Towards scalable quantum computers: nano-design and simulations of quantum register Sergei Ya. Kilin1, Alexander P. Nizovtsev1, Alexander S. Maloshtan1, Alexander L. Pushkarchuk2, Vadim A. Pushkarchuk3, Semen A. Kuten4, Fedor Jelezko5 and Joerg Wrachtrup5 1
B.I.Stepanov Institute of Physics NASB, Independence Ave, 68, 220072 Minsk, Belarus, e-mail:
[email protected] 2 Institute for nuclear problems, BSU, Bobruiskaia, 11, 220050 Minsk, Belarus, 3 BSUIR, P. Brovki 6, 220013 Minsk, Belarus 4 Iinstitute of Physical Organic Chemistry NASB, Surganova 13, 220072 Minsk, Belarus, 5 3. Institute of Physics, University of Stuttgart, Pfaffenwaldring, 57, 70550 Stuttgart, Germany ABSTRACT
Quantum information technology (QIT) is extremely fast developing area strongly connected with achievements in modern physics. We present a review of recent achievements in implementation of solid-state scalable quantum processors with special emphasize on diamond-based quantum hardware. Keywords: Quantum information, quantum computers, diamond-based spintronics.
1. INTRODUCTION The field of quantum informatics (QI) is extremely fast growing towards its possible implementations. New achievements in investigations of different candidates for prototypes of QI hardware (qubits, gates, scalable quantum processors, quantum memories and repeaters, etc.) are emerging in a short time scale, making leading edge of the investigations wider and more flexible. Different fields of science and technology contribute to the QI technologies with unknown final leader of possible commercial implementations. Together with general review of QI hardware we discuss in more details the solid-state implementations of QC and more specifically the diamond-based systems1-4. Paramagnetic defects in diamond like the NV color center can be individually observed and their spin state can be optically detected1-4. Even at room temperature, they have coherence times sufficient for non-trivial coherent spin manipulation. In the material, the NV center electron spin coherence time T2 is limited by local magnetic field fluctuations induced by flip-flops of electron or nuclear spins of nearby defects. Recent observations have demonstrated T2 ~350 μs at room temperatures in isotopically pure diamond. Nuclear spins in semiconductors are also envisaged as carriers of quantum information owing to their very long coherence time, lasting up to seconds. In the diamond crystal lattice, nuclear spins like 13C carbon can be naturally coupled via hyperfine interaction (hfi) to the electron spin of neighboring NV center. This coupling leads to a hyperfine structure in the color center energy levels4 which has been proposed5 as the basis of a scalable quantum processor built with the help of two-qubit quantum gates, formed by the NV color center and a set of neighboring 13C nuclear spins. It was shown6 that optical read-out of the electron spin state indeed gives access to the single nuclear spin state and allows to implement a two-qubit CROT quantum logical gate. As a further step, few 13C nuclei located in the first and the next coordination shells around individual NV center can be detected and manipulated by means of microwave (MW) and radiofrequency (RF) fields7. It means that their states can be used as register for quantum logical gates (QLG)8. To implement different QLG with this register, full understanding of its spin properties is necessary. Recently, coherent coupling between electronic spin of the NV center and nuclear spin of proximal 13C atom (I=1/2) have been studied experimentally9,10 opening the door for coherent manipulation by the spin state of proximal 13C nuclear spins and entangling them with the electron spin of the NV center. It was concluded in9,10 from simple analysis that both isotropic and anisotropic parts of the hyperfine interaction (ihfi and ahfi) and principal values of the hyperfine tensor between NV center and 13C atoms contribute to the extent of the entanglement. It is one of the aims of this paper to simulate the spin properties of such quantum register using complementary methods of quantum chemistry and spin-Hamiltonian. The other aim is to simulate various diamond nanostructures which are used to optimize extraction of the light emitted by NV centers.
