Keywords: Pulsed EPR; Rabi oscillations; Spin decoherence; Inhomogeneous microwave field; Longitudinal radiofrequency field; Rabi resonance.

Transcription

1 Suppression of eletron spin deoherene in Rabi osillations indued by an inhomogeneous mirowave field A.P. Saiko, R. Fedaruk, and S.A. Markevih Sientifi-Pratial Materials Researh Centre NAS of Belarus, Minsk, Belarus Institute of Physis, University of Szzein, 70-45, Szzein, Poland Е-mail: Abstrat. The deay of Rabi osillations provides diret information about oherene of eletron spins. When observed in EPR experiments, it is often shortened by spatial inhomogeneity of the mirowave field amplitude in a bulk sample. In order to suppress this undesired loss of oherene, we propose an additional dressing of spin states by a weak longitudinal ontinuous radiofrequeny field. The Gaussian, osine and linear distributions of the mirowave amplitude is analyzed. Our alulations of the Rabi osillations between the doubly dressed spin states show that for all these distributions the maximum suppression of the inhomogeneity-indued deoherene is ahieved at the so-alled Rabi resonane when the radio-field frequeny is in resonane with the Rabi frequeny of spins in the mirowave field. The manifestations of suh suppression in the published EPR experiments with the bihromati driving are disussed. The realization of the Rabi resonane using the radiofrequeny field ould open new possibilities for separating the ontributions of relaxation mehanisms from those due to the inhomogeneous driving in spin deoherene. Keywords: Pulsed EPR; Rabi osillations; Spin deoherene; Inhomogeneous mirowave field; Longitudinal radiofrequeny field; Rabi resonane.. Introdution Coherent manipulation of two-level spin systems (qubits) by eletromagneti fields is attrating interest in pulsed eletron paramagneti resonane (EPR) and in its many appliations inluding quantum information tehnologies [,]. Paramagneti ions and defets in diamagneti solid matrixes are among the promising andidates for solid-state qubits [3-5]. Rabi osillations [6] (or transient nutations [7]) represent the basi phenomenon used for oherent manipulation of spin states. The oupling between the driving eletromagneti field and qubits is haraterized by the Rabi frequeny. The deay of Rabi osillations (transient nutations) provides diret information about the oherene time of the oupled field-qubit system. A detailed understanding of proesses destroying the quantum oherene is of entral importane for quantum omputation. In EPR experiments, the deay of Rabi osillations may results from the intrinsi deoherene in spin systems indued by ouplings to the environment as well as from extrinsi deoherene indued by flutuations and inhomogeneities of external magneti or mirowave fields [3,4]. Spatial inhomogeneities or flutuations of the driving field an be a soure of the so-alled driven deoherene of the Rabi osillations [4]. In partiular, the effet of inhomogeneity in the driving amplitude over a spin ensemble inreases with inreasing this amplitude (the Rabi frequeny) and an signifiantly shorten the deay of the Rabi osillations. Changing the struture of the

