Spin-orbit-induced strong coupling of a single spin to a nanomechanical resonator
András Pályi, P. R. Struck, Mark Rudner, Karsten Flensberg, Guido Burkard
SSpin-Orbit-Induced Strong Coupling of a Single Spin to a Nanomechanical Resonator
Andr´as P´alyi,
1, 2
P. R. Struck, Mark Rudner, Karsten Flensberg,
3, 4 and Guido Burkard Department of Physics, University of Konstanz, D-78457 Konstanz, Germany Department of Materials Physics, E¨otv¨os University, H-1517 Budapest POB 32, Hungary Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Niels Bohr Institute, University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen, Denmark
We theoretically investigate the deflection-induced coupling of an electron spin to vibrationalmotion due to spin-orbit coupling in suspended carbon nanotube quantum dots. Our estimatesindicate that, with current capabilities, a quantum dot with an odd number of electrons can serveas a realization of the Jaynes-Cummings model of quantum electrodynamics in the strong-couplingregime. A quantized flexural mode of the suspended tube plays the role of the optical mode andwe identify two distinct two-level subspaces, at small and large magnetic field, which can be usedas qubits in this setup. The strong intrinsic spin-mechanical coupling allows for detection, as wellas manipulation of the spin qubit, and may yield enhanced performance of nanotubes in sensingapplications.
Recent experiments in nanomechanics have reachedthe ultimate quantum limit by cooling a nanomechani-cal system close to its ground state [1]. Among the vari-ety of available nanomechanical systems, nanostructuresmade out of atomically-thin carbon-based materials suchas graphene and carbon nanotubes (CNTs) stand out dueto their low masses and high stiffnesses. These propertiesgive rise to high oscillation frequencies, potentially en-abling near ground-state cooling using conventional cryo-genics, and large zero-point motion, which improves theease of detection [2, 3].Recently, a high quality-factor suspended CNT res-onator was used to demonstrate strong coupling be-tween nanomechanical motion and single-charge tunnel-ing through a quantum dot (QD) defined in the CNT[4]. Here, we theoretically investigate the coupling of asingle electron spin to the quantized motion of a dis-crete flexural mode of a suspended CNT (see Fig.1),and show that the strong-coupling regime of this Jaynes-Cummings-type system is within reach. This couplingprovides means for electrical manipulation of the electronspin via microwave irradiation, and leads to strong non-linearities in the CNT’s mechanical response which maypotentially be used for enhanced functionality in sensingapplications [5–7].In addition to their outstanding mechanical properties,carbon-based systems also possess many attractive char-acteristics for information processing applications. Thepotential for single electron spins in QDs to serve as theelementary qubits for quantum information processing[8] is currently being investigated in a variety of systems.In many materials, such as GaAs, the hyperfine interac-tion between electron and nuclear spins is the primarysource of electron spin decoherence which limits qubitperformance (see e.g., [9]). However, carbon-based struc-tures can be grown using starting materials isotopically-enriched in C, which has no net nuclear spin, thus prac-tically eliminating the hyperfine mechanism of decoher- nanotube u(z)x z electronsupport gate
FIG. 1: Schematic of a suspended carbon nanotube (CNT)containing a quantum dot filled with a single electron spin.The spin-orbit coupling in the CNT induces a strong couplingbetween the spin and the quantized mechanical motion of theCNT. ence [10], leaving behind only a spin-orbit contribution[11, 12]. Furthermore, while the phonon continuum inbulk materials provides the primary bath enabling spinrelaxation, the discretized phonon spectrum of a sus-pended CNT can be engineered to have an extremelylow density of states at the qubit (spin) energy splitting.Thus very long spin lifetimes are expected off-resonance[13]. On the other hand, when the spin splitting is nearlyresonant with one of the high-Q discrete phonon “cav-ity” modes, strong spin-phonon coupling can enable qubitcontrol, information transfer, or the preparation of entan-gled states.The interaction between nanomechanical resonatorsand single spins was recently detected [14], and has beentheoretically investigated [15, 16] for cases where thespin-resonator coupling arises from the relative motionof the spin and a source of local magnetic field gradients.Such coupling is achieved, e.g., using a magnetic tip on avibrating cantilever which can be positioned close to anisolated spin fixed to a nonmoving substrate. Creating a r X i v : . [ c ond - m a t . m e s - h a ll ] J un !"