Twelfth International Workshop on Nanodesign Technology and Computer Simulations, edited by Alexander I. Melker, Vladislav V. Nelayev, Proc. of SPIE Vol. 7377, 737711 © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.837010 Proc. of SPIE Vol. 7377 737711-1
2. Ab-initio SIMULATION OF DIAMOND-BASED QUANTUM REGISTER To start the work with this register we have to know the characteristics of hfi splitting to be able to control correctly the individual qubits from the register. One of the important tools to characterize this ability is quantum chemistry calculations of spin density distribution in hydrogen-terminated nano-diamond clusters. The nitrogen atom and vacancy forming the [NV]- defect center were placed in the center of the clusters. Calculations of the optimized geometrical structure of clusters as well as calculations of spin density distributions were carried out within frameworks of the semi-empirical quantum-mechanical PM3 and DFT methods using the GAMESS11 and CAChe12 software packages. The calculations have been performed for singly negatively charged clusters in the triplet ground state. The first coordination shell of the [NV]- defect consisted of six carbon atoms with three of them being the nearest neighbors (NN) to the vacancy and the remaining three being NN to the nitrogen atom of the center. More distant carbon atoms, the next nearest neighbors (NNN), form the second, the third etc. coordination spheres around the center. (n)
describing super-hyperfine interactions (shfi) between Here we are mainly interested in calculation of tensors A electronic spin S=1 of the NV center and nuclear spins I=1/2 of various (n-th) isotopic 13C atoms disposed in different positions around the NV center. Usually, the shfi tensors
A
(n)
(x,y,z)(n) coordinate system (PACS) which is specific for each
are presented as diagonal tensors in the principle axes (n)
13
(n)
С atoms. Diagonal elements Axx , Ayy
(n)
Azz
of the
(n)
diagonalized A tensors are usually termed as hyperfine constants for the n-th nucleus. Orientations of the specific PACS (x,y,z)(n) with respect to the “molecular” coordinate system are determined by respective directional cosines. As is well know, the tensors A(n) can be decomposed into an isotropic (Fermi-contact) and an anisotropic (dipole-dipole) parts:
A
(n)
(n)
= Aiso +
the nucleus ( a
(n)
A
(n) aniso
(n)
where the first term Aiso =
a
(n)
I
is proportional to the spin density localized at the place of
is the isotropic shfi constant) while the second part resulting from dipole-dipole interaction of localized
nuclear spins with delocalized electronic spin is described by the traceless tensor with diagonal elements ( −b 2b
(n)
(n)
,- b
(n)
,
). The aforementioned software packages allow calculations of both isotropic and anisotropic contributions to the (n)
shfi tensors A for every carbon atoms in the considered NV-containing carbon clusters along with calculations of respective directional cosines. Previously13,14 we have studied rather small clusters C27H36[NV]-, C33H36[NV]- and C36H42[NV]- and found that in accordance with the known ESR data15 as well as with the other ab initio calculations16 the spin density is localized mainly in the region of the dangling bonds of the carbon atoms being NN to the vacancy. In accordance with our aim to study quantum register, consisting of the NV center and few nuclear spins of 13C located at different distances from the center here we have simulated larger cluster, more specifically, the С69H84[NV]- cluster which allows us to calculate shfi tensors for C atoms being more distant from the NV center. To unify and simplify the analysis of the considered spin system NV+ few 13C from the early beginning we calculated (n)
for various 13С atoms in the only one coordinate system wherein the Z axis was along the [111] the shfi tensors A direction of the diamond lattice, while X and Y axes were fixed arbitrary. To be more specific we took the Y axis to lie in the plane passed through the nitrogen atom of the NV center and one of the carbon atom being NN to the vacancy of the center. In this case the two other nearest-to-vacancy carbon atoms were disposed nearly symmetrically about the Z-Y plane. Note that the chosen coordinate system X,Y,Z is the principle axes coordinate system of the electronic spin of the (n)
tensors, calculated in the PACS NV center where the Z axis is the С3V symmetry axis of the center. Evidently, the A (X,Y,Z) were not diagonal but these were exactly the tensors which can be used in spin-Hamiltonian-based calculations (n)
(n)
(see below) of eigenstates and eigenvalues of the considered system of interacting spins. Principle values Axx , Ayy (n)
Azz
of the respective shfi tensors can be easily founded by diagonalization of the calculated
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A
(n)
tensors.
