Multiqubit entanglement is extremely important to perform truly secure quantum optical communication and computing operations. However, the efficient generation of long-range entanglement over extended time periods between multiple qubits randomly distributed in a photonic system remains an outstanding challenge. This constraint is mainly due to the detrimental effects of decoherence and dephasing. To alleviate this issue, we present engineered epsilon-near-zero (ENZ) nanostructures that can maximize the coherence of light–matter interactions at room temperature. We investigate a practical ENZ plasmonic waveguide system, which simultaneously achieves multiqubit entanglement in elongated distances, extended time periods, and, even more importantly, independent of the emitters' positions. More specifically, we present efficient transient entanglement between three and four optical qubits mediated by ENZ with results that can be easily generalized to an arbitrary number of emitters. The entanglement between multiple qubits is characterized by computing the negativity metric applied to the proposed nanophotonic ENZ configuration. The ENZ response is found to be substantially advantageous to boost the coherence between multiple emitters compared to alternative plasmonic waveguide schemes. Finally, the super-radiance collective emission response at the ENZ resonance is utilized to design a high fidelity two-qubit quantum phase gate that can be used in various emerging quantum computing applications.

Quantum entanglement1 lies at the heart of quantum teleportation, quantum cryptography, and other quantum information processing protocols that are expected to have paramount importance in secure quantum optical communications. However, it is usually achieved at extremely short distances and under very short time periods mainly due to the weak and incoherent interactions between qubits.2 To overcome this limitation, the coupling of various quantum emitters (also known as qubits) to dissipative plasmonic reservoirs, such as plasmonic waveguides, nanospheres, and nanoantennas, has been proposed,3,4 resulting in significantly enhanced and coherent dipole–dipole interactions. These plasmonic quantum electrodynamic systems can serve as efficient platforms to achieve effective and long-lived qubit–qubit entanglement.5 Nevertheless, the vast majority of relevant quantum entanglement studies are focused on the most usual case where the plasmonic system interacts with only two quantum emitters.6–12 Interestingly, only a few studies13–15 have been focused on the entanglement of multiple emitters coupled to simplified plasmonic systems based on nanoparticles or arrays of them. The metric of concurrence, which is typically used to characterize the entanglement between two qubits, is not applicable to three or more qubits and more complicated quantum optical metrics, such as negativity16 and genuine multipartite entanglement,17 should be employed. In addition, the emitters' spatial positions along simplified plasmonic systems (nanoparticles) are crucial to the efficient quantum entanglement performance.18 In plasmonic nanoparticles or waveguides sustaining surface plasmons, each qubit has to be accurately placed at a specific location where the density of optical states is locally maximized.19,20 Currently, however, large field enhancement is achieved in plasmonic systems only over very limited spatial extents. In addition, the development of the envisioned integrated quantum photonic circuitry will require efficient coupling between quantum emitters and nanophotonic structures, where, ideally, the coupling will need to be independent of the emitters' spatial locations. Addressing these grand challenges will be the key to enable efficient long-range multiqubit entanglement.

To overcome these limitations, metamaterials exhibiting epsilon-near-zero (ENZ) permittivity response have attracted increased attention21–24 due to their unique and fascinating features to enable uniform field enhancement in elongated regions leading to numerous potential applications in boosting optical nonlinearities,25–29 integrated photonic devices,30,31 and efficient optical interconnects.32 Here, we investigate an array of engineered plasmonic waveguides that exhibit an effective ENZ response at their cutoff frequency.25 Strong coupling of the incident light inside each waveguide is achieved, accompanied by a large field enhancement and uniform phase distribution,33 a combination of properties ideally suited to boost the coherent interaction between different emitters embedded in the nanochannels.34 

Recently, a variety of quantum optical effects, such as enhanced spontaneous emission,35 boosted super-radiance,36,37 resonance energy transfer and bipartite entanglement,12 have been achieved by using such ENZ plasmonic configurations. In this Letter, we present efficient multipartite entanglement between three and four qubits mediated by ENZ plasmonic waveguides with results that can be easily generalized to an arbitrary emitter number. The entanglement of multiple emitters coupled to other plasmonic waveguide systems is also explored and compared to ENZ. It is explicitly proven that only the ENZ system can substantially generate prolonged entanglement among multiple quantum emitters independent of their positions in the nanowaveguide. Moreover, it is theoretically predicted that the ENZ plasmonic waveguide can realize a deterministic quantum phase gate between two qubits with high fidelity and long separation distances due to the super-radiance collective emission state at the ENZ resonance. A way to alleviate the inherent losses of metals, and, as a result, reduce decoherence and dephasing, based on the introduction of gain in the nanowaveguides that subsequently increases the fidelity of the proposed quantum phase gate is also presented.

