Introduction

The mission of the Quantum Staging Group (QSG) is to provide leadership and guidance for the Materials Research Society (MRS) in promotion of materials science for development of quantum information sciences and quantum sensing. The QSG provides advice and support to international leaders in the field of quantum information science and engineering on symposium and workshop development and other activities including development of publications that promote materials science for quantum technologies. It was formed in 2018 and has held a number of workshops and panel discussions linked to both the Spring and Fall meetings.

While the QSG provides support for quantum materials development across all platforms, including superconductors, ion traps, topological systems, and more, the particular focus of this workshop was solid-state atom-like systems, including impurity-based qubits in semiconductors and defects in two-dimensional (2D) and three-dimensional (3D) materials systems.

This article provides a summary of the QSG workshop titled “Challenges in Advancing Our Understanding of Atomic-Like Quantum Systems: Theory and Experiment,” that was held at the 2022 MRS Spring Meeting in Hawaii. The workshop was held in conjunction with the meeting symposium QT07-Atomic and Molecular Quantum Systems and Defect Engineering for Quantum Technologies. The aim of the workshop was to bring together scientists with experimental and theoretical expertise in materials for quantum technologies to discuss key near-term challenges that need to be addressed to further promote and accelerate development of solid-state atom-like systems with applications in quantum technologies.

The main emphasis was on having both theoreticians and experimentalists come together to provide their perspectives on the opportunities and challenges associated with the underlying materials science needed to develop and control atom-like quantum entities across a range of platforms that form qubits, arrays of qubits, and long-range communication between qubit systems. As a result, the workshop promoted greater awareness of where key focus is required and what tools are needed to make real progress. There were four themes addressed in the workshop and summarized next: “Are there unifying perspectives on the relevant length scales for quantum systems? How to span from 1 nm to 1 micron?,” “Decoherence at the interface: How can quantum nanoscience address materials challenges in quantum information technologies?,” “Are there unifying perspectives on the role of electrical noise in atomic-like systems in materials?”, and “What are the predictive challenges for atomic-like quantum systems?”

Unifying perspectives on spanning nanometer to micron-length scales

The first discussion was led by S. Coppersmith and focused on how materials questions that are critical to further progress keep evolving because of the tremendous advancements made in the development of qubits in Si/SiGe quantum dots. In 2022, three groups published reports of experiments achieving one- and two-qubit gate fidelities exceeding 99.5 %1,2,3 as well as successful operation of a six-qubit processor.4 With the successful demonstration of gate fidelities high enough to be compatible with the requirements of the current generation of quantum error correction protocols, the question of how to achieve extremely high yields of devices with this level of gate performance becomes critical. In the context of the entire workshop, this session highlighted recent progress on materials issues that are enabling substantial improvements in yields of devices that host qubits with high-fidelity gates. Both talks in the session highlighted how new key questions emerge as the field progresses and figuring out how to scale up to large numbers of qubits becomes a pressing concern.

The work in References 1,2,3,4 was performed using quantum dots hosted in silicon/silicon-germanium (Si/SiGe) heterostructures, and M. Friesen’s talk in the session focused on this materials system and on new understanding of how to fabricate quantum dots in which the lowest-lying excited states are reliably found to be spin states, rather than conduction-band valley states. The presentation began by introducing quantum dot (QD) spin qubits as systems that have the advantage of long-lived spin states, all-electrical control, and compatibility with existing classical electronics due to the choice of materials they are embedded in and made of. Widely employed GaAs QDs have the drawback of predominant nuclear spins, and thus, in recent years, the focus has shifted to Si and Ge materials.5 The state-of-the-art one- and two-qubit gates show fidelities of >99.96 and \(99.5\%\).

A major challenge in Si arises due to the sixfold orbital degeneracy of the lowest conduction band.7 By applying biaxial strain, the fourfold xy and the twofold z valleys split by about 0.2 eV (Figure 1). A further valley splitting of <0.2 meV, between the low-lying \(+z\) and \(-z\) valley eigenstates, can be achieved by strongly confining the quantum dot within the quantum well.8 However, the different spin states may be Zeeman split by about 0.2 meV, leaving the valley splitting as the defining energy scale for the qubits, rather than the spin splitting.

Figure 1
figure 1

Valley splitting in a SiGe/Si/SiGe quantum well. Silicon conduction-band valleys are sixfold orbitally degenerate in bulk. Applying strain in a SiGe/Si/SiGe quantum well reduces the low-lying orbital degeneracy to twofold. Quantum-well confinement (green), combined with Zeeman splitting \(E_Z\), fully lifts the ground-state degeneracy to give the valley splitting \(E_\text {VS}\). Here, \(|g\rangle\) and \(|e\rangle\) refer to ground and excited valleys. However, details of the quantum-well interface could result in a low valley splitting (\(E_\text {VS}<E_Z\)), and an undesired level ordering, in which the ground-state manifold is not spin-like.

