Han
Guo
a,
Claire E.
Dickerson
a,
Ashley J.
Shin
a,
Changling
Zhao
b,
Timothy L.
Atallah
a,
Justin R.
Caram
*a,
Wesley C.
Campbell
*b and
Anastassia N.
Alexandrova
*a
aDepartment of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, CA 90095, USA. E-mail: jcaram@chem.ucla.edu; ana@chem.ucla.edu
bDepartment of Physics and Astronomy, University of California, Los Angeles, Los Angeles, CA 90095, USA. E-mail: wes@physics.ucla.edu
First published on 21st September 2020
Quantum information processors based on trapped atoms utilize laser-induced optical cycling transitions for state preparation and measurement. These transitions consist of an electronic excitation from the ground to an excited state and a decay back to the initial ground state, associated with a photon emission. While this technique has been used primarily with atoms, it has also recently been shown to work for some divalent metal hydroxides (e.g. SrOH) and alkoxides (e.g. SrOCH3). This extension to molecules is possible because these molecules feature nearly isolated, atomic-like ground and first-excited electronic states centered on the radical metal atom. We theoretically investigate the extension of this idea to a larger scale by growing the alkyl group, R, beyond the initial methyl group, CH3, while preserving the isolated and highly vertical character of the electronic excitation on the radical metal atom, M. Theory suggests that in the limit as the size of the ligand carbon chain increases, it can be considered a functionalized diamond (or cubic boron nitride) surface. Several requirements must be observed for the cycling centers to function when bound to the surface. First, the surface must have a significant band gap that fully encapsulates both the ground and excited states of the cycling center. Second, while the surface lattice imposes strict limits on the achievable spacing between the SrO– groups, at high coverage, SrO– centers can interact, and show geometric changes and/or electronic state mixing. We show that the coverage of the diamond surface with SrO– cycling centers needs to be significantly sub-monolayer for the functionality of the cycling center to be preserved. Having the lattice-imposed spatial control of SrO– placements will allow nanometer-scale proximity between qubits and will eliminate the need for atom traps for localized cycling emitters. Our results also imply that a functionalization could be done on a scanning microscope tip for local quantum sensing or on photonic structures for optically-mediated quantum information processing.
Here, we present a new concept for molecular qubits that are designed to be scalable and packed at high density via surface modification, potentially produced by self-assembly of organic molecules. By attaching an optical cycling qubit moiety to a chemically defined crystal surface, we combine initialization and state readout based on optical cycling that trapped atoms provide, with the potential for the nanoscale lithographic control that surfaces furnish. In this way, this “chemically-trapped ion” platform borrows strengths from both gas-phase and solid state devices in an effort to build a concept that can leverage the best features of each for building the next generation of quantum devices.
Our calculations suggest that increasing the size and complexity of the ligands may slightly disrupt the closure of the optical cycle. However the size and complexity effects may be small and can be compensated by additional repumping lasers, if the ligand is spatially and electronically separated from the OCC.6 This raises the possibility that an OCC can be chemically tethered to a substrate in vacuo while retaining its ability to spontaneously emit a series of photons upon state-selective excitation, providing readout of the initial internal quantum state of the molecule.
Beyond OCCs’ roles in supplying an isolated transition in the molecule, they also provide a natural basis for storing and using quantum information. A quantum state stored in a pair of long-lived sublevels of the electronic ground state (such as hyperfine structure) serves as a qubit and a state-selective optical cycle would allow excitation of only one of these qubit eigenstates (the “bright state”) for laser-induced fluorescence, while the other eigenstate (the “dark state”) would not emit fluorescence photons. Lack of collected fluorescence therefore constitutes detection (and heralds preparation) of the dark state, while detection of fluorescence indicates the bright state. As such, the optical cycling property allows SPAM of the molecular qubit, in a scheme analogous to those used in trapped ion quantum computers.7,8
Taken together, the tantalizing possibility that OCCs might be realized on surfaces and even patterned to produce multi-OCC interactions argues for a theoretical treatment of whether molecules hosting optical cycling centers can be attached to surfaces while retaining both chemical and physical properties. We approach this by expanding on the complexity of alkyl chain R-groups that are amenable to optical cycling. For strontium (Sr) as our example of an alkaline earth metal, we suggest that diamond and cubic boron nitride (cBN) may serve as a good substrate for OCC anchoring and derive some general trends. We then explore how multiple OCCs may interact chemically or spectroscopically on a surface. We close by speculating on several schemes by which surface bound OCCs may provide next generation quantum systems through nanoscale control and patterning.
