Universal Quantum Transducers Based on Surface Acoustic Waves
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Citation Schuetz, M. J. A., E. M. Kessler, G. Giedke, L. M. K. Vandersypen, M. D. Lukin, and J. I. Cirac. 2015. Universal Quantum Transducers Based on Surface Acoustic Waves. Physical Review X 5, 3. doi:10.1103/physrevx.5.031031.
Published Version doi:10.1103/PhysRevX.5.031031
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Universal Quantum Transducers Based on Surface Acoustic Waves
M. J. A. Schuetz,1 E. M. Kessler,2,3 G. Giedke,1,4,5 L. M. K. Vandersypen,6 M. D. Lukin,2 and J. I. Cirac1 1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany 2Physics Department, Harvard University, Cambridge, Massachusetts 02318, USA 3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02318, USA 4Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain 5Ikerbasque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain 6Kavli Institute of NanoScience, TU Delft, P.O. Box 5046, 2600 GA Delft, Netherlands (Received 22 April 2015; published 10 September 2015) We propose a universal, on-chip quantum transducer based on surface acoustic waves in piezoactive materials. Because of the intrinsic piezoelectric (and/or magnetostrictive) properties of the material, our approach provides a universal platform capable of coherently linking a broad array of qubits, including quantum dots, trapped ions, nitrogen-vacancy centers, or superconducting qubits. The quantized modes of surface acoustic waves lie in the gigahertz range and can be strongly confined close to the surface in phononic cavities and guided in acoustic waveguides. We show that this type of surface acoustic excitation can be utilized efficiently as a quantum bus, serving as an on-chip, mechanical cavity-QED equivalent of microwave photons and enabling long-range coupling of a wide range of qubits.
DOI: 10.1103/PhysRevX.5.031031 Subject Areas: Condensed Matter Physics, Quantum Physics, Quantum Information
I. INTRODUCTION the surface of a solid and are widely used in modern electronic devices, e.g., as compact microwave filters The realization of long-range interactions between [14,15]. Inspired by two recent experiments [16,17], where remote qubits is arguably one of the greatest challenges the coherent quantum nature of SAWs has been explored, towards developing a scalable, solid-state spin-based quan- here we propose and analyze SAW phonon modes in tum information processor [1]. One approach to address piezoactive materials as a universal mediator for long- this problem is to interface qubits with a common quantum bus that distributes quantum information between distant range couplings between remote qubits. Our approach qubits. The transduction of quantum information between involves qubits interacting with a localized SAW phonon stationary and moving qubits is central to this approach. A mode, defined by a high-Q resonator, which in turn can be particularly efficient implementation of such a quantum bus coupled weakly to a SAW waveguide (WG) serving as a can be found in the field of circuit QED where spatially quantum bus; as demonstrated below, the qubits can be separated superconducting qubits interact via microwave encoded in a great variety of spin or charge degrees of photons confined in transmission line cavities [2–4]. In this freedom. We show that the Hamiltonian for an individual way, multiple qubits have been coupled successfully over node (for a schematic representation see Fig. 1) can take on ℏ 1 relatively large distances of the order of millimeters [5,6]. the generic Jaynes-Cummings form ð ¼ Þ, Fueled by dramatic advances in the fabrication and ω manipulation of nanomechanical systems [7], an alternate q σz ω † σþ σ− † Hnode ¼ 2 þ ca a þ gð a þ a Þ; ð1Þ line of research has pursued the idea of coherent, long- range interactions between individual qubits mediated by where σ~ refers to the usual Pauli matrices describing the mechanical resonators, with resonant phonons playing the qubit with transition frequency ω and a is the bosonic role of cavity photons [8–13]. q operator for the localized SAW cavity mode of frequency In this paper, we propose a new realization of a quantum ω 2π ∼ transducer and data bus based on surface acoustic waves c= GHz [18]. The coupling g between the qubit and (SAWs). SAWs involve phononlike excitations bound to the acoustic cavity mode is mediated intrinsically by the piezoproperties of the host material, it is proportional to the electric or magnetic zero-point fluctuations associated with a single SAW phonon and, close to the surface, can reach Published by the American Physical Society under the terms of ∼ 400 the Creative Commons Attribution 3.0 License. Further distri- values of g MHz, much larger than the relevant bution of this work must maintain attribution to the author(s) and decoherence processes and sufficiently large to allow for the published article’s title, journal citation, and DOI. quantum effects and coherent coupling in the spin-cavity
2160-3308=15=5(3)=031031(30) 031031-1 Published by the American Physical Society M. J. A. SCHUETZ et al. PHYS. REV. X 5, 031031 (2015)
electromagnetic devices: (i) SAW resonators, the mechani- cal equivalents of Fabry-Perot cavities, with low-temper- ature measurements reaching quality factors of Q ∼ 105 even at gigahertz frequencies [22–24], and (ii) acoustic waveguides as analog to optical fibers [14]. (4) For a given frequency in the gigahertz range, due to the slow speed of sound of ∼103 m=s for typical materials, device dimen- sions are in the micrometer range, which is convenient for fabrication and integration with semiconductor compo- nents, and about 105 times smaller than corresponding electromagnetic resonators. (5) Since SAWs propagate FIG. 1. SAW as a universal quantum transducer. Distributed elastically on the surface of a solid within a depth of Bragg reflectors made of grooves form an acoustic cavity for approximately one wavelength, the mode volume is surface acoustic waves. The resonant frequency of the cavity is intrinsically confined in the direction normal to the surface. 2 determined by the pitch p, fc ¼ vs= p. Reflection occurs Further surface confinement then yields large zero-point effectively at some distance inside the grating; the fictitious fluctuations. (6) Yet another inherent advantage of our mirrors above the surface are not part of the actual experimental system is the intrinsic nature of the coupling. In piezo- setup, but are shown for illustrative purposes only. Red arrows indicate the relevant decay channels for the cavity mode: leakage electric materials, the SAW is accompanied by an electrical ϕ through the mirrors, internal losses due to, for example, surface potential , which has the same spatial and temporal imperfections, and conversion into bulk modes. Qubits inside and periodicities as the mechanical displacement and provides outside of the solid can be coupled to the cavity mode. In more an intrinsic qubit-phonon coupling mechanism. For exam- complex structures, the elastic medium can consist of multiple ple, recently, qubit lifetimes in GaAs singlet-triplet qubits layers on top of some substrate. were found to be limited by the piezoelectric electron- phonon coupling [25]. Here, our scheme provides a new system as evidenced by cooperativities [19] of C ∼ 10–100 paradigm, where coupling to phonons becomes a highly (see Sec. IV and Table I for definition and applicable valuable asset for coherent quantum control rather than a values). For ω ≈ ω , H allows for a controlled map- liability. q c node In what follows, we first review the most important ping of the qubit state onto a coherent phonon super- features of surface acoustic waves, with a focus on the position, which can then be mapped to an itinerant SAW associated zero-point fluctuations. Next, we discuss the phonon in a waveguide, opening up the possibility to different components making up the SAW-based quantum implement on-chip many quantum communication proto- transducer and the acoustic quantum network it enables: cols well known in the context of optical quantum SAW cavities, including a detailed analysis of the achiev- networks [13,20]. able quality factor Q, SAW waveguides, and a variety of The most pertinent features of our proposal can be different candidate systems serving as qubits. Lastly, as summarized as follows. (1) Our scheme is not specific exemplary application, we show how to transfer quantum to any particular qubit realization, but—thanks to the states between distant nodes of the network under realistic plethora of physical properties associated with SAWs in conditions. Finally, we draw conclusions and give an piezoactive materials (strain and electric and magnetic outlook on future directions of research. fields)—provides a common on-chip platform accessible to various different implementations of qubits, comprising II. SAW PROPERTIES both natural (e.g., ions) and artificial candidates, such as quantum dots (QDs) or superconducting qubits. In particu- Elastic waves in piezoelectric solids are described by lar, this opens up the possibility to interconnect dissimilar systems in new electroacoustic quantum devices. ρüi − cijkl∂j∂luk ¼ ekij∂j∂kϕ; ð2Þ (2) Typical SAW frequencies lie in the gigahertz range, closely matching transition frequencies of artificial atoms ϵij∂i∂jϕ ¼ eijk∂i∂kuj; ð3Þ and enabling ground-state cooling by conventional cryogenic techniques. (3) Our scheme is built upon an where the vector uðx;tÞ denotes the displacement field (x established technology [14,15]: Lithographic fabrication is the Cartesian coordinate vector), ρ is the mass density, techniques provide almost arbitrary geometries with high and repeated indices are summed over ði; j ¼ x; y; zÞ; c, ϵ, precision as evidenced by a large range of SAW devices and e refer to the elasticity, permittivity, and piezoelectric such as delay lines, bandpass filters, resonators, etc. tensors, respectively [26]: they are largely defined by In particular, the essential building blocks needed to crystal symmetry [27]. For example, for cubic crystals interface qubits with SAW phonons have already been such as GaAs, there is only one nonzero component for the fabricated, according to design principles familiar from permittivity and the piezoelectric tensor, labeled as ϵ and
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−1 2 e14, respectively [26]. Since elastic disturbances propagate follows a generic ∼A = behavior, where A is the effective much slower than the speed of light, it is common practice mode area on the surface. The estimate given in Eq. (4) to apply the so-called quasistatic approximation [27], agrees very well with more detailed calculations presented where the electric field is given by Ei ¼ −∂iϕ. When in Appendix C. Several other important quantities that are considering surface waves, Eqs. (2) and (3) must be central for signal transduction between qubits and SAWs supplemented by the mechanical boundary condition that follow directly from U0: The (dimensionless) zero-point there should be no forces on the free surface (taken to strain can be estimated as s0 ≈ kU0. The intrinsic piezo- be at z ¼ 0 with zˆ being the outward normal to the surface), electric potential associated with a single phonon derives that is Tzx ¼ Tzy ¼ Tzz ¼ 0 at z ¼ 0 (where Tij ¼ from Eq. (3) as ϕ0 ≈ ðe14=ϵÞU0 [31]. Lastly, the electric cijkl∂luk þ ekij∂kϕ is the stress tensor), and appropriate field amplitude due to a single acoustic phonon is electrical boundary conditions [26]. ξ0 ≈ kϕ0 ¼ðe14=ϵÞkU0, illustrating the linear relation If not stated otherwise, the term SAW refers to the between electric field and strain characteristic for piezo- prototypical (piezoelectric) Rayleigh wave solution as electric materialspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi[8]. In summary, we typically find 2 theoretically and experimentally studied, for example, U0 ≈ 2 fm= A ½μm , yielding U0 ≈ 2 fm for micron- in Refs. [16,17,26,28] and used extensively in different scale confinement (cf. Appendix D). This is comparable electronic devices [14,15]. It is nondispersive, decays to typical zero-point fluctuation amplitudes of localized exponentially into the medium with a characteristic pen- mechanical oscillators [32]. Moreover, for micron-scale etration depth of a wavelength, and has a phase velocity surface confinement and GaAs material parameters, we vs ¼ ω=k that is lower than the bulk velocities in that obtain ξ0 ≈ 20 V=m, which compares favorably with typ- medium, because the solid behaves less rigidly in the ical values of ∼10−3 and ∼0.2 V=m encountered in cavity absence of material above the surface [27]. As a result, it and circuit QED, respectively [2]. cannot phase match to any bulk wave [14,29]. As usual, we For the sake of clarity, we have focused on piezoelectric consider specific orientations for which the piezoelectric materials so far. However, there are also piezomagnetic field produced by the SAW is strongest [14,29], for materials that exhibit a large magnetostrictive effect. In example, a SAW with a wave vector along the [110] that case, elastic distortions are coupled to a (quasistatic) direction of a (001) GaAs crystal (cf. Refs. [16,26] and magnetic instead of an electric field [33,34]; for details, see Appendixes A and B). Appendix D. For typical materials, such as Terfenol-D, the magnetic field associated withpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a single phonon can be 2 A. SAWs in quantum regime estimated as B0 ≈ ð2–6Þ μT= A ½μm . Finally, we note that composite structures comprising both piezoelectric In a semiclassical picture, an acoustic phonon associated and piezomagnetic materials can support magnetoelectric with a SAW creates a time-dependent strain field, surface acoustic waves [35,36]. skl ¼ð∂luk þ ∂kulÞ=2, and a (quasistatic) electrical potential ϕðx;tÞ. Upon quantization, the mechanical dis- placement becomes an operator that can be expressed III. SAW CAVITIES AND WAVEGUIDES uˆ x Pin terms of the elementary normal modes as ð Þ¼ A. SAW cavities v x † n½ nð Þan þ H:c: , where anðanÞ are bosonic annihila- To boost single-phonon effects, it is essential to increase tion (creation) operators for the vibrational eigenmode n U0. In analogy to cavity QED, this can be achieved by v x and the set of normal modes nð Þ derives from the confining the SAW mode in an acoustic resonator. The Wv x −ρω2v x Helmholtz-like equation nð Þ¼ n nð Þ associated physics of SAW cavities has been theoretically studied and Rwith Eqs. (2) and (3). The mode normalization is given by experimentally verified since the early 1970s [14,37]. Here, 3xρv x v x ℏ 2ω d nð Þ · nð Þ¼ = n [25,30]. An important figure we provide an analysis of a SAW cavity based on an of merit in this context is the amplitude of the mechanical on-chip distributed Bragg reflector in view of potential zero-point motion U0. Along the lines of cavity QED [2],a applications in quantum information science; for details, simple estimate for U0 can beR obtained by equating the see Appendix E. SAW resonators of this type can usually be 3 _ 2 classical energy of a SAW ∼ d xρu with the quantum designed to host a single resonance fc ¼ ωc=2π ¼ vs=λc energy of a single phonon, that is, ℏω. This leads directly to (λc ¼ 2p) only and can be viewed as an acoustic Fabry-Perot pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi resonator with effective reflection centers, sketched by U0 ≈ ℏ=2ρvsA; ð4Þ localized mirrors in Fig. 1, situated at some effective penetration distance into the grating [14]. Therefore, the where we used the dispersion relation ω ¼ vsk and the total effective cavity size along the mirror axis is Lc >D, intrinsic mode confinement V ≈ Aλ characteristic for where D is the physical gap between the gratings. The total κ ω κ κ SAWs. The quantity U0 refers to the mechanical amplitude cavity linewidth ¼ c=Q ¼ gd þ bd can be decomposed associated with a single SAW phonon close to the surface. into desired (leakage through the mirrors) and undesired It depends on only the material parameters ρ and vs and (conversion into bulk modes and internal losses due to
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25 ≲ 5λ κ κ ≳ 20 D c) and gd= bd (illustrated by the circle in (a) (b) Fig. 2); accordingly, the probability for a cavity phonon to leak through themirrors(rather than into the bulk for example) 20 κ κ κ ≳ 95% is gd=ð gd þ bdÞ . Note that the resulting total cavity linewidth of κ=2π ¼ fc=Q ≈ ð1–3Þ MHz is similar to the 15 ones typically encountered in circuit QED[6].Tocomparethis to the effective cavity-qubit coupling, we need to fix the effective mode area of the SAW cavity. In addition to the 10 longitudinal confinement by the Bragg mirror (as discussed ˆ above), a transverse confinement length Ltrans (in direction y) can be provided, e.g., using waveguiding, etching, or (similar tocavityQED)focusingtechniques[14,39,40].Fortransverse ≈ 1–5 μ confinement Ltrans ð Þ m and a typical resonant cavity λ ≈ 1 μ 1 wavelength c m, the effective mode area is then ≈ 40–200 μ 2 1 2 3 4 5 A ¼ LtransLc ð Þ m . In the desired regime κ κ ≫ 1 gd= bd , this is largely limited by the deliberately low reflectivity of a single groove; accordingly, the cavity mode FIG. 2. Characterization of a groove-based SAW cavity. leaks strongly into the grating such that Lc ≫ D (a) Quality factor Q for N ¼ 100 (dashed blue curve) and (cf. Appendix E for details). While up to now we have 300 N ¼ (red solid curve) grooves as a function of the normal- focused on this standard Bragg design (due to its experimen- h=λ Q ized grove depth c. For shallow grooves, is limited by tally validated frequency selectivity and quality factors), let us leakage losses due to imperfect acoustic mirrors (Qr regime, gray area), whereas for deep grooves conversion to bulk modes briefly mention potential approaches to reduce A and thus increase single-phonon effects even further. (i) The most dominates (Qb regime); compare asymptotics (dash-dotted lines). (b) Ratio of desired to undesired decay rates κ =κ . The stronger straightforwardstrategy(thatisstillcompatiblewiththeBragg gd bd λ Q is dominated by Q , the higher κ =κ . Here, w=p ¼ 0.5, mirror design) is to reduce c as much as possible, down to the r gd bd λ D ¼ 5.25λ , and f ¼ 3 GHz; typical material parameters for maximum frequency fc ¼ vs= c that can still be made c c ’ LiNbO3 are used (cf. Appendix E). resonant with the (typically highly tunable) qubit s transition frequency ωq=2π; note that fundamental Rayleigh modes with f ≈ 6 GHz have been demonstrated experimentally surface imperfections etc.) losses, labeled as κ and κ , c gd bd [41]. (ii) In order to increase the reflectivity of a single groove, respectively; for a schematic illustration, compare Fig. 1.For one could use deeper grooves. To circumvent the resulting the total quality factor Q, we can typically identify three increased losses into the bulk [cf. Fig. 2(b)], freestanding distinct regimes (cf. Fig. 2): For very small groove depths λ ≲ 2% structures (where the effect of bulk phonon modes is reduced) h= c , losses are dominated by coupling to SAW could be employed. (iii) Lastly,alternative cavity designs such modes outside of the cavity, dubbed as the Qr regime κ ≫ κ as so-called trapped energy resonators make it possible to ð gd bdÞ, whereas for very deep grooves, losses due to strongly confine acoustic resonances in the center of plate conversion into bulk modes become excessive (Qb regime, resonators [42]. κ ≪ κ gd bd). In between, for a sufficiently high number of grooves N, the quality factor Q can ultimately be limited by B. SAW waveguides internal losses (surface cracks, defects in the material, etc.), Not only can SAWs be confined in cavities, but they can referred to as the Q regime ðκ ≪ κ Þ.ForN ≈ 300,we m gd bd also be guided in acoustic waveguides [14,43].Two find that the onset of the bulk-wave limit occurs for λ ≳ 2 5% dominant types of design are (i) topographic WGs, such h= c . , in excellent agreement with experimental as ridge-type WGs, where the substrate is locally deformed findings [37,38]. With regard to applications in quantum using etching techniques, and (ii) overlay WGs (such as information schemes, the Qr regime plays a special role in strip- or slot-type WGs), where one or two strips of that resonator phonons leaking out through the acoustic one material are deposited on the substrate of another to mirrors can be processed further by guiding them in acoustic form core and clad regions with different acoustic veloc- SAW waveguides (see below). To capture this behavior ities. If the SAW velocity is slower (higher) in the film than κ κ quantitatively, we analyze gd= bd (cf. Fig. 2): for in the substrate, the film acts as a core (cladding) for the κ κ ≫ 1 gd= bd , leakage through the mirrors is the strongest guide, whereas the unmodified substrate corresponds to the decay mechanism for the cavity phonon, whereas the cladding (core). An attenuation coefficient of ∼0.6 dB=mm undesired decay channels are suppressed. Our analysis has been reported for a 10 μm-wide slot-type WG, defined shows that, for gigahertz frequencies fc ≈3 GHz, N ≈100, by Al cladding layers on a GaAs substrate [39,40]. This 3 and h=λc ≈2%, a quality factor of Q ≈ 10 is achievable, shows that SAWs can propagate basically dissipation free together with an effective cavity confinement Lc ≈ 40λc (for over chip-scale distances exceeding several millimeters.
