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PHYSICAL REVIEW X 8, 021052 (2018)

Displacemon Electromechanics: How to Detect Interference in a Nanomechanical

K. E. Khosla,1,2 M. R. Vanner,3,4 N. Ares,5 and E. A. Laird6,5,* 1Center for Engineered Quantum Systems, The University of Queensland, Brisbane, Queensland 4072, Australia 2School of Mathematics and , The University of Queensland, Brisbane, Queensland 4072, Australia 3QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom 4Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom 5Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom 6Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

(Received 30 October 2017; revised manuscript received 28 March 2018; published 24 May 2018)

We introduce the “displacemon” electromechanical architecture that comprises a vibrating nanobeam, e.g., a carbon nanotube, flux coupled to a superconducting . This platform can achieve strong and even ultrastrong coupling, enabling a variety of quantum protocols. We use this system to describe a protocol for generating and measuring quantum interference between trajectories of a nanomechanical resonator. The scheme uses a sequence of qubit manipulations and to cool the resonator, to apply two effective gratings, and then to measure the resulting interference pattern. We demonstrate the feasibility of generating a spatially distinct state of containing more than 106 nucleons using a vibrating nanotube acting as a junction in this new superconducting qubit configuration.

DOI: 10.1103/PhysRevX.8.021052 Subject Areas: Condensed Physics, Mesoscopics, Quantum Physics

I. INTRODUCTION Nanomechanical span this mesoscopic 106 1016 The is a fundamental tenet of scale ranging from about to amu and therefore quantum , and it is essential to understand a wide provide an attractive route to extend the scale over which range of quantum phenomena. As the scale of quantum quantum effects can be observed. Recently, cooling to the [15–17], as well as elements of quantum objects increases, the experimental consequences of this behavior such as squeezing [18–20] and coherent qubit principle become increasingly hard to isolate. Is there a coupling [15,21,22], has become accessible with mechanical scale at which this tenet begins to break down? The resonators of this scale. Superconducting charge or flux strongest tests of superposition come from matter- coupled to mechanical resonators offer a path to between trajectories of large . achieving coupling strength exceeding the qubit decoherence Remarkably, interference can be measured using molecules 104 rate, allowing the study of mechanical quantum states of mass exceeding atomic mass units (amu) [1,2]. The [23–25]. Quantum control of mechanical systems also offers ability to create unambiguous superpositions on a meso- a significant potential for quantum memories and for state scopic scale would allow tests of quantum collapse theories transfer [26,27], as well as for mass and force sensing – and gravitational decoherence [3 7], ultimately addressing [28,29]. Progress towards observing mechanical superposi- experimentally the question of why we fail to see super- tion states has been made in both optomechanics and positions in everyday life [8]. This has inspired numerous electromechanics, and mechanical interference fringes have challenging proposals to detect interference of larger recently been observed at a classical level [30]. The obser- particles [9–11] via optomechanical coupling [12],orby vation of quantum interference, however, remains outstand- using levitated nanodiamonds [13,14]. ing and is a key goal of this paper. Here, we introduce the “displacemon,” a device that enables strong coupling between a nanomechanical resona- *[email protected] tor and a superconducting qubit. We show how to create an effective that leads to an interference Published by the American Physical Society under the terms of pattern in the resonator’s displacement. The scheme works the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to by using a sequence of manipulations on the qubit to create the author(s) and the published article’s title, journal citation, an effective grating with a fine pitch and therefore a and DOI. large displacement. In molecular interference

2160-3308=18=8(2)=021052(15) 021052-1 Published by the American Physical Society KHOSLA, VANNER, ARES, and LAIRD PHYS. REV. X 8, 021052 (2018) experiments, the diffraction grating is typically an etched (a) (b) membrane; however, van der Waals interactions with the slits ΔΦ make this hard to extend to large particles [1]. More advanced B|| implementations use optically defined gratings; the pitch, which sets the momentum separation of the diffracted beams, is then limited by the optical [31]. In our scheme, Nanotube the pitch is limited neither by an optical wavelength nor by Flux the size of the resonator, but by the qubit-resonator coupling tuning strength. As we will show, this allows for diffraction gratings with a pitch narrower than the ground-state . ΔΦ Our proposed device uses a vibrating nanobeam that is flux coupled to a superconducting qubit, through which all Qubit manipulations and measurements are performed. As a nano- B Qubit beam that optimally combines high mechanical , || drive/readout low dissipation, and the ability to couple strongly to super- conducting quantum devices, we propose a suspended (c) Displacement X (nm) (d) X/XZP carbon nanotube. Previous proposals for quantum motion -5 0 5 -1000 0 1000 in nanotubes [32,33] have been based on coupling to a 2 4

4 λ

/2 qubit; however, the requirement on the qubit is π (MHz) stringent [34]. Here, using realistic parameters derived from (MHz/pm) 0 0 X (GHz)

π experiments, we show how to construct a superconducting )/d /2 π q qubit that can achieve strong and coherent coupling ω /2 q -2 -4 to the mechanical motion of a carbon nanotube. Exciting 0 ω -1ΔΦ/2Φ 1 d( -1ΔΦ/2Φ 1 recent progress in superconducting electromechanics has 0 0 provided empirical evidence for quantum behavior up to a FIG. 1. Device for strong qubit-mechanical coupling. (a) Elec- 1014 mass scale of about amu [15,35], though the observa- trical schematic. The device is a gradiometric transmon qubit, tion of mechanical interference fringes from a quantum biased by flux difference ΔΦ between the SQUID loops. With a superposition state remains outstanding. Our approach here suspended nanotube acting as at least one junction (shown here for focuses on generating effective mechanical diffraction gra- the right junction), the displacement modifies ΔΦ and therefore tings to measure quantum interference fringes in a mechani- modulates the qubit levels. (b) Arrangement of the qubit, cal object greater than 106 amu. The largest scale at which vibrating nanotube, flux tuning coil, and drive/readout cavity such fringes were observed is currently set by . The in-plane magnetic field Bjj introduces strong coupling ΔΦ interferometry experiments [2], and this proposal enables the between the vibrations and . (c) Qubit frequency as a function mass scale of objects exhibiting quantum interference fringes of flux difference, with parameters as in the text. Solid lines assume equal Josephson coupling in the two SQUID junctions; the dotted to be extended by nearly 3 orders of magnitude. line assumes 30% asymmetry (see Appendix A). Curves are plotted as a function of flux (bottom axis) and, equivalently, of II. MODEL displacement (top axis). (d) Qubit displacement sensitivity (left In general, strongly coupling a mechanical resonator to a axis) and mechanical coupling rate (right axis) as a function of flux. The bias point that achieves the assumed coupling is indicated by a qubit is challenging because the best qubits are engineered to vertical dashed line. be insensitive to their environment [36]. We propose a design that is robust against electrical and magnetic noise while still (SQUID), so the qubit frequency is set by the flux differ- achieving strong mechanical coupling. We envisage a super- ence ΔΦ between the two loops. This flux difference is conducting qubit of the concentric transmon design [37] in tuned primarily by means of a variable perpendicular field which at least one of the junctions is a vibrating nanotube ΔB⊥ðtÞ, while mechanical coupling is flux mediated using (Fig. 1). Nanotube resonators offer unique advantages for – a static in-plane field Bjj [45 47]. This type of concentric studying quantum motion [38]: (i) The zero-point transmon is insensitive to uniform magnetic fields, which is typically greater than 1 pm, much larger than other has two advantages for this proposal: The qubit can be mechanical resonators; (ii) the resonant frequency is suffi- operated coherently away from a flux sweet spot [37], and ciently large to allow near-ground-state thermal occupation, any misaligned static flux will not perturb the energy levels. suppressing thermal decoherence [39,40]; (iii) a nanotube Both these facts are favorable for strong nanomechanical can act as a Josephson junction [41–43]; and (iv) ultraclean coupling. Because this variant of the transmon is designed 6 devices offer mechanical quality factors greater than 10 , for strong coupling to nanomechanical displacement, we which provide long-lived states [44]. refer to it as a displacemon. In this section, we derive the In this design [Fig. 1(a)], the two junctions form a displacemon Hamiltonian and estimate the parameters for a gradiometric superconducting quantum interference device feasible device.

