Coupling single-molecule magnets to quantum circuits

The magnetic configuration of a SMM is mainly determined by exchange couplings between the ions that form its magnetic core and by their interactions with the crystal field. The former give rise to multiplets with well-defined spin values, while the latter generate a magnetic anisotropy, thus also a zero-field splitting within each multiplet. Here, we consider only the ground state multiplet S and neglect its quantum mixing with excited multiplets. This approximation, widely used to describe the physics of SMMs, is known as the 'giant spin approximation'. The effective spin Hamiltonian of a SMM reads then as follows:

$$\begin{equation} {\cal H}_{\rm s} = \sum_{k,l} B_{k}^{l}O_{k}^{l}-g_{S}\mu_{\rm B} \left( B_{X}S_{X} + B_{Y}S_{Y} + B_{Z}S_{Z} \right), \end{equation} \tag{ 1 }$$

where O^l_k are Stevens effective spin operators [26], B^l_k are the corresponding magnetic anisotropy parameters, g_S is the gyromagnetic ratio and B_X, B_Y and B_Z are the components of an external magnetic field along the molecular axes X, Y and Z. The molecular symmetry and structure determine which anisotropy parameters are non-zero as well as their relative intensities.

One of the simplest situations corresponds to a spin with Ising-like second-order anisotropy, which corresponds to B⁰₂ < 0 and all other terms being zero, i.e. to a spin Hamiltonian

$$\begin{equation} {\cal H}_{\rm s} = B_{2}^{0} \left[ 3S_{Z}^{2} - S(S+1) \right] - g_{S}\mu_{\rm B} \left( H_{X}S_{X} + H_{Y}S_{Y} + H_{Z}S_{Z} \right). \end{equation} \tag{ 2 }$$

As figure 1 shows, such a 'diagonal' anisotropy splits the S multiplet into a series of doublets, associated with eigenstates | ± m〉 of S_Z. As a function m, the energy shows then a characteristic double-well potential landscape. Off-diagonal anisotropy terms (i.e. those having l ≠ 0 in equation (1)) induce quantum tunnelling, across the magnetic anisotropy barrier, between states | + m〉 and | − m〉 and remove their initial degeneracy by a quantum tunnel splitting Δ_m(0). The degeneracy can also be removed by the application of an external magnetic field $\vec {H}$. Energy splittings can be tuned, to some extent, by varying the intensity and orientation of $\vec {H}$ (see figures 2–4). In particular, close to B_Z = 0, the splitting between the first excited and ground states $\hbar \omega _{12} \simeq \sqrt {[\Delta _{S}(\vec {B})]^{2} + \xi _{S}^{2} }$, where $\Delta _{S}(\vec {B})$ is the ground doublet field-dependent quantum tunnel splitting and ξ_S = 2g_Sμ_BB_ZS is the magnetic bias. The magnetic field enables also the initialization of the SMM state. For S = 10, and at T = 0.1 K, the thermal population of the ground state becomes ≳99.99% for μ₀H_Z ≳ 34 mT.

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It is worth mentioning here that equation (1) applies also to, e.g. NV-centres in diamond, which have S = 1 and a zero-field splitting determined by second-order anisotropy terms with B⁰₂ ≃ 2.88 GHz (0.144 K) and $B_{2}^{2}/B_{2}^{0} \lesssim 3.5 \times 10^{-3}$. Therefore, the theoretical framework that follows will enable us to compare both situations.

The coupling between a spin, described by the Hamiltonian ${\cal H}_{\mathrm { s}}$, and a superconducting quantum circuit, described by ${\cal H}_{\mathrm { q}}$, is governed by the Zeeman interaction

$$\begin{equation} {\cal H} = {\cal H}_{\rm q} + {\cal H}_{\rm s} - ( \vec{W}^{{\rm (q)}} V_{\rm q})\vec{S}, \end{equation} \tag{ 3 }$$

where $\vec {W}^{\mathrm { (q)}} = g_{S}\mu _{\mathrm { B}}\vec {B}^{\mathrm { (q)}}$ is proportional to the magnetic field $\vec {B}^{\mathrm { (q)}}$ generated by the superconducting circuit at the spin position and V_q is an operator acting on the circuit's variables.

