four nearestneighbour sites.
CP process under light irradiation
Figure 2a shows the _{cp} dependence of Spolarization, M_{S}(_{cp}) under various levels of light power, P_{IR}, which exhibits new local maxima (peaks) that are not observed in the dark. The maxima form a number of series (α, β, γ...), and the maximal position in each series shifts towards smaller values of _{cp} as the light power is increased. For example, the series 'β' shown by the red arrows starts with a broad maximum around _{cp}=3.1 ms at P_{IR}=100 mW, which shifts towards smaller values of _{cp} as P_{IR} is increased and eventually merges into a peak at _{cp}=0.8 ms at 166 mW. These maxima represent new polarization transfer processes that appear under light irradiation. Figure 3 shows a threedimensional representation of Figure 2a obtained by interpolating the data in between. The continuous shift of the maximum in each series can be readily confirmed.
This phenomenon can be explained by discrete increments in the number of Snuclei (^{71}Ga) involved: in the dark (P_{IR}=0 mW), the number of nuclei participating in the process is small and oscillatory behaviour is observed, as shown in Figure 1. Light illumination causes a series of Snuclei batches (α, β, γ...) to participate in sequence. Their contributions are successively added to the signal intensity one after another, whereas the speed of transfer from I to Snuclei in each series increases with the light power (that is, the maximal position shifts towards smaller values of _{cp}). As the number of nuclei involved is further increased, the cross relaxation is expected to approach the exponential relaxation behaviour given by equation (1)^{20}. That is, we see here the transition from oscillatory behaviour in the 'dark' towards exponential relaxation with increasing light power. This is a unique case in that such a transition can hardly be observed in ordinary NMR measurements.
Another factor responsible for the phenomenon is the optical pumping effect^{8,9}. It contributes to the Smagnetization (^{71}Ga) through the enhancement of the Imagnetization (^{75}As) caused by the polarization transfer from the optically oriented electrons for the duration of light irradiation _{L}=60 s. Figure 2b shows the lightpower dependence of the Imagnetization (^{75}As), which increases linearly with light power. This enhancement is partly responsible for the increase of the Smagnetization with increasing light power, as seen in Figures 2a and 3.
The _{cp} dependence of M_{s} is expressed as,
where M_{I}^{dark}(_{L}) in the first term on the righthand side represents the Imagnetization in the dark portion of the sample recovered during _{L}=60 s and represents the polarization transfer process to Sspins in the first nearest neighbour sites through J_{IS}. This process is essentially the same as that in Figure 1. On the other hand, in the second term is the Imagnetization in the illuminated area generated by the optical pumping effect during _{L}=60 s, whose P_{IR} dependence is shown in Figure 2b. This magnetization is transferred to Sspins through the two processes shown in the brackets: the same process as that in the dark, , and additional ones induced by light irradiation, . The latter are caused by the optically induced heteronuclear indirect coupling, . We will discuss characteristics of this coupling in the next section.
Discussion
The new coupling is presumably mediated by photoexcited electrons. It is known that electrons in metals can mediate indirect nuclear spin–spin couplings, a process referred to as the RKKY interaction^{22}. In the present case, however, is observed in semiconductors where no intrinsic Fermi surfaces exist. Moreover, the lifetimes and spin relaxation times of the photoexcited electrons usually fall in the range between pico and nanoseconds, which is more than six orders of magnitude smaller than that of the CP process, that is, milliseconds. It is intriguing that two phenomena with such different time scales are coupled with each other.
The strength of in each batch (α, β, γ...) can be evaluated by the crossrelaxation time T_{IS}^{opt} in . According to Demco et al.^{17,20,23}, the crossrelaxation time is expressed as
where _{c} is the correlation time of the CP and,
is the second moment of the I–S heteronuclear spectrum due to . Equation (5) indicates that M_{2}^{IS} is proportional to ()^{2}. Provided that is proportional to P_{IR} and that _{c} is independent of , equations (4) and (5) lead to the conclusion that 1/T_{IS}^{opt} is proportional to P_{IR}^{2}. The actual value of T_{IS}^{opt} in each batch is determined from the analysis of the functional form of . Here we evaluate it from the maximal position; T_{IS}^{*} is defined as the contact time at which the maximum is formed. Figure 2c shows 1/T_{IS}^{*} for the three series of maxima (α, β and γ) plotted against P_{IR}^{2}. The graph suggests that 1/T_{IS}^{*} is roughly proportional to P_{IR}^{2}, implying that is proportional to P_{IR}.
