1. Introduction

Achieving strong light-matter interaction has been one of the focal points of modern quantum science for the past four decades. With a wide range of potential applications including single photon nonlinear optics, [1–4] quantum simulations of many body systems, [5–9] quantum networks, [10–12] quantum repeaters [13] and quantum computation, [14–17] researchers have sought to engineer the largest possible coupling strengths between photons held in an optical cavity and atomic systems. For the cavity quantum electrodynamics (cQED) optical setups studied so far however, reaching the strong coupling regime with optical photons is extremely difficult and typically requires either individual spins and ultra-small cavities or an ensemble of identical spins coupled to larger cavities.

Due to the wide range of applications and the constraints which hinder traditional cQED methods, enormous effort has been directed towards the implementation of hybrid cavities in cQED. In particular, photonic crystal and whispering gallery type cavities have been recently used to achieve relatively strong coupling [1, 18–25]. Whispering gallery resonators have become increasingly popular due to their relatively small mode volumes, large quality factors and their potential to strongly couple distant spins within the same mode [26]. While sub-GHz coupling rates have been achieved using micro-disc/ring resonators in the infrared band [21], much work has focused on the toroidal and spherical resonators due to their huge quality factors (Q ∼ 10¹⁰) [27], and their capability to reach strong coupling at optical wavelengths. Optical coupling strengths on the order of MHz have been achieved using such resonators [19, 20]. In the case of spherical resonators this had required a slight oblatness in the resonator to couple the atom to a single WGM [20, 22, 23]. In this work we propose a system which will use the many symmetries present in spherical WGM resonators to achieve coupling strengths orders of magnitude larger than in traditional setups and which can provide a scalable architecture to fashion large scale strongly-coupled cQED arrays (Figs. 1(b)–1(c)).

 

Fig. 1 (a) A spherical resonator supporting many degenerate rotated WGMs (green tubes) coupled to two antipodal spins (red spheres) and (b) a depiction of the extension of the model into one and (c) two dimensional arrays.

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To achieve enhanced cQED coupling we consider the familiar $\sqrt{N}$ enhancement between a single optical cavity mode and N spins. This enhancement is due to quantum interference and requires that no information exists that can differentiate which spin emits/absorbs into/from the cavity mode. We now take this familiar setup and invert it to consider a single spin coupled to N optical cavity modes. To retain the enhancement via quantum interference one must again ensure that all the optical modes are indistinguishable with respect to the spin. Arranging a large number of intersecting orthogonal degenerate optical cavity modes is practically impossible using traditional types of Fabry-Pérot cavities. By considering the spatial cavity mode profile it is possible to engineer N ≤ 5 near-degenerate cavity modes yielding only a minor enhancement in the collective coupling [28]. In this work we propose an experimentally accessible cQED arrangement to upscale N, and the collective coupling, by several orders of magnitude and use this to strongly couple distant spins over tens of microns. We achieve this by coupling the spin to a large ensemble of degenerate optical WGMs in a microsphere resonator where the spins are spatially located at the antipodes (Fig. 1(a)). Each of the degenerate modes correspond to a fundamental WGM of a spherical resonator, which has been rotated about the x and z axes.

2. Multi-mode Jaynes-Cummings model

The Jaynes-Cummings model has been used to successfully describe the interactions between a single spin and a single mode of light inside a resonator for the past five decades [29]. The Hamiltonian is typically expressed as,

The enhancement to the coupling rate can be observed by introducing the collective operator, ${\widehat{\mathrm{\Sigma }}}_{+}=\frac{1}{\overline{g}\sqrt{N}}{\sum }_{i=1}^N{g}_i{\widehat{\sigma }}_{+}^i$, where ḡ is the root mean squared of the coupling rates and ${\widehat{\mathrm{\Sigma }}}_{-}={\widehat{\mathrm{\Sigma }}}_{+}^{†}$. Rewriting Eq. (2) in terms of these operators reduces the interaction term, for ω_c = ω_i, to

While the TC model has been thoroughly studied, little work has been focused on the consideration of N distinct, degenerate modes. For the most part, this lack of consideration is due to experimental difficulties in creating such a system with traditional cavities, e.g. Fabry-Pérot cavities. Nevertheless, the consideration of many degenerate modes coupled to a single spin should behave in an analogous manner to the TC model. To show this, we extend Eq. (1) to consider N modes,

