Controllable Coupling Between Flux Qubit and Nanomechanical Resonator by Magnetic Field

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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   6   0   7   1   8   0  v   2   [  c  o  n   d  -  m  a   t .  m  e  s  -   h  a   l   l   ]   1   M  a  r   2   0   0   7 Controllable Coupling between Flux Qubitand Nanomechanical Resonator by Magnetic Field Fei Xue, Y. D. Wang, and C. P. Sun Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China  H. Okamoto, H. Yamaguchi, and K. Semba NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan  (Dated: June 26, 2018)We propose an active mechanism for coupling the quantized mode of a nanomechanical resonatorto the persistent current in the loop of superconducting Josephson junction (or phase slip) flux qubit.This coupling is independently controlled by an external coupling magnetic field. The whole systemforms a novel solid-state cavity QED architecture in strong coupling limit. This architecture can beused to demonstrate quantum optics phenomena and coherently manipulate the qubit for quantuminformation processing. The coupling mechanism is applicable for more generalized situations wherethe superconducting Josephson junction system is a multi-level system. We also address the practicalissues concerning experimental realization. PACS numbers: 74.78.Fk, 85.85.+j, 03.65.Lx I. INTRODUCTION In recent years, great advances in improving the co-herence of superconducting qubit have made it a promis-ing candidate for the physical realization of quantum in-formation processing. Single qubit Rabi oscillation andRamsey fringe have been observed and two qubits en-tanglement is also achieved. Meanwhile, as an artificialtwo-level atom, superconducting qubit is adjustable ( flux, bias voltage, etc.) and scalable. These featuresare favorable for quantum state engineering. A numberof protocols are proposed to engineer the superconduct-ing qubit to form a quantum network. Among them, avery intriguing and successful example is the circuit QEDarchitecture [1]. By coupling the Cooper pair box (chargequbit) to the quantized field of a coplanar superconduct-ing transmission line, a macroscopic solid-state analog of cavity QED is realized on chip. Most recently, vacuumRabi oscillations are observed in a coupling system of 3-JJ flux qubit and LC circuit [2]. Quantum optical phe-nomena in traditional cavity QED can be demonstratedin this solid state composite system. Furthermore, dueto its special structure, it offers a number of advantages,such as strong coupling and easy controllability. Thus,some protocols that cannot be realized previously in theoptical cavity QED now becomes possible [3, 4]. The circuit QED experiments motivate us to investi-gate the possibility of substituting other quantum solidstate devices for the transmission line. It is much desir-able to couple Josephson junction qubit to a device withlow energy consuming and small size. If the strong cou-pling and easy controllability can also be achieved, we getanother favorable cavity QED structure. A possible can-didate of this solid state device is the nanomechanical res-onator (NAMR). Nanomechanical resonators of GHz os-cillation have already been observed. It is supposed thatthe nanomechanical resonator enter the quantum regimeat the attainable temperature of the dilution refrigerator.The schemes of coupling Josephson charge qubit or phasequbit to NAMR have already been proposed. Based onthese coupling mechanisms, several quantum state engi-neering protocols are put forward [5, 6, 7, 8, 9]. However, due to the difficulty to reach quantum regime of NAMR,those protocols have not been implemented experimen-tally yet. On the other hand, the coupling mechanism of NAMR and flux qubit is also an attractive problem sincethe flux qubit is supposed to have longer coherence timeas it is less affected by the charge fluctuation in the struc-ture. To our best knowledge, this has not been studied indetail previously. Here, we present a novel mechanism of coupling NAMR to flux qubit. As we present below, thecoupling strength between the NAMR and the flux qubitcan be adjusted conveniently by a coupling