2 mirowave field in a avity or dereasing the volume of bulk samples not always an be used to overome spatial driving-field inhomogeneities. Deoherene arising from an inhomogeneity of polarizing magneti field (i.e. from variations in the loal magneti field ating on eah spin) an be reversed by a spin-eho tehnique [8]. Similarly, so-alled rotary ehoes permit refousing the Rabi osillations dephased by inhomogeneity in the driving field amplitude [9]. The rotary eho signals are generated by periodi pulse hanges of the driving field phase by [9,0]. The rotary eho an also be formed by pulse variations in the resonant frequeny of spins using the Zeeman effet []. However, ompliated pulse sequenes of the exiting fields are required in this tehnique. On the other hand, a double dressing of spin qubits by a bihromati field ontaining a strong transverse mirowave (MW) and weak longitudinal radiofrequeny (RF) omponents an also signifiantly extend their observed oherene times. Inreasing in the deay time of Rabi osillations between the doubly dressed states in a omparison with that between the singly dressed states was observed for spin ensembles in EPR [-5] and NMR [6], when the radio frequeny was in resonane with the Rabi frequeny in the MW field. This seondary resonane used for the seond dressing of spin states is sometimes termed rotary saturation [7], nutation resonane [4] or Rabi resonane [8,9]. It has been observed also for a single spin qubit [0]. However, the observed improvement in oherene times has not been studied in detail, though a ompletely different regime of interations with the environment and a refousing of the MW field inhomogeneity were mentioned as the possible origin of the observed improvement. Suh refousing differs essentially from the spin-loking method in whih other relaxation dynamis is used to prolong the spin oherene time. In partiular, keeping a system in long-lived singlet states by spin-loking has been proposed to inrease the overall lifetime of the system []. Reently, it has been shown [] that the double dressing modifies the dephasing and dissipative proesses. This theoretial approah desribes the dissipative dynamis of spin qubits assuming that there are no spatial inhomogeneities or flutuations in the amplitudes of the MW and RF fields over a spin ensemble. The obtained analytial results an be useful in studies of the oherent dynamis at spatial inhomogeneity in the amplitude of the MW field. In the present paper, we theoretially demonstrate that the double dressing of eletron spin states by the bihromati MW and RF field an suppress the dephasing effet of inhomogeneity of the MW field aross the sample and enhanes the observed deay times of Rabi osillations towards their values limited by the spin relaxation proesses. Suh suppression is illustrated for experimental onditions orresponding to the EPR experiments

3 with the Rabi osillations between the doubly dressed spin states of paramagneti nitrogen enters in diamond. We find that the maximum suppression of the inhomogeneity-indued deoherene is ahieved at the Rabi resonane and is feasible even when the RF phase is random. Partiularities of different types of the MW field inhomogeneity desribed by the osine, Gaussian and linear distributions are onsidered. The manifestation of suppression of the observed spin deoherene at the Rabi resonane in some published EPR experiments is disussed.. Deay of Rabi osillations between doubly-dressed spin states at a homogeneous driving In this setion we briefly review the theory developed in previous studies on the Rabi osillations in the doubly dressed spin qubit [4,,3]. We onsider a spin qubit in three fields: a strong MW one oriented along the x axis of the laboratory frame together with weak RF and stati magneti fields, both direted along the z axis (Fig.a). Fig.. (a) Field onfiguration in the laboratory frame. (b) Moving from the bare states in the homogeneous stati magneti field B 0 to the doubly dressed states using the bihromati (MW + RF) field. The resonant ( mw 0 ) MW field dresses eah of initial spin states and splits into two states separated by the Rabi frequeny B. The RF field with 3

4 dresses eah of dressed states and splits into new two states separated by the Rabi frequeny B. The inhomogeneity B auses the distribution B of the Rabi frequenies and the broadening of levels dressed by the MW field. The effet of the inhomogeneity may be redued by making the RF field amplitude B muh larger than B. The Hamiltonian of the qubit an be written as H H H ( t) H ( t), () 0 z where H0 0s is the Hamiltonian of the Zeeman energy of the qubit in the stati magneti field 0 B, ( )os mw z H t s s t and H ( t) s os( t ) are the Hamiltonians of the qubit interation with linearly polarized MW and RF fields, respetively. Here 0 B0, is the eletron gyromagneti ratio, B and B are the Rabi frequenies, B and B, mw and are the respetive amplitudes and frequenies of the MW and RF fields, is the phase of the RF field, the MW phase being set to zero;, z s are omponents of the spin operator, desribing the state of the qubit and satisfying the ommutation relations: [ s, s ] s z z, [ s, s ] s (RWA) is used for the interation between the qubit and the MW field.. The rotating-wave approximation The dynamis of the qubit is desribed by the master equation for the density matrix i H, i t (in the following we take ). The superoperator desribing deay proesses is defined as z D[ s ] D[ s ] D[ s ], (3) where D[ O] OO O O O O, and are the rates of the transitions from the exited state of the qubit to its ground state and vie versa, and is the dephasing rate. Using the Krylov Bogoliubov Mitropolsky nonseular perturbation theory on the parameter /, from Eqs. () (3) we an obtain the density matrix rot [] in the singly rotating frame (SRF), whih rotates with frequeny mw around the z axis of the laboratory frame. The MW field is muh stronger than the RF field,. Moreover, we assume that /, also,, and,,,, where 0 mw, and () 4