For the present purposes, the spin can be treated as a two-level system. This is possible by focusing only on those two spin levels whose energy difference is in (near) resonance with the circuit's transition frequency ℏω. More specifically, we choose the spin ground state |G〉 and one excited state |E〉. Two possible choices, relevant to real SMMs, are shown in figure 1. We define the spin transition frequency

$$\begin{equation} \hbar\omega_{{G,E}} \equiv \langle E|{\cal H}_{\rm s}|E \rangle - \langle G|{\cal H}_{\rm s}|G\rangle \end{equation} \tag{ 4 }$$

and the transition matrix element

$$\begin{eqnarray} \hbar g &\equiv& \langle G|\vec{W}^{(\mathrm{q})}\vec{S} |E \rangle \nonumber\\ &=& W^{(\mathrm{q})}_{X} \langle G|S_{X} |E \rangle + W^{(\mathrm{q})}_{Y} \langle G|S_{Y} |E \rangle+ W^{(\mathrm{q})}_{Z} \langle G|S_{Z} |E \rangle. \end{eqnarray} \tag{ 5 }$$

Achieving strong coupling requires that the SMMs can be tuned to resonance with the circuit, i.e. that ℏω_G,E ≃ ℏω for a given |E〉, and that the relevant matrix element of the Zeeman interaction is sufficiently large. In the remainder of this section, we discuss how matrix elements 〈G|S_I|E〉, with I = X,Y,Z, thus also g, depend on the choice of state |E〉 as well as on the magnetic anisotropies and experimental conditions that can be met with real SMMs. The actual coupling g depends also on the magnetic field generated by a given circuit, thus on its design and geometry. These aspects will be considered in sections 3 and 4 below.

2.3.1. Transitions between zero-field split levels

In this and the next subsections, we consider a generic S = 10 SMM with ${\cal H}_{\mathrm { s}}$ described by equation (1) and second-order anisotropy terms only. This situation applies to some of the best known SMMs, such as Fe₈, shown in the inset of figure 1, or even Mn₁₂. A first choice, reminiscent of the situation met with NV-centres in diamond, is to identify |E〉 with state |3〉, as shown in figure 1. Neglecting B²₂, which plays a minor role here unless it is comparable to B⁰₂ and B_Z ≃ 0 (see figure 2), this situation corresponds to |G〉 ≃ | + S〉 and |E〉 ≃ | + S − 1〉 for B⁰₂ < 0 and to |G〉 ≃ |0〉 and |E〉 ≃ | + 1〉 for B⁰₂ > 0. The splitting ℏω₁₂ = ℏω₁₂(0) + gμ_BB_Z, where ℏω₁₂(0) = 3(2S − 1)B⁰₂ in the former case and ℏω₁₂(0) = 3B⁰₂ in the latter case. Relevant transition matrix elements correspond to transverse spin components S_X and S_Y and can be calculated analytically

$$\begin{equation} g \propto \frac{1}{2} \sqrt{\left(S-m_{\rm gs}\right)\left( S + m_{\rm gs} + 1 \right)}, \end{equation} \tag{ 6 }$$

where m_gs is the S_z eigenvalue of the ground state. Depending on the sign of the anisotropy, equation (6) gives

$$\begin{eqnarray*} B_{2}^{0} > 0 & \Rightarrow & m_{\rm gs} = 0 \Rightarrow g \propto \frac{1}{2}\sqrt{S (S+1)},\\ B_{2}^{0} < 0 & \Rightarrow & m_{\rm gs} = +S \Rightarrow g \propto \frac{1}{2}\sqrt{2S}. \end{eqnarray*}$$

High couplings are therefore achieved for high-spin S materials, optimally with B⁰₂ > 0. Furthermore, g is but weakly affected by external magnetic fields (see figure 2). A difficulty associated with this choice of basis is that the zero-field splittings of high-spin SMMs, such as Fe₈ or Mn₁₂, are often very large (e.g. ℏω₁₂(0) ≃ 114 GHz for Fe₈) as compared with the typical resonance frequencies of either superconducting resonators [27, 28] (ω/2π ≃ 1–40 GHz) or gap-tunable FQs [29] (for which ω/2π ≃ 1–10 GHz).

