Abstract
The term ‘molecular magnet’ generally refers to a molecular entity containing several magnetic ions whose coupled spins generate a collective spin, S (ref. 1). Such complex multispin systems provide attractive targets for the study of quantum effects at the mesoscopic scale. In these molecules, the large energy barriers between collective spin states can be crossed by thermal activation or quantum tunnelling, depending on the temperature or an applied magnetic field^{2,3,4}. There is the hope that these mesoscopic spin states can be harnessed for the realization of quantum bits—‘qubits’, the basic building blocks of a quantum computer—based on molecular magnets^{5,6,7,8}. But strong decoherence^{9} must be overcome if the envisaged applications are to become practical. Here we report the observation and analysis of Rabi oscillations (quantum oscillations resulting from the coherent absorption and emission of photons driven by an electromagnetic wave^{10}) of a molecular magnet in a hybrid system, in which discrete and wellseparated magnetic clusters are embedded in a selforganized nonmagnetic environment. Each cluster contains 15 antiferromagnetically coupled S = 1/2 spins, leading to an S = 1/2 collective ground state^{11,12,13}. When this system is placed into a resonant cavity, the microwave field induces oscillatory transitions between the ground and excited collective spin states, indicative of longlived quantum coherence. The present observation of quantum oscillations suggests that lowdimension selforganized qubit networks having coherence times of the order of 100 μs (at liquid helium temperatures) are a realistic prospect.
Main
In the context of quantum computing, it was recently discussed how the decoherence of molecular magnet spin quantum bits could be suppressed, with reference to the discrete low spin clusters V_{15} and Cr_{7}Ni (ref. 7; see also refs 8 and 14). In both systems, their low spin states cause weak environmental coupling^{7}, making them candidates for the realization of a longlived quantum memory. Measurement of the spin relaxation time τ_{2} in Cr_{7}Ni was subsequently reported and found to be interestingly large^{15,16}; however, the important Rabi quantum oscillations were not observed, probably because electronic and nuclear degrees of freedom were too strongly linked to each other. As these oscillations have until now only been observed in nonmolecular spin systems (see, for example, refs 17–20), it has remained an open question whether quantum oscillations could in principle be realized in molecular magnets^{7,8}. This question is now answered by our observation of quantum oscillations of the Rabi type in V_{15}. The main reason for this success lies in the fact that the important pairwise decoherence mechanism^{7,8} associated with dipolar interactions could be strongly reduced.
Before discussing the observed quantum oscillations, we first briefly describe the magnetic/electronic structure of the species as determined experimentally. Following the synthesis of the quasispherical mesoscopic cluster anion nearly two decades ago (ref. 11), the properties of this molecule have received considerable attention (see, for example, refs 1, 11, 14, 21–25). The V_{15} cluster with an ∼1.3 nm diameter exhibits an unique structure with layers of different magnetizations: a large central triangle is sandwiched by two smaller hexagons^{11} (Fig. 1). The 15 S = 1/2 spins are coupled by antiferromagnetic superexchange and Dzyaloshinsky–Moriya (DM) interaction^{13,21,22,23,24,25} (see also refs 26, 27) through different pathways, which results in a collective low spin ground state with S = 1/2 (refs 12, 13, 24, 25).
Energy spectrum calculations for the full cluster spin space give two S = 1/2 (spin doublet) ground states slightly shifted from each other by DM interactions, and an S = 3/2 (spin quartet) excited state; these states are ‘isolated’ from a quasicontinuum of states lying at energy E/k_{B} ≈ 250 K above the S = 3/2 excited state. These lowlying energy states can be obtained with a good accuracy using the generally accepted threespin approximation (valid below 100 K), in which the spins of the inner triangle are coupled by an effective interaction mediated by the spins of the hexagons^{12,13,21,22,23,24,25} (Fig. 2 and Methods; J_{0} and J′ are shown in Fig. 1b).