In order to study the dynamics of this system with the inclusion of loss we must solve the master equation. However, for N ≥ 10, this can be computationally demanding and is not practical for the very large values of N we intend to consider. An approximate method often used to evolve such a system involves solving the Schrödinger equation with damping accounted for by a non-Hermitian Hamiltonian [31],

After solving the Schrödinger equation in the interaction picture for a set of degenerate modes in the single excitation basis we obtain a set of coupled first order differential equations which closely resemble those obtained when working with the TC model [32] and can be solved analytically. To further establish this similarity we also performed a numerical comparison which is shown in the Appendix. It is now clear that the proposed N-mode system provides an identical enhancement to the coupling rate. The remaining challenge is to devise a physical system where this can be achieved. We will see that the WGMs of a spherical resonator may provide us with such a system.

3. Whispering gallery modes of a spherical optical resonator

Following the rationale above we seek a system where many degenerate optical modes can couple to a single spin. Within spherical resonators total internal reflection confines light resulting in an ensemble of optical modes. These modes are called whispering gallery modes. As we are interested in the interaction between a single spin and an ensemble of WGMs we require understanding of both the spatial intensity distribution, frequency and the polarization of the WGM electric fields within a microsphere. The WGM fields of a microsphere are governed by the Helmholtz equation in spherical coordinates. Analytic solutions to this equation have been thoroughly studied [33], and exist in the form of Vector Spherical Harmonics (VSH). These solutions correspond to transverse magnetic (TM) and transverse electric (TE) polarizations of light. They are typically characterised by three mode numbers (q, l, m) which respectively relate to the number of intensity maxima in the; radial, polar and azimuthal directions. Here we consider a microsphere of radius a and refractive index n_s which is suspended in a medium of refractive index n_m. As we intend to place the spin near the surface of the microsphere we seek a set of mode numbers which correspond to modes propagating around the equator of the sphere. This particular set of mode numbers is called the fundamental mode and occurs when q = 1 and l = m = l_max, where l_max, can be approximated using the Schiller expansion [34]. Typically l_max is directly proportional to the sphere’s radius and the wavenumber of the WGM, k. The frequency of the WGMs essentially depends only on the q and l mode numbers. Modes with identical values of q and l but different azimuthal mode numbers, m, are degenerate for perfectly spherical resonators. However, if the eccentricity of the sphere, ∊_s, is non-zero this degeneracy is broken resulting in a detuning between modes with different azimuthal mode numbers. The detuning can be approximated by,

In cQED applications of WGM resonators the polarization of the modes is either ignored or assumed to be perfectly orthogonal to the direction of propagation [36–39]. However, this approximation is not entirely valid. It has been recently shown in both bottleneck and cylindrical resonators that the TM modes are not completely transversal [40, 41]. To address this assumption in microspheres we calculate the electric field associated to a WGM of wavelength λ = 637 nm, the zero phonon line of NV centers, within an intensity maximum along the sphere’s equator. We consider a fused-silica microsphere of radius a = 32.72 μm and refractive index n_s = 1.46. By examining the magnitude of each vectorial component of the electric fields associated to the TE and TM WGMs, Fig. 2, it is clear that in the case of TE modes the electric field is predominately θ̂ directed. However, this is not the case for the TM modes. For the TM mode we find that the electric field is comprised of an uneven proportion of both r̂ and ϕ̂ components. This not only means that the electric field is not orthogonal to the direction of propagation, ϕ̂, but that the field itself is partially circularly polarized in two dimensions. This suggests that counter propagating modes will have different polarizations, that is, a clockwise (CW) propagating mode will have E ∝ r̂ + iϕ̂ while a counter-clockwise (CCW) propagating mode will have E ∝ r̂ − iϕ̂. Such polarization means that TM modes will interact with Δm_s = ±1 optical transitions in an atom, depending on the direction of propagation [40].

 

Fig. 2 The norm of the TM and TE field components for a 32.72 μm sphere with n_s = 1.46 suspended in air which supports a WGM of wavelength λ = 637 nm. A special radial position is depicted, ‘Transversal Point’, where the azimuthal component of the TM mode electric field is zero and the WGM is completely transversal.

