Generation of frequency entanglement with an effective quantum dot-waveguide two-photon quadratic interaction

TL;DR

The paper addresses the challenge of creating frequency entanglement between independent single photons without post-selection. It develops an ab initio effective two-photon quadratic Hamiltonian by adiabatically eliminating frequency-dependent one-photon transitions in a quantum-dot–waveguide system, and analyzes the resulting FrEnGATE using Markovian scattering theory. The results show Gaussian, controllable joint spectra along both the sum and difference of photon frequencies, with a trade-off between entanglement quality (Schmidt number, entropy) and generation efficiency; entanglement up to a Schmidt number near 5 and about 15% success is demonstrated, and the scheme extends to frequency-qudit states. This measurement-free approach offers a scalable route to robust time–frequency entanglement for quantum communication and processing, with practical applicability to integrated photonic platforms and potential enhancements via waveguide chirality and Purcell enhancement.

Abstract

Light-matter interactions with quantum dots have been extensively studied to harness key quantum properties of photons, such as indistinguishability and entanglement. In this theoretical work, we exploit the atomic-like four-level structure of a quantum dot coupled to a waveguide to model a shaping frequency entangling gate (FrEnGATE) for single photons. Our approach is based on the identification of input frequencies and an atomic level structure for which frequency-dependent one-photon transitions are adiabatically eliminated, while frequency-dependent two-photon transitions are resonantly enhanced. The frequency entanglement performance of the gate is analyzed using a Schmidt decomposition for continuous variables, revealing a trade-off between entanglement generation efficiency and entanglement quality. We further demonstrate the use of the FrEnGATE for the generation of entangled frequency qudit states.

Figures (7)

