Quantum computing exploits the complex many-particle quantum wavefunction and the laws of quantum mechanics for information processing. Taking advantage of the superposition principle of quantum states for parallel computing, it is becoming a superior system in the processing of enormous information at high efficiency beyond ordinary computers. Although a number of qubits, such as atom- and ion-trap based qubits, superconducting charge and flux qubits, spin- and charge-based quantum dots, nuclear spin, and photons, have been proposed and developed for quantum computing so far, the existing quantum systems remain bulky and fragile^1,2. Specifically, an ideal quantum computer has a very harsh requirement, i.e., the quantum effect and internal quantum operation has to be protected by a closed box which is isolated from the rest of the universe. This is mainly ascribed to the fact that the disturbance caused by the environmental noises, termed as decoherence, will quench the quantum effect and turn the quantum system into a classical one. Thus, it is of vital importance to protect quantum states from the dissipative environment and operation errors.

Among various efforts for quantum system protection, the motivation of decoherence-free subspace (DFS) is to encode quantum information utilizing specific quantum states that are intrinsically less susceptible to environmental noise, rather than using quantum error correction to detect and eliminate certain errors induced by the environment^3⇓⇓-6. For quantum states in the DFS, which are termed “dark states” (and the counterparts outside the DFS are termed bright states), the influences from the environment on different qubits cancel each other and make the whole system free from decoherence^7⇓-9. It is noted that the decay rate of a dark state is highly relevant to the interqubit distance r_ij, since the influence of vacuum fluctuations on different qubits cancels each other only when $r_{ij}\ll k_0^{-1}$, where, $k_0=2\pi /\lambda$ and $\lambda$ is the emission wavelength of the quantum emitters⁸. However, in the existing DFS systems, the emitters are placed at fixed positions in a cavity or along an optical waveguide, between the cavity mirrors in a linear trap or in the nodes of a standing light field. The pumping of the different emitters has to meet the requirement of “anti-phase-matching” rule, i.e., ${\overrightarrow{k}}_L·{\overrightarrow{r}}_{12}=(2n+1)\pi$ (${\overrightarrow{k}}_L$ is the wave-vector of the pump), so as to maximize the excitation of the dark state and avoid the excitation of the bright state. Thus, the interqubit distance is at wavelength scale and the dipole-dipole interaction can be ignored^10⇓⇓-13. In other words, the DFSs prepared in these works are only decoherence-free from certain dissipation channels (e.g., the cavity or waveguide mode induced dissipation channels), while are not decoherence-free from the vacuum environment (i.e., the free space other than the cavity/waveguide or other photonic structures). It remains a significant yet difficult task to modulate the state vectors in the DFS at deep sub-wavelength (DSW) interqubit distance for preparation of perfect dark states that are free from spontaneous decay into the free space^14⇓-16.

The key to address this fundamental problem is to break the traditional anti-phase-matching rule in the excitation of dark states. In the current work, a hybrid plasmonic quantum emitter (HPQE) was designed by coupling a dissipative plasmonic nanocavity (PNC) to two qubits located in the vicinity of the PNC. The PNC provides a new pumping approach to excite the two emitters through the asymmetric near-field hot spots by controlling the incident light polarization, which allows strong interqubit dipole-dipole interaction at the DSW scale and induces collective dissipative coupling to suppress the vacuum spontaneous decay. Moreover, the high loss of the PNC is harnessed to improve the decay rate of the bright state, which enables the scheme of using dissipation to remove the unwanted bright state while keeping the population of the dark state nearly unaffected¹⁷. These benefits enable the preparation of a nearly perfect DFS decoherence-free from both the cavity mode and the vacuum environment (>96% suppression of the vacuum spontaneous decay). More interestingly, a switching of pumping light polarization turns the antisymmetric superposition state $|D〉$ in the DFS into its counterpart, the symmetric superposition state $|B〉$, the second order correlation function at zero delay ($g^{\left(2\right)}\left(0\right)$) of which shows a wide-range tunability down to near zero. The HPQE in the current work provides an all-solid-state, robust, compact, and reconfigurable quantum platform for future quantum computing chips without rigorous environmental requirement.

As schematically illustrated in Fig. 1a, the HPQE system composed of a gold nanotriangle and two quantum qubits is located at the tips. The PNC is dominated by the electric dipole resonance, which exhibits a central resonance frequency ωa and a damping rate κ (see Supplementary Material 1). Each qubit i is simplified as a closed two-level system with a ground state $|g_i〉$ and an excited state $|e_i〉$, the latter of which exhibits a transition frequency ${\omega }_i$ and a vacuum spontaneous emission rate ${\gamma }_i$. The PNC and the qubits can be described using the lowering (raising) operators, $a\left(a^{†}\right)$ and ${\sigma }_i\left({\sigma }_i^{+}\right)$, respectively, and the interaction strength between the PNC and the ith qubit is denoted by gi. To manipulate the qubits, the system is pumped by a monochromatic laser $E=E_0\cos \left({\omega }_{L}t\right)$.