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The two remaining properties of WGMs which are essential in cQED are the mode volume and the quality factor. For the WGMs of microspheres an approximate expression of the mode volume has been derived, $V_{\mathit{mode}}\approx 3.4{\pi }^{3/2}{\left(\frac{\lambda }{2\pi n_s}\right)}^3{l}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\sqrt{l-m-1}$, which is valid only for q = 1 [42]. The quality factor is also a crucial parameter in cQED experiments. To calculate the quality factor all loss channels must be considered. In general there are many loss channels for WGMs, however, in the case of microspheres the dominant losses are that due to surface scattering and material absorption [42, 43]. This gives an approximate form of the spherical WGM quality factor,

As discussed above, typically in microsphere cQED experiments one breaks the perfect spherical symmetry to lift the degeneracy between WGMs with different azimuthal mode numbers and couple spins to only one mode. Instead we consider maintaining spherical symmetry and consider an ensemble of degenerate WGMs which all intersect at the north and south poles of the microsphere where the spins are located. Each of these WGMs corresponds to a rotated fundamental mode. For this we must find expressions for the fields of such rotated modes and determine how many of these a sphere can support while maintaining orthogonality. To clearly perform these rotations we write the spherical harmonics in ket notation, that is, we express Y_lm as |l, m〉. As we are only interested in the case where m = l = l_max our kets become |l, l〉. The states |l, l〉 can be arbitrarily rotated using Euler angles via R̂(α,β,γ) ≡ R_z(α)R_x(β)R_z(γ), through the use of the Wigner D function [45], ${|l,l〉}^{\prime }={\sum }_{m^{\prime }=-l}^l|l,m^{\prime }〉D_{m^{\prime },l}^l(\alpha ,\beta ,\gamma )$, where $D_{m^{\prime },m}^l$ is the Wigner D function. This expression can be extended to VSHs to achieve expressions for the electric fields of the rotated modes [45,46]. We can now examine the orthogonality between the rotated WGMs. As the spins are located at the poles of the sphere, the fundamental WGM, which sits in the x–y plane, must be rotated about the x-axis by angle $\frac{\pi }{2}$. This first rotation ensures that the fundamental mode intersects both of the spins. To obtain expressions for each of the rotated modes, a rotation of η_i about the z-axis can be performed where 0 ≤ η_i ≤ 2π and 1 ≤ i ≤ N, see Fig. 3. It is important to notice here that each of the rotated modes identically intersects the spins.

 

Fig. 3 Depiction of the rotations which are performed to obtain expressions for the rotated WGMs. First the fundamental WGM, which lays in the x–y plane, (green tube) is rotated about the x-axis by $\frac{\pi }{2}$ (red tube). The mode now intersects the spin (black sphere) which is located on the z-axis. Next, rotations about the z-axis by angle η_i are performed which generate the i^th rotated WGM of the ensemble (blue tube).

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With expressions for the rotated WGM fields achieved, the orthogonality between them can be studied. To do so, in accordance with the rotations discussed above, a state that has been rotated arbitrarily by an angle η_i about the z-axis can be written,

4. WGM-spin coupling

When considering the coupling between a mode of light and a single spin the orientation of the spins optical transition dipole moment is often assumed to be aligned with the mode’s electric field. To achieve such alignment in most cases is quite challenging experimentally and consequently maximal coupling rates are not achieved. To study the effect of the dipole orientation on the coupling strength, we use the standard definition,

 

Fig. 4 The north pole of the spherical resonator where a single spin (red sphere) is located. The energy level diagram of the spin is presented where the emission of a π transition into a super position of σ₊ and σ₋ circularly polarised light is depicted. The two circular polarization correspond to two counter propagating fundamental WGMs.

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There are two possible approaches one can make to remove the dependence of the coupling on the handedness of the light. One way to remove this dependence becomes apparent when considering Fig. 2. At a specific depth inside of the sphere, the azimuthal, or, in the case of the rotated modes, polar component of the TM field is negligible so the field is linearly polarised in the radial direction. If the spins are placed at this location they will interact with the N WGMs identically via a π polarised transition and hence the coupling strength will benefit from a $\sqrt{N}$ enhancement. The second approach involves placing the spins at the surface of the resonator where they will interact simultaneously with σ₊ and σ₋ polarised WGMs. As mentioned previously, it has been observed that the non-transversal TM fields of the WGMs couple to degenerate Δm_s = ±1 transitions in an atom, depending on the “handedness” of the mode’s polarisation, see Fig. 4 [40]. If the first spin is initialised in the m_s = 0 level of the optically excited state while the second spin and the resonator are initialised in the ground state, the spin simultaneously couples to both CW (σ₊) and CCW (σ₋) WGMs. This interaction can be described by the Hamiltonian,