(n)
(n)
The histograms in Fig. 1 show the shfi constants Axx , Ayy
(n)
Azz , calculated by DFT nethod using the MINI/3-21G
basis set and B3LYP functional, for different nuclei in the cluster C69H84[NV]-. It is seen from this histogram that along with the C5, C6, and C7 carbon atoms, which are the NN to the vacancy, nine other more distant carbon atoms have (n)
(n)
(n)
Azz .
Principal value Axx (MHz)
appreciable values of shfi constants Axx , Ayy
3 NN (C5 C6 C7) (~125.8-126.3 MHz)
125 100
Axx
75 50 25 0
9 NNN C atoms (~7.0-8.2 MHz)
N atom (~-2.0 MHz)
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71
Number of the atom in the cluster
Principal value AYY (MHz)
125
3 NN C atoms (C5 C6 C7) (~126.0-126.5 MHz)
100
Ayy
75 50
b)
N atom (~ -2.0 MHz)
9 NNN C atoms (~ 7.2 - 8.4 MHz)
25 0
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71
Principal value Azz (MHz)
a)
3 NN (C5 C6 C7) (~193.3-193.7 MHz)
125 100
Azz
75 N atom (~-2.0 MHz)
50
9 NNN C atoms (~11.1 - 12.9 MHz)
25 0
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71
Number of the atom in the cluster
Number of atom in cluster
c)
d) (n)
(n)
Fig. 1. Graphical representation of the relaxed C69H84[NV]- cluster ( fig. 1a) and calculated shfi constants Axx , Ayy (n)
Azz
for various nuclei in the cluster ( figs. 1b-1d). The numbers on atoms in Fig. 1a correspond to the numeration
of atoms in the following histograms: the number of the N atom is 1; the number of the vacancy is 71; the C5, C6 and C7 are the carbon atoms, being NN to the vacancy, which have the highest values of all shfi constants. Passivating H atoms are not shown in fig. 1a. In the Table 1 we are presenting the calculated principal values Axx, Ayy, Azz of the shfi tensors for those C atoms in the С69H84[NV]- cluster which have rather high shfi with electronic spin of the canter. For comparison, we show also in this Table the experimental data (values of Axx[15], Ayy[15], Azz[15] taken from article15) and theoretical results obtained from supercell calculations16 (valies Axx[16], Ayy[16], Azz[16]). All values are in MHz.
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Table 1. Calculated principal values Axx, Ayy, Azz of the shfi tensors for those C atoms in the С69H84[NV]- cluster Atom C (NN) (C5) C (NN) (C6) C (NN) (C7) C (NNN) (C33) C (NNN) (C36) C (NNN) (C43) C (NNN) (C46) C (NNN) (C47) C (NNN) (C48) C (NNN) (C67) C (NNN) (C69) C (NNN) (C70)
Axx Ayy Azz Axx [15] Ayy [15] Azz[15] 125.8 126.0 193.3 125.7 125.9 193.4 ±123 ±123 ±205 126.3 126.5 193.7 8.0 8.3 12.7 8.2 8.3 12.8 6.9 7.0 11.1 7.2 7.3 11.5 8.3 8.5 13.0 ±15.0 ±15.0 ±15.0 7.9 8.1 12.5 7.0 7.2 11.3 7.9 8.1 12.5 8.2 8.4 12.9
Axx [16] 109.5 (3 equi-valent C atoms) 13.5 (6 equi-valent C atoms) 12.8 (3 equi-valent C atoms)
Ayy [16] 110.2 (3 equi-valent C atoms) 14.2 (6 equi-valent C atoms) 12.8 (3 equi valent C atom)
Azz[16] 185.4 (3 equi-valent C atoms) 19.4 (6 equi-valent C atoms) 18.0 (3 equi-valent C atoms)
These findings are in a reasonable agreement with an available experimental data and show that rather distant 13C nuclear spins around NV--center can be used as qubits to implement quantum register with optical access. The simulated hfi parameters have been used in the analysis of spin properties of quantum registers consisting of few 13C atoms around the NV center.
