Excitation and coherent control of spin qudit modes with sub-MHz spectral resolution
V. A. Soltamov C. Kasper A. V. Poshakinskiy A. N. Anisimov E. N. Mokhov A. Sperlich S. A. Tarasenko P. G. Baranov G. V. Astakhov V. Dyakonov


apsrev













[E-mail: ]astakhov@physik.uni-wuerzburg.de 

[E-mail: ]dyakonov@physik.uni-wuerzburg.de

^1Ioffe Institute, 194021 St. Petersburg, Russia  
 
^2Experimental Physics VI, Julius-Maximilian University of Würzburg, 97074 Würzburg, Germany 
 
^3National Research University of Information Technologies, Mechanics and Optics, St Petersburg, Russia., 197101 St. Petersburg, Russia  

^4Peter the Great St. Petersburg Polytechnic University, St. Petersburg, 195251, Russia 


  
Quantum bit or qubit is a two-level system, which builds the foundation for quantum computation, simulation, communication and sensing <cit.>. Quantum states of higher dimension, i.e., qutrits (D = 3) and especially qudits (D = 4 or higher),  offer significant advantages.  
Particularly, they can provide noise-resistant quantum cryptography <cit.>, simplify  quantum logic <cit.> and improve quantum metrology <cit.>. Flying and solid-state qudits have been implemented on the basis of photonic chips <cit.> and superconducting circuits  <cit.>, respectively.  However, there is still a lack of room-temperature qudits with long coherence time and high spectral resolution. The silicon vacancy centers in silicon carbide (SiC) with spin S = 3/2 are quite promising in this respect <cit.>, but until now they were treated as a canonical qubit system <cit.>. Here, we apply a two-frequency protocol to excite and image multiple qudit modes in a SiC spin ensemble under ambient conditions. Strikingly, their spectral width is about one order of magnitude narrower than the inhomogeneous broadening of the corresponding spin resonance. By applying Ramsey interferometry to these spin qudits, we achieve a spectral selectivity of 600 kHz and a spectral resolution of 30 kHz.  As a practical consequence, we demonstrate absolute DC magnetometry insensitive to thermal noise and strain fluctuations.





Excitation and coherent control of spin qudit modes with sub-MHz  spectral resolution
    V. Dyakonov^2
    December 30, 2023
=====================================================================================



 

In our experiments, we focus on the silicon vacancies (V_Si) in two main hexagonal polytypes of SiC, namely 6H and 4H, with natural and modified isotope abundance. The 6H- and 4H-^28SiC samples, purified from the ^29Si isotope, were grown by the seeded physical vapour transport method <cit.>. The ^29Si isotope abundance of 1%, as established by the electron paramagnetic resonance (EPR) spectroscopy (Supplementary Fig. S1), is 4.5 times smaller than that in natural SiC crystals. 


The V_Si centers in SiC possess a specific spin S = 3/2 fine structure <cit.>.  Compared to comon spin-1/2 qubits, these centers provide additional functionality for quantum applications because they can be in multiple superpositions of four basis states in the Hilbert space with the spin projections m_S = ± 3/2, ± 1/2. Similar to atomic orbitals, an ensemble of such spin qudits can be described by 15 linearly independent spherical multipoles, with three components being the spin dipole 𝒫, five components being the spin quadrupole  𝒟 and seven components being the spin octupole  ℱ <cit.>. In the simplest case, when the external magnetic field 𝐁 is applied along the symmetry axis  (c-axis of the SiC lattice), the density matrix ρ is diagonal and can be written in the form ρ = ℐ / 4 + p_0 𝒫_0 + d_0 𝒟_0+ f_0 ℱ_0. Here, ℐ, 𝒫_0, 𝒟_0, and ℱ_0 are linearly-independent 4 × 4 diagonal matrices (table <ref>). To describe our experimental results, we use all (15 + unit ℐ) basis  matrices (Supplementary Theory). When qudit multipoles are excited, they decay independently, and in the spherical approximation, there are only three relaxation times: that of the spin dipole (T_p), quadrupole (T_d), and octupole (T_f) (table <ref>). 



