A Inductance simulations and calculations

Prospective two orders of magnitude enhancement in direct magnetic coupling of a single-atom spin to a circuit resonator

Abstract

We report on the challenges and limitations of direct coupling of the magnetic field from a circuit resonator to an electron spin bound to a donor potential. We propose a device consisting of a lumped-element superconducting resonator and a single donor implanted in enriched Si. The resonator, in contrast to coplanar waveguide resonators, includes a nano-scale spiral inductor to spatially focus the magnetic field from the photons within. The design promises approximately two orders of magnitude increase in the local magnetic field, and thus the spin to photon coupling rate , compared to the estimated coupling rate to coplanar transmission-line resonators. We show that by using niobium (aluminum) as the resonator’s superconductor and a single phosphorous (bismuth) atom as the donor, a coupling rate of =0.24 MHz (0.39 MHz) can be achieved in the small photon limit. For this truly linear cavity quantum electrodynamic system, such enhancement in is sufficient to enter the strong coupling regime.

pacs:

Silicon-based spin qubits, including gate-defined quantum dot Hanson et al. (2007); Loss and DiVincenzo (1998); Petta et al. (2005) and single-atom Zwanenburg et al. (2013); Pla et al. (2012) devices, use the spin degree of freedom to store and process quantum information, and are promising candidates for future quantum electronic circuits. The electronic or nuclear spin is well decoupled from the noisy environment, resulting in extremely long coherence times Saeedi et al. (2013); Tyryshkin et al. (2012); Steger et al. (2012) desirable for fault-tolerant quantum computing. Single-atom spin qubits offer additional advantages over quantum dot qubits such as longer coherence due to strong confinement potentials, and are expected to have better reproducibility by nature. Therefore, it is no surprise that silicon-based single-atom spin qubits hold the record coherence times of any solid state single qubit Muhonen et al. (2014). However, this attractive isolation from sources of decoherence comes at the price of relatively poor coupling to the control and readout units. This causes relatively long qubit initialization times, degraded readout fidelity, and weak spin-spin coupling for multi-qubit gate operations Awschalom et al. (2013).

One simple way to enhance the coupling rate is to increase the ac magnetic field from the external circuit. In this regard, superconducting circuit resonators are attractive due to their relatively large quality factors, ease of coupling to other circuits, their capability of generating ac magnetic fields by carrying relatively large currents, and monolithic integration with semiconductor devices.

For many years, superconducting microwave resonators have had extensive applications that range from superconducting qubit manipulation Wallraff et al. (2004) and inter-qubit coupling Majer et al. (2007) to dielectric characterization Sarabi et al. (2016). One of the most commonly used superconducting quantum computing architectures is one based on cavity quantum electrodynamics (cQED) Blais et al. (2004), in which a 2D (circuit based) or 3D cavity is employed to initialize, manipulate and readout the superconducting qubit. Superconducting circuit cavities have not yet found a similar prevalence in spin qubit circuits due to the fact that the magnetic field of a typical superconducting resonator and the spin magnetic moment are relatively small, leading to an insufficient spin-photon coupling strength for practical purposes. The direct magnetic coupling of coplanar resonators to donor electrons in silicon Eichler et al. (2017); Imamoğlu (2009); Tosi et al. (2014) and diamond nitrogen-vacancy centers Kubo et al. (2010); Amsüss et al. (2011) can only achieve a maximum single-spin coupling rate of a few kHz.

Several methods have been proposed to enhance the coupling of a single spin to a photon within a superconducting circuit resonator. It is easier to couple a photon to quantum dot spin qubits than to single-atom spins because in the former, the spin dynamics can be translated into an electric dipole interacting with resonator’s electric field Petersson et al. (2012); Viennot et al. (2015); Hu et al. (2012). Such architectures, however, require hybridization of the spin states with charge states, coupling charge noise to the spin and thus affecting its coherence. For the single-atom spin qubits, indirect coupling to a resonator via superconducting qubits Kubo et al. (2011); Twamley and Barrett (2010) can enhance the coupling, but imposes nonlinearity on the circuit complicating cQED analysis, and also introduces loss from Josephson junction tunnel barriers Simmonds et al. (2004); Martinis et al. (2005) and magnetic flux noise Kumar et al. (2016).

