Exciton-assisted optomechanics with suspended carbon nanotubes

We propose a framework for inducing strong optomechanical effects in a suspended carbon nanotube based on deformation potential exciton-phonon coupling. The excitons are confined using an inhomogeneous axial electric field which generates optically active quantum dots with a level spacing in the milli-electronvolt range and a characteristic size in the 10-nanometer range. A transverse field induces a tunable parametric coupling between the quantum dot and the flexural modes of the nanotube mediated by electron-phonon interactions. We derive the corresponding excitonic deformation potentials and show that this interaction enables efficient optical ground-state cooling of the fundamental mode and could allow us to realise the strong and ultra-strong coupling regimes of the Jaynes-Cummings and Rabi models.

We analyze a framework for optical manipulation of the motional state of a suspended carbon nanotube based on deformation potential exciton-phonon coupling. The excitons are confined using an inhomogeneous axial electric field which generates optically active quantum dots with a level spacing in the milli-electronvolt range. A transverse field induces a tunable parametric coupling between the quantum dot and the flexural modes of the nanotube. We show that this interaction enables efficient optical ground-state cooling of the fundamental mode and will allow access to quantum signatures in its motion.
Optical transducers underpin a host of high precision measurement techniques and recent developments in optomechanics suggest that they may enable quantum limited control of a macroscopic mechanical degree of freedom [1]. Given the versatileness of mechanical non-linearities this would provide an alternative to atomic systems for fundamental tests of quantum mechanics and the development of quantum technologies [2]. Paradigmatic goals in this direction are the preparation of a mechanical resonator in its quantum ground state [3,4,5,6] and the demonstration of quantum signatures in its dynamics [7,8]. These endeavors are seriously hampered by the mechanical quality of typical materials. In this respect suspended single-walled carbon nanotubes (CNTs) [9] are emerging as a unique candidate. Indeed recent transport experiments in these systems have demonstrated strong coupling of charge to vibrational resonances [10,11] and ultra-high mechanical frequency quality-factor (f Q) products [12]. These developments raise the question of which are the prospects for optical manipulation of motional degrees of freedom in CNTs. The standard paradigm in optomechanics is based on an optical cavity whose frequency is modulated by the motion of one of its mirrors or of a dielectric object inside it via radiation pressure effects [1]. However this approach becomes inefficient for resonators with deep subwavelength dimensions and low polarizabilities like CNTs.
Here we propose a solution to this conundrum based on an alternative way of inducing coherent optomechanical transduction which exploits the unique properties of excitons in semiconducting CNTs [13,14,15,16]. The role of the optical cavity is played by an excitonic resonance of the CNT that couples parametrically to the motion via deformation potential electron-phonon interactions [17]. Homodyne detection of the output field of the two level emitter afforded by the excitonic resonance allows then to perform a continuous measurement of the mechanical amplitude. This procedure, which could be implemented using the differential transmission technique [30], is analogous to cavity-assisted schemes [1] and equivalent to ion-trap measurements. We envisage a suspended CNT where the center of mass (CM) of the exciton is localized via the spatial modulation of the Stark-shift induced by a static inhomogeneous electric field. We analyze a tip electrode configuration that effectively engineers a pair of tunable optically active nanotube quantum dots (NTQDs) with excitonic level spacing in the meV range corresponding to a confinement length below 10nm. The quantum confinement is induced by the inhomogeneity in the field component along the CNT axis E . In turn the normal component E ⊥ can be used to induce a tunable parametric coupling between the exciton and the flexural motion of the CNT. This allows for optical ground-state cooling of the fundamental mode at an ambient temperature in the Kelvin range. A major advantage of this mechanical resonator-NTQD system with respect to prior scenarios [4] is the possibility of realizing a mechanical analogue of the strong-coupling regime of cavity-QED [18] with an "optomechanical coupling" in the 100MHz range. Furthermore, this coherent coupling can be switched on and off on demand which offers rich possibilities for deploying quantum-optical schemes to demonstrate quantum signatures in the motion [19].
