Optical detection and storage of entanglement in plasmonically coupled quantum-dot qubits

Recent proposals and advances in quantum simulations, quantum cryptography, and quantum communications substantially rely on quantum entanglement formation. Contrary to the conventional wisdom that dissipation destroys quantum coherence, coupling with a dissipative environment can also generate entanglement. We consider a system composed of two quantum-dot qubits coupled with a common, damped surface plasmon mode; each quantum dot is also coupled to a separate photonic cavity mode. Cavity quantum electrodynamics calculations show that upon optical excitation by a femtosecond laser pulse, entanglement of the quantum-dot excitons occurs, and the time evolution of the $g^{\left(2\right)}$ pair correlation function of the cavity photons is an indicator of the entanglement. We also show that the degree of entanglement is conserved during the time evolution of the system. Furthermore, if coupling of the photonic cavity and quantum-dot modes is large enough, the quantum-dot entanglement can be transferred to the cavity modes to increase the overall entanglement lifetime. This latter phenomenon can be viewed as a signature of entangled, long-lived quantum-dot exciton-polariton formation. The preservation of total entanglement in the strong-coupling limit of the cavity–quantum-dot interactions suggests a novel means of entanglement storage and manipulation in high-quality optical cavities.

Figure 1

Schematic of the system. (a) Two quantum-dot (QD) qubits are embedded into optical cavities and coupled with plasmonic modes in a neighboring metal nanoparticle. The chosen setup with two individual cavities, each of them enclosing a single QD, enables one to separately tune the QD-photon and QD-plasmon coupling strengths. (b) Graphical representation of our model: The two-level QDs are coupled with plasmonic modes and photon cavity modes; gray arrows show the respective coupling. In our calculations, the QDs and plasmons are excited by a laser pulse with full width at half maximum of 20 fs. We show that QD qubit entanglement, defined as Wootters' concurrence $C$, can be both detected via the $g^{\left(2\right)}$ pair correlation function of the cavity photon and stored in high-quality optical cavities.

Figure 2

The dynamics of the system for weak QD-cavity photon coupling, $\xi =0.268$: QD concurrence $C\left(t\right)$ and the same-time cavity photon pair correlation function $g_{ij}^{\left(2\right)}\left(t\right)$ (main plot), and the QD population (inset). The system is optimally assembled with the QD-plasmon coupling constants ratio of $1/\sqrt{3}$ that maximizes the QD entanglement; $\hslash g_s^i=30$ and 17.3 meV for $i=1$ and 2, respectively; $\hslash g=1\mathrm{meV}$; $\hslash {\gamma }_s=150\mathrm{meV}$; the QD decay and dephasing rates are $0.05\mu \mathrm{eV}$ and $8.6\mu \mathrm{eV}$; the respective photon decay and dephasing rates are 0.1 meV and $8.6\mu \mathrm{eV}$; the transition dipole moments for the surface plasmons and QDs are $d_s=4000$ D and $d_i=13$ D; the energy level spacing of the QD and cavity photon systems is $\hslash \omega =2.05$ eV (Refs. [16, 17, 42]). The maximum electric field in the driving pulse is reached at $t=36.3$ fs.

Figure 3

Oscillatory dynamics of the system in the strong-coupling regime $\xi =2.68$. (a) Population of QD 1 and 2 and photon population in cavities 1 and 2. Inset shows the total population in the system as a function of time $t$. (b) QD concurrence and photon correlation functions. Vertical arrows in the inset approximate the moments at which the QD concurrence reaches the maxima, as shown in the main plot. The simulations were done for the same parameters as in Fig. 2, but the QD-cavity photon coupling constant $g$ was increased $10\times$.

Figure 4

(a) Photon and total entanglement and (b) the photon cross-correlations for the model where the number of photon levels is restricted to $N_{\mathrm{ph}}=2$. The results are obtained for the strong-coupling regime with the same parameter set as in Fig. 3. It is seen in (a) that the squared fidelity $F^2\left(t\right)$ of the photon state relative to the maximally entangled Bell state ${\mathrm{\Psi }}^{-}$ follows the photon concurrence. It is evident from (a) that change in the photon level number $N_{\mathrm{ph}}$ from 2 to 4 does not result to significant changes in fidelity $F\left(t\right)$ thus, the photons are entangled at $N_{\mathrm{ph}}>2$. As is also seen in (b), the cross-correlation function $g_{12}^{\left(2\right)}$ calculated for $N_{\mathrm{ph}}=2$ and 4 shows similar qualitative patterns with sharp peaks positioned at the moments when the QD entanglement reaches the maximum values in Fig. 3 (arrows).

