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# Increasing the Hilbert space dimension using a single coupled molecular spin

## Abstract

Quantum technologies are expected to introduce revolutionary changes in information processing in the near future. Nowadays, one of the main challenges is to be able to handle a large number of quantum bits (qubits), while preserving their quantum properties. Beyond the usual two-level encoding capacity of qubits, multi-level quantum systems are a promising way to extend and increase the amount of information that can be stored in the same number of quantum objects. Recent work (Kues et al. 2017), has shown the possibility to use devices based on photonic integrated circuits to entangle two qudits (with “d” being the number of available states). In the race to develop a mature quantum technology with real-world applications, many possible platforms are being investigated, including those that use photons, trapped ions, superconducting and silicon circuits and molecular magnets. In this work, we present the electronic read-out of a coupled molecular multi-level quantum systems, carried by a single Tb2Pc3 molecular magnet. Owning two magnetic centres, this molecular magnet architecture permits a 16 dimensions Hilbert space, opening the possibility of performing more complex quantum algorithms.

## Introduction

For some decades, significant efforts have been invested in quantum information research, with the promise to revolutionise the way information is stored and processed. The strength of quantum computing lies in the possibility of using a coherent superposition of states, and interference between them, which enables a class of algorithms that are not accessible to classical computers. To achieve this goal, it is first mandatory to coherently manipulate and read-out a two-state system (qubit), as demonstrated in photonic devices1, trapped ions2,3, superconducting circuits4, electronic spins in two-dimensional electron gases (2DEG)5,6, phosphorous atom impurities in silicon7,8,9,10, nitrogen-vacancy (NV) centres11,12, or molecular systems13,14, among others. The second step consists of the interconnection of N entangled qubits, spanning the number of states to 2N, while preserving their quantum properties, in order to allow information processing, as demonstrated for example in superconducting circuits15,16,17, two-dimensional electron gases5,6, silicon devices18 and magnetic molecules19,20. Towards the aim of scalability, the number of interconnected qubits has to be increased, as demonstrated by the 16-qubit processor of the IBM Q-experience21, a 72-qubit quantum chip announced by Google22, a 19-qubits computer tested by Rigetti23, and a silicon spin-based 2-qubits processor obtained by QuTech24. More recently, even a NASA-Google partnership has claimed quantum supremacy25. Among these results, qudits, have been proposed as an alternative path towards the enlargement of the number of available quantum states for computation26,27,28,29, as qudits allow to increase the number of states for the same number of quantum systems. Recently, the coupling of qudits has even been obtained using photons30. In their work, Kues et al. demonstrated an approach in which photons, possessing multiple high-purity frequency modes, are generated on an integrated micro-chip. With d = 10, they confirmed a 100-dimension Hilbert space, and validated this platform by measuring Bell inequality violations and performing quantum state tomography.

Towards this alternative, magnetic molecules possessing magnetic memory, better known as Single Molecule Magnets (SMMs), are a promising spin qudits platform13,14,31. These systems have the advantage of being synthesised in thousands of identical units and can be integrated in devices through bottom-up approaches, retaining their intrinsic magnetic properties32,33,34. SMMs can also be tailored through the use of different metal ions and ligands compositions for their integration in devices. Likewise, SMMs have been shown to exhibit quantum phenomenon such as Quantum Tunnelling of the Magnetisation (QTM)35,36,37,38, quantum phase interference39,40, and coherent spin manipulation with coherence times of the order of 1 μs41. This is the case of a bis(phthalocyaninato)terbium(III) SMM (TbPc2), which has been integrated into transistor-like configurations42,43,44,45 as well as in carbon-nanotube transistors46,47. Different studies have shown its magnetic resilience when deposited on surfaces, and the Tb3+ ion redox stability34. In the TbPc2, the π-electron system on the ligand eases the electric conduction, enabling electric read-out42,44 of the nuclear spin I = 3/2 carried by the Tb3+ ion. With coherence time of the order of 300 μs, the coherent manipulation of the different nuclear spin states was demonstrated, leading to the implementation of the Grover algorithm using this nuclear spin qudit45. This strategy clearly highlights the potential of qudits, as long coherence times and quantum algorithms have been achieved without requiring inter-qubit coupling. In this context, increasing the Hilbert space dimension is an appealing option for implementing more complex quantum algorithms or universal reversible quantum logic gates, such as the Fredkin or Toffoli gates.