021052-2 DISPLACEMON ELECTROMECHANICS: HOW TO DETECT … PHYS. REV. X 8, 021052 (2018)

A. Mechanical resonator Assuming equal critical current Ic in the two junctions, this We consider the nanotube as a beam of length l and Josephson energy is diameter D and focus our studies on its fundamental πΔΦ 0 vibrational mode [38–40,44,48–50]. The mechanical res- EJ ¼ EJ cos ; ð2Þ † † 2Φ0 onator Hamiltonian is Hm ¼ ℏΩa a, where a ðaÞ is the creation (annihilation) for the resonator. Typically, where ΔΦ is the flux difference between the two loops, 0 the restoring force for a clamped nanotube is dominated by E ¼ I Φ0=π is the maximum Josephson energy, and ’ J c the beam s tension T p[51]ffiffiffiffiffiffiffiffiffiffiffiffi, so the mechanical angular Φ0 ¼ h=2e is the flux quantum. Ω π μ frequency is p¼ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi=lÞ ðT= Þpandffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the zero-point ampli- The flux difference can be tuned both directly, via a ℏ 2 Ω ℏ 2π μ −1=4 B⊥ tude is XZP ¼ ½ =ð m Þ ¼ ½ =ð Þð TÞ , where perpendicular magnetic field , and via the displacement using a static in-plane field B .Wehave μ ¼ πρSD is the mass per unit length and ρS ¼ jj 8 × 10−7 kg m−2 is the sheet density of graphene. The ΔΦ ¼ AΔB⊥ þ 2β0lB X; ð3Þ displacement profile as a function of axial coordinate Z is jj pffiffiffi ˜ 2 π ≡ † XðZÞ¼X sin½ð ZÞ=l, where X ða þ a ÞXZP is the where A is the area of one SQUID loop. Since quite small displacement coordinate. This profile is normalized so perpendicular fields suffice to tune the qubit frequency over that the root-mean-square displacement is equal to X its full range, we envisage an on-chip coil to modulate [51]. The flux coupling is proportional to the area swept ΔB⊥ðtÞ as a function of time t [37,54]. Substituting Eq. (2) out by the nanotube, which is equal to β0lX, where β0 ≡ into the definition of ω gives R pffiffiffi q l ˜ ½1=ðlXÞ 0 XðZÞdZ ¼½ð2 2Þ=π is a geometric coupling dω πβ0lB sin πΔΦ=2Φ0 coefficient [46]. q −ω 0 jj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ q 2Φ ; ð4Þ Nanotube resonators can also be fabricated without dX 0 j cos πΔΦ=2Φ0j tension, so the restoring force is dominated by the beam’s pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ω 0 ¼ 8E0E =ℏ is the maximal qubit frequency. rigidity [48,49]p. Inffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi this limit, the mechanical frequency is q J C The dependence of ω on X gives rise to an electro- Ω ¼ð22.4=l2Þ ED2=8μ and the coupling coefficient is q 3 mechanical coupling, resulting in the Hamiltonian [23,36] β0 ¼ 0.831, where E ≈ D × 1.09 × 10 Pa m is the exten- ω sional rigidity. ℏΩ † ℏ q σ ℏλ † σ H ¼ a a þ 2 z þ ðtÞða þ a Þ z; ð5Þ B. Qubit λ 2 ω where ðtÞ¼ðXZP= Þd q=dX [from Eq. (4)] is the qubit- The qubit consists of a pair of superconducting electro- mechanical coupling strength, dynamically controlled des coupled through the SQUID junctions. The qubit through the field ΔB⊥ðtÞ. Hamiltonian is [52] Achieving coherent interaction between the qubit and the resonator requires the quantum coherent coupling regime, 4 ˆ − 2 − φˆ Hq ¼ ECðn ngÞ EJ cos ; ð1Þ where the maximum accessible coupling λ0 exceeds both the mechanical rethermalization rate κ ¼ k T=ðℏQ Þ and the where E is the charging energy, E is the SQUID th B m C J qubit decoherence rate γ ¼ 1=T2. Here, Q is the mechani- Josephson energy, and nˆ and φˆ are the overall charge m cal quality factor and T2 is the qubit coherence time. The (expressed in Cooper pairs) and across the junctions, large zero-point motion makes nanotube resonators particu- with ng being the offset charge. Here, we have neglected the larly favorable for achieving this regime. Taking device qubit inductance, which makes a small contribution on the parameters from simulation and experiment (Appendix A) energy levels [37]. In the transmon limit EJ ≫ EC, we can leads to Ω=2π ¼ 125 MHz, ωq=2π ¼ 2.19 GHz, and approximate Eq. (1) by an effective Hamiltonian ffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ0=2π ¼ 4.2 MHz, with the flux dependence shown in ≈ 1 ℏω σ ω p8 ℏ Hq 2 q z, where q ¼ EJEC= is the qubit fre- Fig. 1. This is favorable for achieving the quantum coherent σ quency and z is the standard Pauli matrix, acting on the coupling regime since both κ =2π and γ=2π are typically less − th qubit ground state j i and the jþi. Qubit than 1 MHz. rotations, initialization, and projective are To create well-separated mechanical superpositions, a now well-established techniques, using capacitive coupling stronger condition is desirable; the qubit should precess to a cavity in a circuit appreciably within an interval during which the resonator architecture [37,53]. can be considered stationary. This is the ultrastrong coupling regime, where λ0 > Ω [47]. It is possible that a C. Strong and ultrastrong coupling device similar to that of Fig. 1 could access this regime (see Strong and tunable coupling between the qubit and the Appendix A). However, here we instead suppose that mechanical resonator is achieved by flux coupling to the effective ultrastrong coupling is engineered by modulating SQUID loops, which tunes the qubit Josephson energy. ΔB⊥ðtÞ at the mechanical frequency (see Sec. III B). In this