The electronic structure of a semiconducting CNT can be understood in terms of graphene rolled into a cylinder [20]. In the absence of a magnetic field there is a single bright level: the singlet bonding direct exciton |KK * + |K K * (the conjugated wavefunctions correspond to the hole) [21]. Threading a small Aharonov-Bohm flux φ AB renders the antibonding state |KK * − |K K * weakly allowed so that its spontaneous emission rate Γ can be tuned. Thus we focus on the E 11 direct excitons |KK * ± |K K * , whose zero-field splitting lies in the meV range [15], and consider their deformation potential (DP) coupling to the low frequency phonons corresponding to the compressional (stretching) and flexural (bending) branches for φ AB = 0 [33]. To obtain a tractable model for the excitonic wavefunction suitable for analyzing exciton CM confinement and the excitonsoft phonon coupling we adopt: (i) the k · p graphene zone-folded scheme following Ref. 21 but neglecting intersubband transitions [22], and (ii) an envelope function approximation within each subband. For the latter we arXiv:0911.1330v1 [cond-mat.mes-hall] 6 Nov 2009 adopt the parameterization developed in Ref. 22 but take the Bloch function at K, K as determined by (i) and the assumption of electron-hole symmetry. This leads to the following singlet direct exciton wavefunctions: where the envelope functions |F nm , |F nm satisfy F nm (z e , z h ) = F nm (z h , z e ) -note that the inter-valley mixing preserves the total momentum and angular momentum. The indices n, m correspond to the quantization of the single particle azimuthal momenta (n = 0, ±1, ±2, . . .) so that n = m = 0 for E 11 excitons. The associated subband electronic 1D Bloch functions are given by |K n,± .
The symmetric tip electrode configuration sketched in Fig. 1 with voltages V 1 and V 2 allows independent tuning of E ⊥ and E as the reflection symmetries imply that they are determined respectively by (V 1 − V 2 )/2 and (V 1 + V 2 )/2. For the parameters that allow to confine the exciton's center of mass (CM), E ⊥ (z) and E (z) can be regarded as constant across the CNT's cross section and the length scale over which they vary appreciably is much 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 weak perturbation ∝ e ±iϕ that only induces intersubband virtual transitions.
More precisely, within each pair of subbands n, m (i) We consider E (z) much weaker than the critical field to ionize the exciton so that ψ nm± |Ĥ int |ψ nm± is much smaller than the binding energy. The latter allows to derive an effective Hamiltonian for the ground state manifold of the quasi-1D hydrogenic series associated to n, m by adiabatic elimination of the corresponding excited manifolds. This effective Hamiltonian for the CM motion has a potential part whose leading contribution is second order inĤ int and yields the effective confining potential: V . In a classical picture the field polarizes Effective potential (a.u.) the exciton that thus experiences a force proportional to the gradient of the squared field. The characteristic level spacing of the hydrogenic series implies that the excitonic polarizability satisfies α is the exciton Bohr radius and n−m (q) an appropriate dielectric function. In particular 0 (0) = ≈ 7 corresponds to the intrinsic permittivity along the CNT [24]. As shown in Fig. 1 we have calculated the field generated by this electrode configuration for typical parameters using FEM. These result for a (9, 4) tube [22], in a zero point motion σ CM (z CM ) which can be taken to be Gaussian for the ground state (henceforth |ψ 00± ) [22].
We now consider the effect of E ⊥ 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 E 12 , E 21 , E 13 , E 31 . We find that the contri-butions from E 12 and E 21 vanish identically and obtain |ψ z), and l labels a complete set of envelope functions for the E 13 and E 31 manifolds [36] -⊥ ≈ 1.6 denotes the intrinsic relative permittivity normal to the CNT axis [24].
