Spin-bearing molecules as optically addressable platforms for quantum technologies

2.1 Chromophore-radical (C-R^·)-based systems

In the case of (C-R^·)-based systems, a series of steps following the photoexcitation of a chromophore in the ground state lead to the formation of a high-multiplicity spin state. This state can then be polarized under an external magnetic field – termed as electron spin polarization (ESP). For such polarized states, long enough electron spin relaxation time (T ₁ ^e) is essential for implementing QIP protocols [30], [31]. In general, excitation of a suitable organic chromophore – for example, one featuring long-lived triplet state (T ₁) such as fullerene (C₆₀; 145 µs and C₇₀; 12 ms at RT in toluene) [32] – leads first to the formation of a singlet excited state (S ₁) followed by conversion into a triplet state by intersystem crossing (ISC). Appending a stable radical-bearing entity with S = 1/2 in the chromophore skeleton enhances the rate of the singlet-to-triplet ISC, therefore the yield of chromophore-based T ₁ state. This mechanism is known as radical-enhanced intersystem crossing (EISC) [30]. The states involved are denoted using the notations D _y and Q, where D and Q represents overall spin multiplicity, i.e. doublet and quartet respectively, while y represents the ground (y = 0) or excited (y = 1 or 2) states. EISC from the D ₁ state leads then to the formation of doublet (D ₂) and quartet states (Q), see Figure 2 for details. Under magnetic field, a non-equilibrium spin population or polarization is observed in the Q state (Figure 2(b)). Qubits and higher-order qubits – qudits – can then be accessed, and quantum superposition states can be created using MW excitation. The optical readout is carried out by monitoring the luminescence and absorption of these excited states produced by exchange coupling [33]. In these systems, qubit states based on the stable radical-based D ₀ state can be also realized taking advantage of polarization transfer from the high-multiplicity excited state to the ground D ₀ state (Figure 2(b)). In essence, two different types of qubits based on the Q and D states can be obtained in the same C-R^· system.

Figure 2:

Chromophore-radical (C-R^·)-based systems. (a) Structure of a chromophore-radical system in which optical polarization and readout of qubits have been demonstrated [33]. Radical bearing (tris(2,4,6-trichlorophenyl)methyl-carbazole) (TTM-1C_Z) and anthracene chromophore are shown in blue and black, respectively. (b) Photophysical process involved in the C-R^· system leading to the formation of the high-multiplicity quartet (Q) state. Excitation of the C-R^· system into the D ₁ state followed by enhanced intersystem crossing (EISC) leads to the formation of D ₂ and Q states. In a strong external magnetic field, the D ₂ and Q states are separated in energy by 3J _TR, where J _TR is the exchange interaction leading to polarization into the spin quartet state. Superposition states can then be created in the Q sublevels, represented in the figure by blue double-headed arrows, using MW pulses. Those can then be optically readout leveraging the population difference in the Q sublevels. Qubits based on D ₀ state can also be obtained by taking advantage of the polarization transfer from the Q state to the D ₀ state [34].

Table 1 provides a representative compilation of Q- and D-based electronic coherence time (T ₂ ^e) values. The impetus for harnessing the utility of the C-R^· systems as quantum materials are based on time-resolved electron spin resonance (Tr-ESR) studies [35]. Those demonstrate the presence of spin-polarized excited quintet states in fullerene- (C₆₀) [36], porphyrin- [37] and anthracene-based systems [38] tethered with nitroxide-based radicals and transition metalloporphyrins [39], [40], [41]. Subsequently, it has been demonstrated that the rates, sign, and magnitude of polarization transfer to the ground state can be tuned by changing the position of a radical bearing group on the chromophore [34], and by engineering the distance between the radical and chromophore [42]. In a seminal study dealing with a PDI-TEMPO based, (PDI and TEMPO stand for 1,6,7,12-tetra(4-tert-butylphenoxy)perylene-3,4:9,10-bis(dicarboximide, and 2,2,6,6-tetramethylpiperidine-1-oxyl, respectively) C-R^· system, Wasielewski, Richert, and co-workers demonstrated that photogenerated quartets in a PDI-based C-R^· system can be used to host electron spin qubits with T ₂ ^e = 1.8 μs at 80 K [20] (Entry 1; Table 1). They have also demonstrated that higher-order qudits can be obtained by leveraging electron-nuclear spin hyperfine interactions. Later studies dealing with PDI-based C-R^· systems substantiated that spin coherence times (T ₂ ⁿ ) in the µs regime can be obtained even at RT [43] (Entry 2; Table 1). Moreover, spin-polarization from the quintet state can be transferred to the radical-based doublet state [44] (Entry 3; Table 1), rendering access to a polarized spin qubit involving the ground D ₀ state. In the examples discussed before, optical readout of qubit states is however inhibited by the non-luminescent nature of the radical bearing entities [45]. By designing a C-R^· system in which both chromophore (anthracene) and radical bearing entities (tris(2,4,6-trichlorophenyl)methyl-carbazole) are luminescent (Figure 2; entry 4, Table 1), Friend, Evans, and co-workers demonstrated that efficient readout of qubit states can be achieved [33], [46]. Optical polarization and readout are mediated by the close energetic proximity between the triplet state of anthracene and doublet excited state of the carbazole-based radical in the C-R^· system. In this system, excited states combining high-multiplicity with near unity quantum yield can be obtained.

