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Abstract
This work demonstrates the efficient tuning of incoherent and coherent coupling between emitters embedded in an epsilon-near-zero (ENZ) waveguide coated with a multilayer graphene. As a result, a tunable two-qubit quantum phase gate based on the ENZ waveguide is realized at the cutoff frequency. Furthermore, due to the vanishingly small permittivity of the ENZ waveguide, all incoherent coupling between any two identical emitters located in the central area of the slit approaches a maximum, enabling near-ideal bipartite and multipartite entanglement. The coherent coupling between emitters is much larger at an operating frequency far from the ENZ resonance frequency than at the cutoff frequency, and the coherent coupling and resulting energy transfer efficiency can also be effectively tuned by the Fermi level of graphene. These results demonstrate an efficiently tunable electro-optical platform for quantum devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Surface plasmons supported by nanostructures have captured significant interest due to their ability to confine sub-wavelength light near the surface of nanostructures [1–5]. Numerous efforts have been made to achieve efficient interaction between emitters and nanostructures [6–8] and the plasmon-mediated coupling between emitters [9–12]. The traditional plasmonic system, on the other hand, has strict requirements on the location of the emitters to improve light-matter and emitter-emitter interactions. Because of their vanishingly small index, epsilon-near-zero (ENZ) materials overcome the shortcomings of plasmonic systems and preserve their ability to localize light [13–17]. Due to the infinite effective wavelength, ENZ waveguides enable nearly uniform field localization, allowing emitters to obtain adequate interactions at any distance and location. Using this property, intriguing effects and interesting applications such as nonlinear effects [18], doping and anti-doping effects [19–22], bipartite and multipartite quantum entanglements [23–25], electromagnetic cloaking [26–28], and more can be expected.
Controllable quantum devices relying on tunable coupling lie at the heart of controlled access to quantum operations and quantum dynamics [29–33]. The tunable two-qubit quantum gate, for example, plays an essential role in scalable quantum information processing [34–36]. Given its wide spectral range from terahertz (THz) to visible frequencies, monolayer graphene has been widely used in various photonic devices in recent years [37–43]. The fact that the Fermi level and surface conductivity of two-dimensional (2D) graphene can be sensitively tuned by an applied voltage underpins its widespread use in quantum devices. Because monolayer graphene has a limited conductance, some quantum devices use multilayer graphene to increase the tuning range [44].
In general, emitters operating near the cutoff frequency of ENZ waveguides exhibit strong incoherent coupling and weak coherent coupling [24]. The Kramers-Kronig relation is satisfied between coherent and incoherent coupling [45,46], and an intrinsic trade-off exists between them, that is, maximal coherent coupling always corresponds to minimal incoherent coupling, and vice versa. Because plasmonic waveguides have numerous resonant modes, it is natural to ask whether strong coherent coupling can exist in operating regions far from the ENZ frequency. Coherent coupling has been demonstrated to be essential for coherent energy transfer [47–49], so it is interesting to explore large coherent coupling and energy transfer efficiency far from the cutoff frequency.
This work investigates the coupling and dynamic properties of emitters embedded in the slit of a periodic ENZ waveguide. The results demonstrate an efficiently tunable coupling between emitters embedded in an ENZ waveguide coated with multilayer graphene. As an application, by changing the Fermi level of graphene, a tunable two-qubit quantum phase gate is realized. In addition, at the cutoff frequency, we demonstrate near-ideal bipartite and tripartite entanglement between arbitrarily arranged emitters in a finite spatial range. At an operating region far from the cutoff frequency, a large coherent coupling can exist between emitters, and the Fermi level of graphene can also effectively tune the coherent coupling strength and the efficiency of coherent energy transfer.
2. Coherent and incoherent couplings between emitters in ENZ waveguides
As shown in Fig. 1(a), the structure used in this work comprises a series of periodic rectangular slit waveguides filled with a dielectric material with permittivity $\varepsilon =2.2$. The medium surrounding the slit waveguide is silver, whose dielectric constant is taken from experimental data [50]. In this work, the imaginary part of the dielectric function of silver medium is also taken into the calculations, thus the damping is inevitable in the dynamics of emitters. The permittivity of silver is sufficiently negative at infrared and optical frequencies of our interest, the modal distribution is very similar to that of an ideal TE$_{10}$ mode, known as quasi-TE$_{10}$ mode [51]. In its dominant quasi-TE$_{10}$ mode, this structure can maintain ENZ resonance near the cutoff frequency [52]. The slit width and height are $w=200$ nm and $h=40$ nm, respectively, the waveguide length is $l=1$ $\mu \rm {m}$, and the grating period is $a=b=400$ nm. The simulation of the electric field in this work is done using COMSOL multiphysics. The dispersion relation guides us to the operating frequency at which ENZ occurs, which can be obtained by solving the dispersion equation of a rectangular waveguide [51,53], and the calculated result is presented in Fig. 1(b), where the red circle represents the ENZ region in which $f_1 \approx 295$ THz.
