Exciton-assisted optomechanics with suspended carbon nanotubes

We now turn to the analysis of the quantum confinement of the exciton's CM motion induced by an applied inhomogeneous electrostatic field. The symmetric tip electrode configuration sketched in figure 1 with voltages V₁ and V₂ allows independent tuning of E_⊥ and E_∥, as the reflection symmetries imply that their magnitudes are determined, respectively, by (V₂ − V₁)/2 and (V₂ + V₁)/2. In order to generate a Stark-shift-induced excitonic quantum dot, we consider parameters such that the length scale over which E_⊥ and E_∥ vary appreciably is much larger than the CNT radius R (i.e. they can be regarded as constant across the CNT's cross-section) and larger than the excitonic Bohr radius. It follows that for sufficiently weak magnitudes (see below): (i) the effect of E_∥ is dominated by intrasubband virtual transitions whose effect on the CM motion can be treated adiabatically, while (ii) E_⊥ leads to a perturbation ∝e^±iφ that only induces intersubband virtual transitions.

More precisely, the effect of the external fields on the exciton can be described by the following interaction Hamiltonian:

$$\begin{equation} \hat{V} = \hat{V}_\parallel + \hat{V}_\perp \end{equation} \tag{ 5 }$$

with

$$\begin{equation} \hat{V}_\parallel = e \left[U(0,\hat{z}_{\mathrm{h}})-U(0,\hat{z}_{\mathrm{e}})\right] \end{equation} \tag{ 6 }$$

and

$$\begin{equation} \hat{V}_\perp = \frac{e}{\epsilon_\perp} \left[\hat{x}_{\mathrm{e}}\, E_\perp(\hat{z}_{\mathrm{e}}) -\hat{x}_{\mathrm{h}}\, E_\perp(\hat{z}_{\mathrm{h}})\right]. \end{equation} \tag{ 7 }$$

Here U(x,z) is the external electrostatic potential (in the plane y = 0), so that $E_\parallel (z)=-\frac {\partial U}{\partial z}(0,z)$ and $E_\perp (z)=-\frac {\partial U}{\partial x}(0,z)$, and _⊥ ≈ 1.6 denotes the intrinsic relative permittivity normal to this axis [49]. Given that the depolarization effect only affects the perpendicular field and the fact that the typical level spacing between the relevant excitonic states arising from a given pair of subbands n, m is smaller than the intersubband energy spacing, we can assume that the Stark-shift-induced CM confinement will be dominated by $\hat{V}_\parallel$ and neglect $\hat{V}_\perp$ in this subsection⁶. Then, for each pair of subbands n, m, we perform the substitution

$$\begin{equation} {z}_{{\rm e/h}}={z}_{\mathrm{CM}}\mp\frac{\mu}{m^{(n/m)}}\,{z}' \end{equation} \tag{ 8 }$$

with inverse

$$\begin{equation} z_{\mathrm{CM}}=\frac{m^{(n)}z_{\mathrm{e}}+ m^{(m)}z_{\mathrm{h}}}{m^{(n)}+ m^{(m)}}, \quad z'=z_{\mathrm{h}}-z_{\mathrm{e}}, \end{equation} \tag{ 9 }$$

where m⁽ⁿ⁾ (m^(m)) is the effective mass of subband n (m) and μ = m⁽ⁿ⁾m^(m)/(m⁽ⁿ⁾ + m^(m)) is the excitonic reduced mass. Here the indices n, m refer to the K point and the corresponding envelope function is enough to determine the wavefunction (cf equation (4))—note that an analogous substitution can be performed for the K' point and $m'^{(-n)}=m^{(n)}$. The aforementioned smoothness of the fields implies that $\langle \hat{z}_{\mathrm {CM}}^2\rangle \gg \langle \hat{z}'^{2}\rangle$ so that substituting equation (8) into (6) and Taylor expanding we obtain a linear potential

$$\begin{equation} \hat{V}_\parallel = e \hat{z}'\frac{\partial U}{\partial z}(0,\hat{z}_{\mathrm{CM}}) + \mathcal{O} \left[\hat{z}'^{2}\right]. \end{equation} \tag{ 10 }$$

We now consider E_∥(z) much smaller than the critical field to ionize the exciton so that the relevant intrasubband matrix elements of $\hat{V}_\parallel$ are much smaller than the binding energy. The latter warrants for the ground state manifold of the excitonic hydrogenic series [50] associated with n, m, a description in terms of an effective Hamiltonian by adiabatic elimination of the corresponding excited manifolds. Naturally, the corresponding states have well-defined parity. Thus, this effective Hamiltonian for the CM motion has a potential part whose leading contribution is second order in $\hat{V}_\parallel$. From equation (10), we obtain for this effective confining potential

$$\begin{equation} V_{\mathrm{eff}}^{(nm)}(z_{\mathrm{CM}})=-\alpha_X^{(nm)}E_\parallel^2(z_{\mathrm{CM}}), \end{equation} \tag{ 11 }$$

where

$$\begin{equation} \alpha_X^{(nm)}=e^2 \sum_{l\neq0} \frac{|\langle f_{{\mathrm{eh}},0}^{(nm)}|\hat{z}'|f_{{\mathrm{eh}},l}^{(nm)}\rangle|^2}{E^{(nm)}_l-E^{(nm)}_0} \end{equation} \tag{ 12 }$$

is the corresponding excitonic polarizability. Here we have introduced the wavefunctions {|f^(nm)_eh,l〉} on the relative coordinate z' for the hydrogenic series l = 0,1,... arising from n, m. In a classical picture, the field polarizes the exciton that thus experiences a force proportional to the gradient of the squared field. If one considers the characteristic level spacings

$$\begin{equation} E^{(nm)}_l-E^{(nm)}_0\sim \frac{e^2}{8\pi\epsilon_0\epsilon_{n-m}(0)\sigma_{\mathrm{eh}}^{(nm)}} \end{equation} \tag{ 13 }$$

and the order of the relevant matrix elements ∼σ^(nm)_eh (where σ^(nm)_eh is the exciton Bohr radius and _n−m(q) an appropriate dielectric function [43]), it follows from equation (12) that the excitonic polarizability is of order

$$\begin{equation} \alpha_X^{(nm)}\sim 8\pi\epsilon_0 \epsilon_{n-m}(0){\sigma_{\mathrm{eh}}^{(nm)}}^{3}.\! \end{equation} \tag{ 14 }$$

Here ₀(0) = _∥ ≈ 7 corresponds to the intrinsic relative permittivity along the CNT axis [49].