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Typically, one-dimensional WG designs have been inves- 2t H ¼ δSz þ g c ðSþa þ S−a†Þ; ð5Þ tigated, but—to expand the design flexibility—one could dot ch Ω use multiple acoustic lenses in order to guide SAWs around a bend [29]. where δ ¼ Ω − ωc specifies the detuning between the qubit and the cavity mode, and S ¼j ih∓j (and so on) refer to IV. UNIVERSAL COUPLING pseudospin operators associated with the eigenstates j i of the DQD Hamiltonian (cf. Appendix F). The Hamiltonian A. Versatility Hdot describes the coherent exchange of excitations To complete the analogy with cavity QED, a nonlinear between the qubit and the acoustic cavity mode. The ϕ F 2 element similar to an atom needs to be introduced. Here, strength of this interaction gch ¼ e 0 ðkdÞ sin ðkl= Þ is we highlight three different exemplary systems, illustrating proportional to the charge e and the piezoelectric potential the versatility of our SAW-based platform. We focus on associated with a single phonon ϕ0. The decay of the SAW quantum dots, trapped ions, and nitrogen-vacancy (NV) mode into the bulk is captured by the function FðkdÞ (d is centers, but similar considerations naturally apply to other the distance between the DQD and the surface; see promising quantum information candidates such as super- Appendix B for details), while the factor sin ðkl=2Þ reflects conducting qubits [7,8,17,44], Rydberg atoms [45],or the assumed mode function along the axis connecting the electron spins bound to a phosphorus donor atom in silicon two dots, separated by a distance l. For (typical) values of ≈ λ 2 250 ≈ 50 ≪ λ [41]. In all cases considered, a single cavity mode a, with l c= ¼ nm and d nm c, the geometrical F 2 frequency ωc close to the relevant transition frequency, factor ðkdÞ sin ðkl= Þ then leads to a reduction in cou- is retained. We provide estimates for the single-phonon pling strength compared to the bare value eϕ0 (at the coupling strength and cooperativity (cf. Table I), while surface) by ap factorffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of ∼2 only. In total, we then obtain ≈ 2 μ 2 ≈ more detailed analyses go beyond the scope of this work gch GHz= A ½ m . For lateral confinement Ltrans 1–5 μ ≈ and are subject to future research. ð Þ m, the effective mode area is A ¼ LtransLc ð20–100Þ μm2. The resulting charge-resonator coupling ≈ 200–450 1. QD charge qubit strength gch ð Þ MHz compares well with values obtained using superconducting qubits coupled to localized A natural candidate for our scheme is a charge qubit nanomechanical resonators made of piezoelectric material, embedded in a lithographically defined GaAs double where g ≈ ð0.4–1.2Þ GHz [7,8], or superconducting reso- quantum dot (DQD) containing a single electron. The nators coupled to Cooper pair box qubits ðg=2π ≈ 6 MHzÞ DQD can well be described by an effective two-level [3], transmon qubits ðg=2π ≈ 100 MHzÞ [48], and indium ϵ system, characterized by an energy offsetpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiand interdot arsenide DQD qubits ðg=2π ≈ 30 MHzÞ [49]. Note that, in Ω ϵ2 4 2 tunneling tc yielding a level splitting ¼ þ tc [46]. principle, the coupling strength gch could be further The electron’s charge e couples to the piezoelectric enhanced by additionally depositing a strongly piezoelec- potential; the deformation coupling is much smaller than tric material such as LiNbO3 on the GaAs substrate [16]. the piezoelectric coupling and can therefore safely be Moreover, with a LiNbO3 film on top of the surface, neglected [47]. Since the quantum dot is small compared nonpiezoelectric materials such as Si or Ge could also be to the SAW wavelength, we neglect potential effects used to host the quantum dots [50]. The level splitting ΩðtÞ coming from the structure making up the dots (hetero- and interdot tunneling tcðtÞ can be tuned in situ via external structure and metallic gates); for a detailed discussion, see gate voltages. By controlling δ one can rapidly turn on and Appendix I. Performing a standard rotating-wave approxi- off the interaction between the qubit and the cavity: For an δ ≪ ω τ π 2 2 Ω mation (RWA) (valid for ;gch c), we find that the effective interaction time ¼ = geff ðgeff ¼ gchtc= Þ on system can be described by a Hamiltonian of Jaynes- resonance ðδ ¼ 0Þ, an arbitrary state of the qubit is Cummings form, swapped to the absence or presence of a cavity phonon;
TABLE I. Estimates for single-phonon coupling strength g and cooperativity C. We use A ¼ð1–5Þ μm×40λc, 3 T ¼ 20 mK [17], (conservative) quality factors of Q ¼ð1; 1; 3; 1Þ × 10 , and frequencies of ωc ¼ 2πð6; 1.5; 2 × −3 ⋆ 10 ; 3Þ GHz for the four systems listed. For the spin qubit T2 ¼ 2 μs [21], and for the trapped ion scenario, 150 μ gionðCionÞ is given for d ¼ m due to the prolonged dephasing time farther away from the surface (Cion improves with increasing d, even though gion decreases, up to a point where other dephasing start to dominate). Further details are given in the text.
Charge qubit (DQD) Spin qubit (DQD) Trapped ion NV center Coupling g (200–450) MHz (10–22.4) MHz (1.8–4.0) kHz (45–101) kHz Cooperativity C 11–55 21–106 7–36 10–54
031031-5 M. J. A. SCHUETZ et al. PHYS. REV. X 5, 031031 (2015) i.e., ðαj−iþβjþiÞj0i → j−iðαj0i − iβj1iÞ, where jni with a strongly piezomagnetic material. Equivalently, labels the Fock states of the cavity mode. Apart from this building upon current quantum sensing approaches SWAP operation, further quantum control techniques known [56,57], one could use a diamond nanocrystal (typically from cavity QED may be accessible [51]. Note that below we ∼10 nm in size) in order to get the NV center extremely generalize our results to spin qubits embedded in DQDs. close to the surface of the piezomagnetic material and thus maximize the coupling to the SAW cavity mode; 2. Trapped ion compare Fig. 3(a) for a schematic illustration. In the B presence of an external magnetic field ext [58], the The electric field associated with the SAW mode system is described by does not only extend into the solid, but, for a free surface, in general there will also be an electrical potential decaying exponentially into the vacuum above the surface H ¼ DS2 þ γ B · S þ ω a†a NV z XNV ext c ∼ exp ½−kjzj [26]; cf. Appendix B. This allows for cou- g ηα Sα a a† ; pling to systems situated above the surface, without any þ NV NV ð þ Þ ð7Þ α x;y;z mechanical contact. For example, consider a single ion of ¼ charge q and mass m trapped at a distance d above the surface of a strongly piezoelectric material such as LiNbO3 where D ¼ 2π × 2.88 GHz is the zero-field splitting, γ or AlN. The electric dipole induced by the ion motion gNV ¼ NVB0 is the single-phonon coupling strength, ηα couples to the electric field of the SAW phonon mode. The and NV is a dimensionless factor encoding the ori- dynamics of this system are described by the Hamiltonian entation of the NV spin with respect to the magnetic stray field of the cavity mode. For d ≪ λc, a rough ω † ω † † † ηα Hion ¼ ca a þ tb b þ gionðab þ a bÞ; ð6Þ estimate shows that at least one component of NV is of order unity [34]. For a NV center close to a Terfenol-D where b refers to the annihilation operator of the ion’s layer of thickness h ≫ λ ,wefindg ≈ 400 kHz= ω pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c NV motional mode and t is the (axial) trapping frequency. μ 2 A ½ m .p Thus,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as compared to direct strain coupling The single-phonon coupling strength is given by 2 ϕ ϕ η ≲200 Hz= A ½μm , the presence of the piezomagnetic gion ¼ qx0kc 0FðkcdÞ¼q 0FðkcdÞ LD. Apart from the exponential decay FðkdÞ¼exp ½−kd , the effective cou- layer is found to boost the single-phonon coupl- pling is reduced by the Lamb-Dicke parameter η ¼ ing strength by 3 orders of magnitude; this is in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LD agreement with previous theoretical results for a static 2πx0=λ , with x0 ¼ ℏ=2mω , since the motion of the c t setting [34]. ion is restricted to a region small compared with the SAW wavelength λc. For LiNbO3, a surface mode area of 1–5 μ 40λ 9 þ A ¼ð Þ m× c, the commonly used Be ion, (a) and typical ion trap parameters with d ≈ 30 μm and ω 2π ≈ 2 ≈ 3–6 7 t= MHz [52], we obtain gion ð . Þ kHz. ’ Here, gion refers to the coupling between the ion s motion and the cavity. However, based on Hion, one can, in principle, generalize the well-known protocols operating on the ion’s spin and motion to operations on the spin and the acoustic phonon mode [53]. (b)
3. NV center Yet another system well suited for our scheme are NV centers in diamond. Even though diamond itself is not piezoactive, it has played a key role in the context of high- FIG. 3. (a) Schematic illustration for coupling to a NV center frequency SAW devices due to its record-high sound via a piezomagnetic (PM) material (see text for details); surface velocity [14]; for example, high-performance SAW reso- grooves (not shown) can be used to provide SAW phonon confinement. (b) SAWs can be generated electrically based on nators with a quality factor of Q ¼ 12 500 at ωc ≳ 10 GHz were experimentally demonstrated for AlN/diamond standard interdigital transducers (IDTs) deposited on the surface. heterostructures [54,55]. To make use of the large Typically, an IDT consists of two thin-film electrodes on a piezoelectric material, each formed by interdigitated fingers. magnetic coupling coefficient of the NV center spin When an ac voltage is applied to the IDT on resonance (defined γ ¼ 2π × 28 GHz=T, here we consider a hybrid device NV by the periodicity of the fingers as ωIDT=2π ¼ vs=pIDT, where vs composed of a thin layer of diamond with a single is the SAW propagation speed), it launches a SAW across the (negatively charged) NV center with ground-state spin S substrate surface in the two directions perpendicular to IDT implanted a distance d ≈ 10 nm away from the interface fingers [14,15,17].