Fig. 1. (a) Diagram of periodic slit waveguide. The waveguide length is $l=1$ $\mu \rm {m}$, and the grating period $a=b=400$ nm, the slit width and height are $w=200$ nm and $h=40$ nm. (b) Dispersion of parallel plate waveguides. The red circle represents the ENZ region at $f_1 \approx 295$ THz. (c) Spontaneous decay rate $\gamma _{11}$ of emitter placed in the slit. The resonance frequencies are marked with $f_i$. (d), (e) Electric field distribution of a single dipole placed in the center of the slit (indicated by white dashed circles) at ENZ frequency ($f_1=295$ THz) and non-ENZ frequency ($f_2=324$ THz). (f), (g) Normalized coherent coupling $g_{12}/\gamma$ and incoherent coupling $\gamma _{12}/\gamma$ of two dipoles inserted symmetrically on the central axis of the slit as a function of their separation distance $d$ [as shown in the inset of panel (f)] under the ENZ frequency ($f_1=295$ THz) and non-ENZ frequency ($f_2=324$ THz).
The slit waveguide shown in Fig. 1(a) possesses rich resonant modes, which can be detected by the peaks of a dipole’s decay rate $\gamma$, as shown in Fig. 1(c). Obviously, the ENZ frequency $f_1$ is one of the resonance frequencies, and the other resonance frequencies are labeled as non-ENZ frequencies. The field distributions at ENZ and non-ENZ frequencies differ completely. The electric field in the $x\text {-}z$ plane generated by a single dipole placed in the center of the slit (indicated by white dashed circles) is shown in Figs. 1(d) and 1(e), which also represent the cases of ENZ frequency $f_1=295$ THz and non-ENZ resonance frequency $f_2=324$ THz, respectively. At the ENZ frequency, the field generated by the dipole in the slit is relatively uniform, whereas it fluctuates greatly in space at the non-ENZ frequency. This means that the coupling between two ENZ dipoles is insensitive to their spatial arrangement.
Coherent and incoherent coupling are the two mechanisms for coupling two emitters. The coherent coupling between two emitters is known as dipole-dipole coupling, and incoherent coupling is the interference term of spontaneous emission, also known as dissipative coupling [54]. The coherent dipole-dipole coupling $g_{ij}$ between the $i$th and $j$th emitters is given by [55]
In Eqs. (1) and (2), the Green tensor $\overset {\leftrightarrow }{\mathop {G}}\,\left ( \vec {r}_i,\vec {r}_j,\omega \right )$ represents the electric field at $\vec {r}=\vec {r}_i$ and originates from the emitter located at $\vec {r}=\vec {r}_j$ with emitter transition frequency $\omega$. The direction of the dipole moments in this work are all taken as parallel to the $y$ axis, and the magnitude of the dipole moments are all assumed to be unity. Therefore, the Green’s functions in Eqs. (3) and (4) can be obtained by calculating the electric field [45]:
Figures 1(f) and 1(g) present the coupling strength of two dipoles inserted on the central axis of the slit as a function of their separation distance $d$ [as shown in the inset of Fig. 1(f)] under the ENZ frequency $f_1=295$ THz and non-ENZ frequency $f_2=324$ THz, respectively. The two dipoles are symmetric about the center to ensure that $\gamma _{11}=\gamma _{22}=\gamma$. The solid black and dashed red lines in Figs. 1(f) and 1(g) represent the relative coherent coupling $g_{12}/\gamma$ and incoherent coupling $\gamma _{12}/\gamma$, respectively. Comparing the two panels shows that the coherent and incoherent coupling along with the change of $d$ are relatively flat at the ENZ frequency, whereas both couplings oscillate with $d$ at the non-ENZ frequency. In addition, at the ENZ frequency, the coherent coupling $g_{12}/\gamma$ is close to zero, whereas the incoherent coupling $\gamma _{12}/\gamma$ is almost maximal.