This result provides a clue to the nature of the series of Snuclei batches (α, β, γ...). Equation (5) indicates that the condition is fulfilled only when all I–S pairs participating in the summation of the righthand side share the same , such that can be taken out of the summation Σ. Therefore, each series may be assigned to the polarization transfers to a batch of Sspins at the same distance from the Ispin at the origin. Figure 3 illustrates this situation. For example, the αmaxima may be assigned to the second nearestneighbour Sspins, the βmaxima to the third, and so forth. It is reasonable that equivalent Sspins in the same order situated at the same distance from the Ispin share the same scalar coupling with it. Moreover, the contributions from nuclei in the same order to the correlation time are partially cancelled out^{17,20,23}. Therefore, the assumption that _{c} is independent of may be a reasonable approximation.
The result also provides us with information about the reach of . The dotted red line in Figure 3 shows the P_{IR} dependence of M_{s}(_{cp}) at a constant _{cp}. One finds a plateaulike feature around 80 mW, which indicates that the Sspins at the third nearestneighbour sites participate in the process only when P_{IR}>80 mW; that is, can reach farther than D and J_{IS}couplings, as the latter two may not substantially reach the thirdnearest neighbours.
Features such as the external switchability and the long reach may add variety to the qubit arrangement in semiconductorbased NMR quantum computers. Figure 4 shows an example. In an array of nuclear–spin qubits, a few lattice points apart from each other, the Dcouplings are out of reach, so that the qubits are decoupled. Then, the present mechanism provides the external control of internuclear couplings. As the light power is increased, the reach of couplings is extended. This would further enhance the flexibility of qubit arrangements.
This mechanism is compatible with most schemes proposed for semiconductorbased NMR quantum computing. The implantation of nuclear spin arrays in semiconductors has been studied using various techniques such as scanning probe microscopes, ion beams, and isotope engineerings^{24,25,26}. These techniques will provide promising technologies to implement the schemes shown in Figure 4.
In conclusion, we have discovered an optically switchable indirect nuclear spin–spin coupling, which is manifested in CP experiments with GaAs under light illumination. As the strength of this coupling is externally controllable through light power, we expect it to have an essential role in the quantum gate operations of solidstate NMR quantum computers in the future.
Methods
Optical pumping double resonance NMR system and the sample
The CP experiments were performed at 10 K using an opticalpumping doubleresonance NMR system developed for this study. The system consists of an NMR spectrometer, a laser system and a cryostat loaded in a 6.34T superconducting magnet. A custombuilt doubleresonance probe is installed inside the cryostat. The sample used in this study is an undoped semiinsulating GaAs singlecrystal wafer with a thickness of 600 μm and a crystal orientation of (100) (Mitsubishi Chemical). It is mounted on a sample stage made of sapphire located at the probe head and set with the surface normal to the magnetic field. The sample stage is cooled through thermal contact with the cold head of the cryostat. Infrared light emitted from a Ti:Sapphire laser is delivered to the cryostat through a polarizationmaintaining fibre^{27}. It is converted to circularly polarized light by a quarterwave plate at the probe head, and applied to the sample in parallel to the magnetic field. The spot size at the sample surface is about φ5 mm.
Pulse sequence
The pulse sequence used is shown in Figure 5. The magnetization of Ispins, saturated by the first comb pulse, is regenerated during the time interval _{L} and transferred to Sspins through the CP immediately thereafter. The infrared light is irradiated at a constant strength P_{IR} throughout the sequence. The photon energy of 1.50 eV (near the band gap) and the helicity σ^{−} were selected so that the optical pumping NMR signal enhancements for both ^{71}Ga and ^{75}As were nearly maximal. The experiment in the dark (Fig. 1) was obtained with the same pulse sequence as that in Figure 5, with the exception that P_{IR}=0 mW.
Additional information
How to cite this article: Goto, A. et al. Optical switching of nuclear spin–spin couplings in semiconductors. Nat. Commun. 2:378 doi: 10.1038/ncomms1378 (2011).
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Acknowledgements
We acknowledge C. Takizawa for technical assistance. We also acknowledge High Magnetic Field Station (Tsukuba Magnet Laboratory), NIMS, and Niki Glass for their support. This work was partially supported by JST PRESTO program. K.H. was supported by a GrantinAid for Scientific Research (No. 20540323) from the Japan Society for the Promotion of Science.
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A.G. conceived and designed the experiments. All the authors jointly designed the system for the experiments, and A.G. and S.O. constructed it. A.G. carried out the main experiments. All the authors were involved in the analyses. A.G. wrote the paper, with the help of the coauthors.
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Goto, A., Ohki, S., Hashi, K. et al. Optical switching of nuclear spin–spin couplings in semiconductors. Nat Commun 2, 378 (2011). https://doi.org/10.1038/ncomms1378
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DOI: https://doi.org/10.1038/ncomms1378
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