Electric Field Generation and Control of Bipartite Quantum

Article pubs.acs.org/IC

Electric Field Generation and Control of Bipartite Quantum Entanglement between Electronic Spins in Mixed Valence Polyoxovanadate [GeV14O40]8− Andrew Palii,*,†,‡ Sergey Aldoshin,† Boris Tsukerblat,*,§ Juan José Borràs-Almenar,∥ Juan Modesto Clemente-Juan,*,⊥ Salvador Cardona-Serra,⊥ and Eugenio Coronado⊥ †

Institute of Problems of Chemical Physics, Chernogolovka, Moscow Region, Russia Institute of Applied Physics, Academy of Sciences of Moldova, Chisinau, Moldova § Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel ∥ Departament de QuímicaInorganica, Universidad de Valencia, c/Vicent Andrés Estellés, s/n, 46100 Burjassot, Valencia, Spain ⊥ Instituto de Ciencia Molecular, Universidad de Valencia, Polígono de la Coma, s/n, 46980 Paterna, Spain ‡

ABSTRACT: As part of the search for systems in which control of quantum entanglement can be achieved, here we consider the paramagnetic mixed valence polyoxometalate K 2 Na 6 [GeV 14 O 40 ]·10H 2 O in which two electrons are delocalized over the 14 vanadium ions. Applying a homogeneous electric field can induce an antiferromagnetic coupling between the two delocalized electronic spins that behave independently in the absence of the field. On the basis of the proposed theoretical model, we show that the external field can be used to generate controllable quantum entanglement between the two electronic spins traveling over a vanadium network of mixed valence polyoxoanion [GeV14O40]8−. Within a simplified two-level picture of the energy pattern of the electronic pair based on the previous ab initio analysis, we evaluate the temperature and field dependencies of concurrence and thus indicate that the entanglement can be controlled via the temperature, magnitude, and orientation of the electric field with respect to molecular axes of [GeV14O40]8−.

1. INTRODUCTION The study of quantum entanglement is an emerging field of contemporary research closely connected to the basic issues of the quantum mechanical description of matter and fascinating applications such as quantum computing, cryptography, and communication.1−5 Recently, the problem of entanglement has become a challenge of molecular magnetism6 because magnetic clusters have been proposed as novel objects for spin-based quantum computing.7−18 Advantages in the engineering of the relevant metal complexes provided by supramolecular chemistry have been demonstrated by the design of the coupling between molecular spin qubits and the study of entangled states in antiferromagnetic Cr 7 Ni rings {such as [NH 2 Pr 2 ][Cr7NiF8(O2CCMe3)16] and [NH2Pr2][Cr7NiF8(O2CCMe3)15(O2CC5H4N)]}. A thorough discussion can be found in refs 15, 16, and 19−21 and a review in ref 22. Development in this area led to the tailor-made molecules for spin-based CNOT and SWAP quantum gates,23 two-qubit molecular spin quantum gates based on bimetallic lanthanide clusters,24 and the modular design of molecular qubits.25 Entanglement is an essentially quantum mechanical concept, which means that a wave function of a complex system cannot be factorized into states belonging to the constituent © 2017 American Chemical Society

components. This means that in a complex system consisting of two parts it is impossible to describe one part without knowledge of the state of the second part (see detailed descriptions in refs 1, 2, and 26−28). The S = 0 state of a pair of spin-1/2 particles (|↑↓⟩ − |↓↑⟩)/√2 is traditionally used in textbooks as an example of entangled states. Thus, the investigation and control of the entanglement are interrelated with the magnetic coupling in dimeric metal clusters involving this state along with the spin-triplet one. The MS = 0 state (|↑↓⟩ + |↓↑⟩)/√2 belonging to the spin triplet is also entangled, although the entanglement of the spin-1/2 pair in a mixed triplet state is zero (see a detailed discussion in ref 14). Molecular spin clusters were identified as suitable objects for testing the entanglement through the study of magnetic phenomena and inelastic neutron scattering. The individual qubits in these systems are the spin-1/2 magnetic centers (encoding binary information), while the coupling between the spin qubits occurs via exchange interaction (mostly antiferromagnetic), creating thus the entangled states. That is why much effort has been applied to the detection and study of entanglement in the Received: April 18, 2017 Published: August 3, 2017 9547

DOI: 10.1021/acs.inorgchem.7b00991 Inorg. Chem. 2017, 56, 9547−9554

Article

Inorganic Chemistry

formed by 12 diamagnetic VV ions and 2 VIV ions having S = 1/2 spins. As was theoretically demonstrated in refs 32 and 33, an external electric field is able to efficiently control the spin−spin coupling in this MV system, and therefore, the V14 POM is quite attractive in the context of controllable quantum entanglement. In particular, the V14 POM is advantageous over the MV tetramer31 because V14 POM comprises a pair of a distant weakly coupled spins that can be associated with qubits. In this work, we demonstrate that the external homogeneous electric field of a suitable direction can be used as a source of controllable quantum entanglement between weakly coupled electronic spins in the MV V14 POM. The temperature and field dependence of the concurrence and quantum entanglement are analyzed.