To experimentally visualize the V_Si spin qudit modes, we use optically detected magnetic resonance (ODMR) as described elsewhere <cit.> (see also Supplementary Methods and Fig. S2). Optical pumping results in a preferential population of either the m_S = ± 3/2 or m_S = ± 1/2 states (depending on SiC polytype and the V_Si crystallographic site). Following table <ref>, only the spin quadrupole 𝒟_0 is excited under these conditions, with either d_0 > 0 or d_0 < 0, respectively. Application of a strong microwave (MW) field at a fixed frequency ν_pump mixes the m_S = ± 3/2 and m_S = ± 1/2 states, resulting in the excitation of other qudit modes as well. They are probed as relative difference of the spin-dependent photolumenescence Δ PL / PL  while sweeping the frequency of a second (weak) MW field ν_probe.

Figure <ref>(a) shows an ODMR resonance associated with the V3(V_Si) center in 6H-^28SiC <cit.>. The ODMR resonance without the pump MW field (black curve) reveals the quadrupole spin splitting 2D = 26.8 MHz between the ± 1/2 and ± 3/2 states. Application of a strong, pump MW field at ν_pump = 26.8  MHz saturates the spin transitions, which is seen as a spectral hole burning <cit.>. To model it, we assume that the ODMR resonance is inhomogeneously broadened and agregates many homogeneous spin packets with different resonance frequencies, as schematically shown by the thin lines in Fig. <ref>(b). The pump MW field excites qudit modes in particular spin packets, which are detected as a reduction of the ODMR signal at certain frequencies. In zero magnetic field, all the excited modes are degenerate and manifest themselves as a single spectral hole. Remarkably, the hole spectral width can be much narrower than the inhomogeneous linewidth and in certain cases we observe 250  kHz (Supplementary Figs. S3 and S4).  



To increase the sensitivity, we modify the detection scheme. The MW pump is now modulated on/off and the ODMR spectrum is detected using a lock-in amplifier. The pump-induced changes of the ODMR spectrum are presented in Fig. <ref>(c) for several magnetic fields B_z applied along the c-axis of SiC. Several qudit modes ν_mod are now clearly detected and their spectral positions relative to ν_pump depend on B_z. 

To understand the spectrum of spin qudit modes, we first consider the inhomogeneous broadening caused by magnetic fluctuations (for instance, due to nuclear fields). This mechanism was shown to sufficiently influence the inhomogeneous broadening of the nitrogen-vacancy (NV) defects in diamond <cit.>. Upon application of the magnetic field along the c-axis, a single spin packet is characterized by four resonances shifting linearly with the magnetic filed as 2D ±γ B_z and 2D ± 2 γ B_z  <cit.>. Here, γ =  28  MHz / mT is the gyromagnetic ratio. The corresponding fan chart is shown by the red lines in Fig. <ref>(d). The line thickness represents the coupling strength to the MW field. Another spin packet feels another local magnetic field δ B and the fan chart is shifted along the vertical axis in Fig. <ref>(d). The mean value δB̅ defines the inhomogeneous ODMR linewidth. 

Assume that the pump MW field is in resonance with one of the spin transitions of a particular spin packet (the solid circle in Fig. <ref>(d)). Then the other three spin resonances of the same spin packet will be effected due to the excitation and relaxation of spin multipoles, leading to the appearance of qudit modes  (the open circles in Fig. <ref>(d)). Their spectral position relative to the pump frequency is symmetric in respect to the zero field splitting 2 D. In case when ν_pump = 2 D, the excited qudit modes remain degenerate even when B_z ≠ 0 (the square in Fig. <ref>(d)), which contradicts to the experimental observation of Fig. <ref>(c).