Figure 1: (a) Schematic of the circuit showing the resonator galvanically coupled to the coplanar waveguide (CPW). is the resonator’s capacitance to ground. , , and show the microwave input power, output power, geometric inductance and the parasitic inductance of the resonator circuit, respectively. This arrangement of and is valid in the limit of . (b) Layout of the device showing resonator’s capacitor and inductor . Blue and orange show the top and bottom superconducting layers, respectively. The area within the red dashed square is magnified. The spiral inductor within the green dashed square is further magnified to show (c) the single spiral geometry (red dot represents the spin). Dimensions are nm, compatible with standard electron beam lithography, and nm. The crystalline Si (x-Si) growth is required only in the volume surrounding the spin, and amorphous or polycrystalline Si is acceptable everywhere else. The relatively weak magnetic field (directions shown represent magnetic fields at the location of the spin) from the via determines the best choice for the direction of the static magnetic field, , where is the magnetic field from the spiral inductor. This ensures that the total ac field is perpendicular to . Current within the spiral is shown by green in- or out of plane vectors.

In this paper, we show that replacing the coplanar transmission line resonator with a lumped-element circuit resonator that includes a spiral inductor, can lead to a dramatic enhancement in the spin-photon magnetic coupling rate of approximately two orders of magnitude. As we will discuss, this improvement is a result of employing a lumped-element trilayer design along with nano-scale spiral loops which effectively localize the resonator’s magnetic field at the location of the spin, eliminating the need for an Oersted line Laucht et al. (2015). We also show that this coupling rate is enough to take the spin-resonator hybrid system to the strong coupling regime, where is larger than or approximately equal to the resonator decay rate . Due to the relatively small total inductance , large capacitance and hence small impedance of the device, coplanar interdigitated capacitances are insufficient to achieve the desired operation frequency . Therefore, the proposed device geometry, compatible with standard micro- and nanofabrication techniques, includes a trilayer (parallel-plate) capacitor with a deposited insulating layer. Trilayer capacitors, however, give rise to a lower resonator quality factor than that of a typical coplanar geometry. Nevertheless, as our calculations show, the increase in resulted from employing a trilayer lumped-element design is large enough to overcome the limited , allowing for strong spin-photon coupling.

The lossless dynamics of a spin- system with spin transition frequency coupled with rate to a cavity resonant at , is described by the Jaynes-Cummings Hamiltonian, Jaynes and Cummings (1963). Here, and are photon and spin operators, respectively. The spin-photon magnetic coupling rate is obtained as where is the electron g-factor, is the Bohr magneton, and is the root mean square (RMS) local magnetic field at the location of the spin with photons on resonance. and are the ground and excited spin states, respectively, that couple to the microwave field. Finally, the spin rotation speed can be enhanced with a larger local RMS magnetic field for any arbitrary .

The schematic and layout of the proposed circuit are shown in Figs. 1(a) and 1(b-c), respectively, where a LC resonator is coupled to a coplanar waveguide (CPW) through direct (galvanic) connection through a coupling inductor and the donor is within the resonator’s deliberate inductor . The galvanic coupling, employed in previous experiments Vissers et al. (2015), can help to achieve the desired CPW-resonator coupling rates especially when the resonator impedance is significantly different from the CPW’s characteristic impedance . The capacitance is provided using a trilayer capacitor. The RMS current through the inductor at an average photon number on resonance is , where is the resonance frequency and denotes the total inductance within the resonator circuit. Since the desired ac magnetic field from the spiral loop(s) is proportional to , it is clear that the inductance (or impedance ) must be minimized and photon frequency must be maximized for the maximum magnetic field. Therefore, it is necessary to study the sources of inductance and obtain operation frequency limitations.