The spin hamiltonian of V_{15} can be written as:
where D_{ij} is the antisymmetric vector of the DM interaction associated with the pair ij, and A is the hyperfine coupling constant of the ^{51}V isotope (see below). The six components of D_{ij} can be expressed in terms of two parameters, namely D_{Z} (perpendicular to the plane) and D_{XY} (inplane). The DM interaction removes the degeneracy of the two lowlying doublets and produces a first order zerofield splitting (plus small second order corrections)^{22,23,24,25}. The excited (quartet) state shows only a second order splitting caused by a small intermultiplet mixing through the inplane component of DM coupling, that is, (refs 24, 25). The energy separation between the doublet states and quartet state is given by (refs 13, 21–25). Figure 2 shows the level scheme calculated by diagonalization of the hamiltonian (equation (1)), with only one free parameter D_{Z} ≈ 43 mK adjusted to fit the positions of the measured resonances (a value close to that obtained from magnetization data^{13,21,24}), and D_{XY} = 0, a choice conditioned by the fact that the transverse DM component has a negligible effect on resonance fields below 0.5 T (this is important in the calculation of transition probabilities only). To ensure legibility, hyperfine interactions are not included in Fig. 2 (they simply broaden the levels).
A new hybrid material, based on the use of a cationic surfactant—DODA —as an embedding material for the anionic clusters, was developed for the present work (see Methods). The related frozen system contains V_{15} clusters integrated into the selforganized environment of the surfactant. The clusters—prepared according to ref. 11—were extracted from aqueous solution into chloroform by the surfactant DODA present in large excess. The surfactants, which wrap up the cluster anions, are amphiphilic cations, with their long hydrophobic tails pointing away from the cluster anions, enabling solubility in chloroform. The procedure ensures that the cluster anions cannot get into direct contact with one another; they are clearly separated by the surfactants (mean distance ∼13 nm).
Electron paramagnetic resonance (EPR) experiments were performed on this hybrid material at ∼4 K using a Bruker E580 Xband continuouswave (CW) and pulsed spectrometer operating at 9.7 GHz. The CWEPR spectrum, recorded at 16 K on a frozen sample, corresponds precisely to that obtained in the solid state in a previous study^{12}. In particular, the resonance field shows the same profile and linewidth (∼30 mT), compatible with the gtensor values of a single crystal ( and ). The measured transition width W ≈ 35 mT is directly connected with the energy E occurring in the expression of decoherence calculated for a multispin molecule^{7,8} (see below). Note that this transition width W should be associated with S = 3/2, the EPR spectrum being dominated by the excited quartet.
Rabi oscillations were recorded using a nutation pulse of length t, followed (after a delay greater than τ_{2}) by a π/2π sequence. Experimental results showed two different types of Rabi oscillations, corresponding to the resonant transitions 1, 2 and 3 for S = 3/2 spins, and 4, 5, 6 and 7 for S = 1/2 spins, here called ‘3/2’ and ‘1/2’, respectively (Fig. 3b and a, respectively). Although both types of oscillation are associated with the same collective degrees of freedom of the clusters, they show very different behaviour. In particular, the first type of Rabi frequency compares well with that of a single spin3/2 system, whereas the Rabi frequency of the second type is much smaller than that of a single spin1/2. This is a consequence of selection rules: the transition type ‘3/2’ is always allowed, whereas the transitions 5 and 7 of the ‘1/2’ type occur only due to transverse DM interactions or/and breaking of the C_{3} symmetry^{25} (Methods). Therefore we obtained Rabi oscillations with quite different frequencies, and , and a small ratio of transition probabilities (or intensities) R < 6 × 10^{2} (Fig. 3, Methods). When the transition ‘1/2’ is excited (by a single excitation pulse), a whole spectrum of Rabi oscillations is generated. The frequency of the detected oscillation depends on the characteristics of the detection pulse, such as its length or its amplitude (Fig. 3). This spectrum is due to the presence of an avoided level crossing and the special selection rules; these are caused by the uniaxial anisotropy introduced by the DM interactions in the spinfrustrated (orbitally degenerate) ground state giving the overlapping transitions 4–7 (Fig. 2). The glassy character of the investigated frozen material is also relevant here; this material contains different cluster orientations, leading to a distribution of transverse field components, which gives a scattering of the coefficients of the states entering in the twolevel wavefunctions and and therefore a distribution of the Rabi frequencies (Fig. 2 and Methods). Whereas the splitting of the excited quartet state in a magnetic field is almost isotropic, the distribution function of the associated Rabi frequency is very narrow.