We now consider another inhomogeneous broadening mechanism, caused by the variation of the zero field splitting around the mean value 2D̅ = 26.8  MHz. This mechanism is natural to expect, given that we investigate isotopically-purified  6H-^28SiC with low content of the spin-carrying ^29Si isotopes. In this case, the fan chart of the field-dependent ODMR lines associated with different spin packets are shifted along the horizontal axis in Fig. <ref>(e). The positions of the excited qudit modes are given by ν_mod = ν_pump± s γ B_z with s = 1,2,3,4. They depend on the magnetic field strength, which is in qualitative agreement with the experiment of Fig. <ref>(c). 



Figure <ref>(a) visualizes the magnetic field evolution of the V3(V_Si) spin qudit modes in 6H-^28SiC. The observed behavior is more complex compared to the qualitative consideration above.  Not all qudit modes shift linearly with B_z, they have different strength and some of them are not even detected. To understand this behavior, we have developed a theory of qudit mode excitation in inhomogeneously broadened spin ensembles.  

We start from the effective spin Hamiltonian of the V_Si centers in a simple axial model H = D ( S_z^2 - 5/4) + γ𝐒𝐁.  In small magnetic fields that we consider, γ B ≪ D, the eigenstates are given by 


    E_“±3/2" = +D ±3/2γ B_z  ,
       E_“±1/2" = -D ±1/2γ√(B_z^2 + 4 B_⊥^2) ,


where B_z and B_⊥ are the magnetic field components parallel and perpendicular to the c-axis, respectively. 
Due to the mixing of the spin states by B_⊥ and a low trigonal pyramidal local symmetry of the V_Si centers in SiC, all four transitions between the “± 1/2" and “± 3/2" spin states are allowed <cit.>. 

Their spectral positions read 


    ν_ mod^(s,s') = ν_ pump + 3 s γ B_z + s' γ√(B_z^2+4 B_⊥^2) ,


where s,s' = 0,± 1. Equation (<ref>) describes the spectral hole burning at ν_ pump (s,s'=0) and  8 qudit modes. 

We now analyze the strength of the qudit modes coupling to the MW field and  how excitations between these modes are transfered. At room temperature, when the energy of thermally excited phonons is much higher than the spin splitting, all spin multipoles of table <ref> are involved into the spin relaxation process. If spin relaxation occurs due to fluctuating magnetic fields, only the transitions with Δ m_S = ± 1 are allowed, as was also assumed in the earlier works <cit.>. Application of this constraint to the spin relaxation matrix (Supplementary Eq. (S14)) yields the ratio between the multipole relaxation times T_p = 3 T_d = 6 T_f. Remarkably, the spin relaxation rates between the + 3/2 ↔ + 1/2 and - 3/2 ↔ -1/2 states are not equal to that  between the +1/2 ↔ -1/2 states, and we obtain for them (1/2)T_d^-1 and (2/3)T_d^-1, respectively. 

Following the analytical solution of the rate equations (Supplementary Eq. (S11)), the strength of the modes (s,s')=(+1,+1), (+1,-1), (-1,+1) and (-1,-1) is proportional to 5T_d -T_p -4T_f. Therefore, they vanish in the case if the aforementioned ratio T_p = 3 T_d = 6 T_f holds. As a consequence, only 4 out of 8 possible qudit modes are observed in Fig. <ref>(a). The calculated evolution of the qudit modes with B_z  is shown in Fig. <ref>(b) (see Supplementary Theory). Here, we assume the Gaussian distribution of the zero-field splitting, f(D) ∝exp[(D-D̅)^2/( δ D^2)], with the mean value 2D̅ = 26.8  MHz corresponding to the V3(V_Si) center  in 6H-SiC. 
A perfect agreement with the experimental data is achieved for B_⊥ = 60 μT, accounting for a uncompensated perpendicular component of the external magnetic field. 