5 ( ). In the SRF, the time-dependent absorption of the bihromati field by the / qubit an be written as rot rot t = dd sin ( e )sin( t ) (4) i t t t e t sin sin sin sin( ) os sin os sin( ) 4 t sin (os ) sin( ) (os ) sin( ) t 8 t t sin sin ( ) sin ( ) t os sin (os )sin ( ) t (os )sin ( ) t e, where sin /, os /, [ sin ], BS, /4 BS, sin sin /, os /, ( ) f (, ), ( ) (, ) f, 3,, ( ) /, f (, ) sin ( sin )sin dd ( ) / ( )os os / (if 0, ). The longitudinal, T, and transverse (oherene), T, relaxation times of the bare qubit in the laboratory frame are represented by the respetive relaxation rates as /T and /T. Equation (4) desribes the Rabi osillations of the qubit in the bihromati field and shows that in the SRF the Rabi osillations our at three frequenies, and. These osillations result from the multiphoton transitions between the doubly dressed states of the qubit (Fig. b) [3]. The predited frequenies of the Rabi osillations were onfirmed experimentally [3,4,0]. However, dephasing and relaxation of these states have not been studied experimentally in details until now. It follows from Eq. (4) that the doubly-dressed relaxation rates and depend on a detuning of the MW frequeny from the qubit resonant frequeny as well as on a detuning between the radio frequeny and. At 0 the expressions for and redue to those known for the singly dressed qubits [7]. 3. Deay of Rabi osillations between doubly-dressed spin states under an inhomogeneous MW driving Besides dephasing and relaxation proesses, inhomogeneities of the stati magneti field as well as the MW and RF fields aross the sample an destroy the oherene of qubits inreasing the deay of the Rabi osillations. In partiular, an inhomogeneity in the stati 5

6 magneti field results in the distribution of resonant frequenies of qubits. It is known that near resonane ( mw 0 ), the effet of inhomogeneity B of the stati magneti field in the deay of Rabi osillations may be redued to an insignifiant level by making B muh larger than B [7, 4]. In this ase the effetive field ating on the spins in the rotating frame is almost the same as B, and the inhomogeneity elimination of inhomogeneity resonane. Note that in this method inhomogeneity B is only a seond-order effet. The B weakens with inreasing the detuning from the B is eliminated by a different proess than in the spin-eho method whih reverses dephasing of spin preession due to inhomogeneity B [8]. However, at large B, inhomogeneity B of the MW field beomes important and may result in the more rapid deay of Rabi osillations than it is expeted from relaxation proesses. In the following, we assume that the stati magneti field is homogeneous and we onsider the time evolution for qubits with a homogeneous EPR line in an inhomogeneous MW field. At the resonant MW field ( mw 0 ), inhomogeneity B may be expeted to result in the deay of Rabi osillations in a time of order of / B. At the double resonane in the bihromati field ( mw 0, ), the effet of the inhomogeneity B may be redued by making the RF field amplitude B muh larger than B. In this ase in the doubly rotating frame, whih rotates with frequeny around the singly rotating frame, the homogeneous field B is exatly aneled by the fititious field /. In the presene of inhomogeneity B the anelation is inomplete, as illustrated in Fig. a. However, if B B, the effetive field / depends only slightly on the value of ( / B ). Far from resonane ( ) the effetive field / has a muh larger B omponent along B than at resonane. Consequently, the effet of inhomogeneity B beomes stronger (Fig. b). Fig.. The formation of effetive field / in the doubly rotating frame in the presene of an 6