antiferromagnetic dimers of metal ions or more complicated weakly coupled molecular moieties [as in the (Cr7Ni)2 dimer26]. The controlled generation of quantum entanglement is one of the main requirements for implementing quantum information processing.1,2 In particular, the ability to assemble weakly interacting magnetic molecules to generate controlled entanglement has been discussed.15,16 In the context of the challenging tasks of molecular spintronics, the electric field has been proposed as an external stimulus that is desirable for the control of spin states in polar magnetic molecules.29 Magnetoelectric coupling and spin-induced electrical polarization in metal−organic magnetic chains were considered in ref 30. Vitally important advantages of the electric field as a tool for the control over properties of spin systems in comparison with other factors, for example, magnetic field, are discussed in ref 30. Conceptually, the polar molecules have, in general, different dipole moments (and polarizabilities) in high-spin and low-spin states, so that the Stark effect can cause spin crossover.30 This underlying idea was applied to the pair of two identical magnetic ions coupled by the superexchange interaction that is modulated by an electric field, resulting in the crossover of spintriplet and spin-singlet levels.30 Mixed valence (MV) systems are especially interesting in the context of the possibility of the electric field control of entanglement due to the mobility of the itinerant electrons whose degrees of localization are sensitive to the field. In fact, in the localized spin systems, the effect of the electric field appears due to field-induced corrections of the highly excited levels in Anderson’s model of kinetic exchange. As distinguished from the localized spin systems, in MV clusters a large electric dipole moment can be easily induced by mixing of the two levels of opposite parity originating from electron delocalization. Bearing in mind this consideration, we have demonstrated the possibility of electric field control of quantum entanglement in a square-planar MV molecule containing a pair of itinerant electrons.31 In searching for systems in which the control of quantum entanglement can be achieved, we consider here the MV polyoxoanion [GeV14O40]8− (see refs 32 and 33), hereunder abbreviated as V14 polyoxometalate (V14 POM). The metal network of V14 consists of the central square (X−Y plane) that is sandwiched between the two square pyramids (with Z directed along the common C4 axis, although the actual symmetry is lower than the tetragonal one) as shown in the schematic drawing in Figure 1. This doubly reduced system is

2. TWO-LEVEL MODEL FOR THE V14 POLYOXOANION IN A TRANSVERSE ELECTRIC FIELD As distinguished from many other doubly reduced MV POMs, the two extra electrons in the V14 system are strongly localized. To gain the Coulomb repulsion energy, they are located mostly at the remote sites.32,33 Thus, the two electrons are separated from each other by a large distance and therefore are almost magnetically independent. As a result, the ground state of the reduced polyoxoanion containing two unpaired electrons represents a paramagnetic mixture of the S = 0 and S = 1 states in accordance with the observed magnetic behavior of this system.34 Actually, the ground level is slightly split into the low-lying spin singlet and excited spin triplet, but the splitting is only ∼0.2 cm−1 (see ref 32) and so can be almost always neglected, excluding that at very low temperatures. As a result, no entanglement can exist at reasonable temperatures as schematically shown in Figure 1a, which illustrates decoupled spins. However, under the action of the electric field applied along the molecular axes, the two electrons are able to overcome the interelectronic Coulomb repulsion and approach to a distance at which the antiferromagnetic exchange between them becomes efficient. This leads to a stabilization of the diamagnetic ground state (S = 0), and in this way an electric field is expected to generate the quantum entanglement between the two electronic spins considered as spin qubits (Figure 1b). To quantify this qualitative view, the following t−J model Hamiltonian32 can be applied to reproduce the electronic energy pattern of the MV V14 POM subjected by the action of an external electric field: Ĥ =

∑ (εi + riε)nî + i

1 2

∑ i,j

Uijnî nĵ

(i ≠ j)

+

1 2

∑ i,j,σ (i ≠ j)

tij(cî+σ cĵ σ + cj+̂ σcî σ ) −

∑ i,j

⎛ 1⎞ Jij nî nĵ ⎜sî sĵ − ⎟ ⎝ 4⎠

(i ≠ j)

(1)

where the index i in the first term runs over dxy-like magnetic orbitals of all metal centers and the factor 1/2 in the second and third terms is introduced to exclude doubling of the summations. The first term in eq 1 describes the one-center contributions in which εi is the energy of the dxy orbital of site i at zero electric field, εi + riε is the corresponding orbital energy in the applied homogeneous electric field ε, ri is the position vector of site i, and n̂i is the occupation number operator for