To further confirm the conclusions of our model, we investigate the qudit modes in a stronger magnetic field B_z = 211  μ T when the Zeeman splitting is larger than inhomogeneous broadening. A precise analysis  of the ODMR spectra <cit.> shows that  B_ = 73  μ T remains almost the same. Figure <ref>(a) shows relative positions of the  V_Si spin sublevels calculated for this field configuration. The ODMR spectrum consists of four lines [the lower curve in Fig. <ref>(b)] and the corresponding spin transitions enumerated from ν_1 to ν_4 are shown by the arrows in Fig. <ref>(a).  The inner transitions ν_1,2 are stronger than the outer transitions ν_3,4, in accordance with earlier studies <cit.>. 

Now, we burn a spectrally narrow hole at ν_1 (ν_pump = 21.8  MHz), as shown in Fig. <ref>(b). As expected from our model, excitation of qudit modes leads to the emergence of spectral holes at the ν_2, ν_3  and ν_4 transitions [see the upper curve in Fig. <ref>(b)]. Their spectral positions shift linearly with ν_pump, keeping the frequency difference the same. This is particularly pronounced in another sample with much larger inhomogeneous broadening, isotopically purified 4H-^28SiC (Supplementary Fig. S5). We hence conclude that by varying ν_pump within the ν_1 ODMR line, different spin packets are selected. Remarkably, there is another signal at ν_5 in Fig. <ref>(b), which has the opposite sign and appears only if the ν_1 qudit mode is excited. The same behavior is also observed in 4H-SiC <cit.> (see also Supplementary Fig. S6). These general properties are used to implement spectrally selective coherent control of qudit modes. 

We start with driving the ν_1 transition. Figure <ref>(c) shows Rabi oscillations when the driving power is relatively low W_1 = 11  dBm. The corresponding π-pulse duration of 1.2  μ s provides sub-MHz spectral selectivity. We then spectrally select a spin packet using a long π pulse at ν_1 and coherently drive it at the ν_5 resonance followed by a second long π pulse at ν_1 for the readout [inset of Fig. <ref>(d)]. A high driving power W_5 = 33  dBm yields fast Rabi oscillations as presented in Fig. <ref>(d). For such a power, the π / 2 pulse is 80  ns, corresponding to a bandwidth of approximately 10  MHz. This bandwidth is wide
enough to encompass the ν_5 linewidth. 



To demonstrate advantages of the qudit modes for quantum information processing and sensing, we perform two-frequency Ramsey experiments using the protocol presented in  Fig. <ref>(a) (and Supplementary Fig. S2).  When the probe frequency ν_probe = 6.7  MHz is  equal to the ν_5 qudit mode frequency, the signal represents free induction decay [Fig. <ref>(b)]. We note that though the other, ν_3 resonance lies within the bandwidth, the population of the m_S = -3/2 and m_S = +1/2 is equal after the ν_1 π-pump pulse [Fig. <ref>(a)] and this resonance is not driven. This is also confirmed by the absence of pronounced fringes in the free induction decay of Fig. <ref>(b). 

 When ν_probe = 11.7  MHz is detuned  from the ν_5 mode, Ramsey fringes are clearly detected [Fig. <ref>(c)].  A fit  of these dynamics to cos (2 π f_R τ) exp(- τ / T_2^*) gives the coherence time of spin packets T_2^* = 357 ± 24  ns. For comparison, we also perform standard Ramsey measurements using single MW frequency, but do not observed fringes because of inhomogeneous broadening and non-selective excitation (Supplementary Fig. S7). 

Solid circles in Fig. <ref>(d) represent the fast Fourier transform (FFT) of the experimental data from Fig. <ref>(c). The fitting of these data to a Lorentz function yields the frequency of the Ramsey fringes f_R = 4.51 ± 0.03  MHz, with the corresponding spectral resolution of 30  kHz. Following Eq. (<ref>), the effective magnetic field can be measured with high accuracy as 

    B_eff =  ν_probe - f_R/γ√(1 + 3  sin^2 (θ) ) .