We now examine the sensitivity of the enhanced coupling strength to miss-positioning and miss-alignment of the two antipodal spins. Such imperfections generate some level of distinguishability of the modes as they couple with the spin and hence can reduce the enhancement. There are two sources leading to distinguishability between the modes. The first stems from the spatial dependence of the electric field intensity. If the two spins are not located exactly at the antipodes of the resonator the interaction strengths with the ensemble of modes is no longer homogeneous, due to slight difference in the electric field intensities presented by each WGM to the spins. In Fig. 5(a) we plot the latitudinal position dependence on the fundamental WGM field strength and from this we see that, in the case of a 32.72 μm fused-silica microsphere, the intensity of the TM WGM electric field only deviates by ∼ 2% at latitudinal distances ±330 nm away from the field maximum. This means that no significant decrease in the coupling strength will be observed if the spins are located within this region, which will be confirmed in later simulations. Such precise positioning of nanodiamonds has been achieved through the use of an AFM tip where nanometre precision is attainable [47]. Similarly, provided the spin lies within this region, the polarisation of the modes remain essentially constant, see Fig. 5(b). If the spin’s dipole moment is not radially aligned with the electric field (i.e. if d̂ ∝ r̂ + θ̂ + ϕ̂), there may be coupling to both the TE and TM modes. Further, the TE modes are not degenerate with the TM modes, typically ω_TM − ω_TE ≈THz in our setup. This means for small linewidth spins, if the TM modes are on resonance with the spin, the TE modes are not. The effect of misalignment is then simply a reduction of the TM field coupling strength by a factor d̂ê_TM.

 

Fig. 5 (a) The latitudinal variation of the TM WGM field intensity and (b) of the polarisation about the maximum intensity. The calculations were performed using a 32.72 μm fused-silica microsphere supporting a WGM of wavelength 637 nm. (c) The enhanced coupling rate as a function of microsphere radius.

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Indistinguishability of the WGMs can also be destroyed if scatterers are introduced into the setup. Here there are two possible scatterers that must be considered. The first is the spin itself. The solid state spins discussed above reside within nanodiamond particles which can be as small as 10 nm in diameter. The second scatterer stems from experimental requirements in studying spherical WGM resonators. Typically in experiments spherical WGM resonators are attached to the ends of fibre tips. The introduction of such a tip to the proposed setup will cause scattering of any modes which intersect the tip. In either case, such scattering not only reduces the quality factor of the resonator but also results in inter-mode coupling. For large scatterers inter-mode coupling can cause an undesired degeneracy breaking of counter-propagating modes, destroying mode indistinguishability [48]. If nanodiamonds as small as 10 nm in diameter are considered the effects of inter-mode coupling can be neglected as the inter-mode interaction strength, $g_M=4\pi r_{\mathit{ND}}^3\frac{\omega }{2V_M}\frac{s^2-1}{s^2+2}$, is small, resulting in negligible detuning between the two counter-propagating modes [36]. In the case of the fibre tip, the degeneracy breaking can be limited in the proposed WGM-spin setup if the diameter of the tip is smaller then the width of the fundamental WGM, $D_{\mathit{tip}}<W_{\frac{1}{2}}$. If this condition is met the setup can be arranged such that inter-mode degeneracy splitting will only effect two of the N counter-propagating modes. Thus if the spherical resonator is attached to a fibre tip the maximal number of identical WGMs is N − 2.

Up until now we have assumed perfect spherical symmetry, however, in realistic experiments oblateness must be considered. In [49] a microsphere with an eccentricity of 0.001 was reported which corresponds to a deviation of less then 1Å in the radius of the sphere. This small oblateness gives rise to a 100 MHz overall spread of the resonance frequencies of the WGMs, see Eq. 6. To address this issue, we consider a spin with a linewidth large enough to encapsulate each of the detuned WGMs. For a microsphere of eccentricity 0.001 this requires a spin with linewidth γ > 100 MHz. A spin with such a linewidth will interact with each of the WGMs identically as the resonant frequencies of the modes each lie within the linewidth of the spin. We can now calculate the maximum enhancement which can be achieved if a spin is located at the transversal point, within the sphere. Here we will only consider the case of an ideally aligned spin, that is when d̂·r̂ = 1, which is positioned in the center of the WGM field maximum (|E|/E_max = 1). Using Eq. (13) and the fact that $g_E=\sqrt{N}g$ we find that coupling strengths on the order of GHz can be reached using spheres less then 100μm, Fig. 5(c).