Herein, in the quest of molecular spin systems with larger accessible Hilbert space, we present the first electrical read-out of the nuclear spin states, carried by a coupled multi-level system, achieved with a single-molecular unit of Tb3+ triple-decker phthalocyanine, incorporated in a three-terminal transistor (Fig. 1a). We show that the magnetic moment of the two Tb3+ ions effectively couples with the electronic structure of the Pc ligands allowing the electronic read-out of the nuclear spin states. More importantly, we observe the indirect coupling between the nuclear spin states residing on the Tb3+ ions. We emphasise here that there is no entanglement between the two electronics or nuclear spins, but a larger Hilbert space dimension (d = 16) can be achieved via indirect interaction between the nuclear spins. Indeed, these results could represent an alternative for the realisation of more complex quantum algorithms using a single-molecular multi-level system.

## Results

### Tb2Pc3 single-molecule magnet

to probe our coupling scheme, we chose a tris(phthalocyaninato)di(terbium(III)) Tb2Pc3 SMM, composed of two Tb3+ ions and three phthalocyaninato (Pc) ligands. Both terbium ions sit on a distorted square antiprism geometry. As the TbPc2 analogue, the total angular momentum J = L + S = 6 characterises the [Xe]4f8 electronic shell of each Tb3+ metal ion. Note that the strong spin-orbit coupling separates the J = 6 state from the J = 5 by 2900 K, hence, at common operating temperatures, solely the J = 6 is considered. At a molecular level, the ligand field in which the Tb3+ is located, created by the Pc ligands, leads to a doublet electronic ground state with strong uniaxial anisotropy. To calculate the Zeeman diagram for the Tb2Pc3 molecule, a ligand field Hamiltonian for C4 geometry is expressed in the Stevens operators formalism:

$${H}_{{lf}}^{i}={{\langle }}{r}^{2}{{\rangle }}{u}_{2}{A}_{2}^{0}{O}_{2}^{0}+{{\langle }}{r}^{4}{{\rangle }}{u}_{4}({A}_{4}^{0}{O}_{4}^{0}+{A}_{4}^{4}{O}_{4}^{4})+{{\langle }}{r}^{6}{{\rangle }}{u}_{6}({A}_{6}^{0}{O}_{6}^{0}+{A}_{6}^{4}{O}_{6}^{4})$$
(1)

and the Zeeman Hamiltonian,

$${H}_{{{{{\mathrm{Zeeman}}}}}}^{i}={g}_{J}{\mu }_{0}{\mu }_{B}{J}_{{||}}^{i}\cdot {B}_{{{{\rm{||}}}}}\,$$
(2)

where in (1) ui are the Stevens factors, $${O}_{i}^{j}\,$$ are the Stevens operator and $${A}_{i}^{j}$$ <rj> are the ligand field parameters. In (2) gJ, J and $${{{{\boldsymbol{B}}}}}_{{{{\boldsymbol{||}}}}}$$ are respectively the gyromagnetic ratio of Tb3+, the electronic moment operator of Tb3+ and the applied field vector. The presence of $${O}_{i}^{j}$$ provides a strong anisotropy to Tb2Pc3, causing an energy gap of ca. 400 K between the mJ = $${|}\!{\pm} \!6\rangle$$ and the mJ = $${|}\!{\pm}\!5\rangle$$ electronic states. µ-SQUID48 and ab-initio CASSCF49 studies have revealed the anisotropy of the mJ = $${|}\!{\pm}\!6\rangle$$ doublet state of the Tb3+ ions to have an easy axis perpendicular to the Pc planes.