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Although valley degeneracy can be lifted by means of a strong quantum-well confinement potential, the possibility of low-lying valley excited states arises because the valley minima occur at k vectors far from the zone center. In particular, small valley splittings can occur when there are no features in the confinement potential with large enough k vectors to couple the valleys. This problem is challenging to address theoretically because it is inherently multiscale–atomic-scale details of the quantum-well interface as well as the details of germanium concentration fluctuations in consecutive atomic layers play a critical role in determining the valley splitting. This understanding of the physics and the origins of the problem provides concrete avenues for develo** heterostructure designs and growth processes that result in consistently high valley splittings.

The main challenge for achieving reproducibility of the valley splittings across a quantum dot device arises from unavoidable atomic-scale disorder. The dominant types of disorder in modern Si/SiGe quantum wells include (1) single-atom steps at interfaces, which can suppress valley splitting significantly (around 80%);9,10,11 (2) Ge concentration fluctuations in the SiGe portion of the device, as Ge in Si forms a random alloy, resulting in a wide range of splitting between 0 and 250 μeV;12 and (3) diffuse interfaces that are not perfectly sharp but spread out over several atomic layers, which can significantly suppress the splitting.13 In many cases, valley splittings must be at least as large as the thermal energy scale in a dilution refrigerator (\(k_BT \approx\) 10–20 μeV), to enable high-fidelity initialization and readout. This is particularly true for qubit encodings known as Loss–DiVincenzo qubits14 (i.e., a single-electron spin in a quantum dot). In other spin qubit encodings such as singlet-triplet qubits15 (i.e., two electrons in a double quantum dot), quantum dot hybrid qubits16 (i.e., three electrons in a double quantum dot), or exchange-only qubits17 (i.e., three electrons in a triple quantum dot), large valley splittings are not necessarily a prerequisite for forming a well-defined qubit. However, good control and reproducibility of the valley splitting and other device parameters are always required for scaling up to many qubits.18 The wide range of observed valley splittings, which vary significantly from dot to dot even within a single wafer, therefore poses a significant challenge for these systems. Although this variability can be mitigated slightly through carefully engineered heterostructures and gate control, it cannot be completely eliminated.

Deterministic solutions to the problem of valley splitting variability have recently been explored. The most promising of these is an approach known as the “Wiggle Well,”19 wherein the z valleys are coupled by introducing specially chosen Ge concentration oscillations in the interior of the quantum well. This method has the potential of coupling valleys in the same Brillouin zone (short-period Wiggle Well) or in neighboring Brillouin zones (long-period Wiggle Well). The former structure will be challenging to grow in the laboratory, but it holds the promise of providing large, reproducible valley splittings 1.2 meV, which would certainly enable good quality spin qubits. To date, however, only the long-period Wiggle Well has been grown and measured.19 Simulations confirm that the valley splitting enhancements provided by this structure are quite weak, and are overwhelmed by random-alloy disorder fluctuations of SiGe within the quantum well.

An alternative, nondeterministic approach to suppressing the variability of the valley splitting is to introduce uniform Ge into the quantum well.13 In this case, natural Ge concentration fluctuations over the extent of a quantum dot enhance valley splitting, on average, by approximately mimicking the effect of a short-period wiggle well. A side effect of this statistical enhancement is the simultaneous increase of the variability of the valley splitting as a function of lateral position, due to increased overlap of the electron with the random alloy. To overcome this variability, simulations show that the ability to electrostatically control the position of the quantum dot over about 20 nm provides a mechanism for boosting the valley splitting to acceptable levels for the vast majority of devices on a given chip.20

In conclusion, Friesen emphasized the urgent need to develop atomic-scale growth and simulation techniques, to provide the level of control and uniformity needed for large-scale qubit implementations.

In the second talk in this session, C. Déprez presented recent results on a different materials system, holes in Ge/SiGe heterostructures, and discussed challenges in the scaling up of these hole spin qubits in germanium (see Figure 2a). Quantum dots with Ge holes have some advantages over Si electrons, because this materials system has no low-lying valley states, and it has strong spin–orbit coupling that obviates the need to fabricate lithographically patterned micromagnets in order to implement gate operations. Déprez presented state-of-the-art results in the context of the fabrication and the operation of a four-qubit processor21 and of a sixteen-quantum dot device with shared gates.22 The talk discussed physical effects that can limit the operation of large spin-qubit processors in Ge, such as variations in the hole g-factor caused by its strong dependence on the local electric field, both intrinsic and externally applied. While the applied external field used to manipulate the qubits is largest, electric fields from defects also affect qubit frequencies and need to be accounted for. The impact of nonuniformities of the confining potential on the operation of larger spin qubit devices was also discussed, and Déprez showed that they can be mitigated substantially by appropriate protocols for ram** gate voltages.

Semiconductors such as GaAs, Si/SiGe heterostructures, and SiMOS were employed for electron spin qubits and now SiMOS and Ge/SiGe are used for hosting hole spin qubits. Ge/SiGe heterostructures are particularly interesting for quantum information applications.23 They were reported to host a 2D hole gas having a high mobility and they have a low charge noise level.24,25,26,27 Further, holes in planar germanium have low effective mass, strong spin–orbit coupling, and an anisotropic and tunable g-factor. In the last few years, operations of all-electrically controlled single spin qubits28,29 and of singlet-triplet qubits30,31,32 based on hole quantum dots in planar germanium have been shown. They have been extended to implement a four-qubit quantum circuit21 and a quantum simulation32 using a 2D array of germanium quantum dots.