3. Near-ideal entanglements among location-insensitive emitters
Maximal incoherent coupling implies an ideal situation in which the entanglement can theoretically last an infinite time [56]. The dynamics of emitters in a weakly coupled environment can be solved using the Lindblad master equation, where the Born–Markov and rotating-wave approximations can be used. After tracing out the environment, the master equation of emitters can be written as [56–58]
The two-qubit entanglement can be measured with entanglement concurrence [59]
The entanglement among emitters arranged in a line has already been studied [24,25]. Previous research in this work shows that the distribution of the electric field in the $x\text {-}z$ plane is very uniform at the ENZ frequency. Therefore, even if the emitters deviate from the central axis, a large incoherent coupling can be expected. As shown in Fig. 2(a), we put four identical emitters in the $x\text {-}z$ plane, and the distance between emitters 2 and 3 and emitters 3 and 4 is 100 nm. Due to the symmetry, we need only consider the three incoherent couplings $\gamma _{23}$, $\gamma _{24}$, and $\gamma _{34}$, as illustrated in Fig. 2(b). All are very close to 1 at the ENZ frequency. In fact, the coupling between any two emitters placed in this space is nearly maximal incoherent coupling.
Fig. 2. (a) $x\text {-}z$ diagram of slit waveguide. (b) Normalized incoherent coupling $\gamma _{ij}/\gamma$ as a function of transition frequency. (c) Two-qubit entanglement $C_{ij}$ as a function of time. The initial state is selected as $|e_ig_j\rangle$. (d) Three-qubit (selected dipoles 2, 3, 4) entanglement negativity $N_3$ as a function of time. The initial state is selected as $|g_2e_3g_4\rangle$ and $|\psi _2(\frac {\pi }{4},0)\rangle \otimes |\psi _3(\frac {\pi }{3},\pi )\rangle \otimes |\psi _4(\frac {\pi }{4},0)\rangle$. The green dot-dashed lines in panels (c) and (d) represent the non-ENZ case.
Figure 2(c) shows three pairwise entanglements as functions of time, and the initial state is $|e_ig_j\rangle$. The maxima of the three entanglements are all about 0.5, and their evolution can perdure for a very long time, indicating a near-ideal situation [56]. In fact, if we omit the dissipation induced by silver medium, the resonance peaks of the single emitter decay rate showed in Fig. 1(c) should be very sharp, and the entanglement dynamics showed in Fig. 2 will not decay. However, the damping effect induced by metal is inevitable, and even the ENZ case can only slow down the damping, thus the entanglements are near-ideal.
The three-qubit entanglement $N_3$ is sensitive to the initial state. Figure 2(d) shows the evolution of $N_3$ with two initial states, $|g_2e_3g_4\rangle$ and $|\psi _2(\frac {\pi }{4},0)\rangle \otimes |\psi _3(\frac {\pi }{3},\pi )\rangle \otimes |\psi _4(\frac {\pi }{4},0)\rangle$, with $|\psi (\theta,\varphi )\rangle =\cos \theta |g\rangle +e^{i\varphi }\sin \theta |e\rangle$. The latter is a coherent state; it is the optimal initial state we discovered through numerical optimization. More importantly, the entanglement among emitters randomly arranged in the area denoted by the dashed box in Fig. 2(a) is close to ideal, which means that the ideal entanglement is insensitive to the location of the emitters. For comparison, we plot the entanglements of two and three emitters under the non-ENZ case with green dot-dashed lines in Figs. 2(c) and 2(d). With the initial states $|e_1g_4\rangle$ and $|e_1g_2g_4\rangle$, both the bipartite and multipartite entanglement rapidly decay to zero.