021052-3 KHOSLA, VANNER, ARES, and LAIRD PHYS. REV. X 8, 021052 (2018) modulated frame (similar to the toggled frame obtained by scheme consists of applying a π burst at the bare qubit repeatedly flipping the qubit [55]), the resonator is effec- frequency ωq [Fig. 2(a)]. If the resonator is near its tively frozen and the required coupling condition is relaxed equilibrium position, this results in a qubit flip. By con- λ κ γ to 0 > th; . ditioning on this outcome (i.e., utilizing only those runs of the experiment where this qubit outcome is measured), the III. GENERATING AND MEASURING resonator state is constrained to a narrow window [Fig. 2(b)]. MECHANICAL QUANTUM INTERFERENCE A single operation of this type cools only one quadrature of the motion because resonator states with high momen- To realize the nanomechanical interferometer, we propose tum may still pass the window. To cool the orthogonal a series of operations and measurements on the qubit; this quadrature, the same selection should be applied a quarter provides the necessary nonlinearity to generate mechanical superposition states. The core idea is that the state of the of a mechanical period later [57], which filters out high- mechanical resonator is constrained by the qubit measure- energy states that pass the first selection step [Fig. 2(c)]. ment outcome in the same way that the state of a particle is The combination of these two pulse sequences therefore constrained by passing through a diffraction screen. By prepares the resonator close to its ground state, at the price concatenating a series of qubit rotations and measurements, of accepting only a fraction of the measurement runs. the resonator can be cooled, diffracted, and measured. B. Diffracting the resonator A. Cooling the resonator The effective diffraction grating for the resonator (Fig. 3) The first step is to prepare the resonator close to its ground is implemented using Ramsey interferometry to generate a state. A mechanical frequency of 125 MHz requires a bath periodic spatial filter [58,59]. To understand how the temperature below about 5 mK for a thermal occupation less grating arises, consider the operator UðtÞ than unity. Such temperatures are achievable but challenging generated from Eq. (5). As shown in Ref. [59], this time- with cryogenic cooling [56]. At a more accessible cryostat ordered unitary is temperature of 33 mK, the initial thermal occupation is ¯ 5 − ω 2 σ † n ¼ . To approach the ground state from a thermal state, we UðtÞ¼RðΩtÞe ½ði qtÞ= z ðDðαÞj−ih−jþD ðαÞjþihþjÞ; propose a cooling scheme utilizing the qubit as a thermal filter (Fig. 2). Following initialization to the jþi state, the ð6Þ

α †−α − θ † (a) (b) where DðαÞ¼e a a and RðθÞ¼e i a a are resonator displacement and rotation operators, respectively, and )

X Before ( P After Initialize Measure (a) (c) t -30 30

X (pm) ) Qubit drive

X Before ( (c) (i) (ii) (iii) P After Initialize Measure

X -10 10 t X (pm) Qubit drive t (b)

FIG. 2. Cooling the resonator using the qubit. (a) Pulse sequence (see text). Gaussian pulse shaping is chosen for near-optimal filtering. (b) Effect of this sequence, conditioned on the qubit outcome j−i, on an initial mechanical distribution (i) (ii) (iii) with a thermal distribution n¯ ¼ 5. (Burst duration is τπ ¼ 38 ns with λ0=2π ¼ 4.2 MHz.) The pulse sequence has the effect of passing the wave function through a narrow slit. The filter FIG. 3. Using pulsed qubit operations to construct an effective function is obtained by numerically solving the time evolution diffraction grating. (a) Pulse sequence. (b) Evolution of the qubit generated by Eq. (5) with an additional π-pulse term gðtÞσx=2. on the Bloch sphere, during (i) preparation, (ii) precession at a (c) Cartoon showing the effect of two pulse sequences, offset by a rate set by the displacement-dependent qubit frequency shift quarter of a mechanical period. Only the lowest-energy trajecto- Δωq, and (iii) projection. (c) Effect of this pulse sequence on the ries (solid line) survive both measurements, effectively cooling ground-state mechanical distribution. Conditioning the resonator. For the purpose of this cartoon, the duration of the on the qubit measurement outcome jþi, the resonator wave two pulse sequences is compressed; in fact, each pulse sequence function is multiplied by a periodic filter [Eq. (9)], analogous to a lasts for several mechanical periods. grating. Here, we have taken jαj¼1.9.

021052-4 DISPLACEMON ELECTROMECHANICS: HOW TO DETECT … PHYS. REV. X 8, 021052 (2018) Z pffiffiffi t π iΩt0 0 0 α ¼ i e λðt Þdt ð7Þ α ¼ i pffiffiffiffiffiffiffi λ0τλ: ð11Þ −∞ 4 ln 2 is the amplitude of the coherent displacement. The super- In the following, we take α ≈ 1.9i, thus achieving the position of displacement operators in Eq. (6) applies equal desired momentum separation. A price to pay for this and opposite momentum kicks [assuming ReðαÞ¼0 from modulation is that the qubit and resonator are susceptible to here on [60]] to the resonator depending on the state of the decoherence over the full duration of the envelope. With qubit. This is analogous to standing-wave gratings in our parameters, we require τλ ≈ 130 ns, corresponding to interferometry, which, when decomposed into left- and N ≈ 17 mechanical periods. This interaction time is short right-propagating beams, can be understood to impart enough that the evolution of the resonator-qubit system is superposed positive and negative impulses to . well approximated as unitary. (See Appendix C for model- We now describe the protocol that realizes this super- ing of qubit dephasing and mechanical decoherence.) position of momentum kicks [Figs. 3(a) and 3(b)]. Following initialization of the qubit in the excited state C. Nanomechanical interferometry jþi, a microwave burst applied to the qubit generates a pffiffiffi We now show how a sequence of grating operations can π=2 rotation, preparing a superposition − = 2. ðjþi þ j iÞ be combined to create an interferometer (Fig. 4). The effect of The mechanical interaction then causes the qubit state to precess at the displacement-dependent rate ΔωqðtÞ¼ 2 λ τ π 2 f XðtÞ=XZPg ðtÞ. After an interval R, a second = burst is applied, followed by a σz measurement. Conditioning on the qubit outcome gives a measurement operator that acts on the mechanical system,

ϒ ϕ ≡ Π† Π ð Þ hj ϕUðtÞ 0jþi

RðΩtÞ † ϕ ω D α i þi qtD α ¼ 2 ½ ð Þe ð Þ: ð8Þ

Here, Πϕ denotes a π=2 qubit rotation with phase ϕ, and is the result of the σz measurement. The (unnormalized) state of the resonator after the interaction is   ConditionCondition cos ϕ onon ψ → ϒ ψ R Ω α X ψ j iM j iM ¼ ð tÞ j j þ 2 j iM; ð9Þ sin XZP ConditionCondition on where cosð·Þ or sinð·Þ correspond to finding the qubit in the RecordRecord resultresult excited or ground state, respectively [61]. The resonator wave function is thus projected onto an effective diffraction FIG. 4. Protocol for detecting nanomechanical interference. π α Following cooling (not shown), the first grating operation, with grating with pitch XZP=j j [Fig. 3(c)]. Since the only amplitude α1, diffracts the resonator wave function into a difference between conditioning on the ji outcomes is a superposition of left-moving and right-moving components. relative change in phase of the effective grating, either After an interval τ1 of free evolution, here set as a quarter of measurement outcome may be used to define the grating. the mechanical period TM, a second grating operation is applied We refer to the Ramsey sequence followed by conditioning with amplitude α2. This leads to recombination of the two on the qubit measurement outcome as a grating operation. components an eighth of a mechanical period later. To measure Its effect is to split the resonator wave function into a the resulting interference, a third (unconditioned) Ramsey se- τ 8 superposition of left-moving and right-moving branches. quence is applied after time 2 (here set to TM= ). The resulting α ϕ A well-separated superposition, with both branches qubit return probability pþð 3; Þ probes the mechanical inter- displaced by more than the zero-point amplitude, requires ference fringes. The main panel shows a simulated resonator ψ 2 jαj ≳ 1. To achieve this with our parameters, we require that spatial density j ðXÞj , beginning from the ground state, plotted as a function of displacement and time, with the grating λðtÞ is modulated at the mechanical frequency: operations and the final Ramsey measurement operation indi- λ λ Ω cated schematically as filters. The resonator Wigner distributions ðtÞ¼ 0gðtÞ cos t; ð10Þ and marginals (see Fig. 5) are shown as insets just before each filter. To illustrate the continuing periodic evolution of the where gðtÞ is a Gaussian envelope function with a maxi- resonator wave function, the spatial density beyond the final mum of unity and a full width at half maximum of Ramsey measurement is plotted as it would be probed by τλ ≫ 1=Ω. Equation (7) then gives applying the measurement instead at a later time.