The g-factor in germanium depends strongly on the local electric field28,30,31,33 and is highly anisotropic with \(\vert g_{\parallel }\vert \approx 0.1-0.4\) and \(g_{\perp }\approx 5-15\),21,24,25,28,29,30,31,33,34,35 as illustrated in Figure 2b–c. These properties affect the operation of hole spin qubits and could challenge their scaling up. The strong dependence of the g-factor on the electric field enhances the effect of charge noise: it converts fluctuations of the electric environment into fluctuations of the Zeeman splitting, which affects coherence of single spin qubits. This dependence also leads to significant variations of the g-factor values inside the same device. It can even change sign for in-plane magnetic field configurations.31 This spoils the initialization of singlet-triplet \(S-T_0\) qubits as it leads to leakages to the \(T^-\) states. The g-factor variability also implies that the dynamics of singlet-triplet qubits within the same chip can be quite different. It complicates the practical operation of devices composed of multiple singlet-triplet qubits. For single spin qubits, large variations of the g-factors within a single device could also lead to an overhead in the control electronics used to drive the qubits.

Figure 2
figure 2

g-factor anisotropy in germanium spin qubit devices. (a) Schematic of a germanium hole quantum dot hosting a single spin qubit. Holes are accumulated beneath the plunger gate and confined using the barrier gates. An external magnetic field B is applied to split the spin states and to define a qubit. (b, c) Examples of g-factor evolution with the magnetic field orientation. The field orientation is described in the basis (xyz) formed by the principal axes of the g-tensor. The effective g-factor is then given by \(\sqrt{(g_x B_x)^2+(g_y B_y)^2+(g_z B_z)^2}/\vert B\vert,\) where \((g_x,g_y,g_z)\) are the g-factor values along the principal axes.36 The graphs are generated using typical values for \((g_x,g_y,g_z)\) of (0.1, 0.4, 10).

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Furthermore, the anisotropy of the g-factor also impacts the coherence of hole qubits via the hyperfine interaction. In 2D hole systems, the hyperfine interaction is also highly anisotropic such that the hole spin mostly couples to the z-components (out-of-plane) of the nuclear spins.37,38 Thus, spin qubits in natural germanium are usually operated with an in-plane magnetic field. In practice, the external field is always slightly misaligned with the in-plane direction. Hence, the strong g-factor anisotropy magnifies the effects of this misalignment and amplifies the effect of hyperfine couplings affecting qubit coherence.33,39 Note that the hyperfine interaction could be suppressed by using isotopically purified germanium, which would lead to an increase of the qubit coherence.39,40

To reduce dephasing and errors due to g-factor properties, solutions need to be found that allow to minimize the local variations of the electric field and the inhomogeneities in the confining potentials. One approach to achieve this is to increase the material uniformity, for example, by lowering the number of defects in the heterostructures or in the oxide layers, and in improving nanofabrication processes, for instance by using techniques that allow to fabricate gates having more uniform and more regular shapes. This would reduce the variations in the confining electric fields between quantum dots and thus in the g-factor values. Likewise, we could envision to adapt the design of quantum dots and engineer the confining potentials in order to make the g-factor less sensitive to electric field fluctuations.

Alternatively, one could take advantage of the g-factor properties to facilitate operation of hole spin qubit processors. Theoretical investigations41,\(T_1\) and \(T_2\) that originates from impurities or defects. Temperature dependence will enter, for example, through the time scale for trapped charge hop**. Additional contributions to this spectral function would come from phonons, through direct, Raman, or Orbach processes that arise from the absorption, emission, or scattering of phonons in the crystal via the spin–orbit interaction. These relaxations are temperature-dependent and thus in turn decide the working temperature regime of the qubit, which can be dependent on the Debye temperature of the host crystal. Materials with high Debye temperature are therefore preferred for qubit operation at elevated temperatures, especially for quantum sensing. Phonons can also impact the optical structure, broadening lines, and reducing radiative efficiency. This is important when narrow wavelength, coherent photons are desired to mediate quantum entanglement between spins at a distance. Here, the common solution is simply to operate in appropriate cryogenic environments, where consideration needs to be made for the scalability of the necessary cryostat technology. Other possibilities for reducing this noise are creating a phononic bandgap, engineering the size or dimensionality of the crystal, or tuning the qubit’s susceptibility to phonons via strain engineering.61

Magnetic noise is also substantially influenced by the crystal’s chemistry and formation energies for defects. This noise is induced by the nuclear spins and unpaired electron spins residing in the host crystal, as the magnetic moments of these can fluctuate. These fluctuations cause uncontrolled noise that leads to loss of coherence. The dominant source of intrinsic magnetic field noise in a crystal without defects is often the nuclear spin of isotopes of a material, and their effects can be readily simulated by employing the fully quantum mechanical cluster correlation expansion (CCE) approach,62 aiding in understanding decoherence and accurately predicting coherence times. This technique has even been used to screen more than 12,000 host materials for spin qubits in terms of coherence, breaking down contributions element-by-element, thus creating a new periodic table for spin coherence.63 For systems with magnetically tunable optical lines, such noise can also impact the coherence of the emitted photons. Growth processes often influence the occurrence of unpaired electron spins; for example, in diamond doped with NV’s, there is commonly a substantial presence of substitutional nitrogen that  provides a spin bath decohering the NV.

