Electrically tunable quantum interfaces between photons and spin qubits in carbon nanotube quantum dots
aa r X i v : . [ qu a n t - ph ] M a y Electrically tunable quantum interfaces between photons and spin qubits in carbonnanotube quantum dots
Ze-Song Shen and Fang-Yu Hong Department of Physics, Center for Optoelectronics Materials and Devices,Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China (Dated: September 18, 2018)We present a new scheme for quantum interfaces to accomplish the interconversion of photonicqubits and spin qubits based on optomechanical resonators and the spin-orbit-induced interactionsin suspended carbon nanotube quantum dots. This interface implements quantum spin transducersand further enables electrical manipulation of local electron spin qubits, which lays the foundationfor all-electrical control of state transfer protocols between two distant quantum nodes in a quantumnetwork. We numerically evaluate the state transfer processes and proceed to estimate the effect ofeach coupling strength on the operation fidelities.
PACS numbers: 03.67.Hk, 07.10.Cm, 73.63.Fg
I. INTRODUCTION
In quantum information processing, quantum inter-faces (QIs) play a fundamental role in distributing quan-tum states between individual nodes of large quantumnetworks, which also offers potential applications for scal-able and distributed quantum computation [1, 2]. Somepromising setups related to quantum networks have beenintroduced to fulfill information distribution protocols ei-ther “on-chip” or over long distances [3–5]. They serve toaccomplish transferring, swapping, and entangling qubitswithin the building blocks. The prototype quantum in-terface (QI) coherently interconverting so-called “flying”optical qubits and “stationary” matter qubits was pro-posed by Cirac et al. [6], using cavity-assisted Ramanprocesses. Qubits are stored in hyperfine states of atomsand then transformed into the number states of a photonwavepacket and vice versa [7].In view of substantial coherence and control of nanoengi-neered solid-state qubits, recently Stannigel et al. [4]presented an optical interface with optomechanical res-onators [8] to address the incompatibility between lightand qubits. The scheme makes use of a magnet-mediatedspin qubit [9] and a charge qubit that employs electro-static interactions. Analogous magnetic cantilevers towhich the spin qubit couples could be utilized for co-herent sensing of mechanical resonators [10, 11]. In thetwo interfaces above, laser pulses are used as the control-ling tool, which requires bulky setups and makes themsomewhat inconsistent in large-scale quantum informa-tion processing. Here we propose an electrically tunableinterface based on a suspended carbon nanotube quan-tum dot (CNTQD) for implementation.CNTQDs are currently being investigated in a varietyof systems, such as vibrational state control [12], ciruitQED [13], and coupling to cavity photons [14, 15]. Thepotential for carbon-based systems to be used in quan-tum information processing has been widely explored byexploiting different degrees of freedom in CNTs and CN-TQDs as quantum bits. For example, quantum informa- tion can be stored in the motional degrees of freedom ofnanomechanical devices [16] by creating single quantumexcitations (phonons) in a resonator [17]. It is also reli-able to utilize the spin-orbit interaction with the valleydegrees of freedom in CNTQDs [18–20].We describe in this paper a quantum interface to real-ize the all-electrical control of coherent interconversionbetween optical qubits and the single electron spins inCNTQDs. In our work, we take advantage of the spindegrees of freedom. It is feasible to modify the couplingstrength between the electron spin and the vibrationalmotion of a suspended CNT by shifting the effective cen-ter of the quantum dot using a longitudinal electric field[21]. Our setup proves to be robust and well-controlledowing to the intrinsic spin-mechanical coupling in CN-TQD, which could otherwise be used to mechanicallymanipulate the electron spin [22] supported by the in-herent strong spin-orbit interaction [23].In addittion, the outstanding mechanical properties ofCNT -low mass and noise [24], high quality factor, andwidely tunable resonance frequencies [5, 25], make it at-tractive to be used as a resonator in a QI. Meanwhile,long spin life time of electrons in CNTQDs can be ab-tained off-resonance [13]. In the resonant case though,the setup can accomplish qubit control and tranfer whilegaining high operation fidelities, with the reduced chargenoise-induced dephasing from the suspended CNT [13]and further elemination of the hyperfine contributions toqubit decoherence.