7 inhomogeneity B in the MW field amplitude. (a) Spins are exatly at resonane. (b) Spins are far from resonane. An inhomogeneity in the MW field amplitude gives rise to the distribution of the Rabi frequenies of qubits. The distribution of the Rabi frequenies aross the sample is a omplex funtion of the sample sizes and the struture of the MW field in the resonator. Often this distribution is unknown in detail. In some ases it an be approximated by the Gaussian funtion g( ) exp ( ), where is the entral Rabi frequeny and is the width of the distribution. We also study the Rabi osillations for other types of an inhomogeneity in B desribing the distribution of by osine and linear funtions. 4. Results and disussion In our alulations we use the relaxation parameters that are harateristi for suh defets in solids as the isolated neutral substitutional nitrogen (P enter) and the negatively harged nitrogen vaany (NV) enter in diamonds. The long-lived Rabi osillations of these defets an be observed in the pulsed EPR even at room temperature. Their spin oherene time T is hanged from a few miroseonds in nitrogen-rih diamond to a few milliseonds in ultrapure diamonds [5,5]. The NV enters are among the most promising andidates for solid-state qubits operating under ambient onditions [5]. Figure 3a displays the Rabi osillations exited by the homogeneous MW field in the absorption EPR signal at the resonant MW driving, mw 0. The Rabi osillations our at the frequeny and their deay is aused by the relaxation proesses. The inhomogeneous MW field with B, gives rise to the distribution of the Rabi frequenies of qubits that result in the very fast deay of Rabi osillations. The absorption signals averaged numerially over the distribution g( ) are represented in Fig. 3a by the red and green lines for g( ) approximated by the Gaussian and osine funtions, respetively. 7

8 Fig. 3. The Rabi osillations exited by the homogeneous (grey lines) and inhomogeneous (red and green lines) MW field without (a) and with the resonant (b) and non-resonant () RF driving. The parameters are = 0.0 MHz, 0, T = 5 s, and T = ms. The resonant ( ) and non-resonant ( ( ) / 0.5 MHz) RF field with the amplitude / 0. MHz is used. The distribution of is approximated by the Gaussian (red lines) and osine (green lines) funtions. For the Gaussian distribution 0.08 MHz. For the osine distribution the relative hange of the MW amplitude is about The RF phase is random. In order to demonstrate the suppression of deoherene in the Rabi osillations using the seond dressing of spin states by a longitudinal ontinuous RF field, we assume in our alulations that the stati magneti field and the RF field are homogeneous and that the deay rate of the Rabi osillations is mainly due to the inhomogeneity of the MW field amplitude. Moreover, the RF field amplitude B B and B is muh larger than the inhomogeneity B. Figures 3b and 3 demonstrate the Rabi osillations in the doubly resonant bihromati field when the frequeny of the RF field is equal or near to the Rabi frequeny. The amplitude B of the seond driving is 50 times of magnitude weaker than the MW amplitude B (i.e. 8