!" Kx Kz S B $ $ B ! ! T " B ! T " E ! m e V " FIG. 2: Energy levels of the four dimensional (due to spinand valley) orbital ground state subspace of the QD, as afunction of the magnetic field parallel ( B (cid:107) ) and perpendic-ular ( B ⊥ ) to the CNT axis. The boxed areas indicate theworking regime for the spin qubit (S) and Kramers qubit(K), the latter being operated either in a longitudinal (Kz)or perpendicular (Kx) magnetic field. Parameter values [30]:∆ so = 170 µ eV, ∆ KK (cid:48) = 12 . µ eV, µ orb = 330 µ eV / T. strong, well-controlled, local gradients remains challeng-ing for such setups. In contrast, as we now describe, inCNTs the spin-mechanical coupling is intrinsic , suppliedby the inherent strong spin-orbit coupling [17–20] whichwas recently discovered by Kuemmeth et al. [21].Consider an electron localized in a suspended CNTquantum dot (see Fig. 1). Below we focus on the case ofa single electron, but expect the qualitative features to bevalid for any odd occupancy (see Ref. [22]). We work inthe experimentally-relevant parameter regime where thespin-orbit and orbital-Zeeman couplings are small com-pared with the nanotube bandgap and the energy of thelongitudinal motion in the QD. Here, the longitudinaland sublattice orbital degrees of freedom are effectivelyfrozen out, leaving behind a nominally four-fold degen-erate low-energy subspace associated with the remainingspin and valley degrees of freedom (see Refs. [9, 24]).A simple model describing the spin and valley dynam-ics in this low-energy QD subspace, incorporating thecoupling of electron spin to deflections associated withthe flexural modes of the CNT [25, 26], was introducedin Ref. [27]. In principle, the deformation-potentialspin-phonon coupling mechanism [11] is also present.The deflection coupling mechanism is expected to domi-nate at long phonon wavelengths, while the deformation-potential coupling should dominate at short wavelengths(see discussion in [27]). For simplicity we consider onlythe deflection coupling mechanism, but note that the ap-proach can readily be extended to include both effects.The Hamiltonian describing this system is [10, 24, 27] H = ∆ so τ ( s · t )+∆ KK (cid:48) τ − µ orb τ ( B · t )+ µ B ( s · B ) , (1) where ∆ so and ∆ KK (cid:48) denote the spin-orbit and interval-ley couplings, τ i and s i are the Pauli matrices in valleyand spin space (the pseudospin is frozen out for the stateslocalized in a QD), t is the tangent vector along the CNTaxis, and B denotes the magnetic field. Note that thespin-orbit coupling has contributions which are diagonaland off-diagonal in sublattice space [18–20, 22]. Whenprojected onto to a single longitudinal mode of the quan-tum dot, the effective Hamiltonian given above describesthe coupling of the spin to the nanotube deflection at thelocation of the dot [24].For a nominally straight CNT we take t pointing alongthe z direction, giving s · t = s z and B · t = B z . Herewe find the low-energy spectrum shown in Fig. 2. Thetwo boxed regions indicate two different two-level systemsthat can be envisioned as qubit implementations in thissetup: we define a spin qubit [8] (S) at strong longitudi-nal magnetic field, near the value B ∗ of the upper levelcrossing, and a mixed spin-valley or Kramers (K) qubit[10], which can be operated at low fields applied either inthe longitudinal (Kz) or perpendicular (Kx) directions.We now study how these qubits couple to the quantizedmechanical motion of the CNT. For simplicity we con-sider only a single polarization of flexural motion (alongthe x -direction), assuming that the two-fold degeneracyis broken, e.g., by an external electric field. A general-ization to two modes is straightforward.A generic deformation of the CNT with deflection u ( z )makes the tangent vector t ( z ) coordinate-dependent. Ex-panding t ( z ) for small deflections, we rewrite the cou-pling terms in Hamiltonian (1) as s · t (cid:39) s z + ( du/dz ) s x and B · t (cid:39) B z + ( du/dz ) B x . Expressing the deflec-tion u ( z ) in terms of the creation and annihilation op-erators a † and a for a quantized flexural phonon mode, u ( z ) = f ( z ) (cid:96) √ ( a + a † ), where f ( z ) and (cid:96) are the wave-form and zero-point amplitude of the phonon mode, wefind that each of the three qubit types (S, Kx, Kz) ob-tains a coupling to the oscillator mode which we describeas H (cid:126) = ω q σ + g ( a + a † ) σ + ω p a † a +2 λ ( a + a † ) cos ωt. (2)Here the matrices σ , are Pauli matrices acting on thetwo-level qubit subspace, and we have included a term de-scribing external driving of the oscillator with frequency ω and coupling strength λ , which can be achieved bycoupling to the ac electric field of a nearby antenna [4].Below we describe the dependence of the qubit-oscillatorcoupling g on system parameters for each qubit type (S,Kx, or Kz). The derivation of Eq. (2) is detailed in [24].For the spin qubit (S), the relevant two-fold degree offreedom is the spin of the electron itself. Therefore inEq.(2) we have σ = s z and σ = s x , and the qubitlevels are split by the Zeeman energy, measured relativeto the value B ∗ where the spin-orbit-split levels cross, (cid:126) ω q = µ B ( B − B ∗ ). A spin magnetic moment of µ B isassumed, and B ∗ ≈ ∆ so / µ B for ∆ KK (cid:48) (cid:28) ∆ so . For thequbit-resonator coupling, we find g = ∆ so (cid:104) f (cid:48) (cid:105) (cid:96) / √
2, in-dependent of B . Here, (cid:104) f (cid:48) (cid:105) is the derivative of the wave-form of the phonon mode averaged against the electrondensity profile in the QD.For a symmetric QD, positioned at the midpoint of theCNT, the coupling matrix element proportional to (cid:104) f (cid:48) (cid:105) vanishes for the fundamental and all even harmonics (theopposite would be true for the deformation-potential cou-pling mechanism). The cancellation is avoided for a QDpositioned away from the symmetry point of the CNT,or for coupling to odd harmonics. Here, for concreteness,we consider coupling of a symmetric QD to the first vi-brational harmonic of the CNT. Using realistic param-eter values [4, 21, 29, 30], L = 400 nm, (cid:96) = 2 . so = 370 µ eV, ∆ KK (cid:48) = 32 . µ eV, µ orb = 1550 µ eV / T,and ω p / π = 500 MHz, we find g/ π ≈ .