4. Tunable incoherent and coherent couplings
4.1 Tunable incoherent coupling and tunable quantum phase gate
The maximal incoherent coupling can be used to realize a two-qubit quantum phase gate [54,63] in addition to achieving steady entanglement. Consider two identical three-level emitters, labeled $1$ and $2$. The resonant levels are $|e\rangle$ and $|g\rangle$, with $|a\rangle$ serving as a nonresonant auxiliary level. Assume that both emitters produce spontaneous emission $\gamma$ in the environment, and that their incoherent coupling is $\gamma _{12}=\gamma _{21}$. The spontaneous decay rate of the cross state $|\pm \rangle =(|e_1\rangle \pm |e_2\rangle )/\sqrt {2}$ is $\gamma _{\pm }=\gamma \pm \gamma _{12}$. When $\gamma _{12}/\gamma \approx 1$, we have $\gamma _+ \approx 2\gamma > \gamma$ and $\gamma _{-} \approx 0$, implying that $|+\rangle$ is superradiant and $|-\rangle$ is subradiant, as shown in Fig. 3(a). We then introduce two resonant drive fields that act on two emitters with Rabi frequencies $\Omega _1$ and $\Omega _2$, respectively. As a result, the effective Rabi frequencies for $|\pm \rangle$ are $\Omega _{\pm }=(\Omega _1 \pm \Omega _2)/\sqrt {2}$. Following a similar proposal from the literature [54,63], we can perform a $2\pi$ Rabi oscillation on the system, resulting in a $\pi$ phase change in the state $|g_1g_2\rangle$. Therefore, if we prepare the initial state at $(|a_1a_2\rangle +|a_1g_2\rangle +|g_1a_2\rangle +|g_1g_2\rangle )/2$, it finally becomes $(|a_1a_2\rangle +|a_1g_2\rangle +|g_1a_2\rangle -|g_1g_2\rangle )/2$ after the aforementioned operations. This is a two-qubit deterministic quantum phase gate. The fidelity of the two-qubit quantum phase gate can be calculated using the formula $1-\sqrt {1-\gamma _{12}/\gamma }$ [63].
Fig. 3. (a) Realization of a two-qubit quantum phase gate. (b) Schematic diagram of coating graphene on the upper and lower surfaces of the slit. (c) Single-emitter decay rate of different Fermi levels as functions of emitter frequency. (d), (e) Incoherent coupling $\gamma _{12}/\gamma$ and corresponding fidelity at different Fermi levels near the ENZ frequency. The distance between two dipoles is fixed at $d=200$ nm. (f) Incoherent coupling parameter $\gamma _{12}/\gamma$ and corresponding fidelity as a function of Fermi level $E_f$. (g) Real and imaginary parts of the surface conductivity of 2D graphene. (h) Electric-field distribution of a dipole located at the center of the slit with transition frequency $f=286.5$ THz.
To achieve tunable coupling and tunable quantum gates, four layers of graphene are coated on the upper and lower sides of the slit, as shown in Fig. 3(b). The conductivity of 2D graphene can be described by intraband and interband contributions [64,65],
where the intraband transition contribution is given byFigure 3(c) depicts the single-emitter decay rate as a function of frequency for different Fermi levels. Changing the Fermi level shifts the resonant frequency. As a result, as illustrated in Figs. 3(d) and 3(e), the peak of incoherent coupling $\gamma _{12}/\gamma$ and the corresponding fidelity can also be tuned. This result implies that, if we fix a frequency, such as the frequency indicated by the vertical dotted lines in Figs. 3(d) and 3(e), we can easily modulate the fidelity of the quantum phase gate by tuning the Fermi level. We plot in Fig. 3(f) the incoherent coupling $\gamma _{12}/\gamma$ and the corresponding fidelity as a function of Fermi level $E_f$, where the maximal tuning range of $\gamma _{12}/\gamma$ is 0.6–0.9, and the corresponding fidelity can be tuned from less than 0.38 to 0.75. The reason for the large tuning range is shown in Fig. 3(f), where the surface conductivity of graphene changes abruptly near $E_f=0.6$ eV, resulting in a sudden change of incoherent coupling $\gamma _{12}/\gamma$ and the corresponding fidelity. Figure 3(c) shows that the ENZ resonance frequency is redshifted after coating with graphene, so determining whether the field remains spatially uniform within the new structure requires further investigation. Figure 3(h) shows the electric field distribution of a dipole under a new ENZ frequency $f=286.5$ THz. Clearly, the field distribution is very similar to the case without graphene shown in Fig. 1(d). Therefore, a tunable two-qubit quantum phase gate based on two location-insensitive emitters is demonstrated.