Moreover, both Tb3+ ions in the dimer are coupled through a ferromagnetic interaction, therefore, the Hamiltonian considering the ligand field and the interaction is:

$${H}_{{{{{{\mathrm{Tb}}}}}}_{2}}={H}_{{{{{\mathrm{lf}}}}}}^{i}+{H}_{{{{{\mathrm{Zeeman}}}}}}^{i}-C{J}_{1}{J}_{2}$$
(3)

where the first accounts for the ligand field for each Tb3+ ion, the second term is the Zeeman term and the third term represents the coupling between the electronic spin of the Tb3+ ions. Figure 1b shows the Zeeman diagram corresponding to the diagonalization of the (2J + 1)·(2J + 1) matrix. At zero field, the ferromagnetic ground state $${|}\!{\pm}\!6,{\pm}\!6\rangle$$ and the antiferromagnetic excited state $${|}\!{\pm}\!6,\mp 6\rangle$$ are separated from the first excited state $${|}\!{\pm}\! 5,{\pm}\!5\rangle$$ by ΔE ≈ 400 K (Fig. 1b). Note that at our experimental electronic temperatures (≈80 mK) only the mJ = $${|}\!{\pm}\! 6,{\pm}\! 6\rangle$$ is populated.

The presence of the transverse terms $${O}_{4}^{4}$$ and $${O}_{6}^{4}$$ in the ligand field Hamiltonian induces mixing of the $${|}\!+\!6,+6\rangle$$ and $${|}\!-\!6,-6\rangle$$ states via $${J}_{4}^{\pm }$$, resulting in an opening of the anti-crossing between the $${|}\!{\pm}\! 6,{\pm}\! 6\rangle$$ states. At very low temperatures, this opening acts as a relaxation pathway for the temperature-independent relaxation of the magnetisation, i.e., QTM process. At the anti-crossings the tunnelling probability is governed by the Landau–Zener equation:

$${P}_{m,m^{\prime} }=1-{\exp }\left(-\frac{\pi {\Delta }_{m,m^{\prime} }^{2}}{2{{\hslash }}{g}_{J}{{{\mu }}}_{{{{\mathrm{B}}}}}\left|m-m^{\prime} \right|{{{\mu }}}_{0}{{{\mathrm{d}}}}{H}_{\parallel }/{{{{\mathrm{d}}}}t}}\right)$$
(4)

where µ0 is the vacuum permeability, µB is the Bohr magneton, m and m′ are the two states involved ($$\left|m-m^{\prime} \right|$$ = 24 for Tb2Pc3) and $${\Delta }_{m,m^{\prime} }^{2}$$ is the anti-crossing splitting energy, which is contingent upon the coupling between the J1 and J2 states. Calculation of $${\Delta }_{m,m^{\prime} }^{2}$$ for different ferromagnetic coupling values C is presented in Fig. 1d. The red dashed line represents the crossing between the ferromagnetic and antiferromagnetic states48. From this value, we obtain $${\Delta }_{m,m^{\prime} }$$ = 0.6 mK. This large value of the anti-crossing energy results in a Landau–Zener probability Pm,m = 1 even for magnetic field sweeping rates of 103 T/s. As the maximum magnetic field sweeping rate accessible for the experiment was lower than 100 mT/s, reversal of the magnetisation is only governed by QTM, and no direct relaxation from the ferromagnetic state to the antiferromagnetic one was observed.

Up to this point, solely the electronic states of Tb2Pc3 have been considered. From now on, we consider the nuclear spin I = 3/2 carried by each Tb3+ ions. Considering the strong hyperfine interaction, each $${|}\!{\pm}\!6\rangle$$ ground state is split into four states unevenly separated corresponding to mI = ±3/2, ±1/2. The effect of the nuclear spins and quadrupolar interaction is considered in the following Hamiltonian:

$${H}_{{{{{\mathrm{hf}}}}}}^{i}={A}_{{{{{\mathrm{hf}}}}}}{I}^{i}{J}^{i}+P{({I}^{i})}^{2}$$
(5)

where A = 30.9 mK and P = 14.4 mK are the hyperfine interaction and the quadrupolar constants respectively, considered to be the same for both Tb3+ sites48. As nuclear spins have magnetic moments ~2000 times smaller than electronic spins, direct nuclear spin interaction can be safely neglected. In contrast, the nuclear spins can be indirectly coupled via the dipolar interaction between the Tb3+ ions, resulting in 4·4 = 16 possible nuclear spin states. Hamiltonian in Eq. (6) accounts for all interactions in Tb2Pc3.