With θ = 19^∘  from the data of Fig. <ref>(a,b), we obtain B_eff = 223 ± 1  μ T. To analyze the spectral selectivity of different spin packets, we marginally vary the external magnetitic field strength and the pump/probe frequencies. The results are summarized in Table <ref>, where measurement No. 1 corresponds to the case discussed above. We then increase B and change ν_pump  to select the same spin packet (measurement No. 2 in Table <ref>). The corresponding FFT of the Ramsey fringes is represented by open squares in Fig. <ref>(d) and a lorentzian fit (the dashed line) yields f_R = 3.92 ± 0.05  MHz and B_eff = 242  ± 2  μ T. A comparison of measurements No. 1 and No. 2 demonstrates the spectral selectivity of spin packets to be approximately 600  kHz. We note that a unusually narrow ODMR linewidth of 500  kHz in SiC  was observed earlier in two-frequency experiments, but its origin has not been discussed <cit.>.  



Measurements No. 1 and No. 3 in Table <ref> demonstrate that the effective magnetic field B_eff seen by different spin packets, which are selected by different ν_pump, is the same within the error bars.  This is a manifestation that the inhomogeneous broadening is caused mostly by local variations of the zero-field splitting 2 D rather than magnetic fluctuations. Remarkably, onother paar of measurements (No. 2 and No. 3) in Table <ref> show that the magnetic field strength can be measured without calibration of the zero-field splitting, which can be used to implement absolute (i.e., immune to thermal noise and strain inhomogeneity) DC magnetometry <cit.> .  

To summarize, we have demonstrated coherent manipulation of spin qudit modes in isotopically-purified SiC at room temperature. We have also developed a theory describing the excitation and detection of these modes in inhomgeneously broadened systems and shown that qudits are characterized by multiple relaxation times. These findings can lead to dipole-coupled networks <cit.>, unconditional electron-nuclear spin registers  <cit.> and spectral selection of highly-coherent individual spins <cit.>, particularly in nanocrystals <cit.>. Our results hence open new possibilities to improve the sensitivity of quantum sensors and execute nontrivial quantum protocols in dense spin ensembles. 



This work has been supported by the German Research Foundation (DFG) under Grants DY 18/13 and AS 310/5,  by the ERA.Net RUS Plus program and the German Federal Ministry of Education and Research (BMBF) within project DIABASE, as well as by the Russian Science Foundation under Agreement #16-42-01098 and RFBR #16-02-00877. V.A.S. was supported by the Alexander von Humboldt-Foundation. A.V.P. also acknowledges the support by the RF President Grant SP-2912.2016.5 and the "BASIS" Foundation.




27
natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL 


[Awschalom et al.(2013)Awschalom,
  Bassett, Dzurak, Hu, and Petta]Awschalom:2013in
authorD. D. Awschalom,
  authorL. C. Bassett,
  authorA. S. Dzurak,
  authorE. L. Hu, and
  authorJ. R. Petta,
  journalScience volume339,
  pages1174 (year2013).

[Kaszlikowski et al.(2003)Kaszlikowski,
  Oi, Christandl, Chang, Ekert, Kwek, and Oh]Kaszlikowski:2003js
authorD. Kaszlikowski,
  authorD. K. L. Oi,
  authorM. Christandl,
  authorK. Chang,
  authorA. Ekert,
  authorL. C. Kwek, and
  authorC. H. Oh,
  journalPhysical Review A volume67,
  pages656 (year2003).

[Lanyon et al.(2008)Lanyon, Barbieri,
  Almeida, Jennewein, Ralph, Resch, Pryde, O'Brien, Gilchrist, and
  White]Lanyon:2008gvc
authorB. P. Lanyon,
  authorM. Barbieri,
  authorM. P. Almeida,
  authorT. Jennewein,
  authorT. C. Ralph,
  authorK. J. Resch,
  authorG. J. Pryde,
  authorJ. L. O'Brien,
  authorA. Gilchrist,
  and authorA. G. White,
  journalNature Physics volume5,
  pages134 (year2008).