9 ). The signals were alulated with and without taking into aount effets of inhomogeneity in. We assume that there are no random flutuations in the amplitude of the MW field and average a large number of the osillations assuming that the RF phase is random. In aordane with Eq. (4), at the bihromati driving the Rabi osillations our at frequenies and. At the Rabi resonane ( ), and the deay rate of the Rabi osillations in the doubly resonant bihromati field is learly slower than the one in the monohromati field. Figure 3 shows the osillations at detuning the radio frequeny from. Besides an inrease of the value of, detuning from the resonane results in inreasing the deay rate of the Rabi osillations and at large values of the detuning the deay rate reahes its value observed in the MW field. There is no suppression of deoherene aused by the MW inhomogeneity. The deay of Rabi osillations as a funtion of detuning for different types of inhomogeneity of the MW field is shown in Fig. 4. For the relative hange of the MW amplitude of about (i.e. 60 khz), eah type of suh inhomogeneity results in the omparable deay of the Rabi osillations in the MW field. At the bihromati driving, the B -indued deay rate is muh stronger sensitive to the detuning from the Rabi resonane than the deay rate in the homogeneous MW field. The deoherene from the all onsidered types of MW inhomogeneity is suppressed only at. The suppression is dereases with detuning ausing the faster deay. In partiular, if there is a Gaussian distribution of the MW field amplitude of width 0.08 MHz and a detuning of the radio frequeny ( ) / 0.5 MHz, the fastest deay time of the osillations roughly dereases by a fator of 6.7 (Fig. 4). At the homogeneous MW field and the same detuning, the hange in the deay times is about 7% (Fig. 4a). Note that for the homogeneous MW field at the detuning / = 0.5 MHz and 0.9 MHz the deay rate is even smaller than the one is at the exat Rabi resonane (Fig. 4a). That is due to the different -dependene of the deay rates and the amplitudes for the osillations at 9 and []. For the symmetri distribution of, the deay is the same for the different sign of detuning (Fig. 4b and ). For the asymmetri distribution of, the deay beomes dependent on the detuning sign (Fig. 4d). The presented results demonstrate that for the all onsidered types of the MW inhomogeneity the doubly resonant bihromati field suppresses the dephasing effet of the inhomogeneity and inreases the deay of the Rabi osillations towards its value limited by the relaxation times.

10 Beause the used RF field is weak ( ), the effet of its inhomogeneity is also muh weaker than the one from the B -inhomogeneity. Therefore, it is negleted in our desription. The inhomogeneities of the MW field annot influene the homogeneity of the RF field beause the relatively low-frequeny ontinuous RF field is produed by the additional modulation oil. Unlike the longitudinal magneti-field modulation, at the phase or frequeny modulation of the MW field, the inhomogeneity in B leads to the inhomogeneity in B. Fig. 4. The deay of Rabi osillations exited by the bihromati driving as a funtion of detuning for different types of inhomogeneity of the MW field. The MW field is homogeneous (a). The distribution of is approximated by the osine (b), Gaussian () and linear (d) funtions and is shown in insets. Other parameters are the same as in Fig. 3. As we found in Ref. [], in the homogeneous MW and RF fields the amplitude of the Rabi osillations at eah frequeny ( and ) is strongly dependent on the RF field phase []. This dependene an be used to realize onditions when the effet of inhomogeneity B will be suppressed with the minimal distortion of the Rabi osillations in the MW field. Fig. 5 illustrates the effet of the RF field phase in the inhomogeneous MW 0

11 field. At the exat resonane and the RF field phase 0, the signals at the frequenies are only due to the Bloh-Siegert effet in the RF field and they are muh smaller than the signal at []. For small values of ( ), the ontribution of the osillations at the frequenies to the summarized signal is negligible. In this ase the RF field suppresses the dumping of the Rabi osillations due to inhomogeneity B and the osillations our at the frequeny without their modulation by the frequeny. Moreover, the deay of the Rabi osillations is pratially the same as the one observed at the homogeneous MW driving (Fig. 5a). At the exat resonane and the RF phase /, the B -indued dumping of the Rabi osillations is also suppressed (Fig. 5b). However, in this ase the osillations our at frequenies and their deay is something faster than the one at the homogeneous MW driving. That is beause the deay rate of the osillations at frequenies is faster than the deay rate of the osillations at. It is interesting that for the Gauss distribution of the double dressing reovers the deay rate to its value in the homogeneous MW field. For 0 and /, detuning from the resonane destroys the suppression of deoherene (Fig. 5 and d). Fig. 5. The Rabi osillations for two different values of the RF phase at the homogeneous