magnetic fieldand turned on and off within the coherence time of fluxqubit. Since the coupling magnetic field is independent of the single qubit operation, it is possible to make it strongenough even for GHz oscillation. Therefore our proposalis in principle possible to approach the “strong couplingregime” of cavity QED at attainable temperature of dilu-tion refrigerator. This coupling system acts as an analogof cavity QED system with more flexibility. We expectthat it enables various applications to quantum informa-tion processing and quantum state engineering.This paper is organized as follows: in Sec.II, we brieflyreview the setup of 3 Josephson-junction flux qubit andNAMR as well as their experimental progresses. Thenwe get into the coupling mechanism for flux qubit andNAMR. This coupling mechanism can be equally ap-plied to rfSQUID flux qubit and phase slip flux qubit. InSec.III, the spectrum of the coupling system is presentedin the “weak coupling” and the “strong coupling” limitsrespectively. The readout and quantum nondemolitionmeasurement for the flux qubit is studied in Sec.IV. Wealso consider the application of this coupling mechanismin quantum computation in Sec.V and generalize the cou-pling system to beyond spin-boson model in Sec.VI. The  2possible problems on experimental realization and theirsolutions are given in Sec. VII. In Sec.VIII, some discus- sions and remarks are included. II. COUPLING NANOMECHANICALRESONATOR AND FLUX QUBIT BY ATRANSVERSE MAGNETIC FIELDA. The 3-junction flux qubit Josephson charge qubit system has been used to cou-ple with NAMR. Here, we study another superconduct-ing qubit system – 3 Josephson junction (3-JJ) flux qubit[10, 11, 12, 13, 14]. In contrast with charge qubits, the flux qubit is far less sensitive to charge fluctuations. Es-timations show that the flux qubit have a relatively highquality factor [11, 13]. The configuration of flux qubit consists of a superconducting loop with three Josephson junctions, and the Josephson coupling energy is muchlarger than the charging energy for each junction. Thequantum state of this system is mainly determined bythe phase degree of freedom. The Josephson energy of the three Josephson junction loop reads U   ( ϕ 1 ,ϕ 2 ) =  −  E  J   cos ϕ 1 − E  J   cos ϕ 2 −  αE  J   cos(2 πf   − ϕ 1 − ϕ 2 ) ,  (1)where the constraint of fluxoid quantization has alreadybeen taken into account. Here,  E  J   is the Josephsoncoupling energy of two identical junction and  ϕ 1 ,  ϕ 2 are phase differences across the two junctions respec-tively. The Josephson energy of the third junction is αE  J  ,  f   = Φ f  / Φ 0  with Φ f   the external flux applied in theloop and Φ 0  =  h/ 2 e  the flux quantum. In the vicinityof   f   = 0 . 5, if   α >  0 . 5, a double-well potential is formedwithin each 2 π × 2 π  cell in the phase plane and the twolowest stable classical states have persistent circulatingcurrents  I   p  = 2 eE  J    1 − (1 / 2 α ) 2 /    with opposite direc-tions. Therefore, the flux qubit is also called persistentcurrent qubit. Within the qubit subspace spanned by {| 0  , | 1 } ( | 0  and | 1  denote clockwise and counterclock-wise circulating states respectively), the Hamiltonian of the qubit system reads as H  f   =  ω f  σ z  + ∆ σ x  = Ω˜ σ z ,  (2)with  ω f   =  I   p Φ 0  ( f   − 0 . 5) is the energy spacing of thetwo classical stable states and ∆ the tunneling split-ting between the two states, Ω =   ω 2 f   + ∆ 2 and ˜ σ z  =cos θσ z  + sin θσ x , tan θ  = ∆ /ω f  . The offset of   f   from0 . 5 determines the level splitting of the two states andthe barrier for quantum tunneling between the statesstrongly depends on the value of   α . If the third junctionis replaced by a dc SQUID, both  f   and  α  are tunable inexperiments by the applied flux or the microwave current[10, 11]. B. The nanomechanical resonator The flexural modes of thin beams can be described bythe so-called Euler-Bernoulli equations [15]. In our pro-posal only the fundamental flexural mode of the NAMRis taken into account. All the other modes have amuch smaller coupling to flux qubit and can be ne-glected [16, 17]. In this case, the NAMR is modeled as a harmonic oscillator with a high-Q mode of frequency  ω b .The Hamiltonian without dissipation reads [6, 7] H   =  p 2 z 2 m  + 12 mω 2 b z 2 .  (3)In pursuing the quantum behavior of macro scale objectthe nano scale mechanical resonator plays an importantrole. At sufficient low temperature the zero-point fluc-tuation of nano mechanical resonator will be comparableto its thermal Brownian motion. The detection of zero-point fluctuation of the nano mechanical resonator cangive a direct test of the Heisenberg’s uncertainty prin-ciple. With a sensitivity up to 10 times the amplitudeof the zero-point fluctuation, LaHaye  et al.  have experi-mentally detected the vibrations of a 20-MHz mechanicalbeam of tens micrometers size [18]. For a 20-MHz me-chanical resonator its temperature must be cooled below1 mK to suppress the thermal fluctuation. For a GHzmechanical resonator a temperature of 50 mK is suffi-cient to effectively freeze out its thermal fluctuation andlet it enter quantum regime. This temperature is alreadyattainable in the dilution refrigerator.The lithographic technology for NAMR is rather ma-ture. The important advantages of NAMR are the poten-tially higher quality factor and frequency comparable tosuperconducting qubit. Ever since the early demonstra-tion of a radio frequency mechanical resonator at Caltech[19], great advances have been made. The attainable fre-quencies for the fundamental flexural modes can reach590 MHz for the doubly-clamped SiC mechanical res-onator of the size 1 × 0 . 05 × 0 . 05  µ m [20] and 1 GHzoscillation frequency has also been measured [21]. It isargued that quantized displacements of the mechanicalresonator were observed despite of some opposite opin-ions [22]. For a 1  µ m beam a quality factor  Q  of 1700has been observed at a frequency of 110 MHz [23]. In acarefully designed antenna shape, Gaidarzhy  et al.  haveachieved Q  = 11000for 21 MHz oscillation at the temper-ature of 60 mK and  Q  = 150 for 1 . 49 GHz oscillation atthe temperature of 1K with a comparatively large doubleclamped beam [24]. The significantly small size ( ∼  µ m)of the NAMR is also favorable for incorporating it in thesuperconducting qubit circuit. C. The composite system with tunable coupling To achieve a “strong” interaction, the coupling dynam-ical variable usually should be the dominant one in thedynamics of the composite system. For the Josephson  3 yzx   f   0 FIG. 1: (Color on line) A 3-Josephson-junction flux qubitloopis located in the x-y plane and a NAMR is integrated in theloop(indicated by a green line). The z-direction oscillationof the NAMR couples to the current in the flux qubit loopby a transverse magnetic field  B 0  in y-direction. Anothertunable magnetic flux Φ f   penetrates this loop tunes the freeHamiltonian of the 3-JJ system. phase qubit [25, 26], the phase degree of freedom domi- nates the dynamics and the bias current coupled with thephase is modified by the dilatational motion of the piezo-electric dilatational resonator [27]. While for the Joseph-son charge qubit, the coupling mechanism is that theresonator displacement modifies the effective bias chargeof a Cooper-pair box [7, 8, 17, 28]. These previous inves- tigations enlighten us to consider the coupling betweenpersistent current in superconducting flux qubit loop andthe motion of nano mechanical resonator.Since the Josephson coupling energy of each junctionin the flux qubit is much larger than that in the chargequbit, the persistent current in the loop could be abouthundreds of nano ampere [12] in contrast with the criticalcurrent of the charge qubit (usually about 20 ∼ 50 nA).The magnitude of this persistent current naturally leadsus to consider the magnetomotive displacement actua-tion and sensing technique [15, 19]. It is well known when a current passes through a beam with conducting mate-rial, the perpendicular arrangeed of an external magneticfield and the direction of the current generates a Lorentzforce in the plane of the beam. This is just the actuationpart of the magnetomotive technique. Meanwhile, the re-sulted displacement of the beam under the Lorentz forcegenerates an electromotive force, or voltage, which servesas measurement. Thus, if