The proposed plasmonic waveguide design is shown in Fig. 1(a), where a narrow rectangular slit of width w, height t ≪ w, and length l is carved inside a silver (Ag) screen and loaded with a homogeneous dielectric medium with relative permittivity ε=2.2. This configuration consists the unit cell of an array of waveguides with periodicities equal to a and b. The waveguide's width w is appropriately designed to tune the cutoff frequency of its dominant quasi-transverse electric (quasi-TE) mode to be equal to 295THz, where an effective ENZ response can be achieved25 (more details in the supplementary material). We consider three identical z-oriented quantum emitters, such as quantum dots (QDs), located inside the plasmonic waveguide channel with equal separation distances d to obtain a fair comparison with the response of other plasmonic waveguides, as presented later. Each emitter can be regarded as a qubit with two energy states: ground |g and excited |e, and we assume that only one qubit is excited initially. We assume that the emitters are sufficiently far from each other, so that interatomic Coulomb interactions can be ignored. All the states of interest and transitions between them are illustrated in Fig. 1(b), where |Ui represents the excited state of the ith emitter when all the other emitters are in the ground state, and |ggg is the state where all emitters are in the ground state. For simplicity, the three identical qubits are assumed to have the same transition frequencies, chosen to be equal to the ENZ cutoff frequency. Under the Born–Markov and rotating wave approximations and assuming weak excitation approximation at the weak coupling regime, a master equation for the three-qubit system, represented by the reduced density matrix ρ(t), can be derived and solved numerically (more details provided in the supplementary material).

FIG. 1.

(a) Geometry of the ENZ plasmonic waveguide with z-oriented emitters embedded in the nanochannel. (b) Quantum states diagram of three emitters.

FIG. 1.

(a) Geometry of the ENZ plasmonic waveguide with z-oriented emitters embedded in the nanochannel. (b) Quantum states diagram of three emitters.

Close modal

Once the density matrix ρ(t) is computed, the entanglement of three or more qubits cannot be characterized by the conventional concurrence function,12 but needs to be quantified by means of alternative more complicated quantum optical metrics, such as the entanglement entropy,38 state negativity,15 or genuine multipartite entanglement measurements.13 More specifically, for a quantum system consisting of two qubits A and B, the negativity of a state, which is based on the Peres–Horodecki criterion39 for entanglement, is defined as40 

NAB(ρ)=max(0,2σneg(ρTA)),
(1)

where σneg(ρTA) is the sum of the negative eigenvalues of the partial transpose (ρTA) of the density matrix (ρ) with respect to qubit A.41 For N-qubit quantum systems, the negativity metric can be generalized and defined as NA1A2An(ρ)=(NA1A2AnNA2A1A3AnNAnA1An1)1/n, where each Ai denotes a qubit. Negativity greater than zero is a sufficient inseparable (entanglement) condition for multipartite quantum systems, indicating that each qubit is inseparable (entangled) from each other.15 Moreover, we also compute the genuine multipartite entanglement as an additional metric to further quantify the multipartite entangled state with results shown in the supplementary material. Note that both metrics lead to similar results.

Hence, we calculate the transient negativity of the ENZ plasmonic waveguide system when using three qubits and compare the results to the simple free space and other commonly used plasmonic waveguide configurations, such as finite groove and rod.8 The positions of the three emitters in each waveguide system are illustrated in the right insets of Fig. 2, where they are oriented vertically in the ENZ and rod waveguides and horizontally along the groove to maximize their coupling with the ENZ and surface-plasmon mode, respectively. The dipole moments of all emitters are identical, since similar qubits (such as QDs) are expected to be loaded in the plasmonic waveguides. The emission frequency of each emitter corresponds to the ENZ cutoff (295THz). In addition, the normalized electric field patterns when one emitter is placed inside or along each waveguide are shown in the right insets of Fig. 2, where it is proven that only the ENZ waveguide has a homogeneous enhanced field distribution. The groove parameters are L=235nm and θ=10, while the rod has radius R=25nm. The waveguide lengths are chosen to be 1μm for both rod and ENZ cases and 1.4μm for groove waveguide to ensure the distance between two field antinodes at the groove's end is also 1μm. We study two separation distances for each adjacent emitters: d=200nm and d=400nm, and consider all potential excitation situations in each qubit, which are Q1 (or Q3 due to symmetry) initially excited, i.e., ρ11(0)=1, and Q2 initially excited, i.e., ρ22(0)=1.