We focus now on E₁₁ excitons so that n = m = 0. To determine the electrostatic potential generated by the aforementioned symmetric tip electrode configuration, we have performed finite element method (FEM) calculations with COMSOL multiphysics for relevant parameters (section 3.1). We assume for simplicity cylindrical symmetry around the tips' axis so that the ground electrode corresponds to a cylindrical shell with diameter approximately equal to the CNT length L (figure 1). To model this configuration with electrodes of length L_T ≫ L, we consider Neumann boundary conditions at |x| = L_* with L_* smaller but comparable to L_T (i.e. E_x = 0 for |x| = L_* ∼ 1 μ). The effective confining potential V_eff is then estimated from equations (11) and (14). As expected it vanishes at the origin and is given by a double well with minima at |z| = z₀, comparable to the distance between the tip electrodes (i.e. the tailored confinement generates a QD molecule). For typical parameters, it is permissible to neglect inter-well tunnelling and use in our estimates the harmonic approximation for the ground state of each well. We consider as a representative example a (9,4) CNT, E_∥(z₀) = 9.6 Vμm⁻¹ and z₀ = 21 nm (figure 1). This results in an excitonic CM zero point motion σ⁽⁰⁰⁾_CM ∼ 9 nm (corresponding to σ⁽⁰⁰⁾_eh ≈ 1.5 nm [48] and m_CM = 0.2m_e [44], with m_e the electron mass in vacuum) and a level spacing ΔE ∼ 2 meV—where $\frac {\partial ^2E^2_\parallel }{\partial z^2}(z_0)$ is taken from the FEM calculation. Alternatively, heuristic considerations imply that the curvature at the well minima is of order |V^(nm)_eff(z₀)|/z²₀ which, together with equations (11) and (14), yields

$$\begin{equation} \sigma_{\mathrm{CM}}\sim\frac{\sqrt{\hbar z_0/E_\parallel(z_0)}}{2[2\pi m_{\mathrm{CM}} \epsilon_0\epsilon_\parallel \sigma_{\mathrm{eh}}^{3}]^{1/4}} \end{equation} \tag{ 15 }$$

and

$$\begin{equation} \Delta E\sim \frac{\hbar E_\parallel(z_0)}{2 z_0}\sqrt{\frac{2\pi\epsilon_0\epsilon_\parallel \sigma_{\mathrm{eh}}^{3}}{m_{\mathrm{CM}}}}, \end{equation} \tag{ 16 }$$

where we have omitted the subband superscripts. Given the subwavelength separation between the wells, for each doublet only the symmetric state is bright so that the degeneracy is irrelevant. Thus, the wavefunctions for relevant E_∥ are well approximated by the form given in equation (4) with a suitable envelope function

$$\begin{equation} F_{nm}(z_{\mathrm{e}},z_{\mathrm{h}})=f_{\mathrm{eh}}^{(nm)}(z_{\mathrm{h}}-z_{\mathrm{e}})f_{\mathrm{CM}}\left({\textstyle\frac{m^{(n)}z_{\mathrm{e}}+ m^{(m)}z_{\mathrm{h}}}{m^{(n)}+ m^{(m)}}}\right) \end{equation} \tag{ 17 }$$

(cf equation (9)), and for the ground state of the E₁₁ bonding (antibonding) manifold, henceforth denoted by |ψ₀₀₊〉 (|ψ₀₀₋〉), we take both functions f_CM and f_eh to be Gaussian [48].

Finally, more precise criteria to warrant the above adiabatic treatment are given by

$$\begin{equation} ({\mathrm{i}})\quad |E_\parallel(z_0)|\ll e/8\pi\epsilon_0\epsilon_\parallel\sigma_{\mathrm{eh}}^2\equiv\mathcal{E}_* \quad {\rm and}\quad ({\rm ii}) \quad z_0\gg\sigma_{\mathrm{eh}}, \end{equation} \tag{ 18 }$$

which follow, respectively, from requiring that the relevant matrix elements of $\hat{V}_\parallel$ be much smaller than the corresponding unperturbed energy differences (equation (13)) and that the variation of the fields be smooth enough⁷. This second condition (ii) is necessary, together with (i), to ensure that

$$\begin{equation} \Delta E\ll E^{(nm)}_l-E^{(nm)}_0\quad\rm{or\ alternatively}\quad \sigma_{\mathrm{CM}}\gg\sigma_{\mathrm{eh}}, \end{equation} \tag{ 19 }$$

which justifies the Taylor expansion leading to equation (10), and can be obtained using equations (11), (13)–(16) together with the assumptions σ_eh ∼ 4π_0∥ℏ²/μe² and μ ∼ m_CM.

In what follows, we will first consider the influence of the transverse field E_⊥ on the ground state of the E₁₁ axially polarized exciton, which will amount to a slight hybridization with the cross-polarized excitonic states, and then proceed to derive the corresponding excitonic DPs in the presence of the applied fields.

To this end, we treat the effect of $\hat{V}_\perp$ on |ψ_00±〉 to lowest order in perturbation theory. In principle, the linear correction |ψ⁽¹⁾_00±〉 involves contributions from all four excitonic manifolds for which |n| = 1, m = 0 or n = 0, |m| = 1, namely E₁₂, E₂₁, E₁₃ and E₃₁. It is straightforward to determine the necessary single-particle matrix elements of $\hat{x}$ within our model for the excitonic wavefunctions, provided that the 2D 'lattice contributions' (on the folded graphene sheet) are first reduced to momentum matrix elements which can be read out from the k·p Hamiltonian (1). To this effect, one can first consider the CNT as a 1D lattice so that there is only a 'lattice contribution' to the matrix elements. Given that the operators $\hat{x}$ and $\hat{p}_x$ are well behaved on the corresponding Bloch functions (which have finite range in the transverse directions) we can apply the standard relation between their matrix elements

$$\begin{equation} \langle K_{m,\pm}| \hat{x}|K_{n,\pm}\rangle=\frac{\mathrm{i}\hbar\langle K_{m,\pm}| \hat{p}_x|K_{n,\pm}\rangle}{m_{\mathrm{e}}(\epsilon_{n,\pm}-\epsilon_{m,\pm})}, \end{equation} \noindent \tag{ 20 }$$

reducing the problem to the evaluation of matrix elements of $\hat{p}_x$⁸—analogous relations hold for states at K'. We now use: (i) $\hat{p}_x=\cos \,\hat{\varphi }\,\hat{p}_{\mathrm {r}}-\sin \,\hat{\varphi }\,\hat{p}_\varphi$, (ii) that Π-orbitals have reflection symmetry with respect to the graphene sheet so that they do not carry $\hat{p}_{\mathrm {r}}$ and (iii) that for Bloch functions near the Dirac points $\hat{p}_\varphi \to m_{\mathrm {e}} v_{\mathrm {F}}\hat{\sigma }_{x'}-\frac {\mathrm {i}\hbar }{R} \frac {\hat{\partial }}{\partial \varphi }$. These, together with equations (3) and (20), imply

$$\begin{equation} \langle K_{m,\pm}|\hat{x}|K_{n,\pm}\rangle= \frac{\mp\mathrm{ i} \langle K_{m,\pm}|\sin\,\hat{\varphi}\big(R\hat{\sigma}_{x'} - \frac{\mathrm{i}\hbar}{m_{\mathrm{e}}v_{\mathrm{F}}}\frac{\hat{\partial}}{\partial\varphi}\big) |K_{n,\pm}\rangle}{|n-\nu/3| -|m-\nu/3|}. \end{equation} \noindent \tag{ 21 }$$