56 MHz, irre-spective of the magnetic field strength B along the CNT.For the Kramers qubits (Kx and Kz), both ω q and g depend on B . The qubit splitting for the Kxqubit is controlled by the perpendicular field, (cid:126) ω q = µ B (2∆ KK (cid:48) / ∆) B x , while for the Kz qubit, it is controlledby the longitudinal field (cid:126) ω q = ( µ B + µ orb (∆ so / ∆)) B z ,where ∆ = (cid:112) ∆ + 4∆ KK (cid:48) denotes the zero-field split-ting between the two Kramers pairs. Resonant couplingoccurs when ω q = ω p . This condition sets the relevantvalue of B x ( B z ) in the case of the Kx (Kz) qubit; theparameters above yield B x ≈
103 mT ( B z ≈ . (cid:126) g = − ( (cid:104) f (cid:48) (cid:105) (cid:96) / √ µ orb ∆ so / ∆ + µ B ∆ / ∆ ) B x , while for theKz qubit it scales with the longitudinal field, (cid:126) g =( (cid:104) f (cid:48) (cid:105) (cid:96) / √ µ orb KK (cid:48) ∆ so / ∆ ) B z . Using the values of B x and B z obtained above, we estimate couplings of g/ π ≈ .
49 MHz for the Kx qubit, and g/ π ≈ .
52 kHzfor the Kz qubit. Thus the coupling for the Kx qubit iscomparable to that of the spin qubit, while the couplingof the Kz qubit is much weaker. Therefore, we restrictour considerations to the spin and Kx qubits below.Ref. 4 reports the fabrication of CNT resonators withquality factors Q ≈ , Q = 63 ,
000 forthe following estimate. Together with the oscillator fre-quency ω p / π = 500 MHz, this value of Q implies anoscillator damping rate of Γ ≈ · s − (cid:28) g . Becauseof the near-zero density of states of other phonon modesat ω q , it is reasonable to assume a very low spontaneousqubit relaxation rate γ . These observations suggest thatthe so-called “strong coupling” regime of qubit-oscillatorinteraction, defined as Γ , γ (cid:28) g , can be reached withCNT resonators.To quantify the system’s response in the anticipatedparameter regime, we study the coupled qubit-oscillatordynamics using a master equation which takes into ac-count the finite lifetime of the phonon mode as well asthe non-zero temperature of the external phonon bath. For weak driving, λ (cid:28) ω p , and ω p ≈ ω q ≈ ω (cid:29) g ,we move to a rotating frame and use the rotating waveapproximation (RWA) to map the Hamiltonian, Eq.(2),into Jaynes-Cummings form [31] H RWA (cid:126) = ˜ ω q σ + g ( aσ + + a † σ − )+ ˜ ω p a † a + λ ( a + a † ) , (3)where ˜ ω i = ω i − ω . Including the nonunitary dynamicsassociated with the phonon-bath coupling, the masterequation for the qubit-oscillator density matrix ρ reads:˙ ρ = − i (cid:126) [ H RWA , ρ ] + ( n B + 1)Γ (cid:0) aρa † − { a † a, ρ } (cid:1) + n B Γ (cid:0) a † ρa − { aa † , ρ } (cid:1) , (4)where n B = 1 / ( e (cid:126) ω p /k B T −
1) is the bath-mode Bose-Einstein occupation factor, and k B is the Boltzmann con-stant.Because of the phonon damping, in the long-time limitthe system is expected to tend towards a steady state,described by the density matrix ¯ ρ . We study thesesteady states, found by setting ˙ ρ = 0 in Eq.(4), usingboth numerical and semiclassical analytical methods. InFigs. 3a,c we show the steady-state phonon occupationprobability distribution P ( δω, n ) as a function of thedrive frequency–phonon frequency detuning δω = − ˜ ω p and the phonon occupation number n , for the case wherethe qubit and oscillator frequencies are fixed and degen-erate, ω q = ω p (see caption for parameter values). Pan-els a and c compare the cases with and without qubit-oscillator coupling. In Figs. 3b and 3d we show the aver-aged phonon occupation number ¯ n ( δω ) = (cid:80) n nP ( δω, n ),which is closely related to the mean squared resonatordisplacement in the steady state: X = x = (cid:96) (¯ n + ).For g (cid:54) = 0, we observe a splitting of the oscillator reso-nance, which is characteristic of the coupling to the two-level system, and can serve as an experimental signatureof the qubit-oscillator coupling. For drive frequenciesnear the split peaks, the phonon number distribution isbimodal (Fig. 3f) showing peaks at n ≈ n ,indicating bistable behavior (see below).For strong excitation, where the mean phonon occupa-tion is large, we expect a semiclassical approach to cap-ture the main features of the system’s dynamics [11, 33].Extending the approach described in [11] to include dis-tinct values of the qubit, oscillator, and drive frequencies, ω q , ω p , and ω , we derive semiclassical equations of mo-tion for the mean spin and oscillator variables (see [24]).The steady-state values of the mean squared oscillatoramplitude obtained from the resulting nonlinear systemare shown in Fig. 3e. In the vicinity of the split peak wefind two branches of stable steady-state solutions, indica-tive of bistable/hysteretic behavior [4]. The semiclassicalresults in Fig. 3e are in correspondence with the phononnumber distribution in Fig. 3c, and explain its bimodalcharacter. Similar oscillator instabilities have been used ∆Ω Π MHz n g ∆Ω Π MHz n X p m ∆Ω Π MHz n Pg ∆Ω Π MHz n X p m ∆Ω Π MHz n n P n ∆ B mT ∆ Ω Π M H z X pm T ∆ B mT ∆ Ω Π M H z X pm T (a) (c) (e) (g)(b) (d) (f) (h) FIG. 3: Response of the spin-oscillator system. (a) Phonon number probability distribution P ( n, δω ), (b) average phononoccupation ¯ n and root mean squared displacement X of the uncoupled driven CNT resonator ( g = 0), as functions of the drivefrequency–oscillator frequency detuning δω = ω − ω p . The parameters are T = 50 mK, ω p / π = 500 MHz, Γ = 5 · s − and λ/ π = 0 .