The dynamics of the hybrid system in vacuum was modelled with the Lindblad’s master equation. The master equation, as well as the total Hamiltonian and the Lindblad superoperator modeling the spontaneous decay can be written as^18,19.

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Where, ${\Delta }_a={\omega }_a-{\omega }_L$ and ${\Delta }_i$ denote the pumping detuning of PNC and the ith qubit, respectively. ${\gamma }_{ij}{|}_{i\ne j}={\gamma }_{12},{\gamma }_{ii}={\gamma }_i$ are used for abbreviations. It is noted that the problem in the frame rotating was dealed with the laser frequency ${\omega }_L$ under rotating-wave approximation.

Considering that the interqubit distance ${\overrightarrow{r}}_{12}$ is much shorter than the emission wavelength, the vacuum induced dipole-dipole interaction strength ${\Omega }_{12}=-G\left(k_0{r}_{12}\right)\sqrt{{\gamma }_1{\gamma }_2}$ and collective effect induced dissipation rate ${\gamma }_{12}=F\left(k_0{r}_{12}\right)\sqrt{{\gamma }_1{\gamma }_2}$ cannot be ignored^20⇓-22. They can be calculated by the following formula:

Where, $\widehat{p}={\widehat{p}}_1={\widehat{p}}_2$ and ${\widehat{r}}_{12}={\widehat{r}}_1-{\widehat{r}}_2$ are unit vectors along the qubit transition dipole moments and the line between qubits, respectively. ${\gamma }_{12}/\sqrt{{\gamma }_1{\gamma }_2}=F\left(k_0{r}_{12}\right)$ and ${\Omega }_{12}/\sqrt{{\gamma }_1{\gamma }_2}=-G\left(k_0{r}_{12}\right)$ are plotted in Fig. 1b, from which it can be concluded that these collective terms exhibit very small values when the interqubit separation is larger than the mean qubit emission wavelength while the values increase dramatically with decreasing separation when $r_{12}<\lambda /2$. In the current system, since $r_{12}/\lambda \approx 0.1$, and $\widehat{p}\parallel {\widehat{r}}_{12}$ is set for simplicity, which leads to $F\left(k_0{r}_{12}\right)\approx 0.96$ and $-G\left(k_0{r}_{12}\right)\approx 7.13$.

The HPQE is composed of two qubits and a PNC, and both of them should be taken into consideration. However, what we are interested in is the subsystem of the qubits, and thus, we traced off the degree of freedom of the PNC to simplify the theoretical design. Adiabatic elimination of the cavity bosonic operators was applied in the weak qubit-plasmon coupling regime, during which the information of inter-qubit dynamics was well preserved, and the influence from the PNC were taken into consideration in the effective coefficients in the Hamiltonian and the Lindbladian. In the strong coupling regime, such an elimination will not work and the whole coupled system must be taken into consideration²³. However, the main result of the current work is still applicable since the proposal is based on the decay rate contrast of the collective states, which still persists in the strong coupling regime (see Supplementary Material 2).

The simplified Hamiltonian and Lindbladian after adiabatic elimination could be expressed as follows^19,24,25. (see Supplementary Material 2 for derivation of effective parameters):

It is noted that these equations show the same form in the absence of the PNC, just with the coefficients modified. If the two qubits are identical and symmetrically located (which is our case throughout the rest of the article if not specified), we have ${\gamma }_1={\gamma }_2=:\gamma$, $g_1=g_2=:g$, and

where, ${\Gamma }_{\kappa }=4g^2/\kappa$ is the spontaneous decay rate enhancement of a single emitter induced by the Purcell effect²⁶. In ther following numerical calculations, we choose $\gamma =35\mathrm{\mu }\mathrm{e}\mathrm{V}$ and $\kappa =100\mathrm{meV}$, detailed information could be seen in Supplementary Material 1^{27, 28}.