2.3.2. Transitions between 'spin-up' and 'spin-down' states: photon induced quantum tunnelling

2.3.3. Single ion magnets versus SMMs

The previous results show that, because of their high spin values, the coupling of SMMs to a rf magnetic field can attain very high values. However, the also high magnetic anisotropy barriers (they tend to increase with S) and correspondingly small quantum tunnel splittings can pose some important technical difficulties: the use of very high frequencies to attain resonant conditions with zero-field split levels (≳ 110 GHz for Fe₈ or ≳220 GHz for Mn₁₂) or the need of applying strong and very accurately aligned magnetic fields if one focuses on transitions within the tunnel split ground state doublet. In addition, achieving a pre-designed control over relevant parameters (spin and magnetic anisotropies) is a difficult task, if feasible at all, with polynuclear clusters.

Mononuclear SMMs (or single ion magnets (SIMs) [30–32]) are, by contrast, much simpler: they consist of just one magnetic ion, often a lanthanide, encapsulated inside a non-magnetic shell of ligand molecules. These materials can be seen as the molecular analogues to diluted lanthanide salts, which are also seen as promising spin qubits [33]. An advantage of molecular SIMs over these materials is that the local coordination of the magnetic ion can be modified by adequately choosing the nature and structure of the ligand shell, thus providing a rich playground for the rational design of their spin Hamiltonian [34].

Some specific examples can help to understand how the problems mentioned above can be overcome with the use of simpler molecules. Here, we discuss some possibilities offered by two families of SIMs based on polyoxometalate complexes (see figure 5). If one seeks to reduce the magnetic anisotropy, thus also the zero-field splitting ℏω₁₃, the use of Gd³⁺ ions is a good option because of their close to spherical electronic configuration. In addition, the sign and intensity of the magnetic anisotropy are determined, to a large extent, by the local coordination [34]. In the elongated GdW₁₀ molecule, for instance, the preferred magnetization axis points along the molecular axis Z, giving rise to a ground state m_gs ± 7/2 separated from the first excited doublet ±5/2 by a small zero-field splitting. This leads to a rather convenient value for the resonance frequency ω₁₃/2π ≃ 20 GHz. For the donught-shaped GdW₃₀ molecule, the anisotropy is even weaker and of opposite sign (i.e. B⁰₂ > 0), thus Z becomes a hard magnetization axis and the easy axis (Y ), lying within the molecular plane, is determined by the presence of strong off-diagonal anisotropy terms. At zero field, the splitting is ℏω₁₃ ≃ 6.4 GHz, thus in tune not only with coplanar resonators but also with gap-tunable FQs. In fact, all spin levels of GdW₃₀ lie within a frequency band of about 20 GHz, accessible to most superconducting circuits.

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For lanthanide ions other than Gd³⁺, the magnetic anisotropy and the ground state depend also on intrinsic electronic structure of the ion itself. A particularly interesting situation is found for TbW₃₀. The combination of second- fourth- and sixth-order diagonal anisotropy terms gives rise to a ground state doublet with m_gs = ± 5 [35]. More importantly, the fivefold molecular symmetry allows the presence of a strong B⁵₆O⁵₆ term, which efficiently mixes these spin states. The result is a two-level spin system characterized by a large zero-field quantum tunnel splitting Δ_S(0) ∼ 60 GHz and therefore a high transition matrix element g. More importantly, g is very robust against the action of external magnetic fields (see figures 3 and 4).

Coplanar resonators are microwave devices that consist of a λ/2 section of a coplanar waveguide (CPW) that is coupled to external feed lines via gap capacitors. A schematic diagram of such a device is shown in figure 6. The fundamental mode resonant frequency is determined by the length of the resonator through the equation $f_0 = \frac {c}{\sqrt {\epsilon _{\mathrm { eff}}}}\frac {1}{2l}$. Here _eff is the effective dielectric constant of the CPW and depends on the waveguide geometry and the dielectric constants of the surrounding media [36]. As with transmission lines, the electromagnetic mode is described as a voltage and current wave where the current in the centre line is equal and opposite to the current in the ground plates.

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Making the resonator out of superconducting materials, such as Nb or NbTi, and using low loss dielectric substrates, such as sapphire, helps to reduce the losses in the system and allows the reduction of the resonator cross section down to the micrometer level while maintaining quality factors of up to 10⁵–10⁶ [37–39].