Figure 1. Schematic drawing of POM K2Na6[GeV14O40]·10H2O containing two extra electrons predominantly localized at distant vanadium sites: (a) zero electric field when the two spin qubits are unentangled and (b) entanglement generated by the transverse electric field. 9548


Article

Inorganic Chemistry

Figure 2. Low-lying energy levels of [GeV14O40]8− calculated as functions of the electric field at (a) Z, (c) X, and (e) Y directions of the field. The order of S = 0 and S = 1 labels corresponds to the order of levels. Panels b (Z direction), d (X direction), and f (Y direction) show only the two lowlying levels. The ground level with S = 0 is taken as a reference one.

included in the definition of the exchange Hamiltonian is omitted here to avoid the doubling of the summation. The question of the extent to which the t−J model Hamiltonian is applicable to the present case of two electrons per 14 vanadium sites remains. At less than half-filling, which is true in the present case, the inequality U ≫ tij35,36 is fulfilled reliably because the maximal transfer integral in the V14 POM proves to be smaller than 0.5 eV,37 while the value of U for vanadium ions was estimated to be >3 eV.38 Thus, the t−J model Hamiltonian seems to be well justified for the system being studied. The calculations of the energy levels performed with the Hamiltonian (eq 1) with the parameters found from the ab initio calculations (for details, see refs 32 and 33) have demonstrated that the effect of the electric field crucially depends on its direction. Thus, when the electric field is applied along the molecular Z axis the ground state remains almost paramagnetic up to very high critical value of the field of ∼6.5 V/nm (Figure 2a,b). At this point, the magnetic properties change abruptly because the spin singlet becomes the ground state due to field-induced antiferromagnetic coupling. In

this site. The second term describes the intersite interelectron Coulomb repulsion, with Uij being the corresponding Coulomb repulsion integrals. The third term describes the one-electron transfer, where tij values are the one-electron hopping integrals, ciσ+ and ciσ are the creation and annihilation operators, respectively, and σ is the spin projection. The summation in this term of eq 1 is extended over the jumps of the electrons from the occupied sites to the unoccupied ones. By definition of the t−J model, the double occupancy of the sites is excluded from the Hamiltonian because the on-site Coulomb repulsion energy U is assumed to be much larger than all intersite Uij and transfer parameters. The doubly occupied states participate in the second-order perturbation theory with respect to parameter tij/U (providing U ≫ tij), which gives rise to the effective isotropic exchange interaction between the ions with halfoccupied orbitals. This is taken into account by the last term in eq 3 that describes the magnetic exchange between the electrons occupying sites i and j, where ŝi and ŝj are the electronic spin operators and Jij values are the exchange integrals. It is assumed that only the nearest-neighbor sites are magnetically coupled. Note that the factor 2 that is usually 9549


Article

Inorganic Chemistry

Figure 3. (a) Numbering of the vanadium sites. Populations of the sites in the ground spin-singlet state as functions of the electric field in the (b) X and (c) Y directions of the field.

contrast, when the field is applied along the X and Y axes, the lowest levels demonstrate a gradual field dependence as shown in Figure 2c−f. It is to be noted that because of the presence of the rhombic component of the crystal field the X and Y directions are not equivalent. We will assume that the field is directed along X or Y axes (transverse field) as only in these directions are the generation and control of quantum entanglement possible. A remarkable feature of the energy patterns in both X (Figure 2c) and Y (Figure 2e) directions of the electric field is the fact that the patterns consist of pairs of closely located levels, with the ground state being the spin singlet and the excited state being the spin triplet. The scale in panels c and e of Figure 2 masks the gap between these two levels. To make this gap visible, we have plotted in panels d and f of Figure 2 only the two low-lying levels on a smaller scale. It is seen that the singlet−triplet gap gradually increases with the field, with this increase being more pronounced for the Y direction of the field. To visualize the effects of the electric field directed along the X and Y axes, we have presented in Figure 3 the electronic populations of sites (numbered as shown in Figure 3a) in the ground spin-singlet state as functions of the electric field. The field directed along the X axis tends to depopulate two V4 sites and populate two V1 sites (Figure 3b), thus allowing the two electrons to interact antiferromagnetically via superexchange mediated by the oxygen bridge linking the two V1 sites. Similarly, the electric field in the Y direction leads to a depopulation of two V4 sites with simultaneous population of two V2 sites (Figure 3c), which are also exchange coupled through the oxygen bridge. Therefore, the transverse electric field induces antiferromagnetic coupling and thus the entanglement between the two electronic spins. An important peculiarity of the energy patterns at a transverse electric field (Figure 2c,e) is that for all reasonable values of the electric field the low-lying part of the energy pattern comprises two isolated levels (S = 0, and S = 1), while the rest of the spin levels are significantly higher in energy.