the doubly-clamped nano beamcoated with superconducting material is incorporated inthe superconducting qubit loop, the persistent current in-duces a Lorentz force with opposite directions for clock-wise and counterclockwise current. The oscillation of theNAMR is modulated by these Lorentz force. In this way,the quantized harmonic oscillation mode of the beam iscoupled to the quantum state of the flux qubit system.And this is just the coupling mechanism considered inour paper.Our proposal is illustrated in Fig.1. A 3-JJ system isfabricated on the x-y plane. The external applied mag-netic flux Φ f   is enclosed in the loop modulated by thecontrol lines (the lines are not plotted). The 4-JJ ver-sion of flux qubit system can also be used here to allowthe modulation of the effective Josephson energy of thethird junction and hence the tunneling amplitude of thetwo current states. One side of the loop (indicated bythick (green) rod) is suspended from the substrate andclamped at both ends. This can be fabricated with adoubly-clamped nanomechanical beam coated with su-perconductor or with the superconductor itself as the me-chanical resonator. A magnetic field  B 0  is applied in the y -direction. As we discussed above, the circulating sup-percurrent under the magnetic field generates a Lorentzforce in the z-direction. The magnitude of the force is B 0 I   p L , with  L  the effective length of the resonator along x -direction ( L  =  ξL 0  and  L 0  is the actual length of theresonator,  ξ   a factor depending on the oscillation mode[29], for the fundamental oscillation mode of a doublyclamped beam  ξ   ≈ 0 . 8). This force results a forced termin the Hamiltonian, which reads  H  fb  =  Fz  =  B 0 I   p Lz .With the two-level approximation of 3-JJ loop and thesinglemode boson approximation, the coupling is writtenas H  fb  =  g  a + a †  σ z ,  (4)for  z  ∼ a + a † . Here, g  =  B 0 ( t ) I   p Lδ  z  (5)and  δ  z  =     / 2 mω b  is the amplitude of zero point motionin z-direction of the NAMR, with  m  the effective massof the resonator,  ω b  the frequency of the fundamentalflexural mode;  a  ( a † ) is the creation (annihilation) op-erator of the mode of the flexural motion in z-direction. σ z  =  | 0  0 |−| 1  1 |  is the Pauli matrix defined in thebasis of   {| 0  , | 1 } . We see that this interaction  H  fb  ac-tually couples the two systems. Together with the freeHamiltonian of flux qubit and NAMR, the Hamiltonianof whole system reads H   =  ω b a † a + ω f  σ z  + ∆ σ x  + g  a + a †  σ z .  (6)An important advantage of this coupling mechanismis the convenient controllability. As seen from Eq.(5),the coupling constant is directly dependent on the ap-plied coupling magnetic field  B 0 . Thus, both the magni-tude and sign of the coupling constant can be modified.What’s more important is that the control parameter  B 0 in coupling coefficient Eq.(5) is independent from the pa-rameters of free Hamiltonian, such as bias voltage andexternal magnetic flux Φ f  . This means the free Hamilto-nian and the interaction Hamiltonian can be manipulatedindependently. This full controllability is a rather favor-able feature for quantum state engineering and quantuminformation processing protocols. It is in contrast withthe coupling of charge qubit and NAMR, where the cou-pling strength is controlled by the bias voltage which isalso the crucial parameter to determine the energy spac-ing of the charge qubit. For example, for bang-bang  4cooling of NAMR by charge qubit [8], the bias voltageshould be set to certain value to induce desirable damp-ing. Therefore the on-and-off of the interaction betweenthe qubit and NAMR can only approximately controlledby detuning and this can result in harmful reheating of the NAMR. But in our present coupling mechanism, boththe coupling coefficient and the energy spacing are inde-pendent. Thus, this “bang-bang” cooling protocol shouldbe implemented more reliable by the flux qubit and theNAMR with above coupling mechanism.To estimate the coupling strength, we use the follow-ing parameters in refs. [7, 30, 31]:  I   p  = 660 nA,  L 0  = 3 . 9 µ m,  ω b  = 100 MHz,  δ  z  = 2 . 