Multimode Storage of Quantum Microwave Fields in Electron Spins over 100 ms

We report long coherence times (up to 300 ms) for near-surface bismuth donor electron spins in silicon coupled to a superconducting microresonator, biased at a clock transition. This enables us to demonstrate the partial absorption of a train of weak microwave fields in the spin ensemble, their storage for 100 ms, and their retrieval, using a Hahn-echo-like protocol. Phase coherence and quantum statistics are preserved in the storage.

Figure 1

Quantum memory. (a) The proposed architecture consisting of a quantum processor coupled via a coaxial cable to a quantum memory from electron spins. (b) Circuit implementation: an ensemble of $N$ electron spins are inductively coupled to a lumped resonator which is capacitively coupled to a microwave line. Weak coherent data pulses in our demonstration, travel along the lines, are stored by the spin ensemble, and then released on demand by a control pulse. The coupling strength of an individual spin to the resonator is $g_0$. (c) Device setup: The resonator is fabricated on top of the silicon substrate in a superconducting aluminum thin film to minimize internal losses. It consists of a capacitor shunted by a $5\mu \mathrm{m}$-wide inductance wire, to which the spins of bismuth donors implanted around a depth $\sim 100\mathrm{nm}$ are inductively coupled. A magnetic field $B_0$ is applied along the inductor.

Figure 2

Clock transition in bismuth and its characterization. (a) Energy level diagram $|F,m⟩$ of electron spins from bismuth at small magnetic fields. Green lines indicate the four levels $|4,-1⟩,|4,0⟩,|5,-1⟩$, and $|5,0⟩$ involved in the two quasidegenerate CT. (b) The ESR-allowed transition frequency of bismuth donor spins in silicon between 7.3–7.4 GHz, and the measured resonator frequency ${\omega }_0$ as a function of $B_0$. The CT with $d\omega /d{B}_0=0$ can be seen at $B_0=27\mathrm{mT}$. Inset: zoom around CT to show degenerate transition and expected strain broadened spectrum in frequency. (c) Measured echo detected spectroscopy around the CT (green symbols, left axis), calculated one (black line, left axis, see Ref. [38]), and two-pulse echo coherence time $T_2^E$ versus $B_0$ (red bars, right axis). (d) Measured echo area decay at CT (symbols) and corresponding exponential fit (line) yielding $T_2^E=0.3\pm 0.05\mathrm{s}$. Magnitude detection is employed to circumvent phase noise from the measurement setup. (e) Measured (symbols) and simulated (line) Rabi oscillations at the CT. Each $A_e$ is plotted as a function of the amplitude $\beta$ of the first pulse. (f) Measured spin relaxation at the CT using an inversion recovery sequence (symbols), and corresponding exponential fit (line) yielding $T_1^E=5.5\pm 0.5\mathrm{s}$.

Figure 3

Storage and retrieval of weak pulses at the clock transition ($B_0=27\text{mT}$). (a) Measured intracavity field $\alpha$ averaged over ${10}^3$ repetitions at a repetition rate of 200 ms (spins saturated, dashed curve) and of 16 s (spins polarized, solid curve). (b) Measured (left panel, $4\times {10}^3$ averages) and simulated (right panel) reflected field on resonance ($\sqrt{{\kappa }_c}\alpha -{\beta }_{\mathrm{in}}$) and echo at a time $\ll T_2^{\mathrm{E}}$. The simulation assumes a spectral density of $N/(\mathrm{\Gamma }/2\pi )={10}^6\text{spins}/\mathrm{MHz}$ and a fixed $g_0/2\pi =40\mathrm{Hz}$. For panels (a),(b) there are 240 photons in the input pulse. (c) Top (bottom): a train of weak microwave fields, containing 240 (24) photons at the input and their retrieval after the refocusing $\pi$ pulse. The numbering highlights that retrieval maintains the phase relation with respect to the input field. The retrieved fields are multiplied by ${\zeta }^{-1}\approx 24$, where $\zeta =4C({\kappa }_C/\kappa )$ is the theoretical efficiency of the retrieved field at a time $\ll T_2^E$. Note that the input fields are measured at a frequency shifted by 5 MHz from ${\omega }_0/2\pi$ to enable a direct comparison with the retrieved fields.

Figure 4

Noise statistics of retrieved echo. (a) Average of ${10}^3$ single shot traces of reflected signals acquired with a repetition rate of 16 s. The input and retrieved field contain 240 and 0.25 photons, respectively. In this run, the internal losses were slightly larger than in Fig. 3, leading to a lower value of $\zeta$ and a lower echo amplitude. (b) Histograms of signal quadratures before the refocusing $\pi$ pulse and on the echo. Each histogram sample is acquired by integrating signals for $100\mu \mathrm{s}$ weighted by a normalized gaussian mode shape describing the echo. Solid curves are calculated Gaussians of the same area.