Figure 5

Storage of the entanglement of QDs in high-finesse cavities in the strong-coupling regime, compared to QDs in an open geometry. As is seen in the main plot, at $t=4814$ fs, the concurrence of QDs in the cavities is $\approx 4.58\times$ greater that that with no cavities. The QD decay rates are $500\mu \mathrm{eV}$ (main plot) and $50\mu \mathrm{eV}$ (inset); the cavities' quality factor is $Q={10}^6$; the photon and QD dephasing rates are $8.6\mu \mathrm{eV}$; the cavity photon energy is $\hslash \omega =2.05$ eV; the cavity photon decay rate is $Q^{-1}\hslash \omega =2.05\mu \mathrm{eV}$ [16, 17, 36, 42, 52]. The strength of the QD-photon interactions is marked in the plot.

Figure 6

(a) Population (main plot), QD concurrence, and cavity photon correlations (inset) for equal plasmon coupling $\mathrm{\Delta }{g}^s=0$. It is seen in the inset that the QD concurrence is $C\left(t\right)\ll 1$ and, at the same time, the pair cross-correlation function for the cavity photons tends to $g_{12}^{\left(2\right)}\left(t\right)\approx 1$ for $t>10$ fs. The QD-plasmon interaction strength is the same for both dots, $\hslash g_s^1=\hslash g_s^2=30\mathrm{meV}$; $\hslash g=1\mathrm{meV}$; $\hslash {\gamma }_s=150\mathrm{meV}$; the QD decay and dephasing rates are $0.05\mu \mathrm{eV}$ and $8.6\mu \mathrm{eV}$; the respective photon decay and dephasing rates are 0.1 meV and $8.6\mu \mathrm{eV}$; the transition dipole moments for the surface plasmons and QDs are $d_s=4000$ D and $d_i=13$ D; the energy level spacing of the QD and cavity photon systems is $\hslash \omega =2.05$ eV.

Figure 7

Normalized correlation function of cavity photons $g_{ab}^{\left(2\right)}\left(t\right)$, Eq. (C25) [solid (red) curve] and quantum-dot concurrence $C$, Eq. (C31), analytically calculated for the state $\left|\mathrm{\Psi }\right(t)\rangle$ given in Eq. (7) at $\varepsilon =0.1$. In the figure, the correlation function $g_{ab}^{\left(2\right)}\left(t\right)$ is divided by a factor of 75 for better visibility. It is seen that the peaks at the photon $g_{ab}^{\left(2\right)}\left(t\right)$ curve are positioned at the same moments of time, at which the QD concurrence reached its maxima, in agreement with the results of our simulations in Fig. 3.

Figure 8

Quantum dynamics of plasmonically coupled QDs and cavity photons under the action of a long laser pump. The time duration of the laser pulse is $\mathrm{\Delta }t=720$ fs, the characteristic pulse formation, and decay time is $\delta =10$ fs. The maximum electric field in the pulse is $E_{\max }=2.5\times {10}^6$ V/m. (a) Population of the first (QD1) and second (QD2) quantum dots as functions of time $t$ (left scale). The orange curve shows the shape of the envelope of the laser pulse $E_0\left(t\right)d_i$ where $d_i=13$ D is the transition dipole moment of QD and $E_0\left(t\right)$ is given in Eq. (D1) (right scale). It is seen that after the transient oscillations are damped, the QD populations tend to a steady-state value of 0.5 when the laser pump is turned on and then decrease with time after the pump is turned off. (b) The QD concurrence (left scale) and the cavity photon cross-correlation function $g_{12}^{\left(2\right)}$ (right scale). The orange curve shows the laser pulse envelope curve (in arb. units). It is seen that the QD concurrence is $C\left(t\right)=0$ when the pump pulse is on and then begins to increase $\approx 180$ fs after the driving pulse is switched off. The photons are emitted with $g_{12}^{\left(2\right)}\approx 1$ when the pulse is switched on. The photons antibunch, $g_{12}^{\left(2\right)}<1$, when the QD entanglement is formed after the driving pulse is turned off. In the simulations, the QD-plasmon coupling strength are 30 meV and 17.3 meV for QD1 and QD2, respectively; the QD-photon coupling strength is 10 meV; the plasmon decay rate is 150 meV; the QD decay and dephasing rates are $0.05\mu \mathrm{eV}$ and $8.6\mu \mathrm{eV}$, respectively; the respective photon decay and dephasing rates are 0.1 meV and $8.6\mu \mathrm{eV}$; the transition dipole moments for the surface plasmons and QDs are $d_s=4000$ D and $d_i=13$ D; the energy level spacing of the QD and cavity photon systems is $\hslash \omega =2.05$ eV.