Observation of Tunneling-Assisted Highly Forbidden Single-Photon Transitions in a ${\mathrm{Ni}}_4$ Single-Molecule Magnet

Forbidden transitions between energy levels typically involve violation of selection rules imposed by symmetry and/or conservation laws. A nanomagnet tunneling between up and down states violates angular momentum conservation because of broken rotational symmetry. Here we report observations of highly forbidden transitions between spin states in a ${\mathrm{Ni}}_4$ single-molecule magnet in which a single photon can induce the spin to change by several times $\hslash$, nearly reversing the direction of the spin. These observations are understood as tunneling-assisted transitions that lift the standard $\mathrm{\Delta }m=\pm 1$ selection rule for single-photon transitions. These transitions are observed at low applied fields, where tunneling is dominated by the molecule’s intrinsic anisotropy and the field acts as a perturbation. Such transitions can be exploited to create macroscopic superposition states that are not typically accessible through single-photon $\mathrm{\Delta }m=\pm 1$ transitions.

Figure 1

Spin-state energy-level diagram for one conformational state (“black”—see below) of ${\mathrm{Ni}}_4$. Energies of various levels are shown as a function of magnetic field, calculated by diagonalizing the molecule’s spin Hamiltonian. The diagram illustrates the levels’ behavior when $\theta =30°$. Arrows indicate the major transitions observed in this study: $\text{Black}=\text{allowed}$, $\mathrm{orange}=\text{forbidden}$. The two orange arrows are labeled with $\star$ and $+$, the designations used throughout this article. Inset: Molecular unit of $[\mathrm{Ni}(\mathrm{hmp})(\mathrm{dmb})\mathrm{Cl}{]}_4$ [18], where hmp is the anion of 2-hydroxymethylpyridine and dmb is 3,3-dimethyl-1-butanol. Color code: green, chloride; cyan, nickel(II); black, carbon; red, oxygen; blue, nitrogen. Hydrogens have been omitted for clarity.

Figure 2

Absorption ESR spectra at 1.8 K for several angles ${\theta }_H$. The spectrum for ${\theta }_{\mathrm{H},\mathrm{ref}}=26.6°$ shows actual $Q$ values. All other spectra have been shifted vertically by an amount proportional to ${\theta }_H-{\theta }_{\mathrm{H},\mathrm{ref}}$. Spectra from three different crystals are combined in this figure. Each spectrum has been shifted slightly horizontally to account for inductive effects due to sweeping $H$ (see Ref. [28]).

Figure 3

Resonance positions in $B-\theta$ space. The points are the peak positions from Fig. 2 after correcting for the effects of dipole fields [28]. The lines are the results of simulations after fitting the observed spectra. Black and red correspond to different conformational states of the molecule with correspondingly different anisotropy constants. Solid curves indicate allowed transitions and dashed curves correspond to forbidden transitions. The small shift seen in the calculated results at $\sim 40°$ arises from use of different samples at angles above and below this value and associated differences in the direction ($\varphi$) of the transverse field in the samples’ hard planes [28].

Figure 4

Decomposition of eigenstates involved in the forbidden transitions in the $m$ basis [cf. Eq. (2)]. Values of $|c_m|$ were calculated by diagonalizing the spin Hamiltonian at the fields corresponding to the (a) $\star$ and (b) $+$ transitions, setting $\theta =30°$. Blue circles (orange squares) indicate the values of $|c_m|$ for the lower (upper) state involved in each transition. Insets schematically show the double-well potentials for the associated transitions, marked with red arrows.

Figure 5

Oscillator strength (OS) and $D_{\mathrm{RFI}}$ for one of the transitions studied as a function of field. Here $|i⟩\approx |m=3⟩$ and $|f⟩=|E_3⟩$ are the second- and third-lowest energy eigenstates (cf. Fig. 1), respectively. As the field increases, the angle $\theta$ is adjusted to maintain resonance of the transition with the radiation frequency. For this pair of levels, the transition is forbidden (allowed) at small (large) fields with a crossover at the field of the anticrossing. The inset shows a parametric plot of ${\mathcal{F}}_{\psi }$ vs OS, illustrating how, near the anticrossing, one quantity rises as the other falls, but both can be substantial over some region. Similar calculations for the transition between $|E_2⟩$ and $|E_4⟩$, the second and fourth energy eigenstates, show complementary behavior [28].