In terms of engineering the host crystal to suppress magnetic noise, one can change the dimensionality using 2D systems,64 or isotopically purify the material.65 On the other hand, static and dynamic Hamiltonian engineering is extremely powerful for increasing coherence. For static protection, one needs a coupling between two subsystems (e.g., electrons and nuclear spins). This coupling hybridizes levels and causes anticrossing, which can be, to first order, insensitive to perturbations. For example, by tuning the magnetic field of certain hyperfine coupled systems, the qubit can be operated at a “clock” transition with a zero-first-order susceptibility to magnetic noise, and therefore increased coherence. For dynamic protection, one varies the number and type of quantum control pulses and drives, which is more universal and easy to implement in well-controlled systems. Such decoupling has been used to create the longest electron spin coherence in a solid with thousands of pulses.66 In addition, if two-level systems are driven strongly, new dressed states are created, which are protected from noise. These two techniques can also be combined, creating a universally protected, highly coherent spin subspace.67 From a materials science perspective, the coherence of these qubits can also be used as a probe of the nanoscale dynamics of spin, charge, and sound. In addition, using the previously mentioned techniques to extend coherences from elimination of one source of noise can often reveal new, underlying physical mechanisms in the material. Finally, the stated opportunities for coherence protection arise from exact details of crystal and defect site symmetry, hyperfine coupling, and the ability to grow custom materials with variable spin density and dimensionality, for example.

For magnetic noise at interfaces, proximity is key. At about 20 nm or closer, spin coherence becomes dominated by surface effects.120 to probing magnetotactic bacteria121 and cellular organization in embryogenesis.122 These applications rely on the high sensitivity, enabled by long spin coherence times, and the ability to position the NV probe within the nanoscale vicinity of a desired target.

The sensitivity of these quantum sensors scales with the square root of the qubit coherence time.123 For nanoscale sensing, the qubit sensor usually needs to be brought into close proximity with the sensing target, which means that dephasing due to surface noise becomes an important factor. Hence, understanding and engineering electric and magnetic noise sources associated with the diamond surface has become a central factor in extending the coherence of diamond qubit sensors and, with that, the range of potential applications.

Two orthogonal, but complementary, strategies have been successfully explored to drastically increase coherence for near-surface NV centers. The first approach relies on material engineering to create surfaces with a reduced density of charges traps and spins. This can be achieved by controlling the surface termination through carefully designed annealing steps and limit exposure to crystallographic damage during processing.116,124 Alternatively, surface defect occupation and their dynamics can also be controlled by engineering dielectric constants of the surrounding materials125 and/or through the creation of heterostructures that lead to a depletion of surface defects.126 These approaches have shown to substantially increase qubit coherence by as much as a factor of 100. The second approach relies on dynamical decoupling to reduce the adverse effects of surface noise on the spin qubit. Specifically, many sources of surface noise are highly non-Markovian, enabling an efficient use of dynamical decoupling. Using a Carr–Purcell–Meiboom–Gill sequence with up to 1000 microwave π pulses, coherence times for near-surface NV centers can be extended from a few microseconds to values close to 100 μs.125 These dynamical decoupling techniques enable the reconstruction of the noise power spectrum experienced by the NV center.127 Connecting these experimentally observed noise power spectra with established materials characterization techniques, such as x-ray photoelectron spectroscopy (XPS), near-edge x-ray absorption fine structure (NEXAFS), and ultraviolet photoelectron spectroscopy (UPS), could enable a deeper understanding of the microscopic mechanisms responsible for NV dephasing at interfaces.

The second speaker in this session, A. Laucht, reported on microwave-induced electron spin resonance frequency shifts in silicon spin qubit devices. Spin qubits in silicon have shown great promise as the platform to host a quantum processor, as evidenced by their high (>99%) 1- and 2-qubit gate fidelities.1,2,3,128,129,130 In order to ensure the viability of this platform when scaling to greater numbers of qubits, it is important to fully understand and characterize the noise sources influencing these devices at the microscopic level.

One of the most prevalent noise sources in silicon spin qubit devices is magnetic noise, caused by the flip-flop** of nonzero spin isotopes of silicon, such as \(^{29}\)Si, that are present in the vicinity of the qubit. This source of noise has been significantly reduced in recent years with the advancement of isotopic enrichment, which has resulted in the quantity of \(^{29}\)Si present in qubit devices being reduced from approximately 47,000 ppm, down to only 50 ppm. At such low \(^{29}\)Si concentrations, the qubit no longer experiences a net fluctuating magnetic field from the surrounding nuclei, but rather is strongly hyperfine-coupled to only a few individual \(^{29}\)Si nuclei in the qubit’s vicinity.131,132 This has two benefits: first, the strong coupling of these \(^{29}\)Si nuclei allow for their accurate characterization and, hence, the ability to initialize the nuclei into some desired state, or even use them as a qubit in their own right133 or as quantum memory.134 The second benefit is that the strong hyperfine coupling between the qubit and the individual \(^{29}\)Si has the effect of detuning the \(^{29}\)Si nuclei from one another in energy. With a sufficient degree of detuning, such that the detuning is greater than the coupling between the nuclei, the flip-flop** between the nuclei is suppressed, resulting in the elimination of magnetic fluctuations experienced by the qubit. This is an effect known as nuclear freezing.135 However, if this hyperfine coupling is removed, for example, through the physical removal of the electron during readout, then the spin bath is “unfrozen” and fluctuations of the strongly coupled \(^{29}\)Si nuclei can manifest as non-Markovian noise experienced by the qubit.