An extension of the experiments shown in Fig. 3 to other values of the applied field showed that Rabi oscillations could be detected for each value of the applied field below 500 mT, while the transitions are inhomogeneously broadened. Figure 4 gives the result of a systematic investigation, consisting of the measurement of the spinecho intensity at time t = 0 in a sweeping magnetic field. Two broad resonance distributions are observed, which correspond to the Rabi oscillations ‘3/2’ and ‘1/2’ of Fig. 3b and a, respectively, which were measured near the maxima H_{3/2} ≈ 357 mT and H_{1/2} ≈ 335 mT of the curves of Fig. 4. Whereas the nearly symmetrical type ‘3/2’ distribution shows resonances which are optimally excited by pulse durations and powers similar to those generally used for isolated 3/2 spins, the asymmetrical type ‘1/2’ distribution shows resonances requiring larger power and pulse length, confirming much smaller transition probabilities. The observed inhomogeneous widths (∼50 ± 10 mT) result from the existence of different transitions—that is 1 to 3 and 4 to 7 shifted by the longitudinal field components associated with the glassy character of the frozen solution. The width of the resonance of type ‘1/2’ (Fig. 4) fits the transition fields calculated from the hamiltonian (equation (1)) for the resonances 4 to 7 with limiting angles and π/2 (Fig. 2), whereas the width of the resonance of type ‘3/2’ is simply given by the unique resonance field of transitions 1 to 3 (Fig. 2 a). In both cases, the ^{51}V hyperfine interactions contribute equally to the resonance widths.
To conclude, it was possible to entangle the 15 spins of a molecular magnet—a complex system which, formally speaking, entails a Hilbert space of dimension D_{H} = 2^{15} (Methods)—with photons by performing pulse EPR experiments on a frozen solution of randomly oriented and well separated clusters. Despite the complexity of the system^{11,12,13,14,21,22,23,24,25} (involving in a formal consideration dozens of cluster electrons and nuclear spins of ^{51}V, ^{75}As and ^{1}H), longlived Rabi oscillations^{10} were generated and selectively detected. An analysis, based on the widely used threespin approximation of V_{15} (refs 12, 13, 21–25; the related interactions are mediated by the 12 other spins) gives a global interpretation of the results.
The observed coherence on the microsecond timescale seems to be mainly limited by the bath of nuclear spins. Each V_{15} cluster is correspondingly weakly coupled to 36 firstneighbour protons of the six DODA methyl groups distributed around the cluster, and to two water protons at the cluster centre. According to the charge (6) of V_{15}, six cationic DODA surfactants are relevant, with their positively charged parts (six dimethyl groups) attached to the O atoms of the cluster surface (see also ref. 28); the corresponding neutral hybrid just leads to the solubility in the organic solvent. The distance from the H atoms of a methyl group to a V^{IV} is ∼0.45 nm. For this typical spin–proton distance, the halfwidth of the gaussian distribution of the coupling energy of a cluster/surfactant unit is E ≈ 3.5 mK, giving, for the level separation Δ ≈ 0.4 K (Fig. 2), the coherence time^{7,8} . The contribution of more distant neighbouring protons should reduce this value to a few microseconds. Regarding the decoherence effect from ^{51}V, the transition width W ≈ 35 mK gives E = W/2 ≈ 17 mK and , suggesting that the observed decoherence of the S = 3/2 resonances is almost entirely caused by the ^{51}V nuclear spins. The observed larger coherence time of the S = 1/2 transitions is presumably due to their smaller hyperfine coupling. In spite of the relatively high temperature of the measurement, the phonons’ decoherence^{7,8} is strongly lowered due to the low spin and anisotropy values involved in the electron–phonon^{29,30} coupling , giving , that is, . Finally, the pairwise decoherence mechanism originating from electronic dipolar interaction^{7}, which is usually considered as the most destructive, is nearly negligible, owing to the strong dilution of the clusters that results from the surfactant environment. This allows weak dipolar interactions only (∼0.5 μK) and very large coherence times (). A comparison of the different decoherence mechanisms suggests that coherence times greater than 100 μs should be obtained in molecular magnets at liquidhelium temperatures if nuclearspinfree molecules and deuterated surfactants are used.
The control of complex coherent spin states of molecular magnets—in which exchange interactions can be tuned by well defined chemical changes of the metal cluster ligand spheres—could finally lead to a way to avoid the ‘roadblock’ of decoherence. This would be particularly important in the case of selforganized one or twodimensional supramolecular networks, where well separated magnetic species could be addressed selectively, following different schemes already proposed for the molecular magnet option.