For a resonator, equation (3) takes the following form [40]:

$$\begin{eqnarray} \nonumber {\cal H} & = & {\cal H}_{\rm s} + {\cal H}_{\rm res} - \vec{\mu}\cdot \vec{B}^{\rm (q)} \nonumber\\ & = & {\cal H}_{\rm s} +\hbar a^\dagger a +\hbar g(\vec r_j) \sigma_x (a -a^\dagger), \end{eqnarray} \tag{ 7 }$$

where we have projected onto the basis formed by the two relevant SMM states |G〉 and |E〉, σ_x is the Pauli matrix acting on this basis and the coupling strength

$$\begin{equation} g(\vec r_j) = g_S \mu_{\rm B}| \langle {\rm G}| \vec{b}_{\rm rms} (\vec r_j)\vec{S} | {\rm E}\rangle| \end{equation} \tag{ 8 }$$

with $\vec {b}_{\mathrm { rms}} (\vec r_j)$ the root mean square value of the field $\vec {b}$ generated by the vacuum current (see below). The position $\vec r_j$ matches the spin location. Through this section we will assume that the magnetic sample is centred at the maximum magnetic field strength generated by the resonator, i.e. at the midpoint of the resonator.

The coupling per spin is usually small (of the order of a few 100 Hz, see below) and losses can easily overcome the coherent coupling. Therefore, we will also consider the coupling to an ensemble, e.g. a crystal, of N SMMs. For this, we sum (8) over each spin at position $\vec r_{\mathrm { j}}$. It is convenient to introduce the collective spin operator

$$\begin{equation} b^{\dagger}=\frac{1}{\sqrt{N}\bar{g}}\sum_{j}^{N}g_{j}^{*}\sigma_{j}^{+}, \end{equation} \tag{ 9 }$$

where $\bar {g}$ is the average coupling, defined as $\bar {g}^{2} \equiv \sum _{j} \left | g_{j} \right |^{2}/N$. In the low polarization level $\langle \sum \sigma _j^\dagger \sigma ^-_j \rangle {\ll } N$ these operators approximately fulfil bosonic commutation relations, [b,b^†] ≈ 1 [41]. Equation (7) then becomes approximately equal to the Hamiltonian of two coupled resonators

$$\begin{equation} {\cal H} = {\cal H}_{\rm q} + {\cal H}_{\rm s} - \hbar g_{N} (b^{\dagger} + b) (a^\dagger + a) \end{equation} \tag{ 10 }$$

with an effective coupling given by

$$\begin{equation} g = g_s\frac{\mu_{\rm B}}{h}\sqrt{n\int_V | \langle G|\vec{b}_{\rm rms}\cdot\vec{S}|E\rangle|^2\mathrm{d}V}, \end{equation} \tag{ 11 }$$

where we have replaced the sums by integrals and assumed a uniform density n. Let us emphasize that equation (11) leads to a $\sqrt {N}$ enhancement of the effective coupling with respect to that of a single spin.

In order to calculate the collective coupling of SMMs to a single photon, one needs to evaluate the magnetic field generated by the rms of the vacuum current fluctuations, I_rms. This current can be found considering that the zero point energy of the resonator is shared equally between the electric and magnetic fields

$$\begin{equation} \frac{\hbar \omega}{4} = \frac{1}{2}LI^{2}_{\rm rms} \quad \Rightarrow \quad I_{\rm rms}=\omega \sqrt{\frac{\hbar\pi}{4Z_0}}, \end{equation} \tag{ 12 }$$

where L = 2Z₀/(πω) is the lumped inductance of the resonator [42] and Z₀ is the characteristic impedance of the transmission line segment that forms the resonator. Taking a standard value of Z₀ ≃ 50 Ω, we find that the vacuum current fluctuations are of about 8 nA GHz⁻¹. For $\omega /2 \pi \lesssim 37\,{\mathrm { GHz}}$, the current in a resonator is therefore somewhat smaller than the current I_p ∼ 0.3 μA circulating via a FQ close to its compensation point, so we can then expect smaller couplings.

We use the Comsol multiphysics ac/dc module to calculate the field distribution given the resonator geometry and currents. We model only a cross section of the resonator so the calculated fields are approximated by those generated by an infinite conductor length. This approximation holds as long as the SMM crystals are placed close to the resonator centre and the crystal length is much shorter than that of the resonator itself, which ranges from 0.5 to 10 mm for the frequencies of relevance here.