Table 1:

Electron spin coherence times of chromophore-radical (C-R^·)-based molecular systems.

Entry	Molecule	Medium	T 2 e	Temperature	Ref.
1	PDI-TEMPO	Frozen toluene solution	1.8 µs (Q)	80 K	[20]
2	PDI-TEMPO-d 17-15 N	Polymer matrix	1.1 µs (D 0) 0.7 µs (Q)	300 K	[43]
		Frozen toluene solution	4.4 µs (D 0) 2.6 µs (Q)	80 K
3	tpPDI-BDPA-d 16 a	2-methyl tetrahydro-furan (mTHF) glassy matrix	2.1 µs (D 0) 2.8 µs (Q)	85 K	[44]
4	Anthracene-(tris(2,4,6-trichlorophenyl)methyl-carbazole) (TTM-1Cz-An)	Frozen toluene solution and PMMA matrix	10 µs (Q)	295 K	[33]
5	p-d 5 -PDI-CHAMPO-d 25 derivatives b	Frozen toluene solution	9.1 µs (D 0) 4.2 μs (Q)	85 K	[48]
6	PDI-biph-tetrathiaryltrityl c	Frozen toluene solution	4.7 µs (D 0) 3 μs (Q)	80 K	[47]

1. ^aBDPA stands for 3-bisdiphenylen-2-phenylallyl. ^bCHAMPO: methyl groups of TEMPO are replaced by spirocyclohexyl groups. ^cOne among the three C-R^· systems featuring long coherence time is shown; see ref. [47] for two other examples.

As discussed before, chemistry can be leveraged to tune the nature of polarization transfer, exchange coupling between chromophore and radical, and electron spin coherence times. T ₂ ^e enhancement following isotope substitution and chemical structure modifications have been demonstrated for PDI-TEMPO systems [43], [48]. In particular, methyl group tunnelling in TEMPO radical contributes to decoherence in PDI-TEMPO systems. It has been demonstrated that by replacing methyl groups with cyclohexyl groups – TEMPO versus CHAMPO in Table 1 – T ₂ ^e of chromophore-centred Q and radical-centred D states can be increased (Entry 5; Table 1). The chemical nature of radical bearing system also influences the coherence time. A comparison between the T ₂ ^e values reported for PDI-based chromophore systems featuring TEMPO, BDPA, and trityl radicals reveals that the PDI-trityl (Entry 6; Table 1) system shows longer T ₂ ^e relative to its two other counterparts [47]. Such enhancement is due to the mitigation of hyperfine interactions in the trityl radical relative to the TEMPO and BDPA systems. It has also been reported that the medium – polymer matrix versus frozen toluene solution – in which C-R^· systems are diluted and concentration of the systems, directly modulate the T ₂ ^e value [49]. The above points emphasize the fact that several criteria need to be optimized to progress toward implementing QIP schemes using these molecular systems.

2.2 Polycyclic aromatic systems capable of undergoing singlet fission

Entangled triplet excitons of high spin-multiplicity obtained as a consequence of SF have also been demonstrated as qubits. Strong exchange coupling between triplet pairs (Figure 3) gives rise to an exchange coupled quintet state (⁵TT; S = 2). Similar to the C-R^· case, the application of magnetic field brings the spin-population out of equilibrium, ultimately leading to spin polarization. Notably, the coupling of two triplet states producing an entangled pair of high multiplicity represents the fact that higher-order qudits, involving five states, can be obtained from SF relative to the three states associated with a triplet state located on a single-molecule [50], [51], [52], [53]. Till date, the utility of this ⁵TT has been demonstrated in pentacene- and tetracene-based systems (Figure 3(a)). Two seminal studies unambiguously elucidated the presence of a ⁵TT state by studying tetracene- [50] and pentacene-based [51] molecular systems at low temperatures. The later study demonstrates SF intramolecularly in pentacene as against the interdimer tetracene pair studied in the former case. Intermolecular ⁵TT pair lifetime of 3 µs and T ₂ ^e of 0.87 µs have been obtained at 10 K [50] for TIPS-tetracene where TIPS stands for triisopropylsilylethynyl group) = (Entry 1; Table 2). Although these first studies dealing with the ⁵TT pair focus more on elucidating the formation and dynamics of the ⁵TT pair, the observation of Rabi oscillations elucidate the utility of these intramolecularly generated ⁵TT pairs for QIP [50], [51].