Figure 5
figure 5

Excitation of two-level fluctuators via microwave drive. Two-level fluctuators in the gate oxide get excited via the same microwave drive that is used to control spin qubits in silicon.136 This can lead to a shift in the wave function of the electron and consequently to a change in its g-factor and resonance frequency, resulting in contextual noise during the measurement. This effect has been observed in \(^{31}\)P donor spin qubits, as well as in SiMOS and SiGe quantum dot spin qubits.

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There is, however, another noise source observed in these devices, which is contextual in nature (see Figure 5); for example, it depends on how the measurement is performed: Noise originates from a shift in the resonance frequency of the spin, induced by the application of the oscillating pulse used to control the qubit, either at radiofrequency or microwave frequency. This pulse-induced resonance shift has been independently observed in a number of devices from groups around the world.4,134,137,138,139,140 Although the exact characteristics of this effect differ depending on the exact nature of the device used, there appear to be unifying features observed across the different silicon platforms such as donors and quantum dots in Si/SiGe and SiMOS:

  1. 1.

    The resonance frequency shift occurs over some nonzero time scale after the application of a control pulse and saturates within a few microseconds. The resonance frequency returns to its original value over hundreds of microseconds, after the pulse has ceased. The finite time scale of this resonance shift differentiates this effect from AC Stark shift, which is instead an instantaneous shift in frequency during the application of an off-resonant pulse.

  2. 2.

    The magnitude of the resonance frequency shift depends on the power and duration of the pulse and does not require the pulse to be in resonance with the qubit.

  3. 3.

    The resonance shift appears to be affecting the Zeeman term in the qubit Hamiltonian, with some evidence to suggest that the effect is caused by a change in the gyromagnetic ratio of the spin.

  4. 4.

    This effect leads to a reduction in the coherence of the qubits and can even be the dominant source of infidelity in gate operations.4

The time scale over which this resonance shift occurs, along with the dependence on the power and duration of the applied pulse, potentially indicate that this is a temperature-related effect, caused by heating of the device with the application of a control pulse.139 However, further investigation by the team at UNSW revealed a nontrivial dependence of the magnitude of the pulse-induced frequency shift on device temperature and applied magnetic field, indicating that a more complex phenomenon than a direct temperature dependence could be at play.

Although a microscopic understanding of this effect is still lacking, there are two prevalent theories regarding its origin. The first theory is that the oscillating drive results in local heating of the device, which in turn, leads to a variation in strain that affects the conduction-band edge.139 The second theory is that the drive instead leads to the excitation of charge traps and two-level fluctuators in the gate oxide, which in turn, leads to a variation in the electrostatic environment. Both of these effects ultimately result in the deformation of the electronic wave function and, hence, lead to a change in the gyromagnetic ratio through spin–orbit coupling.141,60 From a theoretical point of view, the ability to quantitatively predict and calculate such complex, application-specific properties, including room-temperature, spin-dependent optical contrast, has only recently become possible. The two speakers in this session described the current state of the art and outstanding challenges for this goal.

The first speaker in this session, A. Gali, focused on the physics of defect qubits, in particular, on the role of intersystem crossing (ISC) as a critical phenomenon and a topic of continued interest in research. The full optical spin-polarization process in an NV center comprises an ISC mechanism, characterized by nonradiative transitions between states of different spin multiplicity (i.e., between the singlet states and triplet states [see Figure 6]).

Figure 6
figure 6

Schematic diagram of the optical spin polarization of a nitrogen-vacancy center in diamond. (a) Structure of nitrogen-vacancy defect in diamond. (b) The schematic level diagram is typical at elevated temperatures for the \({}^3E\) excited state. The \(m_S=\pm 1\) states could split by either strain, electric fields, or constant external magnetic fields. Optical spin polarization starts with spin-selective intersystem crossing (brown curved arrows) from the 3E triplet excited state toward the \({}^1A_1\) singlet excited state upon illumination. The electron decays to the \({}^1E\) singlet shelving state where the dominant relaxation path is toward the \(m_S=0\) state of the \({}^3A_2\), so that level will be predominantly populated after several optical cycles as depicted by the large circle. The dominant spin-selective route (green arrow) between \({}^1E\) and \({}^3A_2\) is caused by strong electron–phonon coupling between \({}^1A_1\) and 1E.

