Proposal for detecting a single electron spin in a microwave resonator

We propose a method for detecting the presence of a single spin in a crystal by coupling it to a high-quality factor superconducting planar resonator. By confining the microwave field in the vicinity of a constriction of nanometric dimensions, the coupling constant can be as high as 5–10 kHz. This coupling affects the amplitude of the field reflected by the resonator and the integrated homodyne signal allows detection of a single spin with unit signal-to-noise ratio within few milliseconds. We further show that a stochastic master equation approach and a Bayesian analysis of the full time-dependent homodyne signal improves this figure by $\sim 30\%$ for typical parameters.

Figure 1

Proposed setup. (a) Superconducting $LC$ resonator consisting of two pads (capacitor $C$) and wire (inductance $L$) placed in a three-dimensional microwave cavity. (b) Nanometer-scale constriction made at the center of the inductance wire, below which a single spin is located at 15 nm. (c) Cross section of the structure. (d) Schematic of the considered measurement circuit. The signal leaking out of the cavity is first amplified by a quantum-limited Josephson parametric amplifier (JPA), followed by cryogenic low-noise HEMT and room-temperature amplifiers.

Figure 2

The two spin systems studied in this paper. (a) Schematic of a NV center in diamond crystal. Here ${}^{15}\mathrm{N}$, which has a nuclear spin $I=1/2$, is assumed. (b) Energy levels of ${}^{15}\mathrm{N}$-vacancy centers as a function of external magnetic field. Here the external magnetic field $B_0$ is assumed to be parallel to the orientation of [110], so a factor $cos\alpha$ is taken into account (see the text). (c) Schematic of a bismuth donor in silicon (Bi:Si). (d) Energy levels of Bi:Si as a function of the bias field $B_0$.

Figure 3

Magnetic field $\delta B$ generated by vacuum fluctuations of the current $\delta i$ in a constriction 20 nm wide and 10 nm thick. (a) Magnetic-field map analytically derived from the Biot-Savart law for a conductor with uniform current density and rectangular cross section. Here $\delta B$ is given for $\delta i=35$ nA, as in the NV center case. (b) Cut at a distance of 15 nm from the constriction. For $\delta i=35$ nA, we can expect $\delta B=0.33\mu \mathrm{T}$. The red line is the result of the exact analytical formula and the black dashed line shows the approximation ${\mu }_0\delta i/2\pi \sqrt{x^2+y^2}$. (c) Cut along the vertical crossing through the constriction. The green solid line is the exact formula and the black dashed line is ${\mu }_0\delta i/2\pi \sqrt{x^2+y^2}$.

Figure 4

Resonator geometries, with the nanowire depicted in red. (a) For an NV center spin, the central wire is $30\mu \mathrm{m}$ wide and 0.96 mm long. Two large pads of $3\times 1 {\mathrm{mm}}^2$ ensure the coupling to the cavity and 72 pairs (only half are drawn) of $20-\mu \mathrm{m}-\mathrm{wide}$ fingers spaced by $20\mu \mathrm{m}$ are used as additional capacitance to bring the impedance down to $Z_r=15.3\mathrm{\Omega }$. (b) For bismuth in silicon, the central wire is $50\mu \mathrm{m}$ wide and 0.83 mm long. Two large pads of $1.4\times 0.18 {\mathrm{mm}}^2$ ensure the coupling to the cavity and six pairs of $50-\mu \mathrm{m}-\mathrm{wide}$ fingers spaced by $50\mu \mathrm{m}$ are used as additional capacitance to bring the impedance down to $Z_r=26.5\mathrm{\Omega }$.

Figure 5

Integrated currents when there is a spin interacting with the resonator (represented with blue) and when there is no spin (represented with gray). The simulated voltage signals were obtained with the parameters $g=2\pi \times 10\times {10}^3$ rad/s, $\kappa =4.6\times {10}^5 {\mathrm{s}}^{-1}, {\gamma }_{\varphi }={10}^4 {\mathrm{s}}^{-1}$, and $\eta =0.5$. In (a) the solid lines correspond to values of the integrated current for single simulations and the dotted lines to mean values for 3000 simulations. The distributions of the ensembles of simulations at time $t=20{\tau }_1\simeq 5$ ms are shown in (b), where the solid lines are Gaussian fits to each distribution. The threshold value ${\zeta }_c$ is shown as the vertical dotted line.

Figure 6

The probability of inferring that a spin is interacting with the superconducting resonator $P\left[\text{spin}\right|Y\left(t\right)]$. (a) Probabilities as a function of time when $dY\left(t\right)$ is a measurement current from a simulation with a spin (blue curve) and when there is no spin (black curve). Also shown is the normalized distribution of the values of $P\left[\text{spin}\right|Y\left(t\right)]$ for 3000 simulations at measurement times (b) $t=0.4{\tau }_1$, (c) $t=4{\tau }_1$, and (d) $t=12{\tau }_1$. Blue histograms correspond to simulations with a spin and black histograms to simulations with no spin interacting with the resonator. The parameters of the simulation are specified in the figure.

Figure 7

Time-dependent probability of wrongly assigning the presence or absence of a spin based on the continuous measurement. The probabilities are shown on a logarithmic scale for different values of the detector efficiency $\eta$ indicated next to the arrows that connect the smooth curves depicting Eq. (23), based on the integrated measurement current, and the more noisy curves, based on the Bayesian analysis. The red smooth and noisy curves correspond to the experimentally realistic detector efficiency value $\eta =0.5$. The interaction and damping parameters are the same as in Fig. 6.