Even for dc currents, the current density distribution in a superconductor is not uniform and different from that of a normal conductor. In real superconductors, the superconducting current density decays exponentially with the distance to its surface and the decay constant is the London penetration depth λ_L. For Nb, λ_L ≃ 80 nm at 4 K and increases as temperature increases towards the critical temperature (T_c ≃ 9 K) [43]. Since the thickness of the superconducting lines we consider (see figure 6) is of this order or smaller, we need to simulate the current distribution carefully to take this effect into account. As a first approximation, we use the skin effect of standard conductors to produce the current profiles in the superconducting regions, i.e. we use alternating currents and tune the frequency ω_ac and material parameters (conductivity σ and magnetic permeability μ) in the simulation to make the skin depth of the conductor

$$\begin{equation} \lambda_{\rm skin} = \sqrt{\frac{2}{\sigma\omega_{\rm ac}\,\mu}} \end{equation} \tag{ 13 }$$

equal to λ_L = 80 nm.

Taking all this into account, we simulate the magnetic field distribution for the geometry shown in figure 6. A typical magnetic field distribution is shown in figure 7. As expected, the superconducting current and magnetic field concentrate near the edges of the centre line and the inner edges of the ground planes. Using these magnetic field distributions and the matrix element values calculated for each SMM sample, it is possible to obtain the coupling strength from equation (11) for crystals of varying dimensions. An appealing aspect of many SMMs crystals (including those considered here) is that the magnetic anisotropy axes of all molecules are aligned with respect to each other. This enables orienting them so that the fields from the resonator can induce the desired transitions. Each sample and each choice of computational basis has a different optimal orientation of the magnetic anisotropy axes (X,Y,Z) with respect to the resonator coordinate system (x,y,z). In our simulations, the axis with the largest absolute value of the transition matrix element points along the x-axis of the resonator (i.e. horizontal, see figures 6 and 7) while the second largest is placed along the y-axis (i.e. perpendicular to the resonator). This is because the integral of b²_X entering in equation (11) is slightly larger than the b²_Y integral, thus leading to also slightly larger g_N. We also calculate the collective coupling of NV-centres in diamond crystals. In this case, one has to average over the four different orientations of their magnetic anisotropy axes. In all these calculations, we consider crystals of fixed length and width (both equal to 40 μm, see figure 6) and study how g_N depends on the crystal thickness (thus also the number of spins), from 100 nm to 75 μm.

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The results are shown in figure 8 for several SMMs and different choices of their computational basis |G〉 and |E〉 as well as for NV-centres in diamond. We see that the coupling first increases with crystal thickness and then saturates once the crystal is thicker than about 10–15 μm. This behaviour reflects the decay of $\vec {b}$ with the distance y from the resonator surface. It shows that only a very thin layer of spins significantly contributes to g_N and emphasizes the importance of carefully placing the sample on top of the device. As would be expected, the dependence on the crystal thickness is essentially the same for all samples.

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It can be seen that, because of their specific characteristics, the coupling to SMM crystals can be very large, much larger indeed than the coupling to NV-centres in diamond crystals of equivalent size. The largest couplings g_N ≃ 2–3 GHz are found for transitions between tunnel split states of Tb₃₀ and between states 1 and 3 of Fe₈. Yet, these transitions are characterized by very high resonance frequencies ω₁₂ ∼ 60 GHz and ω₁₃ ≃ 114 GHz, respectively. Very large couplings (g_N ≃ 0.5 GHz) are also found for transitions between tunnel split states of e.g. Fe₈, for which ω₁₂ can be tuned by applying external magnetic fields (see figure 4). However, one then has to deal with rather strong (≳ 2 T) and very accurately aligned (typically within less than 0.5°) magnetic fields (see figure 4). For this reason, it might be experimentally simpler to work with Gd-based SIMs, for which ω₁₃ lies between 6.4 and 20 GHz, and whose transition matrix elements are more robust against the action of external magnetic fields.