Geometry optimizations of OCCs bound to surfaces were performed using the Perdew–Burke–Ernzerhof (PBE) functional,29,30 with the projector augmented-wave (PAW) potentials,31,32 implemented in the DFT-based Vienna ab initio simulation package (VASP).31,33–36 The hybrid functional HSE0637 is used for single point energy calculations, density of states calculations and the Bader charge analysis.38,39 HSE06 has been extensively benchmarked against other functionals and performs well in predicting band gaps that fall within 1–5 eV.40,41 A 1 × 1 × 1 Monkhorst–Pack k-point grid centered at Γ-point and a plane-wave cutoff energy of 400 eV are employed, with a convergence criterion of 10−6 eV for the SCF relaxation. The k-point grid is considered sufficient due to the large size of the unit cell and the significant separation between OCCs that we probe, expected to produce flat bands, accurately modeled with just the Γ-point calculation. A five-layer 2 × 2 supercell with the exposed (111) facet, in periodic boundary conditions, represents the cubic diamond surface as an infinite slab, in which the two surfaces are covered with H atoms. In this supercell 160 C atoms in total, of which 64 are surface-exposed (or 32 per face), all capped with H before SrO deposition. The dimensions of the supercell are 14 Å × 14 Å × 20 Å, which we found to be sufficient to prevent the interaction between the periodic images. The top-three layers of C atoms and the surface H atoms are relaxed during the geometry optimizations. The optimizations are considered converged when the residual forces are less than 0.01 eV per atom.
We start from the hypothesis that the alkyl group R, in SrOR, can be increased beyond methyl, CH3. We apply the TD-DFT method to a series of SrOR molecular radicals, R, with increasing numbers of carbon atoms, CH3, C4H9, C13H21 (Fig. 1). Our goal is to progressively build up to the local structure of a diamond surface. We focus on the changes in geometries within the SrO part of the molecular radicals in the ground and first excited states, the excitation energies, and the Frank–Condon factors, FCFs. All the cluster models can maintain a local linear structure, with the Sr–O–C angles of 179.8°–179.9° for the ground state, and 179.6°–180.0° for the excited state. This bond angle is characteristic also of SrOH, and suggests a strongly ionic bond character between Sr and the O atom. As shown in Table 1, the two states in these models are aligned within 0.02–0.03 Å in terms of the equilibrium Sr–O bond length, and larger R sizes correspond to slight deviations from the linear character.
Molecule | ΔrSr–O (Å) | ΔE (eV) | FCF |
---|---|---|---|
SrOH | −0.017 | 1.750 | 0.9651 |
SrOCH3 | −0.018 | 1.754 | 0.9556 |
SrOC4H9 | −0.021 | 1.738 | 0.9163 |
SrOC13H21 | −0.029 | 1.721 | 0.7819 |
The three cluster models exhibit similar charge distribution on the Sr and O atoms from the natural population analysis (NPA),44 due to the similar electron withdrawing strength of the alkoxide groups bound to Sr. The natural transition orbitals (NTOs) in all cluster-bound SrO have the same shape as those in SrOH, with electron density localized on the Sr atom (Fig. 1). This suggests that the perturbation from the R group is minor. However, we observe that, as the cluster size increases, a small amount of electron density appears on the O atom and some on the C atoms, which will make the potential energy surfaces (PESs) of the ground () and first excited (Ã) states, less parallel, and thus the FCF less diagonal (deviating more form the highly diagonal FCF in SrOH5,6). This is because more modes are contributing to the off-diagonal FCFs. In addition, the contribution from the Sr–O stretching mode is slightly increased, consistent with our observation that the Sr–O bond length change is increased, and thus the PESs of the two states are less parallel along this mode as we move to larger model systems. However, the decrease in FCF, due to the size effect, is not significant, and SrO will still have a reasonably good FCF. Note as well that the diamond-supported system will have a very different reduced mass.
In SrOC13H21, there is a very small amount of electron density on the two C atoms that are not directly bound to O, but are only 2.7 Å apart from the O atom. These carbon atoms will conform to the top-layer atoms on the diamond surface, which are 2.9 Å away from O in Fig. 2. Consistently, we find that the distance between the O atom and these C atoms changes by 0.004 Å from the 2A′ to the Ã2A′′ state, and that constitutes the second largest geometric changes in SrOC13H21. Therefore, when the cycling center is deposited onto the diamond surface, the orbitals of these two C atoms may have weak couplings to the O and Sr atoms and affect the FCF. On the other hand, the chemical nature of these atoms in the diamond surface is not identical to that in a finite hydrocarbon, and thus, this small involvement of the C atoms of the C13H21 system in the key transition does not necessarily preclude the possibility of the infinite surface to be a viable OCC platform.
The TD-DFT calculations show that the first vertical excitation energy is slightly decreased as the size of the R group increases (Table 1), agreeing well with previous experimental measurements42 (Table S3, ESI†). Given this trend, the diamond-bound SrO is expected to have slightly lower excitation energy than SrOH.