Subsequently, from equations (2), (3) and (21), and using that ν = ± 1 implies

$$\begin{eqnarray} &&\begin{array}{ll} \displaystyle \mathrm{sgn} [3n-\nu]|_{n=\mp\nu} =\mp\nu, \quad \mathrm{sgn} [3n-\nu]|_{n=0} =-\nu ,\\ \displaystyle \frac{\mathrm{i}}{2\pi}\int_0^{2\pi}\!{\rm d}\varphi\, {\mathrm{e}}^{{\mathrm{i}} \nu\varphi}\sin\varphi = -\frac{\nu}{2}, \quad \int_0^{2\pi} \!\frac{{\rm d}\varphi}{2\pi}\, {\mathrm{e}}^{{\mathrm{i}}\frac{4\nu}{3}\varphi}\sin\varphi \,\frac{\partial}{\partial\varphi}\mathrm{e}^{-\mathrm{i}\frac{\nu}{3}\varphi}= \frac{1}{6}, \end{array} \end{eqnarray} \noindent \tag{ 22 }$$

and that

$$\begin{equation} \hat{\sigma}_{x'}|K_{0,\pm}\rangle=\mp\nu|K_{0,\pm}\rangle, \quad \hat{\sigma}_{x'}|K_{-\nu,\pm}\rangle=\mp\nu|K_{-\nu,\pm}\rangle, \end{equation} \noindent \tag{ 23 }$$

we obtain

$$\begin{equation} \langle K_{-\nu,\pm}|\hat{x}|K_{0,\pm}\rangle =\frac{R}{2} \pm \frac{\hbar}{6 m_{\mathrm{e}}v_{\mathrm{F}}}, \quad \langle K_{+\nu,\pm}|\hat{x}|K_{0,\pm}\rangle = 0 . \end{equation} \noindent \tag{ 24 }$$

In turn, equations (2) and (3) imply that the substitution ν → − ν yields, for the states |K'_n,±〉 and |K'_m,±〉, expressions analogous to equations (21)–(24) so that

$$\begin{equation} \langle K'_{+\nu,\pm}|\hat{x}|K'_{0,\pm}\rangle =\frac{R}{2} \pm\frac{\hbar}{6 m_{\mathrm{e}}v_{\mathrm{F}}}, \quad \langle K'_{-\nu,\pm}|\hat{x}|K'_{0,\pm}\rangle = 0. \end{equation} \tag{ 25 }$$

Finally, from equations (4), (7), (24) and (25) and using that $\hat{x}$ is invariant under time reversal, it is straightforward to obtain

$$\begin{eqnarray} &&\fl \Big\langle\psi^{(0)}_{-\nu0\mp}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle =-\Big\langle\psi^{(0)}_{0-\nu\mp}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle= \frac{e R}{2\epsilon_\perp} \langle F_{-\nu0}|E_\perp(\hat{z}_{\mathrm{e}})|F_{00}\rangle, \nonumber \\ &&\fl \Big\langle\psi^{(0)}_{-\nu0\pm}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle = \Big\langle\psi^{(0)}_{0-\nu\pm}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle= \frac{\hbar e}{6 \epsilon_\perp m_{\mathrm{e}}v_{\mathrm{F}}} \langle F_{-\nu0}|E_\perp(\hat{z}_{\mathrm{e}})|F_{00}\rangle,\\ &&\fl \Big\langle\psi^{(0)}_{\nu0\mp}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle = \Big\langle\psi^{(0)}_{0\nu\mp}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle = \Big\langle\psi^{(0)}_{\nu0\pm}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle = \Big\langle\psi^{(0)}_{0\nu\pm}\Big|\hat{V}_\perp\Big|\psi^{(0)}_{00\pm}\Big\rangle = 0,\nonumber \end{eqnarray} \tag{ 26 }$$

where we have used

$$\begin{equation} F_{nm}(z_{\mathrm{e}},z_{\mathrm{h}})=F'_{nm}(z_{\mathrm{h}},z_{\mathrm{e}}) \end{equation} \tag{ 27 }$$

and that for a suitable choice of phases (equation (17))

$$\begin{equation} F_{nm}(z_{\mathrm{e}},z_{\mathrm{h}})=F_{mn}(z_{\mathrm{h}},z_{\mathrm{e}}). \end{equation} \tag{ 28 }$$

Thus, the contribution from |ψ⁽⁰⁾_ν0±〉 (|ψ⁽⁰⁾_0ν±〉) corresponding to E₁₂ (E₂₁) vanishes identically and, to first order in perturbation theory, we obtain for the admixture of cross-polarized excitons to the states |ψ_00±〉 induced by E_⊥

$$\begin{eqnarray} &&\fl \Big|\psi_{00\pm}^{(1)}\Big\rangle\approx - \xi\sum_l \left(\Big|\psi^{(0)}_{-\nu0\mp,l}\Big\rangle-\Big|\psi^{(0)}_{0-\nu\mp,l}\Big\rangle +\zeta\Big|\psi^{(0)}_{-\nu0\pm,l}\Big\rangle +\zeta\Big|\psi^{(0)}_{0-\nu\pm,l}\Big\rangle\right)\langle F_{-\nu0,l}|E_\perp(\hat{z_{\mathrm{e}}})|F_{00}\rangle, \end{eqnarray} \noindent \tag{ 29 }$$

where ξ ≡ eR/2_⊥(E₁₃ − E₁₁), ζ ≡ ℏ/3m_ev_FR and l labels a complete set of envelope functions for the E₁₃ and E₃₁ manifolds. Here, we have assumed for the latter that those having appreciable overlap with |F₀₀〉 correspond to excitons with energies well approximated by the lowest one (E₁₃ = E₃₁) and have neglected the splitting between the bonding and antibonding manifolds. Note that these manifolds are likely to consist of delocalized electron–hole pair-like states as is the case with the E₃₃ band, so that the analysis in section 2.1 may not apply [51]. However, our treatment of E_⊥ relies only on the validity of equations (4) and (28) irrespective of the specific form of the envelope functions F_−ν0,l, and on satisfying the weakness criterion |ξE_⊥(z₀)|² ≪ 1.

