Engineering coherent interactions in molecular nanomagnet dimers

Proposals for systems embodying condensed matter spin qubits cover a very wide range of length scales, from atomic defects in semiconductors all the way to micron-sized lithographically-defined structures. Intermediate scale molecular components exhibit advantages of both limits: like atomic defects, large numbers of identical components can be fabricated; as for lithographically defined structures, each component can be tailored to optimize properties such as quantum coherence. Here, we demonstrate what is perhaps the most potent advantage of molecular spin qubits, the scalability of quantum information processing structures using bottom-up chemical self-assembly. Using Cr7Ni spin qubit building blocks, we have constructed several families of two-qubit molecular structures with a range of linking strategies. For each family, long coherence times are preserved, and we demonstrate control over the inter-qubit quantum interactions that can be used to mediate two-qubit quantum gates.


Introduction
An information processing device whose elements are capable of storing and processing quantum superposition states (a quantum computer) would support algorithms for useful tasks such as searching 1 and factoring 2 that are much more efficient than the corresponding classical algorithms 3 , and would allow efficient simulation of other quantum systems 4 . One of the key challenges in realizing a quantum computer lies in identifying a physical system that hosts quantum states sufficiently coherently, and provides appropriate interactions for implementing logic operations 5 . Among the molecular spin systems that have been proposed as qubit candidates are N@C 60 6,7,8,9 , organic radicals 10,11 , and molecular magnets 12,13,14 . We proposed exploiting molecular magnets based on heterometallic antiferromagnetic rings 15,16 . These systems exhibit a number of favourable features supporting their application as components of a quantum computer: flexibility in their chemical composition allows control over both the total ground state spin (by modifying the heteroatom) and the carboxylate ligands 17 ; their welldefined internal magnetic excitations may offer mechanisms for efficient singlequbit manipulations 15 ; and the ground state spin is highly coherent 18 , particularly when the chemical structure is optimised 19 .
In the context of molecular spin qubits, the simplest conceivable multiqubit structure is a molecular dimer. This observation has motivated various efforts to synthesise dimers including, for example, N@C 60 N@C 60 20,21 , radical radical 10,11 , N@C 60 molecular magnet 22 , and molecular magnet molecular magnet 23,24,25,26 . However, the design of a dimer specifically to host twoqubit experiments should take account of the importance of three key time scales relative to one another: T 2 , the individual qubit phase relaxation time, must be longest; h / J , which is characteristic of the duration of twoqubit gates (where J is the interqubit interaction energy and h is Planck's constant) should be intermediate; and the singlequbit manipulation time should be the shortest. In practice, phase relaxation times in heterometallic antiferromagnetic rings are in the 1 to 10 μ s range at low temperatures 18,19 and they can be manipulated in a typical pulsed electron spin resonance (ESR) apparatus on the 10 ns timescale; thus an interaction offering h / J in the 100 ns range could be exploited, for example, in a multiqubit experiment to generate controlled entanglement. In this Article we report the synthesis of two families of dimers of the antiferromagnetic ring Cr 7 Ni and pulsed electron spin resonance (ESR) experiments probing the spin coherence times and the intradimer magnetic interactions. We demonstrate that the modular nature of our synthetic approach provides independent control of the molecular nanomagnet components and of the chemical coupling between them. This allows us the flexibility to optimise the physical properties of the dimers with respect to the three key time scales.

Synthesis of multiqubit molecular structures
The molecular structures of the dimers that we studied are shown in Figure 1. The monomer components are based on Cr 7 Ni rings in which intraring nearestneighbour antiferromagnetic coupling gives rise to a welldefined S =1/2 ground state. Each dimer family employs a different variety of Cr 7 Ni ring, allowing very different approaches to chemical dimerisation.
The first family ( 1A , 1C and 1D in Figure 1) is a collection of hybrid [3]rotaxanes 27 . In each of these compounds two Cr 7 Ni rings are threaded by a rigid organic molecule. There is no covalent bonding between the two rings and the magnetic interaction is expected to be purely dipolar (through space), and modulated by both the length of the threading molecule and the orientation of the dimer with respect to an external magnetic field.
For both threads there are three aromatic rings between the two amines, and they differ only in the stopper on the thread. The reaction to make the [3]rotaxanes involves adding the organic thread to a solution of hydrated chromium trifluoride dissolved in pivalic acid (HO 2 C t Bu); the heterometallic rings then grow around each ammonium cation and the ring is completed by addition of a nickel precursor. We made four related [3]rotaxanes (see Figure 1 and  The Xband spectrum of 2A shows more structure on the lowfield edge; we believe that this due to partial resolution of hyperfine structure. In both, there is a broadening arising from unresolved hyperfine interactions.