5. Results and discussion

With interaction between an individual spin and the ensemble of WGMs established we will now focus on the scalability of the proposed enhanced coupling with realistic spins. In particular we will focus on the optical transition of the nitrogen vacancy (NV) center in nanodiamond (ND) as these solid state systems exhibit many properties favourable to scaling and experimental realisation. To investigate the scalability of our system we will examine the dependence of the spheres radius on several parameters which are commonly used to analyse strong coupling. To perform such calculations we require values for the spontaneous emission rate γ of the spin and the optical damping rate of the WGMs, κ = 2πω/Q. Typically the linewidth of the NV center in a ND ranges from 2π × 10 to 2π × 1 GHz [50]. Here cryogenic temperatures will be considered where ξ = 1 using and NV with linewidth γ = 2π × 200 MHz. Our results, Fig. 6, show that the strong coupling scales very nicely with the radius of the sphere and that the regime can be easily met for 100 μm spheres.

 

Fig. 6 Ultra-strong coupling of a single spin to the collective optical modes as a function of the microsphere radius (a) n₀ photon saturation number; (b) L denotes the visibility of the vacuume Rabi splitting; (c) P Purcell factor; (d) C Cooperativity. For strong coupling we require g_E > κ, γ; P ≫ 1; L ≫ 1; n₀ ≪ 1; C ≫ 1. The calculations were performed using a fused-silica microsphere with γ = 2π × 200 MHz at cryogenic temperatures.

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Finally, we perform simulations of the proposed system to observe effective interactions between distant antipodal spins. We find that the excitation is transferred between the two spins, which are located ∼ 65.4μm apart, with ∼84% fidelity after ∼200 ps, Fig. 7. In this case each spin corresponds to a nanodiamond of radius r_ND ∼ 5 nm, thus scattering effects are negligible. The effect of inhomogeneous WGM-spin interactions on the enhanced coupling strength was also simulated in Fig. 7. In the case of a randomly sampled 2% decrease in the maximum coupling strength only a minor decrease to the enhanced coupling strength can be observed. To compare against previous experiments we have also simulated the effective interactions via a single WGM which shows significantly lower frequency oscillations.

 

Fig. 7 Simulations of the total spin occupation probability, $\text{Tr}\left[\left({\sum }_i^N{\widehat{\sigma }}_z^i\right)\widehat{\rho }\right]$, for two antipodal spins coupled to a single WGM (green curve) and an ensemble of WGMs in a fused-silica resonator with homogeneous/inhomogeneous coupling (black/red curves) and two clusters of 2070 antipodal spins within a diamond resonator (blue curve). The simulations for the silica resonator were performed using g = 2π × 250 MHz, κ = 2π × 157 kHz, γ = 2π × 200 MHz and a 2% random coupling inhomogeneity. For the diamond resonator g = 2π × 334 MHz, κ = 2π × 109 kHz and homogeneous coupling was considered.

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This result can be further enhanced by considering two antipodal clusters of spins. We perform simulations of such a system by extending multi-mode Hamiltonian, Eq. (4), to consider M identical spins. Before performing such simulations we estimate the maximal number of spins which can be contained within the transversal point, where a maximal enhancement will be achieved. To do so, we first assume that each of the spins are separated by d = 50nm, to avoid spin-spin interactions. In order to assure the coupling is identical for each of the spins, the largest volume they can occupy is limited by λ³. The maximum number of spins in each cluster is then,

6. Conclusions

In this work a hybrid quantum system designed to achieve strong coupling between distant spins was proposed. The proposed system consisted from a spherical WGM resonator in which spins were placed at the antipodes. It was shown that a spherical WGM resonator of radius a could be used as a $N~\sqrt{a}$ degenerate mode cavity, where each of the modes corresponded to a rotated fundamental WGM. Analogous to the effective enhancement to the coupling strength achieved in the TC model, it was also shown that the many mode extension can provide an identical enhancement. However, achieving this enhancement required that the interaction between the spin and each of the rotated WGMs was identical, which essentially required that the spin was placed at a specific depth inside the sphere. Simulations were then performed showing that enhanced coupling strengths on the order of GHz could be achieved and used to efficiently transport excitations between spins separated by 65.4 μm. Overall the proposed WGM-spin system was shown to provide an excellent, experimentally feasible and relatively scalable platform for achieving strong coupling which is applicable in many quantum technologies. Further, the results show that in principle the WGM-spin system can be used to construct large, strongly coupled, cQED arrays in which each spin can be individually addressed with optical light.