$${H}_{{{{{{\mathrm{Tb}}}}}}_{2}}={H}_{{{{{\mathrm{lf}}}}}}^{i}+{H}_{{{{{\mathrm{Zeeman}}}}}}^{i}+{H}_{{{{{\mathrm{hf}}}}}}^{i}-C{J}_{1}{J}_{2}$$
(6)

Numerical diagonalization of the Hamiltonian in Eq. (6) results in a ferromagnetic ground state with multiple level-crossings in the range $${B}_{\parallel }$$ = ±46 mT (Fig. 2a). The square regions in Fig. 2b indicate the anti-crossings. In these regions, electronic spins reversal is allowed via QTM, while preserving the nuclear spin states. As calculated, a total of 10 anti-crossings allow the reversal of the electronic spins. Nevertheless, at three of these positions the electronic spins reversal occurs at the same magnetic field $${B}_{\parallel }$$, therefore, only seven positions in the magnetic field where QTM occurs should be observed.

the nuclear spin states read-out of the Tb2Pc3 molecular magnet was studied via transport measurements through the molecule while sweeping the external magnetic field in two directions. The quantum dot is defined by the Pc ligands which are known to be good electrical conductors due to their π delocalised links. Moreover, the overlap between the 4f shell of the Tb3+ ions and the π delocalised links of the Pc ligands give rise to an exchange coupling44. Our detection scheme is then based on an exchange interaction between the electronic spin of the Tb3+ ions and the nearby Pc read-out quantum dot. The coupling between the two systems results in a spin-dependent conductance through the read-out dot and establishes an all-electrical single spin detection. Sweeping back and forth the external magnetic field BX in the direction of the coil, the differential conductance is simultaneously recorded using a digital lock-in amplifier. Additionally, a constant external magnetic field BY is applied along the other direction of the coil. Considering a single magnetic field sweep, we observe an abrupt jump in the differential conductance as presented in Fig. 3a. This abrupt variation is due to the reversal of the electronic spins carried by the two Tb3+ ions42,43,44,45,46,47,50. Sweeping BX back and forth, a hysteresis is clearly observed in Fig. 3a. By measuring the difference of the differential conductance Δ(dI/dV) while sweeping back and forth the magnetic field BX, for different values of BY, we observe different regions where abrupt jumps are measured (Fig. 3a). From now on, we will focus on the magnetic field region highlighted by the green double arrow in Fig. 3b. Other regions correspond to other molecular systems for which the nuclear spin signatures were not resolved.

Figure 3c presents the histogram compiling the differential conductance jump positions for 6000 back and forth magnetic field BX sweeps at 50 mT/s with an additional constant BY = 1 T. Note that we were not able to sweep the external magnetic field in the direction of the easy axis of the Tb2Pc3 molecular system due to a missing coil in the Z-direction of our experimental setup. As presented in Fig. 3c, two main features are observed: (i) peaks in the histogram with a high amplitude and roughly equally spaced, (ii) peaks in the histogram with a smaller amplitude and width and unequally spaced from the previous ones.

(i) We first focus on the main feature corresponding to the peaks in the histogram with a high amplitude and relatively large width (counts framed in grey in Fig. 3c). Seven peaks are observed, corresponding to the ten anti-crossings where tunnelling is allowed as expected from the theoretical predictions. Note here that the experimental values do not correspond to the QTM positions expected from Eq. (6); consequence of the miss-alignment of the easy axis of the Tb2Pc3 molecule with our applied magnetic field. As a result, while the positions of the QTM in a parallel magnetic field are expected within $${B}_{\parallel }$$ = ±46 mT, the measured positions using an external magnetic field occur within BX = ±55 mT.

(ii) Two additional peaks are clearly visible at magnetic field BX higher than +55 mT and lower than −55 mT. First, the amplitude of these two peaks is lower compared to the amplitude of the main seven peaks. Moreover, the magnetic field spacing is lower than the spacing obtained for the main seven peaks. Therefore, these additional peaks in the histogram could be attributed to signatures of another molecular system we were not able to distinguish. This phenomenon will need further investigations and is a probable source of the large envelope observed in Fig. 3c, d. It is important to state that additional peaks should also be present in between those two, but the amplitude is too low to be observable. However, in the following studies, quantitative measurements could be approximative, as an accurate determination of the different nuclear spin states population would require not to take into account these additional peaks events, with a much lower amplitude, that could not be discriminated from the main seven peaks.