II. SETUP
To describe the mechanisim of electrical control of oursystem, we first model the spin of an electron localizedin a suspended CNTQD, the relevant mechanical oscilla-tor mode, and the coupling between these two degrees offreedom.In the presence of a longitudinal magnetic field of mag-nitude around B ∗ = ∆ so g s µ B , the QD can serve as a spin microtoroidalcavitytapered fiber (a) CNT B | | spin qubit E z κG qubit α out (t)α in (t) MTC λγ c γ r γ q (b) . FIG. 1. (color online) (a) Schematic quantum interface toachieve robust qubit-light interversion using the spin degreeof freedom in a CNTQD: An electron spin ( ↑ , ↓ ) is extrinsi-cally coupled to the evanescent optical field in a microtoroidalcavity (MTC) through the intrisinc spin-mechanical coupling,and flying photon quibts are coupled in and out the cavitythrough a tapered fiber. (b) Illustration of the interactionsand the decoherence sources in the system. See text for de-tails. qubit coupled to the mechancial resonator mode with theHamiltonian [23] H so = ω q σ z λ (cid:0) b † + b (cid:1) ( σ + + σ − ) + ω p b † b, (1)where ∆ so denotes the spin-orbit coupling, g s is the g factor of the electron spin, σ z is the Pauli operator, σ ± are the raising and lowering operators for the spin qubitwith a tunable splitting ω q , and ω p is the frequency ofthe flexual mode of the CNT described by the anni-hilation operator b . For simplicity, we assume ~ = 1throughout this paper. The spin-phonon coupling λ inFig. 1(b) is defined as λ = ∆ so h f ′ i µ / √
2, with h f ′ i being the derivative of the phonon waveform averagedagainst the electron density profile n in the QD and µ being the zero-point fluctuation amplitude of the phononmode [23]. We classify the electron density profile as n ( z ) ≃ exp [ − A ( z − z c ) ] by assuming a parabolic trap-ping potential and the ground state of the electron in theCNTQD [26]. The parameter A is associated with the ef-fective electron mass m ∗ and the characteristic frequency ω of the harmonic oscillator, and z c is the center of thepotential well.To realize the manipulation of the coupling strength λ electrically, we can modulate the electron distributionin the QD by adding and tuning an electric field E z ( t )along the z direction of CNT without modifying the en-ergy level spacing of the QD [21, 27].Now we proceed to discuss the framework with the unionof the spin-oscillator system we described above by con-sidering a single node of an optical network as illustratedin Fig. 1(a).The total Hamiltonian is given by [29, 30] H = H + H I , (2) H = 12 ω q σ z + ω p b † b + ∆ c c † c + Z ∞ ωa † ω a ω dω, (3) H I = (cid:18) λ σ + b + H.c. (cid:19) + (cid:0) Gc † + G ∗ c (cid:1) (cid:0) b † + b (cid:1) (4)+ Z ∞ dω (cid:0) κca † ω + H.c. (cid:1) , (5)where a ω is the destruction operator for the mode of fre-quency ω in the optical quantum channel, and c is theannihilation operator for the cavity mode, respectively.The detuning ∆ c = ω c − ω L − | G | /ω p and the couplingG can be regulated by the field amplitude and the fre-quency ω L of local driving lasers. The cavity mode isassumed to be coupled to the field in the tapered fiberwith a constant κ = p γ/ π in Fig. 1(b).Corresponding to the rotating-wave approximation(RWA), we simplify the interaction Hamiltonian by drop-ping the energy unconserving terms and abtain˜ H I = λ σ + b + Gc † b + Z ∞ dω r γ π ca † ω + H.c. . (6)Under the condition k B T ≪ ω p , thesystem has two invariant Hilbert sub-spaces with the bases {| ↓ , , i| vac i} and {| ↑ , , i| vac i , | ↓ , , i| vac i , | ↓ , , i| vac i , | ↓ , , i a † ω | vac i} ,respectively. Here in | k, l i , k and l denote the number ofphonons in the flexual mode and the number of photonsin the cavity, respectively. | vac i denotes