We study two sets of materials, one that uses niobium as superconductor and phosphorous as the donor (Nb/P), and one with aluminum as superconductor and bismuth as donor (Al/Bi). Fabrication is considered to be slightly better established with Al, whereas Nb has a much higher critical magnetic field of T, which is the highest among the single-element superconductors. Through the Zeeman effect, for the Nb/P set, the uncoupled electron spin-up () and spin-down () states are split by the photon frequency GHz corresponding to . For the Al/Bi set, we consider operating at mT (below the critical magnetic field of Al), and use the splitting of the spin multiplets ( GHz) of the Bi donor. These splittings arise from the strong hyperfine interaction between the electron () and Bi nuclear spin () Wolfowicz et al. (2012), leading to a total spin and its projection along . By employing the transition corresponding to the largest matrix element (0.47), and a static magnetic field of mT, the multiplet degeneracy at is lifted by more than 20 MHz, enough to decouple the nearby transitions Bienfait et al. (2016); Mohammady et al. (2010).

For both Nb/P and Al/Bi sets, is simulated and shown in Fig. 2(a) as a function of number of spiral loops, (see supplemental material sup () for details of inductance calculations and simulations). Increasing from 1 to 2 increases , but, for , the competing effect of larger due to larger loop radii suppresses and lowers . This effect is clearly demonstrated in Fig. 2(b) as the optimum for both material sets, where the vacuum fluctuation’s coupling rates for the Al/Bi and Nb/P configurations are obtained as MHz and 0.17 MHz, respectively. If we consider a double layer (2S) spiral inductor (see supplemental material sup ()), these coupling rates approach MHz and 0.24 MHz, respectively. To the knowledge of the authors, these values are approximately two orders of magnitude larger than a previously proposed architecture Tosi et al. (2014). As we report in the supplemental part sup (), this new regime of can lead to significant improvements in spin qubit manipulation rates and readout fidelities, and potentially enables new experiments such as individual spin spectroscopy.

Figure 2: Plots of the (a) total inductance and (b) spin-photon coupling rate versus the number of spiral loops for different combinations of materials and geometries, i.e. superconducting aluminum and Bismuth donor (Al/Bi), superconducting niobium and phosphorous donor (Nb/P), single spiral and double spiral (2S) geometries (see supplemental material for 2S geometry sup ()).

We now discuss a relatively simple proof-of-principle experiment. We consider a low-impedance resonator on top of a Bi-doped substrate or Si layer and propose to measure the spectrum in the small photon limit while the electron spin transitions are tuned across the resonator bandwidth. To further simplify this experiment, a single spiral loop can be employed with dimensions compatible with standard photolithography. In this simplified experiment with no single donor implantation or e-beam lithography requirements, one can choose Al as superconductor and thus the kinetic inductance within the circuit is negligible with micron-scale dimensions of the inductor loop. When spins are incorporated and exposed to the relatively uniform magnetic field inside the inductor loop, the collective coupling rate from the spin ensemble becomes Imamoğlu (2009). This collective coupling rate would yield more clear spin-resonator interaction features in the spectra for this first proof-of-principle version of the device compared to the single donor device. In our proposed device, tuning of the spin(s) can be performed using magnetic (Zeeman) or electric (Stark, see Fig. 1) fields, or a combination of both.

Since the spin ensemble spectroscopy device is micron-scale, high-frequency simulations using a electromagnetic simulation software become feasible. The layout of the device is shown in Fig. 3(a). We consider galvanic coupling to the CPW and assume a resonator internal quality factor of . The simulated transmission through the CPW shows a characteristic resonance circle with diameter of 0.5. This indicates that, because of the galvanic coupling and despite , critical coupling () can be achieved (see Fig. 3(b)).

From the high-frequency simulation, the geometric inductance of the loop is extracted as pH. Despite the significantly larger inductor dimensions of the ensemble device, its impedance is similar to that of the nanoscale (single-atom) device for , primarily due to the smaller contribution of the kinetic inductance in the ensemble device. Therefore, we conclude that the CPW can also be sufficiently coupled to the single-atom device with a nanoscale spiral inductor.

Figure 3: (a) Layout of the simulated Al resonator coupled to an ensemble of Bi spins (ground plane not shown). Here, the dielectric thickness for the capacitor is assumed to be nm. (b) Resonator transmission simulation around its resonance frequency GHz. Spectroscopy simulation of the hybrid resonator-spin ensemble device near for (c) zero spin-resonator detuning and (d) with tunable static magnetic field near the degeneracy field .