We now present the study of the probability of the QTM events considering the states degeneracy and the positions of the anti-crossings. For both coupled Tb3+ ions, a total of 16 states is expected. As described earlier, out of these 16 states, 10 are not degenerate, while three of the anti-crossings occur at the same magnetic field. As a result, a total of seven states are truly observable. The nuclear spins states distribution can be obtained considering the 16 states as follows: the two most external peaks at ±55 mT (positions #1 and #7) have a probability P = 1/16 each, while the two immediate neighbours at ± 38 mT (positions #2 and #6) and ±20 mT (positions #3 and #5) account for P = 2/16 and P = 3/16 respectively. Finally, the events at zero magnetic field (position #4) have P = 4/16, which corresponds to the double electronic spin reversal for 4 different nuclear spins. The results are summarised in Table 1.

### Coupled nuclear spins dynamics

The dynamic of the nuclear spins was studied analysing the same 6000 back and forth magnetic field sweeps. A correlation cartography of the identified abrupt conductance jump positions is presented in Fig. 4a. The abscissa corresponds to the reversal of the electronic spins during the field sweep, while the ordinate corresponds to a reversal during the next sweep. The white dashed lines indicate the anti-crossing positions for which events are predicted. In order to ease the reading, a Gaussian filter is used to bring out the density of the local events. The utility of this correlation cartography is twofold: (i) first we obtain a better visualisation of the nuclear spins states; (ii) we can visualise to what extent the quantum system remains in the same nuclear spins state in between two sweeps through the anti-crossing regions.

Using the correlation cartography, it is possible to obtain an estimation of the lifetime of the nuclear spin states compared to the measurement time (≈3 s). As the two axes correspond to the trace and retrace BX magnetic field sweeps, two subsequent measurements with the same nuclear spin states are situated on the diagonal of the correlation map, whereas the off-diagonal positions correspond to nuclear spin state relaxation or excitation. As presented in Fig. 4, the correlation cartography does not exhibit an excess of events on the diagonal. It reveals a nuclear spins states lifetime shorter than the measurement time. Moreover, taking into account that the seven anti-crossing regions have different degeneracies, we renormalized the event counts as per the cumulated degeneracy of the corresponding abscissa and ordinate anti-crossings. The result is presented in the histogram Fig. 4b. We observe a higher population of the up left and bottom right blocks, clearly demonstrating the quantum system relaxation in between two consecutive measurements.

A more detailed insight into the effective temperature of the nuclear spins states has been obtained by analysing the population of the states for 6000 forward magnetic field (sweeps from negative to positive magnetic field values). We present in Fig. 5a the renormalisation of the populations of the different nuclear spins states as per each anti-crossing degeneracy. Due to a low lifetime compared to the measurement time, the occupancy of states exhibits a relaxation towards a thermal equilibrium, as shown by the measurement of a higher population for the nuclear spin states with lower energies. To obtain a quantitative analysis of the temperature of the nuclear spins, we used Maxwell–Boltzmann statistics:

$$\frac{{N}_{i}}{N}=\frac{{g}_{i}{e}^{\frac{-{E}_{i}}{{k}_{{{{\mathrm{B}}}}}T}}}{Z(T)}\,{{{\rm{with}}}}\,Z(T)=\mathop{\sum }\limits_{i=1}^{n}{e}^{\frac{-{E}_{i}}{{k}_{{{{\mathrm{B}}}}}T}}$$

where Z(T) is the system partition function, kB is the Boltzmann constant, T is the temperature, Ei are the energies separation compared to the lowest energy level, and gi their respective degeneracies. Ni is the population of each state, with the total population $$N=\mathop{\sum }\nolimits_{i=1}^{n}{N}_{i}$$. Note that Ni/gi is the normalisation used previously for the correlation cartography. From this study, we obtain an effective temperature T ≈ 290 mK of the nuclear spins, which is consistent with the one obtained on a TbPc2 molecular magnet, measured with a comparable experimental set-up. We believe that the short lifetimes could be a result of flow current through the Tb2Pc3 unit, which induces a larger FWHM, and thus decreases the fidelity of the system50.