021052-5 KHOSLA, VANNER, ARES, and LAIRD PHYS. REV. X 8, 021052 (2018) a single grating operation [Eq. (9)] with phase ϕ ¼ 0 and amplitude α ¼ α1 is to divide the resonator’s wave function (a) (d) (g) α ℏ into two components with added momentum j 1j =XZP. A second grating operation with the same amplitude and phase, applied after a duration τ1 ¼ π=2Ω corresponding to a quarter period of free evolution, further splits the branches of the superposition, allowing quantum interference between recombined branches to be observed. After a second evolu- (b) (e) (h) tion time τ2, the interference can be detected using a third Ramsey sequence. In this step, there is no conditioning; the ϕ α probability pþð ; 3Þ for the qubit to return to state jþi is measured as a function of the phase ϕ and amplitude α3 of the Ramsey sequence. This probability is (c) (f) (i) † p ðϕ; α3Þ¼Tr½ϒ ϒ ρ þ Z þ þ m  X0 ϕ 0 2 α 0 ¼ dX cos j 3j þ 2 PðX Þ; ð12Þ XZP where ρm is the describing the state of the resonator immediately before the third grating, with position PðXÞ. FIG. 5. Wigner distributions of the resonator’s state at different Scanning the phase of the third Ramsey sequence is values of τ2. (a)–(c) Maximum interference from an initial pure analogous to scanning the position of the third grating in a state with unitary evolution. (d)–(f) Loss of interference due to molecular interferometer [2], and the signature of interfer- initial thermal occupation n¯ ¼ 5 (with unitary evolution). – ence is a sinusoidal dependence on ϕ. In fact, Eq. (12) can (g) (i) Loss of interference with nonunitary evolution equivalent be understood as a Fourier decomposition in which each to adding 0.05 (g) or 0.1 (h,i) thermal . The initial state α for panels (g) and (h) is the vacuum state, and the initial state of choice of j 3j probes the component of PðXÞ with wave ¯ 5 2 α ≡ panel (i) is a thermal state with n ¼ phonons. Blue and green number j 3j. From here on, we use x X=XZP ¼ða þ † traces show the position and momentum marginals, respectively, a Þ as a dimensionless . normalized to the same maximum. Each plot uses jαj ≈ 1.9, and Our goal now is to use pþ to distinguish quantum the scale has been normalized to the range 1. interference from classical fringes that might appear in the ’ resonator s probability distribution PðxÞ. Classical fringes ¯ 0 – might arise, for example, from the shadow of the diffraction the ground state, n ¼ [Figs. 5(a) 5(c)], but is washed out if the resonator is initially in a thermal state, leaving only gratings, or from Moir´e patterns. To recognize the quantum – interference, we plot the resonator Wigner distribution at the orthogonally sliced pattern visible [Figs. 5(d) 5(f)]. During the evolution time τ2, the Wigner distribution different times τ2 after the second grating operation, rotates [Figs. 5(b), 5(c), 5(e), and 5(f)] so that the choosing jα1j¼jα2j ≈ 1.9 (Fig. 5). The effect of applying the first grating operation is to “slice” the position dis- interference fringes oscillate between the position and 2 momentum marginals (plotted as blue and green traces, tribution by multiplying by cos ðjαjxÞ and to prepare a superposition of two momentum states (see second inset in respectively). Interference patterns arise when two lobes of Fig. 4, plotting the state after the first grating, rotated by a coherent quantum superposition overlap in position space, for example, the northeast and southeast lobes in one quarter period). Since the second grating operation is ≈ 3 applied a quarter period after the first, it acts along an Fig. 5(a) interfering around x , or the north and south ≈ 0 orthogonal axis, leading to the “quantum compass” state of lobes in Fig. 5(c) interfering around x . The wave Fig. 5(a) [62]. These compass states are widely studied numbers present in the position marginal (which is mea- theoretically and find applications in, e.g., sensing and sured by the third grating) are proportional to the momen- tum separation of lobes in phase space, geometrically metrology [63] and quantum codes with multicomponent pffiffiffi superpositions [64]. illustrating the 2 ratio between wave-number components The compass state can be intuitively understood as present in Figs. 5(a) and 5(c). follows. The quarter-period rotation after the first grating The interference fringes arising when the resonator is turns the momentum superposition state into a position initially prepared in its ground state can be compared with superposition. Each branch of the superposition then passes those arising from an initial thermal state (n¯ ¼ 5). If the width of the initial thermal state [as in Figs. 5(d)–5(f)]is the grating, generating its own momentum superpositions pffiffiffi and resulting in the four-lobe compass state. The compass larger than the superposition size ( n¯ > jαj), then the state is clearly visible if the resonator is initially prepared in vertical slicing of the grating is no longer accompanied by a

021052-6 DISPLACEMON ELECTROMECHANICS: HOW TO DETECT … PHYS. REV. X 8, 021052 (2018) distinct momentum superposition but rather by an increase in the momentum variance [as seen by the broader- (a) (b) (c) than-Gaussian position and momentum distributions in Fig. 5(e)], and this results in a checkerboard pattern. We can now see the distinction between quantum and classical fringes appearing in the marginal distributions. The shadow of the gratings is dominated by components close to the wave number 2jα1j [Figs. 5(a) and 5(d)]. The Moir´e patterns arising from the checkerboard have com- 2 α FIG. 6. The probability to find the qubit in its excited state, ponentsffiffiffi close to at most two wave numbers, j 1j and α p pþð 3Þ, can be used to probe different wave-number components 2 2jα1j (see Appendix D). By contrast, the quantum in the position probability distribution PðxÞ. As the resonator interference pattern [Figs. 5(a)–5(c)] has multiple wave- evolves (increasing Ωτ2), different wave numbers are present in Ωτ α numbers components at each 2, as seen in the position PðxÞ (see Fig. 5). The probability pþð 3Þ is plotted for the marginals. Furthermore, quantum interference appears for resonator initially in (a) a pure state, (b) a thermal state of n¯ ¼ 5 all evolution times τ2, whereas classical fringes are washed phonons, and (c) a pure state, but with decoherence (represented out [Fig. 5(e)] except at particular fractions of the mechani- by n0 added phonons) after the second grating. cal period. Hence, for this protocol, a marginal PðxÞ with multiple wave-number components, observed at all rotation namely, damping of all features in p , including the pffiffiffiffi þ angles Ωτ2, indicates quantum interference. 0 classical fringes, for jα3j > n . This mechanical interfer- We now show that this interferometer is a sensitive probe ometry scheme could thus be used as a specific probe of for , which damps the interference quantum decoherence during the mechanical evolution. P x fringes in ð Þ and therefore destroys the signature of We can compare the degree of macroscopicity in this quantum coherence in p . To model decoherence follow- þ proposal with other quantum superposition experiments. ing the second grating operation, we consider weak 6 With parameters from Appendix A,a3 × 10 amu reso- thermalization of the state, resulting in a decohered state nator achieves a maximum spatial superposition of 12 pm. (superscript d), Applying the macroscopicity measure μ introduced in Z μ 14 0 −jβj2=n0 Ref. [65], the nanotube reaches ¼ . (assuming a ðdÞ 2 e † fringe visibility of 0.8 and coherence time of 100 oscillator ρm ¼ d β DðβÞρmD ðβÞ; ð13Þ πn0 periods). By this measure, the superposition is more