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≈ 0 1 ≈ 10 B. Decoherence whereas the spin-spin coupling is gdr . gNV kHz. 1 In the analysis above, we ignored the presence of Therefore, T2 ¼ ms is already sufficient to approach the ≫ κ −1 decoherence, which in any realistic setting will degrade strong-coupling regime, where gdr dr;T2 . the effects of coherent qubit-phonon interactions. In this D. Coherent driving context, the cooperativity parameter, defined as C ¼ 2 κ ¯ 1 Finally, we note that, in all cases considered above, one g T2=½ ðnth þ Þ , is a key figure of merit. Here, T2 refers ¯ could implement a coherent, electrical control by pumping to the corresponding dephasing time, while nth ¼ −1 the cavity mode using standard interdigital transducers ðexp ½ℏω =k T − 1Þ gives the thermal occupation num- c B (IDTs) [14,15,17]; compare Fig. 3(b) for a schematic ber of the cavity mode at temperature T. The parameter illustration. The effect of the additional Hamiltonian C compares the coherent single-phonon coupling strength Ξ ω † g with the geometric mean of the qubit’s decoherence rate Hdrive ¼ cos ð IDTtÞ½a þ a can be accounted for by −1 replacing the cavity state by a coherent state; that is, ∼T and the cavity’s effective linewidth ∼κðn¯ þ 1Þ;in 2 th a → α. For example, in the case of Eq. (5), one could direct analogy to cavity QED, C>1 marks the onset of then drive Rabi oscillations between the states jþi and j−i coherent quantum effects in a coupled spin-oscillator with the amplified Rabi frequency Ω ¼ gα. system, even in the presence of noise, cf. Ref. [10] and R Appendix H for a detailed discussion. To estimate C,we V. STATE TRANSFER PROTOCOL take the following parameters for the dephasing time T2. For system (i), T2 ≈ 10 ns is measured close to the charge The possibility of quantum transduction between SAWs degeneracy point ϵ ¼ 0 [46]. In scenario (ii), motional and different realizations of stationary qubits enables a decoherence rates of 0.5 Hz are measured in a cryogeni- variety of applications including quantum information cally cooled trap for an ion height of 150 μm and 1 MHz achitectures that use SAW phonons as a quantum bus to motional frequency [52]. Since this rate scales as ∼d−4 couple dissimilar and/or spatially separated qubits. The 4 [53,59], we take T2½s ≈ 2ðd ½μm =150Þ . Lastly, for the most fundamental task in such a quantum network is the NV center (iii), T2 ≈ 0.6 s is demonstrated for ensembles implementation of a state transfer protocol between two of NV spins [60] and we assume an optimistic value of remote qubits 1 and 2, which achieves the mapping α 0 β 1 ⊗ 0 → 0 ⊗ α 0 β 1 T2 ¼ 100 ms, similarly to Ref. [61]. The results are ð j i1 þ j i1Þ j i2 j i1 ð j i2 þ j i2Þ. In anal- summarized in Table I. We find that C>1 should be ogy to optical networks, this can be accomplished via experimentally feasible, which is sufficient to perform a coherent emission and reabsorption of a single phonon in a quantum gate between two spins mediated by a thermal waveguide [13]. As first shown in the context of atomic mechanical mode [9]. QED [20], in principle perfect, deterministic state transfer can be implemented by identifying appropriate time- C. Qubit-qubit coupling dependent control pulses. Before we discuss a specific implementation of such a When placing a pair of qubits into the same cavity, ≫ 1 transfer scheme in detail, we provide a general approximate the regime of large single spin cooperativity C allows result for the state transfer fidelity F. As demonstrated in for coherent cavity-phonon-mediated interactions and detail in Appendix H, for small infidelities one can take quantum gates between the two spins via the effective − þ −1 interaction Hamiltonian Hint ¼ gdrðS1 S2 þ H:c:Þ, where F ≈ 1 − ε − CC ð8Þ 2 δ ≪ gdr ¼ g = g in the so-called dispersive regime [4]. For the estimates given in Table I, we restrict ourselves as a general estimate for the state transfer fidelity. Here, 3 to the Qr regime with Q ≈ 10 , where leakage through the individual errors arise from intrinsic phonon losses ∼ε ¼ −1 −1 acoustic mirrors dominates over undesired (nonscalable) κ =κ and qubit dephasing ∼C ∼ T2 , respectively; the κ ≫ κ bd gd phonon losses ð gd bdÞ. However, note that small-scale numerical coefficient C ∼ Oð1Þ depends on the specific experiments using a single cavity only (where there is no control pulse and may be optimized for a given set of need for guiding the SAW phonon into a waveguide for experimental parameters [62]. This simple, analytical result further quantum information processing) can be operated in holds for a Markovian noise model where qubit dephasing the Qm regime (which is limited only by internal material is described by a standard pure dephasing term leading 5 losses), where the quality factor Q ≈ Qm ≳ 10 is maxi- to an exponential loss of coherence ∼ exp ð−t=T2Þ and mized (and thus overall phonon losses minimal). agrees well with numerical results presented in Ref. [62]. As a specific example, consider two NV centers, both For non-Markovian qubit dephasing an even better scaling ≈ 100 coupled with strength gNV kHz to the cavity and with C can be expected [9]. Using experimentally achiev- in resonance with each other, but detuned from the resonator. able parameters ε ≈ 5% and C ≈ 30, we can then estimate Since for large detuning δ the cavity is only virtually F ≈ 90%, showing that fidelities sufficiently high for κ populated, the cavity decay rate is reduced to dr ¼ quantum communication should be feasible for all physical 2 2 −2 5 ðg =δ Þκ≈10 κ≈1kHz (for fc ¼ 3 GHz, Q ¼ 2 × 10 ), implementations listed in Table I.
031031-7 M. J. A. SCHUETZ et al. PHYS. REV. X 5, 031031 (2015) X X In the following, we detail the implementation of a L ρ 2κ D ρ − δ z ρ noise ¼ bd ½ai i i½Si ; : ð10Þ transfer scheme based on spin qubits implemented in i¼1;2 i gate-defined double quantum dots [63]. In particular, we D ρ ρ † − 1 † ρ consider singlet-triplet-like qubits encoded in lateral QDs, Here, ½a ¼ a a 2 fa a; gPis a Lindblad term with where two electrons are localized in adjacent, tunnel- jump operator a and HSðtÞ¼ iHiðtÞ,whereHiðtÞ¼ coupled dots. As compared to the charge qubits discussed − † g ðtÞ½Sþa þ S a describes the coherent Jaynes- above, this system is known to feature superior coherence i i i i i Cummings dynamics of the two nodes. The ideal time scales [64–67] which are largely limited by the cascaded interaction is captured by L , which contains relatively strong hyperfine interaction between the elec- ideal the nonlocal coherent environment-mediated coupling tronic spin and the nuclei in the host environment [65], transferring excitations from qubit 1 to qubit 2 [76], resulting in a random, slowly evolving magnetic while L summarizes undesired decoherence proc- (Overhauser) field for the electronic spin. To mitigate this noise esses: We account for intrinsic phonon losses (bulk-mode decoherence mechanism, two common approaches are conversion, material imperfections, etc.) with a rate κ (i) spin-echo techniques, which allow us to extend spin bd and (nonexponential) qubit dephasing. Since the nuclear coherence from a time-ensemble-averaged dephasing time ⋆ spins evolve on relatively long time scales, the electronic T ≈ 100 ns to T2 ≳ 250 μs [67], and (ii) narrowing of the 2 spins in quantum dots typically experience non- nuclear field distribution [65,68]. Recently, real-time adap- Markovian noise leading to a nonexponential loss of tive control and estimation methods (that are compatible ⋆ coherence on a characteristic time scale T2 givenbythe with arbitrary qubit operations) have allowed to narrow the σ ⋆ ⋆ ⋆ width of the nuclear field distribution nuc as T2 ¼ nuclear spin distribution to values that prolong T2 to T2 > pffiffiffi 2 σ 2 μs [21]. For our purposes, the latter is particularly = nuc [21,65]. Recently, a record-low value of σ 2π 80 attractive as it can be done simply before loading and nuc= ¼ kHz has been reported [21], yielding an transmitting the quantum information, whereas spin-echo extended time-ensemble-averaged electron dephasing ⋆ 2 8 μ techniques can be employed as well, however, at the expense time of T2 ¼ . s. In our model, to realistically account of more complex pulse sequences (see Appendix G for for the dephasing induced by the quasistatic, yet unknown δ details). In order to couple the electric field associated with Overhauser field, the detuning parameters i are sampled δ the SAW cavity mode to the electron spin states of such a independently from a normal distribution pð iÞ with DQD, the essential idea is to make use of an effective electric zero mean (since nominal resonance can be achieved via dipole moment associated with the exchange-coupled spin the electronic control parameters) and standard deviation σ states of the DQD [69–72]. As detailed in Appendix G,we nuc [68];seeAppendixGfordetails.InAppendixJ,wealso then find that in the usual singlet-triplet subspace spanned by provide numerical results for standard Markovian dephas- the two-electron states fj⇑⇓i; j⇓⇑ig, a single node can well ing, showing that non-Markovian noise is beneficial in be described by the prototypical Jaynes-Cummings terms of faithful state transfer. L 0 Hamiltonian given in Eq. (1). As compared to the direct Under ideal conditions where noise ¼ , the setup is analogous to the one studied in Ref. [20] and the same time- charge coupling gch, the single phonon coupling strength g is reduced since the qubit states jli have a small admixture symmetry arguments can be employed to determine the of the localized singlet hS02jliðl ¼ 0; 1Þ only. Using typical optimal control pulses giðtÞ for faithful state transfer: if a ≈ 0 1 ≈ 200 phonon is emitted by the first node, then, upon reversing parameterspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values, we find g . gch MHz= 2 the direction of time, one would observe perfect reabsorp- A ½μm [73]. In this system, the coupling gðtÞ can be tion. By engineering the emitted phonon wave packet such tuned with great flexibility via the tunnel-coupling t and/or c that it is invariant under time reversal and using a time- the detuning parameter ϵ. reversed control pulse for the second node g2 t g1 −t , The state transfer between two such singlet-triplet qubits ð Þ¼ ð Þ the absorption process in the second node is a time-reversed connected by a SAW waveguide can be adequately copy of the emission in the first and therefore in principle described within the theoretical framework of cascaded perfect. Based on this reasoning (for details, see Ref. [20]), quantum systems, as outlined in detail, for example, in we find the explicit, optimal control pulse shown in Refs. [13,20,74,75]: The underlying quantum Langevin Fig. 4(c). equations describing the system can be converted into an To account for noise, we simulate the full master effective, cascaded master equation for the system’s density equation numerically. The results are displayed in matrix ρ. For the relevant case of two qubits, it can be Fig. 4(a), where for every random pair δ ¼ðδ1; δ2Þ the written as ρ_ ¼ L ρ þ L ρ, where ideal noise fidelity of the protocol is defined as the overlap between the ψ ρ target state j tari and the actual state after the transfer ðtfÞ; † † F ψ ρ ψ F¯ L ρ − κ − ρ that is, δ ¼h tarj ðtfÞj tari. The average fidelity of the ideal ¼ i½HSðtÞþi gdða1a2 a2a1Þ; protocol is determinedR by averaging over the classical þ 2κ D½a1 þ a2 ρ; ð9Þ ¯ gd noise in δ; that is, F ¼ dδ1dδ2pðδ1Þpðδ2ÞF δ. Taking an
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since nuclear spins are largely absent in isotopically (c) 1 purified 28Si.
VI. SUMMARY AND OUTLOOK 0.9 (a) In summary, we propose and analyze SAW phonons in piezoactive materials (such as GaAs) as a universal 0.8 quantum transducer that allows us to convert quantum information between stationary and propagating realiza- 0.7 tions. We show that a sound-based quantum information architecture based on SAW cavities and waveguides is very versatile, bears striking similarities to cavity QED, and can (b) 0.6 serve as a scalable mediator of long-range spin-spin interactions between a variety of qubit implementations, allowing for faithful quantum state transfer between remote 0.5 0 2 4 6 8 10 qubits with existing experimental technology. The pro- posed combination of techniques and concepts known from quantum optics and quantum information, in conjunction FIG. 4. (a) Average fidelity F¯ of the state transfer protocol for a with the technological expertise for SAW devices, is likely pffiffiffi coherent superposition jψi¼ðj0i − j1iÞ= 2 in the presence of to lead to further, rapid theoretical and experimental quasistatic (non-Markovian) Overhauser noise, as a function of progress. σ the root-mean-square fluctuations nuc in the detuning parameters Finally, we highlight possible directions of research going δ 1 2 κ κ 0 κ κ iði ¼ ; Þ,for bd= gd ¼ (solid line, circles) and bd= gd ¼ beyond our present work. First, since our scheme is not 10% (dash-dotted line, squares). (b) After n ¼ 100 runs with specific to any particular qubit realization, novel hybrid ¯ random values for δi, F approximately reaches convergence. The systems could be developed by embedding dissimilar σ κ 0 2 … 10 % curves refer to nuc= gd ¼ð ; ; ; Þ (from top to bottom) systems such as quantum dots and superconducting qubits κ κ 10% for bd= gd ¼ . (c) Pulse shape g1ðtÞ for first node. into a common SAW architecture. Second, our setup could also be used as a transducer between different propagating effective mode area A ≈ 100 μm2 as above and Q ≈ 103 to quantum systems such as phonons and photons. Light can be κ κ ≈ 5% coupled into the SAW circuit via (for example) NV centers be well within the Qr regime where bd= gd ,we ≈ κ ≈ 20 or self-assembled quantum dots, and structures guiding both have g gd MHz. For two nodes separated by millimeter distances, propagation losses are negligible photons and SAW phonons have already been fabricated and κ =κ ≈ 5% captures well all intrinsic phonon experimentally [39,40]. Finally, the SAW architecture opens bd gd up a novel, on-chip test bed for investigations reminiscent of losses during the transfer. We then find that for realistic quantum optics, bringing the highly developed toolbox of undesired phonon losses κ =κ ≈ 5% and σ =2π ¼ bd gd nuc quantum optics and cavity QED to the widely anticipated 80 kHz (such that σ =κ ≈ 2.5%) [21], transfer fidel- nuc gd field of quantum acoustics [11,16,17,82]. Potential applica- ities close to 95% seem feasible. Notably, this could be tions include quantum simulation of many-body dynamics improved even further using spin-echo techniques, such [83], quantum state engineering (yielding, for example, ≈ 102 ⋆ that T2 T2 [67]. Therefore, state transfer fidelities squeezed states of sound), quantum-enhanced sensing, F 2 3 > = as required for quantum communication [77] sound detection, and sound-based material analysis. seem feasible with present technology. Near-unit fidel- ities might be approached from further optimizations of ACKNOWLEDGMENTS the system’s parameters, the cavity design, the control pulses, and/or from communication protocols that correct We thank the Benasque Center for Science for great for errors such as phonon losses [78–80]. Once the working conditions to start this project. M. J. A. S., G. G., transfer is complete, the system can be tuned adiabati- and J. I. C. acknowledge support by the DFG within SFB cally into a storage regime that immunizes the qubit 631, and the Cluster of Excellence NIM. E. M. K. acknowl- against electronic noise, and dominant errors from edges support by the Harvard Quantum Optics Center and hyperfine interaction with ambient nuclear spins can be the Institute for Theoretical Atomic and Molecular Physics. mitigated by standard, occasional refocusing of the spins [67,69]. Alternatively, one could also investigate silicon L. M. K. V. acknowledges support by a European Research dots: while this setup requires a more sophisticated Council Synergy grant. Work at Harvard was supported by heterostructure including some piezoelectric layer (as NSF, Center for Ultracold Atoms, CIQM, and AFOSR studied experimentally in Ref. [41]), it potentially bene- MURI. M. J. A. S. thanks M. Endres, A. Gonzalez-Tudela, ⋆ fits from prolonged dephasing times T2 > 100 μs [81], and O. Romero-Isart for fruitful discussions.
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APPENDIX A: CLASSICAL DESCRIPTION 2. Mechanical boundary condition OF NONPIEZOELECTRIC SURFACE The free surface at z ¼ 0 is stress free (no external forces ACOUSTIC WAVES are acting upon it), such that the three components of stress In this appendix, we review the general (classical) across z ¼ 0 shall vanish; that is, T13 ¼ T23 ¼ T33 ¼ 0. theoretical framework describing SAW in cubic lattices, This results in the boundary conditions such as diamond or GaAs. We derive an analytical solution ∂ for propagation in the [110] direction. The latter is of uk Tizˆ ¼ cizklˆ ¼ 0 at z ¼ 0: ðA6Þ particular interest in piezoelectric systems. The classical ∂xl description of a SAW is explicitly shown here to make our work self-contained, but follows standard references such 3. Cubic lattice as Ref. [26]. For a cubic lattice (such as GaAs or diamond), the elastic tensor c has three independent elastic constants, gen- 1. Wave equation ijkl erally denoted by c11;c12, and c44; compare Table II. The propagation of acoustic waves (bulk and surface Taking the three direct twofold axes as the coordinate waves) in a solid is described by the equation axes, the wave equations then read ∂2 ∂2 ∂2 ∂2 ∂Tij ux ux ux ux ρüðx;tÞ¼ ; ðA1Þ ρ ¼ c11 þ c44 þ i ∂x ∂t2 ∂x2 ∂y2 ∂z2 j ∂2 2 uy ∂ uz u þðc12 þ c44Þ þ ðA7Þ where denotes the displacement vector with ui being ∂x∂y ∂x∂z the displacement along the Cartesian coordinate xˆi ðxˆ1 ¼ x;ˆ ˆ ˆ ˆ x2 ¼ y; x3 ¼ zˆÞ, ρ gives the mass density, and T is the stress (and cyclic permutations), while the mechanical boundary tensor; Tij is the ith component of force per unit area conditions can be written as ˆ x perpendicular to the xj axis. Moreover, is the Cartesian coordinate vector, where in the following we assume a ∂uz ∂ux ˆ ˆ T13 ¼ c44 þ ¼ 0; ðA8Þ material with infinite dimensions in x; y and a surface ∂x ∂z perpendicular to the zˆ direction at z ¼ 0. The stress tensor obeys a generalized Hooke law (stress is linearly propor- ∂ ∂uz uy tional to strain) T23 ¼ c44 þ ¼ 0; ðA9Þ ∂y ∂z Tij ¼ cijklukl; ðA2Þ ∂ ∂uz ∂ux uy T33 ¼ c11 þ c12 þ ¼ 0; ðA10Þ ∂z ∂x ∂y where the strain tensor is defined as at z ¼ 0. In the following, we seek solutions that propagate 1 ∂ ∂ uk ul along the surface with a wave vector k ¼ kðlxˆ þ myˆÞ, ukl ¼ þ : ðA3Þ θ θ θ 2 ∂xl ∂xk where l ¼ cosð Þ, m ¼ sinð Þ, and is the angle between the xˆ axis and k. Following Ref. [84], we make the ansatz
Using the symmetry cijkl ¼ cijlk, in terms of displacements 0 1 0 1 ux U we find B C B C −kqz ikðlxþmy−ctÞ @ uy A ¼ @ V Ae e ; ðA11Þ ∂u k uz W Tij ¼ cijkl ; ðA4Þ ∂xl such that Eq. (A1) takes the form of a set of three coupled TABLE II. Material properties [26] for both diamond and GaAs. The elastic tensor c has three independent parameters, wave equations, given in units of ½1010 N=m2 , while the piezoelectric tensor e has a single independent parameter e14 for cubic materials (units of ∂2 2 ρ̈ x − uk 0 C=m ). uið ;tÞ cijkl ∂ ∂ ¼ : ðA5Þ xj xl 3 c11 c12 c44 ρ ½kg=m e14 The elasticity tensor c obeys the symmetries c ¼ c ¼ Diamond 107.9 12.4 57.8 3515 0 ijkl jikl GaAs 12.26 5.71 6.00 5307 0.157 cijlk ¼ cklij and is largely defined by the crystal symmetry.