3. SPIN HAMILTONIAN SIMULATION OF SPIN PROPERTIES OF QUANTUM REGISTER ON 13С NUCLEAR SPINS NEARBY THE NV CENTER Large shfi between electronic spin S=1 of ground-state 3A of the single NV center and some nearby 13С nuclear spins along with spin-selective photophysics of the NV center17 provide the unique opportunity to initiate and read-out the individual nuclei spin states optically at room temperature, while manipulating them with microwave/radiofrequency pulse series1,2,4. Recently such spin systems have been studied experimentally10 by optically detected magnetic resonance (ORMR) method on single NV centers in 13С-enriched diamond samples (8.4% and 20.7% content of 13С isotope). The ODMR spectra have been monitored for single NV centers having one, two and three 13С atoms in NN positions to the vacancy exhibiting characteristic splitting of ~130 MHz due to strong shfi between electronic spin and the nuclear spins. Moreover, additional characteristic lines in ODMR spectra with shfi splitting of ~13 MHz have been observed in these diamond samples which have been ascribed to the single NV centers coupled to one or two 13С atoms in more distant positions (NNN C atoms). To elucidate the spin properties of such systems of coupled spins and describe these new experimental ODMR data we consider spin Hamiltonian
H = ge βe S B + S D S + ∑n S A( n ) I( n ) ,
(1)
where the first term describes the interaction of electronic spin S of the NV center with external magnetic field B (ge =2.0028 is the electronic g -factor of the NV center, β e is Bohr magneton) and the second one takes into account zerofield splitting of the ground triplet state of the center. D is diagonal tensor with elements DXX=D/3, DYY=D/3, DZZ=-2D/3, D=2.88 GHz 15 in the PACS (X,Y,Z) of the center, wherein the Z axis is the С3V symmetry axis aligned along the [111] direction of diamond lattice. The third term in Eq. (1) describes shfi of the electronic spin of the NV center with neighboring 13С nuclear spins, where A(n) are shfi tensors for various 13С atoms. For NV centers having one 13С atom in the nearest-to-vacancy position the isotropic a=150.5 MHz and anisotropic b=27.2 MHz shfi constants have been deduced from ESR experiment 15 along with the respective directional cosines [0.6269 -0.5509 -0.5509] of the z-axis of the PACS for this 13C with respect to the NV PACS. Note that the angle between the above z-axis of the 13C PACS and the Z-axis of the NV PACS consists of 105.910 in coincidence with the conclusion of the work10. For the other two 13C atoms being also located in the nearest-to-vacancy positions the analogous cosines values [-0.5509 0.6269 -0.5509] and [-0.5509 -0.5509 0.6269] were deduced from the above EPR data by analysis which took into account the diamond lattice geometry, C3V symmetry of the center and magnetic equivalence of the three 13C atoms from the first shell. Knowing them we transformed the diagonal shfi matrices with elements Axx=Ayy=a-b=123.3 MHz and Azz =a+2b=204.9 MHz into the NV PACS and got respective non-diagonal shfi matrices
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A(n). After substitution of these matrices into the spin Hamiltonian (1) it was diagonalized numerically to found eigenenergies and eigenstates of spin systems consisting of the NV electronic spin interacting with one, two and three nuclear spins 13C atoms located in the nearest-to-vacancy positions. The spin wavefunctions thus obtained along with the conventional selection roles for EPR and NMR transitions have been used to calculate ODMR spectra for these cases. Fig. 2 and 3 show that the results of our simulations are in very good accordance with respective experimental ODMR spectra obtained both at zero (Fig. 2) and non-zero (Fig. 3) external magnetic field. Detailed analysis of these spectra from the viewpoint of transitions between spin sublevels will be done elsewhere. Analogous simulations of the ODMR spectra have been done using instead of the shfi A(n) tensors in the NV PACS determined by the above geometrical analysis those ones obtained by direct ab initio simulation of hydrogen-terminated carbon cluster C69H84[NV]- (see section 1). Again the details of comparison of the two approaches will be discussed elsewhere. Here we just would like to certify that we found the results be practically indistinguishable. Simulation 13
ODMR signal intensity (a.u.)