The Hamiltonian describing the interaction between electrons and low frequency phonons has two distinct terms: (i) a true DP contribution diagonal in sublattice space corresponding to an energy shift of the Dirac point and (ii) a bond-length change contribution off-diagonal in sublattice space [17]. For the aforementioned exciton states we find that the electron-hole and K-K symmetries imply that finite couplings only arise from (ii): Hereτ i andσ i are, respectively, Pauli matrices in valley (K-K ) space and in sublattice (A-B) space, and g 2 is the off-diagonal DP. Given that the relevant phonon wavelengths λ are much larger than the CNT radius R we can adopt a continuum shell model [17] withû ij (r) the corresponding Lagrangian strain and use the lowest orders in R/λ [thin rod elasticity (TRE)]. Both compressional and flexural deformations have the structure of a local stretching so that the strain components satisfy u ϕz = 0, ∂z 2 for flexural modes and u zz = ∂φc ∂z for compressional modes where φ f /c are the 1D fields [25] and σ ≈ 0.2 is the CNT Poisson ratio [26]. Then Eq. (1), the aforementioned approximation for |ψ (1) 00± , and the single particle Hamiltonian (2) allow us to obtain the lowest order contributions in the electric field to the interaction Hamiltonian H QD−ph between the exciton states |ψ 00± and low frequency phonons where we have exploited the completeness of {|F −ν0,l }. Henceforth, we consider parameters for which the flexural and compressional branches are expected to present resonances with a free spectral range larger than the optical linewidth of the zero phonon line (ZPL) of the transition associated to the state |ψ 00− . We focus on laser excitation of the latter near resonant with the ensuing lowest-frequency flexural phonon red sideband. We consider a bridge geometry (cf. Fig. 1) with a length L short enough that the relative strength of these phonon sidebands is weak [cf. Eq. (5)]. Hereafter, we use the formalism developed in Ref. 25 and adopt a resonator-bath representation with the resonator mode (annihilation operator b 0 , angular frequency ω 0 , and quality factor Q) corresponding to the fundamental in-plane flexural resonance that we intend to manipulate and laser cool to the ground state. In turn the bath modes include the 3D substrate that supports the CNT coupled to the other nanotube vibrational resonances. Hence we insert in the effective field operatorsφ f /c in Eq. (3) the resonator-bath mode decomposition, i.e.
Here φ 0 (z) is the normalized resonator 1D eigenmode and µ is the linear mass density of the CNT, while u x,q (z) [u z,q (z)] is the x [z] component of the CM displacement of the CNT cross section at z for the bath mode corresponding to the scattering eigenmode q and ρ s is the substrate's density.
Thus in a polaronic (shifted) representation [4], the Hamiltonian for the laser driven NTQD coupled to the resonator mode, to the phonon bath (annihilation operators b q ), and to the radiation field (annihilation operators a k and couplings g k ) reads H = H sys + H int + H B with ( = 1): Here B ≡ e η(b0−b † 0 ) , δ is the laser detuning from the ZPL and Ω the Rabi frequency, we have introduced Pauli matrix notation for the optical pseudospin (σ z = 1 corresponds to |ψ 00− and σ z = −1 to the empty NTQD), applied a shift to the phonon modes q, and adopted a rotating frame at the laser frequency ω L . The parameter η, which characterizes the strength of the exciton-resonator coupling (e −η 2 /2 is the Frank-Condon factor) is given by where we have introduced the effective field E ⊥ ≡ ( √ L/q 2 0 ) F 00 | ∂ 2 φ0 ∂z 2 (ẑ e )E ⊥ (ẑ e )|F 00 , q 0 is the TRE phonon wavevector for the resonator mode, h = 0.66Å is the effective thickness for the continuum shell model [26], E = 1TPa is the CNT Young modulus, and σ G = 7.7 × 10 −7 Kgm −2 is the mass density of graphene. We focus on parameters such that η < 0.2. Its favorable scaling as √ L is a direct consequence of the quadratic flexural dispersion. Note that the perturbative treatment of E ⊥ underpinning Eq. (5) implies ξE ⊥ 1. Finally, the couplings ζ q and λ q to the bath modes lead, respectively, to the resonator mode's phonon tunneling dissipation and to pure dephasing of the NTQD. The RWA for the ζ q is justified given their weakness and η 1. These conditions and the anharmonicity of the flexural spectrum also imply that the flexural λ q can be neglected and the pure dephasing is dominated by the compressional branch [25].