0 macroscopic than already achieved in state-of-the-art where n is the number of thermal phonons effectively near-field molecular interferometry experiments [66] added to the resonator, causing loss of quantum coherence. (μ ¼ 12.1) or in a micromembrane near the ground state The loss of coherence is equivalent to convolving P x with pffiffiffiffi ð Þ [17] (μ ¼ 11.5), and it is comparable to other proposals, a Gaussian of width n0, thereby exponentially damping pffiffiffiffi including a satellite-based [67] oscillations of wave number jαj > n0 (Appendix C). (μ ¼ 14.5)ora105 amu Talbot-Lau interferometer [68] Figures 5(g)–5(i) plot the effect of loss of coherence (μ ¼ 14.5). However, because of its low mass, a nanotube between the second and third gratings, assuming an initial has a lower measure than proposals for superpositions of a ground state (g) and (h), and an initial (n¯ ¼ 5) thermal state micromirror [9] (μ ¼ 19.0) or nanosphere interference [10] (i). The decoherence is modeled only after the second (μ ¼ 20.5) [69]. grating, so without thermalization, panels (g) and (h) would Before concluding, we contrast our scheme with conven- coincide with panel (c), and panel (i) would coincide with tional optical interferometers and current matter-wave panel (f). The plots show that even the addition of a fraction interferometers. The interference fringes of Fig. 5(c) are of a phonon drastically suppresses the interference pattern loosely analogous to Mach-Zehnder interference because in PðxÞ and the corresponding signature in pþ. the mechanical wave function is split and then recombined. Finally, in Fig. 6, we show explicitly how these two However, the full position distribution is a result of all effects—thermal occupation before the interferometry and four components of the superposition state, whereas con- decoherence during the interferometry—degrade the quan- ventional interferometers usually combine only two optical tum signatures in pþ. In the ideal case [Fig. 6(a)], there are paths. In matter-wave experiments, a successful current several values of α3 at each time τ2 that give nontrivial technique is Talbot-Lau interferometry [2], where a of pþ. In contrast, beginning in a thermal state sequence of gratings is used to generate and observe with n¯ ¼ 5 phonons [Fig. 6(b)], all fringes are washed out, near-field diffraction patterns. In our scheme, the quarter with the exception of the shadow of the grating (at α3=α2 p¼ffiffiffi period of mechanical free evolution between the gratings is 1 and Ωτ2 ¼ 0, π=2) and the Moir´e pattern (at α3=α2 ¼ 2 equivalent to far-field diffraction, so the key physics is quite and Ωτ2 ¼ π=4). Loss of coherence during the interferom- different. We further highlight that the mechanical state etry [Fig. 6(c)] leads to a qualitatively different behavior, prepared after the second grating is the “quantum compass

021052-7 KHOSLA, VANNER, ARES, and LAIRD PHYS. REV. X 8, 021052 (2018) state” introduced by Zurek [62]; this scheme allows the World Charity Foundation, and the Royal Academy of features of this exotic to be explored. Engineering. K. K. would like to thank the Department of Furthermore, this work opens an avenue to implement Materials, University of Oxford for their hospitality during other pulse sequences that create grating geometries for the initial stages of this work. mechanical-state engineering.

IV. CONCLUSION APPENDIX A: DEVICE PARAMETERS AND IMPLEMENTATION We have introduced the displacemon electromechanical system, which provides sufficiently strong coupling to We assume device parameters based on a mixture of generate and detect quantum interference of a massive object experiment and simulation as follows. We take the param- containing a quarter of a million atoms. Using device eters of the nanotube and the junctions from the nanotube parameters based on current technology, our qubit-resonator SQUID device of Ref. [43]. For a resonator with length 800 Ω 2π displacemon can achieve effective ultrastrong coupling using l ¼ nm, the frequency was measured as = ¼ 125 2 5 a modulated coupling scheme. A similar interferometry MHz, which, with estimated diameter D ¼ . nm 5 10−21 3 106 scheme could also be applied to other kinds of solid-state and mass m¼ × kg¼ × amu, leads to XZP ¼ qubits coupled to high-quality resonators, such as spin qubits 3.7 pm, typical of nanotube resonators. In each SQUID ≈ 12 coupled to nanotubes [32,33], diamond defects coupled to junction, a critical current Ic nA was achieved, 0 12 cantilevers [55], or piezoelectric resonators coupled to implying EJ=h ¼ GHz. superconducting qubits [70,71]. However, the parameters For the qubit, the charging energy is set by the electrode estimated for the proposed device of Fig. 1 may be geometry, which is a design choice. Finite-element capaci- particularly experimentally favorable for this implementa- tance simulation for the device of Fig. 1(b), with qubit tion because they imply that the coupling exceeds both the diameter 340 μm, gives EC=h ¼ 0.2 GHz, typical of qubit 0 ≫ thermal decay rate of the resonator and the typical dephasing devices and well into the transmon regime EJ EC ω0 2π rate of a qubit (see Appendix E). Importantly, our scheme [74]ffiffiffiffiffiffiffiffiffiffiffiffiffiffi. The maximal qubit frequency is then q= ¼ does not rely on degeneracy between the qubit and the p 8EJEC=h ¼ 4.4 GHz, and the calculated qubit energy resonator [15], nor on qubit coherence over the lifetime of levels are shown in Fig. 1(c). the mechanical superposition [72]. 0 5 For the in-plane magnetic field, we assume Bjj ¼ . T, Using this ultrastrong coupling, we have shown that it is which nanotube SQUIDs can withstand [43]. The operating possible to extend matter-wave interferometry to nano- flux point should then be chosen to maximize λ while still mechanical resonators, opening up a range of new devices maintaining a qubit frequency compatible with microwave that can be used to study quantum physics at the mesoscale. resonators. We assume flux bias ΔΦ=Φ0 ¼ −0.84, leading Furthermore, interferometry performed on the wave func- 0 to a qubit frequency ωq ¼ 2π × 2.19 GHz ¼ ωq=2 [dashed tion of a mechanically bound resonator is qualitatively vertical line in Figs. 1(c) and 1(d)]. With symmetric distinct from existing free-particle interferometry tech- junctions, and assuming that the restoring force on the niques. An important advantage of nanomechanical reso- nanotube is dominated by tension, the coupling is then nators for quantum tests is that they can readily be extended λ=2π ¼ 4.2 MHz. Since the coupling can be reduced by to probe much larger than in molecular vapors or tuning ΔΦ towards zero, we take this as the maximum even levitated nanoparticles. Although the nanotube reso- coupling strength λ0. nator considered here does not have enough mass to In a realistic device, we must take account of asymmetry seriously challenge the interesting parameter regime of between the junctions. Denoting the two critical currents objective collapse theories, a similar protocol could be by I 1 and I 2, with δ ≡ 2ðI 1 − I 2Þ=ðI 1 þ I 2Þ being extended to more massive objects still well within the range c c c c c c the asymmetry parameter, we have E ¼ of nanomechanics. This research direction could allow for pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 0 2 πΔΦ 2Φ δ2 4 2 πΔΦ 2Φ testing specific theories of quantum collapse [8],asan EJ cos ð = 0Þþð = Þsin ð = 0Þ, with a alternative to proposals based on single- optome- corresponding modification to Eq. (4) [47,52]. This asym- λ chanics [9] or levitated nanoparticles [11,72]. Multiple metry leads to a small reduction in 0 [Figs. 1(c) and 1(d)]. resonators coupled to the same qubit (such as the pair of For these parameters, the device would be in the strong λ ℏ 1 nanotube junctions in Fig. 1) could allow for generation of coupling regime ( 0 >kBT= Qm, =T2) for a compara- ≳ 15 entanglement between massive objects, leading to Bell tests tively modest resonator quality factor Qm and ≳ 120 of mechanical resonators [73]. T2 ns. Accessing the ultrastrong coupling regime (λ0 > Ω) is more challenging but may be possible [47]:If the suspended length could be increased to l ≈ 2.9 μm and ACKNOWLEDGMENTS the tension reduced to zero while keeping other parameters We thank P. J. Leek and E. M. Gauger for discussions. We unchanged, the coupling would be λ0=2π ≈ Ω=2π ≈ acknowledge FQXi, EPSRC (EP/N014995/1), Templeton 8 MHz. However, in the simulations, we do not make this