To explore the DFS in the HPQE, the product state formed by state ${|a〉}_1$ of qubit 1 and state ${|b〉}_2$ of qubit 2 was denoted with $|ab〉$, i.e., ${|a〉}_1\otimes {|b〉}_2=:|ab〉$. Due to the energy exchange between qubits^29,30, the Hamiltonian Eq. (6) are diagonalized to yield four orthogonal collective eigenstates, viz. $|E〉=|ee〉$, $|G〉=|gg〉$, $|D〉=(|eg〉-|ge〉)/\sqrt{2}$, and $|B〉=(|eg〉+|ge〉)/\sqrt{2}$. When the interqubit distance $r_{12}\to 0$, it is generally claimed that the singlet state $|D〉$ constitutes a one-dimensional DFS in a two-qubits situation (see Supplementary Material 3 for detail), i.e., $|D〉$ is free from the environment induced dissipation. This result is obvious according to the quantum jump theory, i.e., a quantum jump takes place in the time interval $[t,t+\mathrm{d}t]$ with probability $P\left(t\right)=〈\psi \left(t\right)|\gamma S^{+}{S}^{-}|\psi \left(t\right)〉\mathrm{d}t$ ³¹ (Here, $S^{+}={\sigma }_1^{+}+{\sigma }_2^{+}$ and $S^{-}={\sigma }_1+{\sigma }_2$). It follows that if an initial state $|\psi 〉$ fulfils $S^{-}|\psi \left(0\right)〉=0$, it will remain undisturbed by the quantum jump at any time.

With $r_{12}\ne 0$ (even if $r_{12}$ is small), the dark state $|D〉$ exhibits a decay rate. For an intuitive understanding, the evolution of the state population was calculated with the effective master equation in Supplementary Material 4 and the following equations could be obtained:

From these equations, the spontaneous decay rates of [Math Processing Error]|D〉 and [Math Processing Error]|B〉 are derived to be

It can be inferred that the state $|D〉$ is decoherence-free from the cavity-induced decay channel and the local spontaneous decay rate ${\gamma }_{12}$ is partly cancelled by the collective-effect-induced decay rate $r_{12}\to 0$. When $r_{12}\to 0$, ${\gamma }_{12}\to \gamma$ and $|D〉$ is perfectly isolated from any dissipation, which is in accordance with the result of the quantum jump theory. As the interqubit distance is at the DSW scale in the HPQE, the spontaneous decay rate of $|D〉$ is sufficiently small. Thus, $|D〉$ is termed the “dark state”.

Attention should also be paid to the symmetric superposition state $|B〉$. As the counterpart of $|D〉$, the spontaneous decay rate of $|B〉$ is enhanced by both the collective effect and the Purcell effect from the PNC: ${\tilde{\gamma }}_B=\gamma +{\gamma }_{12}+2{\Gamma }_{\kappa }$. Thus, it is termed “bright state”. It is noted that the dark and bright states do not arise from the use of a plasmonic structure. But in the current system, it can be seen that, the bright state exhibits a higher decay rate and tends to be “brighter” with presence of the PNC, while the decay rate of dark state remains unchanged. Consequently, the terms “bright” and “dark” are adopted due to the fact that both of them exhibit different decay rates and they behave differently to the cavity induced loss channel. The different behavior of the bright and dark states to the dissipation endows them with a drastic decay rate contrast, for example, at the setup of a weak PNC-qubit coupling strength of $g=0.1\kappa$ (see Supplementary Material 1 ³²), $g=0.1\kappa$ and ${\tilde{\gamma }}_D\approx 1.36\mathrm{\mu }\mathrm{e}\mathrm{V}$. This contrast is of great significance when using dissipations to remove the unwanted $|B〉$ state for entanglement generation, as will be discussed later.

The dependency on pumping polarization in the preparation of DFS provides the possibility to switch the quantum electrodynamics of the HPQE all-optically. Besides the asymmetric pumping discussed above, we also investigate the quantum electrodynamics of the HPQE under symmetric pumping. The energy level shift of the bright state induced by the qubit-qubit interaction will be cancelled by the pump laser detuning if $-\tilde{\Delta }\approx {\tilde{\Omega }}_{12}\gg \tilde{\gamma },{\tilde{\gamma }}_{12},\tilde{\eta }$. In this situation, the transition $|G〉\to |B〉$ is in resonance while the transition $|B〉\to |E〉$ is rarely excited because of energy mismatch. The system behaves like a single two-level system composed of the ground state $|G〉$ and the excited state $|B〉$, as shown in Fig. 2b. Such an energy level anharmonicity mimics the nonlinearity of the Jaynes-Cummings ladder^42⇓-44, where the excitation of emitter-cavity system by the first photon blocks the transmission of the second photon, while the energy level nonlinearity arises from interqubit interaction instead of qubit-photon strong coupling in the plexcitonic systems^45⇓⇓-48.

Moreover, we calculated $g^{\left(2\right)}\left(0\right)$ to show the quantum signature of the correlated emission from the coupled-qubit system ⁴⁹ (see details in supplementary material 8):

Along with the weak pumping condition $\stackrel{\sim }{\eta }=0.1\tilde{\gamma }$ and zero cavity-qubit detuning, the dependency of $g^{\left(2\right)}\left(0\right)$ on qubit-pump detuning and different coupling strength is illustrated in Fig. 4a, which unambiguously demonstrated the heuristic discussion above.