The Hamiltonian describing the interaction between electrons and low-frequency phonons has two distinct terms: (i) a DP contribution diagonal in sublattice space corresponding to an energy shift of the Dirac point,

$$\begin{equation} \skew3\hat{H}_{\mathrm{e{{\raise -1pt\hbox{--}}}ph}}^{(1)} = g_{1} \!\left[ \hat{u}_{\varphi\varphi} (\hat{\bar{r}}) + \hat{u}_{zz} (\hat{\bar{r}})\right], \end{equation} \tag{ 30 }$$

and (ii) a bond-length change contribution off-diagonal in sublattice space [27, 28] that in the Dirac picture emulates a gauge field,

$$\begin{eqnarray}&&\fl \skew3\hat{H}_{\mathrm{e{{\raise -1pt\hbox{--}}}ph}}^{(2)}= g_2 \!\left\{\left( \cos\,\! 3 \theta \,\hat{\sigma}_{x'} \hat{\tau}_3 - \sin\,\!3\theta \,\hat{\sigma}_{y'} \right)\left[ \hat{u}_{\varphi\varphi} (\hat{\bar{r}}) - \hat{u}_{zz} (\hat{\bar{r}})\right] \!-\! \left( \sin\,\! 3 \theta \,\hat{\sigma}_{x'} \hat{\tau}_3 + \cos\,\!3\theta \,\hat{\sigma}_{y'} \right) 2\hat{u}_{\varphi z} (\hat{\bar{r}})\!\right\}.\nonumber\\ \end{eqnarray} \noindent \tag{ 31 }$$

Here g₁ ≈ 30 eV is the DP constant, g₂ ≈ 1.5 eV the off-diagonal electron–phonon coupling constant [27, 28], and $\hat{x},\hat{y}$ only act on the 'orbital' part of the 1D Bloch functions (equation (2)). The single-particle Hamiltonians (30) and (31) naturally lead to the following excitonic two-particle Hamiltonian⁹:

$$\begin{equation} \skew3\hat{H}_{\mathrm{X{{\raise -1pt\hbox{--}}}ph}}=\hat{\mathbb{P}}_{\mathrm{e}}\skew3\hat{H}_{\mathrm{e{{\raise -1pt\hbox{--}}}ph}} \hat{\mathbb{P}}_{\mathrm{e}}\otimes \hat{\mathbb{I}}_{\mathrm{h}} - \hat{\mathbb{I}}_{\mathrm{e}}\otimes\hat{\mathbb{P}}_{\mathrm{h}}\skew3\hat{H}_{\mathrm{e{{\raise -1pt\hbox{--}}}ph}} \hat{\mathbb{P}}_{\mathrm{h}}, \end{equation} \tag{ 32 }$$

where $\skew3\hat{H}_\mathrm {e{{\raise -1pt\hbox{--}}}ph}=\skew3\hat{H}_\mathrm {e{{\raise -1pt\hbox{--}}}ph}^{(1)}+\skew3\hat{H}_\mathrm {e{{\raise -1pt\hbox{--}}}ph}^{(2)}$ and $\hat{\mathbb {P}}_{\mathrm {e}}$ ($\hat{\mathbb {P}}_{\mathrm {h}}$) denotes the projector on the conduction (valence) band.

Given that the relevant phonon modes have wavelengths, 2π/q, that are much larger than the CNT radius, R, one can describe them using a continuum shell model [27, 52], with $\hat{u}_{ij}(\hat{r})$ the corresponding Lagrangian strain operator, and keeping only the lowest orders in qR [27, 52], which amounts to the use of thin rod elasticity (TRE) [53]. Within this approximation, both compressional and flexural deformations have the structure of a local stretching so that the strain components satisfy u_φz = 0, u_φφ = −σu_zz with the latter given, respectively, by

$$\begin{equation} u_{zz}=-R\,\cos\,\varphi \,\frac{\partial^2 \phi_{\rm f}}{\partial z^2}\, \quad \mathrm{and} \quad \,u_{zz}=\frac{\partial \phi_{\rm c}}{\partial z}, \end{equation} \tag{ 33 }$$

where ϕ_f(z) [ϕ_c(z)] is the flexural (compressional) 1D field [53] that satisfies the Euler–Bernoulli (classical-wave) equation and σ = 0.2 is a Poisson ratio for the CNT [52]. Finally, we consider the lowest-order contributions in the external field (zeroth order in ϕ_AB) to the interaction between the lowest bright exciton states |ψ_00±〉 ≈ |ψ⁽⁰⁾_00±〉 + |ψ⁽¹⁾_00±〉 of the NTQD and low-frequency phonons, i.e.

$$\begin{equation} \Big\langle\psi_{00\pm}\Big|\skew3\hat{H}_{\mathrm{X-ph}}\Big|\psi_{00\pm}\Big\rangle \approx \Big\langle\psi_{00\pm}^{(0)}\Big|\skew3\hat{H}_{\mathrm{X-ph}}\Big|\psi_{00\pm}^{(0)}\Big\rangle + 2\Re\! \left[\Big\langle\psi_{00\pm}^{(0)}\Big|\skew3\hat{H}_{\mathrm{X-ph}}\Big|\psi_{00\pm}^{(1)} \Big\rangle\right]. \end{equation} \tag{ 34 }$$

Naturally, the axial angular momentum is preserved by coupling to compressional phonons and modified by interactions with flexural phonons, so that only the first or the second term on the rhs of equation (34) contributes, respectively, to the corresponding NTQD DPs (equations (30)–(33)). Then, equations (2), (4), (22), (23), (27)–(34) and

$$\begin{equation} \nu\!=\!\pm1\quad\Rightarrow\quad \int_0^{2\pi} \!\frac{{\rm d}\varphi}{2\pi} \,\mathrm{e}^{\pm \mathrm{i}\nu\varphi}\!\cos\,\varphi=\frac{1}{2} \end{equation} \tag{ 35 }$$

allow us to obtain

$$\begin{eqnarray} &&\fl \langle\psi_{00\pm}|\skew3\hat{H}_{\mathrm{X-ph}}|\psi_{00\pm}\rangle \approx 2\nu g_2 (1+\sigma) \cos\,3\theta\,\Bigg\langle F_{00}\Bigg|\frac{\partial \hat{\phi}_c}{\partial z}(\hat{z}_{\mathrm{e}})\Bigg|F_{00}\Bigg\rangle +2\xi R \left[g_1(1-\sigma) \right.\nonumber\\ &&\left. + \,\nu \zeta g_2 (1+\sigma) \cos\,3\theta\right]\Bigg\langle F_{00}\Bigg|\frac{\partial^2\hat{\phi}_{\rm f}}{\partial z^2}(\hat{z}_{\mathrm{e}}) E_\perp(\hat{z}_{\mathrm{e}})\Bigg|F_{00}\Bigg\rangle, \end{eqnarray} \tag{ 36 }$$

which is independent of the excitonic wavefunction's symmetry (bonding versus antibonding). Here, we have exploited the completeness of the basis {|F_−ν0,l〉}. The 'off-diagonal' coupling to flexural phonons, i.e. the term proportional to ζ, constitutes a negligible correction given that ζ = α_Fca_B/3v_FR ≲ 0.1 (for stable CNT radii [52]) and g₂/g₁ ≈ 0.05.

