Measuring Spin⋅⋅⋅Spin Interactions between Heterospins in a Hybrid [2]Rotaxane

Abstract Use of molecular electron spins as qubits for quantum computing will depend on the ability to produce molecules with weak but measurable interactions between the qubits. Here we demonstrate use of pulsed EPR spectroscopy to measure the interaction between two inequivalent spins in a hybrid rotaxane molecule.

Molecular electron spins are potential qubits for quantum information processing (QIP). [1] Much recent work has been dedicated to showing that the phase memory times of S = 1/2 molecules can be controlled, and extended to the point where multiple spin manipulations will be possible. [2] One of the key next steps is to study the interactions between such spins. Recently,A rdavan et al. have shown using double electronelectron resonance (DEER) spectroscopy that the interaction between electron spins in {Cr 7 Ni}r ings can be controlled to fall within the range needed for two qubit gates. [3] These studies were performed on dimers of {Cr 7 Ni}rings [4] where the S = 1/2 spins in each half are identical.
Systems containing different spins also have great potential for QIP as there is then the possibility of manipulating each spin separately.T his underpins the g-engineering idea proposed by Takui and co-workers for organic radicals [5] and work employing heterometallic lanthanide dimers. [6] Large differences in g-values could also be used as am eans to implement entangling two qubit gates. [7] Quantifying weak interactions between very different spins is challenging.I n cases where the interaction can easily be measured, for example by magnetometry or continuous-wave (CW) electron paramagnetic resonance (EPR) spectroscopy, [8] the interaction will be too large to permit implementation of both one-and two-qubit gates in the same molecule. [3] CW EPR spectroscopy can still be au seful tool to investigate dipolar interactions;byobserving the line broadening of aN triplet in at wo-qubit structure,Z hou et al. showed the existence of ad ipolar interaction that they could estimate in conjunction with spin density calculations. [9] On the other hand, DEER spectroscopy, [10] while ap owerful tool to measure weak interactions by directly manipulating two weakly interacting spins with specific microwave pulses,i s limited as the bandwidth of the microwave source must encompass the resonant frequency of both electron spins.
Here we report at wo-qubit assembly comprising an organic radical within the thread of ah ybrid [2]rotaxane containing a{Cr 7 Ni}ring:this is aheterospin system where the constituent spins possess vastly different g-values.T oquantify the weak interaction between the spins we use "Relaxation Induced Dipolar Modulation" (RIDME) spectroscopy, [11] which has been developed in structural biology to measure distances between dissimilar spins. [12] To make the [2]rotaxane an organic thread was synthesized containing as econdary amine and terminating in an aldehyde (see the Supporting Information for details).  octagon. Thed istance between the oxygen atom of the TEMPO group (which bears the majority of the nitroxide spin density [13] )and the metal ions of the ring varies from 17.07 to 18.14 ,w hile the distance to the amine nitrogen atom around which the ring is templated is 16.97 .T he ring is approximately perpendicular to the thread. While the thread is rather rigid, the NÀCl inkage between the TEMPO group and the adjacent phenyl ring places the O-atom off the thread axis;free rotation about this bond would produce acircle with ar adius of approximately 2.59 .T he distance between the radical and the ring is too great for any interaction to be observed by magnetometry or by CW EPR spectroscopy (see Figure S2).
Thef ield-swept echo-detected (FSED) EPR spectrum of 1 at Q-band and 5K shows two well-separated transitions: as harp resonance split by 14 N-hyperfine centered at ca. g = 2.007 for the TEMPO radical and am uch broader and axial resonance for the {Cr 7 Ni}r ing centered at ca. g = 1.779 ( Figure 2);these values are typical for the two spins. [14,15] RIDME benefits from the two spins having different longitudinal (spin-lattice) relaxation times, T 1 :in1 the nitroxide has avery long T 1 n (0.2 sat10K [16] )and the {Cr 7 Ni}ring has ac omparatively short T 1 r (ca. 1 msa t5K [2a] ). Thes pin echo of the more slowly relaxing spin is measured and the modulation of this echo caused by the spontaneous flipping of the more rapidly relaxing spin allows the spin···spin interaction to be quantified. TheR IDME sequence requires pulsed EPR resonances to be measured only at the resonant frequency of the slowly relaxing spin which makes it useful for heterospin systems.R IDME is also less orientation selective than DEER and therefore can benefit from more intense signals and spectral simplicity.