031031-10 UNIVERSAL QUANTUM TRANSDUCERS BASED ON … PHYS. REV. X 5, 031031 (2015) where the decay constant q describes the exponential decay of the surface wave into the bulk and c is the phase velocity. Plugging this ansatz into the mechanical wave equations can be rewritten as MA ¼ 0, where 0 1 2 2 2 2 c11l þ c44ðm − q Þ − ρc lmðc12 þ c44Þ lqðc12 þ c44Þ B 2 2 2 2 C M ¼ @ lmðc12 þ c44Þ c11m þ c44ðl − q Þ − ρc mqðc12 þ c44Þ A; ðA12Þ 2 2 lqðc12 þ c44Þ mqðc12 þ c44Þ c11q − c44 þ ρc and A ¼ðU; V; iWÞ. Nontrivial solutions for this homo- where we introduce the quantities geneous set of equations can be found if the determinant M of vanishes, resulting in the so-called secular equation 2 2 − 2 − ρ 2 c11m þ c44ðl qrÞ c mqrðc12 þ c44Þ detðMÞ¼0. The secular equation is of sixth order in q;as ξ ¼ ; r 2 − ρ 2 all coefficients in the secular equation are real, there are, in mqrðc12 þ c44Þ c11qr c44 þ c 2 2 2 general, three complex-conjugate roots q1;q2;q3, with the mq ðc12 þ c44Þ lmðc12 þ c44Þ η r phase velocity c and propagation direction θ as parameters. r ¼ 2 2 ; ðA15Þ c11q − c44 þ ρc lq ðc12 þ c44Þ If the medium lies in the half-space z>0, the roots with r r negative real part will lead to a solution that does not converge as z → ∞. Thus, only the roots that lead to and vanishing displacements deep in the bulk are kept. Then, 2 2 2 2 the most general solution can be written as a superposition lm c12 c44 c11m c44 l − q − ρc ζ ð þ Þ þ ð r Þ of surface waves with allowed qr values as r ¼ : lq ðc12 þ c44Þ mq ðc12 þ c44Þ X r r −kq z ikðlxþmy−ctÞ ðux;uy;iuzÞ¼ ðξr; ηr; ζrÞKre r e ; ðA16Þ r¼1;2;3
ðA13Þ Note that for each root qr and displacement ui there is an associated amplitude. The phase velocity c, however, is the q q c; θ where, for any r ¼ rð Þ, the ratios of the amplitudes same for every root qr, and needs to be determined from can be calculated according to the mechanical boundary conditions as we describe below. Similarly to the acoustic wave equations, the boundary U V iW r r r conditions can be rewritten as BðK1;K2;K3Þ¼0, where Kr ¼ ξ ¼ η ¼ ζ ; ðA14Þ r r r the boundary condition matrix B is
0 1 lζ1 − q1ξ1 lζ2 − q2ξ2 lζ3 − q3ξ3 B C B ¼ @ mζ1 − q1η1 mζ2 − q2η2 mζ3 − q3η3 A; ðA17Þ
lξ1 þ mη1 þ aq1ζ1 lξ2 þ mη2 þ aq2ζ2 lξ3 þ mη3 þ aq3ζ3
pffiffiffi with a ¼ c11=c12. Again, nontrivial solutions are found for diagonal as xˆ0 ¼ðxˆ þ yˆÞ= 2. Subtracting the second detðBÞ¼0. The requirements detðMÞ¼0, detðBÞ¼0, row from the first in M, one finds that the common 2 2 together with Eq. (A14) constitute the formal solution factor ðc11 − c12Þ=2 − c44q − ρc divides through the first of the problem [84]; detðMÞ¼0 and detðBÞ¼0 may be row, which then becomes 1; −1; 0 . Thus, U V and the 2 2 ð Þ ¼ seen as determining c and q , and Eq. (A14) then gives wave equations can be simplified to M110ðU; iWÞ¼0, the ratios of the components of the displacement. In the where following, we discuss a special case where one can eliminate the q dependence in detðBÞ¼0, leading to an 2 2 q 0 − ρ − pffiffi explicit, analytically simple equation for the phase velocity c11 c c44q 2 ðc12 þ c44Þ M110 ¼ ffiffiffi ; c, which depends only on the material properties. p 2 2 2qðc12 þ c44Þ c11q − c44 þ ρc 4. Propagation in [110] direction ðA18Þ The wave equations simplify for propagation in high- symmetry directions. Here, we consider propagation in the 0 2 2 pffiffiffi with c11 ¼ðc11 þ c12 þ c44Þ= . Then, the secular equa- [110] direction, for which l ¼ m ¼ 1= 2; we define the tion detðM110Þ¼0 is found to be
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0 2 2 2 2 2 2 ðc −ρc −c44q Þðc44 −ρc −c11q Þþðc12 þc44Þ q ¼ 0; 11 20 ðA19Þ 15 2 2 yielding the roots q1;q2. We choose the roots commensu- rate with the convergence condition yielding the general GHz 10 ansatz 5 X 0 ux0 Ur − 0− ¼ e kqrzeikðx ctÞ; ðA20Þ iuz r¼1;2 iWr 0 0.0 0.2 0.4 0.6 0.8 1.0 pffiffiffi k m 0 with ux ¼ uy ¼ ux0 = 2. The amplitude ratios γr ¼ 0 ω 1 2 3 iWr=Ur can be obtained from the kernel of M as FIG. 5. Dispersion relation n ¼ vnk of the three ðn ¼ ; ; Þ Rayleigh-type SAW modes for propagation along xˆ0∥½110 . If not c12 c44 stated otherwise, we refer to the lowest frequency solution as the γ0 þ r ¼ qr 2 ; ðA21Þ SAW mode (solid line). c44 − c11ðX þ qr Þ 2 2 0 which is cubic in X ρc =c11. If not stated otherwise, we where X ¼ ρc =c11. In the coordinate system fxˆ ; zˆg, the ¼ mechanical boundary conditions read consider the mode with the lowest sound velocity, referred to as the Rayleigh mode; compare Fig. 5.
∂u ∂u 0 Using the secular equation given in Eq. (A19) and z þ x ¼ 0 ðz ¼ 0Þ; ðA22Þ ∂x0 ∂z the mechanical boundary conditions, the ansatz given in Eq. (A20) can be simplified as follows: The roots com-
∂u 0 ∂u patible with convergence in the bulk are complex con- x z 0 0 0 c12 þ c11 ¼ ðz ¼ Þ: ðA23Þ jugate, i.e., q ≡ q1 q γ ≡ γ γ ∂x0 ∂z ¼ 2, and therefore, 1 ¼ 2. Then, using the first row in the boundary condition matrix For the ansatz given in Eq. (A20), they can be reformulated [compare Eq. (A24)], we can deduce 0 0 as B110ðU1;U2Þ¼0, with 0 −iφ 0 iφ U1 ¼ Ue ;U2 ¼ Ue ; ðA26Þ 0 0 γ1 − q1 γ2 − q2 B where 110 ¼ c11 0 c11 0 : ðA24Þ 1 q1γ 1 q2γ þ c12 1 þ c12 2 γ − q e−2iφ ¼ − : ðA27Þ The requirement detðB110Þ¼0 can be written as γ − q
2 2 2 2 q1½c12 þ ρc þ c11q1 ½c12ðc44 − ρc Þþc11c44q2 2 2 2 2 − q2½c12 þ ρc þ c11q2 ½c12ðc44 − ρc Þþc11c44q1 ¼0:
From the symmetry of this equation it is clear that one can remove a factor ðq1 − q2Þ leading to c12 c44 2 2 c44 c12 þ X − X þ c44q1q2 þ c12 − X c11 c11 c11 2 2 c12 × ðq1 þ q2 þ q1q2Þ − c44q1q2 þ X ¼ 0: c11
2 2 2 2 Using simple expressions for q1q2 and q1 þ q2 obtained from Eq. (A19), one arrives at the following explicit equation for the wave velocity c [26,84]: FIG. 6. Depth dependence of the (normalized) vertical dis- 0 placement uz=U along xˆ ∥½110 for a Rayleigh surface acoustic 0 − 2 2 0 1 − c11 c11c11 c12 − 2 c11 − wave propagating on a (001) GaAs crystal. The acoustic X 2 X ¼ X X ; amplitude decays away from the surface into the bulk on a c44 c11 c11 characteristic length scale approximately given by the SAW ðA25Þ wavelength λ ¼ 2π=k ≈ 1 μm.
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In summary, we find the following solution [26]: includes the polarization produced by the strain. Therefore, Newton’s law becomes −qkz−iφ ikðx0−ctÞ ux0 ¼ Uðe þ H:c:Þe ; ∂2u ∂2ϕ −qkz−iφ ikðx0−ctÞ ρ̈ k iuz ¼ Uðγe þ H:c:Þe ; ðA28Þ ui ¼ cijkl þ ekij : ðB2Þ ∂xj∂xl ∂xj∂xk where the material-dependent parameters c, q, γ, and φ are For an insulating solid, the electric displacement Di must determined by Eq. (A25), Eq. (A19), Eq. (A21), and satisfy Poisson’s equation ∂Di=∂xi ¼ 0, which yields Eq. (A25), respectively. For the GaAs parameters given in Table II, we get c ¼ 2878 m=s, q ¼ 0.5 þ 0.48i, 2 2 ∂ uj ∂ ϕ γ ¼ −0.68 þ 1.16i, and φ ¼ 1.05, respectively. The cor- e − ϵ ¼ 0;z>0; ðB3Þ ijk ∂x ∂x ij ∂x ∂x responding (normalized) transversal displacement is dis- i k i j played in Fig. 6. Δϕ ¼ 0;z>0: ðB4Þ APPENDIX B: SURFACE ACOUSTIC WAVES IN PIEZOELECTRIC MATERIALS b. Mechanical boundary conditions In a piezoelectric material, elastic and electromagnetic In the presence of piezoelectric coupling the mechanical waves are coupled. In principle, the field distribution can boundary conditions [compare Eq. (A6)] generalize to be found only by solving simultaneously the equations of ∂ ∂ϕ both Newton and Maxwell. The corresponding solutions uk Tizˆ ¼ cizklˆ þ ekizˆ ¼ 0 at z ¼ 0: ðB5Þ are hybrid elastoelectromagnetic waves, i.e., elastic ∂xl ∂xk waves with velocity vs accompanied by electric fields, 5 and electromagnetic waves with velocity c ≈ 10 vs accom- Using the symmetries cijkl ¼ cjikl and ekij ¼ ekji, it is easy panied by mechanical strains. For the first type of wave, the to check that this is equivalent to Eq. (41) in Ref. [26]. magnetic field is negligible, because it is due to an electric field traveling with a velocity vs much slower than the c. Electric boundary condition speed of light c; therefore, one can approximate Maxwell’s equations as ∇ × E ¼ −∂B=∂t ≈ 0, giving E ¼ −∇ϕ. In addition to the stress-free boundary conditions, Thus, the propagation of elastic waves in a piezoelectric piezoelectricity introduces an electric boundary condition: material can be described within the quasistatic approxi- The normal component of the electric displacement needs mation, where the electric field is essentially static com- to be continuous across the surface [39]; that is, pared to electromagnetic fields [27]. The potential ϕ 0þ 0− and the associated electric field are not electromagnetic Dzðz ¼ Þ¼Dzðz ¼ Þ; ðB6Þ in nature but rather a component of the predominantly where by definition Dz ¼ ezjkˆ ∂uj=∂xk − ϵzjˆ ∂ϕ=∂xj. mechanical wave propagating with velocity vs. Outside of the medium ðz<0Þ, we assume vacuum; thus, ϵ −ϵ ∂ϕ ∂ 1. General analysis Dz ¼ 0Ez ¼ 0 out= z, where the electrical potential ’ Δϕ 0 has to satisfy Poisson s equation out ¼ . The ansatz a. Wave equation ϕ ikðx0−ctÞ Ωkz The basic equations that govern the propagation of out ¼ Aoute e ðB7Þ acoustic waves in a piezoelectrical material connect the D Δϕ − 2 Ω2 2 ϕ 0 mechanical stress T and the electrical displacement with gives out ¼ð k þ k Þ out ¼ . Thus, for proper the mechanical strain and the electrical field. The coupled convergence far away from the surface z → −∞,we constitutive equations are take the decay constant Ω ¼ 1; accordingly, the electrical potential decays exponentially into the vacuum above the ∂u ∂ϕ surface on a typical length scale given by the SAW T ¼ c k þ e ; ij ijkl ∂x kij ∂x wavelength λ ¼ 2π=k ≈ 1 μm. Therefore, for the electrical l k −ϵ ϕ ∂ϕ ∂u displacement outside of the medium, we find Dz ¼ 0k . −ϵ j Lastly, the electrical potential has to be continuous across Di ¼ ij ∂ þ eijk ∂ ; ðB1Þ xj xk the surface [26], i.e.,
ϵ ϕ 0þ ϕ 0− where e with ðeijk ¼ eikjÞ and are the piezoelectric and ðz ¼ Þ¼ outðz ¼ Þ; ðB8Þ permittivity tensor, respectively. Here, Hooke’slawis extended by the additional stress term due to the piezo- which allows us to determine the amplitude Aout.In electric effect, while the equation for the displacement Di summary, Eq. (B6) can be rewritten as
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∂ ∂ − ϵ ∂ϕ ∂ ϵ ϕ 0 ðezjkˆ uj= xk zjˆ = xj þ 0k Þjz¼0 ¼ : ðB9Þ piezoelectric coupling, one can obtain analytical expres- sions for the strain and piezoelectric fields. Here, we summarize the results for SAWs propagating along 2. Cubic lattice xˆ0∥½110 of the zˆ∥½001 surface following Refs. [26,28]. a. Specific analysis for cubic systems Since the piezoelectric coupling e14 is small, it follows For a cubic, piezoelectric system there is only one from Eq. (B11) that ϕ will be order e14 smaller than the independent nonzero component of the piezoelectric tensor mechanical displacements u; that is, called e14 [26,27]. With this piezoelectric coupling, the e14 wave equations are given by four coupled partial differ- ϕ ∼ u: ðB16Þ ential equations, ϵ ∂2 ∂2 ∂2 ∂2 This results in additional terms in the wave equations that ux ux ux ux 2 8 2 ρ ¼ c11 þ c44 þ ðB10Þ are of order ∼e =ϵ ≈ 10 N=m . Since the elastic con- ∂t2 ∂x2 ∂y2 ∂z2 14 stants are 2–3 orders of magnitude bigger than this piezo- ∂2 2 2 electric term, the wave equations [Eq. (B10)] and (cyclic uy ∂ uz ∂ ϕ þðc12 þ c44Þ þ þ 2e14 ; versions for uy;uz) will be solved by the nonpiezoelectric ∂x∂y ∂x∂z ∂y∂z 2 solution with corrections only at order e14. The nonpiezo- electric solution derived in detail in Appendix A can be ∂2 ∂2u ∂2 summarized as ϵΔϕ 2 ux y uz ¼ e14 ∂ ∂ þ ∂ ∂ þ ∂ ∂ ; ðB11Þ y z x z x y −qkz−iφ ikðx0−vtÞ ux0 ¼ 2URe½e e ; Δ and cyclic permutations for uy and uz. Here, is the uy0 ¼ 0; Laplacian and ϵ is the dielectric constant of the medium. 0 u −2iURe γe−qkz−iφ eikðx −vtÞ; B17 For a cubic lattice, the mechanical boundary conditions at z ¼ ½ ð Þ z ¼ 0 explicitly read where the sound velocity v for the Rayleigh mode follows ∂ ∂ ∂ϕ from the smallest solution of uz ux 0 T13 ¼ c44 ∂ þ ∂ þ e14 ∂ ¼ ; ðB12Þ x z y 2 0 2 2 2 ðc44 − ρv Þðc11c − c − c11ρv Þ 11 12 ∂ ∂ ∂ϕ 2 4 0 2 uz uy ¼ c11c44ρ v ðc − ρv Þ; ðB18Þ T23 ¼ c44 þ þ e14 ¼ 0; ðB13Þ 11 ∂y ∂z ∂x 0 with c ¼ c44 þðc11 þ c12Þ=2. The decay constant q is a ∂ ∂ ∂ 11 uz ux uy solution of T33 ¼ c11 þ c12 þ ¼ 0; ðB14Þ ∂z ∂x ∂y 0 2 2 2 2 ðc11 − ρv − c44q Þðc44 − ρv − c11q Þ while the electrical boundary condition [compare the 2 2 0 general relation in Eq. (B9)] leads to þ q ðc12 þ c44Þ ¼ : ðB19Þ ∂ γ φ ∂ux uy ∂ϕ Lastly, the parameters , can be obtained from e14 þ − ϵ þ ϵ0kϕ ¼ 0: ðB15Þ ∂y ∂x ∂z z¼0 c12 c44 q γ −q γ ð þ Þ −2iφ − ¼ 2 2 ;e ¼ : ðB20Þ In general, the wave equations can be formulated into a c44 −c11q −ρv γ −q 4 × 4 matrix M; the condition det M ¼ 0 can then used to find the four decay constants. In addition, the mechanical Now, based on the nonpiezoelectric solution given in and electrical boundary conditions can be recast to a 4 × 4 Eq. (B17), the potential ϕ is constructed such that both boundary condition matrix B, from which one can deduce the wave equation in Eq. (B11) and the electrical boundary the allowed phase velocities of the piezoelectric SAW by condition in Eq. (B15) are solved. In the fxˆ0; yˆ0; zˆg solving det B ¼ 0. coordinate system they read explicitly 2 2 ∂ ux0 ∂ uz b. Perturbative treatment ϵΔϕ ¼ e14 2 þ ; ðB21Þ ∂x0∂z ∂x0∂x0 For materials with weak piezoelectric coupling (such as GaAs), the properties of surface acoustic waves are ∂ ∂ϕ ux0 primarily determined by the elastic constants and density of 0 ϵ0kϕ e14 − ϵ : B22 ¼ þ ∂ 0 ∂ ð Þ the medium. Then, within a perturbative treatment of the x z z¼0