FIG. 2.

Negativity for three qubits placed in the ENZ, cylindrical rod, and groove plasmonic waveguides or in free space. The inter-qubit separation distances are (a) d=200nm and (b) d=400nm. The initially excited qubit is Q2 with ρ22(0)=1 or Q1 with ρ11(0)=1. The x-axes are normalized to γ22, i.e., the computed emission decay rate of Q2 in each scenario. Right insets: plasmonic waveguide schematics and normalized electric field patterns when an emitter is placed inside or along each waveguide.

FIG. 2.

Negativity for three qubits placed in the ENZ, cylindrical rod, and groove plasmonic waveguides or in free space. The inter-qubit separation distances are (a) d=200nm and (b) d=400nm. The initially excited qubit is Q2 with ρ22(0)=1 or Q1 with ρ11(0)=1. The x-axes are normalized to γ22, i.e., the computed emission decay rate of Q2 in each scenario. Right insets: plasmonic waveguide schematics and normalized electric field patterns when an emitter is placed inside or along each waveguide.

Close modal

As shown in Fig. 2, a transient entangled state (corresponding to negativity larger than zero values) can be created in all cases but decays under different timescales. The negativity values are dependent on the initial state of each qubit, e.g., the negativity under Q2 initially excited is always larger than when Q1 is initially excited in the same waveguide system. Note that the transient negativity mediated by the ENZ plasmonic waveguide is superior to other plasmonic waveguides and free space, even for relatively large separation distance (d=400nm). This is mainly due to the enhanced homogeneous field mode [Fig. 2 upper right inset and supplementary material, Fig. S1(b)] at the cutoff frequency that spreads across the entire ENZ plasmonic waveguide geometry, resulting in constantly large absolute values in the mutual interaction decay rates γij(i,j=1,2,3) and much lower dipole–dipole interaction rates gij (see the supplementary material), which are ideal conditions to achieve enhanced entanglement.12 Interestingly, the tripartite entanglement mediated by the groove waveguide for d=200nm [green lines in Fig. 2(a)] has much lower negativity values than the other waveguide cases. The poor performance of the groove waveguide is somewhat surprising, since a strong surface-plasmon mode is excited along its length (Fig. 2 lower right inset). This problem is primarily due to the standing wave field distribution of the surface-plasmon mode that results in nonuniform decay rate enhancement along its surface.20 Specifically, Q2 is located at the center (antinode area) of the groove when d=200nm, achieving a rather large decay rate γ22104GHz, while Q1 and Q3 are located around the field nodes leading to relative small decay rates γ11=γ3313GHz. Hence, three qubits cannot be simultaneously entangled to each other due to γ11=γ33γ22, naturally leading to small tripartite entanglement. In contrast, the negativity mediated by the rod waveguide for d=200nm [blue lines in Fig. 2(a)] can rapidly reach very high values accompanied by strong oscillations that are fully dissipated after a very short time duration. This situation usually happens in photonic-crystal or microcavity mediated entanglement42,43 due to the large dipole–dipole interactions (gijγii). As an example, in the rod waveguide case (Fig. 2 middle right inset), the Q2 is fixed at the center (node region) and has low decay rate γ225GHz, much smaller than the dipole–dipole interaction rate g1335GHz. Note that the entanglement is always studied for qubits placed inside one nanochannel of the ENZ waveguide system, but similar results will be obtained if the qubits are placed in each periodic nanochannel, since the structure's periodicity will not affect its ENZ operation.25 

Therefore, commonly used plasmonic waveguide configurations, such as groove and rod, achieve low three-qubit entanglement that strongly depends on the spatial position of each emitter, obviously a severe disadvantage for their practical application in quantum technologies since it is extremely difficult to accurately position emitters in nanoscale areas. However, the proposed ENZ plasmonic waveguide can generate efficient and prolonged multiqubit entanglement without being affected by the emitters' separation distance due to the uniform field amplitude along the entire nanochannel. Hence, the emitters can be randomly distributed inside the ENZ nanochannel, which is a much more practical scenario that will help toward the experimental verification of the concept.