(а) 0 С in the first shell
13
(b) 1 С in the fisrt shell
13
(в) 2 С in the first shell
13
(г) 3 С in the first shell
2600
2700
2800
2900
3000
3100
MW frequency (МGz)
a) b) Fig. 2. Comparison of the ODMR spectra experimentally measured at zero external magnetic field B=0 (Fig. 2a) for NV+n(NN)13C (n=0,1,2,3) with respective theoretical ODMR spectra obtained using the shfi tensors for few 13C atoms calculated with EPR experimental PACS values a=150.5 MHz and b=27.3 MHz for the shfi parameters.
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Fig. 3. Upper red curves - ODMR spectra experimentally measured for the NV+n(NN)13C (n=0,1,2,3) at non-zero magnetic field В=78.5 gauss (B||[111]) in comparison with respective theoretical ODMR spectra obtained using the shfi tensors for few 13C atoms calculated with EPR experimental PACS values a=150.5 MHz and b=27.3 MHz for the shfi parameters. The above geometrical analysis was possible only due to the availability of EPR data for shfi of electron spin of the NV center with the nearest-to-vacancy nuclear spins 13C. Analogous data for nuclear spins 13C located in more distant sites of the diamond lattice are absent and the only way to obtain such information is the ab initio simulation described in the Section 1. The results of such a simulation in comparison with available experimental data are shown in Figs. 4-6 for various cases of the presence of 13C nuclear spins in different positions near the NV center. CW ODMR signal intensity (a.u.)
0,05 0,00
-0,05 -0,10 -0,15 -0,20 -0,25 -0,30 -0,35
2600
2800
3000
MW frequency (MHz) 13
Fig. 4. ODMR spectrum for the case of no C atom in NN position and one 13C atom in NNN position. Experimental splitting is ~13 MHz, calculated - ~ 9 MHz
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3200
CW ODMR signal intensity (a.u.)
0,05 0,00
-0,05 -0,10 -0,15 -0,20 -0,25 -0,30 -0,35
2600
2800
3000
3200
MW frequency (MHz)
CW ODMR signal intensity (a.u.)
Fig. 5. ODMR spectrum for the case of no 13C atom in NN position and two 13C atom in NNN position
0,00
-0,05
-0,10
-0,15
-0,20
-0,25
2600
2800
3000
3200
MW frequency (MHz) Fig. 6. ODMR spectrum for the case of one 13C atom in NN position and one 13C atom in NNN position One can see from Figs. 4-6 that the calculated ODMR spectra are in reasonable qualitative coincidence with experimental ones but quantitatively the simulated spectra show smaller shf splitting - ~ 9 MHz instead of the experimental splitting is ~13 MHz. The reason for this divergence is evidently in the size of the simulated carbon cluster C69H84[NV]- where more distant NNN 13C atom having rather high shfi with electronic spin were disposed practically at the border of the cluster. Preliminary results of spin Hamiltonian simulation of the above experimental ODMR spectra made using shfi parameters calculated by ab initio DFT for larger cluster C84H78[NV]- show better coincidence with respect of hf splitting available in the ODMR spectra (for theoretical splitting due to the NNN C atoms we got the value ~11 MHz).
4. NANO-DESIGN OF PHOTON AND SPIN INTERFACE To be able to reach single spins optically a good interface between photons and NV centers has to be created. We have proposed to use photonic structures in diamonds to manipulate locally by photons inside the structures 18. These structures can be created in diamond by a Focused Ion Beam (FIB) assisted lift-off technique 19. We start from noting an electron dynamics in a double-well potential. In symmetrical potentials two lowest electron states have opposite symmetries resulting in small energy splitting. Electron preparation in superposition of these states results in tunneling between the wells. Tunneling time depends on the level splitting. But tunneling is suspended if electron prepared in one of those states. Similar considerations can be transfered to photons inside structured dielectric. An example of photonic crystal-like system composed from diamond layers with two NV centers implanted into the defects is presented on Fig. 7. The system is irradiated from the left side. The PBGs positions can be estimated from
Proc. of SPIE Vol. 7377 737711-7
refraction index
infinite periodic structure with the same refraction indexes and layers thickness. Defect in geometrical structure of system results in formation of resonances inside PBG. There are few typical defect mode profiles. In some defect modes field intensity is localized in both defects or distributed inside interdefect layers. A number of outer diamond layers was determined from the following considerations. A "good" enough resonance should arise at left defect, an intensity of resonance should be larger than intensity inside input layers. There is no "good" resonance with larger number of input and output layers and smaller number of layers.