We find that for all environmental couplings the Born-Markov approximation is valid and after eliminating the bath phonon modes and the radiation field, we obtain a master equation for the NTQD coupled to the resonator with a Hamiltonian contribution given by H sys and a dissipative contribution of Lindblad form with collapse operators: √ Γ σ − , γ D /2 σ z , ω 0 n(ω 0 )/Q b † 0 , and ω 0 [n(ω 0 ) + 1]/Q b 0 . Here n(ω 0 ) is the thermal equilibrium occupancy at the ambient temperature and γ D is the phonon-induced dephasing rate. Other relevant sources of dissipation beyond those considered in Hamiltonian (4) can be incorporated by adopting modified values of Q [12] and Γ [37]. The dephasing rate γ D is determined by the low frequency behavior of the phonon spectral density J(ω) = π q |λ q | 2 δ(ω − ω q ) (with q ∈ compressional branch). For a bridge geometry the scattering modes derived in Ref. 25 result in an Ohmic spectral density J(ω) = 2πα con ω, for ω much smaller than the fundamental compressional resonance ω c ω 0 , which naturally leads to γ D = 2πα con k B T / . In turn, it is straightforward to determine that the "confined" dimensionless dissipation parameter satisfies α con = α/πQ c , where Q c is the clamping-loss limited Q-value of the fundamental compressional resonance [25] and α the dissipation parameter that would result for an infinite length. The latter can be calculated using Eq. (3) and reads: α = g 2 2 √ σ G (1 + σ) 2 cos 2 3θ/2π 2 R(Eh) 3/2 . It depends on the chirality and may approach unity for small radius zigzag tubes. Thus, when the exciton linewidth is dominated by electron-phonon interactions [27] the phonon confinement in our structure will reduce it by at least the factor πQ c 1. Finally, α con 1 warrants the Born-Markov approximation in the treatment of the pure dephasing in the relevant regime γ D Γ/2.
In complete analogy with the Lamb-Dicke limit 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's Fock states. This incorporates both the mechanical dissipation and the dissipative effects induced by the scattering of laser light. The latter result in cooling and heating with rates η 2 A ∓ that read the same as in Ref. 4 with the quantum dot Liouvillian L QD including now the pure dephasing γ D . As γ D → 0 the steady state occupancy for Q → ∞, i.e. the quantum backaction limit A + /(A − − A + ), becomes independent of Ω (in stark contrast to atomic laser cooling) and reduces to the same expression valid for the cavity-assisted backaction cooling [5] with the cavity decay rate 1/τ replaced by the spontaneous emission rate. In the resolved sideband regime and for the optimal detuning δ = −ω 0 , this fundamental limit yields (Γ/4ω 0 ) 2 .
In the un-shifted representation, for δ = 0 and after a π/2 rotation of the pseudospin aroundŷ, H sys reduces to the Jaynes-Cummings model with the spin degree of freedom afforded by the NTQD states dressed by the laser field. The spin-oscillator coupling and resonance condition are given, respectively, by g = ηω 0 /2 and Ω = ω 0 . Thus, given 1/Q η reaching the strong coupling regime depends on satisfying ηω 0 > Γ/2. This regime is akin to the parametric normal mode splitting in cavity optomechanics [28] and 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 allows to emulate the adiabatic passage scheme used in Ref. 19 for performing QND measurements of the oscillator's energy. This would enable the observation of motional quantum jumps.
In conclusion, we set forth a scheme for optomechanical manipulation of nanotube resonators via the deformation potential exciton-phonon interaction. This provides a high-performance alternative to radiation-pressure based schemes [1] for an ultra-low mass and high frequency nanoscale resonator leading to large backaction-cooling factors and opening a direct route to the quantum behavior of a "macroscopic" mechanical degree of freedom [2]. Most importantly, these breakthroughs rely on a lifetime-limited zero phonon line much narrower than the smallest CNT linewidths reported so far [13]. Indeed, the envisaged NTQDs will allow to suppress the two most likely linewidth-broadening mechanisms, namely: inhomogeneous broadening and phonon-induced dephasing [27], by providing a controlled electrostatic environment and strong confinement of low frequency phonons. Furthermore, a doped version of these NTQDs will enable a tunable spin-photon interface [29].
IWR acknowledges helpful discussions with N. Qureshi and A. Bachtold.