Thef ive-pulse RIDME [12a] sequence was used at Q-band (ca. 34 GHz) at 5K (Figure 3). This sequence is dead-timefree and uses only one frequency centered on the observer spin (the nitroxide, Figure 2). First, aCarr-Purcell Method A sequence (p/2ÀtÀp)t ips and re-focuses the spins in the xy plane.T he third pulse,a fter variable time t,c onverts this refocusing transverse magnetization into longitudinal magnetization. Thes ystem is then allowed to evolve freely for atime T on the order of the longitudinal relaxation time T 1 r of the ring (Figure 3). During this time T,t he ring spin will flip spontaneously with ap robability of 1 = 2 {1Àexp(ÀT/T 1 r )}. [11] A final refocusing p/2 and p pulse series is used to obtain ar efocused virtual echo (RVE). Thet ime between the 3rd and 4th pulses (T)i sh eld constant and both pulse positions are incremented at constant rate.T he echo intensity is detected as af unction of t (Figure 3). If the ring spin flips during the evolution time T,t he resonance frequencyo ft he nitroxide shifts by the interaction frequency,thus modulating the final echo amplitude.T he interval time T used was 4000 ns,which is around four times T 1 r under these conditions ( Figure S5). Measurement at Q-band suppresses electron spin echo envelope modulation effects under these conditions ( Figure S6).
Removing ab ackground function of the form exp(Àkt) where k = 1.8 10 5 ns À1 gives the experimental form factor that displays amodulation depth of 0.287 (Figure 4a,raw data is given in Figure S7). Fourier transform gives the frequencyspace spectrum which contains two peaks;t he separation of these two peaks is ad irect measure of 2 D, D being the magnetic interaction which is around 8MHz (Figure 4b).
Fitting the data with DeerAnalysis [17] using ap oint dipole model based on two localized S = 1 = 2 species is wholly inadequate due to the spin density distribution in the ground state of the {Cr 7 Ni}r ing. [18,19] To go beyond this limitation we have simulated the data using an in-house code SSD ("Spatial Spin Density"), which calculates the form factor directly using aspatially distributed dipolar model [20] considering the perturbation from the flipping spin of the ring during the time interval T.T he spin projection coefficients for the S = 1 = 2 ground state of {Cr 7 Ni} were obtained using the "Irreducible Tensor Operator" (ITO) technique with the PHI [21] program. This includes the full microscopic Hamiltonian, [15] and reproduces the spin distribution measured by NMR spectroscopy. [19] Using the structure and including random rotations of the TEMPO and ring about the thread (Figure 4c), we directly fit the RIDME form factor by introducing the standard deviation for aG aussian distribution of the ring-nitroxide distance, s.
While this one-parameter model can give agood fit to the data ( Figure S8), asignificantly better fit (Figures 4a and b) is obtained when allowing for an isotropic exchange parameter (J =+0.15 MHz) or as mall change in the average ringnitroxide distance (ca. À0.1 ). Both of these models give excellent simulations of the data, revealing that the dipolar interactions perpendicular and parallel to the thread axis are D perp = 9(2) MHz (0.0003 cm À1 )a nd D para = À18(3) MHz (0.0006 cm À1 ), respectively.I np revious work where we looked at the interaction between ar ing and as ingle Cu II center [8] an exchange interaction of around 0.02 cm À1 was  This magnetic interaction would give agate time of 125 ns, which falls in the correct range to implement atwo-qubit gate with 1. [3] Unfortunately,w hile RIDME is very good at measuring the interaction between different spins it only addresses one of the two spins.T herefore,i fh eterospin systems were to be used in quantum algorithms there remains an eed for EPR spectrometers capable of pulsing at two distinct frequencies.

Experimental Section
Thec rystallographic data of 1 were recorded on aB ruker Prospector CCD diffractometerw ith Cu Ka radiation (l = 1.5418 ). Thes tructurew as solved by direct methods and refined against F 2 using SHELXTL. CCDC 1502519 containst he supplementary crystallographic data for this paper.T hese data can be obtained free of charge from TheC ambridge Crystallographic Data Centre.P ulsed EPR spectroscopy was measured on 0.002 mm solutions of 1 compound in anhydrous toluene.P ulsed Q-band EPR measurements (Inversionr ecovery,E SEEMa nd RIDME traces) were recorded using an E580 Bruker spectrometer( 3W)e quipped with aS pinjet AW Gand 2mmdielectric resonator.Experimental details are found in the SupportingI nformation.