Figure 3:

High spin-multiplicity excited states generated from singlet fission (SF) for QIP applications. (a) Molecular structures of TIPS-tetracene and pentacene dimer in which SF-mediated generation of entangled ⁵(TT) pair have been demonstrated. (b) Mechanism of ⁵(TT) pair formation. Excitation of singlet pair leads to the formation of singlet excited state that decays into ¹(TT) and then ⁵(TT) states. In the presence of magnetic field, electronic spins are polarized in the ⁵(TT)₀ state. By applying a MW pulse of suitable frequency, coherence between the ⁵(TT)₀ and ⁵(TT)₁ states, for example, can be created. The absorption a and emission e from the ⁵(TT)₀ state can then be used as readout. (c) Transient ESR spectrum of TIPS-tetracene at 10 demonstrating the presence of the ⁵(TT) quintet pair. The outer components are from the triplet state associated with single TIPS-tetracene molecules. The Q− and Q+ terms represent emission and absorption from the ⁵(TT)₀ state. (d) Rabi oscillations associated with the TIPS-tetracene in thin-film and microcrystalline states. The notations Q ⁺ and T ⁺ represent oscillations arising from the ⁵(TT) quintet and triplet states, respectively. (e) Fourier-transform of the oscillations shown in figure (d). Oscillations from the ⁵(TT) quintet state have higher frequency than those from the triplet state, serving as the unambiguous proof for the presence of the high spin multiplicity quintet state. Figures (c), (d), and (e) are reproduced with permission from ref. [50].

Table 2:

Electron spin coherence times of molecular systems hosting ⁵TT state derived from SF.

Entry	Molecule	Medium	T 2 e	Temperature	Ref.
1	Triisopropylsilylethynyl (TIPS)-tetracene	Thin-film and microcrystals	∼1 µs	10 K	[50]
2	tricyclohexylsilylethynyl (TCHS)-tetracene	Single crystals	3/10 µs	10 K	[52]
3	TIPS-bipentacene	Intramolecular dimer in dilute 2-methyltetrahydrofuran (mTHF) glass	1.4 µs	75 K	[53]
4	4,4′-(pentacene-6,13-diyl)dibenzoic acid (PDBA)	PDBA-based metal-organic framework	0.150 µs	300 K	[54]

In a quest to optimize the dynamics of ⁵TT pair formation, Wasielewski and co-workers designed a new tetracene derivative – TCHS-tetracene, where TCHS is tricyclohexylsilylethynyl – and showed that by engineering the packing in the crystal lattice via molecular design, coherence times can be improved [52]. In particular, T ₂ ^e = 3 µs is observed at 10 K (Entry 1; Table 2), and the authors have also demonstrated that the T ₂ ^e can be further increased to 10 µs by employing dynamical decoupling (Entry 2; Table 2) [52]. A recent study dealing with a covalently linked dimer elucidates that a T ₂ ^e in the order of 1.4 µs can be obtained at 40 K [53] (Entry 1; Table 3). The utility of chemistry in generating pentacene-based ⁵TT systems for QIP applications have been further demonstrated in a recent study dealing with a pentacene-based MOF. Due to the rigid nature of the MOF backbone, mechanisms such as triplet-triplet annihilation and exciton diffusion have been mitigated resulting in the observation of 150 ns coherence time at RT [54] (Entry 4; Table 2).

Table 3:

Electron spin coherence times of D-B-A molecular systems.

Entry	Molecule	Medium	T 2 e	Temperature	Ref.
1	Tetrathiafulvalene (TTF; donor)-1,8-dicarboximide (ANI; bridge/chromophore)-pyromellitimide (PI; acceptor)	4-cyano-4′-(n-pentyl)-biphenyl (5CB)a	1.8 µs	85 K	[55]
2	2,6,6-tetramethylbenzo[1,2-d:4,5-d′]bis([1, 3]dioxole) (donor)-4-aminonaphthalene-1,8-imide (acceptor/chromophore) and α,γ-bisdiphenylene-β-phenylallyl (radical; partially deuterated)b	Butryonitrile glass	0.89 µs (D·+) 1.16 µs (R·)	85 K	[22]
3	Pyrene (Py)-Napthalene dianhydride (A)	Single-crystal	7.1 µs	20 K	[56]
4	Napthalenediimide (NDA; acceptor)-DNA (bridge)-tetrathiofulvalene (TTF; donor)	Deuterated buffer solution	4 µs	85 K	[57]
5	5,12-diazatetracene (DAT) chromophore in a metal-organic framework (MOF)c	MOF dispersed in paraffin	0.98 µs	RT	[31], [58]

1. ^aNematic liquid crystal. ^bStructure shown in Figure 4(a). ^cSBCT-active, see [31] for two additional examples.

2.3 D-B-A systems

Within the category of electron donor (D) - bridge (B) - acceptor (A) pairs, photogenerated qubits have also been demonstrated [59], [60]. In a typical D-B-A system, electron-rich and -deficient fragments are connected via a bridge (B). Molecular distance engineering between D and A fragments can be leveraged to modulate the interactions between the fragments. Optical excitation of one of the components in a D-B-A system leads to the formation of a charge-separated state composed of an excited state radical-ion-pair (for example, D^·+-B-A^·−) coupled via spin-exchange and dipolar interactions. Those are often referred to as a spin-qubit pairs (SQPs) in the context of QIP. In SQPs, state polarization is achieved under external magnetic fields and the resultant absorptive and emissive transitions can be probed by performing time-resolved (pulsed) ESR studies. The mechanism of optical spin polarization in D-B-A systems is described in details Figure 3(b).