[Shlyakhov et al.(2018)Shlyakhov,
  Zemlyanov, Suslov, Lebedev, Paraoanu, Lesovik, and
  Blatter]Shlyakhov:2018dg
authorA. R. Shlyakhov,
  authorV. V. Zemlyanov,
  authorM. V. Suslov,
  authorA. V. Lebedev,
  authorG. S. Paraoanu,
  authorG. B. Lesovik,
  and authorG. Blatter,
  journalPhysical Review A volume97,
  pages022115 (year2018).

[Kues et al.(2017)Kues, Reimer, Roztocki,
  Cortés, Sciara, Wetzel, Zhang, Cino, Chu, Little et al.]Kues:2017db
authorM. Kues,
  authorC. Reimer,
  authorP. Roztocki,
  authorL. R. Cortés,
  authorS. Sciara,
  authorB. Wetzel,
  authorY. Zhang,
  authorA. Cino,
  authorS. T. Chu,
  authorB. E. Little,
  et al., journalNature
  volume546, pages622 (year2017).

[Neeley et al.(2009)Neeley, Ansmann,
  Bialczak, Hofheinz, Lucero, O'Connell, Sank, Wang, Wenner, Cleland
  et al.]Neeley:2009ic
authorM. Neeley,
  authorM. Ansmann,
  authorR. C. Bialczak,
  authorM. Hofheinz,
  authorE. Lucero,
  authorA. D. O'Connell,
  authorD. Sank,
  authorH. Wang,
  authorJ. Wenner,
  authorA. N. Cleland,
  et al., journalScience
  volume325, pages722 (year2009).

[Riedel et al.(2012)Riedel, Fuchs, Kraus,
  Väth, Sperlich, Dyakonov, Soltamova, Baranov, Ilyin, and
  Astakhov]Riedel:2012jq
authorD. Riedel,
  authorF. Fuchs,
  authorH. Kraus,
  authorS. Väth,
  authorA. Sperlich,
  authorV. Dyakonov,
  authorA. Soltamova,
  authorP. Baranov,
  authorV. Ilyin, and
  authorG. V. Astakhov,
  journalPhysical Review Letters volume109,
  pages226402 (year2012).

[Mizuochi et al.(2002)Mizuochi, Yamasaki,
  Takizawa, Morishita, Ohshima, Itoh, and Isoya]Mizuochi:2002kl
authorN. Mizuochi,
  authorS. Yamasaki,
  authorH. Takizawa,
  authorN. Morishita,
  authorT. Ohshima,
  authorH. Itoh, and
  authorJ. Isoya,
  journalPhysical Review B volume66,
  pages235202 (year2002).

[Kraus et al.(2014a)Kraus,
  Soltamov, Riedel, Väth, Fuchs, Sperlich, Baranov, Dyakonov, and
  Astakhov]Kraus:2013di
authorH. Kraus,
  authorV. A. Soltamov,
  authorD. Riedel,
  authorS. Väth,
  authorF. Fuchs,
  authorA. Sperlich,
  authorP. G. Baranov,
  authorV. Dyakonov, and
  authorG. V. Astakhov,
  journalNature Physics volume10,
  pages157 (year2014a).

[Widmann et al.(2015)Widmann, Lee,
  Rendler, Son, Fedder, Paik, Yang, Zhao, Yang, Booker
  et al.]Widmann:2014ve
authorM. Widmann,
  authorS.-Y. Lee,
  authorT. Rendler,
  authorN. T. Son,
  authorH. Fedder,
  authorS. Paik,
  authorL.-P. Yang,
  authorN. Zhao,
  authorS. Yang,
  authorI. Booker,
  et al., journalNature Materials
  volume14, pages164 (year2015).