By assuming a uniform inside the inductor loop that is applied to spins within 500 nm from the substrate surface with doping density of cm, we find MHz using the method described in Ref. Bienfait et al. (2016b). We use a theoretical model previously developed for a different quantum device with similar physics to simulate the spectroscopy of the spin ensemble Sarabi (2014). Figures 3(c-d) show the spectroscopy simulation results using for the resonator. As done previously Sarabi et al. (2015), one can extract important parameters of this hybrid spin-resonator system directly from the spectroscopy data. These parameters include , the resonator and the relaxation time of the spin ensemble.

In summary, we have proposed and designed a novel device that enhances the coupling of a single atom spin to the magnetic field of a circuit resonator by approximately 100 times compared to the previously proposed architectures that use coplanar transmission line resonators. This dramatic improvement is a result of using a low impedance, lumped element resonator design and a spiral inductor geometry. We showed the possibility of entering the strong coupling regime necessary for practical purposes, i.e., spectroscopic measurements and qubit realization. As shown in the supplemental part sup () using the principles of cavity quantum electrodynamics, this large can lead to a significantly enhanced spin relaxation rate desired for qubit initialization, tens of megahertz spin rotation speed during manipulation without the need for an Oersted line, and superb dispersive readout sensitivity. Moreover, this architecture can be useful for coupling distant qubits using cavity photons for the realization of multi-qubit gates.

The authors thank G. Bryant, D. Pappas, T. Purdy, M. Stewart, K. Osborn, J. Pomeroy, C. Lobb, A. Morello, C. Richardson, R. Murray and R. Stein for many useful discussions.

Supplementary Material for

“Prospective two orders of magnitude enhancement in direct magnetic coupling of a single-atom spin to a circuit resonator” Prospective two orders of magnitude enhancement in direct magnetic coupling of a single-atom spin to a circuit resonator

Bahman Sarabi, Peihao Huang and Neil M. Zimmerman

National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA

Appendix A Inductance simulations and calculations

The total inductance within the resonator consists of the geometric inductance of the spiral inductor giving rise to the magnetic field that couples to the donor electron spin, and parasitic inductance which does not contribute to and only limits it. consists of the trace inductance arising from the length of the spiral trace, and some mutual inductance between the loops such that . The parasitic inductance arises from the kinetic inductance of the spiral , kinetic inductance within capacitor plates and the geometric self-inductance of the capacitor such that . Since does not create any magnetic fields at the location of the spin and only limits through , we want to minimize it. Below, we describe our estimation of all of the aforementioned inductances.

In order to obtain an accurate estimation of and , we performed calculations as well as software simulations. We study two different device geometries, one using a single spiral inductor and the other using a double spiral (2S) inductor shown in Figs. 1(c) and S1, respectively. refers to the number of loops in a single spiral layer regardless of the geometry, e.g., for both Figs. 1(c) and S1. In general, the single spiral design is expected to be easier to fabricate at the expense of smaller compared to the double-spiral geometry. This results in four configurations under consideration, i.e. Al/Bi, Al/Bi-2S, Nb/P and Nb/P-2S.

Figure S1: Layout of the double-spiral (2S) device in the vicinity of the nanoscale spiral inductor. Here, nm and other dimensions are identical to those in Fig. 1(c).

is approximately calculated and also separately simulated. In the calculations, for simplicity, we assume that each loop of the spiral inductor is truly circular and estimate using the geometric mean distance (GMD) method to the second order in the ratio of the conductor diameter to the loop radius Grover (2004). This independent loop approximation (ILA) ignores the mutual inductance within the spiral loops. However, one should note that contributes to and is not parasitic. Nevertheless, in addition to the ILA, we also simulated the spiral geometry in FastHenry 3-D inductance extraction program Kamon et al. (1994), which accounts for the mutual inductances within the spiral geometry. Figure S1 shows a comparison between the simulation and the approximate calculation, where the latter ignores . Clearly, constitutes a larger portion of with increasing , but is negligible up to where is optimum (see Fig. 2(b)). The total inductance in Fig. 2(a) is plotted using from FastHenry simulations. However, for simplicity, the ac field contributed by was not taken into account in calculating , resulting in an underestimated in Fig. 2(b).