Typically, T ₂ ^e values in the order of several microseconds have been demonstrated for a range of D-B-A systems – namely, molecular systems tailored with or without stable organic radicals [59], D-A entities tethered with DNA hairpins [61], [57] (Entry 4; Table 3), and organic dye functionalized quantum dot (QD) [62]. In the last case, electron transfer from the dye to the QD generates the SQP. The microsecond-long T ₂ ^e of an entangled spin-pair residing on a D-B-A skeleton (Entry 1; Table 3) was leveraged to demonstrate two qubit gate operations [55]. Addition of a stable radical bearing entity into a D-B-A framework enables the creation of three-spin quantum systems – D^·+-B-A^·−-R^· (Figure 4(a)-right and Entry 2; Table 3) – in which spin polarization can be transferred from one fragment to another fragment, enabling quantum teleportation in the MW regime at nm distances. This has been demonstrated, elucidating short-range quantum interconnectors based on molecules can be realized [22]. Hyperfine interactions between the electron and nuclear spins are one of the factors driving decoherence in D-B-A systems. T ₂ ^e can however be enhanced by mitigating the hyperfine interaction by isotopically enriching molecular skeletons with nuclei that bearing small magnetic moments – for example, replacing hydrogen with deuterium [20], [44], [59]. One general limitation of the molecular systems is the random orientation of the molecules – therefore the g-tensor – with respect to an external magnetic field. This contributes to line broadening of the EPR transitions, typically in the range of GHz, making selective addressing of individual transitions challenging. In this respect, it has been recently demonstrated that spatially oriented triplet excitons can be realized in a co-crystal composed of donor (pyrene) and acceptor (naphthalene dianhydride) entities (Figure 4(a)-left and Entry 3; Table 3). For the system, selective addressing of the spin levels has been reported, as well as T ₂ ^e of 7.1 μs at 20 K [56].

Figure 4:

Electron donor (D)-bridge (B)-acceptor (A) systems as photoinduced spin qubits. (a) Structures of co-crystallized and covalently linked D-A systems. The notations D^·+, A^·− and R^· in the top structure depicts donor-based cationic radical, acceptor-based anionic radical, and the appended stable organic radical, respectively. (b) Mechanism of optical spin polarization in D-B-A systems. Excitation of a D-B-A system in the ground state leads to the formation of the singlet excited state – ¹[*D-B-A], with the donor fragment excited in this specific case. Photo-induced electron transfer (PET) from the donor to the acceptor leads to the formation of the charge-separated state. Due to the spin conserving nature of the PET process, a singlet state – ¹[D^·+-B-A^·−] – is formed initially. A subsequent spin-flip leads to the formation of the triplet spin-pair. In a static magnetic field that is relatively large compared to the spin–spin exchange (J) and dipolar (D) interactions, mixing between the ¹[D^·+-A^·−] and ³[D^·+-A^·−] states produces hybrid electronic levels, as shown in the blue box [61]. Due to the singlet character of the ¹[D^·+-B-A^·−] state, the Ψ_A and Ψ_B levels are initially populated. By using appropriate MW pulses and leveraging the mentioned spin-polarization into the Ψ_A,B states, coherence superposition between the mixed singlet and triplet levels can be created. Addition of a radical bearing entity has been exploited to generate higher order qubits, as in the structure shown in (a, right). (c) To circumvent line-broadening and the consequent qubit addressability issue, co-crystallized D-A systems have been proposed (a, left). Spatially oriented triplet excitons in such systems resulted in the observation of about 7.1 µs coherence lifetime at 20 K. In the top part, the pulse sequence used to determine the coherence lifetime is shown [56]. Figure (c) is reproduced with permission from [56].

Electron-transfer-mediated creation of SQPs can also be generated in molecular systems that can undergo symmetry breaking charge transfer (SBCT). Unlike in the cases of D-B-A systems discussed above, in which electron deficient and rich fragments are involved in the charge transfer process, identical molecular fragments orthogonally connected in a dimer are involved in the SBCT process. Upon light absorption, one of the fragments in an orthogonally connected dimer is excited and a locally excited state is produced. Differences in the local solvent interactions around the dimer creates the driving force required for charge separation, thereby charge separation and stabilization of the SQP is facilitated [28]. Efficient spin polarization, polarization transfer from the triplet to persistent radicals, and long T ₁ ^e and T ₂ ^e of polarized radicals at RT have been demonstrated for SBCT-active metal-organic frameworks (MOF) composed of 5,12-diazatetracene (DAT) chromophore [31], [58] (Entry 5; Table 3). Finally, an interesting approach to increase the utility of D-B-A systems for QIP is the use of a chiral bridge [63]. By harnessing chirality-induced spin selectivity (CISS) effects, strong spin polarization can be achieved, setting the basis for high-fidelity qubit initialization, manipulation and single-spin readout. Some examples of chiral bridges in which strong spin polarization (>90 %) has been reported are supramolecular wires, 2D chiral hybrid lead-iodide perovskites or chiral hybrid copper halides [63].