[Carter et al.(2015)Carter, Soykal, Dev,
  Economou, and Glaser]Carter:2015vc
authorS. G. Carter,
  authorÖ. O. Soykal,
  authorP. Dev,
  authorS. E. Economou,
  and authorE. R.
  Glaser, journalPhysical Review B
  volume92, pages161202
  (year2015).

[Simin et al.(2017)Simin, Kraus,
  Sperlich, Ohshima, Astakhov, and Dyakonov]Simin:2017iw
authorD. Simin,
  authorH. Kraus,
  authorA. Sperlich,
  authorT. Ohshima,
  authorG. V. Astakhov,
  and authorV. Dyakonov,
  journalPhysical Review B volume95,
  pages161201(R) (year2017).

[Tairov and Tsvetkov(1978)]Tairov:1978hd
authorY. M. Tairov and
  authorV. F. Tsvetkov,
  journalJournal of Crystal Growth volume43,
  pages209 (year1978).

[Simin et al.(2016)Simin, Soltamov,
  Poshakinskiy, Anisimov, Babunts, Tolmachev, Mokhov, Trupke, Tarasenko,
  Sperlich et al.]Simin:2016cp
authorD. Simin,
  authorV. A. Soltamov,
  authorA. V. Poshakinskiy,
  authorA. N. Anisimov,
  authorR. A. Babunts,
  authorD. O. Tolmachev,
  authorE. N. Mokhov,
  authorM. Trupke,
  authorS. A. Tarasenko,
  authorA. Sperlich,
  et al., journalPhysical Review X
  volume6, pages031014 (year2016).

[Tarasenko et al.(2018)Tarasenko,
  Poshakinskiy, Simin, Soltamov, Mokhov, Baranov, Dyakonov, and
  Astakhov]Tarasenko:2017ky
authorS. A. Tarasenko,
  authorA. V. Poshakinskiy,
  authorD. Simin,
  authorV. A. Soltamov,
  authorE. N. Mokhov,
  authorP. G. Baranov,
  authorV. Dyakonov, and
  authorG. V. Astakhov,
  journalphysica status solidi (b) volume255,
  pages1700258 (year2018).

[Kraus et al.(2014b)Kraus,
  Soltamov, Fuchs, Simin, Sperlich, Baranov, Astakhov, and
  Dyakonov]Kraus:2013vf
authorH. Kraus,
  authorV. A. Soltamov,
  authorF. Fuchs,
  authorD. Simin,
  authorA. Sperlich,
  authorP. G. Baranov,
  authorG. V. Astakhov,
  and authorV. Dyakonov,
  journalScientific Reports volume4,
  pages5303 (year2014b).

[Christle et al.(2015)Christle, Falk,
  Andrich, Klimov, Hassan, Son, Janzén, Ohshima, and
  Awschalom]Christle:2014ti
authorD. J. Christle,
  authorA. L. Falk,
  authorP. Andrich,
  authorP. V. Klimov,
  authorJ. u. Hassan,
  authorN. T. Son,
  authorE. Janzén,
  authorT. Ohshima, and
  authorD. D. Awschalom,
  journalNature Materials volume14,
  pages160 (year2015).

[Falk et al.(2013)Falk, Buckley,
  Calusine, Koehl, Dobrovitski, Politi, Zorman, Feng, and
  Awschalom]Falk:2013jq
authorA. L. Falk,
  authorB. B. Buckley,
  authorG. Calusine,
  authorW. F. Koehl,
  authorV. V. Dobrovitski,
  authorA. Politi,
  authorC. A. Zorman,
  authorP. X. L. Feng,
  and authorD. D.
  Awschalom, journalNature Communications
  volume4, pages1819 (year2013).