To recapitulate, in the present section we have used an envelope function approximation based on the graphene zone-folded description to analyse how weak inhomogeneous electrostatic fields affect the axially polarized direct excitons of a wide-gap semiconducting CNT. We have shown that such perturbative fields can induce quantum confinement of the CM motion dominated by the axial component and characterized by a level spacing in the milli-electronvolt range, and have derived the effective DP for the lowest-energy direct-exciton states of the resulting optically active quantum dot (NTQD). This approximation for the excitonic DP in the presence of applied electrostatic fields, given by equation (36), constitutes a central result of this paper. It relies on: (i) the aforementioned envelope function approximation, (ii) the existence of a weakly confined Wannier-exciton state |ψ₀₀〉—which requires the adiabatic conditions on E_∥(z) detailed in equation (18), (iii) neglecting the dispersion of the relevant cross-polarized excitonic manifold and (iv) the weakness criterion on the transverse field |ξE_⊥(z₀)|² ≪ 1.¹⁰ This effective excitonic DP involves coupling to both compressional (i.e. stretching) and flexural phonons. The latter is linear in the transverse field E_⊥, which induces a hybridization with the cross-polarized excitons. Interestingly, within our envelope function approximation (i), the 1D subband Bloch functions alternate between the two orthogonal eigenstates of the sublattice pseudospin $\hat{\sigma }_{x'}$ as the corresponding eigenenergy is increased—cf equations (2) and (3). This leads to the selection rules embodied in equations (24) and (25), which imply that the contribution from the E₁₂ and E₂₁ excitons vanishes.

To analyse the laser cooling of the resonator mode, we derive first a reduced master equation for the NTQD–resonator system and subsequently a rate equation for its populations. For the relevant regimes, all environmental couplings can be treated within the Born–Markov approximation and, after eliminating the bath phonon modes and the radiation field, the Liouvillian of the driven NTQD coupled to the resonator mode reads

$$\begin{eqnarray}&&\fl \dot{\rho}= -\mathrm{i}\left[H_{\mathrm{sys}},\rho\right]+ {\frac{\Gamma}{2}}\!\left(2\sigma_{-} B\rho B^{\dagger}\sigma_{+}-\rho \sigma_{+}\sigma_{-}-\sigma_{+}\sigma_{-}\rho\right) + \omega_{0} {\frac{n\left(\omega_{0}\right)}{2Q}}\!\left(2 b_0^{\dagger}\rho b_0^{\vphantom\dagger} - b_0^{\vphantom\dagger}b_0^{\dagger}\rho - \rho b_0^{\vphantom\dagger}b_0^{\dagger} \right) \nonumber\\ &&+\,\omega_{0}{\frac{n\left(\omega_{0}\right) +1}{2Q}}\!\left(2 b_0^{\vphantom\dagger}\rho b_0^{\dagger} - b_0^{\dagger}b_0^{\vphantom\dagger}\rho - \rho b_0^{\dagger}b_0^{\vphantom\dagger}\right), \end{eqnarray} \noindent \tag{ 53 }$$

where n(ω₀) is the thermal equilibrium occupancy at the ambient temperature,

$$\begin{equation}\fl \Gamma=2\pi\sum_k|g_k|^2\delta(\omega_L-\delta-\omega_k) \quad\mathrm{and}\quad \frac{\omega_0}{Q}= 2\pi\sum_q|\zeta_q|^2\delta(\omega_0-\omega_q). \end{equation} \tag{ 54 }$$

Other relevant sources of dissipation beyond those considered in the Hamiltonian given by equations (44)–(46), namely internal losses of the CNT resonator and nonradiative recombination of the exciton [24], can be incorporated, respectively, by simply adopting modified values of Q and Γ. We note that the nonradiative recombination rate for an ultra-clean CNT dark exciton has not been measured so far.

In analogy with the Lamb–Dicke limit (LD approximation), we expand up to second order the translation operators B and adiabatically eliminate the NTQD to obtain a rate equation for the populations of the resonator mode's Fock states [57]. The latter reads [32]

$$\begin{eqnarray} &&\fl \dot{P}_n = \left[ \eta^2 A_+ + \omega_0 {\frac{n \left( \omega_0 \right)}{Q}} \right] \left[n P_{n-1} - \left( n+1 \right) P_ n \right] + \left[ \eta^2 A_- + \omega_0 {\frac{n \left( \omega_0 \right) + 1}{Q}} \right]\left[\left( n+1 \right) P_{n+1} - n P_n \right]\nonumber\\ \end{eqnarray} \tag{ 55 }$$

with the cooling (η²A₋) and heating (η²A₊) transition rates per quanta determined by A_∓ = Γ[(δ ∓ ω₀)² + Γ²/4](ω²₀ + Γ² + Ω²/2)Ω²/D, where

$$\begin{eqnarray} &&\fl D\equiv 4\left(\delta^2 + \frac{\Gamma^2}{4} +\frac{\Omega^2}{2}\right) \left\{\big(\omega_0^2 + \Gamma^2\big) \left[\left(\delta-\omega_0 \right)^2 +\frac{\Gamma^2}{4}\right] \left[\left(\delta+\omega_0 \right)^2 + \frac{\Gamma^2}{4}\right] \right. \nonumber\\ &&\left.+ \left[\delta^2 \big(2\omega_0^2 + \Gamma^2\big) - \big(2\omega_0^2 - \Gamma^2\big) \left(\omega_0^2 + \frac{\Gamma^2}{4}\right)\right] \Omega^2 + \left(\omega_0^2 + \frac{\Gamma^2}{4}\right) \Omega^4\right\}. \end{eqnarray} \noindent \tag{ 56 }$$

We note that in this system, in stark contrast with atomic laser cooling, A_∓ → 0 in the limit Ω → ∞—in the polaronic representation this is associated with quantum interference effects between multiple paths [32]. Thus the steady-state occupancy for Q → ∞, i.e. the quantum backaction limit, reduces to

$$\begin{equation} \frac{A_+}{A_--A_+}=-\frac{\left(\delta+\omega_0 \right)^2 + \frac{\Gamma^2}{4}}{4\delta}. \end{equation} \tag{ 57 }$$

Remarkably, this result is independent of the Rabi frequency and coincides with the expression valid for the optomechanical cavity-assisted backaction cooling [58, 59] (equation (6) of [58]) with the cavity decay rate 1/τ replaced by the spontaneous emission rate, which for the optimal detuning $\delta =-\sqrt {\omega _0^2+\Gamma ^2/4}$ yields the fundamental limit $(\sqrt {1+(\Gamma /2\omega _0)^2}-1)/2$. This coincidence can be better understood by performing the adiabatic elimination in the 'original' representation corresponding to equation (42)—a thorough analysis using this representation has been performed by Jaehne et al [60].