the vacuumstate of the optical continum.In the interaction picture, the general state of the systemcould be expressed by the superposition | Ψ i = C | ↓ , , i| vac i + C | Ψ I i , (7)where | Ψ I i = X n c n | µ n i e − iω n t , (8)and c n denotes the amplitudes α ω and β q,r,c of the fourrelevant states | µ n i in the excited Hilbert subspace. Inthis quantum interface, we have the evolution equations˙ c n = − i X m h n | ˜ H I | m i e iω nm t c m . (9)By expressing the qubit-mechanical coupling in thestate amplitudes of the QI system, we abtain the targetcontrol pulse λ = − β q (cid:16) ˙ β r + Gβ c (cid:17) , (10)with β q ( t ) , β r ( t ) , β c ( t ) set according to the specified in-coming/outgoing photon wavepacket. For simiplicity, inthis work we tune this ideal control pulse to drive thereal system with certain decoherences. Note that care-ful designs of the manipulation could help to achieve ahigher fidelity of quantum transmission. III. QUANTUM STATE TRANFER
The mapping process from the stationary spin quibtto the flying photon qubit requires the initial conditions α in ( t i ) = 0, β q ( t i ) = 1, β r ( t i ) = 0, β c ( t i ) = 0. Followingthe driven evolution passage: β q λ s ( t ) −−−→ β r G −→ β c κ −→ α ω ,the coherent tranfer of the quantum amplitude is fin-ished, and vice versa. With the normalization under-stood, the outgoing photon wavepacket ˜ α out ( t ) can bearbitrarily specified to contain an average number ofsin θ : sin θ R t f t i | ˜ α out ( t ) | dt = sin θ . At the remotefuture t f → + ∞ , the photon generation is completed, i.e. , β r ( t f ) = β c ( t f ) = 0, hence β q ( t f ) = e iφ cos θ .The general photon-generation process in the QI reads( C | ↓i + C | ↑i ) | vac i λ s ( t ) −−−→ C | ↓i| vac i (11)+ C (cid:0) e iφ cos θ | ↑i| vac i + sin θ | ↓i| ˜ α out i (cid:1) . (12)When the emission operation is completed, θ = π/ α ω ( t f ) = 0. Then, the spin qubit is mapped onto theflying qubit by the conversion( C | ↓i + C | ↑i ) | vac i λ s ( t ) −−−→ | ↓i ( C | vac i + C | ˜ α out i ) . (13)The absorption of an incoming photon at an receivingnode is typically the reverse of the above.Also, the sending manipulation can be mediated suchthat θ < π/
2. The initia state | ↑i| vac i is then trans-formed into an entangled state | ↑i| vac i λ ′ s ( t ) −−−→ e iφ cos θ | ↑i| vac i + sin θ | ↓i| ˜ α out i . (14)If we manage to absorb the above entangled photon, wewould have an entanglement between the sending node 1and the receiving node 2: | ↑i | ↓i λ ′ s ( t ) −−−→ λ r ( t ) e iφ cos θ | ↑i | ↓i + sin θ | ↓i | ↑i . (15) IV. NUMERICAL SIMULATION
Here we evaluate the performance of the QI by takingsome necessary decoherences into account. The phonon α o u t ( t ) −4 −3 −2 −1 0 1 2 3 4 5−1−0.8−0.6−0.4−0.20 t/ µ s λ / π [ M H z ] (b)(c) β c Im β r (a) β q FIG. 2. (color online)Generation of a single photon with theshape of a Gaussian wavepacket ˜ α ( t ) = exp( − Γ t ). (a) Thetime evolution of state amplitude β q ( t ), β c ( t ) and the imag-inary β r ( t ) in the QI. (b) The generated photon wavepacket(solid line) in comparision with the target one (dashed line).(c) The coupling strength λ ( t ) determinining the control-ling gate potential for the sending node. The parametersare Γ = 0 . π × √ γ/ π = 5MHz, G/ π = 1 . γ r / π = 2 . γ c / π = 2 . γ q = 0 . mode operator obeys the following quantum Langevinequation ddt b = − i [ b, H ] − γ r b − √ γ r ζ ( t ) , (16)where γ r denotes the mechanical damping rate and thenoise operator ζ satisfies h ζ ( t ) i = 0 in thermal equilib-rium. At low temperatures k B T ≪ ω p , we have ddt (cid:10) b † b (cid:11) = − i Dh b † b, H − i γ r b † b iE . (17)Under the same assumption, the heating of the nanotuberesonator from the thermal bath reservoir is negligibleand we find the effective interaction Hamiltonian to be˜ H effI = ˜ H I − i γ r b † b . Meanwhile, the mechanical damp-ing rate can be derived from γ r = ω p Q m with Q m beingthe quality factor of the mechanical resonator. Here, wetake the resonance frequency ω p / π as 360MHz and thequality factor Q m = 140 ,