Herein we have presented the electrical transport measurements of a Tb2Pc3 molecule embedded in a transistor, resulting in the first electronic read-out of a single coupled molecular spin. The read-out procedure consists of the detection of abrupt conductance jumps induced by the QTM of the electronic spins, in agreement with theoretical predictions. The seven measured QTM positions in magnetic fields are a direct consequence of the indirect coupling of the nuclear states residing in each Tb3+ ion via the electronic interaction. Moreover, we estimated a low lifetime compared to our measurement time which has been observed and ascribed to the current flowing through the molecular unit50. These two aspects can be certainly enhanced by device design. Our results clearly demonstrated the potential of coupled multi-level systems to span the Hilbert space using a single-molecular magnet, an important characteristic for the realisation of more complex quantum algorithms. Furthermore, the uneven separation of the different coupled nuclear spins would in principle allow the manipulation of each individual coupled state. These two characteristics may path the way to the realisation of more complex algorithms such as Fredkin or Toffoli.

## Methods

### Nanofabrication

Gold nano-constrictions acting as source and drain electrodes are fabricated through e-beam angles depositions, on the top of a gate composed of a metallic gold part and a high-k hafnium dioxide. It has to be noticed that the obtained electrodes arrow-like geometry enables to reduce the screening effect of the gate electric field51. The molecules were diluted in a solvent and later deposited at room temperature after a chemical cleaning (reactive-ion etching) of the sample surface. Afterwards, the electromigration process was achieved at low temperatures (4 K). The electromigration process consists of applying a difference of potential in between two electrodes forming a nano-constriction, so that the conduction electrons put in motion the gold ions, until a nano-interstice is opened52. In order to control the interstice size, the potential difference has to be precisely controlled with a real-time electronic. Thereby, when the conductance is below the so-called quantum of conductance (G0 = 2e2/h), the process is stopped in around 10 μs.

### Cryogenic measurement

If the first conduction measurements between the source and drain electrodes were showing nano-metre size quantum dot signatures, the magnetisation reversal measurements were performed in a 3He/4He dilution refrigerator so the sample temperature was below 100 mK. The cryostat used for this work was equipped with two coils, generating up to 5.5 and 1.25 T with a maximum sweep rate up to 100 mT/s, and allows the sample to rotate with respect to them, providing a 3D control of the applied magnetisation. A more detailed description of the sample fabrication and experimental set-up can be found in Chaps. 4 and 5 in ref. 53.

## Data availability

The data that supports the findings of this study are available from the corresponding authors upon request.

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## Acknowledgements

We thank E. Eyraud, C. Thirion, T. Crozes and L. Abbassi for help with cryogenic set-up, maintenance, data collection and treatment, and sample preparation. We also thank S. Lumetti, M. Nale, M. Urdampilleta and J. Viennot for helpful discussions. H. B. kindly thanks the 2015 AGIR-POLE-PEM programme for PhD funding. ﻿We acknowledge the DFG-TR 88 “3Met” (project A8) and the Karlsruhe Nano Micro Facility (KNMF, www.kit.edu/knmf) for the provision of access to instruments at their laboratories. EMP thanks the Panamanian National System of Investigators (SNI, SENACYT) for support. W.W. thanks the A. v. Humboldt foundation and the ERC grant MoQuOS No. 741276.

## Funding

Open Access funding enabled and organized by Projekt DEAL.

## Author information

Authors

### Contributions

H.B. prepared all samples, conducted the measurements and analysed the data. F.B., M.R. and W.W. conceived the idea and supervised the project. The data were discussed and analysed by H.B., E.B., F.B. and W.W. E.B. created the software to conduct the measurements in the device. The synthesis and bulk characterisation of the Tb2Pc3 molecule was conducted by E.M.-P. and M.R. The manuscript was written by H.B., E.M.-P. and F.B. and discussed and analysed by all authors.

### Corresponding authors

Correspondence to Wolfgang Wernsdorfer or Franck Balestro.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Biard, H., Moreno-Pineda, E., Ruben, M. et al. Increasing the Hilbert space dimension using a single coupled molecular spin. Nat Commun 12, 4443 (2021). https://doi.org/10.1038/s41467-021-24693-6

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• DOI: https://doi.org/10.1038/s41467-021-24693-6