Therefore, we can restrict ourselves to the consideration of only these two lowest levels that can be described by the following effective exchange Hamiltonian: Ĥ eff = −2Jeff s1̂ s 2̂

(2)

where the exchange parameter is defined as Jeff = −δ0−1/2. The singlet−triplet gap δ0−1 (and consequently Jeff) is a function of the electric field as follows from panels d and f of Figure 2. We thus arrive at the conclusion that the V14 POM subjected to the action of the transverse electric field can be regarded as an effective Heisenberg dimer with spins of 1/2 (Bleaney−Bowerstype dimer). Hereafter, we will see that the two-level structure of the low-lying part of the energy pattern considerably simplifies the evaluation of quantum entanglement. A quite different kind of spin switching mechanism takes place at the longitudinal electric field. The analysis of the lowlying energy levels (Figure 2a) and site populations allows us to conclude that in this case the exchange coupling is of minor importance. Indeed, an effective antiferromagnetic coupling between the two spins (as well as spin switching) is induced by the electron delocalization in conjunction with the intersite Coulomb repulsion. Just this mechanism has been proposed for the explanation the spin pairing in two-electron-reduced POMs with Keggin and Wells−Dawson structures39,40 in which the two electrons are spatially well separated and for this reason the exchange interaction is inoperative. Indicative of the presence of this mechanism in the system under consideration is the avoided crossing of two spin-singlet levels at the critical field (Figure 2a) when the two spin singlets are strongly mixed and, as a result, the ground singlet is strongly stabilized with respect to the spin triplet. Such a three-level picture is drastically different from the two-level scheme arising at the transverse electric field. This shows that the superexchange coupling induced by the electric field cannot be regarded as an actual mechanism of spin switching at longitudinal field. 9550


Article

Inorganic Chemistry

Figure 4. Entanglement calculated as a function of (a) the temperature at different values of εX of the electric field shown in the plot and (b) electric field εX at different temperatures shown in the plot.

Figure 5. Entanglement calculated as a function of (a) the temperature at different values of εY of the electric field shown in the plot and (b) electric field εY at different temperatures shown in the plot.

v=

⎛ J ⎞ 1 exp⎜ eff ⎟ Z ⎝ 2kBT ⎠

p=

⎛ J ⎞ ⎛ J ⎞ 1 exp⎜ − eff ⎟ cosh⎜ eff ⎟ Z ⎝ 2kBT ⎠ ⎝ kBT ⎠

w=

⎛ J ⎞ ⎛ J ⎞ 1 exp⎜ − eff ⎟ sinh⎜ eff ⎟ Z ⎝ 2kBT ⎠ ⎝ kBT ⎠

3. EVALUATION OF QUANTUM ENTANGLEMENT For the two-spin-qubit system under consideration, the entanglement of formation is related to concurrence C as follows:1 ⎡1 E = h⎢ (1 + ⎣2

⎤ 1 − C 2 )⎥ ⎦

(3)

where the function h(x) is the von Neumann entropy for a binary probability distribution: h(x) = −x log 2(x) − (1 − x) log 2(1 − x)

In the particular case of the density matrix given by eq 5, the concurrence can be evaluated with the aid of the following expression:14,41

(4)

The concurrence is determined by the density matrix. In the particular case of a Bleaney−Bowers-type dimer that is described by the effective kinetic exchange Hamiltonian (eq 2), the thermal equilibrium density matrix has the following form in the basis |↑↑⟩,| ⟨↑↓|),|↓↑⟩,|↓↓⟩: ⎛v ⎜ ⎛ Heff ⎞ ⎜ 0 1 ρ(T ) = exp⎜ − ⎟= Z ⎝ kBT ⎠ ⎜ 0 ⎜ ⎝0

0 0 0⎞ ⎟ p w 0⎟ w p 0⎟ ⎟ 0 0 v⎠

C = 2 max{|w| − v , 0}

(8)

which represents a particular case of the Wootters formula42,43 for bipartite (two-qubit) entanglement. As follows from eqs 6−8, in the present case of antiferromagnetic effective exchange coupling (Jeff < 0), the concurrence can be found to be 2 |Jeff |

( ), T < T , C(T ) = 1 + 3 exp(− ) 1 − 3 exp −

kBT

2 |Jeff |

E

kBT

(5)

C(T ) = 0, T ≥ TE

where kB is the Boltzmann constant and ⎛ J ⎞ ⎛ 3J ⎞ Z = 3 exp⎜ eff ⎟ + exp⎜ − eff ⎟ ⎝ 2kBT ⎠ ⎝ 2kBT ⎠

(7)

(9)

where TE = 2|Jeff | /kB ln(3)

(6)

is the partition function. In eq 5, the following notations are used:

(10)

is the critical temperature above which the entanglement disappears. 9551


Article

Inorganic Chemistry

Figure 6. (a) Field dependencies of the critical temperature of entanglement (solid lines) and field dependencies of the temperature of the maximum magnetic susceptibility (dashed lines). (b) χ vs T curves calculated for 1 and 2 V/nm magnitudes of the field shown in the plot. All dependencies are calculated at X (blue lines) and Y (red lines) directions of the field.

where NA is Avogadro’s number, μB is the Bohr magneton, and g is the g factor. It immediately follows from the comparison of eqs 9 and 11 that

4. RESULTS AND DISCUSSION Figure 4a presents the temperature dependencies of quantum entanglement evaluated with the aid of eq 9 at different values of the electric field directed along the X axis. In the lowtemperature limit, the system is fully entangled (E = 1) even at weak electric field εX because in this case only the spin singlet is thermally populated. Then, with an increase in temperature, the value of E decreases and disappears at critical temperature TE that increases with an increase in the electric field. For a sufficiently strong field (when εX = 2 V/nm in Figure 4a), the system remains entangled up to ≈6.5 K because of the increase in the singlet−triplet energy gap caused by the electric field that prevents the thermal population of the excited spin triplet and depopulation of the ground singlet. In contrast, at a weak electric field, entanglement disappears at very low temperatures because in this case the singlet−triplet gap is small. Figure 4b shows the dependence of the entanglement on electric field εX calculated at different temperatures. It is seen that the higher the temperature, the stronger the electric field required to create entanglement. Thus, at 7 K, the system remains fully unentangled (E = 0) up to εX ≈ 1.2 V/nm, while at 0.055 K, it is almost fully entangled (E ∼ 1) even in the absence of an electric field. This is evidently due to the fact that at such a low temperature energy gap Δ1−0 significantly exceeds thermal energy kBT even at a zero electric field. Similar dependencies are obtained for the field directed along the Y axis (Figure 5), but in this case, higher critical temperatures of entanglement can be reached because of the more pronounced effect of the electric field on the singlet−triplet energy gap (this can be seen by comparing panels d and f of Figure 2). Thus, at the strong field εY = 2 V/nm, the system remains entangled up to approximately 14 K, while for the X direction, the entanglement already disappears at ∼6.5 K. An additional illustration of this fact is provided by comparison of the field dependencies of critical temperature TE calculated for different directions of the electric field (solid lines in Figure 6a). It is seen that for a sufficiently strong electric field (exceeding the value of ≈1.1 V/nm) the Y direction of the field is preferable for generating quantum entanglement. For the experimental observation of these dependencies, one can use the relationship between the entanglement and the magnetic susceptibility. Thus, the molar magnetic susceptibility of the effective Bleaney−Bowers dimers is given by χ (T ) =

−1 ⎛ 2J ⎞⎤ 2NAg 2μB 2 ⎡ ⎢3 + exp⎜ − eff ⎟⎥ kBT ⎢⎣ ⎝ kBT ⎠⎥⎦

C(T ) = 1 −

3kBTχ (T ) NAg 2μB 2

(12)

The χ(T) curve passes through the maximum that is shifted to the higher temperature with the increase in the field. This is clearly seen in Figure 6b, which shows χ(T) curves calculated at the electric fields directed along X and Y axes and having 1 and 2 V/nm magnitudes for each direction. The maximum χ(T) occurs at Tmax =

2|Jeff | kB[1 + W (3/e)]

(13)

where W(x) is the Lambert-W function defined by the relation Wew = x. Then, as follows from eqs 10 and 13, the ratio TE/Tmax is independent of Jeff (and hence also independent of the electric field) and equal to TE/Tmax = [1 + W (3/e)]/ln(3) ≈ 1.4596

(14)

This is also seen from the comparison of the calculated Tmax(εγ) and TE(εγ) dependencies depicted in Figure 6a. Because the field dependence of T max can be extracted from the experimental data on magnetic susceptibility, this kind of experiments can provide also the field dependence of TE. By using eq 12, one can find also dependence E(εγ,T) from experimentally derived dependence χ(εγ,T). It follows from the previously reported results32,33 and the discussion in section 2 that at the critical value of the field along the Z axis the system abruptly changes from unentangled over a full temperatures range (except extremely low temperatures) to fully entangled at high temperatures. This critical field is ∼6.5 V/nm, which is too high to be experimentally attainable and moreover exceeds the dielectric strength. Therefore, one can conclude that the electric field directed along the Z axis (as distinguished from the X and Y directions) is inappropriate to generate and control the quantum entanglement. In addition, the two-level picture used above, simplifying calculation of quantum entanglement in the presence of the field directed along the X or Y axis, loses its validity along the Z direction of the field in the vicinity of its critical value. At this point, we must consider the validity of approximation of an isolated charged molecule. The influence of the counterions in this system was already discussed by some of the authors in the previous article.33 The density functional theory (DFT) calculations showed that there was not a