000 with the resulting decayrate γ r / π = 2 . γ r and the cavity leakage γ c are small. The long spin lifetime in the QD is also expected off-resonance due to thelow density of states at the spin energy splitting for the (a) β q β q α out −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 200.20.40.60.8 t/ µ s λ / π [ M H z ] −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −0.8−0.6−0.4−0.20 (b) λ ′ λ FIG. 3. (color online) Distribution of the target entanglement √ | ↑i | ↓i + √ | ↓i | ↑i between the sending node 1 and thereceiving node 2. (a) The evolution of state amplitude β q ( t )in node 1 and β q ( t ) in node 2, both are mediated by theintermediate photon wavepacket α out ( t ). (b) The driving fre-quency λ ′ ( t ) for the sending node and λ ( t ) for the receivingone. The parameters are γ / π = 0 . G / π = 0 . γ / π = 4 . G / π = 1 . γ r / π = 2 . γ c / π = 2 . γ q = 0 . phonon spectrum of a suspended CNT. Therefore, tak-ing into account the intrinsic spin qubit dephasing γ q andthe cavity decoherence, the system dynamics can then bedescribed by the effective non-unitary Hamiltonian˜ H eff = H + ˜ H effI , (18)˜ H effI = ˜ H I − i γ r b † b − i γ c c † c − i γ q | ↑ih↑ | . (19)Correspondingly the evolution equations of the QI areexpressed in the following complete form:˙ β q = − i λ β r − γ q β q , (20)˙ β r = − i (cid:18) λ ∗ β q + G ∗ β c (cid:19) − γ r β r , (21)˙ β c = − iGβ r − √ γα in ( t ) − γ β c − γ c β c , (22)= − iGβ r − √ γα out ( t ) + γ β c − γ c β c , (23)where we have assumed the resonance condition∆ c = ω p = ω q for the convenience of simulation. Nextwe use a simple Gaussian wavepacket ˜ α ( t ) = exp( − Γ t ) with normalization understood, to simulate numericallythe functions of the QI.The generation process of a single photon is illustratedin Fig. 2. During this operation, the control strength − λ needs to be smaller than the maximum one tunable bymediating the longitudinal electric field strength [21] toperform this manipulation.The parameters used here are within the reach ofpresent state-of-the-art techniques [23, 28]. Significantly,the pulse duration is rather short, about 6 µs , due to thestrong coupling γ .Note also that meticulous designs of the couplingstrength γ , G and λ can help to achieve faster sendingand receiving manipulations while ensuring high fideli-ties. The related estimation is shown in Fig. 4. FromFig. 3, the bell state √ | ↑i | ↓i + √ | ↓i | ↑i is pre-pared with β q , β q evolving according to the intermedi-ate pulse wavepacket α out ( t ). The corresponding drivingfrequencies are depicted in Fig. 3(b).Finally, we investigate the effects of coupling strength γ , G , shown in Figs. 4(a) and 4(b), on the fidelity ofstimulating a single photon and transferring a local en-tanglement: √ | ↑i| vac i + √ | ↓i| ˜ α out i . In the stimula-tion process, both γ and G have a favorable influence onthe fidelity. Minor decoherences serve to amplify this re-lation. Interestingly though, the coupling strength γ inthe preparation of the local entanglement has a negativeeffect on the fidelity. This could be better understoodif we account for the superposition characteristic of en-tanglement. Fig. 4(c) shows the influence of the qubitdephasing, which is the predominant source of decoher-ence in a typical three-level quantum interface, on thefidelity F ′ of tranferring the target entangled state. V. CONCLUSIONS
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