Electron spin resonance experiments
In both compounds the CW spectra are similar to the spectra of the corresponding monomers 18,31 , indicating that in these compounds the interring magnetic interaction is small compared to the ESR linewidth in energy. This is consistent with the requirement that twoqubit gates should take longer than singlequbit manipulations, a first indication that these dimers are more suitable for quantum information processing experiments than earlier heterometallic ring dimers 24 . Xband pulsed ESR experiments yield spin coherence times, T m (given in Table 1) that are broadly consistent with the times measured for monomers 19 , offering reassurance that the formation of complexes and dimers is not intrinsically detrimental to the quantum phase coherence.
The form of the CW spectra imposes an upper bound on the intradimer magnetic interaction, but to measure this interaction precisely in this regime requires coherent methods; double electron electron resonance (DEER, also known as pulsed electron double resonance, PELDOR) 32,33,34 is the established method for so doing. Although originally applied to inorganic model systems 32 , in recent years, DEER has been applied extensively to biological systems 35 .
By modifying pairs of sites in proteins with spin labels such as nitronyl nitroxides, DEER can be used to measure the dipolar interaction, and therefore the distance, between the labels 36 .
DEER techniques exploit the fact that, if there is a magnetic interaction between two spin centres, the coherent precession rate of one of the centres is modified by an inversion of the spin state of the other centre. In practice this is achieved through pulsed ESR sequences using two frequencies, such as the standard fourpulse DEER sequence shown in Figure 3(a) 33 . The first two pulses, π /2 τ 1 π at frequency ν 1 , set up a Hahn echo 34 at a time 2 τ 1 on one of the spins. As this spin continues to precess, a "pump" π pulse at a second frequency ν 2 at time T (relative to the ν 1 echo) inverts the second spin, thus modifying the effective magnetic field (and therefore the precession rate) experienced by the first spin. A subsequent refocusing π pulse at frequency ν 1 a time τ 2 after the first echo generates a second echo at a time 2 τ 1 +2 τ 2 , with an amplitude that depends on the time at which the ν 2 inversion pulse was applied and on the strength of the interaction between the two spins. Observing this echo as a function of the time, T , of the pump pulse reveals oscillations at the frequency corresponding to the magnetic interaction energy between the two centres. A threepulse variant, shown in Figure 3(b) 32 , has the advantage of being shorter and having fewer pulses, thus enhancing the amplitude of the measured echo, particularly for systems in which the spin coherence times are short. The disadvantage is that for short times T , the ν 2 pump pulse overlaps the coherencegenerating ν 1 π /2pulse, distorting the spectra around the time T = 0 [33].
This description of DEER depends on some heterogeneity between the two spin centres in the molecule under study, so that the frequencies ν 1 and ν 2 are resonant with different centres. We performed DEER experiments at low temperatures (2.5 K, to maximise spin coherence times) on dilute frozen solutions of dimers (0.1 0.2 mM, to minimise interdimer dipolar interactions).
In this case differences between the two heterometallic rings may arise from conformational flexibility. In the first family, the [3]rotaxane dimers, there may be variation in the position of the Ni centre between the two rings, or some slight bending of the stiff threading molecule. In the second family, the covalently bound dimers, the rings could rotate about the axis defined by the linker. In either case, the distortions of individual rings (giving rise to g strains) or variations in configurations of hyperfinecoupled nuclei (resulting in variations of the effective local magnetic field) offer asymmetry between the two rings within a dimer.
The CW spectra in Figure 1 extend over a magnetic field range of about 30 mT at Xband (about 9.5 GHz), corresponding to an absorption spectrum at a fixed magnetic field covering about 750 MHz, as plotted in Figure 3(c). Intervals within this energy range can be identified with orientationally selected subpopulations of rings. Thus, microwave pulses at different frequencies within the spectrum excite orientations selectively, shown as coloured regions on spheres in Figure 3(d). We anticipate that, notwithstanding a degree of flexibility, the orientations of two rings within a single dimer should be reasonably well correlated. With this in mind, we choose to separate ν 1 and ν 2 by a frequency that is small compared to the total width of the spectrum, so that they excite neighbouring orientational subpopulations of rings.
In practice, the response of the microwave resonator used in the experiment is not uniform over the frequency range of the absorption line. Instead we used fixed frequencies for ν 1 and ν 2 , and we achieved orientation selection by adjusting the external magnetic field to bring the appropriate part of the absorption spectrum into resonance with the applied pulses. We used detection pulses (at frequency ν 1 ) of 40 ns and pump pulses ( ν 2 ) of 24 ns, separated in frequency by 80 MHz. Figure 4 shows typical DEER spectra taken on compound 1A . The left panel shows the amplitude of the final echo as a function of the time, T , of the pump pulse for three different orientation selections. Open circles are raw data; the solid curves are filtered using standard tools (the Matlab package DeerAnalysis 37 ) to smooth the data and to remove oscillations (electron spin echo envelope modulation, or ESEEM 18,34 ) associated with couplings to proton nuclear moments. To the right of each trace, a spherical intensity map indicates the range of orientations probed at the given applied magnetic field and experimental detection frequency.