The presented theoretical calculations to quantify multiqubit entanglement can also be extended to four-qubit or even N-qubit scenarios. A relevant schematic is shown in Fig. 3(a), where we consider four identical quantum emitters Qi mediated by a plasmonic waveguide dissipative reservoir. The state transitions are depicted in Fig. 3(b), where all emitters operate at the ENZ cutoff frequency. The more complicated master equation of the four-qubit system is derived in the supplementary material. Once the density matrix ρ(t) is computed, the quadripartite entanglement characterized by negativity can be calculated by generalizing the metric of Eq. (1) to four qubits. Assuming that Q2 is initially excited and each inter-qubit separation distance is d=300nm, we plot in Fig. 3(c) the transient negativity for ENZ, finite groove, and rod plasmonic waveguides along with free space. The negativity values are again superior for an extended time duration only with the ENZ system, meaning that an enhanced quadripartite entanglement is realized. Again, this interesting effect is mainly due to the homogeneous field enhancement along the entire nanochannel at the ENZ resonance. On the contrary, the finite groove and rod waveguides exhibit nonuniform standing wave field distributions, similar to a Fabry–Pérot cavity modes. As a result, four qubits are less likely to be efficiently entangled to each other; therefore, the associated negativity is lower and decays faster than the ENZ system. Note that the current results can be extended to even larger qubit numbers, subject that they can fit along each waveguide, by generalizing the negative metric given by Eq. (1).

FIG. 3.

(a) Schematic of the ENZ plasmonic waveguide loaded with four qubits with similar distance. (b) The corresponding quantum states diagram. (c) Negativity of four qubits placed in ENZ, groove, and rod plasmonic waveguides, or free space. The x-axis is normalized to γ22, i.e., the computed emission decay rate of Q2 in each scenario.

FIG. 3.

(a) Schematic of the ENZ plasmonic waveguide loaded with four qubits with similar distance. (b) The corresponding quantum states diagram. (c) Negativity of four qubits placed in ENZ, groove, and rod plasmonic waveguides, or free space. The x-axis is normalized to γ22, i.e., the computed emission decay rate of Q2 in each scenario.

Close modal

Finally, we present the realization of a two-qubit quantum phase gate based on the ENZ plasmonic waveguide that exhibits high fidelity. This quantum phase gate is expected to work by employing the large decay difference between super-radiant and subradiant states18 that exists in the presented ENZ nanowaveguide system.12,37 As shown in Fig. 4(a), we consider two three-level Λ-type atoms embedded inside the ENZ waveguide and assume that only the |e|g transition can be coupled to the ENZ mode through external pumping with strength equal to the effective Rabi frequencies Ω1 and Ω2. |s is a non-resonance auxiliary level of the Λ-type emitter that does not couple to the ENZ mode. The schematics of the collective states and associated energy levels are given in Fig. 4(b), where |+=(|ge+|eg)/2 and |=(|ge|eg)/2 are symmetric and antisymmetric states with collective decay rates γ+=γ11+γ12 and γ=γ11γ12, respectively. It was recently shown37 that the ENZ waveguide at the cutoff frequency has γ11γ12 along the entire channel, meaning that the symmetric state |+ is super-radiant, whereas the antisymmetric state | is subradiant (γ+γ). Note that a pure super-radiant emission and a near-zero subradiant decay rate, independent of the qubits' positions, can be sustained only by the presented ENZ waveguide.37 

FIG. 4.

(a) Two Λ atoms (Q1 and Q2) embedded inside the ENZ nanochannel. The |e|g transitions resonantly couple to the ENZ mode. (b) Diagram of the two Λ atoms energy levels. (c) Gate fidelity as a function of inter-qubit separation distance when passive and active ENZ plasmonic waveguides or free space are used.

FIG. 4.

(a) Two Λ atoms (Q1 and Q2) embedded inside the ENZ nanochannel. The |e|g transitions resonantly couple to the ENZ mode. (b) Diagram of the two Λ atoms energy levels. (c) Gate fidelity as a function of inter-qubit separation distance when passive and active ENZ plasmonic waveguides or free space are used.