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0.7 We show that they agree very well with the simple estimate given in the main text. Finally, we provide details on the 0.6 material parameters used to obtain the numerical estimates. 0.5 Our first approach follows closely the one presented in 0.4 Ref. [32]. The analysis starts out from the mechanical z displacement operator in the Heisenberg picture: 0.3 X − ω 0.2 ˆ i nt uðx;tÞ¼ ½vnðxÞane þ H:c: : ðC1Þ 0.1 n 0.0 To obtain the proper normalization of the displacement 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 profiles, we assume a single-phonon Fock state,Q that is, z Ψ † 0 … 0 1 0 … 0 j i¼anjvaci¼j ; ; ; n; . i, where jvaci¼ nj in FIG. 7. The dimensionless function FðzÞ determines the decay is the phonon vacuum, and compute the expectation of the electrical potential away from the surface into the bulk; the value of additional field energy above the vacuum Emech, characterisitc length scale is approximately set by the SAW defined as twice the kinetic energy, since for a mechanical wavelength λ ¼ 2π=k ≈ 1 μm. Inset: Density plot of the (nor- mode half of the energy is kinetic, the other one potential 0 malized) electric potential Re½ϕ =ϕ0 ¼ −FðkzÞ sin ðkx − ωtÞ [32]. We find 0 along xˆ ∥½110 for a Rayleigh surface acoustic wave propagating Z on a (001) GaAs crystal at t ¼ 0. 2ω2 3rρ r v r v r Emech ¼ n d ð Þ nð Þ · nð ÞðC2Þ
2ρ ω2 v r 2 One can readily check that this is achieved by the form ¼ V n max½j nð Þj ; ðC3Þ proposed in Refs. [26,28], where the last equality defines the effective mode volume ikðx0−vtÞ for mode n. Setting U0 max v r , and assuming the iϕ0FðkzÞe z>0 ¼ ½j nð Þj ϕ ¼ ðB23Þ phonon energy as E ¼ ℏω , we arrive at the general kz ikðx0−ctÞ 0 mech npffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aoute e z< ; result for a phonon mode, U0 ¼ ℏ=2ρVωn; this confirms ϕ ϵ ϕ F 0 the simple estimate given in the main text. where 0 ¼ðe14= ÞU and Aout ¼ i 0 ð Þ. Here, we introduce the dimensionless function FðkzÞ, which deter- mines the length scale on which the electrical potential 1. Explicit example generated by the SAW decays into the bulk. It is given by Next, we provide a calculation based on the exact analytical results derived in Appendixes A and B. In what −αkz −kz FðkzÞ¼2jA1je cos ðβkz þ φ þ ξÞþA3e ; follows, we assume that, in analogy to cavity QED, cavity confinement leads to the quantization kn ¼ nπ=Lc, where ðB24Þ 2 A ¼ Lc is the effective quantization area. In a full 3D −iξ model, A L L , where L is related to the spread of the with A1 ¼jA1je , q ¼ α þ βi, and ¼ x y y transverse mode function as discussed (for example) in γ − 2q Ref. [14]. For simplicity, here we take Lx ¼ Ly. Surface A1 ¼ ;A3 q2 − 1 wave resonators can routinely be designed to show only 2 one resonance k0 [14]. Within this single-mode approxi- − ϵ φ ϵ −iφ ϵ −iφ mation, based on results derived in Appendix A for a SAW ¼ ϵ ϵ f cos þ Re½A1qe þ 0Re½A1e g: þ 0 traveling wave, we take the quantized mechanical displace- ðB25Þ ment describing a SAW standing wave along the axis xˆ0 ¼ð110Þ as For AlxGa1−xAs, we obtain the following parameter values 0 1 ≈ 1 59 −3 1 α ≈ 0 501 0 (compare Ref. [26]): jA1j . , A3 ¼ . , . , χ0ðzÞ cos ðk0x Þ β ≈ 0 472 φ 1 06 ξ −0 33 B C . , ¼ . , and ¼ . . The electric potential 0 † uˆðx ;zÞ¼U0@ 0 A½a þ a : ðC4Þ for this parameter set is shown in Fig. 7. 0 ζ0ðzÞ sin ðk0x Þ
APPENDIX C: MECHANICAL Here, the functions χ0ðzÞ and ζ0ðzÞ describe how the SAW ZERO-POINT FLUCTUATION decays into the bulk, Here, we provide more detailed calculations and esti- −Ω k0z χ0 z 2e r Ω k0z φ ; mates for the mechanical zero-point motion U0 of a SAW. ð Þ¼ cos ð i þ Þ ðC5Þ
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TABLE III. Derived properties for Rayleigh surface waves, for where C is a normalization constant with numerical value both GaAs and diamond. C ≈ 0.45 for GaAs [39]. Therefore, for an effective mode pffiffiffiffi 1 μ −iΘ area of Lc ¼ A ¼ m, we find a single-phonon dis- Decay constant Ω γ ¼jγje φ vs ½m=s δ placement of U0 ≈ 1 fm. This confirms the estimates given GaAs 0.50 þ 0.48i −0.68 þ 1.16i 1.05 2878 1.2 in the main text. Diamond 0.60 þ 0.22i −1.05 þ 0.75i 1.26 11 135 0.44 APPENDIX D: ZERO-POINT ESTIMATES
−Ω In this appendix, we provide details on piezomagnetic ζ 2 γ rk0z Ω φ θ 0ðzÞ¼ j je cos ð ik0z þ þ Þ; ðC6Þ materials and numerical estimates of the zero-point quan- tities for several relevant materials. The results are sum- Ω Ω Ω with material-dependent parameters ¼ r þ i i, marized in Table IV. The underlying input parameters are γ ¼jγj exp ½−iθ , and φ; numerical values are presented given below. in Table III. We note that for GaAs we find Theoretically, piezomagnetic materials with a large mag- ζð0Þ=χð0Þ ≈ 1.33. This is in very good agreement with netostrictive effect are typically described in a 1:1 corre- the numerical values of cx ¼jux=ϕj¼0.98 nm=V and spondence to Eqs. (2) and (3), with the appropriate cz ¼juz=ϕj¼1.31 nm=V as given in Ref. [15]. replacements (using standard notation) E → H, D → B, Normalization of the mode function allows us to determine ϵij → μij, and eijk → hijk [33]. Coupling between mechani- the parameter U0. Performing the integration, we find cal and magnetic degrees of freedom is described by the rffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffi piezomagnetic tensor h, which can reach values as high as ∼700 T=strain [36]; for our estimates we refer to terfenol-D, Ωr ℏ U0 ¼ ; ðC7Þ where h15 ≈ 167 T=strain. The magnetic field associated δ 2ρvsA with a single phonon can then be estimated as B0 ≈ h15s0, where h15 refers to a typical (nonzero) element of h. where the parameter δ depends on the material parameters; For the piezoelectric materials GaAs, LiNbO3, and see Table III. Using typicalp materialffiffiffiffiffiffiffiffiffiffi parameters, we quartz, all material parameters are obtained from Ω δ 0 64 1 17 obtain for GaAs (diamond)pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r= ¼ . ð . Þ and Ref. [27]. Phase velocities for typical cut directions are 2 U0 ≈ 1.2ð1.36Þ fm= A ½μm . This is in very good agree- used; that is, (100)[001] GaAs, Y-Z LiNbO3, and ST ment with the numerical values presented in the main text. quartz. For the piezomagnetic (magnetostrictive) materials CoFe2O4 and terfenol-D, all material parameters are taken from Ref. [35]. We use the phase velocities of the bulk 2. Estimates derived from literature 3 02 103 shear waves given there as vsh ¼ . × m=s and vsh ¼ In Ref. [26], it is shown that the SAW Rayleigh mode 3 1.19 × 10 m=s for CoFe2O4 and terfenol-D, respectively. studied in Appendix A has a classical energy density E This gives a conservative estimate for U0, since Rayleigh (energy per unit surface area) given by modes have phase velocities that are lower than the ones of bulk modes [14]. For example, in the case of CoFe2O4 in E ¼ kU2H; ðC8Þ TABLE IV. Estimates for zero-point fluctuations (mechanical where U is the amplitude of the wave, k the wave vector, amplitude U0, strain s0, electrical potential ϕ0, electric field ξ0 and H a material-dependent factor which is given as H ≈ and magnetic field B0) close to the surface ðd ≪ λÞ for typical 28.2 × 1010 N=m2 for GaAs. By equating the classical piezo-electric and piezo-magnetic (magnetostrictive) materials. energy of the SAW given by EA, where A is the quantiza- Allp valuesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi must be multiplied by the universal scaling factor ℏω 1 μ 2 tion area, with its quantum-mechanical analog Nph (Nph = A ½ m ; thus, they refer to an effective surface mode area 1 μ 2 ϕ ξ is the number of phonons), we estimate the single-phonon of size A ¼ m . Lower (upper) bounds for 0 and 0 comprise minimum (maximum) non-zero element of e with displacement U0 as maximum (minimum) non-zero element of ϵ. We have set rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi k ¼ 2π=μm. Details on cut-directions and material parameters ℏω ℏ 2 vs −21 m are given in the text. U0 ¼ ¼ ≈ 1.05 × 10 pffiffiffiffi ; ðC9Þ kHA HA A −9 U0 [fm] s0½10 ϕ0 ½μV ξ0 [V/m] B0 ½μT pffiffiffiffiffiffiffiffi with U ¼ U0 Nph. This estimate is also found to be in GaAs 1.9 11.7 3.1 19.2 very good agreement with a result given in Ref. [39],as LiNbO3 1.8 11.3 0.9–25.8 5.8–162.2 Quartz 2.75 17.3 2.8–12.0 17.3–75.4 sffiffiffiffiffiffiffiffiffiffi Terfenol-D 2.2 13.8 2.3 2ℏ 2 −21 m CoFe2O4 1.8 11.4 6.3 U0 ¼ C ≈ 1.7 × 10 pffiffiffiffi ; ðC10Þ ρvsA A Diamond 1.17 7.4
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Ref. [36], wave velocities for Rayleigh-type surface waves δf=fc ≈ 2jrsj=π. In practice, the cavity formed by in a piezoelectric-piezomagnetic layered half-space are two reflective gratings can be viewed as an acoustic ≈ 2840 found to be vs m=s
−1 −1 −1 −1 0.4 Q ¼ Qm þ Qbk þ Qr : ðE3Þ
1 2 3 4 5 APPENDIX F: CHARGE QUBIT COUPLED TO SAW CAVITY MODE FIG. 8. Total reflection coefficient jRj as a function of the normalized groove depth h=λc for N ¼ 100 (blue dashed curve) We consider a GaAs charge qubit embedded in a tunnel- and N ¼ 300 (red solid curve). Here, w=p ¼ 0.5, and material coupled double quantum dot containing a single electron. parameters for LiNbO3 are used (see text). In the one-electron regime, the single-particle orbital level
031031-17 M. J. A. SCHUETZ et al. PHYS. REV. X 5, 031031 (2015) R ∼1 xϕ x ψ x ψ x 0 spacing is on the order of meV. Therefore, the system is d ð Þ Lð Þ Rð Þ¼ is satisfied. As shown below, well described by an effective two-level system: The state for a mode function ϕðxÞ of sine form, this condition of the qubit is set by the position of the electron in the maximizes the piezoelectric coupling strength between double-well potential, with the logical basis jLi; jRi cor- the electronic DQD system and the phonon mode. For responding to the electron localized in the left (right) the charge qubit system under consideration, coupling † orbital. The Hamiltonian describing this system reads to the cavity mode is then described by Hint ¼ eða þ a Þ ϕ ϕ ϵ ½ ðxLÞjLihLjþ ðxRÞjRihRj ; here, xi refers to the center σz σx of the electronic orbital wave function φ x of dot i Hch ¼ 2 þ tc ; ðF1Þ ið Þ ¼ L; R and the transverse direction yˆ has been integrated out with the (orbital) Pauli operators defined as σz ¼jLihLj − already. To obtain strong coupling between the qubit and ϕ φ jRihRj and σx ¼jLihRjþjRihLj, respectively. In Eq. (5), the cavity, we assume a mode profile ðxÞ¼ 0 sin ðkxÞ, with a node tuned between the two dots, such that ϵ refers to the level detuning between the dots, while tc ϕðx Þ¼φ0 sin ðkl=2Þ¼−ϕðx Þ; here, l gives the distance gives the tunnel coupling. The levelp splittingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi between the L R Ω ϵ2 4 2 between the two dots. Note that the single phonon eigenstates of Hch is given by ¼ þ tc, with a pure amplitude, defined as φ0 ¼ ϕ0FðkdÞ, with FðkdÞ ≈ 0 tunnel splitting of Ω ¼ 2t at the charge degeneracy point c for d ≫ λ, accounts for the decay of the SAW into the ðϵ ¼ 0Þ; typical parameter values are t ∼ μeV and c bulk. For λ ¼ð0.5–1Þ μm and a 2DEG (where the DQD is ϵ ∼ μeV, such that the level splitting Ω ∼ GHz lies in the embedded) situated a distance d ¼ 50 nm below the sur- microwave regime. At the charge degeneracy point, where face, however, the single-phonon amplitude is reduced by a to first order the qubit is insensitive to charge fluctuations factor of ∼2 only, φ0 ≈ ð0.45–0.52Þϕ0; see Appendix B for ðdΩ=dϵ ¼ 0Þ, the coherence time has been found to be details. Then, the coupling between qubit and cavity reads T2 ≈ 10 ns [46]. We now consider a charge qubit as described above † H ¼ g ða þ a Þ ⊗ σz; ðF3Þ inside a SAW resonator with a single resonance frequency int ch ω ’ ω ≈ Ω c close to the qubit s transition frequency, c , that is where the single-phonon coupling strength is the regime of small detuning δ ¼ Ω − ωc ≈ 0; note that single-resonance SAW cavities can be realized routinely 1.5 μeV ’ g ¼ eϕ0FðkdÞ sin ðkl=2Þ ≈ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðF4Þ with today s standard techniques [14]. Within this single- ch μ 2 mode approximation, the Hamiltonian describing the SAW A ½ m cavity simply reads Here, we assume λ ≈ 2l, such that the geometrical factor † sin ðπl=λÞ ≈ 1 [47]. In principle, the coupling strength g H ¼ ω a a; ðF2Þ ch cav c could be further enhanced by additionally depositing a where a†ðaÞ creates (annihilates) a phonon inside the strongly piezoelectric material such as LiNbO3 on the cavity. The electrostatic potential associated with this GaAs substrate. Moreover, comparison with standard literature shows that the piezoelectric electron-phonon ϕˆ x ϕ x † pffiffiffiffi mode is given by ð Þ¼ ð Þ½a þ a , where the mode ≈ function ϕðxÞ can be obtained from the corresponding coupling strength can be expressed as gpe ¼ PU0 ϵ ϕ ϵ 2 mechanical mode function wðxÞ via the relation eðe14= ÞU0 ¼ e 0, where P ¼ðee14= Þ is a material ϵΔϕðxÞ¼e ∂ ∂ w ðxÞ; here, Δ is the Laplacian, e parameter quantifying the piezoelectric coupling strength kij j k i kij 5 4 10−20 the piezoelectric tensor, and ϵ the permittivity of the in zinc-blende structures [87,88]. Using P ¼ . × material. The electron’s charge e couples to the phonon- for GaAs, the single-phononpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rabi frequency can be ϕˆ estimated as g ≈ 2.87 μeV= A μm2 . This corroborates induced electrical potential . In second quantization,R the pe ½ xϕˆ x ˆ x our estimate for g . piezoelectric interaction reads Hint ¼ ePd ð Þnð Þ, ch † In summary, the total system can be described by the where e is the electron’s charge, nˆðxÞ¼ σψ σðxÞψ σðxÞ † Hamiltonian H ¼ Hch þ Hcav þ Hint: is the electron number density operator, and ψ σðxÞ creates ˆ an electron with spin σ at position x [25]. Since ϕðxÞ varies ϵ † † H ¼ σz þ t σx þ ω a a þ g ða þ a Þ ⊗ σz: ðF5Þ on a micron length scale which is large compared to the 2 c c ch spatial extension ∼40 nm of the electron’s wave function in a QD [65], the electron density is approximately given This corresponds to the generic Hamiltonian for a qubit- by a delta function at the center of the corresponding dots. resonator system [32]. It is instructive to rewrite H in the eigenbasis of H , given by For the DQD system under consideration,P Hint is then ch approximately given by H e ϕˆ x n ; here, x refers int ¼ i ð iÞ i i θ θ to the center of the electronic orbital wave function jþi ¼ sin jLiþcos jRi; ðF6Þ ψ ðxÞ of dot i ¼ L; R. Note that this form of H becomes i int − θ − θ exact if the overlap integral vanishes, that is, if j i¼cos jLi sin jRi; ðF7Þ