12 (grey lines) and inhomogeneous (red and green lines) MW field. The RF phase equals to 0 (a,) and / (b,d). Other parameters are the same as in Fig. 3. To analyze deay kinetis, Fig. 6 shows in the logarithmi sale the deay of the Rabi osillations at the Gaussian distribution of the MW field amplitude without and with the resonant and non-resonant RF driving. If no RF driving, the inhomogeneous MW field results in the fast deay whih is well desribed by the Gaussian envelope exp( at / ) with /a 5.5 μs (line ). Note that in the similar situation suh deay was roughly approximated by a polynomial-type dependene [3,6]. The resonant RF driving suppresses deoherene from the MW inhomogeneity and the exponential deay with /b 48.8 μs ours (line ). This deay time is lose to the value of 49.4 μs aused by the relaxation proesses in the homogeneous MW field. At the non-resonant RF driving, the deay of the Rabi osillations has a omplex form (line 3). Its well-separated fast and slow omponents are approximated by exp( at / ) with /a 6.6 μs (the paraboli part of line 3) and exp( bt) with /b 7 μs, respetively. Fig. 6. The deay of the Rabi osillations at the Gaussian distribution with 0.08 MHz of the MW field amplitude without () and with the resonant () and non-resonant (3) RF driving with the amplitude / 0. MHz and the phase 0. At the non-resonant RF driving ( ) / 0.5 MHz. Other parameters are = 0.0 MHz, 0, T = 5 s, and T = ms. The straight and paraboli lines show the fit to the exponential and Gaussian funtions, respetively. In our paper the Rabi osillations during the long MW pulse are onsidered. Therefore, it is diffiult to ompare diretly this ase with the existing sequenes used to orret the MW

13 field inhomogeneity [7-9]. However, we an estimate fidelity of the spin dynamis omparing the Rabi osillations in the inhomogeneous MW and resonant RF fields with those in the homogeneous MW field (without the RF field). The last osillations an be used as an ideal spin rotation. The four panels show the Rabi osillations after 0, 50, 00, 50 μs and demonstrate the same frequeny at the single MW and bihromati driving even after 50 μs (Fig. 7). The deay in the homogeneous MW field is ditated purely by the relaxation proesses. The deay in the inhomogeneous MW and resonant RF fields is slightly faster due to the residual rotation angle error. Our analytial and numerial alulations give fidelity of Fig. 7. The Rabi osillation in the homogeneous MW field (grey lines) and in the inhomogeneous MW and resonant RF fields (red lines). Other parameters are = 0.0 MHz, 0, T = 5 s, and T = ms, / 0. MHz, 0, 0.08 MHz. As it was mentioned in Introdution, inreasing in the deay time of Rabi osillations was observed for spin ensembles in the pulsed EPR [3-5,30] and NMR [6] experiments with the double driving using the RF modulation of the magneti field. However, the theoretial desription of these experimental results was mainly oriented on the oherent dynamis of the doubly dressed states without the quantitative examination of deoherene. Our approah gives the opportunity to desribe these results taking into aount the modified relaxation between the doubly dressed states as well as the dephasing due to B inhomogeneity. As an example, Fig. 8 depits the simulation of the Rabi osillations in the onditions similar to those in the EPR experiments with the bihromati driving of P enters 3