027 MHz. The same quantities are plotted in (c) and (d) for a resonantly coupled qubit-oscillator system (i.e., ω q = ω p ), with coupling constant g/ π = 0 . X of the resonator amplitude in the coupled spin qubit - oscillator system at (g) T = 0 and (h) T = 50 mK, as functions of magnetic field detuning δB (detuning the qubit frequency away from resonance with the oscillator)and drive frequency–oscillator frequency detuning δω . as the basis for a sensitive readout scheme in supercon-ducting qubits [12], and may potentially be useful formass or magnetic field sensing applications where smallchanges of frequency need to be detected.To predict the oscillator response to be detected viaa charge sensor (see below), we solve for the station-ary state of Eq. (4) directly for a range of driving fre-quencies, qubit-oscillator detunings (set by the magneticfield), and temperatures T . In Figs. 3(g) and 3(h), weshow the T = 0 and T = 50 mK root mean squaredoscillator amplitude X ∝ (cid:112) ¯ n + 1 / B and drive frequency, for the case of a spin(S) qubit. The value δB = 0 corresponds to resonantcoupling ω q = ω p . These results also apply for the Kxqubit, if the magnetic field axis is adjusted appropriately.In the zero-temperature case, only half of the eigenstates (cid:126) ω ± ≈ (cid:126) ω p ∓ (cid:126) g / ( ω p − ω q ) of Eq. (3) can be efficientlyexcited by the drive at fixed δB , giving rise to the upper(lower) feature in Fig. 3g for δB < δB > T (cid:38) (cid:126) ω q , both branches of the Jaynes-Cummings lad-der can be efficiently excited (Fig. 3h). This is a distinctand experimentally accessible signature of the strong cou-pling at finite temperature. Note that the vacuum Rabisplitting is also observed (see arrows in Fig. 3d), but fea-tures arising from nonlinearity in the strongly driven sys-tem dominate by more than 2 orders of magnitude.Displacement detection of nanomechanical systems ispossible using charge sensing [5, 35], where the conduc- tance of a mesoscopic conductor, such as a QD or quan-tum point contact, is modulated via capacitve couplingto the charged mechanical resonator. Furthermore, thequbit state itself can be read out using spin-detectionschemes developed for semiconductor QDs [36], or by adispersive readout scheme like that commonly used insuperconducting qubits coupled to microwave resonators[37]. The dispersive regime can be rapidly accessed by,e.g., tuning the resonator frequency using dc gate pulseswhich control the tension in the CNT [4].In summary, we predict that strong qubit-resonatorcoupling can be realized in suspended CNT QDs withcurrent state-of-the-art devices. The coupling describedhere may find use in sensing applications, and in spin-based quantum information processing, where the CNToscillator enables electrical control of the electron spin,and, with capacitive couplers, may provide long-rangeinteractions between distant electronic qubits [16, 38].Combined with control of the qubit via electron-spin-resonance [39], the mechanism studied here could be uti-lized for ground-state cooling and for generating arbi-trary motional quantum states of the oscillator [15].We gratefully acknowledge helpful discussions with H.Carmichael, V. Manucharyan and P. Rabl. This workwas supported by the OTKA grant PD 100373, the MarieCurie grant CIG-293834 and the QSpiCE program ofESF (AP), DFG under the programs FOR 912 and SFB767 (PS and GB), NSF grants DMR-090647 and PHY-0646094 (MR), and The Danish Council for IndependentResearch — Natural Sciences (KF). Note:
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Spin-Orbit-Induced Strong Coupling of a Single Spin to a Nanomechanical Resonator
Andr´as P´alyi, , P. R. Struck, Mark Rudner, Karsten Flensberg, , and Guido Burkard Department of Physics, University of Konstanz, D-78457 Konstanz, Germany Department of Materials Physics, E¨otv¨os University, H-1517 Budapest POB 32, Hungary Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
APPENDIX A: EFFECTIVE QUANTUM DOT HAMILTONIAN
In the following, we derive Hamiltonian (1) of the main text, which describes the spectrum of a quantum dotformed in a straight, suspended carbon nanotube, in the absence of phonons (i.e. in a static tube). As usual, we startfrom a tight-binding description of a graphene sheet, which is then rolled up with the condition of periodic boundaryconditions. Using the conventions as in Weiss et al. [1], this gives the following Hamiltonian for the longitudinaldegree of freedom H = v F p z σ + ∆ g σ + ˆ t · s τ ( σ ∆ + ∆ ) + V ( z ) . (1)Here z and ˆ t represent the coordinate and unit vector in the direction of the tube, v F is the Fermi velocity, σ i , τ i , and s i are Pauli matrices in sublattice, valley and spin spaces, respectively. Note that to translate between the conventionused here and that used in e.g. Ref. 4, σ must be replaced by τ σ . The energy gap between the valence band andthe conduction band is 2∆ g , where ∆ g = (cid:126) v F ( ν/ R ) + ∆ c with 2∆ c being the curvature induced minigap, which istypically of order 10 meV, but proportional to cos 3 θ , where θ is the chiral angle of the tube. For nominally metallictubes, ν = 0. The spin-orbit interaction has two terms, one that is diagonal in sublattice (the ∆ term) and onewhich is off-diagonal (the ∆ term). The spin-orbit