Methods Summary
When we refer to the threespin approximation of V_{15} (refs 12, 13, 21–25), we consider the three spins located on each corner of the inner triangle (Fig. 1b). However these spins do not interact directly but via the other spins of the cluster. Strictly speaking, each hexagon contains three pairs of spins strongly coupled with J ≈ 800 K (‘dimers’) and each spin of the inner triangle is coupled to two of those pairs, one belonging to the upper hexagon and one belonging to the lower hexagon (J_{1} ≈ 150 K and J_{2} ≈ 300 K). This gives three groups of five spins with resultant spin S = 1/2 (superposition of ‘entangled’ states, coupled through interdimer hexagon superexchange J′ ≈ 150 K), showing that, in fact, the threespin approximation involves all of the 15 spins of the cluster and therefore the Hilbert space has the dimension D_{H} = 2^{15} (D_{H} for the threespin system is 2^{3}). This approximation simplifies the evaluation of the lowlying energy levels of the 15 ‘entangled’ states of the V_{15} cluster. For the S = 1/2 orbital doublet ^{2}E, whose basis functions can be labelled by the quantum number of the total pseudoangular momentum M_{J} = M_{L} + M_{S} , is associated with the pseudoorbital momentum M_{L} = +1 or M_{L} = 1 (refs 24, 25). The allowed EPR transitions satisfy the subsequent selection rules: , , that is for the interdoublet transitions 4 and 6, and , , that is for the weak intradoublet transition 5 whose transition probability is caused by a small intermultiplet mixing through the inplane component of the DM coupling. The intensity of this transition is significantly increased when transition 7 becomes allowed due to a weak deviation from the C_{3} symmetry (Fig. 1). This also leads to an increased zerofield gap where δ is the parameter in the exchange shift δ S_{1}S_{2}.
Online Methods
Sample synthesis
0.04 g (0.0175 mmol) of freshly prepared brown obtained as reported^{10} was dissolved in 20 ml of degassed water. After addition of 25 ml of a (degassed) trichloromethane solution of [DODA]Br (1.10 g/1.75 mmol) the reaction medium was stirred under inert atmosphere. The stirring was continued until the olivebrown coloured aqueous layer turned colourless and the corresponding colour appeared in the organic phase. The organic layer was then quickly separated, put into an EPR tube and frozen to liquid nitrogen temperature. All operations were done in an inert atmosphere.
Comparing Rabi frequencies
The frequency of the Rabi oscillations between two states 1 and 2 is given by^{6,7,8,19}:
Here is the Rabi frequency of a spin 1/2, B_{1} is the amplitude of the a.c. microwave fields, g ≈ 2 the Landé factor, S_{+} the ladder operator and , the wavefunctions associated with these states. The probability of a transition, defined as , is directly connected with its Rabi frequency:
This allows one to evaluate the ratio (R) of the probabilities associated with two transitions (here the ‘3/2’ and ‘1/2’ types) from the measurement of their Rabi frequencies without the knowledge of their wavefunctions:
Using the values of the Rabi frequencies given in Fig. 3, one gets R ≈ (4.5/18.5)^{2} ≈ 5.9 × 10^{2}. The time T_{π/2}, during which the excitation pulse is applied to induce a π/2 rotation, is by definition equal to 1/4Ω_{R} (refs 6, 19), showing that equation (4) is equivalent to:
This gives another way to determine R. Using the T_{π/2} values given in Fig. 4 legend, one gets R ≈ (16/64)^{2} ≈ 6.2 × 10^{2}, which is very close to the first one and shows that the probability associated with the ‘1/2’ type transition is much smaller than the one associated with ‘3/2’.
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Acknowledgements
We acknowledge I. Chiorescu from NHMFLFSU, Tallahassee, USA, for discussions. We thank M.N. Collomb for help in processing samples for EPR measurements, and G. Desfonds for technical support. B.B. and A.M. thank the European Research Council for support through network projects MAGMANet, MolNanoMag, QueMolNa and INTAS; A.M. thanks the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie for support; and B.T. and A.M. thank the German–Israeli Foundation for Scientific Research and Development for support.
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Bertaina, S., Gambarelli, S., Mitra, T. et al. Quantum oscillations in a molecular magnet. Nature 453, 203–206 (2008). https://doi.org/10.1038/nature06962
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