• Figure 1: (a) Sketch of the QD embedded in the waveguide. (b) QD's four-level structure in the circular polarization basis. The energy structure is displayed before the one-photon joint operators $\hat{\xi}_{0X_\pm}^{\sigma \mu}(\omega,t) = \left(\lvert X_\pm \rangle\langle 0 \rvert\otimes \hat{a}_{\sigma\mu}(\omega)\right)(t)$ and $\hat{\xi}_{X_\pm 2X}^{\sigma'\mu'}(\omega',t) = \left(\lvert 2X \rangle\langle X_\pm \rvert\otimes\hat{a}_{\sigma\mu}(\omega)\right)(t)$ adiabatic eliminations. The one-photon transition detunings are $\delta_e(\omega) \equiv \omega-\omega_{X}$ and $\delta_b(\omega')\equiv\omega'-(\omega_{2X}-\omega_X)$. The excitonic transitions $\lvert 0 \rangle\leftrightarrow \lvert X_\pm \rangle$ can be driven by a right-handed and left-handed circularly polarized photon, respectively. The biexcitonic transitions $\lvert X_\pm \rangle\leftrightarrow \lvert 2X \rangle$ can be triggered by a left-handed and right-handed circularly polarized photon, respectively. The fine-structure splitting (FSS) $S$ couples the two excitonic states $\lvert X_\pm \rangle$ together. (c) Effective two-level QD's energy structure after the adiabatic elimination. This adiabatic elimination switches from four one-photon joint transitions operators (a) to two two-photon joint transitions operators $\hat{\xi}_{0X_\pm2X}^{\sigma'\mu'\sigma\mu}(\omega',\omega)$ (b) driving the transition $\lvert 0 \rangle\rightarrow \lvert 2X \rangle$ either through $\lvert X_+ \rangle$ or $\lvert X_- \rangle$. This effectively results in a two-level atomic system. The two-photon joint transition operators $\hat{\xi}_{0X_\pm2X}^{\sigma'\mu'\sigma\mu}(\omega',\omega)$ in (b) are time-independent during the interaction time verifying the adiabatic elimination conditions. Note that the additional $X_\pm$ indices in these joint two-photon transition operators were introduced to indicate the origin of $\hat{\xi}_{0X_\pm2X}^{\sigma'\mu'\sigma\mu}(\omega',\omega)$. They both read $\hat{\xi}_{0X_\pm2X}^{\sigma'\mu'\sigma\mu}(\omega',\omega) = \lvert 2X \rangle\langle 0 \rvert\otimes \hat{a}_{\sigma'\mu'}(\omega')\hat{a}_{\sigma\mu}(\omega)$ regardless of the transition path.
• Figure 2: Effective two-level system where the interaction can be driven through two interaction paths: one where the first photon is $R$-polarized (respectively $L$-polarized) with a frequency far-detuned from $\omega_X$ and the second photon is $L$-polarized (respectively $R$-polarized) with a frequency far-detuned from $\omega_X-\delta_X$. (a) For $\delta_X \neq 0$, the two transition paths are distinguishable and do not interefere leading to a non-zero two-photon coupling term. (b) For $\delta_X=0$, the two interaction paths are now indistinguishable and destructively interfere. This entails a two-photon coupling term equal to zero thus preventing the $\lvert 0 \rangle\leftrightarrow \lvert 2X \rangle$ transition.
• Figure 3: Scattering channels for the two input photons propagating rightward, i.e.. $\{\mu',\mu\}_{\text{in}} = \{+,+\}$, with an initial Gaussian joint spectral amplitude (left panel) along both collective variables $\omega_\Sigma = \omega+\omega'$ and $\omega_{\Delta} = \omega-\omega'$ of width $\alpha = 10^{-6}\omega_{2X}$ and centers $\omega_e =0.5026\omega_{2X}$, $\omega_b=0.4974\omega_{2X}$. The coupling term is taken as Gaussian of width $\beta=4\alpha$. The decay rate is chosen as $\Gamma = 10^{-5}\omega_{2X}\gg\alpha$. There are two kinds of outputs (right panels), the one for which the output photons' auxiliary modes are identical to that of the input (upper right panel), i.e.., $\{\mu',\mu\}_{\text{in}} = \{\mu',\mu\}_{\text{out}}$ and those for which they are not (lower right panel), i.e.., $\{\mu',\mu\}_{\text{in}} \neq \{\mu',\mu\}_{\text{out}}$ . In the first case, of probability $P^{++}\approx 66$% for the chosen physical parameters, the input photons can interfere with the entangled photons scattered out by the QD. This results in an output two-photon distribution that is weakly modified with respect to the input two-photon distribution. In the second case $\{\mu',\mu\}_{\text{in}} \neq \{\mu',\mu\}_{\text{out}}$, which occurs with probability $1-P^{++}\approx 34$% for the chosen physical parameters, only the entangled photons scattered out by the QD contribute to the two-photon distribution. This produces an output two-photon distribution that is reshaped along both collective variables. Along $\omega_\Sigma$, the JSA is filtered by the Lorentzian profile originating from the Markovian approximation. The two-photon coupling term shapes the JSA along the $\omega_\Delta$ collective variable. The exact values of the JSIs have been renormalized for readability. They are presented in arbitrary units (a.u.) on a linear scale.
• Figure 4: Time-scale constraints for the FrEnGATE. The adiabatic elimination of one-photon transitions requires that the interaction time be much longer than the inverse of the one-photon detunings, ensuring that the corresponding joint operators average out to zero and can be neglected (condition (i)). To treat the two-photon joint operators as effectively time-independent during the scattering process, the time must also remain much shorter than the inverse of the two-photon detunings and the spectral linewidths (condition (ii)). Finally, for the scattering theory approach to be valid, the time must exceed $1/\Gamma$ ensuring that the QD has fully decayed to its ground state by the end of the interaction (condition (iii)).
• Figure 5: (a) Modes structure for the input two-photon distribution Gaussian, in both $\omega_\Sigma = \omega+\omega'$ and $\omega_\Delta=\omega-\omega'$ of width $\alpha=10^{-6}\omega_{2X}$ and centers $\omega_e =0.5026\omega_{2X}$, $\omega_b=0.4974\omega_{2X}$. The normalized entanglement entropy and the Schmidt number are computed to be $S_{\text{entropy}}=0.00$ and $K=1.00$ respectively. (b) Modes structure for the output two-photon distribution Gaussian in both $\omega_\Sigma = \omega+\omega'$ and $\omega_\Delta=\omega-\omega'$ of width $\alpha=10^{-6}\omega_{2X}$, centers $\omega_e =0.5026\omega_{2X}$, $\omega_b=0.4974\omega_{2X}$ and $\beta=10\alpha$, respectively. The decay rate is taken as $\Gamma = 10^{-5}\omega_{2X}\gg\alpha$. The normalized entanglement entropy and the Schmidt number are computed to be $S_{\text{entropy}}=0.39$ and $K=4.72$ respectively. The first, third and fifth Schmidt modes' JSIs have been plotted. The exact values of the JSIs have been renormalized for readability. They are presented in arbitrary units (a.u.) on the same linear scale in $\omega/\omega_{2X}$ and $\omega'/\omega_{2X}$ as the (b) left panel. We formally indicate the standard deviations $\alpha$ and $\beta$ although it should be noted that the full widths at half maximum (FWHMs) are rigorously equal to 2$\alpha\sqrt{2\ln2}$ and 2$\beta\sqrt{2\ln2}$, respectively.