021052-8 DISPLACEMON ELECTROMECHANICS: HOW TO DETECT … PHYS. REV. X 8, 021052 (2018) assumption but instead assume that effective ultrastrong In the following, we describe how the cooling envelope coupling is engineered by toggling λðtÞ as in Eq. (10). in Fig. 2 is obtained. Consider the state of a pure qubit- Previous experiments have measured carbon nanotube resonator system, resonators at temperatures smaller than the level spacing Z and therefore presumably in the quantum ground state ψ 0 − 0 0 0 0 [39,40]. However, those experiments did not measure j ðtÞi ¼ c−ðx ;tÞj ;xiþcþðx ;tÞjþ;xidx ; ðB2Þ quantum effects. In this proposal, the strongly coupled 0 qubit provides the nonlinearity that allows non-Gaussian where c−ðx ;tÞ is the of finding the states to be prepared and detected, in a way that goes qubit in the j−i state and the resonator at x0. The beyond measurements using resistive transport [38], Hamiltonian Eq. (B1) generates the equations of motion, SQUID measurements [43], or linear optomechanics _ − λ − [50]. The proposed qubit imposes important experimental cþðx; tÞ¼ i 0xcþðx; tÞ igðtÞc−ðx; tÞ; ðB3Þ challenges. The tunnel coupling between nanotube and _ − λ − superconductor must be large to support a high critical c−ðx; tÞ¼ i 0xc−ðx; tÞ igðtÞcþðx; tÞ: ðB4Þ current. Furthermore, both the superconducting film and the nanotube junction must remain superconducting even in The conditional state of the resonator after the cooling ψ ∝ an in-plane magnetic field, and the qubit must remain measurementR (with outcome jþi) is therefore j im 0 0 0 coherent. The modulated electromechanical coupling dx cþðx ;tfÞjxi, where cþðx ;tfÞ is the solution to scheme considered here requires precise timing of the flux Eqs. (B3) and (B4) after the application of the π pulse, bias pulses to avoid phase errors. However, experiments are 0 ψ 0 with initial conditions c−ðx; Þ¼hxj ð Þim and making progress in these directions: dc transport measure- 0 0 − cþðx; Þ¼ (i.e., the qubit is initially in the j i state). ments show that a suspended nanotube SQUID can be The ∝ is used since the final state is not normalized. 1 operated up to magnetic field Bjj ¼ T [43]. Meanwhile, This generalizes to an initial mixed state of the resonator by pulsed flux bias is an established tool for controlling X transmon qubits [37,54]. ρ 0 − − ⊗ ψ ψ → ð Þ¼j ih j pij iimh ijm Z i APPENDIX B: COOLING VIA X ρ ∝ 0 00 0 0 00 00 QUBIT MEASUREMENT mðtfÞ pi dx dx cþ;iðx Þjx ihx jcþ;iðx Þ; ðB5Þ i † Here, we show that the σzðaþa Þ interaction Hamiltonian can be used to conditionally cool the resonator. Consider the where pi is the classical probability of finding the quantum ψ qubit-resonator interaction in the , with a system in the state j iim. The calculation can be greatly (real) coherent qubit drive gðtÞ with zero detuning to the qubit simplified by noting that the system of differential equa- frequency, tions governing c is linear, and therefore, the solution for 1 0 the initial conditions c− ¼ , cþ ¼ can be used to find ℏ λ σ † σ 0 HI= ¼ ðtÞ zða þ a ÞþgðtÞ x: ðB1Þ c;iðx ;tfÞ for any initial state. The cooling envelope in Fig. 2 is obtained by numeri- The cooling protocol works as follows: First, the qubit is cally solving Eqs. (B3) and (B4), with gðtÞ a Gaussian prepared in the j−i eigenstate; then, the interaction is envelope π pulse, and then calculating the new position switched on (λðtÞ → λ0) and remains time independent for marginal, the duration of the π pulse gðtÞ. At the completion of the π X pulse, the interaction is switched off, and the qubit is ρ ∝ 2 hxj mðtfÞjxi pijcþ;iðxÞj ; ðB6Þ projectively measured in the σz . i The π-pulse cooling can be intuitively understood. If the resonator remains at a fixed position over the duration of where ρmð0Þ is a thermal state. the π pulse (as it must in the interaction picture since a þ a† σ commutes with the Hamiltonian), then the z term in APPENDIX C: DECOHERENCE Eq. (B1) looks like a detuning of the π-pulse drive from the qubit frequency. The probability of a qubit transition Here, we model the effect of decoherence on the inter- from the ground to the excited state under a π pulse depends ference. For our chosen parameters, the effect is estimated to on the detuning and is maximized for zero detuning. be weak because the interaction time τλ ≈ 130 ns is short Therefore, if the resonator is close to its equilibrium compared with other timescales. For a superconducting position, then there is a high probability of the qubit qubit, the decoherence time is typically T2 > 1μs ≫ τλ,so transitioning to the jþi state. However, if the resonator qubit dephasing during the interaction will not significantly is far displaced from equilibrium (i.e., in a high potential change the state. For the resonator, the high quality factor Qm 5 energy state), then the qubit has a low transition probability. suppresses thermal decoherence; assuming Qm ¼ 10 , there