Figure 2 shows the resulting temperature dependence of the optimal steady-state occupancy (n_f) that follows from equation (55), i.e.

$$\begin{equation} \langle b_0^{\dagger}b_0^{\vphantom\dagger}\rangle_{\mathrm{SS}}=\frac{n(\omega_0) + \eta^2 Q A_+/\omega_0}{1 + \eta^2 Q (A_- - A_+)/\omega_0} \end{equation} \noindent \tag{ 58 }$$

minimized with respect to the laser detuning δ and the Rabi frequency Ω, for typical parameters. These correspond, for instance, to Γ = ω₀/2, Q = 1.5 × 10⁵ and η = 0.086. The latter can be realized, for example, with a (9,4) nanotube (equation (40))¹¹ (R = 0.45 nm) of length L = 120 nm (ω₀/2π = 1.67 GHz) and a transverse effective field $\mathcal {E}_\perp =36.8\,{\mathrm {V}}\,\mu {\rm m}^{-1}$ (corresponding to the field configuration in figure 1). For this case, an ambient temperature T = 4.2 K [n(ω₀) = 52] allows cooling to a final (steady state) occupancy n_f = 0.10. In the high-temperature regime (II), the cooling factor is constant and straightforward arguments imply that its optimum is approximately bounded by η²Q ≲ n(ω₀)/n_f ≲ ηQ [32, 60]. We note that for the parameters considered here (figure 1) in this linear regime, the initial effective modulation parameter is no longer small, i.e. η²[n(ω₀) + 1] ≳ 1, and we find for the optimum A₋|_opt ∼ (A₋ − A₊)|_opt ∼ ω₀, so that initially η²[〈b^†₀b₀〉 + 1]A₋ ≳ Γ and equation (56) fails to describe the complete cooling process which will suffer a slowing down. Nonetheless, for these parameters η²[n_f + 1]A₋|_opt/Γ < 0.03 for all n(ω₀) < 10³, and the LD approximation remains valid for the steady state—these aspects have been analysed in detail by Rabl [62].

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The choice of parameters in section 3.1 takes into account the restrictions imposed on the applied electric fields by the need to avoid: deleterious static deflections due to gradient forces [63], field emission and dielectric breakdown (figure 1). For the nanotube the latter is taken care of by the aforementioned restrictions imposed on the fields E_⊥(z₀) ∼ E_⊥(0) and E_∥(z₀) so as to avoid dissociation of the E₁₁ exciton (section 2), while the suspended tip geometry of the electrodes (figure 1) allows for a strong suppression of the field at the dielectric substrate with respect to the maximum field. In fact, static deflections pose the most stringent limitation when trying to maximize the exciton–resonator coupling η, given that they scale as

$$\begin{equation} \Delta x\sim\frac{(\alpha_\parallel+\alpha_\perp)}{\pi^5 E_G} \left(\frac{L}{R}\right)^3E_\perp(0)E_\parallel(z_0) \end{equation} \noindent \tag{ 59 }$$

—here the prefactor is of order unity provided that the transverse dimension of the electrodes is comparable to or smaller than z₀ < L, and we have used the reflection symmetries, which imply that the total electrostatic force on the nanotube scales as 2(α_∥ + α_⊥)E_⊥(0)E_∥(z₀). Note that due to the Duffing geometric nonlinearity associated with a bridge geometry, this static deflection would mainly result in an effective stiffening of the resonator for Δx ≳ R. For the parameters considered in section 3.1, a detailed calculation based on the FEM electric fields (cf section 2.1), the Euler–Bernoulli theory for the CNT flexural response and static polarizabilities [α_⊥ = 13(4π₀Ų), α_∥ = 75(4π₀Ų)] estimated from [64, 65] yields Δx = 0.013 nm (note that R = 0.45 nm). Under these conditions we find that the modification of the resonator's frequency $\omega _0\to \tilde {\omega }_0$ induced by the electric fields and the effect of the Duffing geometric nonlinearity (which lead, respectively, to softening and stiffening) is completely negligible (∼0.1%). In general, this modification also results in a correction to the NTQD modulation parameter given by $\eta \to (\omega _0/\tilde {\omega }_0)^{3/2}\eta$.

The proposed device is compatible with a confocal setup but places stringent requirements on the nanotube positioning needed to enforce the desired reflection symmetries. We note that the latter are not essential for our scheme but just a convenient means of enabling a wider parameter range. If they are relaxed, while it is still possible to tune independently ΔE and η via the two voltages V₁ and V₂, |V₂ − V₁| ≫ |V₂ + V₁| can now result in regions where E_⊥ ∼ E_∥, which reduces the maximum feasible E_⊥ given that E_∥ should not exceed the critical field for the breakdown of an E₁₁ exciton. Nonetheless, relevant couplings could still be achieved (equation (40)). In turn, for such an asymmetric arrangement the NTQD may also couple to the out-of-plane flexural resonances, but this has no impact on the validity of our effective model (equations (44)–(46) with λ_q = 0 for q∈ flexural branch) given that the electrostatic fields and the mechanical coupling to the substrate break the degeneracy between the two flexural branches. A variant of this asymmetric scenario, albeit relinquishing the independent tunability of ΔE and η, is to use a single voltage configuration in which the fields are applied with an STM tip. Another interesting option is to effect the optical excitation of the nanotube using the evanescent field of a tapered fibre, which could allow one for instance to cancel the static deflection by using the optical gradient forces induced by an additional off-resonant laser, thereby lifting the restriction imposed on the length by equation (59) [66].