The couplings need to be compared with spin decoherence frequencies ∼ 1/T₂, where T₂ is the phase coherence time. Experiments performed on crystals of GdW₁₀ and GdW₃₀ [34], and of Fe₈ [44] show that $T_{2} \lesssim 500\,{\mathrm {ns}}$ at liquid helium temperatures and under the best conditions, thus much shorter than T₂ ∼ 1–2 ms of NV-centres [45] at room temperature. Still, the strong coupling limit g_NT₂/2π ≫ 1 should be relatively easy to achieve for all these molecular materials. Furthermore, for some of the examples given in figure 8, g_N can in fact become a sizeable fraction of the resonator frequency, thus opening the possibility to reach and explore the ultra-strong coupling limit with a spin ensemble.

The simulations described in the previous section enable one to estimate also the coupling to a single SMM at any location with respect to the device. For a molecule placed in between the ground and central lines, we find that g ranges between 100 Hz and a few kHz, depending on the particular sample. Notice, however, that the magnetic field is enhanced, up to a factor 5 or so, in narrow regions close to the edges of these lines (remember figure 7). Two distinctive aspects of SMMs, which are not easily found in other qubit realizations, is that they are sufficiently small, with lateral dimensions of the order of 1 nm, to fit inside these regions and that they can be delivered from a solution with very high spatial accuracy by, e.g. using the tip of an atomic force microscope [46]. The magnetic field generated near the central line edges, thus also the coupling to molecules or molecular ensembles located near them, can be further enhanced by fabricating narrow constrictions. Superconducting circuits with dimensions well below 100 nm can be fabricated, and even repaired, by either etching with a focused ion beam or by using the same ion beam to induce the growth of a superconducting material from a gas precursor [47]. Provided that these constrictions are much shorter than the photon wave length, they are expected to have very little effect on the general resonator characteristics.

In order to explore this possibility, we have repeated the above simulations for varying centre line widths, down to 50 nm, while keeping current constant. We then evaluate the coupling to a single SMM located at the point of maximum field on the surface of the centre line and oriented in such a way as to maximize the transition matrix element. The results are shown in figure 9. We see that reducing the width from 14 μm to 50 nm can lead to enhancements of an order of magnitude in the coupling strength. Again, the dependence on the geometry is the same for all samples. The conclusion is that achieving strong coherent coupling of a single SMM (e.g. TbW₃₀) to such nanoresonators requires that the decoherence time T₂ of an individual molecule grafted to a superconducting device can be made significantly longer than 10 μs. Despite the lack of T₂ data for truly isolated molecules, it seems that such coherence times can be reached under adequate conditions, i.e. for molecules having a very low concentration of nuclear spins [48].

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Before moving to the following section, it is worth mentioning here that the potential applications of superconducting resonators or transmission wave guides that maximize the magnetic coupling to very small spin ensembles, or eventually enable detecting single spins, extend well beyond the quantum information research field. For instance, these designs might contribute to the optimization of on-chip electron paramagnetic resonance spectrometers for the characterization of magnetic materials [49].

FQs are superconducting loops interrupted by, almost always, three junctions [50]. When half of a flux quanta passes through the loop, the two lowest eigenstates of the qubit Hamiltonian are symmetric and antisymmetric superpositions of counter propagating persistent currents. Those states define the qubit states, and will be denoted as $\{|\!\circlearrowleft \rangle ,|\!\circlearrowright \rangle \}$. By changing the flux, the qubit is biased to one of those currents. Furthermore, any influence of higher excited levels can be safely neglected for standard qubit parameters. Therefore for our purposes the FQ can be modelled as a two level system:

$$\begin{equation} H_{\rm FQ} = \frac{\epsilon}{2} \sigma_z+\frac{\Delta}{2} \sigma_x \end{equation} \tag{ 14 }$$

with Δ the qubit gap which lies in the GHz regime and  = 2I_p(ϕ_ext − Φ₀/2) the bias term associated with the external flux ϕ_ext and I_p the persistent current in the loop. Here we have chosen the physical basis where the eigenstates are the clockwise and anticlockwise supercurrents: $\{ |\!\circlearrowleft \rangle ,|\!\circlearrowright \rangle \}$.

FQs provide a platform for hybrid structures because their ability to couple to magnetic moments through the field induced by the supercurrents in the loop. Previous studies have focused on considering the coupling to NV-centres [3], with recent experimental realizations showing promising results [29]. Some applications of these hybrid structures to quantum information processing have been recently pointed out [41, 51, 52].