031031-18 UNIVERSAL QUANTUM TRANSDUCERS BASED ON … PHYS. REV. X 5, 031031 (2015) where the mixing angle θ is defined via parameter ϵ ¼ ϵL − ϵR þ ULR − U. In this regime, the θ 2 ϵ Ω ⇑⇑ tan ¼ tc=ð þ Þ. In a rotating-wave approximation relevant electronic levels are defined as jffiffiffiTþi¼j i, δ ≪ ω p ð ;gch cÞ, H then reduces to the well-known T− ⇓⇓ , T0 ⇑⇓ ⇓⇑ = 2, S11 j i¼j i pjffiffiffi i¼ðj iþj iÞ j i¼ Hamiltonian of Jaynes-Cummings form ⇑⇓ − ⇓⇑ 2 † † 0 ðj i j iÞ= , and jS02i¼dR↑dR↓j i, with σσ0 † † 0 2t j i¼dLσdRσ0 j i. H ≈ δSz þ g c ðSþa þ S−a†Þ; ðF8Þ ch Ω Next, Ht describes coherent, spin-preserving interdot tunneling, z where δ ¼ Ω − ωc, S ¼ ðjþihþj − j−ih−jÞ=2, and ∓ X S ¼j ih j. † Ht ¼ tc dLσdRσ þ H:c:; ðG4Þ σ APPENDIX G: SAW-BASED CAVITY QED WITH SPIN QUBITS IN DOUBLE QUANTUM DOTS where tc is the interdot tunneling amplitude. Lastly, HZ In this appendix, we show in detail how to realize the accounts for the Zeeman energies, prototypical Jaynes-Cummings dynamics based on a spin qubit encoded in a DQD inside a SAW resonator. X We consider a double quantum dot (in the two-electron HZ ¼ gμB Bi · Si; ðG5Þ regime) coupled to the electrostatic potential generated by a i¼L;R SAW. The system is described by the Hamiltonian
where g is the electron g factor and μB the Bohr magneton. H ¼ H0 þ H þ H ; ðG1Þ DQD cav int In the presence of a micromagnet or nanomagnet, the two local magnetic fields Bi are inhomogeneous, BL ≠ BR.We where H0, Hcav, and Hint describe the DQD, the cavity, can then write B ¼ B0 þ B ðx Þ, where B0 is the external and the electrostatically mediated coupling between them, i m i homogeneous magnetic field, while B ðx Þ is the micro- respectively. In the following, we discuss the different m i magnet slanting field at the location of dot x . In practice, contributions in detail. i B0 is a few tesla, at least larger than the saturation field of the micromagnet B0 ≳ 0.5 T, while the magnetic gradient 1. Double quantum dot Δ ∥B x − B x ∥ Δ ≈ 100 B ¼ mð RÞ mð LÞ can reach B mT, cor- The DQD is modeled by the standard Hamiltonian responding to an electronic energy scale of jgμBΔBj ≈ 2 μeV [89]. Field derivatives realized experimentally are ∂ ∂ ≈ 1 5 H0 ¼ HC þ Ht þ HZ: ðG2Þ Bm;z= x . mT=nm. Alternatively, the magnetic gra- dient can be realized via the Overhauser field, as exper- Here, HC gives the electrostatic energy imentally demonstrated, for example, in Ref. [67]. X X Note that the Fermi contact hyperfine interactionP H ¼ ϵ n σ þ U n ↑n ↓ þ U n n ; ðG3Þ h S C i i i i LR L R between electron and nuclear spins reads Hhf ¼ i i · i. i;σ i¼L;R Here, hi is the Overhauser field in QD i ¼ L; R. When treating h as a classical (random) variable, H is equiv- where (due to strong confinement) both the left and right i hf alent to H , and thus one can absorb h into the definition dot are assumed to support a single orbital level with energy Z i of the magnetic field B in Eq. (G5); also see Ref. [89]. ϵ ði ¼ L; RÞ only; U and U refer to the on-site and i i LR To facilitate the discussion, we introduce the magnetic interdot Coulomb repulsion, respectively. As usual, niσ ¼ B B B 2 † sum field ¼ð L þ RÞ= and the difference field d d σ and n n ↑ n ↓ refer to the spin-resolved and iσ i i ¼ i þ i ΔB ¼ðBR − BLÞ=2. While B conserves the total spin, total electron number operators, respectively, with the that is, ½BðS þ S Þ; ðS þ S Þ2 ¼0, the gradient field † L R L R fermionic creation (annihilation) operators diσðdiσÞ creat- ΔB does not. We set the quantization axis zˆ along B ¼ Bzˆ. ing (annihilating) an electron with spin σ ¼ ↑; ↓ in the For sufficiently large magnetic field B, the electronic levels z z z ≠ 0 orbital i ¼ L; R. We focus on a setting where an applied with Stot ¼ SL þ SR are far detuned and can be bias between the two dots approximately compensates neglected for the remainder of the discussion. Therefore, z 0 the Coulomb energy of two electrons occupying the right in the following, we restrict ourselves to the Stot ¼ x;y dot; that is, ϵL ≈ ϵR þ U − ULR. Thus, we restrict ourselves subspace. The components ΔB give rise to transitions z 0 to a region in the stability diagram where only (1,1) and out of the (logical) subspace Stot ¼ . Since these processes (0,2) charge states are of interest. All levels with (1,1) are assumed to be far off resonance, they are neglected, charge configuration have an electrostatic energy of leaving us with the only relevant magnetic gradient ϵ ϵ z Eð1;1Þ ¼ L þ R þ ULR, while the (0,2) configuration Δ ¼ gμBΔB =2; compare also Refs. [66,67]. For a sche- 2ϵ has Eð0;2Þ ¼ R þ U. As usual, we introduce the detuning matic illustration, compare Fig. 9.
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transition frequency. Note that the magnetic gradient causes 2 efficient mixing between jT0i and jS11i for Δ ≳ jtc=ϵj.In the regime of interest, the dominant character of the qubit’s levels is j1i ≈ j⇓⇑i, j0i ≈ j⇑⇓i (or vice versa) [66] and the transition frequency is approximately ω0 ≈ 2Δ.For Δ ≈ 1 μeV, the transition frequency ω0 ¼ ϵ1 − ϵ0 ≈ 2 μeV ≈ 3 GHz matches typical SAW frequencies ∼GHz.
2. Coupling to SAW phonon mode FIG. 9. Illustration of the relevant electronic levels under z ≠ 1 consideration. The triplet levels with Stot can be tuned off Along the lines of Appendix F, again we consider a SAW resonance by applying a sufficiently large homogeneous mag- resonator with a single relevant confined phonon mode of netic field. frequency ωc close to the qubit’s transition frequency ω0. For a DQD in the two-electron regime, in the basis of In summary, in the regime of interest H0 simplifies to Eq. (G6) coupling to the resoantor mode can be written as [47] tc H0 ¼ ðjS02ihS11jþH:c:Þ − ϵjS02ihS02j 2 † ⊗ Hint ¼ g0½a þ a jS02ihS02j; ðG9Þ − ΔðjT0ihS11jþH:c:Þ: ðG6Þ φ η η z z z where g0 ¼ e 0 geo. Here, geo is a geometrical factor The eigenstates of H0 within the relevant Stot ¼ SL þ SR ¼ ’ 0 accounting for the DQD s position with respect to the subspace can be expressed as ϕ x φ η mode function ð Þ, defined according to 0 geo ¼ ϕðx Þ − ϕðx Þ. For example, taking a standing-wave jli¼α jT0iþβ jS11iþκ jS02i; ðG7Þ R L l l l pattern along xˆ as demonstrated experimentally in with corresponding eigenenergies ϵ ðl ¼ 0; 1; 2Þ. The spec- Ref. [90], together with a transverse mode function l ˆ trum is displayed in Fig. 10. For large negative detuning restricting the spread in the y direction [40,43], we obtain η 2π λ − 2π λ −ϵ ≫ t , the level j2i is far detuned, and the electronic geo ¼ sin ð xR= Þ sin ð xL= Þ. It takes on its maxi- c η subsystem can be simplified to an effective two-level mum value opt when tuning a node of the standing wave at system comprising the levels fj0i; j1ig, that is, the center between the two dots, that is, xR ¼ l=2, − 2 η 2 π λ xL ¼ l= ; this gives opt ¼ sin ð l= Þ, where l is the ω0 H0 ≈ ðj1ih1j − j0ih0jÞ; ðG8Þ distance between the two dots [47]. As compared to the 2 charge qubit described in Appendix F, there is an additional factor of 2, since here we consider a DQD in the two- which can be identified with a “singlet-triplet-like” logical electron regime, whereas the charge qubit consists of qubit subspace. Here, ω0 ¼ ϵ1 − ϵ0 refers to the qubit’s one electron only. For typical parameters (l ¼ 220 nm, λ ≈ 1 4 μ η ≈ 0 95 . m) as used in Ref. [47], we get opt . , while 6 l ¼ 220 nm, λ ≈ 0.5 μm leads to the largest possible value η ≈ 2 4 of opt . In summary, within the effective electronic two-level 2 subspace fj0i; j1ig, the system is described by the 0 Hamiltonian 2 X ϵ κ2 ˆ κ κ ˆ 0 1 HDQD ¼ ð l þ VpeÞjlihljþ 0 1Vpeðj ih jþH:c:Þ 4 l l¼0;1 6 † þ ωca a; ðG10Þ 10 5 0 5 10 ˆ † where Vpe ¼ g0½a þ a . Applying a unitary transformation to a frame rotating at the cavity frequency ωc according 5Δ † † † FIG. 10. Spectrum of H0 for tc ¼ . The three eigenstates jli to H~ UH U iUU_ , with U exp iω t a a 2 1 DQD ¼ DQD þ ¼ ½ c ð þ are displayed in green dotted ðl ¼ Þ, red solid ðl ¼ Þ, and blue 1 1 1 − 1 0 0 0 2 j ih j 2 j ih jÞ , performing a RWA, and dropping a dashed ðl ¼ Þ lines, respectively. The triplets jT i are assumed to be far detuned by a large external field and are not shown. For global energy shift ϵ~ ¼ðϵ0 þ ϵ1Þ=2, we arrive at the large negative detuning ϵ > −tc, the hybridized levels fj0i; j1ig effective (time-independent) Hamiltonian of Jaynes- can be used as qubit. Cummings form,
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~ δ¯ z þ − † 3.0 HDQD ¼ S þ gQD½S a þ S a ; ðG11Þ 7 where we introduce the spin operators Sþ ¼j1ih0j and 2.5 2 z ¯ 2 6 S ¼ðj1ih1j − j0ih0jÞ=2. Moreover, δ ¼ ω0 − ωc is the
2.0 10 ’ ω 10 2 detuning between the qubit s transition frequency 0 and 0
1 5 the cavity frequency ω , and the effective single-phonon 2 1 c 0 1.5 Rabi frequency is defined as 4 1.0 κ κ η ϕ F ≈ 2κ κ gQD ¼ 0 1 geoe 0 ðkdÞ 0 1gch: ðG12Þ 3 10 9 8 7 6 5 The coupling between the qubit and the cavity mode is detuning mediated by the piezoelectric potential; therefore, it is ’ proportional to the electron s charge e and the single- FIG. 11. The product κ0κ1 directly affects the effective single- ϕ κ κ phonon electric potential 0. Because of the prolonged phonon Rabi frequency gQD=g0 ¼ 0 1 [69], while the difference 2 2 decoherence time scales, here we consider an effective jκ1 − κ0j determines the robustness of the qubit against charge (singlet-triplet-like) spin qubit rather than a charge noise. Here, tc ¼ 5Δ. qubit, such that the coupling gQD is reduced by the (small) admixtures with the localized singlet κl ¼hS02jli. Increasing κ0κ1 leads to a stronger Rabi frequency 2 ⋆ g T Q=ω ≈ 3.8 (compare Table V). Note that C>1 g , but an increased difference in charge configuration QD 2 c QD allows us to perform a quantum gate between two spins jκ2 − κ2j makes the qubit more susceptible to charge noise. 1 0 mediated by a thermal mechanical mode [9]. For typical numbers ðtc ≈5 μeV;ϵ≈−7 μeV;Δ≈1 μeVÞ, −2 2 2 −2 we get κ0κ1 ≈ 5 × 10 , jκ1 − κ0j ≈ 2 × 10 ; see Fig. 11.Forl ≈ 250 nm, λ ≈ 0.5 μm, and d ≈ 50 nm, we 4. Discussion of approximations can then estimate To arrive at the effective Hamiltonian given in Eq. (G11), we make two essential approximations: (i) first, we neglect 200 ℏ ≈ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMHz the electronic level j2i yielding an effective two-level gQD= : ðG13Þ A ½μm2 system, and (ii) second, we apply a RWA leading to a major simplification of HDQD; see Eq. (G11) as compared We take this coupling strength as a conservative estimate, to Eq. (G10). In order to corroborate these approximations, since optimization against the relevant noise sources as we now compare exact numerical simulations of the full done in Ref. [69] yields an optimal point with κ0κ1 ≈ 0.3 system where none of the approximations have been and ω0=2π ≈ 1.5 GHz. Resonance ðδ ¼ 0Þ yields a SAW applied to the simplified, approximate description wavelength of λ ≈ 2 μm; accordingly, for a fixed dot-to- described above. While the dynamics of the former is 250 η ≈ 0 76 dot distance l ¼ nm, geo . (whereas a larger described by the master equation, DQD size l ¼ 400 nm, as used in Ref. [25],gives η ≈ 1 18 250 50 ρ_ − ρ κD ρ geo . ). For l ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinm, d ¼ nm, we then obtain ¼ i½H0 þ Hcav þ Hint; þ ½a ; ðG14Þ ℏ ≈ 600 μ 2 gQD= MHz= A ½ m , which is a factor of 3 with H0, H , and H given in Eqs. (G6), (F2), and (G9), larger than the estimate quoted above. cav int respectively, the latter is described by a similar master
3. Cooperativity TABLE V. Estimates of the single-spin cooperativity C for a In this context, an important figure of merit is the single- ⋆ ≈ 100 2 DQD singlet-triplet qubit with T2 ns, in a SAW cavity at spin cooperativity [61], C ¼ g T2=κðn¯ þ 1Þ, where QD th gigahertz frequencies ωc=2π ≈ 1.5 GHz and cryostat temper- κ ω ¯ ¯ ≈ 0 ¼ c=Q is the mechanical damping rate and nth ¼ atures where nth for both a small, low-Q and a large, high-Q ℏω 1=ðe c=kBT − 1Þ is the equilibrium phonon occupation SAW resonator. The coupling strength gQD could be further increased by additionally depositing a strongly piezoelectric number at temperature T; here, since ℏωc ≫ kBT for ¯ ≈ 0 material such as LiNbO3 on the GaAs substrate, and spin-echo cryostatic temperatures, nth . For singlet-triplet qubits in lateral QDs, T⋆ ≈ 100 ns [64]; using spin-echo tech- (and/or narrowing) techniques allow for dephasing times ex- 2 tended by up to 3 orders of magnitude [21,64]. niques, experimentally this has even been extended to 276 μ 2 T2 ¼ s. Even in the absence of spin-echo pulses, Cavity size A ½μm gQD [MHz] QC ⋆ with a far-from-optimistic dephasing time T2 ≈ 100 ns ≈ 100 μ 2 Small cavity 1 200 10 4.25 [21], for a moderately small cavity size A m ,a Large cavity 500 9 104 8.5 quality factor of Q ¼ 900 is sufficient to reach C ≈