3 2 1 0
5
10
15
20 25 30 coordinate, layers
35
40
45
Fig. 7. System profile. Geometric defects are formed by insertion of two air layers around diamond layer. Light incidences from left. A number of layers between defects affects resonances spectral positions and spatial profiles. Interdefect diamond layers number changing results in resonances moving inside PBG. It worth to note that the resonances demonstrate doublet-like behavior. Decreasing the number of layers between defects moves off resonances from each other and vice versa. That can be qualitatively described by simple dielectric model with two regions with lower refraction indexes. By increasing the number of layers between defects we arrive to resonance with small frequency bandwidth and nontrivial spatiofrequency structure. To work with NV center the system has to provide defects intensity resonances at 637 nm, so the system's parameters is tuned to support a defect mode around 637 nm. The diamond refraction index is 2.4. The air and the diamond optical layers thicknesses are chosen to be equal. Diamond layer thickness was chosen to be 0.205 μm, and air 0.491 μm. Thus, light traveling time through one layer is τ = 1.636 fs. The geometrical defects are formed by the insertion of two air layers around diamond layer. The distance between the geometrical defects centers is 7.66 μm with whole system size of 15.42 μm. There is a PBG occupied the region from 590 nm to 730 nm with two defect modes in the spectrum of the system. The transmitted light calculated by stationary modes technique shows double resonance centered at 637.02 nm (Fig. 8). The doublet structure becomes unresolved when the number of interdefect layers increased. The reflected light shows the spectral dip at the same frequencies. The light which is captured inside the system demonstrates interesting spatial and spectral distribution (Fig. 8b). The captured light is localized in the left defect far from wings of the transmission resonance line, while for the centered of the line the main part of the captured light is concentrated in the right defect. This observation gives us an obvious hint that switching of the driving field frequency from wing to center of the transmission resonance line will produce the captured light localization switching from the left defect to the right one and vice versa. Also, it is possible to control the light localization by frequency modulation of the driving field12. Here we present the results of numerical calculation of temporal evolution of captured light by means of the Green functions method 12. The results strongly support the above mentioned predictions. The system is trapped in the transient state after switching on/off or instant changing of incident light parameters. The transient state duration is about T~0.25 ns. During this time defects field intensities may significant differ from steady state values. The switching dynamics is shown at Fig. 9. One steady state profile converts into another not faster than T. At steady state regime left and right defects intensity ratios strongly depend on wavelength. In the 637.025 nm vicinity right/left the maximal intensities ratio equals 5.85, for 637.0375 nm left/right the maximal ratio equals 5.43. System geometry drastically affects these ratios. For example, by decreasing the number of interdefects layers we can increase ratios. At the same time absolute intensity value decreased. Note, that for spatial addressing accomplished by laser wavelength tuning the resonance transitions of implanted NV centers difference should be in order of 1/80 nm. Probably that might be implemented by external electric field application 14.
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4-
0 0
0.5
-
refIectiony_ transmillion
(a)
1xt
Fig. 8. (a) Transmission and reflection coefficients. (b) Spatio-spectral intensity distribution around 637 nm wavelength. Dark regions correspond to higher intensity. At the transmission resonance center light localizes in right defect, while at wings in left defect. This resonance is coupled to outgoing waves. On the inset corresponding spatial refractive index distribution is sketched
intensity, a.u.
800 600 400 200 0 0
150
300
450 time, ps
600
750
Fig. 9. Light intensities evolution at defects position. Solid line corresponds to left defect, dash line corresponds to the right one. Driven field wavelength is switched instantly from 637.02 nm to 637.04 nm at time moment 327 ps, and switched off after 654 ps. For example, by using our scheme a single spin readout of two distant NV centers can be implemented with single laser light source with no any component mechanical motion. Also, in Raman Adiabatic Passage-like state preparation schemes one of the pulse detuned from resonance can be matched with transmission resonance line. The pulse should be longer than transient state duration time. This results in possibility of spatial selective NV center initial state preparation.
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ACKNOWLEDGEMENT The authors acknowledge the financial support by EC in framework of EQUIND project.
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