7. Appendix

7.1. Non-unitary evolution

Simulating the dynamics of an N-mode M-spin Jaynes-Cummings, or multi-mode Tavis-Cummings (MMTC), system with the inclusion of dissipative effects can be extremely computationally demanding. In this section an alternative approach to the evolution of such a system will be considered and proved equivalent to the standard master equation approach. The Hamiltonian which describes the N-mode M-spin Jaynes-Cummings model can be expressed as,

(16)$${\widehat{H}}_{\mathit{MMTC}}=\hslash \sum _{i=1}^N{\omega }_i{\widehat{a}}_i^{†}{\widehat{a}}_i+\frac{1}{2}\hslash \sum _{i=1}^M{\mathrm{\Omega }}_i{\widehat{\sigma }}_z^i+\hslash \sum _{j=i}^M\sum _{i=1}^N{g}_{ij}\left({\widehat{\sigma }}_{+}^j{\widehat{a}}_i+{\widehat{\sigma }}_{-}^j{\widehat{a}}_i^{†}\right).$$

Typically, evolution of this system with the inclusion of dissipative effects requires solving the full master equation,

(17)$$\dot{\widehat{\rho }}=-\frac{i}{\hslash }\left[{\widehat{H}}_{\mathit{MMTC}},\widehat{\rho }\right]+\sum _{k=1}^M{\gamma }_k\left[{\widehat{\sigma }}_{-}^k\widehat{\rho }{\widehat{\sigma }}_{+}^k-\frac{1}{2}[{\widehat{\sigma }}_{+}^k{\widehat{\sigma }}_{-}^k,\widehat{\rho }\}\right]+\sum _{i=1}^N{\kappa }_i\left[{\widehat{a}}_i\widehat{\rho }{\widehat{a}}_i^{†}+\frac{1}{2}\left\{{\widehat{a}}_i^{†}{\widehat{a}}_i,\widehat{\rho }\right\}\right],$$

where γ_k denotes the spontaneous emission rate of the k^th spin and κ_i the damping rate of the i^th mode. An alternative approach to evolution of this system involves solving the Schrödinger equation with the non-Hermitian Hamiltonian,

(18)$${\widehat{H}}_C=\hslash \sum _{j=1}^N\left({\omega }_j-\frac{i}{2}{\kappa }_j\right){\widehat{a}}_j^{†}{\widehat{a}}_j+\frac{\hslash }{2}\sum _{j=1}^M\left({\mathrm{\Omega }}_j{\widehat{\sigma }}_z^j-i{\gamma }_j{\widehat{\sigma }}_{ee}^j\right)+\hslash \sum _{j=1}^N\sum _{k=1}^N{g}_{ij}\left({\widehat{\sigma }}_{+}^j{\widehat{a}}_k+{\widehat{\sigma }}_{-}^j{\widehat{a}}_k^{†}\right),$$

where σ̂_ee = |e〉〈e| and Ω denotes the spin resonant frequencies. However, as a consequence of this non-Hermiticity, probability is not conserved under such an evolution. To prove that the dynamics obtained using this alternative approach are equivalent to that achieved by solving Eq. (17) the model must be restricted to the single excitation basis,

(19)$$|0〉|e_k〉\equiv |0_1,\dots ,0_N〉|g_1,g_2,\dots ,g_{k-1},e_k,g_{k+1},\dots ,g_M〉,$$

(20)$$|1_k〉|g〉\equiv |0_1,0_2,\dots ,0_{k-1},1_k,0_{k+1},\dots ,0_N〉|g_1,\dots ,g_M〉,$$

(21)$$|0〉|g〉\equiv |0_1,\dots ,0_N〉|g_1,\dots ,g_M〉.$$

In this basis the excitation can be kept track of as here it is possible to assume that any loss of probability is associated with the system evolving into the ground state |0〉|g〉. To begin the proof of equivalence the basis states will be relabelled as,

(22)$$|k〉\equiv |1_k〉|g〉,$$

(23)$$|N+k〉\equiv |0〉|e_k〉,$$

(24)$$|0〉\equiv |0〉|g〉.$$

Now, in terms of the conditional Hamiltonian, the master equation becomes,

(25)$$\dot{\widehat{\rho }}=-i\left[{\widehat{H}}_C\widehat{\rho }-\widehat{\rho }{\widehat{H}}_C^{†}\right]+\sum _{j=1}^N{\kappa }_j{\widehat{a}}_j\widehat{\rho }{a}_j^{†}+\sum _{j=1}^M{\gamma }_j{\widehat{\sigma }}_{-}^j\widehat{\rho }{\widehat{\sigma }}_{+}^j={\widehat{ℒ}}_C\widehat{\rho }+\widehat{𝒥}\widehat{\rho },$$

where the superoperators ℒ̂_C and 𝒥̂ are given by,

(26)$${\widehat{ℒ}}_C\widehat{\rho }=-i\left[{\widehat{H}}_C\widehat{\rho }-\widehat{\rho }{\widehat{H}}_C^{†}\right]\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\widehat{𝒥}\widehat{\rho }=\sum _{j=1}^N{\widehat{𝒥}}_j\widehat{\rho },$$

with,

(27)$${\widehat{𝒥}}_j\widehat{\rho }={\kappa }_j{\widehat{a}}_j\widehat{\rho }{\widehat{a}}_j^{†}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\widehat{𝒥}}_{N+j}\widehat{\rho }={\gamma }_j{\widehat{\sigma }}_{-}^j\widehat{\rho }{\widehat{\sigma }}_{+}^j.$$

For an arbitrary initial state containing only a single excitation, the density operator is,

(28)$$\widehat{\rho }(0)=\sum _{j_1,j_2=1}^{N+M}{\rho }_{j_1{j}_2}|j_1〉〈j_2|.$$

The action of ℒ̂_C on the initial state, ρ̂(0), will evolve it into the state,

(29)$$\widehat{\rho }(t)=e^{{\widehat{ℒ}}_{C}t}\widehat{\rho }(0)=\sum _{j_1,j_2=1}^{N+M}{\rho }_{j_1{j}_2}(t)|j_1〉〈j_2|,$$

while e^ℒ̂_Ct|0〉〈0| = |0〉〈0|. Also, the action of 𝒥̂ on the density operator is,

(30)$$\widehat{𝒥}\widehat{\rho }=\left(\sum _{j=1}^{N+M}{\rho }_{ij}{\mathrm{\Gamma }}_j\right)|0〉〈0|,$$

where,

(31)$${\mathrm{\Gamma }}_j=\{{\kappa }_j\hspace{0.17em}\hspace{0.17em}\text{for}\hspace{0.17em}\hspace{0.17em}j\le N,\hspace{0.17em}\hspace{0.17em}{\gamma }_j\text{for}\hspace{0.17em}\hspace{0.17em}N<j\le N+M,$$

and 𝒥̂|0〉〈0| = 0. By considering the properties of the superoperators, e^ℒ̂_Ct|0〉〈0| = |0〉〈0| and 𝒥̂|0〉〈0| = 0, it is clear that the formal solution to the master equation,

(32)$$\widehat{\rho }(t)=e^{{\widehat{ℒ}}_{C}t}\widehat{\rho }(0)+{\int }_0^{t}d{t}_1{e}^{{\widehat{ℒ}}_C(t-t_1)}\widehat{𝒥}{e}^{{\widehat{ℒ}}_C{t}_1}\widehat{\rho }(0)+{\int }_0^{t}d{t}_2{\int }_0^{t_2}d{t}_1{e}^{{\widehat{ℒ}}_C(t-t_1)}\widehat{𝒥}{e}^{{\widehat{ℒ}}_C(t_2-t_1)}\widehat{𝒥}{e}^{{\widehat{ℒ}}_C{t}_1}\widehat{\rho }(0)+\dots$$

terminates after the second term. This reduces the formal solution to,

(33)$$\widehat{\rho }(t)=e^{{\widehat{ℒ}}_{C}t}\widehat{\rho }(0)+{\int }_0^{t}d{t}_1{e}^{{\widehat{ℒ}}_C(t-t_1)}\widehat{𝒥}{e}^{{\widehat{ℒ}}_C{t}_1}\widehat{\rho }(0),$$

which, after substitution of Eqs. (29, 30), can be expressed as,

(34)$$\widehat{\rho }(t)=e^{{\widehat{ℒ}}_{C}t}\widehat{\rho }(0)+|0〉〈0|{\int }_0^t\sum _{j=1}^{N+M}{\mathrm{\Gamma }}_j〈j|e^{{\widehat{ℒ}}_C{t}_1}\widehat{\rho }(0)|j〉.$$