Close modal

To design an efficient quantum phase gate, we need to introduce a π phase shift on the resulted collective ground state |gg.18,44,45 In general, a 2π Rabi pulse (Ωt=2π) will lead to a total interaction time of t=2π/Ω, which will force the system to undergo a full Rabi oscillation, leading to a π phase shift in the atom-field state.46,47 However, if the decay rate of an excited state is much stronger than the Rabi frequency, the driving field cannot result in Rabi oscillations anymore, and instead, introduces scattering.42 Specifically, the effective driving strengths for |+ and | collective states can be written as Ω+=(Ω1+Ω2)/2 and Ω=(Ω1Ω2)/2 for the two-atom system that is presented here. If we choose atoms with Rabi frequencies Ω1=Ω2, then Ω+=0 and Ω=2Ω1. After selecting Ω to satisfy γ+Ωγ, which is practical in our ENZ system since γ+γ, the transition from |gg to |+ and then |ee will be blocked due to the strong decay γ+, whereas a 2π Rabi oscillation will be performed between the allowed transition |gg| that results in a π phase flip on the state |gg. Furthermore, since |s is a non-resonance auxiliary state, if we start from |gs(or|sg) state, only one atom can interact with the waveguide with strong decay rate γ11γ+/2. The transition |gs|es(or|sg|se) is also blocked due to the driving signal being too weak to excite the atoms.18 Thus, if we initially prepare the system in a starting state |ψinitial=12(|ss+|sg+|gs+|gg), by aid of external classical 2π laser pulses, the final state will turn to |ψfinal=12(|ss+|sg+|gs|gg) due to the resulted π phase shift of the |gg state, meaning that a two-qubit quantum phase gate is realized.

The gate efficiency can be measured by the metric of fidelity F given by the relation: F=1γ/γ11.18 Clearly, our ENZ waveguide, exhibiting a near-zero subradiant decay γ,37 is expected to lead to extremely high gate fidelity, an essential response to achieve an efficient quantum phase gate that can be used in quantum computing and communication systems. Figure 4(c) demonstrates the calculated fidelity as a function of the inter-emitter separation distance in the passive ENZ plasmonic waveguide and free space. The quantum phase gate with high fidelity in free space can be achieved only when the emitters are very closely packed to each other due to the resulted super-radiance phenomenon only in highly subwavelength regions. In the passive ENZ case, where the waveguide shown in the left inset of Fig. 4(c) is loaded with a lossless dielectric material (ε=2.2), a maximum fidelity of nearly 100% can be achieved for extremely subwavelength separation distances between emitters that decreases monotonously when this distance is increased. This is mainly due to the radiative and inherent Ohmic losses of the metal waveguide system leading to decoherence and dephasing. To further alleviate this issue, we insert a layer of an active medium with length lac=200nm in the ENZ nanochannel through choosing a permittivity with a positive imaginary part (ε=2.2+i0.045) (leading to an exceptional point48) that fully compensates the inherent plasmonic losses. This low gain value is practical and can be achieved by ruthenium (Ru) dyes.49 Interestingly, the gate fidelity in the active ENZ waveguide design can be further increased compared to the passive ENZ case and is only limited by radiation losses. Note that d is always larger than lac(d>lac) in Fig. 4(c) meaning that both quantum emitters are always inserted in the lossless passive dielectric material.

In conclusion, a plasmonic waveguide with effective ENZ response has been presented to achieve both transient multiqubit entanglement and two-qubit quantum phase gate functionalities at the nanoscale. Strong and uniform field enhancement is achieved at the ENZ resonance, leading to prolonged and efficient transient entanglement along elongated regions among three and four quantum emitters randomly distributed in this plasmonic waveguide system. These results can be easily generalized to an arbitrary qubit number. Moreover, due to the collective super-radiance decay at the ENZ frequency, the plasmonic waveguide can realize an efficient quantum phase gate between two qubits with high fidelity achieved even for long qubit separation distance.

See the supplementary material for details about the effective epsilon-near-zero resonant response of the proposed plasmonic waveguides, the master equation and associated coupling coefficients in three and four qubit systems, and an additional metric to quantify the multipartite entanglement.

This work was partially supported by the National Science Foundation (Grant No. DMR-1709612), the Nebraska Materials Research Science and Engineering Center (Grant No. DMR-1420645), and the National Science Foundation/EPSCoR RII Track-1: Emergent Quantum Materials and Technologies (EQUATE) under Grant No. OIA-2044049. Y.L. is supported by the National Natural Science Foundation of China (Grant No. 12104233) and the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No. 21KJB140013).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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