To enable single photon emission, $g^{\left(2\right)}\left(0\right)\to 0$ is desirable. Note that a weaker qubit-cavity coupling strength may bring a smaller $g^{\left(2\right)}\left(0\right)$, but it results in a lower excitation and emission efficiency. In other words, there is a trade-off between the purity and the brightness of this effective single photon source. Another approach to modulate $g^{\left(2\right)}\left(0\right)$ is to tune the cavity-qubit detuning ${\Delta }_{aq}$, which changes $\tilde{\Delta }$ and ${\tilde{\Omega }}_{12}$ simultaneously. Fig. 4b shows the evolution of $g^{\left(2\right)}\left(0\right)$ at different ${\Delta }_{aq}$. It can be seen that the minimum of $g^{\left(2\right)}\left(0\right)$ tends to be smaller while the maximum becomes lager with increasing ${\Delta }_{aq}$ (see Supplementary Material 8), indicating an enhanced tunability of $g^{\left(2\right)}\left(0\right)$. However, ${\Delta }_{aq}$ exhibits an upper limit because the coupling strength g (which we assumed to be 0.1κ throughout the analysis) cannot be maintained when the cavity-qubit detuning is too large.

In addition to the tunability of $g^{\left(2\right)}\left(0\right)$, this effective single photon emitter shows its superiority with enhanced brightness. For a single qubit coupled to a PNC (i.e., only qubit 1 exists in the system), the mean photon rate emitted in the steady state is as follows:

where, ${\kappa }_{\text{rad}}$ and ${\kappa }_{\text{nrad}}$ are the dissipation rate of the PNC to the radiative and nonradiative channel, respectively^50,51. All the emitted photons (not only from the nanoantenna) are taken into consideration, and $\tilde{\gamma }$ is used instead of ${\Gamma }_{\kappa }$. For the weak pumping ${\tilde{\eta }}_1=0.1\tilde{\gamma }$, $〈\dot{n}〉\propto \tilde{\gamma }$ and ${\tilde{\eta }}_1\propto \tilde{\gamma }$ can be obtained, i.e., a single emitter coupled to plasmonic antennas achieves an increased spontaneous emission rate^52,53. Moreover, treating the HPQE as one single photon emitter, it shows an effective upper level $|B〉$ with spontaneous decay rate ${\tilde{\gamma }}_B\approx 2\tilde{\gamma }$, and thus the brightness is twice as high as that of a single qubit coupled to a PNC (while the extraction efficiency will remain unchanged). It can be seen from Eq. (22) that the photon emission rate $〈\dot{n}〉$ decreases with an increasing nonradiative decay rate ${\kappa }_{\mathrm{nrad}}$ of the PNC, i.e., the emitters are quenched because of their coupling to the multipolar resonances, which couple to the propagation mode very weakly and are mainly dissipated by Ohmic loss or Landau damping. However, as the multipolar resonances are suppressed at the desired wavelength for the nanotriangle cavity, the configuration shows great superiority of quenching-resistance ^54-56 (see Supplementary Material 1). This is a general strategy and can be applied to all the state-of-the-art single photon emitters for rational control of their brightness.

In summary, different schemes have been proposed for preparing the entangled dark state in the DFS of a HPQE system by removing the unwanted state through dissipation. The proposed dark state provides robust resistance to the environment induced dissipation, opening up a promising way toward faithful and robust on-chip quantum manipulation. The DFS can be optically switched to be a single photon emitter in symmetric pumping. The tunable $g^{\left(2\right)}\left(0\right)$ of the emitter manifests crucial steps for the high quality and tunability of quantum light source technology. Although the discussion is limited in the weak coupling regime, similar phenomena may still persist in the strong coupling regime where the adiabatic approximation is inapplicable. In the current work, the dark state and the bright state are addressable only independently. However, if the quantum information can be retrieved easily from the dark state, for example, by switching it into the bright state, it manifests potential applications such as quantum memory. It is anticipated that the concept of HPQE can be scaled up for application in robust integrated quantum computing system.

Acknowledgments L.L. acknowledges support from the National Key Research and Development Program of China (Grant No. 2020YFA0715000), the National Natural Science Foundation of China (Grant No. 62075111), and L.L. acknowledges the Tsinghua University Initiative Scientific Research Program; H.-B.S. acknowledges support from the National Natural Science Foundation of China (Grant No. 61960206003). Y.L, H.Z., L.L. and H.-B.S. acknowledge support from the State Key Laboratory of Precision Measurement Technology and Instruments.

Declaration of Competing Interest The authors declare no competing interests.