[Sörman et al.(2000)Sörman, Son,
  Chen, Kordina, Hallin, and Janzén]Sorman:2000ij
authorE. Sörman,
  authorN. Son,
  authorW. Chen,
  authorO. Kordina,
  authorC. Hallin, and
  authorE. Janzén,
  journalPhysical Review B volume61,
  pages2613 (year2000).

[Soltamov et al.(2012)Soltamov,
  Soltamova, Baranov, and Proskuryakov]Soltamov:2012ey
authorV. A. Soltamov,
  authorA. A. Soltamova,
  authorP. G. Baranov,
  and authorI. I.
  Proskuryakov, journalPhysical Review Letters
  volume108, pages226402
  (year2012).

[Kehayias et al.(2014)Kehayias,
  Mrózek, Acosta, Jarmola, Rudnicki, Folman, Gawlik, and
  Budker]Kehayias:2014dd
authorP. Kehayias,
  authorM. Mrózek,
  authorV. M. Acosta,
  authorA. Jarmola,
  authorD. S. Rudnicki,
  authorR. Folman,
  authorW. Gawlik, and
  authorD. Budker,
  journalPhysical Review B volume89,
  pages245202 (year2014).

[Simin et al.(2015)Simin, Fuchs, Kraus,
  Sperlich, Baranov, Astakhov, and Dyakonov]Simin:2015dn
authorD. Simin,
  authorF. Fuchs,
  authorH. Kraus,
  authorA. Sperlich,
  authorP. G. Baranov,
  authorG. V. Astakhov,
  and authorV. Dyakonov,
  journalPhysical Review Applied volume4,
  pages014009 (year2015).

[Niethammer et al.(2016)Niethammer,
  Widmann, Lee, Stenberg, Kordina, Ohshima, Son, Janzén, and
  Wrachtrup]Niethammer:2016bc
authorM. Niethammer,
  authorM. Widmann,
  authorS.-Y. Lee,
  authorP. Stenberg,
  authorO. Kordina,
  authorT. Ohshima,
  authorN. T. Son,
  authorE. Janzén,
  and
  authorJ. Wrachtrup,
  journalPhysical Review Applied volume6,
  pages034001 (year2016).

[Fang et al.(2013)Fang, Acosta, Santori,
  Huang, Itoh, Watanabe, Shikata, and Beausoleil]Fang:2013dw
authorK. Fang,
  authorV. M. Acosta,
  authorC. Santori,
  authorZ. Huang,
  authorK. M. Itoh,
  authorH. Watanabe,
  authorS. Shikata, and
  authorR. G. Beausoleil,
  journalPhysical Review Letters volume110,
  pages130802 (year2013).

[Klimov et al.(2015)Klimov, Falk,
  Christle, Dobrovitski, and Awschalom]Klimov:2015bm
authorP. V. Klimov,
  authorA. L. Falk,
  authorD. J. Christle,
  authorV. V. Dobrovitski,
  and authorD. D.
  Awschalom, journalScience Advances
  volume1, pagese1501015
  (year2015).

[Fuchs et al.(2015)Fuchs, Stender,
  Trupke, Simin, Pflaum, Dyakonov, and Astakhov]Fuchs:2015ii
authorF. Fuchs,
  authorB. Stender,
  authorM. Trupke,
  authorD. Simin,
  authorJ. Pflaum,
  authorV. Dyakonov, and
  authorG. V. Astakhov,
  journalNature Communications volume6,
  pages7578 (year2015).

[Muzha et al.(2014)Muzha, Fuchs,
  Tarakina, Simin, Trupke, Soltamov, Mokhov, Baranov, Dyakonov, Krueger
  et al.]Muzha:2014th
authorA. Muzha,
  authorF. Fuchs,
  authorN. V. Tarakina,
  authorD. Simin,
  authorM. Trupke,
  authorV. A. Soltamov,
  authorE. N. Mokhov,
  authorP. G. Baranov,
  authorV. Dyakonov,
  authorA. Krueger,
  et al., journalApplied Physics Letters
  volume105, pages243112
  (year2014).