Results
The right panel shows the Fourier transforms of the timedomain data from the left panels (both raw data, open circles, and filtered data, solid lines).
The dependence of the DEER signal on magnetic field (or, equivalently, orientational subpopulation) is strong, owing to several contributory factors. First, the interring dipolar interaction depends on the orientation, θ , of the dimer interring axis with respect to the magnetic field as (3 cos 2 θ 1). It therefore varies significantly as a function of the dimer orientation and passes through zero at the magic angle, θ ≈ 54°. Second, the magnitude by which the detected echo is modulated in the DEER experiment depends on the proportion of dimers excited initially by ν 1 that is also excited, at the other end, by ν 2 .
In a magnetic field of 377.84 mT (lowest trace, blue), dimers whose interspin axes are perpendicular to the field are predominantly excited. Within this subpopulation the range of interring magnetic dipolar interactions is strongly peaked around a single value, giving rise to clearlydefined oscillations with a period of about 155 ns. The Fourier transform, which represents the distribution of coupling strengths within the selected orientational subpopulation, is correspondingly sharply peaked at about 6.5 MHz. The first minimum in the timedomain data, occurring at about 77 ns, is the duration for inversion of the state of one of the qubits under the influence of the second. This evolution of the state of one qubit conditional on the state of the other is exactly the kind of physical interaction that allows for implementation of multiqubit quantum logic; in our case, the time to the first minimum corresponds to the duration of a twoqubit conditional phase gate. We note that this time satisfies the criteria that we identified above for twoqubit devices: the twoqubit gate time (of order 100 ns) lies between the singlequbit gate time (of order 10 ns) and the phase relaxation time (in the microsecond range).
The subpopulation excited at a magnetic field of 385.84 mT (middle trace, green) is comprised of dimers whose interring axes lie at orientations peaked in between parallel and perpendicular to the external magnetic field. The distribution of orientations includes a significant proportion close to the magic angle at which the dipolar interaction strength goes through zero, resulting in a broad distribution of interaction strengths and an almost monotonic dependence of the time domain DEER signal on T .
At 390.84 mT, ν 1 excites principally rings whose planes are perpendicular to the magnetic field (i.e. those whose effective g factor is close to g ∥ ). Deviations from this orientation lead to a greater contribution to the effective g factor from the larger g ⊥ , and therefore to a higher resonant frequency. Thus, in dimers that are orientationally selected by the excitation of one ring at frequency ν 1 , ν 2 (which is lower in frequency than ν 1 ) is comparatively unlikely to excite the other ring. This gives rise directly to the comparatively weak modulation in the timedomain trace at 390.84 mT. The interspin axes of this subpopulation of dimers are aligned along the magnetic field direction, maximising the orientationdependence of the dipolar interaction (i.e., θ ≈ 0°). Correspondingly, the timedomain signal evolves comparatively rapidly at short times and the Fourier transform contains components at high frequencies.  We note also that our strategy allows for enhancement of the spin coherence times independently of the interring interaction strength. In earlier studies of monomers 19 , we found that deuteration and immobilization of nuclear spins in the structure significantly improve the coherence. Thus, in compounds 1A , 1Bd and 1C , while the similar linkers ensure that h / J is similar for each, deuteration ( 1Bd ) and confinement of hydrogen nuclei to rigid adamantyl groups ( 1C ) result in extensions of the coherence times by factors of about four and three respectively.