(11) 9552


Article

Inorganic Chemistry

(Agreement No.14.W03.31.0001-Institute of Problems of Chemical Physics of RAS, Chernogolovka). This work has been supported by the EU [COST Action CA15128 Molecular Spintronics (MOLSPIN)], the Spanish MINECO (CTQ201452758-P, MAT2014-56143-R, and Excellence Unit Mariá de Maeztu, MDM-2015-0538), and Generalitat Valenciana (Prometeo Programme of Excellence). A.P. thanks the University of Valencia for a visiting grant.

significant influence in treating the bare molecule and when cations are treated as point monopositive charges. On the other hand, the effect of the shielding electric field can be approximately described by introducing the relative permittivity factor (dielectric constant) that is approximately equal to 1.7 (this follows from the fact that in a parallel field the abrupt change in the magnetic properties calculated with the aid of the DFT approach occurs at the electric field that is approximately 1.7 times higher than the value of the field predicted by the model Hamiltonian and ab initio calculation that does not take into account this shielding effect). Assuming that the shielding effect is isotropic, we can use the same value of the permittivity factor for a transverse electric field, so entanglement E found at some value ε neglecting the shielding effect proves to be exactly the same as that found at a larger value of the transverse field (ε′ = 1.7ε) when the shielding effect is taken into account. Still, the transverse electric field of ∼1 V/nm (or 1.7 V/nm in the presence of a shielding effect) that is required to generate entanglement is also quite strong, being at the border of practical feasibility. That is why it is worthwhile to outline a possible way in which this critical field can be decreased. The origin of this lies in the fact that the orbital energy εV4 ≈ 780 meV of the V4 site is significantly lower than that of the V1 and V2 sites (εV1 = εV2 ≈ 1453 meV). As a result, the electric field should overcome not only the Coulomb repulsion to approach the two electrons but also the barrier created by the inequivalence of the orbital energies. One can expect that the electric field required for generating and control of entanglement can be significantly reduced by replacing the V4 ions with Mo or W ions whose orbital energies are known to be higher than those for vanadium ions.33

■

5. CONCLUDING REMARKS In summary, we have analyzed the possibility of generating controllable quantum entanglement between weakly coupled electronic spins in the MV POM [GeV14O40]8− under the action of the external homogeneous electric field that controls the exchange coupling between itinerant electrons. The entanglement was calculated as a function of temperature at different values and orientations of the electric field. We have demonstrated that the conditions for the existence of the entanglement in [GeV14O40]8− are essentially dependent on the direction of the field, with the Y direction being most favorable for tuning quantum entanglement. In this work, we have not considered the time-dependent phenomena related to spin coupling. This will be done elsewhere.