Within the state of the art of both qubit geometry and parameters the coupling to single spins is too weak to overcome the losses. Therefore, in this section we will discuss the coupling between a FQ and a spin ensemble. A schematic representation of a possible layout is depicted in figure 10.

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Following the same reasoning from the previous section, we can again arrive at a Rabi like model:

$$\begin{equation} H = \frac{\Delta}{2} \sigma_x+\omega b^\dagger b+g \sigma_z (b^\dagger + b), \end{equation} \tag{ 15 }$$

where, for simplicity, we have chosen to be at the degeneracy point  = 0 (cf equation (14)). The spin collective modes, b,b^†, were defined in equation (9) and the coupling in (11). In this case, the magnetic field, $b(\vec r_j)$ is generated by the circulating currents in the qubit (cf equation (11)). The current operator can be written as

$$\begin{equation} I=\sum_{m,n=\circlearrowleft,\circlearrowright}|n\rangle\langle n|I|m\rangle\langle m|=I_{\rm p}|\!\circlearrowleft\rangle\langle\circlearrowleft \!|-I_{\rm p}|\!\circlearrowright\rangle\langle\circlearrowright \!|=I_{\rm p}\sigma_{z}\, \end{equation} \tag{ 16 }$$

that justifies the coupling through σ_z. The magnetic field strength, $b(\vec r)$ entering in the formula for g, equation (11), corresponds to the field generated in a loop with a circulating current I_p.

To estimate this coupling, we again perform numerical simulations in Comsol multiphysics assuming a superconducting loop with current I_p. As in the previous section, we simulate the field at a cross section at the centre of the FQ and choose a crystal size of about half the length of the FQ (see figure 10) to avoid edge effects. We also use the skin effect, as before, to simulate the superconducting current distribution (see equation (13)). An example of the field distributions found is shown in figure 7. We complement our numerical studies with an analytical approach. In order to get a tractable and closed formula for the magnetic field generated, we approximate the qubit by two parallel counter currents, I_p. This yields for the magnetic field

$$\begin{eqnarray} &&\fl \mathbf{b} =\frac{\mu_{0}\, I_{\rm p}}{2\,\pi}\left(\frac{1}{\left(x+{w}/{2}\right)^{2}+y^{2}}\left(\begin{array}{@{}c@{}}-y\\ x+{w}/{2} \\ 0 \end{array}\right)-\frac{1}{\left(x-{w}/{2}\right)^{2}+y^{2}}\left(\begin{array}{@{}c@{}} -y\\ x-{w}/{2}\\ 0 \end{array}\right)\right). \end{eqnarray} \tag{ 17 }$$

Both numerical and analytical estimates for g are shown in figure 12. We plot our results as a function of crystal height and as a function of the vertical separation between the crystal and the FQ (S in figure 10). As with the resonator, we observe a saturation of g beyond a certain height. This can also be understood by looking at the dependence on separation S. The field saturation occurs between 1 and 10 μm, i.e. when the magnetic field becomes negligible. We also observe that our simple analytical estimation closely reproduces the numerical results.

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In the case of FQs we have compared two species: GdW₃₀ and NV-centres, with zero field level splittings of 6.4 and 2.8 GHz, respectively. These transition frequencies lie in the range of available qubit tunnelling gaps. The achievement of strong coupling between NV-centres and a FQ has been recently reported in [29]. From the present results we conclude that, as we had anticipated, spin ensembles tend to couple more strongly to FQs than to resonators, cf figures 8 and 12. Also, as in the case of the resonators, the coupling to SMMs is stronger than that to NV-centres. This point is interesting since the qubit-ensemble coupling can reach up to 10% of the qubit natural frequency. For such a coupling strength, the qubit-spin ensemble system enters the so-called ultrastrong coupling limit [53, 54]. This means that the full model (15) is needed to understand the physics. In the usual case of weaker coupling, one can rotate the qubit basis σ_x → σ_z and write the interaction within the rotating wave approximation g(σ⁺a + σ⁻a^†), with σ_± = σ_x ± iσ_y. The latter approximation allows a perturbative treatment. Therefore, SMMs are candidates to observe analogues of light–matter interaction beyond perturbative treatments. We finish by noting that the FQ parameters used here were taken from the experimental paper in [29]. Further optimization of the parameters and of the FQ shape could yield even stronger couplings.