Discussion
From the data presented here, we may draw several conclusions supporting the assertion that Phys. Lett. 110 , 67-72 (1984

Competing financial interests
The authors have no competing financial interests.       where 2 τ was the total duration of the experiment, and T m and x were fit parameters ( x = 1 corresponds to a simple exponential, for which T m ≡ T 2 ) 19 . π /2 and π pulses were, respectively, Page -S1  Chemical shifts are reported in parts per million (ppm) from low to high frequency and referenced to the residual solvent resonance. ESI mass spectrometry and microanalysis were carried out by the services at the University of Manchester
The solid was extracted with CHCl 3 (50 mL), washed with water and dried over anhydrous magnesium sulphate and evaporated. A light yellow liquid was obtained in 78 % yield (1.14 g Isobutylamine (0.69 mL, 6.8 mmol) in methanol (5 mL) was added to a solution of terphenyl-4,4´-dicarbaldehyde (0.5 g, 1.7 mmol) in methanol (30 mL), and the reaction mixture was refluxed for 3 h under nitrogen, then allowed to stir at room temperature overnight. NaBH 4 (10 equivalents) was added and the reaction mixture was stirred for 12 h under nitrogen atmosphere. The reaction was quenched with water and the solvent was evaporated. The solid was extracted with chloroform ( 50 mL), washed with water and dried over anhydrous magnesium sulphate and evaporated. A light yellow liquid was obtained in 80 % yield (
NaBH 4 (5 equivalents) was added and reaction mixture was stirred during 12 h under nitrogen atmosphere. The reaction was quenched with water and the solvent was evaporated. The solid was extracted with chloroform (50 mL), washed with water and dried over anhydrous magnesium sulphate and evaporated. A light yellow liquid was obtained in 80 % yield (1.4 g). Crystal data and refinement parameters are given in Table S1. CCDC 1039429-1039435 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif Page -S10 Continuous wave ESR spectra and echo-detected magnetic field sweeps were recorded at Xband (9.5 GHz). For each compound, spectra were fitted with a simultaneous set of Hamiltonian parameters to obtain an accurate description of the system. The unresolved hyperfine couplings were included in the simulation as an H-strain parameter.
The simulation parameters are tabulated in Table S2. X-band CW data and simulations for 1B, 1C, 1D, 2B and 2C are plotted in Figure S1 (see main manuscript for 1A and 2A). Page -S13

DEER experiments
Double Electron -Electron Resonance (DEER) was measured for all compounds. All DEER experiments used 1 detection pulses (π/2 and π) of 40 ns and 2 pump pulses of length of 24 ns. Compounds 1A, 1Bd, 1C and 2A were measured using 4-pulse DEER. Owing to the shorter coherence times in samples 1D and 2B it was necessary to use a combination of 3-and 4-pulse DEER techniques. In these cases, a 4-pulse trace with a short acquisition window was recorded and carefully combined with a 3-pulse trace with a longer acquisition window using the DEER-stitch method described in Lovett et al., Journal of Magnetic Resonance 223 98 (2012), to reconstruct the zero time on the longer 3-pulse trace. Sample 2C was measured using 3-pulse DEER only.
Using this package, the raw data were smoothed, filtered to remove the effects of electron spin echo envelope modulation (ESEEM) arising from protons, and background-corrected to account for inter-dimer interactions in the three-dimensional homogenous distribution. The sample concentrations were sufficiently small that the background correction was at all times small.
In each of the following figures, open circles represent raw data and solid lines represent smoothed filtered data. At low magnetic fields, rings with effective g-factors close to g⊥ (i.e. dimers with inter-ring axes approximately perpendicular to the field) are excited; as the magnetic field increases, the ring orientations selected approach those with effective g-factors closer to g∥ (i.e. dimers are selected whose axes approach the direction parallel to the field).