031031-21 M. J. A. SCHUETZ et al. PHYS. REV. X 5, 031031 (2015) equation with the coherent Hamiltonian term replaced by 5. Noise sources for the DQD-based system the prototypical Jaynes-Cummings Hamiltonian, a. Charge noise In a DQD device background charge fluctuations and noise in the gate voltages may cause undesired dephasing ρ_ −i δ¯Sz g Sþa S−a† ; ρ κD a ρ; ¼ ½ þ QDð þ Þ þ ½ processes. In a recent experimental study [91], voltage ðG15Þ fluctuations in the interdot detuning parameter ϵ have been identified as the main source of charge noise in a singlet- triplet qubit. Charge noise can be treated by introducing a with ρ referring to the density matrix of the combined Gaussian distribution in ϵ, with a variance σϵ; typically, system comprising the DQD and the cavity mode. Here, we σϵ ≈ ð1–3Þ μeV [89]. The qubit’s transition frequency ω0, also account for decay of cavity phonons out of the however, turns out to be rather insensitive to fluctuations resonator with a rate κ, described by the Lindblad term in ϵ, with a (tunable) sensitivity of approximately −2 D½a ρ ¼ aρa† − 1 fa†a; ρg. As a figure of merit to validate ∂ω0=∂ϵ ≲ 10 ; see Fig. 13. In agreement with experi- 2 ∂ω ∂ϵ ∼ ω approximation (i) we determine the population of the level mental results presented in Ref. [91], we find 0= 0, ω ϵ j2i, that is, Tr½ρj2ih2j , describing the undesired leakage indicating 0 to be an exponential function of . At very negative detuning ϵ, dephasing due to charge noise is out of the logical subspace; ideally, this should be zero. ⋆ practically absent, and T2 will be limited by nuclear noise Note that leakage into the triplet levels jT i could be accounted for along the lines, but they can be tuned far off [91]. Fluctuations in the tunneling amplitude tc can be ω resonance by another, independent experimental knob, the treated along the lines: we find 0 to be similarly ∂ω ∂ ≈ 10−2 external homogeneous magnetic field. The results are insensitive to noise in tc, 0= tc . summarized in Fig. 12: We find very good agreement between the exact and the approximate model, with a b. Nuclear noise: Spin echo ρ 2 2 ∼ O 10−5 negligibly small error Tr½ j ih j ð Þ. This justifies The electronic qubit introduced above has been defined the approximations made above and shows that (in the for a fixed set of parameters ðtc; ϵ; ΔÞ; compare Eq. (G6). regime of interest) the system can simply be described Now, let us consider the effect of deviations from these by Eq. (G15). fixed parameters, H0 → H0 þ δH, where δH can be δ δ δ decomposed as H ¼ Hel þ Hnuc, with
δ δ tc − δϵ (a) (b) Hel ¼ 2 ðjS02ihS11jþH:c:Þ jS02ihS02j; ðG16Þ
δ −δΔ Hnuc ¼ ðjT0ihS11jþH:c:Þ; ðG17Þ
2.10 2.09 (b) 2.08 (c) 2.07 2.06 2.05 2.04 (a)
10 9 8 7 6 5 FIG. 12. Exact numerical simulations of the full (blue, solid detuning line) and approximate (cyan circles) master equations as given in Eqs. (G14) and (G15), respectively. Plots are shown for (a) the FIG. 13. Transition frequency ω0 (blue solid line) and its electronic inversion Sz , (b) the cavity occupation n , and h it h it sensitivity against charge-noise-induced fluctuations in ϵ for (c) the error Tr½ρj2ih2j quantifying the leakage to j2i. The latter −5 ¯ (a) intermediate and (b) large negative detuning; for large is found to be negligibly small ∼10 . We set δ ¼ ω0 − ω ¼ 0. c ω0 ≈ 2Δ ∂ω0=∂ϵ 10 μ ϵ −7 μ Δ 1 μ negative detuning, , and the sensitivity practi- Numerical parameters are tc ¼ eV, ¼ eV, ¼ eV, cally vanishes, leaving nuclear noise as the dominant dephasing η φ 5 2 10−2 μ 4 10−3 μ ≈ geoe 0 ¼ . × eV, such that gQD ¼ × eV process. By occasional refocusing of the spin states, this regime 6 κ 2 MHz. The cavity decay rate is ¼ gQD= , corresponding can be used for long-term storage of quantum information [69]. ≈ 103 to Q . Other numerical parameters: tc ¼ 5Δ.
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δ δϵ g where tc and can be tuned electrostatically and H ¼ QD ½Sþa þ S−a† : ðG20Þ basically in situ. In most practical situations this does eff 2 not hold for δΔ: The primary source of decoherence in this system has been found to come from (slow) fluctua- Thus, in order to cancel the noise term, the effective single- phonon coupling strength is lowered by only a factor tions in the Overhauser field generated by the nuclear spins of 1=2. [21,66,67]. In our model, this can be directly identified with a random, slowly time-dependent parameter δΔ ¼ δΔðtÞ. 0 1 δ 6. Different spin-resonator coupling In the relevantP subspace fj i; j ig, Hnuc is given by δ −δΔ α β δΔ ≈ Hnuc ¼ k;l k l½jkihljþH:c: . Typically, In Appendix G, we show how to realize the prototypical 0 1 μ ≪ ω Jaynes-Cummings Hamiltonian for SAW phonons inter- . eV c is fulfilled, suchP that we can apply a δ ≈ −2δΔ α β acting with a DQD; see Eq. (G11). Alternatively, if one RWA yielding Hnuc l l ljlihlj; physically, δ does not absorb the gradient Δ into the definition of the Hnuc is too weak to drive transitions between the elec- tronic levels j0i and j1i, which are energetically separated qubit basis, one can identify the logical subspace with the by ω0 ≈ ω . Then, in the spin basis used in Eq. (G11), electronic states jT0i and jSi, where jSi is one of the two c 0 we find hybridized singlets (while the other one jS i is far detuned and neglected) [21,66,67,91]. Here, the electronic − ϵ −Δ~ δ δ z Hamiltonian reads H0 ¼ Jð ÞjSihSj ðjT0ihSjþH:c:Þ, Hnuc ¼ ðtÞS ; ðG18Þ ~ where Δ ¼hS11jSiΔ and JðϵÞ describes the exchange interaction. In this regime, the spin-resonator interaction where the gradient noise is given by δðtÞ¼ takes on a form that is well known from other (localized) 2δΔ t α0β0 − α1β1 .For t ; ϵ; Δ ≈ 5; −7; 1 μeV, ð Þð Þ ð c Þ ð Þ implementations of mechanical resonators [9], namely, α0β0 − α1β1 ≈ 0.9. Therefore, when also accounting for nuclear noise as described by Eq. (G18), the full model H ¼ g ða þ a†Þ ⊗ jSihSj; ðG21Þ [compare Eq. (G11)] reads int QD which can be viewed as a phonon-state-dependent force, ~ δ¯ δ z þ − † leading to a shift of the qubit’s transition frequency HDQD ¼½ þ ðtÞ S þ gQD½S a þ S a : ðG19Þ pffiffiffi depending on the position xˆ ¼ða þ a†Þ= 2. Here, the κ2η ϕ F Since the nuclear spins evolve on time scales much longer single-phonon Rabi frequency is gQD ¼ S geoe 0 ðkdÞ, −1 −1 κ than all other relevant time scales, ∼κ ;g [65], the with S ¼hS02jSi. Based on the coupling of Eq. (G21), one Overhauser noise term can be approximated as quasistatic; can envisage a variety of experiments known from quantum that is, δðtÞ¼δ. As experimentally demonstrated in optics: For example, in the limit of vanishing gradient Δ 0 ˆ Ref. [67], the slow (nuclear) noise term ∼δðtÞSz can ¼ , the x quadrature of the phonon mode could serve as be neutralized by Hahn-echo techniques. Here, the dephas- a quantum nondemolition variable, as it is an integral of ing time of the electron spin qubit is extended by more motion of the coupled system of the phonon mode and ⋆ electronic meter. than 3 orders of magnitude from T2 ≈ ð10–100Þ ns to T2 ¼ 276 μs. In the following, assuming nominal resonance δ¯ ¼ 0,we APPENDIX H: GENERALIZED DEFINITION detail a sequence of Hahn-echo pulses that cancels the OF THE COOPERATIVITY PARAMETER undesired noise term and restores the pure, resonant Here, we provide a generalized discussion of the Jaynes-Cummings dynamics: We consider four short time cooperativity parameter C, which, in particular, accounts intervals of length τ, for which the unitary evolution is for losses of the cavity mode other than leakage through the approximately given by Ui ≈ 1 − iHiτ, interspersed by nonperfect mirrors. Furthermore, we derive a simple, three π pulses. First, we let the system evolve with analytical estimate for the state transfer fidelity F in terms δ z þ − † π H1 ¼ S þ gQD½S a þ S a , then we apply a pulse of the parameter C and undesired phonon losses with a z z x x y y ∼κ along xˆ ðS → −S ;S → S ;S → −S Þ such that S → rate bd. S∓ and the system evolves in the second time interval with We consider a single qubit fj0i; j1ig coupled to a cavity −δ z − þ † π mode. The system is described by the master equation H2 ¼ S þ gQD½S a þ S a . Next, we apply a pulse ˆ z → z x → − x y → − y → along z ðS S ;S S ;S S Þ such that S ϱ_ κ κ D ϱ − ϱ Γ D 1 1 ϱ − −δ z − − þ † ¼ð gd þ bdÞ ½a i½HJC; þ deph ½j ih j ; S leading to H3 ¼ S gQD½S a þ S a . Finally, |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} z z x x y y Vϱ a π pulse along yˆ ðS → −S ;S → −S ;S → S Þ is L0ϱ ∓ z applied such that S → −S , giving H4 ¼ δS þ ðH1Þ þ − † gQD½S a þ S a . In summary, the system evolves over 4τ ≈ D ϱ ϱ † − 1 † ϱ a timeP interval of according to Ueff ¼ U4U3U2U1 where ½a ¼ a a 2 fa a; g. The first term describes 1 − τ 1 − 4τ i iHi ¼ iHeff with the effective Hamiltonian decay of the cavity mode. The corresponding decay rate
031031-23 M. J. A. SCHUETZ et al. PHYS. REV. X 5, 031031 (2015) can be decomposed into desired (leakage through the Here, we aim for a theoretical description that singles out mirrors) and undesired (bulk-mode conversion, material the desired trajectories, where phonon emission through the κ κ imperfections, etc.) contributions, labeled as gd and bd, mirrors happens first, from all others. To do so, we rewrite κ κ κ respectively. Thus, we write ¼ gd þ bd. The second Eq. (H2) as þ − † term with (on-resonance) HJC ¼ gðS a þ S a Þ refers to the coherent interaction between qubit and cavity mode, ρ_ − ρ ρ † J ρ J ρ ¼ iH þ i H þ gd þ bd ; ðH4Þ while the last term describes pure dephasing of the qubit with a rate Γ . deph where we define an effective (non-Hermitian) Hamiltonian In the bad cavity limit (where κ ≫ g; Γ ), one can deph H and jump operators according to adiabatically eliminate the cavity mode by projecting the system onto the cavity vacuum, Pϱ ¼ Tr ½ϱ ⊗ ρss ¼ cav cav i i ρ ⊗ jvacihvacj. Standard techniques (perturbation theory − γ 1 1 − κ~ Γ 1 1 H ¼ 2 effj ih j¼ 2 ð þ dephÞj ih j; ðH5Þ up to second order in V, compare Ref. [92]) then yield the effective master equation for the qubit’s density matrix ρ Pϱ J ρ κ~ −ρ þ ¼ Trcav½ only, gd ¼ gdS S ; ðH6Þ
ρ_ κ~D − ρ Γ D 1 1 ρ ¼ ½S þ deph ½j ih j ; ðH2Þ J ρ κ~ −ρ þ Γ 1 1 ρ 1 1 bd ¼ bdS S þ dephj ih j j ih j: ðH7Þ
2 with the effective decay rate κ~ ¼ 4g =κ. Here, we decompose the effective decay rate κ~ as For comparison, the same procedure in standard cavity QED, where κ ¼ κ and Γ D½j1ih1j ρ → γD½S− ρ, gd deph 4 2 4 2 yields the effective master equation for the atom only, κ~ κ~ κ~ g κ g κ ¼ gd þ bd ¼ 2 gd þ 2 bd: ðH8Þ ρ_ ¼ κ~D½S− ρ þ γD½S− ρ. Therefore, the atom decays with κ κ γ an effective spontaneous emission rate tot enhanced by the γ γ κ~ 1 4 2 κγ γ Purcell factor, tot ¼ þ ¼ð þ g = Þ . Comparing Formally solving Eq. (H4) gives good ∼κ~ to bad ∼γ decay channels, here one defines Z the cooperativity parameter in a straightforward way as t C ¼ g2=κγ. This is readily read as the cavity-to-free- ρðtÞ¼UðtÞρð0Þþ dτUðt − τÞJ ρðτÞ; ðH9Þ atom 0 space scattering ratio, since the effective rate at which an excited atom emits an excitation into the cavity is given by J ρ J ρ J ρ κ~ ∼ 2 κ 1 with the total jump operator ¼ gd þ bd and g = .ForCatom > , the atom is then more likely to decay into the cavity mode rather than into another mode † outside the cavity. In cavity QED, large cooperativity UðtÞρ ¼ e−iHtρeiH t; ðH10Þ ≫ 1 Catom has allowed for a number of key experimental demonstrations, such as an enhancement of spontaneous emission [93], photon blockade [94], and vacuum-induced The exact solution given in Eq. (H9) can be iterated, giving J transparency [95]. an illustrative expansion in terms of the jumps . It reads The master equation given in Eq. (H2) describes a Z two-level system subject to purely dissipative dynamics. t ρðtÞ¼UðtÞρð0Þþ dτ1Uðt − τ1ÞJUðτ1Þρð0Þ The dynamics can be fully described in terms of a set of Z Z 0 three simple rate equations for the populations p ¼ t τ2 k τ τ U − τ JU τ − τ k ρ k k 0; 1 and coherence ρ10 1 ρ 0 , summa- þ d 2 d 1 ðt 2Þ ð 2 1Þ h j j ið ¼ Þ ¼h j j i 0 0 rized as p~ ¼ðp1;p0; ρ10Þ, × JUðτ1Þρð0Þþ : 0 1 −κ~ 00 d B C Here, the nth-order term comprises n jumps J with free p~ ¼ @ þκ~ 00Ap;~ ðH3Þ U dt evolution between the jumps. Now, we can single out the 00−γ =2 desired events where the first quantum jump is governed eff J by gd. This leads to the definition where γ ¼ðκ~ þ Γ Þ. This allows for a simple analyti- eff deph ρ U ρ 0 ρ ρ cal solution: For example, for a system initially in the ðtÞ¼ ðtÞ ð Þþ gdðtÞþ bdðtÞ; ðH11Þ excited state ρðt ¼ 0Þ¼j1ih1j, it reads p1ðtÞ¼exp ½−κ~t , 1 − −κ~ ρ 0 ρ p0 ¼ exp ½ t , and 10ðtÞ¼ . where gdðtÞ subsumes all desired trajectories:
031031-24 UNIVERSAL QUANTUM TRANSDUCERS BASED ON … PHYS. REV. X 5, 031031 (2015) Z t 1. ρ τ U − τ J U τ ρ 0 gdðtÞ¼ d 1 ðt 1Þ gd ð 1Þ ð Þ 0Z Z 0.95 t τ2 þ dτ2 dτ1Uðt − τ2ÞJUðτ2 − τ1Þ 0 0 0.9 suc
J U τ ρ 0 p × gd ð 1Þ ð Þþ : ðH12Þ 0.85 We focus on a qubit, initially in the excited state, 0.8 i.e., ρð0Þ¼j1ih1j. Using the relations Uðτ1Þρð0Þ¼ −γ τ −γ τ eff 1 1 1 J U τ ρ 0 κ~ eff 1 0 0 e j ih j, gd ð 1Þ ð Þ¼ gde j ih j, and 1 2 5 10 20 50 100 J U τ − τ J U τ ρ 0 0 cooperativity C gd ð 2 1Þ gd ð 1Þ ð Þ¼ ; ðH13Þ
J U τ − τ J U τ ρ 0 0 FIG. 14. Success probability psuc for a qubit excitation to bd ð 2 1Þ gd ð 1Þ ð Þ¼ ; ðH14Þ leak through the mirror, as a function of the cooperativity C for ε ¼ κbd=κgd ¼ 0 (black solid line) and ε ¼ 5% (blue the qubit’s density matrix evaluates (to all orders in J )to dashed line).