Figure S2: Comparison between the independent loop approximation (ILA) and FastHenry simulations of the spiral geometric inductance versus for different spiral loop counts for the single spiral and double spiral (2S) geometries.

The kinetic inductance of the spiral loop is caused by the kinetic energy of the quasi-particles within the superconductor and hence does not create any magnetic fields. In our design, is the largest part of . Using the Cooper-pair density and mass for a particular superconducting material, the length of the superconducting line and its cross-sectional area , one can estimate Tinkham (1996), where denotes the electron charge. For the single-layer spiral Nb/P resonator with we obtain , linearly proportional to the spiral trace length.

The kinetic inductance within the capacitor plates, fH, is negligible due to the large cross-sectional area of the plates. The geometric self-inductance of the capacitor, calculated using a stripline model, is fH. To confirm this, we simulated the resonator geometry including the capacitor and found fH, in agreement with the calculated value and also negligible for . The relatively small is due to the small capacitor insulating thickness .

In order to achieve the desired resonance frequencies with the relatively small inductances shown in Fig 2(a), relatively large capacitances are required. By using a capacitor insulator thickness of nm, we can keep the square-shaped capacitor dimensions below 200 m, and also suppress capacitor’s geometric inductance which contributes to .

A simulation of as a function of the spiral trace width , spacing and thickness , was performed. By assuming , the results showed that, for the Nb (Al) set, the optimum is obtained when nm ( nm), weakly depending on whether the 1S or the 2S geometry is used. If e-beam lithography resolution of 20 nm is implemented, the Al set can yield a spin-photon coupling rate of MHz.

Appendix B Additional comments on magnetic field and coupling rate calculation

At the location of the spin, is approximated as the sum of the magnetic field from all loops, i.e, where is defined as a characteristic radius which accounts for the spin location with respect to the center of the loops and denotes the vertical spin displacement from this center (see Figs. 1(c) and S1). Note that the assumption of current flowing in the center of the spiral conductor underestimates , and , because in reality the majority of current will flow closer to the superconductor-silicon interface due to the relatively large electric permittivity () of silicon.

For a better understanding of the dependence of on the number of the spiral loops, a naive picture may be helpful to the reader. To the first order for , , and , and , but is a weaker function of than . This suggest that by increasing , and hence increase up to a point where approaches , and begins to drop as , with . Employing a lumped element design provides the required flexibility to reach this optimum and the corresponding .

Appendix C Current density and resonator quality factor

Figure S3(a) shows the vacuum fluctuation’s current density for each configuration, where and are the width and thickness of the spiral traces, respectively (see Figs. 1(c) and S1). Clearly, stays far below the critical current values of Al ( kA/cm Romijn et al. (1982)) and Nb ( kA/cm Huebener et al. (1975)).

The condition for the system to enter the strong coupling regime in the small photon limit is , where and are resonator’s total quality factor and total photon decay rate, respectively. Thanks to the relatively large spin-photon coupling rate, resonator ’s in the range of s are enough to take the system to the strong coupling regime for all four configurations (see Fig. S3(b)). In general, in the limit of low drive power and low temperature, trilayer resonators are significantly lossier than coplanar resonators due to the fact that almost all the photon electric energy is stored within the parallel-plate capacitor dielectric which contains atomic-scale defects. These defects act as lossy two-level fluctuators at small powers and low temperatures, and limit the resonator Martinis et al. (2005). However, loss tangents as small as have been measured for deposited amorphous hydrogenated silicon (a-Si:H) at single photon energies O‘Connell et al. (2008) promising resonator ’s of approximately . More recently, elastic measurements have indicated the absence of tunneling states in a hydrogen-free amorphous silicon film suggesting the possibility of depositing “perfect” silicon Liu et al. (2014), promising even higher trilayer resonators using silicon as capacitor dielectric.

Figure S3: Plots of (a) the vacuum fluctuations current density and (b) the required resonator quality factor to allow strong spin-photon coupling versus the number of spiral loops for different combinations of materials and geometries, i.e. superconducting aluminum and Bismuth donor (Al/Bi), superconducting niobium and phosphorous donor (Nb/P), single spiral and double spiral (2S) geometries.

