Triply Resonant Photonic Crystal Nanobeam Cavities for Unconditional Photon Blockade

Abstract

The development of many scalable quantum technologies requires single-photon nonlinearity, such as single-photon blockade, in solid-state systems. Recently, it has been shown that single-photon Fock states can, in principle, be unconditionally generated using arbitrarily small intrinsic optical nonlinearities in photonic cavities. We investigate the feasibility of such a scheme in achieving photon blockade in an on-chip silicon photonics platform. We show that a triply resonant nanobeam cavity pumped with three monochromatic lasers could achieve such functionalities with quality factors $\sim 10^7$ and effective mode volumes $\sim 10^{-2} μm^3$, for experimentally feasible incident powers. Using quantum optical simulations, we propose an experimental protocol to generate single photons under this scheme. The constraints on the cavity design and experimental conditions are thoroughly explored to determine feasible regimes of operation.

Figures (4)

• Figure 1: a) Schematic of the Experiment $\hbox{--}$ The triply resonant cavity with $\chi^{(3)}$ nonlinearity has three modes at $\omega_{1c}$, $\omega_{2c}$, and $\omega_{3c}$. The one-photon pumping is implemented using a laser at $\omega_1$. For two-photon pumping, two separate lasers can be used at $\omega_{2}$ and $\omega_{3}$ such that $\omega_{2}+\omega_{3} = 2\omega_{1}$. b) Pumping scheme for one and two-photon pumping $\hbox{--}$ the initialization phase lasts a short time $\tau \ll \kappa^{-1}$, where $\kappa$ is the cavity mode decay rate. It consists of three steps for the single-photon drive ($\Lambda_1$) and two steps for the two-photon drive ($\Lambda_2$), which are visible as linear steps with different slopes in the plot. The rationale behind such initialization is discussed in lingenfelter1. $\Lambda_1$ is initially ramped up to a large magnitude, followed by a duration of constant value. $\Lambda_2$, on the other hand, is relatively slowly ramped up in the first step. In the next step, $\Lambda_1$ and $\Lambda_2$ are linearly ramped down and up, respectively, such that the desired displacement as well as the Hamiltonian $\hat{H}_{block}$ is achieved at the same time. During the phase of evolution under $\hat{H}_{block}$, the drives are held at a steady level for a time $t$ on the order of $1/\Lambda_{NL}$. The final phase, displacement to the lab frame, is a reversal of the initialization phase. At the end of the pumping sequence, the photon is collected via a side-coupled waveguide. The emitted photons at $\omega_1$ will be sent to a Hanbury Brown--Twiss interferometer to characterize the second-order time correlation function ($g^{(2)}(\tau)$) of the single photon emission. c) Time evolution of the cavity photon number $\hbox{--}$ The time-dependent average cavity photon number ($\langle n \rangle$) during the evolution under the blockade Hamiltonian $\hat{H}_{block}$ is plotted for multiple values of $\alpha$. The plot here is for the ideal case with no errors in any experimental parameter.
• Figure 2: a) Power dependence of the single-photon pump as a function of the average photon number and mode volume of the cavity. Here, $Q = 10^5$, $\omega_c = 1.21 \times 10^{15} s^{-1}$. b) Power dependence with the mode volume of the cavity for a desired photon number. This is equivalent to a horizontal line in a). c) Power dependence of the single-photon pump as a function of the average photon number and quality factor of the cavity. Here, $V = 0.01\mathrm{\,\mu m^3}$, $\omega_c = 1.21 \times 10^{15}$$\mathrm{Hz}$. d) Same as b), but as a function of the quality factor. The dotted lines represent the parameters assumed for other simulations. e) The time-dependent expected photon number in the displaced frame for three different cavity quality factors and the identical input power of 10 ${\,mW}$.
• Figure 3: Top: Wigner distributions of the state after various initialization conditions. a) Linear cavity, one-photon pump. The state is initialized to a coherent state $\ket{\alpha}$. b) Nonlinear cavity, no two-photon pump. The coherent state is squeezed and shifted slightly. c) Nonlinear cavity, one- and two-photon pumps. The state is moved back to the correct displacement $\alpha$, and the squeezing is less pronounced. Bottom: The individual effect of errors on the final $g^{(2)}(0)$. d) The relationship between $\Delta\alpha$ and $g^{(2)}(0)$ for $\alpha = 2$. e) The error in initialization with an error of $\Delta\alpha = 0.035\alpha$ vs. $\alpha$. f) Under the current protocol, the optimal $g^{(2)}(0)$ given an initialization time $\tau$, assuming $\kappa = 100 \mathrm{\, MHz}$. For e) and f), the blue line represents the loss function that was used for optimization. While there is some mismatch, changing the loss function has not shifted the results significantly. All the subfigures pertain to the standard cavity parameters ($Q = 10^7$ and $V_{eff} = 0.01 \mu m^3$).
• Figure 4: Heatmaps of the relationship between errors in $\Lambda_1$ and $\Lambda_2$ with the resulting $g^{(2)}(0)$. All errors are relative to the magnitude of the drives. Parameters: a) $\Lambda_1$ error during initialization. Multiple minima are present. b) $\Lambda_2$ error during initialization. c) $\Lambda_1$ and $\Lambda_2$ error during evolution in the displaced frame.