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¯ ≈ 5 10−5 weighted by the probability of obtaining a particular reali- are n=Qm × phonons exchanged with the thermal pffiffiffiffiffiffiffi environment every resonator period, or 1 phonon exchanged zation of WðtÞ, where P(WðtÞ) ¼ exp½−W2ðtÞ=2t= 2πt, about every 103 realizations of the interaction. Below, we Z ∞ model the effect of decoherence in detail. ρðtÞ¼ ρ(t; WðtÞ)P(WðtÞ)dWðtÞ: ðC4Þ −∞ ffiffiffi 1. Qubit dephasing − ϕ p Projecting the qubit onto the state ðjþi þ e i j−iÞ= 2 We model dephasing by adding a stochastic frequency gives the unnormalized conditional state of the mechanical shift to the qubit, changing the Hamiltonian [Eq. (5)]to resonator, ω pffiffiffiffiffiffiffi † q † 1 H ¼ ℏΩa a þ ℏ σ þ ℏλðtÞðaþ a Þσ þ ℏ γ=2ξðtÞσ ; ρ ∼ iϕ − ρ −iϕ − 2 z z z m; 2 ðhþj e h jÞ ðtÞðjþi e j iÞ ðC5Þ ðC1Þ 1 ∼ D† α ρ D α D α ρ D† α 4 ½ ð Þ m ð Þþ ð Þ m ð Þ where γ is the qubit dephasing rate and ξðtÞ is a delta- −γt( −iϕD† α ρ D† α iϕD α ρ D α ) correlated white noise term satisfying E(ξðtÞ) ¼ 0 and e e ð Þ m ð Þþe ð Þ m ð Þ ; ðC6Þ E(ξðtÞξðt0Þ) ¼ δðt − t0Þ.Here,Eð·Þ denotes an average over realizations of this stochastic process. Moving into where we have used “∼” because the right-hand side is the interaction picture, the unitary generated by this unnormalized. Separating this into coherent and incoherent Hamiltonian is terms, we find pffiffiffiffiffi −γt − γ 2σ † e † ϕ − ϕ † iWðtÞ = z (D α − − D α ) ρ ∼ D α i D α ρ D α i D α UðtÞ¼e ð Þj ih jþ ð Þjþihþj ; ðC2Þ m; 4 ½ ð Þe ð Þ m½ ð Þe ð Þ R 1 − e−γt where WðtÞ¼ t ξðt0Þdt0 is a stochastic variablewith a mean D† α ρ D α D α ρ D† α : C7 t0 þ 4 ½ ð Þ m ð Þþ ð Þ m ð Þ ð Þ of zero and a variance of t. Since there is classical uncertainty in the realization of WðtÞ, the joint state of the resonator-qubit ϒ ρ ϒ† We notice that the first term is proportional to m , system will be mixed. Because of this classical uncertainty, ϒ ω 0 where is given in Eq. (8) (with q ¼ ). The first term in the measurement operator cannot be understood as mapping Eq. (C7) is exactly the state that one would expect if the pure states to pure states as assumed in Eq. (8). grating protocol worked perfectly, while the second term is We must therefore consider the measurement procedure an incoherent mixture of displacements. We can therefore (used to impose the grating) in the density matrix descrip- understand qubit dephasing as some classical probability tion. Before switching the interaction on, i.e., while λ 0 π 2 that the resonator will coherently pass the grating and ðtÞ¼ , the = pulse changes the jþihþj state of the some probability that will we end up with an incoherent ρ 1 − − qubit to q ¼ 4 ðjþi þ j iÞðhþj þ h jÞ. The state of the mixture. Since the normalization is state dependent, we mechanical resonator is left unchanged in an arbitrary state cannot simply relate the coefficients in Eq. (C7) with direct ρ m. This joint state must be separable because the initial- probabilities. However, we can say that the relative ization of the qubit state at the beginning of the grating probability of introducing an incoherent mixture is operation has the effect of destroying any qubit-resonator ð1 − e−γtÞ=e−γt ¼ eγt − 1 ≈ γt for short times or a low entanglement. ρ dephasing rate. The trace of m; is the probability of finding As the interaction is switched on, the joint state of the the qubit in the ji state, and using Eq. (C6), we may read off system evolves according to 1 e−γt † p ðα; ϕÞ¼ ½eiϕχð−2αÞþe−iϕχð2αÞ; ðC8Þ ρ(t; WðtÞ) ¼ UðtÞρq ⊗ ρmU ðtÞ 2 4 1 pffiffiffiffiffi 2i γ=2WðtÞ ¼ e j−ihþj ⊗ DðαÞρmDðαÞ where χð·Þ¼Tr½Dð·Þρ is the characteristic function 2 D α ρD α 1 pffiffiffiffiffi of the mechanical state [using Tr½ ð Þ ð Þ ¼ −2i γ=2WðtÞ † † D α D α ρ D 2α ρ þ e jþih−j ⊗ D ðαÞρmD ðαÞ Tr½ ð Þ ð Þ ¼Tr½ ð Þ , etc.]. The characteristic 2 function is related to the Wigner distribution via a symplectic 1 þ jþihþj ⊗ D†ðαÞρ DðαÞ Fourier transform, 2 m Z 1 α − α − − ⊗ D α ρ D† α χðαÞ¼ dxdpWðx; pÞeix i ip r ; ðC9Þ þ 2 j ih j ð Þ m ð Þ; ðC3Þ α α α 0 where UðtÞ is the unitary operator in Eq. (C2). Since WðtÞ is where rðiÞ is the real (imaginary) part of .IfReð Þ¼ , unknown, the resulting quantum state at time t must be then

021052-10 DISPLACEMON ELECTROMECHANICS: HOW TO DETECT … PHYS. REV. X 8, 021052 (2018) Z 2 α APPENDIX D: CLASSICAL χð2α Þ¼ dxdpWðx; pÞe ix i i INTERFERENCE PATTERNS Z ¼ dxPðxÞ½cosð2α xÞþi sinð2α xÞ; ðC10Þ To verify that oscillations are in fact due to quantum i i interference, we consider what types of interference pat- terns can be understood using a classical description of where the complex part must vanish as χð·Þ is a real function the resonator. Consider a classical (superscript cl) checker- (for states with π rotational symmetry). This is simply the board phase-space probability density of dimensionless overlap integral between the position probability distribution variables ðx; pÞ, α PðxÞ and a diffraction grating with a pitch j j. Therefore,   2 2 R 1 x þ p 2 2 1 −γt 2 α Pclðx; pÞ¼ exp − cos ðαxÞcos ðαpÞ; ðD1Þ α ϕ e dxPðxÞ cosð xj jÞ ϕ N 2σ2 pð ; Þ¼2 2 cosð Þ; ðC11Þ σ α 2 α where N is a normalization factor and and are the width which exactly probes the j j wave-number components in and wave number of the checkerboard pattern, respectively. PðxÞ, with a reduced amplitude from the qubit dephasing. This is the conditional state after the resonator has passed Thus, we have seen that the effect of qubit dephasing is the first two gratings (of α ¼ α1 ¼ α2) separated by a to introduce some probability of having an incoherent quarter-period rotation. As the distribution rotates in phase mixture of different momentum kicks, thus suppressing any space, we are interested in the wave-number components signatures of interference in the outcomes of qubit mea- present in the reduced probability of PclðxÞ, where surements. Since the duration of the protocol is on the order of about N mechanical oscillations, t ≈ 2π × N=Ω,to Z neglect qubit dephasing, we need γ=Ω ≪ 1=N. For the PclðxÞ¼ Pclðxcosθ þpsinθ;pcosθ −xsinθÞdp ðD2Þ parameters discussed in the main text, this requires γ=2π < 1 MHz. and θ is the phase-space rotation angle. If this probability distribution is now probed by a third grating, then the 2. Loss of resonator coherence probability of finding the qubit in the jþi state is [from α ϕ Eq. (12)] To see that oscillations in pþðj j; Þ are a quantum effect, we consider the effect of adding n0 thermal phonons Z   ϕ to the state of the resonator immediately before the third cl 0 2 0 cl 0 p ðα3; ϕÞ¼ dx cos jα3jx þ P ðx Þ grating [Eq. (13), restated here for convenience], þ 2 2 2 2 Z ∝ e−2σ ðjα3j þ2jα3jjαj( sinðθÞþcosðθÞ)þ2jαj Þ −jβj2=n0 ðdÞ 2 e † 2 2 ρm ¼ d β D½βρ D ½β: ðC12Þ 8jα3jjαjσ cosðθÞ 2 2jαjσ ð2jα3j cosðθÞþjαjÞ 1 πn0 m × ½e þ e þ 2 2 × ½e8jα3jjαjσ sinðθÞ þ 2e2jαjσ ð2jα3j sinðθÞþjαjÞ þ 1 In this case, × cosðϕÞ; ðD3Þ α ϕ ϒ† ϒ ρðdÞ pþðj j; Þ¼Trm½ þ þ m 1 Z the amplitude of which peaks at fΩτ2; jα3=αjg ¼ f2 nπ; 1g −jβj2=n0 pffiffiffi 2 e † † 1 1 π 2 σ ≥ α ¼ d β Tr ½D ½βϒ ϒ D½βρ and fð2 n þ 4Þ ; g for j j. A plot of this function πn0 m þ þ m Z (Fig. 7) looks qualitatively the same as Fig. 6(b), with −β2 0 e r =n the difference being attributable to the superposition of ¼ dx0dβ pffiffiffiffiffiffiffi Pðx0Þ r π 0  n  ϕ 2 α 0 2β × cos j jðx þ rÞþ2 (a) (b) (c) R 1 −4n0jαj2 2 α e dxPðxÞ cosð xj jÞ ϕ ¼ 2 þ 2 cosð Þ; ðC13Þ where Trm denotes a trace over the mechanical degrees of freedom. We therefore see that any loss of coherence FIG. 7. Probability pþ of finding the qubit in the excited state if between the second and third grating reduces the amplitude the resonator were described by a classical probability distribu- −4n0jα2j σ 0 5 of the oscillations in pþ by a factor e . tion using width ¼ . , 1, 5 from (a) to (c).