6 × 10 − 13 m,  Q  = 2 × 10 4 and assume the applied magnetic field to be  B 0  = 5 mT.Then we have  g  ≈  4 . 01 MHz. Hence we see that the“strong coupling” regime for cavity QED is potentiallyrealizable in our scheme. This regime requires the periodof the Rabi oscillation 1 /g , is much shorter than boththe decoherence time 1 /γ   of the two-level system andthe average lifetime 1 /κ  =  Q/ω b  of the “photon” in the“cavity”[32]. For this composite system, the decoherencetime for flux qubit is 1 − 10  µ s and the cavity lifetime isabout 200  µ s, while the Rabi oscillation time 0 . 016  µ s ismuch shorter than the two lifetime scale. For GHz oscil-lation, the quality factor is rather low[24] (1 . 49 GHz with Q  = 150). This corresponding to a much shorter cavitylifetime (about 0 . 1  µ s. However, the coupling strengthcan be increased by larger coupling magnetic field. Forexample, if we take  B 0  = 50 mT, the Rabi oscillation pe-riod 1 /g  ≈  0 . 016  µ s which is still short enough to reach“strong coupling regime”. Therefore, this protocol mightbe promising in dilution refrigerator (several tens of mil-likelvin). D. Phase slip flux qubit and NAMR Most recently, a new type of flux qubit – phase slipflux qubit is proposed based on coherent quantum phaseslip [33, 34]. Phase slip flux qubit is formed by a high-resistance superconducting thin wire instead of theJosephson junctions. The computational basis are alsothe two opposed persistent current states. Our couplingscheme can be equally applicable to this type of fluxqubit. There are two important advantages to consider yzx   f   0 FIG. 2: (Color on line) A superconducting thin wire in theloop acts as the center of phase slip flux qubit and a NAMR. phase slip flux qubit. First, since the superconductingthin wire can be fabricated by a suspended carbon nan-otube, it acts as the phase slip center and the NAMR si-multaneously. The circuit configuration is simplified (seeFig. 2). Secondly, if one use this qubit, one can free fromany fluctuator due to imperfection or two-level systemshidden in the dielectric layer of Josephson junction. III. ENERGY SPECTRUM OF COUPLINGSYSTEM The Larmor frequency of superconducting qubit isabout the order of 10 GHz, while the frequency of NAMRonly reaches several hundred MHz with quality factor10 4 at present stage. Thus, the composite system of fluxqubit and NAMR is in large detuning regime of cavityQED, i.e., the following condition is satisfied g | Ω − ω b | ≪ 1 .  (7)However, the superconducting flux qubit and the NAMRis non-resonant, i.e. Ω  ≫  ω b . This is in contrast withYale’s circuit QED experiment where the Cooper pairbox is resonant with the 1D transmission line [35]. In thefollowing, we discuss the energy spectrum of our modelin two different regimes:  g  ≪ ω b  (denoted as “weak cou-pling”) and  g  ≈  ω b  (denoted as “strong coupling”). Inour proposal, the two regimes can be reached by varyingthe applied coupling magnetic field  B 0 . And the energyspectrum are qualitatively different from each other. Itis notable that Ref.[36] has proved that the dispersivemeasurement back action can be enhanced or reduced bycavity damping respectively in the two regimes. A. “Weak coupling” and sideband spectrum For the parameters  B 0  = 5 mT,  g  = 4 . 01 MHz, both g/ Ω and  g/ω b  are much smaller than 1. In this case, theenergy spectrum can be calculated by Floquet approachor by Fr¨olich transformation. After performing a unitarytransformation on the srcinal Hamiltonian (6), we getthe effective Hamiltonian H  eff1  ≈  ω b a † a  + Ω˜ σ z  + ig 2 sin2 θω b  a 2 − a † 2  ˜ σ y + g 2 sin θ Ω  a + a †  2 (cos θ ˜ σ x  + sin θ ˜ σ z ) .  (8)The spectrum are  nω b + m Ω plus some small off-diagonaltransition terms that are of order  O ( g/ω ) or  O ( g/ Ω).As shown in Fig.3, the energy levels of the two subsys-tems are weakly perturbed by the coupling due to thelarge detuning. By applying microwave pulse to inducethe transition between those levels, the blue sideband( | 00  → | 11  ) and red sideband transition ( | 01  → | 10  )can be observedin addition to the main zero-photontran-sition | 00 →| 10  [37]. This atomic physics phenomenon
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