Generating maximal entanglement between spectrally distinct solid-state emitters

TL;DR

This paper presents a method to generate maximal entanglement between spectrally distinct solid-state emitters using multi-photon scattering, relaxing the need for identical emitters by optimizing optical states.

Contribution

It introduces a novel approach to entangle non-identical solid-state emitters through multi-photon scattering and identifies optimal conditions for entanglement generation.

Findings

Deterministic maximal entanglement achievable with spectrally distinct emitters.

Higher photon numbers can overcome spectral mismatch.

Quasi-monochromatic photons are optimal for entanglement.

Abstract

We show how to create maximal entanglement between spectrally distinct solid-state emitters embedded in a waveguide interferometer. By revealing the rich underlying structure of multi-photon scattering in emitters, we show that a two-photon input state can generate deterministic maximal entanglement even for emitters with significantly different transition energies and line-widths. The optimal frequency of the input is determined by two competing processes: which-path erasure and interaction strength. We find that smaller spectral overlap can be overcome with higher photon numbers, and quasi-monochromatic photons are optimal for entanglement generation. Our work provides a new methodology for solid-state entanglement generation, where the requirement for perfectly matched emitters can be relaxed in favour of optical state optimisation.

Figures

Equations

$$t_{\alpha }(\omega )=\frac{\hslash \omega -E_{\alpha }-i\hslash ({\Gamma }_{\alpha }-{\gamma }_{\alpha })/2}{\hslash \omega -E_{\alpha }+i\hslash ({\Gamma }_{\alpha }+{\gamma }_{\alpha })/2}.$$

$${\mathcal{C}}_{\text{avg}}\equiv \sum _{(p,q)}\text{Pr}(p,q)\mathcal{C}\left({\rho }_{(p,q)}\right).$$

Full text

Quantum technologies promise dramatic improvements in computing and communication by utilizing quantum entanglement between qubits Nielsen and Chuang (2000). Although many promising quantum technology architectures have emerged over the last two decades, none are free from the practical challenges presented by high-fidelity quantum control and scalability. For example, superconducting circuit implementations enjoy excellent coherence properties but operate slowly Wendin (2017), while trapped ion qubits can be prepared with almost unit fidelity but are difficult to scale Kielpinski et al. (2002). Solid state architectures, such as optically coupled spin systems, compete on speed and scalability. They include semiconductor quantum dots and nitrogen-vacancy centres. Large optical non-linearities in solid-state systems are now very common Hallett et al. (2018); Javadi et al. (2015); Sipahigil et al. (2016), and solid-state emitters are readily integrated into complex photonic structures, further enhancing the light-matter interaction Lodahl et al. (2015). However, there are many challenges still to overcome. For example, charge noise and phonon scattering have limited the size of the optical non-linearities observed thus far Hallett et al. (2018).

Another major drawback to solid-state emitters is that the central energies and lifetimes of their transitions are highly dependent on the fabrication process, and vary significantly both across and within samples Michler (2009). Known methods for entangling solid-state qubits require emitters with identical energies to facilitate path-erasure techniques Barrett and Kok (2005); Metz and Barrett (2008). This adds a practically insurmountable overhead to the process of matching multiple solid-state qubits for creating large entangled states Raussendorf and Briegel (2001). Stark shifting and strain tuning the emitter transitions has been employed to tune solid-state emitters onto resonance Stockill et al. (2017); Bernien et al. (2013); Hensen et al. (2015), but this requires a substantial technical overhead and arbitrary emitters in a sample cannot in general be tuned onto resonance. Here, we propose a process for generating entanglement that is robust against spectral variations in the emitters’ transition energies and line-widths. We show that photons, linear optics, and photon counting suffice to create deterministic entanglement between imperfectly matched emitters, revealing a rich underlying structure of multi-photon scattering off two non-identical emitters. While many challenges remain, this work removes a major obstacle to a scalable solid state quantum technology architecture.