In Fig. S1, we see the following optimum combination of materials: i) Amorphous or polycrystalline Si can be deposited on the metal layers; since they have similar permittivities to crystalline Si, the capacitor size will be similar. ii) Since the spin does not have a metal layer below it, one can still deposit crystalline Si and then implant the donor, thus avoiding the fast decoherence which would result from the non-crystalline, non-enriched films Fuechsle et al. (2012). From the fabrication point of view, it is easiest to use the same Si film as the capacitor dielectric which, in general, will be in the polycrystalline form when deposited on the bottom capacitor plate (see Figs. 1(c) and S1). It is noteworthy that using a trilayer design makes the resonator quality factor independent from the surrounding material as almost all the electric field energy is confined within the capacitor dielectric. This is useful for the integration of single electron devices that require lossy oxide layers.

Appendix D Possibility of donor ionization

It has been previously shown that the donor can be ionized in the vicinity of metallic or other conductive structures due to energy band bendings Fuechsle et al. (2012). For Al-Si interface, the relatively small work function difference of -30 meV (4.08 eV for Al and 4.05 eV for Si) is smaller than the donor electron binding energy of -46 meV Jagannath et al. (1981), and hence using aluminum is expected to allow the donor bound state. However, the Nb work function of 4.3 eV causes significant band bending which can result in donor ionization. By biasing the microwave transmission line and hence the resonator with a DC voltage, one can modulate the potential energy arrangement in the neighborhood of the donor and recreate a bound state. If a conduction band electron is required to fill the bound state, solutions such as shining a light pulse using a light emitting diode to create excess electron-hole pairs or using an ohmic path to controllably inject electrons can be employed.

Appendix E Qubit operation

The qubit operation parameters of the single-atom device presented in this paper are adopted from a commonly used cQED approach Blais et al. (2004). To estimate the initialization, manipulation and readout performance, we focus on the Nb/P-2S configuration with MHz, a realistic resonator internal quality factor and an external quality factor easily realizable according to our estimation of capacitive coupling Martinis et al. (2014), or galvanic coupling. We also assume that the resonator frequency and the Zeeman splitting frequency are GHz and GHz, respectively.

The zero-detuning () relaxation time limited by the Purcell effect for this strongly-coupled system is obtained as s Sete et al. (2014) which is several orders of magnitude shorter than the free spin relaxation. Effective initialization of spin systems using resonator coupling has been previously observed Bienfait et al. (2016), and the 2-order of magnitude increase in in our device promises a much faster initialization. The linear dependence of on is a result of strong coupling, which distinguishes it from previously measured weakly coupled systems Bienfait et al. (2016) where .

For spin manipulation, we can Stark shift the spin states using the electrodes shown in Figs. 1(b)-(d), or use magnetic field tuning the perturb the Zeeman energy. For a demonstration of the qubit operation parameters of the device, we chose to operate at , where the strong-coupling Purcell rate becomes Hz with Sete et al. (2014), giving rise to ms. This relaxation time is not far from the measured Hahn-Echo ms for a P donor electron spin in enriched Si, believed to be limited primarily by the static magnetic field noise and thermal noise and not due to the proximity to the oxide layers or other amorphous material Muhonen et al. (2014). Therefore, it is reasonable to assume that the spin time in our device is Purcell limited, hence set by . Note that for the Al/Bi set, the direction of the Stark shift must be such that the transition frequency, already reduced by a relatively small to lift the multiplet degeneracy, is further reduced to avoid exciting the higher frequency multiplet transitions. In this detuned qubit control regime, the microwave drive frequency is GHz where photon number sets the maximum drive power Blais et al. (2004), and is lower than the critical photon number corresponding to the spiral inductor’s critical current. The qubit rotation speed, at on-resonance photon number set by and , is MHz Haroche (1993), corresponding to coherent -rotations.

The spin readout also occurs in the same detuned regime, where cavity frequency is “pulled” giving rise to a kHz separation frequency depending on the spin state. This corresponds to a phase shift of , well above the measurement sensitivity usually considered to be , and suggests an extremely high-fidelity readout. The measurement time to resolve is estimated to be s where Blais et al. (2004) and is the readout photon number (one must use ).

DISCLAIMER: Certain commercial equipment, instruments, or materials (or suppliers, or software, …) are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

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