interaction connects the spin projection along the tube axis to the K, K (cid:48) (valley flavor) quantum number through the prefactor (ˆ t · s ) τ . The microscopic derivation of this Hamiltoniancan be found in Refs. 2–4. Finally, the term V ( z ) describes the confining potential of the quantum dot. It is assumedto be smooth on the atomic scale and hence has no sublattice structure.Furthermore, an applied magnetic field couples to both the spin and orbital degrees of freedom. The coupling tothe orbital degree of freedom appears through the Aharonov-Bohm flux, which modifies the boundary condition ofthe circumferential wave vector and hence changes ∆ g . In total, the Hamiltonian due to magnetic field is H B = H orb + H s , H orb = − µ B ( l/ (cid:126) ) σ τ ˆt · B , H s = 12 gµ B s · B , (2)where l = mv F R .There are several ways to arrive at the Hamiltonian in Eq. (1) in the main text. Here we will assume a hierarchyof energy scales, typical of many experimental realizations of nanotube quantum dots, namely∆ g (cid:29) E L (cid:29) E B , E SO , E KK (cid:48) , (3)where E L is the level spacing due to longitudinal quantization and E B , E SO , and E KK (cid:48) are the energy changes dueto the external magnetic field, spin-orbit coupling, and valley mixing, respectively. This allows us to first solve forthe dot wavefunction in absence of these three contributions and then project onto a single longitudinal mode.In passing we note that in order to get more information about dependence of the orbital magnetic moment andspin-orbit coupling on the number of electrons in the quantum dot, one has to be more precise and use a specific formof the confining potential, e.g. assuming a square well potential, as in Refs. 1, 7. This was done in Refs. 5, 6, wherea method to experimentally extract the two spin-orbit parameters ∆ and ∆ , as well as µ orb , was shown.Now imagine that one has solved for the case without magnetic field, spin-orbit coupling, and mixing between K and K (cid:48) . This gives a set of longitudinal wavefunctions, each one four-fold degenerate due to the spin and valleydegrees of freedom. The energy splitting between these shells is E L (cid:29) E B , E SO , E KK (cid:48) . We label these states by thevalley and spin quantum numbers τ = ± s = ±
1, respectively, which indicate corresponding eigenvalues under τ and s z . Projected onto eigenstates of the spatial coordinates z and c (the circumferential coordinate), the wavefunction in the envelope-function representation has the form[1] (cid:104) ( z, c ) | τ, s (cid:105) = e iτk ⊥ c √ πR φ ( z ) η σ ⊗ χ s ⊗ χ τ , (4)where k ⊥ = − ν R is the wave vector associated with the gap, η σ = √ (1 1) T is a pseudospinor describing thesublattice degrees of freedom, χ s and χ τ describe the spin and valley degrees of freedom, respectively, χ + = (1 0) T and χ − = (0 1) T , and the precise form of the envelope wave function φ ( z ) depends on the confining potential.We can now take matrix elements with respect to H B and H SO = ˆt · s τ ( σ ∆ + ∆ ) and also a term describing themicroscopic disorder that couples valleys: H KK (cid:48) = V KK (cid:48) , where V KK (cid:48) is a short-range disorder potential dependingon the longitudinal and circumferential coordinate operators. (Note that short-range disorder can be systematicallyincorporated into the envelope-function description, see e.g., Refs. 8, 9.) This procedure will produce a Hamiltonianof the form in Eq. (1) in the main text, H = ∆ so τ ( s · t ) + ∆ KK (cid:48) τ − µ orb τ ( B · t ) + µ B ( s · B ) , with ∆ so = 2 (cid:104) + | σ ∆ + ∆ | + (cid:105) , ∆ KK (cid:48) = (cid:104)−| V KK (cid:48) | + (cid:105) , µ orb = ev F R (cid:104) + | σ | + (cid:105) . (5)Here the kets |±(cid:105) stand for the orbital states with τ = ±
1. Note that in general, the valley-mixing term can includeboth τ and τ , but an appropriate unitary transformation in the valley space can be used to put it into the formabove with real ∆ KK (cid:48) . A.1 Derivation of the coupling to vibrations
Next we look at how the vibrations couple to the four states of the quantum dot. The amplitude of the tube is interms of the harmonic oscillator raising/lowering operators given by u ( z ) = f ( z ) (cid:96) √ (cid:0) a + a † (cid:1) , (6)where we focus on a single vibrational mode. The coupling to the spin is via the change of the tangent direction givenby δ ˆt = dudz ˆx , (7)where ˆx is perpendicular to the tube and in the plane of the vibration. The interaction Hamiltonian then becomes H s, vib = δ ˆt · s τ ( σ ∆ + ∆ ) − µ orb τ σ δ ˆt · B . (8)As above, we now project onto a single longitudinal mode, thus taking matrix elements of H s, vib in the basis | τ, s (cid:105) .Such matrix elements involve form factors like (cid:104) τ, s | σ i f (cid:48) ( z ) | τ (cid:48) , s (cid:48) (cid:105) = δ ττ (cid:48) (cid:104) τ, s | σ i f (cid:48) ( z ) | τ, s (cid:48) (cid:105) = δ ττ (cid:48) δ ss (cid:48) F i,τ . (9)At this point we note that the coupling is small for a symmetric dots and even harmonics, because the F factors thentend to cancel, see discussion in the main text. The effective Hamiltonian for coupling between the four states of thequantum dot and the vibration now becomes H s, vib = (cid:96) √ (cid:0) a + a † (cid:1) { s x τ ( F ∆ + F ∆ ) − B x τ µ orb