14 in a single diamond rystal [5,30]. There is a good agreement between the alulated and observed signals. Our alulations explain an inrease of the spin oherene time almost ten times observed in these experiments via the deay of the Rabi osillations. The resonant harater of the improvement in the observed oherene time is a manifestation of the Rabi resonane. For the Е enters in rystalline quartz suh inrease in the spin oherene time was about four times beause the weaker B inhomogeneity was in a volume of the smaller used sample [3,4]. Fig. 8. The Rabi osillations exited by the homogeneous (grey lines) and inhomogeneous (red and green lines) MW field without (a) and with the resonant (b) RF driving. The parameters are.0 MHz, 0, = 0.4 MHz, T = 5 s, and T = ms. The distribution of is approximated by the Gaussian funtion with 0.05 MHz. In some EPR experiments it is diffiult to identify the soure of the deay of Rabi osillations and to separate the ontributions of spin deoherene aused by relaxation and the inhomogeneous driving [3,4]. Our results demonstrate that the use of an additional longitudinal ontinuous RF field during the exitation of Rabi osillations is useful for suh separation. In the onsidered approah for overoming an inhomogeneity in the driving MW amplitude we apply an additional longitudinal ontinuous RF field. The same effetive Hamiltonian an be generated by a weak additional transverse MW driving (see Appendix A). In this ase the differene between two MW frequenies mathes the Rabi frequeny in the strong MW field. In order to avoid the use of an extra MW soure, it is simpler to apply the weak RF field. However, this type of bihromati transverse tehnique may be experimentally easier to implement in ertain lower-field EPR appliations and for ertain sample geometries. 4

15 Reently, the amplitude [3] or phase [3] modulation of the transverse MW driving field was used to dress doubly spin qubits. An order of magnitude improvement in the oherene time for NV enters in diamond was experimentally and theoretially demonstrating with these modulation shemes. It was shown that these dynamial deoupling shemes protet against slow flutuations in the MW driving amplitude. The flutuation of MW field amplitude is muh slower than the relaxation time T in the sheme with a seondorder driving field and it is muh slower than T for high-order driving fields. Both spatial inhomogeneities and slow flutuations in the MW amplitude shorten the deay of the Rabi osillations. Though the theoretial desription of spatial and time hanges in the MW field amplitude is different, the deoherene aused by both these soures an be refoused using a seond weak driving to realize the Rabi resonane. 5. Conlusions We have demonstrated for an eletron spin system that the ombination of the strong resonant MW exitation with a weak longitudinal ontinuous RF driving suppresses deoherene of the Rabi osillations due to the MW field inhomogeneity. For the Gaussian, osine and linear distributions of the mirowave amplitude, it was shown that the maximum suppression is ahieved when the radio frequeny mathes the Rabi frequeny in the MW field. The resonant RF driving leads to the exponential deay of the osillations whih is free from any influene of the MW field inhomogeneity and auses only by the relaxation proesses. At the non-resonant RF driving, the deay has a omplex form with the fast (Gaussian) and slow (exponential) omponents. The fast omponent is mainly due to the MW amplitude distribution. The obtained results allowed us to interpret inreasing in the deay time of Rabi osillations observed previously for spin ensembles in the pulsed EPR experiments with the bihromati driving. The dependene of the deay of the Rabi osillations on the RF frequeny was found to be strongly different in the inhomogeneous and homogeneous MW fields. The high sensitivity of the deay in Rabi osillations to the detuning from the Rabi resonane at the MW field inhomogeneity open new possibilities for separating the ontributions of relaxation mehanisms and the inhomogeneous driving in spin deoherene. Suh disrimination of intrinsi and extrinsi deoherene of spin qubits an be useful in EPR and NMR experiments. The ontinuous RF modulation of the magneti field is easier to realize experimentally than ompliated pulse MW sequenes. This approah an be ombined with various quantum information proessing tasks, eliminating the inhomogeneityindued deoherene. Aknowledgment 5