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The ISC mechanism between the singlet states and the ground-state triplet state in diamond NV centers can be understood by strong electron–phonon coupling between the singlet states, which is a combination of pseudo Jahn–Teller and damped dynamic Jahn–Teller effect.156 Moreover, the temperature dependence of the ISC rate toward the ground-state triplet (i.e., the lifetime of the shelving singlet state) can be understood using a Herzberg–Teller type of ISC mechanism, where the vibronic picture enables to link the shelving singlet state to the triplet ground state.156 The vibronic or polaronic states also play an important role in the ISC process between the excited state triplet and the singlet states.157 The strong electron–phonon coupling is reflected in the temperature dependence of the fluorescence spectrum too, where the fine structure disappears at T  >20 K.157,158

These results highlight the importance of electron–phonon coupling for the properties of defect qubits that frequently occurs due to degenerate electronic states. As a consequence, ab initio modeling of defect qubits often requires to go beyond the Born–Oppenheimer approximation.159 A very special case is the G-center in silicon,160 a single-photon source161,162 with optically read electron spin,163 where the fine structure of the excited state in the GHz energy region can be understood as the fast rotational motion of the silicon interstitial in the defect.164 This defect has low symmetry but the electron–phonon coupling is still an essential feature in understanding its fluorescence.

In particular, electron–phonon coupling is responsible for the temperature dependence of the longitudinal spin-relaxation time (\(T_1\) time). Until recently, this issue could be discussed at phenomenological level (e.g., Reference 59). Due to the advance of the accurate description of defect spins from ab initio simulations, the temperature-dependent \(T_1\) time has been recently studied for the diamond NV center at a microscopic level.

The spin-relaxation rate (\(\Gamma = 1/T_1\)) could be expressed as

$$\begin{aligned} \Gamma = \Gamma _{0}+\Gamma _{1}^{(1)}(T)+\Gamma _{1}^{(2)}(T)+\Gamma _{2}^{(1)}(T)+\dots \text {,} \end{aligned}$$
(1)

where the superscript refers to the order of the spin-phonon interaction (i.e., terms with superscript 1 or 2 are linear or quadratic in the atom displacements, respectively) and the subscript refers to the order in perturbation theory. \(\Gamma _0\) is a sample-dependent constant term arising from spin–spin interactions that is not discussed further. \(\Gamma _1^{(1)}\) describes the absorption or emission of a single resonant phonon by the spin. Because the zero-field splitting energy of the NV ground-state triplet is small in comparison to typical phonon energies in diamond, this process is only relevant at sub-Kelvin temperatures.165 \(\Gamma _{2}^{(1)}\) corresponds to the Raman scattering of low-energy acoustic phonons via first-order interactions. However, it has been recently shown166 that the first-order spin–phonon interactions provide only negligible contributions to Raman scattering for the NV center in diamond. The major effect comes from \(\Gamma _{1}^{(2)}(T)\) (i.e., the quadratic displacements of ions). This has been confirmed by another study,167 which also applies to the negatively charged boron vacancy in hBN. Ab initio theory should be applied to understand the strongly anisotropic spin-relaxation time for the neutral silicon-vacancy (SiV\(^{0}\)) centers in diamond,152 where it is still not clear whether the metastable singlet state or quasilocal vibration modes play a crucial role in the process.

SiV\(^{0}\) is also mentioned from another important perspective. By combining experiments and ab initio calculations, it was proven that optically detected magnetic resonance (ODMR) of the spin triplet ground state could occur via bound exciton states.168 Bound exciton states associated with the effective mass state with hydrogenic excitation series may be not so commonly known in the quantum optics community coming from atomic physics. These excited states often occur in small bandgap semiconductors, but can be present also in wide bandgap materials such as diamond or 4 H-SiC. From an ab initio simulation point of view, the description of the bound exciton states is a great challenge because of the spatial extension of the respective wave function building up the bound exciton states169 that may be treated using a size scaling method with extrapolation to the dilute limit.168

It is noted that there is great progress on ab initio simulations of the dynamics of the defect spins in realistic spin bath environments,169 which can provide spin dephasing and spin coherence times. The spin dynamics of the diamond NV center,170,171,172 SiC defects,73,170 and boron vacancy in hBN64,173,174,175,176 have been thoroughly described via coupled cluster expansion methods.177,178,179 As a next step, the spin–spin interaction associated with realizing qubit gate operations may be studied in the presence of nuclear or electron spin baths to find optimal quantum optics protocols and to guide materials engineering.

The second speaker in this session, Y. **, presented first-principles calculations of the photophysics of quantum defects, in particular in 2D materials. Point defects in 2D materials are prominent candidates for quantum technologies.180 There are several practical advantages, for example, allowing easier integration into smaller solid-state devices because of their atomically thin thickness, possibility for applying large strain, easier manipulation of defects through scanning techniques, and tunability through varying substrates. In particular, point defects in hBN have shown promising room-temperature single-photon emission181 and optical contrast in ODMR,182,183 which points to its potential in single-photon emitters (SPEs) and spin qubits. Defects in transition-metal dichalcogenides (TMDs) are also considered promising candidates, given their much lower nuclei spin concentration than hBN (potentially longer nuclei spin coherence time), although their bandgaps are much smaller, which could lead to coupling between defect and bulk states of the host materials.