Finally, by taking the trace of the above equation and direct substitution into Eq. (34) it can be shown that,

(35)$$\widehat{\rho }(t)=e^{{\widehat{ℒ}}_{C}t}\widehat{\rho }(0)+|0〉〈0|\left(1-\text{Tr}\left[e^{{\widehat{ℒ}}_{C}t}\widehat{\rho }(0)\right]\right),$$

and hence 〈j₁|ρ̂ (t)|j₂〉 = 〈j₁|ρ̂_c(t)|j₂〉 for j₁, j₂ ≠ 0.

7.2. Simulations

As we now have shown that evolution of the system with inclusion of decoherence can be performed by solving the non-Hermitian Schrödinger equation we rewrite the conditional Hamiltonian in the interaction picture,

(36)$${\widehat{H}}_C^I=-\hslash \sum _{j=1}^M\left({\mathrm{\Omega }}_j-i\frac{{\gamma }_j}{2}-\overline{\omega }\right){\widehat{\sigma }}_{gg}^j-i\frac{\hslash }{2}\sum _{j=1}^N{\kappa }_j{\widehat{a}}_j^{†}{\widehat{a}}_j+\hslash \sum _{j=1}^M\sum _{k=1}^N{g}_{kj}\left({\widehat{\sigma }}_{+}^j{\widehat{a}}_k+{\widehat{\sigma }}_{-}^j{\widehat{a}}_k^{†}\right),$$

where σ̂_gg = |g〉〈g| and ω̄ is the average of the mode resonance frequencies. Now by solving the Schrödinger in the interaction picture with the general state

(37)$$|\psi (t)〉=\sum _{k=1}^N{\alpha }_k(t)|0〉|e_k〉+\sum _{k=1}^N{C}_k(t)|1_k〉|g〉,$$

we obtain a set of coupled first order differential equations,

(38)$$i{\dot{\alpha }}_k(t)=-\sum _{j=1}^M\left({\mathrm{\Omega }}_j-i\frac{{\gamma }_j}{2}-\overline{\omega }\right){\alpha }_k(t)+\left({\mathrm{\Omega }}_k-i\frac{{\gamma }_k}{2}-\overline{\omega }\right){\alpha }_k(t)+\sum _{j=1}^N{g}_{jk}{C}_j(t),$$

(39)$$i{\dot{C}}_k(t)=-\sum _{j=1}^M\left({\mathrm{\Omega }}_j-i\frac{{\gamma }_j}{2}-\overline{\omega }\right)C_k(t)-i\frac{{\kappa }_k}{2}{C}_k(t)+\sum _{j=1}^M{g}_{kj}{\alpha }_j(t),$$

In the case of a single spin and many identical modes modes (ω_i = ω_c, g_i = g) this set of equations can be straight forwardly solved for zero detuning (Ω − ω_c = 0) and used to plot the Rabi oscillations (Fig. 8). We also solve for the case of N atoms and a single mode (TC) and compare the results against both the multi-mode case and the JC model (Fig. 8). The results show an identical $\sqrt{N}$ enhancement is achieved by both the MM and TC models over the JC model showing that the dynamics in both systems are essentially identical.

 

Fig. 8 The Rabi-oscillations of the mode in the JC, TC, MM and MMTC models. These simulations were performed with Δ_D = 0 in units of g under ideal conditions (κ_i = γ_i = 0). The simulations of the TC/MM models were performed using 9 identical spins/modes and the initial states ${|\psi (0)〉}_{\mathit{TC}}=\frac{1}{3}{\sum }_{i=1}^9|0,e_i〉$ and |ψ(0)〉_MM = |0, e〉 respectively. In the MMTC simulation 3 identical spins and 3 identical modes were used with the initial state ${|\psi (0)〉}_{\mathit{MMTC}}=\frac{1}{\sqrt{3}}{\sum }_{i=1}^3|0〉|e_i〉$.

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Acknowledgments

We acknowledge Dr James Cresser for constructive discussions regarding the non-Hermitian evolution. This work was partly supported by the ARC Centre of Excellence for Engineered Quantum Systems EQUS (Grant No CE110001013).

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