Fidelity of time-bin-entangled multiphoton states from a quantum emitter

We analyze a scheme for generating multiphoton entangled states by a single solid-state quantum emitter and devise a mathematical framework for assessing the fidelity of the generated state. Within this formalism, we theoretically study the role of imperfections present in real systems on the generation of time-bin encoded Greenberger–Horne–Zeilinger and one-dimensional cluster states. We consider both fundamental limitations, such as the effect of phonon-induced dephasing, interaction with the nuclear spin bath, and second-order emissions, as well as technological imperfections, such as branching effects, nonperfect filtering, and photon losses. The devised framework is applicable to a range of quantum emitters, including semiconductor quantum dots, defect centers in solids, and atoms in cavities.

Figure 1

Ideal protocol. (a) Idealized level structure and (b) driving pulse sequence for the generation of the GHZ states ($\widehat{R}=\widehat{X}$) or cluster states ($\widehat{R}=\widehat{H}$). Laser pulses ${\mathrm{\Omega }}_{\mathrm{O}}$ (green) and ${\mathrm{\Omega }}_{\mathrm{R}}$ (blue) are used for driving the optical transition $|1\rangle \leftrightarrow |2\rangle$ and for the ground-state rotations $|0\rangle \leftrightarrow |1\rangle$, respectively. Following each optical $\pi$-pulse, we wait for the excited state $|2\rangle$ to decay back to the ground state $|1\rangle$ (red wiggly lines). Using $N$ repetitions of the pulse sequence, an entangled state of $N$ photons and the quantum emitter is generated.

Figure 2

Sources of imperfection. Errors considered include (a) level shifting induced by interaction with the nuclear spin bath, (b) phonon-induced pure dephasing, (c) second-order emissions from the resonant and far-detuned levels, and (d) branching errors due to alternative decay paths. See text for explanations of each of the effects.

Figure 3

Measurement setup for detecting time-bin-entangled photons. Measurements in the Z basis are made by passing both early and late photons through the short arm. For measurement in the X basis, the early and late photons are passed through the long and short arms of the interferometer, respectively. The measurement setup delays the early photon by a time equal to the time difference between the two excitation pulses, hence early and late photons arrive at the second beamsplitter at the same time when measured in the X (or Y) basis. The required photon routing can be done either passively by beamsplitters, or actively, e.g., using electro-optic modulators (EOMs). The former is experimentally simple but does not allow for a deterministic selection of the measurement basis as the wave packets are randomly reflected and transmitted at the first beamsplitter. The latter enables low loss measurement at repetition rates compatible with the typically achievable generation rates in our scheme.

Figure 4

Fidelities of the GHZ (circles) and the cluster (dots) states in the presence of phonon dephasing. The first-order approximation (31) for both the GHZ and the cluster state is shown with solid lines and agrees well with the exact solution (30) for a large fidelity $\mathcal{F}\gtrsim 0.8$. Black, red, and blue curves correspond to the dephasing rates ${\gamma }_d/\gamma =0.01, {\gamma }_d/\gamma =0.03$, and ${\gamma }_d/\gamma =0.05$, respectively.

Figure 5

Conditional fidelities of the five-photon GHZ state for a Gaussian-shaped pulse as a function of the pulse length $\gamma T_{\mathrm{FWHM}}$ ($x$ axis) and lifetime of the excited state, $\mathrm{\Delta }/\gamma$ ($y$ axis) at a fixed $\mathrm{\Delta }=2\pi \times 16{\mathrm{ns}}^{-1}$. Panels (a) and (b) correspond to the fidelity with excitation errors ${\tilde{\mathcal{F}}}_{\mathrm{exc}}$ only and to the combined fidelity ${\tilde{\mathcal{F}}}_{\mathrm{comb}}$, respectively. In (b) we have assumed a ratio of $\mathrm{\Delta }/{\gamma }_d=1.7\times {10}^3$, which roughly corresponds to quantum dots in a 2 T magnetic field at a temperature of 1.8 K.

Figure 6

(a) Effect of frequency filtering. Optimized combined fidelities of the GHZ (circles) and the cluster (dots) states as a function of a number of photons with frequency filters such that ${\eta }_2=1, {\eta }_3=0.02$. Solid and dashed lines correspond to, respectively, the GHZ and the cluster state without frequency filtering. (b) Validity of the first-order and square-shaped pulse approximations. Comparison between the combined fidelity of the GHZ state from the exact numerical solution for a Gaussian driving pulse (40) (circles) and its first-order approximation for a square pulse (41) (black solid line), respectively. The exact and first-order fidelities of the cluster state are shown with red dots and dashed lines, respectively. Parameters values are given in the text.

Figure 7

State fidelities with imperfect branching. Numerically calculated fidelities of the GHZ and cluster states are shown circles and dots, respectively. The solid lines show the first-order perturbative expressions for both the GHZ and the cluster states. Black and blue correspond to the fidelities with and without frequency filters, respectively. Here we use numerically simulated branching parameters for quantum dots embedded in a photonic crystal waveguide [52]: ${\beta }_{\perp }=0.05, {\beta }_{\perp }^{\prime }=0.0025, {\beta }_{\parallel }^{\prime }=0.0025, {\beta }_{\parallel }=1-{\beta }_{\perp }-{\beta }_{\perp }^{\prime }-{\beta }_{\parallel }^{\prime }=0.99$, corresponding to a branching ratio of $B\approx 140$.

Figure 8

Total fidelity versus number of photons with all imperfections and frequency filtering taken into account. Circles and dots show the fidelity of the GHZ and cluster states, respectively, and the solid line shows the first-order perturbative approximation of both states (60). Red symbols correspond to the different contributions in Eq. (59), namely, dephasing ${\mathcal{F}}_{\mathrm{ph}}$ ($\times$), excitation errors ${\tilde{\mathcal{F}}}_{\mathrm{exc}}$ ($\square$), and imperfect branching ${\tilde{\mathcal{F}}}_{\mathrm{br}}$ ($+$). Here $\mathrm{\Delta }=2\pi \times 64$ GHz, ${\gamma }_d=0.06{\mathrm{ns}}^{-1}, {\gamma }_{\mathrm{opt}}=5.3{\mathrm{ns}}^{-1}$, and $\beta$-factors are the same as in Fig. 7.