Although numerous candidates in 2D materials have been suggested as single-photon emitters (SPEs) or spin qubits, the most known experimentally confirmed spin defect in hBN is the negatively charged boron vacancy (V\(_B^-\)),175,184 which does not show superior optical properties for quantum applications. For example, the zero-phonon line (ZPL) in the photoluminescence (PL) spectrum is not prominent, accompanied by broad phonon sidebands, in sharp contrast to the dominant ZPL peaks of SPEs in hBN. This indicates that the photoluminescence (PL) lifetime is largely determined by nonradiative processes, in another word, it is not a good SPE intrinsically. The other potentially promising candidates such as carbon-based defects183,185 have not been resolved with the exact chemical composition. Therefore, there is still critical demand for identifying promising SPEs and spin qubit candidates in 2D materials. Important criteria for designing new spin qubits include a stable high spin state, as unpaired electrons are necessary to use the spin as a qubit, sizable zero-field splitting, which is critical to isolate the m\(_s\) ± 1 and 0 sublevels even at B = 0 T. Furthermore, bright optical transitions allow for optical preparation and readout of spin qubits, and possible single-photon emission. Finally, accessible intersystem crossing (ISC) is critical for pure spin state initialization and readout as shown in Figure 7a.

Anisotropic and weak dielectric screening in 2D materials leads to large many-body effects such as electron–hole interactions that impose complexity and outstanding challenges in ab initio 2D defect calculations.186,187 ** discussed recent developments in her group of calculating important physical parameters for excited states of charged defects in 2D materials, using first-principles many-body perturbation theory.186 In particular, the charged defect energy levels are calculated using the GW approximation for accurate electron correlation with a proper 2D charged defect correction that avoids spurious charge interactions.188,189 The radiative lifetime, computed by solving the Bethe–Salpeter equation, includes excitonic effects, which is critical for 2D materials.190 Nonradiative lifetimes include explicit electron–phonon coupling at 2D defects through the effective phonon approximation,191 which has shown success for nonradiative rates of 3D semiconductors.192 Combining radiative and nonradiative lifetime provides the photoluminescence (PL) lifetime, which can be compared to experimental PL measurements directly. Finally, the ISC rate includes electron–phonon coupling and spin–orbit coupling computed from time-dependent DFT.193 With all the information of energy levels and kinetic rates computed above, the multistates diagram can be constructed and spin-dependent PL contrast (ODMR contrast) can be computed. Specifically, by solving a kinetic master equation with computed rates as inputs, under pump light and external magnetic/microwave fields, the excited state occupation of different spin states can be determined, which gives ODMR contrast as a key parameter for optical readout fidelity. Significant challenges remain for intersystem crossing rate calculations, where strong electron correlation of excited state, (dynamical) Jahn–Teller distortion, and spin–orbit coupling need to be calculated accurately simultaneously, which is extremely difficult computationally. A combination of group theory analysis, DFT, and high-level quantum chemistry method may be necessary to solve the problem satisfactorily.

** then discussed an example of defect identification: the carbon dimer defect in hBN,194 where PL lineshape, PL lifetime considering both radiative and nonradiative processes, Huang–Rhys factor, and ZPLs from simulations compare well to available experimental data simultaneously. Therefore, this defect is suggested to be a viable 2 eV SPE in hBN observed experimentally. Besides identifying experimental defects from ab initio calculations, ** also illustrated an example of designing new quantum defects from ab initio calculations.193 For example, the Ti substitution of divacancy, Ti\(_{VV}\), in hBN shows fast ISC rate due to large spin–orbit coupling and electron–phonon coupling. Mo\(_{VV}\) shows reasonably fast ISC and shorter radiative lifetime than Ti\(_{VV}\), which could be promising as optically addressable spin qubit (as shown in Figure 7b). Other examples are optically addressable point defects such as Si\(_{VV}\), which show promising light-emission properties for single-photon emission.

Figure 7
figure 7

Screening of spin defects in semiconducting solids. (a) Schematic of screening criteria and a workflow, where we first search for defects with stable triplet ground state, followed by a sizable zero-field splitting (ZFS), then “bright” optical transitions between defect states required for single-photon emitters (SPEs) or qubit operation by photon, and at the end large intersystem crossing (ISC) rate critical for pure spin state initialization and readout. At the bottom, two candidates of Ti\(_{VV}\) and Mo\(_{VV}\) in hBN were found to be promising based on such criteria. (b) Multiplet structure and related radiative and nonradiative recombination rates of the Ti\(_{VV}\) defect in hBN as the candidate obtained from computational screening, computed at T = 10 K. The radiative process is shown in red with zero-phonon line (ZPL) and radiative lifetime (\(\tau _R\)); the ground-state nonradiative recombination (\(\tau _{NR}\)) is shown as a dashed line in dark blue; and finally, ISC to the singlet state from the triplet excited state is shown in light blue. The zero-field splitting (D) is denoted by the orange line. For the Ti\(_{VV}\) defect, the pseudo-Jahn–Teller (PJT) process is shown with a solid line in dark blue.193

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Spin decoherence, however, is a major challenge in quantum information materials. At very low temperature and large B field, decoherence mainly arises from fluctuating magnetic fields due to nuclear spin flip-flop transitions. At finite temperatures, other effects such as phonons, impurities, and electron–electron interactions through spin–orbit coupling can become dominant. Spin–orbit mediated spin-phonon relaxation in spin qubits can also result in short coherence time. Over the last few years, the ** group has developed methodology based on ab initio density-matrix dynamics,195,196 with quantum electron–phonon, electron–electron, and electron-impurity scattering with self-consistent spin–orbit coupling. ** showed the predicted spin relaxation (\(T_1\)) and ensemble dephasing times (\(T_2\)*) of conducting electrons or holes in disparate solids in excellent agreement with experiments, regardless of dimensionality and symmetry.195,196,197,198