3.1 Cr³⁺ and Cr⁴⁺ complexes

The combination of sub-GHz optical linewidths with long-lived electronic spin states in TM complexes was first reported during the late eighties and nineties using the spectroscopic technique known as persistent spectral hole burning (SHB) [69]. These early studies were mostly focused on Cr³⁺-based complexes – for example, [Cr(en)₃]³⁺ – showing optical homogeneous linewidths (Γ_h ^opt) of the order of 150 MHz for the R1 line of Cr³⁺ at liquid helium temperature [70]. This corresponds to an optical coherence time (T ₂ ^opt) of about 2 ns, according to the relation Γ_h ^opt = 1/πT ₂ ^opt. Alternatives to reduce this broadening, and consequently improve T ₂ ^opt, include approaches such as diluting the complexes (≤0.001 %) in host lattices and leveraging exchange interactions to minimize magnetic noise. For example, intramolecular anti-ferromagnetic coupling in binuclear Cr³⁺ complexes results in an S = 0 ground state leading to the nullification of electron-spin–based dipolar interactions [71]. The possibility of modulating the energy-level structure and the optical and electron spin coherence times through complex and host design has also been reported [69]. In particular, by combining several strategies, narrow optical homogeneous linewidths of 40 MHz were obtained in single crystals of perprotonated and partially deuterated versions of the binuclear [LCr(III)(-OH)₃ – Cr(III)L](ClO₄)₃·H₂O (L = 1,4,7-trimethyl-1,4,7-triazacyclo- nonane) complex [72].

These pioneering studies, many of them authored by Riesen and coworkers [69], [70], [72], [73], remain, to the best of our knowledge, as the state-of-the-art in terms of Γ_h ^opt in TM complexes. However, correlations between the molecular design and qubit figures of merit – for example T ₂ ^e or the ability to initialize the qubit, have not been considered in these studies. In contrast, such correlations have been elucidated for Cr⁴⁺-based complexes [21], [65].

To meet optically addressable electron spin qubit design criteria, Bayliss and co-authors designed three Cr⁴⁺ complexes coordinated by strong-field aryl ligands in a pseudo-tetrahedral configuration. This highly symmetrical environment yields a ground state spin triplet (S = 1) with small zero-field splitting – 1.9–4.2 GHz – while the strong-field is leveraged to obtain the S = 0 state as the lowest-lying electronic excited state (Figure 5(a)). Optical characteristics of three complexes differing in the number and position of the methyl groups (Figure 5(b)) were studied by diluting them in a suitable crystalline host lattice. The minor structural difference results in distinctive ground-state axial (D) and transverse (E) zero-field splitting parameters and optical transition energies for the three complexes. Utility of the complexes as optically addressable electron spin qubits is elucidated by the demonstrations of optical spin state polarization (14 %) and readout under resonant excitation to the excited (S = 0) state, and spin-state manipulations using MW excitation (Figure 5(a)). In particular, optically-detected Rabi oscillations and a T ₂ ^e of 640 ns at 2 mT and 4 K [65] have been observed. This relatively short coherence time was improved to about 10 μs in a subsequent work, harnessing the energy-level-structure tunability through the host environment modification [21]. The engineering of the host crystal structure allows for designing magnetically insensitive – clock – electron spin transitions [16]. Complex 1 in Figure 5(b) was diluted down to 0.5 %–1 % in non-isostructural Sn-based hosts, giving rise to a non-zero transverse zero-field splitting (E ≠ 0) due to symmetry breaking with respect to the isostructural host ‘1’, as well as to lower (host ‘3’), or negligible (host ‘2’) transition energy variation under magnetic field (Figure 5(c)). A significant improvement in T ₂ ^e was in particular reported for the non-isostructural host ‘2’, reaching 10 μs at 4 K [21] (Entry 1; Table 4).

Table 4:

Optically addressable electronic qubits and quantum sensor candidates based on TM complexes.

Entry	Compound	Medium	T 2 e or T m a	Γinh opt b	Wavelength	Ref.
1	Cr(IV)(o-tolyl)4 a	Sn(IV)(4-fluoro-2-methylphenyl)4 crystal	T 2 = 10 μs at 4 K	53.2 GHz at 4 K	1,016 nm	[21]
2	(C6F5)3trenVCNtBu	Diluted into (C6F5)3trenGaCNtBu analoge matrix	T m = 427 ns at 5 K and 6.55 T	∼90 GHz at 4 K	1,237 nm	[66]
3	[(VO)2(bdhb)2]	frozen toluene-d 8/CD2Cl2 matrix	T m = 13 μs at 10 K and 0.34 mT	Not reported	Not reported	[74]
4	[Ni(phen)3](BF4)2]	1 mM solutions in 1:1 H2O:glycerol	T m = 250 ns at 20 K and 1.53 T	∼100 THz at 80 K	940 nm	[67]
5	Ni(ttcn)2](BF4)2	diluted into [Zn(ttcn)2](BF4)2 analogue matrix	T m ∼ 400 ns at 12 K and 0.25 T	Not reported	782 nm	[68]

1. ^a T _m stands for phase memory and encompasses all processes that contribute to decoherence, including T ₂ ^e and the inhomogeneous dephasing time T ₂ ^e *. ^bΓ_inh stands for optical inhomogeneous linewidth.