F } . (10)We see from Eq. (10) that the coupling of the vibrations to quantum dot states have different form factors fromwhat one would get by simply setting (7) into Eq. (1) of the main paper. However, when the energy scales are clearlyseparated as in (3) the eigenstates | τ, s (cid:105) are eigenstates of σ (which can be seen from (1)) and therefore F = F (for the conduction band). Therefore, we do not need to take into account the different form factors in (10), whichsimplifies the analysis and we can write F = F = (cid:104) f (cid:48) (cid:105) . In this language Eq. (10) becomes H s, vib = (cid:96) √ (cid:0) a + a † (cid:1) { s x τ ∆ SO − B x τ µ orb } (cid:104) f (cid:48) (cid:105) , (11)which is the result used in the main text.As mentioned the expression (11) was derived under the assumption that the gap dominates over the longitudinalsize quantization energy, which is valid for few-electron quantum dot. However, it is important to note that one couldeasily extend this to the more general case at higher energies by including the difference in form factors F and F without changing the conclusions and structure of our results qualitatively. APPENDIX B: QUBIT-PHONON COUPLINGS
We treat three different qubit realizations in the main text: the spin qubit (S), the Kramers qubit in a magneticfield perpendicular to the carbon nanotube (CNT) (Kx), and the Kramers qubit in a parallel-to-CNT magnetic field(Kz). Below, we express the three qubit Hamiltonians ¯ H s , ¯ H Kx and ¯ H Kz as functions of system parameters and CNTdeformation. To clarify the correspondence with the Jaynes-Cummings Hamiltonian in Eq. (2) of the main text, welist the formulas for the qubit frequency, the qubit-phonon coupling, as well as numerical estimates for the latter, inTable I. B.1 Spin-phonon coupling
At the finite value B ∗ = ∆ so µ B (cid:118)(cid:117)(cid:117)(cid:116) − KK (cid:48) ∆ (cid:16) µ µ B − (cid:17) (12)of a longitudinally-applied magnetic field, the quantum dot (QD) energy spectrum shows a crossing of the energies ofa pair of spin states belonging to the same valley (see Fig. 2 of the main text). Around this point these two levelsare energetically well separated from the other QD levels. We call this two-level system the spin qubit (S). If thedynamics is restricted to these two levels, it can be described by the following effective Hamiltonian: H s = µ B s z ( B − B ∗ ) + ∆ so dudz s x (13)where we assume that the effect of valley-mixing is negligible, ∆ KK (cid:48) = 0. Averaging over the z coordinate using thecharge density n ( z ) of the electron occupying the CNT QD yields¯ H s ≡ (cid:90) dzn ( z ) H s ( z ) = µ B s z ( B − B ∗ ) + ∆ so s x (cid:90) dzn ( z ) dudz ≡ µ B s z ( B − B ∗ ) + ∆ so s x (cid:28) dudz (cid:29) . (14) B.2 Kramers qubit-phonon coupling in a perpendicular magnetic field
At zero magnetic field, the ground state of the CNT QD, i.e., the ground state of the Hamiltonian H in Eq. (1)of the main text, is formed by a pair of time-reversed states (Kramers pair). The twofold degeneracy is maintainedeven in the presence of spin-orbit interaction and valley mixing. At small enough magnetic field these two states splitup, but they remain energetically well separated from higher-lying states. We call this two-level system the ‘Kramersqubit’ [10] in a perpendicular field (Kx). [Similar considerations hold for the first excited Kramers pair, i.e., the twohigher-lying energy eigenstates of H in Eq. (1) of the main text.] These energetically split states, in the absence ofCNT deformation, will be denoted here as | + (cid:105) and |−(cid:105) . Starting from the Hamiltonian H in Eq. (1) of the main text,averaging over z using the electron density n ( z ), and incorporating the effect of the two higher-lying states on | + (cid:105) and |−(cid:105) via a second-order Schrieffer-Wolff transformation, we find that the dynamics restricted to the Kramers qubit inthe presence of an external magnetic field B = B x ˆ x and CNT deformation u ( z ) is described by the Hamiltonian¯ H Kx = B x (cid:34) σ µ B ∆ KK (cid:48) (cid:112) ∆ + 4∆ KK (cid:48) − σ (cid:28) dudz (cid:29) (cid:32) µ orb ∆ so (cid:112) ∆ + 4∆ KK (cid:48) + µ B ∆ ∆ + 4∆ KK (cid:48) (cid:33)(cid:35) . (15)Here, σ , are the Pauli matrices in the qubit basis, i.e., σ = | + (cid:105)(cid:104) + | − |−(cid:105)(cid:104)−| and σ = | + (cid:105)(cid:104)−| + |−(cid:105)(cid:104) + | . B.3 Kramers qubit-phonon coupling in longitudinal magnetic field
In the absence of CNT deformation, a parallel-to-CNT magnetic field splits two low-energy Kramers doublet of H in Eq. (1) of the main text. We call this two-level system the Kz qubit, and denote the two qubit states as | + (cid:105) and |−(cid:105) in this subsection. Starting from the complete Hamiltonian H in Eq. (1) of the main text, averaging over z Hamiltonian (cid:126) ω q (cid:126) g g/ π (numerical)¯ H s µ B ( B − B ∗ ) ∆ so (cid:104) f (cid:48) (cid:105) (cid:96) √ .
56 MHz¯ H Kx µ B B x KK (cid:48) (cid:113) ∆ +4∆ KK (cid:48) − B x (cid:104) f (cid:48) (cid:105) (cid:96) √ (cid:18) µ orb ∆ so (cid:113) ∆ +4∆ KK (cid:48) + µ B ∆ ∆ +4∆ KK (cid:48) (cid:19) .
49 MHz¯ H Kz − B z (cid:18) µ B + µ orb ∆ so (cid:113) ∆ +4∆ KK (cid:48) (cid:19) B z (cid:104) f (cid:48) (cid:105) (cid:96) √ µ orb KK (cid:48) ∆ so ∆ +4∆ KK (cid:48) .