4.2 Tunable coherent coupling and coherent energy transfer efficiency
To this point, we have only focused on the ENZ mode, which exhibits strong incoherent coupling and weak coherent coupling. However, coherent coupling may in fact have large values far from the ENZ region, as shown in Fig. 4(a). The magnitude of coherent coupling can reach 2.0–3.0 at frequencies $f=312$ and $340$ THz, which is much larger than that at the ENZ frequency. The incoherent coupling in the ENZ region can be tuned by the Fermi level of graphene, as shown in Fig. 3(d). Here we show that the coherent coupling in the non-ENZ region can also be effectively tuned, as shown in Fig. 4(b), while $E_f$ has little effect on the coherent coupling at the ENZ frequency [see gray region in Fig. 4(b)].
Fig. 4. (a) Coherent coupling $g_\text {AD}/\Gamma _\text {D}$ and incoherent coupling $\Gamma _\text {AD}/\Gamma _\text {D}$ as a function of frequency at a distance $d=900$ nm without graphene. The shaded part represents the ENZ region. (b) Effect of Fermi level on coherent coupling $g_\text {AD}/\Gamma _\text {D}$. (c) Populations of acceptor for the case $E_f=0.6$ and 1.4 eV with transition frequency $f=326$ THz. (d) Energy transfer efficiency at different Fermi levels as a function of transition frequency.
During coherent energy transfer, coherent coupling can be used to improve transfer efficiency [67]. Initially, one qubit is excited and the other is in its ground state, denoted by donor (D) and acceptor (A), respectively. The dynamics of two qubits can also be solved by using Eq. (4) and taking $i=\text {D}$, $j=\text {A}$, $\gamma _\text {D}=\Gamma _\text {D}+2\gamma _{\,0}$, and $\gamma _\text {A}=\Gamma _\text {A}+2\kappa$ with $\gamma _{\,0}$ being the donor decay rate in vacuum and $\kappa$ being the charge separation rate [66], and $\Gamma _{\text {D}}$ ($\Gamma _{\text {A}}$) is the donor (acceptor) decay rate in the waveguide. After solving the dynamics of the system, we obtain the energy-transfer efficiency [66,67]
where $P_\text {A}(t)$ is the acceptor population.According to Eq. (12), the donor population plays a key role in energy-transfer efficiency, and increasing the energy-transfer efficiency consists primarily of increasing the area of the donor population over the time axis. Figure 4(c) shows the donor populations for $E_f=0.6$ and 1.4 eV with transition frequency $f=326$ THz. According to Fig. 4(d), the case of $E_f=1.4$ eV has a larger population area and thus a greater efficiency. When comparing Figs. 4(b) and 4(d), the peaks of efficiency correspond to those of coherent coupling, confirming that coherent coupling improves efficiency. Furthermore, by changing the Fermi level $E_f$ from 0.6 to 1.4 eV, the energy-transfer efficiency increases from less than 40${\% }$ to 60${\% }$. This enables tunable and efficient coherent energy transfer between distant emitters.
5. Discussions and conclusion
As we mentioned in the main text, dissipation in metal is unavoidable, so the dynamic evolution of entanglements even in the case of ENZ are only approximately ideal. To further eliminate the damping effect, an all-dielectric photonic crystal with zero-refractive-index [68–70] may be a good candidate, although the locations of emitters need to be selected carefully. In addition, the zero-index media with wavelength infinitely long is sometimes regarded as a finite space with the properties of an infinitely small point [71], this may be the origin of the enhanced and uniform coupling. Furthermore, this work only focuses on the case of straight waveguides, richer spatial couplings between emitters may be obtained in bending waveguides [72,73] with near-zero index.
In conclusion, we demonstrate efficient tuning of incoherent and coherent coupling via a plasmonic ENZ waveguide coated with 2D graphene and followed by a tunable two-qubit quantum phase gate. In a finite spatial region, the incoherent coupling between any two emitters approaches a maximum. In this case, the dominant decay channel is the plasmon, so bipartite and multipartite quantum entanglements among emitters produces a near-ideal situation, indicating that entanglements can last for a very long time. At operating frequencies far from the cutoff frequency, the coherent coupling between emitters may be tuned by the Fermi level and is much larger than that at the cutoff frequency, enabling large and tunable coherent energy transfer efficiency between distant emitters. This tunable and location-insensitive coupling among emitters can find essential applications in controllable quantum devices.
Funding
National Natural Science Foundation of China (11575051, 11904078); Natural Science Foundation of Hebei Province (A2019205266, A2021205020); China Postdoctoral Science Foundation (2020M670683).
Acknowledgments
J.R. acknowledges the funding of project from China Postdoctoral Science Foundation (Grant No. 2020M670683).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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