Fig. 2(c and d) show the electron density of the highest occupied molecular orbital (HOMO), occupied by the unpaired electron, and the doubly-degenerate lowest unoccupied molecular orbital (LUMO) of the SrOH molecule and the SrO group attached to diamond. For both HOMO and LUMO of surface-bound SrO, a majority of electron density is localized on Sr, leading to electronic transition localized on the cycling center. Again, the molecular orbitals of the surface-bound SrO have similar shapes to those of the SrOH radical, suggesting similar distributions of atomic orbital characters. For SrOH in the 2Σ+ state, the orbital occupied by the unpaired electron is dominated by the Sr s character with the remaining contribution from the Sr p orbital, whereas for the Ã2Π state, the Sr p and d orbitals dominate the molecular orbital character.
Given the similarity in geometry, charge distribution and shapes of the HOMO and LUMO between the diamond-surface-bound SrO and the SrOH molecule, we expect SrO to maintain its excitation properties once bound to the diamond surface. We can roughly estimate the excitation energy from the projected density of states (PDOS) (Fig. 2e), to be 1.82 eV for diamond-bound SrO. This is close to the excitation energy in SrOH (1.83 eV), estimated from the PDOS calculation. While the number derived from PDOS is necessarily approximate, the HSE06 functional is known to be quite reliable for the optical gap, as discussed above. Calculating the FCF for the infinite surface is unfortunately technically impossible at the moment, and thus, our results are a proposal, substantiated as best as currently possible. Confirming this obviously requires experimental validation.
Next, we believe there is a rational way to improve the properties of the surface-bound SrO, through changing the support to a close diamond analogue, cubic boron nitride (cBN). cBN has the same number of valence electrons and the same crystal structure as the cubic diamond, and thus, mirrors diamond's properties. However, it additionally features a small amount of charge transfer between B and N. Given that the surface of cBN is expected to be B-terminated, and that SrO– would therefore bind to the boron, the intra-crystal polarization is expected to effectively create a more electron-withdrawing ligand, R, for SrO–. Fig. 3 shows the optimized structure for SrO–cBN from periodic DFT calculations. The Sr–O bond is 2.104 Å on cBN, i.e. 0.021 Å shorter than that on diamond. The Sr–O–B can maintain a linear structure near Sr, with a now-perfect angle of 180.0°. In line with this finding, and our hypothesis regarding cBN, the Sr atom has more positive charge on cBN than that on diamond. The HOMO and LUMO of this system (Fig. 3) are the expected singly-occupied states, very similar to those of SrO– on diamond and SrOH in the gas phase. From the PDOS, the excitation energy of SrO– on cBN is 1.70 eV. Thus, we see that the behavior of OCC is likely again preserved on cBN at this low coverage. While the experimental confirmation of the OCC behavior on both surfaces is warranted, our results suggest that both diamond and cBN could be used to support the SrO– cycling radical centers, and cBN has a potential to be a superior support.
When the cycling center coverage is 1/2 ML, Sr will bind to two O atoms, with bond lengths of 2.33 Å and 2.39 Å, respectively (Fig. 4). The distance between the two Sr centers is 3.57 Å. The HOMO and LUMO are completely delocalized, where the adjacent Sr radical orbitals mix, effectively forming a half-filled σ-band. When the coverage is 1/8 ML, corresponding to the distance between the two cycling centers of 7.13 Å, the Sr–O–C unit can maintain the locally linear structure. However, the orbitals of two neighboring Sr atoms still interact with each other, either polarizing the original orbitals, or mixing and splitting into two mixed states.
We have also studied the systems where two SrO– units are in the same unit cell, and can form dimer geometries. In some cases, one Sr will bind to two O atoms, similar to that in Fig. 4(a), while in other cases the two SrO– groups can maintain the locally linear structures, but their orbitals interact with each other. Fig. 5 shows one case where the distance between two Sr is 3.73 Å, and the distance between the two C atoms that the SrO– groups are bound to is 1.71 Å. The two Sr s orbitals with the same energy form a bonding and an antibonding combinations, while the doubly degenerate p- and d-type orbitals from different Sr form even more mixed orbitals. The charges on the Sr atoms also suggest that they are not chemically identical.
Clearly, these coverages are too high to maintain the clean radical character of the cycling centers in the ground and excited states. Our results indicate that the surface coverage with radical center has to be certainly lower than 1/8 ML, in order for the cycling centers to maintain their local electronic character and the diagonal FCF.
We hypothesize that surface supported optical cycling ligands can be realized through chemical functionalization of the surface. We propose that diamond grown by chemical vapor deposition (CVD) (which is typically H-terminated) can be functionalized with pendant hydroxide groups by low-pressure oxygen plasma treatment.45,46 Following this, reactive Sr can be prepared by exciting the Sr atoms to the metastable state 3P1 with laser excitation.47–49 To control the deposition, the amount of pendant hydroxide can be tuned by modulating the power and duration of the plasma treatment.