■

REFERENCES

(1) Nielsen, M. A.; Chuang, I. L. Quantum computation and quantum information; Cambridge University Press: Cambridge, U.K., 2000. (2) Vathsan, R. Introduction to quantum physics and information processing; CRC Press: Boca Raton, FL, 2016. (3) Bouwmeester, D., Ekert, A., Zeilinger, A., Eds. The physics of quantum information: Quantum cryptography, quantum teleportation and quantum computation; Springer: Berlin, 2000. (4) Kokin, A. A. Solid state quantum computers on nuclear spins; Computer Science Institute, Russian Academy of Sciences: Moscow, 2004 (in Russian). (5) Valiev, K. A. Quantum computers and quantum computations. Usp. Fiz. Nauk 2005, 175 (1), 3−39 [Phys.-Usp. 2005, 48 (1), 1−36]. (6) Gatteschi, D.; Sessoli, R.; Villain, J. Molecular Nanomagnets; Oxford University Press: Oxford, U.K., 2006. (7) Winpenny, R. E. P., Ed. Molecular Cluster Magnets; World Scientific Series in Nanoscience; World Scientific: Singapore, 2011; Vol. 3. (8) Leuenberger, M. N.; Loss, D. Quantum computing in molecular magnets. Nature 2001, 410, 789−793. (9) Meier, F.; Levy, J.; Loss, D. Quantum computing with antiferromagnetic spin clusters. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 68, 134417−134432. (10) Troiani, F.; Ghirri, A.; Affronte, M.; Carretta, S.; Santini, P.; Amoretti, G.; Piligkos, S.; Timco, G.; Winpenny, R. E. P. Molecular engineering of antiferromagnetic rings for quantum computation. Phys. Rev. Lett. 2005, 94, 207208. (11) Affronte, M.; Casson, I.; Evangelisti, M.; Candini, A.; Carretta, S.; Muryn, C. A.; Teat, S. J.; Timco, G.; Wernsdorfer, W.; Winpenny, R. E. P. Linking rings through diamines and clusters: exploring synthetic methods for making magnetic quantum gates. Angew. Chem., Int. Ed. 2005, 44, 6496−6500. (12) Wernsdorfer, W. Molecular magnets: a long-lasting phase. Nat. Mater. 2007, 6, 174−176. (13) Carretta, S.; Santini, P.; Amoretti, G.; Troiani, F.; Affronte, M. Spin triangles as optimal units for molecule-based quantum gates. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 024408. (14) Aldoshin, S. M.; Feldman, E. B.; Yurishchev, M. A. Quantum entanglement in nitrosyl iron complexes. J. Exp. Theor. Phys. 2008, 107, 804−811. (15) Timco, G. A.; Carretta, S.; Troiani, F.; Tuna, F.; Pritchard, R. J.; Muryn, C. A.; McInnes, E. J. L.; Ghirri, A.; Candini, A.; Santini, P.; Amoretti, G.; Affronte, M.; Winpenny, R. E. P. Engineering the coupling between molecular spin qubits by coordination chemistry. Nat. Nanotechnol. 2009, 4, 173−178. (16) Candini, A.; Lorusso, G.; Troiani, F.; Ghirri, A.; Carretta, S.; Santini, P.; Amoretti, G.; Muryn, C.; Tuna, F.; Timco, G.; McInnes, E. J. L.; Winpenny, R. E. P.; Wernsdorfer, W.; Affronte, M. Entanglement in supramolecular spin systems of two weakly coupled antiferromagnetic rings (Purple-Cr7Ni). Phys. Rev. Lett. 2010, 104, 037203. (17) Wernsdorfer, W. Chemistry brings qubits together. Nat. Nanotechnol. 2009, 4, 145−146. (18) Tesi, L.; Lunghi, A. A.; Atzori, M.; Lucaccini, E.; Sorace, L.; Totti, F.; Sessoli, R. Giant spin-phonon bottleneck effects in evaporable vanadyl-based molecules with long spin coherence. Dalton Trans. 2016, 45, 16635−16643. (19) Affronte, M.; Carretta, S.; Timco, G. A.; Winpenny, R. E. P. A ring cycle: studies of heterometallic wheels. Chem. Commun. 2007, 1789−1797.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. ORCID

Boris Tsukerblat: 0000-0002-8182-6140 Juan Modesto Clemente-Juan: 0000-0002-3198-073X Salvador Cardona-Serra: 0000-0002-5328-7047 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS A.P., S.A., B.T., and J.M.C.-J. acknowledge support from the Ministery of Education and Science of the Russian Federation 9553


Article

Inorganic Chemistry

wave function theory and model Hamiltonian calculations. Eur. J. Inorg. Chem. 2009, 2009, 5109−5114. (38) Şaşioğlu, E.; Friedrich, C.; Blügel, S. Effective Coulomb interaction in transition metals from constrained random-phase approximation. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 121101. (39) Borràs-Almenar, J. J.; Clemente-Juan, J. M.; Coronado, E.; Tsukerblat, B. Mixed-valence polyoxometalate clusters. I. Delocalization of electronic pairs in dodecanuclear heteropoly blues with Keggin structure. Chem. Phys. 1995, 195, 1−15. (40) Borràs-Almenar, J. J.; Clemente-Juan, J. M.; Coronado, E.; Tsukerblat, B. Mixed-valence polyoxometalate clusters. II. Delocalization of electronic pairs in 18-site heteropoly blues with wells-Dawson structure. Chem. Phys. 1995, 195, 17−28. (41) Aldoshin, S. M.; Fel’dman, E. B.; Yurishchev, M. A. Quantum entanglement and quantum discord in magnetoactive materials. Low Temp. Phys. 2014, 40 (1), 3−16. (42) Hill, S.; Wootters, W. K. Entanglement of a pair of quantum bits. Phys. Rev. Lett. 1997, 78, 5022−5026. (43) Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 1998, 80, 2245−2249.