16 The work was supported by Belarusian Republian Foundation for Fundamental Researh and by State Program of Sientifi Investigations Physial material siene, new materials and tehnologies, Appendix A. Spin qubit driven by two transverse MW fields We onsider a spin qubit with the ground and exited states driven by two MW fields oriented along the x axis of the laboratory frame. The Hamiltonian of the qubit an be written as H H H ( t) H ( t), (A) 0 z where H0 0s is the Hamiltonian of the qubit in the stati magneti field, i t and H t ( s e h..) i t H t ( s e h..) are the Hamiltonians of the qubit interation with strong and weak MW fields with frequenies and, respetively; and denote the respetive interation onstants, and, z s are omponents of the spin operator. We assumed that, and use the RWA for the interation between the qubit and the MW fields. We also assume that, 0 and, where. The dynamis of the qubit is desribed by the orresponding master equation for the density matrix (see Eqs. () and (3)). z After three anonial transformations: u3 uu uu u3, where u exp( its ), y z u exp( i s ) and u3 exp( its ), the master equation is transformed into,. Here i t H i 4 4 z D[ s ] D[ s ] D[ s ], i t Rapidly osillating ( e i t, e i t i t i t z i t it ( ).. (..) H e e e s h s e e h 4, ( ) /,. ) terms in the Hamiltonian H an be eliminated by the Krylov Bogoliubov Mitropolsky method [33]. Assuming that / and averaging over the period /, we obtain the following effetive Hamiltonian in / : () () () i t eff eff eff with Heff H( t) ( / 4) s e h.. H H H H t () and H up to the seond order z Heff ( i / ) [ d H ( ) H ( ), H ( t)] BSs. In the above, the symbol... eff T denotes time averaging: A( t) A( t) dt, where T / and T 0 /6 is the BS 6

17 Bloh Siegert-like frequeny shift. After the anonial transformation u 4u4, u i t H i with z i ts 4 e, the equation for is transformed into, z H ( ) s (/ 4) ( s s ). The solution of this equation an be written as BS t t (os os os sin ) sin e t i t e s h.. (A) sin os ( t e e t os t) s z. Here ( BS ) ( / ) os (, BS ) /, ( ) ( )sin, 4 the absorption signal is given by / ( )sin. Using Eq. (A), in the SRF, t rot rot e os (os )sin( ) t (os )sin( ) t i 4 t e sin sin( ) t. (A3) Eqs. (4) and (A3) have the same mathematial strutures, if 0, 0, and are replaed by and /, respetively. It means that at the weak transverse MW an be used to suppress spin deoherene in Rabi osillations indued by the inhomogeneous strong MW field. Referenes. A. Shweiger, G. Jeshke, Priniples of pulse eletron paramagneti resonane, Oxford University Press, UK, 00.. T. Takui, L. Berliner, G. Hanson, Eletron spin resonane (ESR) based quantum omputing, Springer, New York, E. Baibekov, I. Kurkin, M. Gafurov, B. Endeward, R. Rakhmatullin, G. Mamin, Coherene times and Rabi osillations in CaWO4:Cr 5+ rystal, J. Magn. Reson. 09 (0) H. De Raedt, B. Barbara, S. Miyashita, K. Mihielsen, S. Bertaina, S. Gambarelli, Quantum simulations and experiments on Rabi osillations of spin qubits: Intrinsi vs extrinsi damping, Phys. Rev. B 85 (0) T. Yamamoto, T. Umeda, K. Watanabe, S. Onoda, M.L. Markham, D.J. Twithen, B. Naydenov, L.P. MGuinness, T. Teraji, S. Koizumi, F. Dolde, H. Fedder, J. Honert, J. Wrahtrup, T. Ohshima, F. Jelezko, J. Isoya, Extending spin oherene times of diamond qubits by high-temperature annealing, Phys. Rev. B 88 (03)

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20 Graphial Abstrat The Rabi osillations exited by the homogeneous (grey lines) and inhomogeneous (red lines) MW field Highlights Suppression of deoherene in Rabi osillations indued by inhomogeneous mirowave field. Weak longitudinal radiofrequeny field is used to suppress spin deoherene. The Gaussian, osine and linear distributions of the mirowave amplitude are analyzed. The maximum suppression of spin deoherene is ahieved at Rabi resonane. A good agreement with the published experimental results is obtained. 0