As the required fields are relatively large (∼10 Vμm⁻¹) and inhomogeneous, it seems natural to also analyse the impact of voltage fluctuations arising from the Johnson–Nyquist noise of the electrodes and/or of the voltage sources used. These induce a fluctuation of the effective potential defining the NTQD (equation (11))

$$\begin{equation} \delta \hat{V}_{\mathrm{eff}}^{(V)}(\hat{z}_{\mathrm{CM}})=-2\alpha_X^{(00)} \frac{E_\parallel^2(\hat{z}_{\mathrm{CM}})}{V_1+V_2} \left(\delta \hat{V}_1+\delta \hat{V}_2\right), \end{equation} \noindent \tag{ 60 }$$

which, given k_BT < ΔE, only affects the exciton as a pure dephasing contribution described by the interaction Hamiltonian $H_p= \frac {1}{2}\sigma _z \hat{V}_p$ with $\hat{V}_p=\langle F_{00}|\delta \hat{V}_{\mathrm {eff}}^{(V)}(\hat{z}_{\mathrm {CM}})|F_{00}\rangle$. Here the white noise variables $\delta \hat{V}_1$ and $\delta \hat{V}_2$ are uncorrelated and determined by $\langle \delta \hat{V}_{1/2}(t)\delta \hat{V}_{1/2}(0)\rangle =2R_Jk_{\mathrm {B}}T\delta (t)$, where R_J and T are the relevant resistance and temperature—for estimating an upper bound we assume without loss of generality that the noise is symmetric and dominated either by the electrodes or the unfiltered sources. Then within the Markov approximation, these voltage fluctuations result in an excess linewidth (full-width at half-maximum (FWHM)) $\gamma _p=\int _{-\infty }^{\infty } \langle \hat{V}_p(t)\hat{V}_p(0)\rangle {\mathrm {d}} t/\hbar ^2$. The latter can be estimated using equations (60), (14) and $\langle F_{00}|E_\parallel ^2(\hat{z}_{\mathrm {CM}})|F_{00}\rangle \sim E_\parallel ^2(z_0)$, leading to

$$\begin{equation} \gamma_p\sim R_Jk_{\rm B}T \left[\frac{32\pi\epsilon_0\epsilon_\parallel}{\hbar \left(V_1+V_2\right)}\right]^2 \sigma_{\mathrm{eh}}^6E_\parallel^4. \end{equation} \noindent \tag{ 61 }$$

Finally, for the envisaged parameters (section 2.1 and figure 1), $R_J<1\,\mathsf{\Omega}$ and T < 300 K, equation (61) yields γ_p/2π ≲ 200 kHz, which is negligible compared to the values for Γ ≳ 100 MHz considered here (sections 3.1 and 3.3). In principle, this voltage noise will also induce a fluctuation of the gradient forces exerted on the CNT (equation (59)), resulting in additional dissipation of the resonator mode. However, this effect is completely negligible for relevant parameters¹².

Another potential source of exciton dephasing is the Brownian motion of the suspended portion of the electrodes, which entails a limit to the suspended length, beyond which performance would be degraded. The convenience of electrode suspension, beyond that imposed by fabrication constraints, resides in the suppression it can induce of the maximum electric field at the neighbouring dielectric free surfaces (E_d−f) with respect to the maximum field. Assuming a cantilever geometry for the suspended portion of the electrode and an electrode–substrate gap size comparable to the suspended length L_e, the suppression factor will scale as (h_e/2L_e)²—where h_e is the transverse dimension of the electrode. For the envisaged device (figure 1), h_e = 30 nm and L_e = 100 nm would already lead to E_d–f ≲ 5 V μm⁻¹. If we assume some generic misalignment of the CNT, both the Brownian motion of the electrode's flexural (in-plane and/or out-of-plane) and compressional modes will contribute a term linear in the relevant displacement X_e to the corresponding fluctuation of the effective confining potential $\delta \hat{V}_{\mathrm {eff}}^{(B)}$. The latter will play a role completely analogous to $\delta \hat{V}_{\mathrm {eff}}^{(V)}$ but with $\delta \hat{V}_{1/2}$ replaced by the {X_e}. In all cases, the relevant fluctuation of the electric field satisfies

$$\begin{equation} \delta E_\parallel(z_0)\lesssim \frac{E_\parallel(z_0)}{z_0} X_{\mathrm{e}} \end{equation} \tag{ 62 }$$

and a suitable model to estimate an upper bound for the corresponding excess linewidth (FWHM) γ_B is afforded by considering only the contributions from the fundamental resonances of each relevant branch (resonant frequency ω_e) which can be treated separately—we assume in what follows that the Brownian motions of different electrodes are uncorrelated. These contributions can be analysed using the exact solution of the ensuing independent-boson model [55]. If we assume Au electrodes with the aforementioned cantilever dimensions, we have for the fundamental flexural mode ω_e ∼ 1 GHz so that Γ ≲ ω_e (cf sections 3.1 and 3.3). There are then two distinct effects of the electrodes' Brownian motion on the optical response of the exciton that could potentially interfere with our scheme: (i) generation of a sideband spectrum and (ii) pure dephasing leading to an excess linewidth of the corresponding ZPL. A critical parameter is the relative weight of the single-phonon sidebands (S) which for each branch, in the relevant regime k_BT ≫ ℏω_e and using equations (11) and (62), can be bounded by

$$\begin{equation} S_{\mathrm{e}}\lesssim k_{\rm B}T\left(\frac{16\pi\epsilon_0 \epsilon_\parallel}{\hbar z_0}\right)^2 \frac{\sigma_{\mathrm{eh}}^6E_\parallel^4}{m_{\mathrm{e}}\omega_{\mathrm{e}}^4}, \end{equation} \tag{ 63 }$$

where m_e is the corresponding effective mass. If we consider the envisaged device parameters (cf section 2.1 and figure 1), the aforementioned 'electrode' cantilever dimensions, the material properties of Au and T = 4.2 K, we obtain from equation (63) adding over all relevant branches of both electrodes: S ≲ 0.07—the ratio of the corresponding polaron shift to the one associated with the CNT resonator (η²ω₀) is ≲0.03 (cf section 3.1). Thus, the influence of (i) on the proposed optomechanical manipulations would be minimal.