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TABLE I. Parameters for a selection of implemented or proposed qubit-resonator coupled systems for carrying out the experiment in the main text. The systems are as follows: (1) this proposed device, (2) proposed levitated NV center in a magnetic field gradient, (3) implemented magnetic cantilever coupled to a NV center, (4) proposed optimization of the device in (3), (5) implemented NV center strain-coupled to diamond cantilever, (6) proposed combination of the Al beam from Ref. [75] with the qubit of Ref. [37], in a field of 10 mT, (7) proposed combination of the SiN membrane from Ref. [76] with the qubit of Ref. [37], in a field of 10 mT. The membrane is taken as supporting one arm of the qubit with length l ¼ 300 μm. Column (8) is the proposed combination of the optimized cantilever from Ref. [55] with the qubit of Ref. [37]. The field from the cantilever is taken as the coupling to an enclosed area of 0.1 μm−2. None of these systems enters the bare ultrastrong coupling regime where λ=Ω > 1. However, it is possible to enter the toggled ultrastrong coupling regime where λ exceeds both the qubit and mechanical dephasing rates.

System 1 2 3 4 5 6 7 8 † † Reference This work [14] [55] [55] [77] [37,75] [37,55] [37,76] Parameters Frequency Ω=2π (Hz) 1.25 × 108 105 8.0 × 104 106 9.19 × 105 1.0 × 106 1.23 × 105 106 Mass m (amu) 3 × 106 8 × 109 7 × 1015 7 × 1015 3 × 1010 7 × 1012 1 × 1017 7 × 1015 5 2 6 4 5 7 6 Quality factor Qm 10 2 × 10 10 1 × 10 6 × 10 4 × 10 10 Temperature T (K) 3.3 × 10−2 1 × 10−3 3 × 102 1 × 10−1 3 × 102 1.3 × 10−1 1.8 × 10−1 1 × 10−1 −6 −4 −4 −3 −6 −6 −6 −6 Qubit T2 2 × 10 10 1.6 × 10 2 × 10 2 × 10 2 × 10 2 × 10 2 × 10 Coupling constant λ=2π (Hz) 4.2 × 106 1.5 × 103 1.6 × 101 2 × 104 1.8 2 × 105 9 × 103 2.4 Dimensionless figures of merit Coupling parameter λ=Ω 3 × 10−2 2 × 10−2 2 × 10−4 2 × 10−2 2 × 10−6 2 × 10−1 7 × 10−2 2 × 10−6 2 −10 1 −9 1 1 −3 Mechanical parameter λ=κth 6 × 10 5 × 10 1 × 10 3 × 10 5 × 10 8 × 10 1 × 10 1 −3 1 −6 −1 −2 −6 Qubit parameter T2λ=2π 8 2 × 10 3 × 10 4 × 10 4 × 10 4 × 10 2 × 10 5 × 10 † In these references, the dephasing time T2 is not directly reported, and we have therefore optimistically used the decoherence time T2 instead. momentum kicks that accompanies the measurement when [2] S. Eibenberger, S. Gerlich, M Arndt, M. Mayor, and J. the resonator is quantized. This confirms that the proba- Tüxen, Matter- of Particles Selected from bility of finding the resonator in the jþi state can therefore a Molecular Library with Masses Exceeding 10,000 amu, be used to distinguish quantum interference patterns Phys. Chem. Chem. Phys. 15, 14696 (2013). [Figs. 6(a) and 6(c)] from classical Moir´e patterns that [3] L. Diósi, Models for Universal Reduction of Macroscopic Quantum Fluctuations, Phys. Rev. A 40, 1165 (1989). arise from a classical probability distribution (Fig. 7). [4] R. Penrose, On Gravity’s Role in Quantum State Reduction, Gen. Relativ. Gravit. 28, 581 (1996). APPENDIX E: OTHER DEVICE [5] M. P. Blencowe, Effective Field Theory Approach to Gravi- IMPLEMENTATIONS tationally Induced Decoherence, Phys. Rev. Lett. 111, 021302 (2013). To assess the experimental feasibility of our scheme, [6] D. Kafri, J. M. Taylor, and G. J. Milburn, A Classical Table I presents parameters of the resonator, the qubit, and Channel Model for Gravitational Decoherence, New J. the coupling strength for various devices that could be used Phys. 16, 065020 (2014). to implement it. The challenge is to achieve ultrastrong [7] I. Pikovski, M. Zych, F. Costa, and C. Brukner, Universal coupling between the qubit and resonator without intro- Decoherence Due to Gravitational Time Dilation, Nat. ducing either rapid dephasing of the qubit or thermal Phys. 11, 668 (2015). decoherence of the resonator. Assuming a toggled cou- [8] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. pling, this requires that the coupling constant λ=2π exceeds Ulbricht, Models of Wave-Function Collapse, Underlying Theories, and Experimental Tests 85 both the qubit dephasing rate 1=T2 and the resonator , Rev. Mod. Phys. , 471 κ (2013). thermal dephasing rate th, as tabulated in the last two rows of Table I. No existing device achieves this, although [9] W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Towards Quantum Superpositions of a , Phys. Rev. an optimized magnetic cantilever coupled to a NV center in Lett. 91, 130401 (2003). diamond would be promising. Thus, the device of Fig. 1 is [10] O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, particularly attractive for investigating mesoscopic quan- N. Kiesel, M. Aspelmeyer, and J. I. Cirac, Large tum interference in nanomechanics. Quantum Superpositions and Interference of Massive Nanometer-Sized Objects, Phys. Rev. Lett. 107, 020405 (2011). [11] J. Bateman, S. Nimmrichter, K. Hornberger, and H. [1] K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and Ulbricht, Near-Field Interferometry of a Free-Falling M. Arndt, Colloquium: Quantum Interference of Clusters Nanoparticle from a Point-like Source, Nat. Commun. 5, and Molecules, Rev. Mod. Phys. 84, 157 (2012). 4788 (2014).

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