Our setup is shown in Fig. 1. Two solid-state emitters each have an $L$-type level structure, with two stable low-lying spin states ($\ket{\uparrow}$, $\ket{\downarrow}$), and a dipole transition that couples a spin state to an excited state $\ket{e}$. The transition energy for emitter $\alpha=1,2$ is $E_{\alpha}$, and the polarisation is circular due to selection rules. The emitters are initially prepared in the product state $(\ket{\uparrow}+\ket{\downarrow})(\ket{\uparrow}+\ket{\downarrow})/2$, and embedded in a waveguide Mach-Zehnder interferometer at c-points, where perfect correlation between propagation direction and circular polarisation occurs Lodahl et al. (2017). Consequently, the emitters scatter circularly polarised light only in the forward direction, as was demonstrated recently using semiconductor quantum dots under an applied magnetic field Söllner et al. (2015); Coles et al. (2016); Lang et al. (2017); Hurst et al. (2018). For a lossless waveguide, the emitter will impart a $\pi$-phase shift to each photon that is on resonance with the transition Fan et al. (2010); Roulet and Scarani (2016); Nysteen et al. (2017); Hurst and Kok (2018). The input to the interferometer is a two-mode Fock state $\ket{n,m}$, and the detectors $D_{1}$ and $D_{2}$ produce classical signatures $(p,q)$ indicating the presence of $p$ and $q$ photons, respectively.

Assuming the emitters are identical, a single monochromatic resonant photon injected into either one of the input arms of the interferometer will scatter from one of the emitters, and after the final beam splitter the light-matter degrees of freedom are in the multi-partite entangled state $\left(\ket{\Phi^{-}}\otimes\ket{1,0}-\ket{\Psi^{-}}\otimes\ket{0,1}\right)/\sqrt{2}$, where $\ket{0,1}$ and $\ket{1,0}$ is the two-mode single photon state at the interferometer output. A photon detector signature $(1,0)$ or $(0,1)$ heralds the maximally entangled spin state $\ket{\Phi^{-}}=\left(\ket{\uparrow,\uparrow}-\ket{\downarrow,\downarrow}\right)/\sqrt{2}$ or $\ket{\Psi^{-}}=\left(\ket{\uparrow,\downarrow}-\ket{\downarrow,\uparrow}\right)/\sqrt{2}$. This method can be made robust to photon loss provided that the emitter detuning is only a fraction of the emitter line-width. Mahmoodian et al. showed how this can form a building block for distributed quantum computing Mahmoodian et al. (2016).

In practice, both the line-widths and transition energies vary significantly between solid-state emitters, and it was generally assumed that this prohibits the creation of perfect entanglement using linear optics and photodetection. In this case, the input photon can no longer be resonant with both emitters simultaneously. With $\hbar\omega$ the single-photon energy, $\Gamma_{\alpha}$ the unidirectional emission rate of emitter $\alpha=1,2$ into the waveguide, and $\gamma_{\alpha}$ the corresponding coupling to non-guided modes, the scattering process is described by the transmission coefficient Rephaeli and Fan (2013)

[TABLE]

We characterise the emitter loss by $\beta_{\alpha}\equiv\Gamma_{\alpha}/(\Gamma_{\alpha}+\gamma_{\alpha})$. For non-zero emitter detuning $\delta\equiv E_{2}-E_{1}$, $t_{\alpha}(\omega)$ ceases to be a $\pi$ phase shift, and for $\beta_{\alpha}<1$, $t_{\alpha}(\omega)$ is no longer a pure phase shift. The setup then does not create maximally entangled states deterministically anymore. Nevertheless, we will now demonstrate how tailoring the optical input state $\ket{n,m}$ into the Mach-Zehnder interferometer leads to deterministic maximal entanglement between two spectrally distinct emitters.