52 kHzTABLE I: Correspondence between the terms of the Hamiltonian in Eq. (2) of the main text and those of the qubit-phononHamiltonians derived for the three different qubits. The parameters used to calculate the last column are L = 400 nm, (cid:96) = 2 . so = 370 µ eV, ∆ KK (cid:48) = 32 . µ eV, µ orb = 1550 µ eV / T. The estimate (cid:104) f (cid:48) (cid:105) = 2 √ /L has been used (see text). using the electron density n ( z ), and applying a second-order Schrieffer-Wolff transformation to describe the effect ofthe higher-lying Kramers pair to the Kz qubit, we find that the dynamics of the latter in the presence of an externalmagnetic field B = B z ˆ z and CNT deformation u ( z ) is described by the Hamiltonian¯ H Kz = B z (cid:34) σ (cid:28) dudz (cid:29) µ orb KK (cid:48) ∆ so ∆ + 4∆ KK (cid:48) − σ (cid:32) µ B + µ orb ∆ so (cid:112) ∆ + 4∆ KK (cid:48) (cid:33)(cid:35) . (16)As before, σ , are the Pauli matrices in the qubit basis, i.e., σ = | + (cid:105)(cid:104) + | − |−(cid:105)(cid:104)−| and σ = | + (cid:105)(cid:104)−| + |−(cid:105)(cid:104) + | . B.4 Deformation of the CNT
All three qubit-phonon Hamiltonians ¯ H s , ¯ H Kx and ¯ H Kz resemble the Jaynes-Cummings Hamiltonian of cavityquantum electrodynamics. This becomes more apparent if we express the z -dependent displacement u ( z ) in terms ofphonon annihilation a and creation a † operators: u ( z ) = f ( z ) (cid:96) √ a + a † ) . (17)Here f ( z ) is a dimensionless function describing the shape of the standing-wave bending phonon mode under consid-eration (normalization: (cid:82) f ( z ) dz = L ) and (cid:96) is the ground-state displacement of that mode.As apparent from Table I, the qubit-phonon coupling vanishes if g ∝ (cid:104) f (cid:48) (cid:105) ≡ (cid:82) dz df ( z ) dz n ( z ) = 0. Therefore, g vanishesif the setup is perfectly left-right symmetric along the CNT ( z ) axis and a bending mode with an even number ofnodes is considered. To have a finite qubit-phonon coupling, either the left-right symmetry of the setup must bebroken or a flexural mode with odd number of nodes should be considered. In the main text and also here we treatthe second case: we investigate the coupling of the first excited flexural phonon (1 node) to the various qubits.The displacement field of the first harmonic of the resonator can be approximated by f ( z ) = −√ (cid:20) πL (cid:18) z + L (cid:19)(cid:21) , (18)where we assume that the CNT is suspended at points z = − L/ z = L/
2. Approximating the charge densitywith a step function symmetrically covering the length fraction ξ of the suspended part of the CNT, we obtain (cid:104) f (cid:48) (cid:105) = − (cid:90) ξL/ − ξL/ dz ξL πL √ (cid:20) πL (cid:18) z + L (cid:19)(cid:21) = 2 √ L sin πξξ (19)The fraction sin( πξ ) /ξ is ∼ ξ is smaller than 1, and therefore we make the approximation (cid:104) f (cid:48) (cid:105) ≈ √ /L in thenumerical estimates appearing in the main text and in Table I. APPENDIX C: SEMICLASSICAL EQUATIONS OF MOTION
In this section we develop semiclassical equations of motion for the coupled qubit-oscillator system, valid in theregime of large oscillator excitation. We follow the procedure of Ref.11, this time allowing for different qubit ( ω q ),0oscillator ( ω p ), and drive ( ω ) frequencies. Our aim will be to find a closed set of equations for the time dependence ofthe expectation values of the oscillator and qubit coordinates, (cid:104) a (cid:105) , (cid:104) σ − (cid:105) , and (cid:104) σ z (cid:105) . We evaluate the time derivativesof these observables using ddt (cid:104)O(cid:105) = Tr[ ˙ ρ O ], with the time-dependence of the density matrix given by Eq.(4) of themain text (below we set (cid:126) = 1),˙ ρ = − i [ H RWA , ρ ] + ( n B + 1)Γ (cid:18) aρa † − { a † a, ρ } (cid:19) + n B Γ (cid:18) a † ρa − { aa † , ρ } (cid:19) , with n B = 1 / ( e ω p /k B T −
1) and H RWA = ˜ ω q σ + g ( aσ + + a † σ − ) + ˜ ω p a † a + λ ( a + a † ). Here, in the rotating frame, thereduced frequencies are given by ˜ ω i = ω i − ω . The qubit-phonon coupling is denoted by g , and the strength of theexternal driving field is denoted by λ .Using the commutation rule aa † = a † a + 1 and the cyclic property of the trace, Tr[ AB ] = Tr[ BA ], we find: (cid:104) ˙ a (cid:105) = ( − i ˜ ω p − Γ / (cid:104) a (cid:105) − iλ − ig (cid:10) σ − (cid:11)(cid:10) ˙ σ − (cid:11) = − i ˜ ω q (cid:10) σ − (cid:11) + ig (cid:104) aσ (cid:105)(cid:104) ˙ σ (cid:105) = − ig (cid:0)(cid:10) aσ + (cid:11) − (cid:10) a † σ − (cid:11)(cid:1) . We close this set of equations by neglecting correlated fluctuations between the qubit and oscillator degrees of freedom,factoring the averages as (cid:104) aσ (cid:105) ≈ (cid:104) a (cid:105) (cid:104) σ (cid:105) and (cid:104) aσ + (cid:105) ≈ (cid:104) a (cid:105) (cid:104) σ + (cid:105) . Using (cid:104) σ + (cid:105) = (cid:104) σ − (cid:105) ∗ and (cid:10) a † (cid:11) = (cid:104) a (cid:105) ∗ , and as inRef.11 defining the complex variables z = (cid:104) a (cid:105) and v = 2 (cid:104) σ − (cid:105) , and a real variable m = (cid:104) σ (cid:105) , we obtain˙ z = − ( i ˜ ω p + Γ / z − i gv − iλ (20)˙ v = − i ˜ ω q v + 2 igmz (21)˙ m = − ig ( zv ∗ − vz ∗ ) . (22)In this representation, the real and imaginary parts of z describe the oscillator coordinate and momentum, the complexvariable v describes the x and y Bloch vector components of the qubit state, and m describes the qubit polarization.The dynamics described by the nonlinear system in Eqs. (20)–(22) can be quite complex. Here we focus on steadystate solutions, ˙ z = 0 , ˙ v = 0 , ˙ m = 0. Setting ˙ v = 0 in Eq. (21), and introducing over-bars to indicate steady statevalues, we obtain ¯ v = 2 g ˜ ω q ¯ m ¯ z. (23)Note that ˙ m = 0 is automatically satisfied under this condition, see Eq. (22).Within the semiclassical description, and in the absence of decoherence acting directly on the spin, the variables v and m describe a vector of unit length, | v | + m = 1. Using Eq. (23) for ¯ v , we find ¯ m = (cid:16) g ˜ ω q | ¯ z | (cid:17) − . Takingthe square root of both sides gives ¯ m ± = ± (cid:18) g ˜ ω q | ¯ z | (cid:19) − / , (24)where the subscript ± indicates two branches of solutions to the square root. Setting ˙ z = 0 in Eq. (20), and usingEqs. (23) and (24), we find that the oscillator amplitude in the steady state satisfies the relation λ | z ± | = (Γ / + ˜ ω p ± g (cid:113) ˜ ω q + 4 g | z ± | . (25)Self-consistent solutions to Eq. (25) can easily be found numerically. In many parameter regimes, multiple solutionsexist due to the non-linearity introduced by the qubit-oscillator coupling. As shown in Fig.3e of the main text, thebranches of stable fixed points match well with the peaks in the phonon number distribution, indicating the utility ofthe semi-classical approach.Along with the presence of multiple solutions, we expected the typical manifestations of multistable behavior, suchas hysteresis and sharp instabilities. The sensitivity to the steady state oscillator amplitude near instability pointswhere stable steady-state solutions disappear may be useful for sensing applications. Closely related behavior hasalready proved quite useful in providing a sensitive read-out mechanism for superconducting qubits.[12]1 APPENDIX D: INTERPRETATION OF RESULTS FOR CHARGE-SENSING-BASED DETECTION
As stated in the main text, the oscillatory motion of the CNT resonator can be detected using a charge sensingscheme in which the conductance of a mesoscopic conductor, such as a QD or quantum point contact, is modulated viacapacitive coupling to the charged mechanical resonator. At a given source-drain bias on the mesoscopic conductor,the current depends on the displacement u of the resonator due to their capacitive coupling. A nonlinear dependenceof the current on the resonator displacement is desired for displacement sensing, I ( t ) ≈ I + I u ( t ) + I u ( t ). Sucha dependence is present in, e.g., a QD tuned to the middle of a Coulomb-blockade peak (corresponds to I = 0 and I < I > I > (cid:104) I (cid:105) through the charge-sensing mesoscopic conductor is sensitive to the steady-state average number ¯ n of phonons in the oscillator. Therefore,the results plotted in Fig. 3b, d, g, h of the main text can be interpreted as being proportional to the measured signal (cid:104) I (cid:105) − I in the charge-sensing setup described above, hence that setup would allow for the experimental confirmationof the predicted features.The steady-state time-averaged current (cid:104) I (cid:105) through the charge-sensing conductor is given by (cid:104) I (cid:105) = lim τ →∞ τ (cid:90) τ dt I ( t ) = I + I lim τ →∞ τ (cid:90) τ dt u ( t ) . (26)The instantaneous square displacement is expressed with the steady-state density matrix ¯ ρ and the position operator x ( t ), both represented in the rotating frame, as u ( t ) = Tr (cid:8) ¯ ρx ( t ) (cid:9) = Tr (cid:40) ¯ ρ (cid:20) (cid:96) √ ae − iωt + a † e iωt ) (cid:21) (cid:41) (27)Substitution of this expression to Eq. (26) yields (cid:104) I (cid:105) − I = I (cid:96) (cid:18) ¯ n + 12 (cid:19) , (28)as stated above. [1] S. Weiss et al., Phys. Rev. B , 165427 (2010).[2] J.-S. Jeong and H.-W. Lee, Phys. Rev. B , 075409 (2009).[3] W. Izumida, K. Sato, and R. Saito, J. Phys. Soc. Jpn. , 074707 (2009).[4] J. Klinovaja, M.J. Schmidt, B. Braunecker, D. Loss, Phys. Rev. B , 0854452 (2011).[5] T. S. Jespersen, K. Grove-Rasmussen, J. Paaske, K. Muraki, T. Fujisawa, J. Nygard, and K. Flensberg, Nat. Phys. , 348(2011).[6] T. Sand Jespersen, K. Grove-Rasmussen, K. Flensberg, J. Paaske, K. Muraki, T. Fujisawa, J. Nygard, Physical ReviewLetters , 186802 (2011)[7] D. V. Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B , 235301 (2008).[8] T. Ando and T. Nakanishi, J. Phys. Soc. Jpn. , 1704 (1998).[9] A. P´alyi and G. Burkard, Phys. Rev. Lett. , 086801 (2011).[10] K. Flensberg and C. M. Marcus, Phys. Rev. B , 195418 (2010).[11] P. Alsing and H. J. Carmichael, Quantum Opt. , 13 (1991).[12] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, Phys. Rev. Lett.105