Regarding open questions for spin relaxation \(T_1\) and coherence time \(T_2\) calculations of spin defects in solids, one of the remaining challenges is the numerical convergence of the electron–phonon calculation for defect supercells, in particular for couplings with low energy or delocalized phonons in spin defect \(T_1\) calculations. The required supercell sizes for converging electron–phonon coupling and energy conservation at meV scale (spin splitting) can be enormous. ** suggests that novel methods such as the embedding method of electron–phonon matrix elements could be a viable pathway to explore. Also, a fully first-principles approach to study \({T_1}\) for spin qubits under macroscopic confinement potentials in large-size nanostructures (e.g., tens of nanometer-size quantum dots) is unavailable. This is significantly beyond the computational feasibility of ab initio calculations. ** suggests that combining model Hamiltonians for nanostructures with ab initio fitted parameters could be a promising pathway for spin \(T_1\) in quantum dots, which realizes an ideal balance between computational cost and accuracy.

High nuclei spin concentration in hBN could be a major source for its spin decoherence at low temperature, possibly posing a fundamental limitation for hBN in quantum applications. Recent work has proposed to control and utilize nuclei spin in hBN through optically detected nuclear magnetic resonance (ODNMR) assisted by point defects, which opens a new frontier with nuclear spins in van der Waals materials for quantum information science.173 Furthermore, recent progress has shown significantly elongated spin coherence time for V\(_B^-\) in hBN through dynamical decoupling techniques. They suppress magnetic noise and extend the spin coherence time by nearly two orders of magnitude, approaching the fundamental \(T_1\) relaxation limit.199

Summary

M. Friesen pointed to the need for improved atomic-level growth control and simulation to allow for reliable qubit scaleup. C. Déprez highlighted the effect of hysteretic changes in the potential landscape on qubit stability and the need to further explore the origin of these hysterectic effects and to develop control mechanisms. C. Anderson highlighted the role of acoustic, magnetic, and electric noise sources and the particular influence of the proximity of interfaces on the stability of defect-based qubits. He also emphasized the need to have detailed knowledge of the structure and the Hamiltonian of the defect to be able to predict the influence of the various noise sources on the qubit operation. T. Zhong highlighted the fact that rare earth ions in solids possess properties, including long coherence times and narrow emission widths that make them leading candidates for quantum memories. However, he also explained that there are only a few demonstrations thus far of individual ions being operated as spin qubits and that considerable development is required in this area. P. Maurer highlighted the information that can be obtained about the influence of the environment on a qubit through operation of the qubit as a sensor and that this can be used to work out better ways to isolate the qubit from its environment. A. Laucht highlighted the role of microwave-field-induced electron spin resonance frequency shifts on the performance of Si qubit devices and the need to develop suitable control mechanisms or to operate the qubits in a different mode, for example, shifting from spin states as the computational basis states to driven/dressed states with always-on control fields. A. Gali and Y. ** both highlighted the enormous progress that has been made in the theoretical modeling of defect-based solid-state qubit candidates in recent years including the dynamics of the systems in realistic environments and that next steps involve modeling of spin–spin interactions during qubit gate operations and the need for greater focus on modeling of sources of decoherence.

Notably, specific materials suggestions will be application specific. When using qubits for sensing at room temperature, the phonon sideband can be used for readout and coherent emission at a fixed wave length is not important. The readout contrast, spin-relaxation time, and coherence time are the most important parameters. Room-temperature operation likely restricts use of host materials with constituting elements in the second row of the periodic table (such as diamond, silicon carbide, and boron nitride) because the spin-relaxation time exponentially decays with the given effective phonon frequency occupied at the given temperature. High enough phonon frequency requires light elements in the crystal. For quantum communication, the coherent emission in ZPL should be strong, stable, and identical, independent from the environment. Quantum computation requires additional conditions such as well-controlled gate operations between qubits (that could pose stringent conditions for the positioning of the qubits in the solid) and high scalability.

A concerted materials science focus and push is needed in order for quantum computing, sensing, and communications to advance to the point that it offers true advantage. This would include both engineering means of identifying and tuning to sweet spots and designing Hamiltonians and fundamental materials innovations in growth and characterization to exploit these sweet spots. Only by combining both of these strategies which tune, understand, grow, and optimize quantum materials will ideal performance be unlocked through exquisitely engineered materials and interfaces that provide access to a broad range of available operating parameters. This is increasingly recognized by funding agencies, as evidenced by materials science focused efforts, for example, by the AFOSR. The contributions to this workshop identify some of the key areas of focus needed to make real progress for the particular qubits systems that were examined here. The Quantum Staging Group will continue to promote activities across a range of qubit platforms that bring practitioners in the field together to focus on the questions that need to be answered. The authors of this article aspire to the goal of providing insights to the materials community to solve underlying problems and find guidance in this work toward new collaborations and collaborators.