In general, a detector signature $(p,q)$ indicates that the two emitters are in a mixed entangled state. We use the concurrence $\mathcal{C}(\rho)$ for a two-qubit state $\rho$ to quantify this entanglement Wootters (1998). Each signature $(p,q)$ occurs with probability $\text{Pr}(p,q)$ and results in an emitter state $\rho_{(p,q)}$, leading to a concurrence $\mathcal{C}({\rho_{(p,q)}})$. We define the average concurrence as

[TABLE]

This is an appropriate figure of merit, since it provides a lower bound for the amount of entanglement expected from a given experiment without post-selection. The entanglement in the two-qubit state can be increased by discarding measurement outcomes corresponding to below-average concurrences. This comes at the expense of the rate of entanglement generation.

The amount of entanglement that can be generated between the two spectrally distinct emitters with a single probe photon is shown in Fig. 2. The single photon protocol is analysed using linear optics transformations Mahmoodian et al. (2016), while a multiphoton input requires taking into account the non-linear nature of the interaction Nysteen et al. (2017) (see SI for details). As expected, for spectrally distinct emitters the average concurrence does not reach its maximal value [Fig. 2(b)]. The amount of entanglement is determined by two competing processes. On the one hand, which-path information for the probe photon must be erased, while at the same time the phase shift induced by the photon scattering event must be maximised. Tuning closer to either emitter increases the relative phase shift but also imparts a degree of path information onto the probe, as the light-matter interaction is now stronger for one of the emitters. For emitters with finite detuning and line-width it is not obvious which photon energy maximises the average concurrence. Three emitter line-widths are shown in Fig. 2(a), and Fig. 2(b) shows the corresponding $\mathcal{C}_{\text{avg}}$. The linewidths shown correspond to emitters with $1$, $0.66$ and $0.33$ ns lifetimes, typical of semiconductor QDs benefiting from modest Purcell enhancements Hughes (2004). Increasing the line-width of the emitters leads to a larger spectral overlap, thereby erasing some of the which-path information and increasing $\mathcal{C}_{\text{avg}}$. Fig. 2(c) shows the optimal frequency of the input photon that maximises $\mathcal{C}_{\text{avg}}$. For narrow line-widths it is preferable to tune the photon energy away from the mean emitter energy ($\hbar\omega-E_{1}=0.5$ $\mathrm{\SIUnitSymbolMicro}$eV for $\delta=1.0$ $\mathrm{\SIUnitSymbolMicro}$eV), and towards resonance with one of the emitters. Though this reduces the concurrence in the state heralded by a click at detector $D_{2}$, it does increase the probability of a successful scattering event.

One may expect that a photon with a wide frequency bandwidth that overlaps with both emitters will improve the entanglement generation. Fig. 2(d) shows the average concurrence for a single probe photon with Lorentzian, Gaussian and square spectral profiles, centred at $\hbar\omega=E_{1}+\delta/2$, as a function of the photon bandwidth. We find that increasing the bandwidth of the input photon only degrades the average concurrence, and a narrow-band probe is always preferable. We attribute this to the reduced temporal extent of the photon at larger bandwidths, which increases the probability of exciting the emitter, and thus the fraction of light emitted incoherently through spontaneous emission. This reduction is particularly noticeable for a Lorentzian wave-packet, where a close spectral match with the emitter increases the excitation probability. We conclude that for given emitter detuning and line-widths, the maximum $\mathcal{C}_{\text{avg}}$ of the single-photon case is limited by the competing requirements of maximising the induced phase shift and path erasure.

