Spin amplification in realistic systems

Abstract

Spin amplification is the process that ideally increases the number of excited spins when one of them is excited initially. We show that by applying optimal control techniques to design classical drive pulse shapes, spin amplification can be achieved in a previously unexplored fast regime, with amplification times comparable to the intrinsic interaction timescale. This is an order of magnitude faster than the previous protocols and makes spin amplification possible even with significant decoherence and inhomogeneity in the spin system. The initial spin excitation can be delocalized over the entire ensemble, which is a more typical situation when a photon is collectively absorbed by the spins. We focus on the superconducting persistent-current artificial atoms and the Rydberg atoms as spins.

Figures (8)

• Figure 1: (a) The idealized spin amplification (SA) operation. If all spins are in the ground state initially, there is no change (top). If one spin is excited initially, all spins become excited (bottom). We consider an equal superposition of all the single excitation states instead of a particular excited spin position as shown in the figure. (b) One of the considered experimental platforms: a 2D array of persistent-current artificial atoms orlando_prb99 as the spin ensemble. The artificial atoms consist of a superconducting loops interrupted by $3$ Josephson junctions (crosses) where one of the junctions is smaller than the other two. The superconducting loops have linear inductance (inductors at the bottom of the loops), and the interactions between the spins are due to the mutual inductance between the loops. The interactions are all-to-all in nature.
• Figure 2: The average parameters of the Hamiltonian \ref{['H_long_range_ising']} for the persistent-current artificial atoms orlando_prb99 as spins: the transition frequencies between the ground and excited state $\omega_{01,\text{avg}}$, the transition frequencies between the first and second excited states $\omega_{12,\text{avg}}$, and the values of the nearest-neighbor couplings $J_{\pm,\text{nn},\text{avg}}$ and $J_{z,\text{nn},\text{avg}}$ ($J_{\pm,jk}$ and $J_{z,jk}$ with the maximal absolute values, respectively). Values for a range of bias fluxes $\Phi_\text{ext}/\Phi_0$ are shown, where $\Phi_0$ is the magnetic flux quantum. The curves are calculated by performing circuit quantization of the persistent-current artificial atoms with the Josephson junction area error of $2\%$ for $10^4$ different realizations of a square $4\times4$ ensemble. The difference in the shown values between using $3\times3$ and $4\times4$ ensembles for averaging is small. The thickness of the curves shows the standard deviation of the parameters. The chosen $\Phi_\text{ext}/\Phi_0=0.5034$ is shown by the black dotted vertical line. The parameters of the persistent-current artificial atoms are close to Ref. lambert_prb16: $E_J/h=300\text{ GHz}$, $E_J/E_C=75$, $\alpha=0.7$ (relative area of the smaller junction). The model is extended to include the linear inductance, and the persistent-current artificial atoms are assumed to be square loops of size $3\text{ }\mu\text{m}\times 3\text{ }\mu\text{m}$ and use $0.2\text{ }\mu\text{m}\times 0.1\text{ }\mu\text{m}$ traces (width $\times$ height). The error in the side lengths of the loops is assumed to be $0.1\%$, and they are spaced $1\text{ }\mu\text{m}$ apart.
• Figure 3: Difference of the final populations $P_1-P_0$ for the spin amplification as a function of the number of spins $N$. The results are for two types of spins: Rydberg atoms and persistent-current artificial atoms (PCAA). For the continuous-wave (CW) PCAA results (cyan squares), the two-frequency parametrization \ref{['two_frequency_Omega']} is used for the Rabi frequency with optimized jones_jota93nlopt$\Omega_1$, $\Omega_2$, $\Delta_{\text{avg},1}$, and $\Delta_{\text{avg},2}$. The other curves show results for the drives found using optimal control. The idealized PCAA results (red crosses) are obtained using the Schrödinger equation without decay and dephasing with the pulse parametrization \ref{['Re_Im_Omega_parametrization']} and lie very close to the line $P_1-P_0=N$. The results using optimal control for PCAA (blue circles) and Rydberg atoms (green triangles) are obtained with the master equation or the quantum trajectory method with the parametrization \ref{['Re_Im_Omega_parametrization_constrained']}. In all cases except the idealized PCAA, the inhomogeneity is modeled by averaging over a number of realizations, and the error bars show the standard deviation of this averaging. The number of inhomogeneity realizations is chosen to be $100$, except for CW PCAA with $N=16$ where only $15$ realizations are used, because each one of them needs a slow global optimization jones_jota93nlopt.
• Figure 4: (a) Populations as functions of time for one inhomogeneity realization of $N=16$ persistent-current artificial atoms as spins. (b) Pulse shapes that drive the dynamics in (a). The pulse shapes are parametrized such that the extremal values of $\text{Re}[\Omega]/(2\pi)$ and $\text{Im}[\Omega]/(2\pi)$ [cf. Eqs. \ref{['Re_Im_Omega_parametrization_constrained']}] are at $\pm 400$ MHz, and the filtering bandwidth is $1/\sigma=4\text{ GHz}$. (c) Average populations as functions of time for all the inhomogeneity realizations of $N=16$ Rydberg atoms as spins. (d) Pulse shapes that drive the dynamics in (c). The Rabi frequency $\Omega/(2\pi)$ is real and positive with the maximum value $3.5$ MHz. The detuning $\Delta/(2\pi)$ is constrained by the extremal values $\pm 20\text{ MHz}$.
• Figure 5: Flux variable definitions. The integration contours that correspond to the flux variables are displaced out of the circuit for clarity, since some of them overlap.