3.2 Ni²⁺, V³⁺ and V⁴⁺ complexes

Optically addressable TM-based spin qubit candidates composed of Ni²⁺ [67], [68], V³⁺ [66] and V⁴⁺ [75], [74] have also been reported (Table 4). Ni²⁺ and V³⁺ systems feature spin triplet (S = 1) ground state while V⁴⁺ complexes present spin doublet (S = 1/2). The design of Ni²⁺ complexes must account for the large spin–orbit constant; therefore, a high sensitivity to ligand field often yields a large zero-field splitting. By leveraging a quasi-octahedral coordination symmetry, Wojnar and co-authors reported two Ni²⁺ complexes – [Ni(phen)₃](BF₄)₂] and [Ni(pyr₃)₂](BF₄)₂] (with phen = 1,10-phenanthroline and pyr₃ = tris-2-pyridyl-methane) – fulfilling the EPR addressability criteria, as well as optical emission occurring between 938 and 944 nm [68] (Figure 6(a)). As far as vanadium is concerned, it features two interesting attributes – intrinsic nuclear spin I = 7/2, potentially useful for long-term quantum storage [3], and infrared (IR) emission lines compatible with fiber optic networks [76]. The proposal in ref. [66] is a trigonal bipyramidal V³⁺ complex: [(C₆F₅)₃trenVCN^tBu], where ‘tren’ stands for tris(2-aminoethyl)amine) (Figure 6(b)). High-frequency EPR addressability (240 GHz) was confirmed in this compound as well as a zero-phonon optical transition at 1,237 nm with a linewidth comparable to the spin zero-field splitting, allowing for a selective optical addressing of the spin transitions under high magnetic field (>6 T) [66]. Another family of vanadium-based complexes that have shown good prospects as electron spin qubits, due to their long T ₂ ^e times, are those based on V⁴⁺ [75]. Among them, the recently-reported vanadyl complex [VO(bdhb)] (with bdhb = 1,3-bis(3,5-dioxo-1-hexyl)benzene) [74] (Entry 3; Table 4), shows EPR addressability in the X-band (9.4 GHz), and is attractive because of its neutral charge, nuclear spin-free donors, and potential for chemical functionalization on the aromatic ring.

Figure 6:

Ni²⁺ and V³⁺ complexes proposed as candidates for optically addressable molecular spin qubits. (a). Molecular structures and energy level diagrams of [Ni(phen)₃](BF₄)₂] (1) and [Ni(pyr₃)₂](BF₄)₂] (2). Green, blue, and gray spheres represent Ni, N, and C atoms, respectively. H atoms are omitted for clarity. The figure is adapted with permission from ref. [67]. (b) Illustration of the energy level scheme for [(C₆F₅)₃trenVCN^tBu], highlighting the spin conserving optical excitation (blue), intersystem crossing (ISC) to the ¹E (S = 0) excited state (gray), and photoluminescence (PL) from the later to the ³A (S = 1) ground state. The molecular structure determined by single-crystal X-ray diffraction is shown below with teal, blue, green, and gray spheres representing V, N, F, and C, atoms, respectively. H atoms are omitted for clarity. Pink circles: Qualitative d-orbital splitting diagram and electronic configurations for ¹E and ³A in ideal C_3v symmetry. The figure is adapted with permission [66].

In conclusion, Ni²⁺, V³⁺ and V⁴⁺ complexes could emerge as attractive TM-based qubit platforms due to their added functionalities – for instance, optical addressability within the IR region, compatible with optical fiber networks. Yet, they are still behind the best Cr⁴⁺ complexes [21], [65] in terms of T ₂ ^e (Entries 2–5; Table 4) and optical spin manipulation capabilities.

Finally, TM complexes exhibiting biocompatibility and chemical stability are also developped for quantum sensing applications. For a molecular system to function as a quantum sensor, it should satisfy the same design criteria laid out for optically addressable molecular spin qubits, i.e. EPR addressability combined with optical initialization and readout capacities. Compounds proposed for quantum sensing include the Ni²⁺ complex [Ni(ttcn)₂](BF₄)_2, featuring a thioether-based ttcn ligand (ttcn = 1,4,7-trithiacyclononane) [68] (Entry 6; Table 4).