In turn, the pure dephasing (ii) is determined by the weak dissipation of these resonances induced by the 3D substrate and is characterized by an Ohmic environmental spectral density. The conditions S ≪ 1 and k_BT ≫ ℏω_e are sufficient to ensure that the dephasing is Markovian and amenable to a treatment similar to the one we have followed for the voltage fluctuations. Each of the relevant fundamental resonances coupled to the substrate satisfies [53]

$$\begin{equation} \int_{-\infty}^{\infty}\langle X_{\mathrm{e}}(t)X_{\mathrm{e}}(0)\rangle{\rm d} t= \left(\frac{2 x_{\mathrm{e}}^{(0)}}{\omega_{\mathrm{e}}}\right)^2\frac{k_{\rm B}T}{\hbar Q_{\mathrm{e}}}, \end{equation} \tag{ 64 }$$

where x⁽⁰⁾_e and Q_e are, respectively, the corresponding zero-point motion and Q-value. Then, following a procedure analogous to the one used for estimating γ_p, we obtain from equations (11), (62) and (64)

$$\begin{equation} \frac{\gamma_B}{2\pi}\lesssim k_{\rm B}T \frac{\sqrt{\rho_{\mathrm{s}}}}{E_{\mathrm{s}}^{3/2}}\left(\frac{16\pi\epsilon_0 \epsilon_\parallel}{\hbar z_0}\right)^2\sigma_{\mathrm{eh}}^6E_\parallel^4. \end{equation} \tag{ 65 }$$

Here, we have also assumed that given the low aspect ratio, the {Q_e} are limited by clamping losses; used the results of [53] (table 1) which imply that $m_{\mathrm {v}}Q_{\mathrm {v}}\omega _{\mathrm {v}}^3=m_{\mathrm {h}}Q_{\mathrm {h}}\omega _{\mathrm {h}}^3\sim m_{\mathrm {c}}Q_{\mathrm {c}}\omega _{\mathrm {c}}^3\sim E_{\mathrm {s}}^{3/2}/\sqrt {\rho _{\mathrm {s}}}$; and added the bounds for all three relevant branches of both electrodes—here v, h and c denote, respectively, the flexural vertical, flexural horizontal and compressional fundamental resonances of the 'electrode' cantilever and the dimensionless prefactors are functions of ν_s. Strikingly, once a cantilever geometry is assumed, the dimensionless prefactor close to unity in equation (65) is independent of any other property of the electrode as it depends only on the Poisson ratio of the substrate. Thus, the magnitude of the suspended length L_e would only begin to play a role in exciton dephasing if it became large enough to ensure that the mechanical dissipation were dominated by internal losses. Finally, we find that for the proposed parameters (cf section 2.1 and figure 1) and typical substrate properties determined by ρ_s = 2 g cm⁻³ and E_s = 100 GPa, equation (65) yields a completely negligible excess linewidth γ_B/2π ≲ 20 kHz ≪ Γ.

Lastly, we have estimated radiation pressure effects and found that the corresponding static deflection and shot noise heating rate are also completely negligible, with typical parameters implying a launched power for a diffraction-limited spot size below 1 nW.

Beyond optical transduction and cooling, the platform introduced here could allow us to realize a mechanical analogue of the strong and ultra-strong coupling regimes of cavity QED. If the driving laser is tuned exactly on resonance, i.e. δ = 0 in equation (42), after a π/2 rotation of the pseudospin around $\hat{y}$ (σ_z → σ_x, σ_x → − σ_z), the system Hamiltonian reduces to the Rabi model [67] (H'_sys → H''_sys)

$$\begin{equation} H''_{\mathrm{sys}}= -\frac{\Omega}{2}\sigma_z + \frac{\eta\omega_0}{2}\sigma_x\left(b_0^{\vphantom{\dagger}}+b_0^{\dagger}\right)+ \omega_0 b_0^\dagger b_0^{\vphantom{\dagger}} \end{equation} \tag{ 66 }$$

with the spin degree of freedom afforded by the NTQD states dressed by the laser field, i.e.

$$\begin{equation} |\pm\rangle'=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle'\pm |\downarrow\rangle'\right)\to|\!\downarrow/\uparrow\rangle''. \end{equation} \tag{ 67 }$$

Here |↑〉' ≡ |ψ₀₀₋〉 corresponds to the relevant single-exciton state and |↓〉' to the empty NTQD. The spin-oscillator coupling and resonance condition are given, respectively, by ηω₀/2 and Ω = ω₀. Thus, given 1/Q ≪ η ≪ 1, reaching the strong coupling regime of the Jaynes–Cummings model (i.e. the Rabi model neglecting counter-rotating terms) depends on satisfying ηω₀ ≳ Γ, which for the aforementioned parameters would require Γ/2π ≲ 100 MHz. Here the precise threshold will differ from the more usual one, ηω₀ = Γ/2, given that the dissipation of the pseudospin induced by the radiation will no longer correspond to a simple relaxation channel. As discussed in section 2 the radiative contribution to Γ can be tuned with the axial magnetic field. We note that as the bright–dark exciton splitting lies in the 100 GHz range (cf section 2) and the radiative decay rate of the bright exciton is in the 10 GHz range [23], Γ ∼ 100 MHz with Ω ∼ ω₀ ∼ 1 GHz is compatible with keeping the excitation of the bright exciton off-resonant.

The strong coupling regime discussed here is akin to the ion-trap realizations of the Jaynes–Cummings model and the parametric normal mode splitting in cavity optomechanics [68] in that the large energy-scale discrepancy between the optical and mechanical domains is bridged by the driving laser, which entails an advantageous frequency upconversion of the output. It offers a wide range of possibilities for the demonstration of quantum signatures in the motion. In particular, a judicious modulation of η locked to pulsed laser excitation could allow us to emulate the adiabatic passage scheme used in [34] for carrying out QND measurements of the oscillator's energy. This would enable observation of motional quantum jumps.

Furthermore, from the analysis in section 3.2 it follows that higher transverse fields $\mathcal {E}_\perp$ could be applied without disrupting the scheme. Then, a tenfold increase in $\mathcal {E}_\perp$ would result in η ∼ 1, allowing us to crossover to the ultra-strong coupling regime of the Rabi model, which has not been realized so far [67]. In this regime where the coupling becomes comparable with or larger than the resonator's level spacing, the ground state is characterized by an appreciable mean phonon occupancy and presents substantial entanglement between the excitonic quantum dot and the resonator [69]. To gain insight into this regime it is instructive to consider the limit η ≫ 1, which is simple to analyse in the 'polaronic' representation by treating the second term in the lhs of equation (44) perturbatively. Thus, one obtains for the ground state of the coupled system

$$\begin{equation} |\phi_0\rangle=\textstyle \frac{1}{2}\left[|+\rangle'\left(|+\!\alpha\rangle +|-\!\alpha\rangle\right)+|-\rangle'\left(|+\!\alpha\rangle -|-\!\alpha\rangle\right)\right], \end{equation} \tag{ 68 }$$

where $|\pm \alpha \rangle =\mathrm {e}^{\pm \frac {\eta \omega _0}{2}(b_0-b_0^\dagger )}|0\rangle$. This corresponds to a maximally entangled state for which measurements of σ_x yield orthogonal Schrödinger cat states of the resonator.