Next, we consider whether two photons can increase the average concurrence. Consider an input state of two identical monochromatic photons $\ket{n,m}=\ket{1,1}$ entering the interferometer. They will evolve into a two-photon noon state $(\ket{2,0}-\ket{0,2})/\sqrt{2}$ via Hong-Ou-Mandel interference on the first beam splitter Hong et al. (1987); Nysteen et al. (2015) and interact with the emitters. Entanglement is then heralded by three detector signatures: two photons in $D_{1}$, two photons in $D_{2}$, or a coincidence count. Using two probe photons leads to a rich structure in the average concurrence, and it is now possible to reach deterministic maximal entanglement for spectrally distinct emitters with finite line-width. The reason for the two-photon advantage can be determined via inspection of Fig. 3(a), where $\mathcal{C}_{\text{avg}}$ is shown as a function of the detuning between the photon energy and the transition energy of the first emitter. In the current example where $\delta=1.0$ $\mathrm{\SIUnitSymbolMicro}$eV, maximum entanglement fidelity occurs for emitters with line-widths of 1.0 $\mathrm{\SIUnitSymbolMicro}$eV and input photons with energy $\hbar\omega=E_{1}+\delta/2$. Comparing this value to Fig. 2(a), this input energy corresponds to the point where the emitter spectra are at half of their maximum intensity. For quasi-monchromatic input states, the imparted phase shift is additive in photon number, i.e., for the two-photon case, each photon imparts a $\pi/2$ phase shift to the emitter and therefore achieves the required $\pi$ phase shift. We consider more general emitter detuning and line-width examples in Fig. 4 and in the Supplementary Information.

We study again the effect of broadband probe photons on the entanglement generation process. In this case the phase shift is no-longer additive in photon number, since the interaction becomes non-linear. When two photons are present, the emitter may be excited, which opens a pathway for stimulated emission, such that the coherence of the wave-packet is maintained. Competition between spontaneous and stimulated emission processes leads to the non-monotonic behaviour in $\mathcal{C}_{\text{avg}}$ presented in Fig. 3(b). The two-photon interaction strength increases when the optical pulse width is broadened to the scale of the emitter line-width Nysteen et al. (2017). Although an increased bandwidth generally reduces $\mathcal{C}_{\text{avg}}$, a local maximum appears close to the emitter line-width. We attribute this to the stimulated emission process, which is maximised for photons that are closely matched to the emitters’ spectral profile. Fig. 3(c) shows the dependence of $\mathcal{C}_{\text{avg}}$ on the loss rate $\beta$. In the regime where it is preferable to tune the input photon to the average emitter energy, the effect of reducing the beta factor of either emitter causes the average concurrence to fall in the same manner as if both beta factors were reduced. In the alternative regime, where it is optimal to tune onto one emitter, a difference in beta factors introduces an asymmetry in the behaviour of the average concurrence and it becomes preferable to tune onto the emitter that is more efficiently coupled to the waveguide. The two-photon entanglement generation process outperforms the single-photon process for all values of $\beta$. In Fig. 3(d) we show how two-photon input states can achieve a significant increase in $\mathcal{C}_{\text{avg}}$ over single photons for larger detunings.

In certain regimes the two-photon process is capable of generating maximal entanglement between spectrally distinct emitters where the single-photon process fails. We studied the robustness of this effect with respect to the system parameters. In Figs. 4(a) and 4(b) we show the maximum $\mathcal{C}_{\text{avg}}$ (optimised over photon frequency $\omega_{\rm opt}$) for the one- and two-photon input states as a function of the emitter detuning and the emitter line-width ratio. The single-photon case outperforms the two-photon case if the emitters are spectrally collocated, or if one of the emitters significant overlaps the other, however narrow the line-width. Crucially, however, by exploiting the multi-photon additivity of the phase shift, a two-photon process can efficiently generate entanglement for any finite detuning without requiring arbitrarily small emitter lifetimes. The converse of this is also true: for any combination of line-widths $\Gamma_{\text{1}}$ and $\Gamma_{\text{2}}$ there exists a non-zero emitter detuning which creates deterministic maximal entanglement given the optimal two-photon input state. In practice, this means a much greater freedom in matching solid-state emitters for entanglement generation in a Mach-Zehnder interferometer than previously thought.