4.1 Europium-based molecular crystals for coherent spin-photon interfaces

Narrow Γ_h ^opt – from few tens of MHz and down to 30 kHz – combined with long nuclear spin relaxation times (T ₁ ⁿ ) – up to several hundreds of seconds – have been reported for molecular europium (Eu³⁺) complexes (Figure 7 and Table 5) in their stoichiometric crystalline form [25], [84], [85], [86]. The transition of interest is the ⁵D₀ → ⁷F₀ transition centered about 580 nm. This transition is forbidden by selection rules and only takes place when the Eu³⁺ center is placed in a low-symmetry coordination environment [87], [88], establishing a first design criteria for Eu³⁺ complexes. Eu³⁺ presents two stable isotopes, ¹⁵¹ and ¹⁵³, both with nuclear spin I = 5/2, giving rise to three doubly degenerated nuclear spin levels at zero magnetic field. The optical and spin transitions of REI are in general less sensitive to their environment than those of transition metal complexes. Yet, as displayed in Table 5, the optical coherence properties of Eu³⁺ complexes can significantly vary from one molecule to another. For instance, at 1.4 K, a Γ_h ^opt of 22 MHz (i.e. 14.5 ns T ₂ ^opt) was observed for a seven coordinate binuclear Eu³⁺ complex (Figure 7(b) and Entry 4; Table 5) [84], while a Γ_h ^opt of only 30 kHz (10.5 μs T ₂ ^opt) was observed for the eight coordinate mononuclear Eu³⁺ complex [Eu(BA)]₄(pip) (BA = benzoylacetonate; pip = piperidin-1-ium) (Figure 7(a) and Entry 1; Table 5), a value several orders of magnitude narrower than the ones observed for other molecular emitters [18], [25], [69]. For the so far studied Eu³⁺ molecules, the obtained Γ_h ^opt is narrower than the ground-state nuclear spin splittings, allowing for optical addressing of the nuclear spin states of ion ensembles within the optical inhomogeneous line Γ_inh ^opt. This was harnessed to probe the nuclear spin relaxation times T ₁ ⁿ , which are in the seconds to hundreds of seconds regime (Table 5 and Figure 8(a)). It is interesting to note that the T ₁ ⁿ values are not directly correlated to the optical Γ_h ^opt values, as observed for the longer T ₁ ⁿ but broader Γ_h ^opt obtained for the Eu(trensal) complex (Entry 2; Table 5). This indicates different limiting underlying mechanisms contributing to optical line broadening and spin relaxation.

Figure 7:

Europium (Eu³⁺) complexes show narrow optical homogeneous linewidths (Γ_h ^opt) and long nuclear spin relaxation times (T ₁ ⁿ ): (a) eight coordinate mononuclear [Eu(BA)]₄(pip) (BA = benzoylacetonate; pip = piperidin-1-ium), (b) seven coordinate binuclear [Eu₂Cl₆(4-picNO)₄(μ ₂-4-picNO)₂]·2H₂O] (4-picNO stands for 4-picoline-N-oxide), (c) seven coordinate [Eu(trensal)] (trensal = 2,2′,2″-tris(salicylideneimino)triethylamine), and (d) ten coordinate [Eu(dpphen)(NO)₃] (dpphen = 2,9-bis(pyrazol-1-yl)-1,10-phenanthroline), with O₈, O₄Cl₃, O₃N₄, and O₆N₄ coordination environments, respectively. (e) Electronic levels of Eu³⁺ – the narrow optical transition occurs between the ⁵D₀ and ⁷F₀ levels – and nuclear spin structure of the two stable isotopes – ¹⁵¹Eu and ¹⁵³Eu – of Eu³⁺ composed of three doubly degenerated levels in the ground and excited states. In the scheme, nuclear spin population polarization by optical pumping followed by spin relaxation causing population equilibrium is represented.

Table 5:

Compilation of spectroscopic properties of Eu³⁺ complexes investigated as optically addressable molecular qubit candidates. Given values are at 1.4 K for [Eu(BA)]₄(pip) and [Eu₂], while at 4.2 K for Eu(trensal) and Eu(dpphen)(NO)₃. Γ_inh ^opt stands for optical inhomogeneous linewidth, T ₁ ^opt for excited state population lifetime, Γ_h for optical homogeneous linewidth, T ₂ ^opt for optical coherence time, with Γ_h ^opt = (πT ₂ ^opt), and T ₁ ⁿ for spin population time). These studies were all carried out in fully concentrated Eu³⁺ molecular crystals.

Entry	Molecule	Site	Γinh opt	T 1 opt	Γh opt	T 2 opt	T 1 n (s)c	Ref.
1	[Eu(BA)]4(pip)	C2v	6.6 GHz	0.54 ms	30 kHza	10.5 μsa	0.43/300	[25]
2	Eu2Cl6(4-picNO)4(μ 2-4-picNO)2]·2H2O	D5h b	50 GHz	0.88 ms	22 MHz	14.5 ns	1.6/>30	[84]
3	Eu(trensal)	C3v	91 GHz	0.35 ms	2.8 MHz	114 ns	7/480	[85]
4	Eu(dpphen)(NO)3	C2v	16 GHz	1.4 ms	3.1 MHz	102 ns	0.3/41	[86]

1. ^aFrom photon echo measurements. The rest of values were estimated from spectral hole widths. ^bSite symmetry for Eu³⁺ in Eu₂Cl₆(4-picNO)₄(μ ₂-4-picNO)₂]·2H₂O identified as distorted pentagonal bipyramidal. See ref. [84] for more details. ^cThe two values correspond to the two distinct time constants observed in the spectral hole decay.