(20) Timco, G. A.; Faust, T. B.; Tuna, F.; Winpenny, R. E. P. Linking heterometallic rings for quantum information processing and amusement. Chem. Soc. Rev. 2011, 40, 3067−3075. (21) Troiani, F.; Affronte, M.; Carretta, S.; Santini, P.; Amoretti, G. Proposal for quantum gates in permanently coupled antiferromagnetic spin rings without need of local fields. Phys. Rev. Lett. 2005, 94, 190501. (22) Ferrando-Soria, J. Cr7Ni Wheels: Supramolecular tectons for the physical implementation of quantum information processing. Magnetochemistry 2016, 2, 36. (23) Luis, F.; Repollés, A.; Martínez-Pérez, M. J.; Aguilà, D.; Roubeau, O.; Zueco, D.; Alonso, P. J.; Evangelisti, M.; Camόn, A.; Sesé, J.; Barrios, L. A.; Aromí, G. Molecular prototypes for spin-based CNOT and SWAP quantum gates. Phys. Rev. Lett. 2011, 107, 117203. (24) Aguilà, D.; Barrios, L. A.; Velasco, V.; Roubeau, O.; Repollés, A.; Alonso, P. J.; Sesé, J.; Teat, S. J.; Luis, F.; Aromí, G. Heterodimetallic [LnLn′] Lanthanide complexes: toward a chemical design of two-qubit molecular spin quantum gates. J. Am. Chem. Soc. 2014, 136, 14215− 14222. (25) Soria, J. F.; Pineda, E. M.; Chiesa, A.; Fernandez, A.; Magee, S. A.; Carretta, S.; Santini, P.; Yrezabal, I. J. V.; Tuna, F.; Timco, G. A.; McInnes, E. J.L.; Winpenny, R. E.P. A modular design of molecular qubits to implement universal quantum gates. Nat. Commun. 2016, 7, 11377. (26) Garlatti, E.; Guidi, T.; Ansbro, S.; Santini, P.; Amoretti, G.; Ollivier, J.; Mutka, H.; Timco, G.; Vitorica-Yrezabal, I. J.; Whitehead, G. F. S.; Winpenny, R. E. P.; Carretta, S. Portraying entanglement between molecular qubits with four-dimensional inelastic neutron scattering. Nat. Commun. 2017, 8, 14543. (27) Lorusso, G.; Troiani, F.; Bellini, V.; Ghirri, A.; Candini, A.; Carretta, S.; Santini, P.; Amoretti, G.; Wernsdorfer, W.; Timco, G.; Winpenny, R. E. P.; Affronte, M. Spin entanglement in supramolecular systems. J. Phys.: Conf. Series 2011, 303, 012033−43. (28) Aromí, G.; Aguilà, D.; Gamez, P.; Luis, F.; Roubeau, O. Design of magnetic coordination complexes for quantum computing. Chem. Soc. Rev. 2012, 41, 537−546. (29) Baadji, N.; Piacenza, M.; Tugsuz, T.; Sala, F.; Maruccio, D.; Sanvito, G. S. Electrostatic spincrossover effect in polar magnetic molecules. Nat. Mater. 2009, 8, 813−817. (30) Scarrozza, M.; Barone, P.; Sessoli, R.; Picozzi, S. Magnetoelectric coupling and spin-induced electrical polarization in metal−organic magnetic chains. J. Mater. Chem. C 2016, 4, 4176−4185. (31) Palii, A.; Tsukerblat, B. Tuning of quantum entanglement in molecular quantum cellular automata based on mixed-valence tetrameric units. Dalton Trans. 2016, 45, 16661−16672. (32) Cardona-Serra, S.; Clemente Juan, J. M.; Gaita-Ariño, A.; Suaud, N.; Svoboda, O.; Coronado, E. Modellingelectricfield control of the spin state in the mixed-valence polyoxometalate [GeV14O40]8−. Chem. Commun. 2013, 49, 9621−9623. (33) Cardona-Serra, S.; Clemente-Juan, J. M.; Coronado, E.; Gaita Ariño, A.; Suaud, N.; Svoboda, O.; Bastardis, R.; Guihéry, N.; Palacios, J. J. Electrically switchable magnetic molecules: inducing a magnetic coupling by means of an external electric field in a mixed-valence polyoxovanadate cluster. Chem. - Eur. J. 2015, 21, 763−769. (34) Bi, L.-H.; Kortz, U.; Dickman, M. H.; Nellutla, S.; Dalal, N. S.; Keita, B.; Nadjo, L.; Prinz, M.; Neumann, M. Polyoxoanion with Octahedral Germanium(IV) Hetero Atom: Synthesis, Structure, Magnetism, EPR, Electrochemistry and XPS Studies on the MixedValence 14- Vanadogermanate [GeVV12VIV2O40]8−. J. Cluster Sci. 2006, 17, 143−165. (35) Gros, C.; Joynt, R.; Rice, T. M. Antiferromagnetic correlations in almost-localized Fermi liquids. Phys. Rev. B: Condens. Matter Mater. Phys. 1987, 36, 381−393. (36) Chao, K. A.; Spalek, J.; Oles, A. M. Kinetic exchange interaction in a narrow S-band. J. Phys. C: Solid State Phys. 1977, 10, L271−L276. (37) Suaud, N.; Masaro, Y.; Coronado, E.; Clemente-Juan, J. M.; Guihéry, N. Origin of the Paramagnetic Properties of the MixedValence Polyoxometalate [GeV14O40]8− Reduced by two electrons: 9554


Electric Field Generation and Control of Bipartite Quantum

Recommend Documents