We extended the entanglement generation process to monochromatic $\ket{n,m}$ Fock states into the interferometer. In Figs. 4(c) and 4(d) we show the maximum $\mathcal{C}_{\text{avg}}$ as a function of the emitter detuning and the emitter line-width ratio for input states $\ket{2,1}$ and $\ket{2,2}$, respectively (for more examples, see the Supplementary Information). There is a marked improvement in the entanglement generation over the single- and two-photon processes, with larger areas of parameter space achieving a near-unity $\mathcal{C}_{\text{avg}}$. Remarkably, this indicates that a wide range of imperfections in the fabrication of two identical emitters can be overcome by optical state optimisation. Note that the $\ket{2,1}$ case inherits features from both the $\ket{1,1}$ and $\ket{1,0}$ processes. It therefore performs well for both spectrally collocated emitters and those with finite detuning. A similar compound structure is visible in Fig. 4(d), where an input state $\ket{2,2}$ shows a double two-photon structure compared to the $\ket{1,1}$ input in Fig. 4(b). A clear trend emerges, where larger spectral emitter detuning can be overcome by higher number input states $\ket{n,m}$ (see Supplementary Information).

There are several practical challenges turning this entanglement generation process into a useful quantum technology. First, a reduced coupling of the emitter to the waveguide mode will reduce the phase shift imparted on the photons, and therefore lower the average concurrence. Second, the photons may be scattered out of the waveguide mode or be lost in the detection process. However, the use of fast, high-efficiency photon number resolving detectors will mitigate this problem, and such detectors are actively developed Divochiy et al. (2008); Heath et al. (2015). Third, the photons must be created in tuneable identical quasi-monochromatic modes. There are a number of ways this can be achieved over a wide frequency range. Spontaneous Parametric Downconversion (SPDC) is inherently tuneable Klyshko et al. (1970); Tanida et al. (2012) and frequency filtering will create the optimal quasi-monochromatic pulses as well as remove unwanted frequency entanglement. The resulting photon generation rate reduction can be mitigated using multiplexing, which has been demonstrated both for SPDC photon sources Joshi et al. (2018) and tuneable quantum dot sources Wang et al. (2017). Alternatively, tuning of single photon pulses is possible via frequency conversion Li et al. (2016). Finally, dephasing will have an impact on the entanglement generation process. The dominant dephasing mechanism for spin-doped solid-state emitters is nuclear spin interactions Merkulov et al. (2002). While this naturally leads to a random precession of the spin ground-state, there are a number of strategies based on dynamical decoupling that may be used to suppress its impact Viola et al. (1999); Press et al. (2010); Kuhlmann et al. (2013). In addition, solid-state emitters are subject to charge fluctuations and phonon interactions. The former leads to spectral wandering occurring on a microsecond timescale, which may be overcome by operating the process on shorter timescales Berthelot et al. (2006). Phonon scattering leads to sidebands Iles-Smith et al. (2017a) that can be removed through frequency filtering or by placing the emitter in an optical cavity Grange et al. (2017); Iles-Smith et al. (2017b).

In conclusion, we presented a robust entanglement generation mechanism between two solid-state qubits embedded in a Mach-Zehnder interferometer. Entangling techniques that use solid-state emitters are well-known to place very stringent requirements on the spectral identity of the emitters Barrett and Kok (2005). Our approach overcomes these restrictions by showing how to tailor multi-photon input states, mitigating a long considered weakness of solid-state emitters. We found that maximal deterministic entanglement between increasingly distinct emitters is possible using higher photon number input states, revealing a rich structure in multi-photon scattering from two emitters with different energies and line-widths. Our work provides a new methodology for solid-state entanglement generation, where the requirement for perfectly matched emitters can be relaxed in favour of optical state optimisation.

Acknowledgements.—

the authors thank A.P. Foster, D. Hallett, and M.S. Skolnick for useful suggestions, and EPSRC for financial support. K.B.J and J.M. acknowledge funding from the Danish Council for Independent Research (DFF-4181-00416). J.I.S. is supported by the Royal Commission for the Exhibition of 1851.

† DLH and KBJ contributed equally to this project.