Figure 8:

Time-dependent hole decay, optical polarisation of nuclear spin states, optical T ₂ extension by dilution and coherent storage in Eu(BA)]₄(pip). (a) Nuclear spin relaxation time (T ₁ ⁿ ) estimated from the decay of the spectral hole depth as a function of time. Inset: detail of a spectral hole. The vertical axis corresponds to transmission. The arrow illustrates the decay of the hole depth with time due to the relaxation of the nuclear spin population. (b) Spin initialization to a single nuclear spin level in [Eu(BA)]₄(pip) [25]. (c) T ₂ ^opt extension by dilution into [Y(BA)]₄(pip), revealing dephasing due to electric dipole-dipole interactions between the Eu³⁺ centres in the fully concentrated [Eu(BA)]₄(pip) molecular crystal. (d) Optical storage of a Gaussian pulse with 0.86 % efficiency using the atomic frequency comb (AFC) quantum memory protocol. Adapted with permission from ref. [25].

The very narrow homogeneous linewidth Γ_h ^opt and long T ₁ ⁿ observed for crystals composed of a mononuclear Eu³⁺ complex [Eu(BA)₄](pip), allowed for further demonstrations of >95 % spin polarization into a single level (Figure 8(b)), coherent optical storage up to 1 µs using the atomic frequency comb (AFC) quantum memory protocol (Figure 8(c)), and controlled ion–ion interactions of use for high-bandwidth two-qubit quantum gates. Impact of the host on T ₂ ^opt was also evidenced by diluting [Eu(BA)₄](pip) in [Y(BA)₄](pip), leading to an increase in T ₂ ^opt by a factor 2–6, depending on the dilution ratio, confirming optical dephasing due to electric dipole interaction between Eu³⁺ ions (Figure 8(d)).

4.2 Enhanced optical addressing and detection of REI molecules

The low sensitivity of REI to environmental perturbations is of use for achieving record-long optical coherence times in REI materials at low temperatures, a feature that distinguishes these materials from other optically addressable spin platforms. It comes however with the drawback of low emission rates for REI-based optical transitions making the readout of single REI difficult. Some degree of Purcell enhancement is therefore a requirement for optically addressing and detecting single REIs in an efficient manner for QIP applications [12], [17], [80], [89]. In the following we provide two examples showing that the natural compatibility of molecules with different photonic devices can be harnessed to obtain enhanced optical addressing and emission rates in REI complexes [90], [91].

In the first example, enhanced photoluminescence (PL) has been reported by Emmanuele and co-authors [90] for the ⁵D₀ → ⁷F₂ transition of Eu³⁺ in the complex [Eu(TTA)₃(DPEPO)], where DPEPO stands for bis(2-(diphenylphosphino)phenyl) ether oxide, via a Fabry–Pérot micro-cavity tuned in resonance with the transition. The Eu³⁺ complexes are deposited onto the bottom mirror of the cavity (Figure 9(a)). A maximum enhancement value of 30 is achieved for a cavity length of 1.2 μm (Figure 9(c) and (d)). From the shortening of the PL decay under cavity coupling, it is also concluded that the spontaneous emission rate is increased by a factor of 140. This emission acceleration is attributed to amplified spontaneous emission from the super-linear dependence of the PL intensity with the excitation power.

Figure 9:

Photonic enhancement of optical addressing and detection of REI molecules. (a) Tunable open microcavity. The photonic structure is formed by two glass mirrors facing each other in the vertical direction and covered by 50 nm of Au and 5 nm of Al₂O₃. A layer of [Eu(TTA)₃(DPEPO)] is deposited onto the bottom mirror of the cavity. Below – optical microscope image of the 200 μm × 300 μm plinth. (b) Cavity length-dependent PL spectra of the Eu³⁺ complex–cavity system at k _|| = 0. The vertical dashed white line corresponds to the most prominent ⁵D₀ → ⁷F₂ transition. (c) PL enhancement factors derived from the white line in (b). Adapted with permission from ref. [90]. (d) Optical setup used to excite a monolayer of Yb³⁺ complexes coated on a tapered single-mode optical fiber. (e) Structure of the complex [Yb³⁺[Zn(II)_MC(QXA)]], where MC and QXA stands for metallo crown and 2-Quinoxaline carboxylic acid, respectively, obtained by XRD (thermal ellipsoid structure presentation with a probability of 50 %; hydrogens are omitted for clarity). (f) PLE spectrum obtained from the TOF coated with a monolayer of the complexes at T < 10 K with a characteristic narrow absorption line around 980 nm attributed to Yb³⁺.

The second example, by Mor and co-authors [91] concerns the evanescent-field coupling of a monolayer of a highly radiative Yb³⁺ complex – [Yb³⁺[Zn(II)_MC (QXA)]], with lifetime >0.95 ms, coated on a tapered single-mode optical fiber (TOF) of ∼500 nm in diameter (Figure 9(d) and (e)). For this, the TOF’s surface was successfully functionalized to allow for the binding of the Yb³⁺ complexes (see ref. [91] for details). Optical addressing of the molecular monolayer was confirmed by monitoring the PL intensity coupled to the fiber as a function of the excitation wavelength, revealing the characteristic absorption maximum of Yb³⁺ ions around 980 nm (Figure 9(f)).