−γ ρ eff t 1 1 ρ ρ ðtÞ¼e j ih jþ gdðtÞþ bdðtÞ; ðH15Þ κ~ gd −γ For an illustration, compare Fig. 14. This shows that ρ 1 − eff t 0 0 gdðtÞ¼γ ð e Þj ih j: ðH16Þ the usual definition and interpretation of the cooperativity eff C holds, provided that ε ≪ 1 is fulfilled. In order to In the long-time limit t → ∞, the system reaches the steady quantify the cooperativity C for SAW cavity modes in ρ → ∞ ρ ρ ρ κ~ γ 0 0 ∼ n¯ ≫ 1 ∼ n¯ ≪ 1 state, ðt Þ¼ gd þ bd, where gd ¼ð gd= effÞj ih j. both the MHz ð th Þ and the GHz ð th Þ The associated success probability p ¼ Tr½ρ for regime, in the main text we take a (conservative) estimate suc gd ≡ 2 ω ¯ 1 faithful decay through the mirrors is then as C g T2Q=½ cðnth þ Þ . For artificial atoms (quan- tum dots, superconducting qubits, NV centers, etc.) with κ~ ∼ gd 1 resonant frequencies GHz, at cryostatic temperatures 2 p ¼ ¼ 2 ; ðH17Þ suc κ~ κ~ Γ κ 1 κ Γ this definition reduces to C ≈ g T2=κ, as discussed gd þ bd þ deph deph κ þ 4 2κ ω 2π ≈ ω gd g gd above, whereas for a trapped ion with t= c= 2π ∼ ≈ 2 ω ¯ MHz, it correctly gives C g T2Q=ð cnthÞ with which is a simple branching ratio comparing the strength of a decreased effective quality factor Q ¼ Q=n¯ ∼κ~ eff th the desired decay channel gd to the undesired ones [10,96,97]. ∼κ~ Γ κ 0 κ κ bd þ deph. In the limit where bd ¼ , i.e., ¼ gd, the For small errors, the expression given in Eq. (H19) ≈ 1 − ε − 1 4 expression for psuc simplifies to can be approximated as psuc =ð CÞ. Since the absorption process is just the time-reversed copy of the 1 C≫1 p 1; H18 emission process in the state transfer protocol for two suc ¼ 1 1 1 ! ð Þ þ 4 C nodes, we can estimate the state transfer fidelity as F ≥ psuc × psuc. For small infidelities, we then find with the usual definition found in the literature, C ¼ 2 2 g =ðκΓ Þ¼g T2=κ; here, κ ¼ ω =Q ¼ðn¯ þ1Þω =Q 1 deph c eff th c F ≳ 1–2ε − ; ðH20Þ is understood to account for thermal occupation of the 2C ¯ environment nth in terms of a decreased mechanical quality factor (compare, for example, Refs. [10,61]). where the individual errors arise from intrinsic phonon ∼ε ∼ −1 ∼ −1 It is instructive to rewrite the general expression for psuc losses and qubit dephasing C T2 , respectively. given in Eq. (H17) as This simple analytical estimate agrees well with numerical results presented in Ref. [62], where (except for noise 1 sources that are irrelevant for our problem) F ≈ 1 − p ; H19 suc ¼ 1 ε 1 1 ð Þ 2 3 ε 1 ε − C −1 ð þ Þ½ þ 4C ð = Þ =ð þ Þ C with a numerical coefficient C ¼ Oð1Þ depending on the specific pulse sequence. ε κ κ with ¼ bd= gd. Based on this definition, it is evident Since ε ≪ 1, this relation can be simplified to −1 that two conditions need to be satisfied in order to reach F ≈ 1 − ð2=3Þε − CC . For the state transferffiffiffi of the → 1 κ 0 p psuc in the regime where bd > : (i) a low undesired coherent superposition jψi¼ðj0i − j1iÞ= 2 as described ε κ κ ≪ 1 loss rate, ¼ bd= gd , and (ii) high cooperativity, in detail in Appendix J, we explicitly verify the linear ≡ 2 κΓ 2 κ ≫ 1 κ κ κ ∼ε ∼1 2ε C ðg = dephÞ¼ðg T2= Þ , with ¼ gd þ bd. scaling and find numerically = for intrinsic phonon
031031-25 M. J. A. SCHUETZ et al. PHYS. REV. X 5, 031031 (2015) losses (compare also Fig. 4, where F ≈ 95% for ε ≈ 10% unchanged with respect to the ungated case outside of σ 0 ∼ε and nuc ¼ ) and take as a simple estimate in Eq. (8). the edges of the gate. Since the QD electrons are confined Using experimentally achievable parameters ε ≈ 5% and to regions outside of the metallic gates, they experience the C ≈ 30, we can then estimate F ≈ 90%. piezoelectric potential as calculated for the ungated case (see Appendix B for details); therefore, our estimates— APPENDIX I: EFFECTS DUE TO THE where this screening effect has been neglected—remain STRUCTURE DEFINING THE approximately valid. (ii) In Ref. [47], the coupling of a QUANTUM DOTS traveling SAW to electrons confined in a DQD has been experimentally studied. Here, in very good agreement with In our analysis of charge and spin qubits defined in the experimental results, the potential felt by the QD quantum dots, we neglect any potential effects arising due electrons has been taken as V ∼ sin ðklÞ (where l is the to the structure defining the quantum dots, that is, (i) the pe lithographic distance between the dots) confirming the heterostructure for the 2DEG and (ii) the metallic top gates sinelike mode profile as used in our estimates. Moreover, for confinement of single electrons. In this appendix, we with l 220 nm and λ 1.4 μm, the parameters used in give several arguments corroborating this approximate ¼ ¼ this experiment perfectly match the ones used in our treatment, showing that the QD structure does not neg- estimates. Intuitively (in the spirit of the standard electric atively influence the cavity or the coupling between qubit λ ≫ and cavity. dipole approximation in quantum optics), since l, the SAW mode cannot resolve the dot structure and thus remains largely unaffected. (iii) In Ref. [16], single-phonon 1. Heterostructure SAW pulses have been detected via a single electron Following the arguments given in Ref. [26], the 2DEG is transistor (SET) directly deposited on the GaAs substrate taken to be a thin conducting layer a distance d away from with time-resolved measurements clearly identifying the the surface of a homogeneous AlxGa1−xAs crystal with coupling as piezoelectric. Similarly to a QD, the SET is ≈ 30% typically x . This treatment is approximately correct defined by metallic gates. Here, a relation between vertical since the relevant material properties (elastic constants, surface displacement and surface charge induced on the densities, and dielectric constants) of AlxGa1−xAs and SET is theoretically derived. Using standard tabulated GaAs are very similar [26]. The mode functions and speed parameter values for GaAs and neglecting any effects of sound are largely defined by the elastic constants [26], due to the presence of the metallic gates, very good which are roughly the same for both AlxGa1−xAs and pure agreement with the experimental results is achieved. In GaAs; for example, the speed of the Rayleigh SAW for particular, based on the results given in Ref. [16],a Al0 3Ga0 7As is v ≈ 3010 m=s, which differs from that of . . s straightforward estimate gives U0 ≈ 30 am for the rather pure GaAs by only 5% [26]. Moreover, the numerical large cavity with A ≈ 106 μm2, whereas Eq. (4) yields a values for the material-dependent parameter H entering the smaller, conservative estimate U0 ≈ 2 am, due to the amplitude of the mechanical zero-point motion U0 accord- averaging over the quantization area A. (iv) The Q factor ing to Eq. (C9) differ by 2% only [26]; accordingly, the of the cavity is not expected to be severely affected by the U0 estimate for our key figure of merit should be rather presence of the metallic gates since metallic Al cladding accurate. Also, the piezoelectric coupling constants are ≈ 0 15 2 layers have been used on a GaAs substrate to show rather similar, with e14 . C=m for pure GaAs and basically dissipation-free SAW propagation over millimeter ≈ 0 145 2 e14 . C=m for Al0.3Ga0.7As [25,26], yielding an distances [39,40]. (v) Finally, there is a large body of ϕ ξ accurate estimate for 0 and 0, respectively. Lastly, the previous theoretical work on electron-phonon coupling in heterostructure is not expected to severely affect the Q gate-defined QDs (see, for example, Refs. [25,87,88,98]) 104 factor, since very high Q values reaching Q> have where any effects due to the structure defining the QDs been observed in previous SAW experiments on AlN/ have been neglected as well. As a matter of fact, our diamond heterostructures [54,55], where the differences description for the electron-phonon coupling emerges in material properties are considerably larger than for the directly from these previous treatments in the limit where heterostructure making up the 2DEG. the continuum of phonon modes is replaced by a single relevant SAW cavity mode (similar to cavity QED, other 2. Top gates bulk modes are still present and contribute to the For the following reasons, we disregard effects due to decoherence of the qubit on a time scale T2). For example, the presence of the metallic top gates. (i) In Ref. [28],a the piezoelectric electron-phonon interaction is given in closed-form analytic solution for the piezoelectric potential Ref. [98] as ϕðx;tÞ accompanying a SAW on the surface of a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GaAs=Al Ga1− As in the presence of a narrow metal gate X ℏ x x β M † has been obtained. In particular, it is shown that ϕðx;tÞ is HGaAs ¼ k½ak;μ þ a−k;μ ; ðI1Þ 2ρVvμk screened right below the gate, but remains practically k;μ
031031-26 UNIVERSAL QUANTUM TRANSDUCERS BASED ON … PHYS. REV. X 5, 031031 (2015)
† where ak;μ creates an acoustic phonon with wave vector k and polarization μ, and Mk refers to the matrix element for electron-phononP P coupling; for free bulk modes, M kr † k ¼ i;j σ hij exp ½i jjidiσdjσ. In agreement with our notation, the coupling constant is β ¼ ee14=ϵ and the square-root factor can be identified with U0. Replacing the sum by a single relevant cavity mode a, we recover the Hamiltonian describing the cavity-qubit coupling with g ∼ βU0 ¼ eϕ0; compare Eq. (F1).
APPENDIX J: STATE TRANSFER PROTOCOL Here, we provide further details on the numerical simulation of the state transfer protocol as described by Eqs. (9) and (10), in the presence of Markovian noise. In line with previous theoretical studies [97], we show that the simple approximate Markovian noise treatment results in a F pessimistic estimate for the noise transfer fidelity . FIG. 16. Quantum state transfer protocol in the presence of We first provide results of the full time-dependent Markovian noise. State transfer fidelity F for a coherent super- pffiffiffi numerical simulation of the cascaded master equation position jψi¼ðj0i − j1iÞ= 2 as a function of the qubit’s given in Eqs. (9) and (10), including an exponential loss dephasing rate Γ for different values of intrinsic phonon Γ 0 deph of coherence for deph > ; see Fig. 15. In contrast to the losses κbd=κgd ¼ 0 (black solid line), κbd=κgd ¼ 5% (blue dashed non-Markovian noise model discussed in the main text, line), and κbd=κgd ¼ 10% (red dash-dotted line). Inset: Pulse the qubits are assumed to be on resonance throughout the shape g1ðtÞ for first node. evolution, that is, δi ¼ 0ði ¼ 1; 2Þ, but experience unde- sired noise as described by Here, the second term refers to a standard Markovian pure X X L ρ 2κ D ρ Γ D z ρ dephasing term that leads to an exponential loss of noise ¼ bd ½ai þ deph ½Si : ðJ1Þ ∼ −Γ 2 1 2 1 2 coherence exp ð depht= Þ. As a reference, we also show i¼ ; i¼ ; L 0 the results for the ideal, noise-free scenario ð noise ¼ Þ, where perfect state transfer is achieved [20]. The results of (a) this type of time-dependent numerical simulations are then summarized in Fig. 16. For optimized, but experimentally ⋆ achievable parameters T2 ≈ 3 μs [21], and accordingly Γ κ ≈ 3% F ≈ 0 85 deph= gd , we then obtain . , in the presence κ κ 5% (b) of realistic undesired phonon losses bd= gd ¼ . This shows that our non-Markovian noise model yields even higher state transfer fidelities than the Markovian noise model. Intuitively, this can be readily understood as follows: The simple Markovian noise model gives a coherence decay ∼ exp ð−t=T2Þ, whereas our non- 2 2 Markovian noise model yields ∼ exp ð−t =T2Þ. Therefore, for Markovian noise, one can estimate the dephasing-induced error on the relevant time scale for ∼κ−1 ∼κ−1 ≈ Γ κ ≪ 1 state transfer as =T2 deph= , whereas non-Markovian noise leads to a considerably smaller ∼ −1 κ 2 FIG. 15. Numerical simulation of state transfer for two error ðT2 = Þ . different initial states 1 (red solid line) and 0 − 1 = pffiffiffi j i1 ðj i1 j i1Þ 2 (blue dashed line) in the presence of Markovian noise. Black (dash-dotted) curves refer to the ideal, noise-free sce- L 0 nario ð noise ¼ Þ, where perfect transfer is achieved, while [1] R. Hanson and D. Awschalom, Coherent Manipulation of colored curves take into account decoherence processes. Single Spins in Semiconductors, Nature (London) 453, 1043 F þ − (a) Transfer fidelity . (b) Excited-state occupation hSi Si it (2008). for first and second qubit. Numerical parameters are [2] R. J. Schoelkopf and S. M. Girvin, Wiring Up Quantum ≥ 0 κ κ κ 5% Γ κ 5% g1ðt Þ¼ gd, bd= gd ¼ ,and deph= gd ¼ . Systems, Nature (London) 451, 664 (2008).
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