Fig. 6 shows three applications that would be enabled by the development of surface-bound, state-selective emitters. For sensing applications, Fig. 6(a) shows a conceptual arrangement whereby a single molecular qubit could be used for spatial imaging of, for instance, local magnetic fields by functionalizing the tip of a scanning tunneling microscope (STM) or an atomic force microscope (AFM). Light for SPAM can be delivered to and collected from the OCC via fibers or optical waveguides built into the tip. A superposition of field-sensitive qubit states will process at a rate determined by the local magnetic field, and readout after free precession can be used to map the field at the atomic scale.
Note, however, that, if AFM is conducted at ambient pressures, the Sr radicals are likely to chemically react with the gases, and quench. For STM, the tip generates an oriented electric field, which at large enough magnitudes may affect charge separation and hinder Franck–Condon factors. Experimentally, an external electric field generated by STM was shown to stabilize or destabilize charge-transfer in organic molecules placed on gold surfaces, based on alignment. The external electric field strengths studied ranged from −1 × 109 to 1 × 109 V m−1.50
To probe how external fields would affect optical cycling, we introduced electrical fields in two ways. A field between −1 × 1010 to 1 × 1010 V m−1 aligned with the Sr–O bond was applied to 1/32 ML Sr–O OCCs on diamond (Table S5, ESI†). In another case, a field of 1 × 108–1 × 1010 V m−1 was applied to two SrO– units located 3.88 Å apart and 10.39 Å from other OCCs on diamond (Table S6, ESI†). In both cases, larger electric fields reduced charge localization on Sr, while the Sr–O bond length would increase or decrease depending on the direction of the field. The effect on charge can be explained by the fact that the field pushes the full electron density in the system, affecting the bonding character on the ground and excited state beyond just the radical-centered HOMO and LUMO. We made a comparison of these results to the cluster model, SrOCH3, with an external electric field applied along the Sr–O bond. As on the surface, the applied field led to charge delocalization and increase or decrease in Sr–O bond length, depending on the field direction. In both field directions, the ground versus excited state geometry difference increases with increasing field magnitude, implying poorer FCFs (Table S7, ESI†). Thus, in STM, large electric fields can adversely affect radical charge localization and FCF of OCCs.
In Fig. 6(b), two qubits in a quantum processor are shown performing a controlled-NOT (CNOT) quantum gate. Each qubit can be defined on a pair of long-lived internal states |0〉 and |1〉 (for instance, hyperfine levels) that have nearly identical electric dipole moments. Even so, inhomogeneities in the local surface features and attachment details are expected to result in strong (in the sense that they fully mix opposite parity states) static electric fields that are different at each qubit's location, which will lead to small shifts that produce a unique splitting for each qubit. This will allow individual addressing of single-qubit gates through frequency resolution of the applied resonant fields, with the size of the splitting required for resolution set by the available interaction time. State preparation is furnished by laser-driven optical pumping through the OCC.
For two-qubit operations, a third, long-lived state (|0〉) in each molecule that does possess a large electric dipole moment (polarized by the local static field of its environment) can be utilized. Frequency resolution can be used to couple only one of the qubit states of any particular qubit to the auxiliary level, which will Stark shift all of the auxiliary states of the qubits around it via the dipole–dipole interaction (any residual differential shift of the qubit states near the dipole, including the target qubit, can be removed through global dynamical decoupling). In this way, a dipole blockade CNOT gate,51 between any two qubits within a blockade radius of one another can be performed that does not depend upon their separation. State readout can likewise be performed using frequency-resolved addressing of each molecule's OCC.
Since the proximity of the qubits can be extremely close (but not closer than 1/32 ML on diamond, in our calculations) the energy scale (and therefore achievable speed) of the gate is expected to be high. For example, for 1 Deybe dipoles, the 1 MHz blockade radius could contain about 1000 qubits at 1/32 ML packing. While the coherence properties of these qubits remain an open experimental question, the basic tools presented here show that there is no fundamental impediment to achieving universal quantum computation with this platform.
As a third example, Fig. 6(c) shows a photonic interconnect for large-scale processing based on surface-bound OCCs. Even qubits that are physically distant can interact strongly with one another due to their strong coupling to photonic cavities built from their substrates, and classical routers can turn interactions between individual qubits on and off by optically connecting their photonic cavities together.
We examine the possibility of constructing pendant radical alkaline earth based OCCs on surfaces. Such structures may bridge surface engineering with the precise quantum control afforded by atomic physics. After validating our computational methods, we find that diamond and cBN could support SrO– based OCCs. We establish the necessary density of OCCs on a surface needed to retain independent electronic character and explore how they might be used in applications such as STMs, universal quantum gates, or as substrates for quantum photonics. We believe this presents a path forward to realizing a novel quantum architecture experimentally.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp04525j |
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