Quantum erasure using entangled surface acoustic phonons

Using the deterministic, on-demand generation of two entangled phonons, we demonstrate a quantum eraser protocol in a phononic interferometer where the which-path information can be heralded during the interference process. Omitting the heralding step yields a clear interference pattern in the interfering half-quanta pathways; including the heralding step suppresses this pattern. If we erase the heralded information after the interference has been measured, the interference pattern is recovered, thereby implementing a delayed-choice quantum erasure. The test is implemented using a closed surface-acoustic-wave communication channel into which one superconducting qubit can emit itinerant phonons that the same or a second qubit can later re-capture. If the first qubit releases only half of a phonon, the system follows a superposition of paths during the phonon propagation: either an itinerant phonon is in the channel, or the first qubit remains in its excited state. These two paths are made to constructively or destructively interfere by changing the relative phase of the two intermediate states, resulting in a phase-dependent modulation of the first qubit's final state, following interaction with the half-phonon. A heralding mechanism is added to this construct, entangling a heralding phonon with the signalling phonon. The first qubit emits a phonon herald conditioned on the qubit being in its excited state, with no signaling phonon, and the second qubit catches this heralding phonon, storing which-path information which can either be read out, destroying the signaling phonon's self-interference, or erased.


I. INTRODUCTION
Quantum mechanics famously uses dual descriptions for quantum objects, representing these as waves or as particles depending on the situation. This is a manifestation of complementarity, and is central to understanding many interferometric experiments. The prototypical example is Young's two-slit experiment [1]: A wave description predicts an interference pattern, while a classical particle-based description, in which the path followed by the particle is known, shows no pattern. For a quantum object passing through a two-path interferometer, an interference pattern is expected, but detecting which path the quantum follows changes this to a non-interfering particle-like description. Since the early days of quantum mechanics, many thought experiments (see e.g. [2,3]) and their experimental realizations have tested the validity and domain of application of these orthogonal representations. These have led to the currently-accepted understanding that the wave or particle nature of a quantum remains undetermined until a measurement occurs.
Among these experiments, a quantum eraser scheme, as proposed by Scully and Drühl [4], investigates whether it is possible to undo the act of determining which path the quantum followed: is it possible to recover an interference pattern that was suppressed by acquisition of which-path information, by "erasing" that information? This can be investigated using a three-step process: (1) observing an interference pattern in a two-path interferometer; (2) acquiring which-path information and observing the corresponding suppression of the interference; and (3) erasing the which-path information and recovering the interference pattern. This test can further be combined with a version of Wheeler's delayed-choice test [3,5], where the act of recombining the paths of an interferometer occurs after the quantum has entered the interferometer, thereby preventing the quantum from "choosing" a wave or particle nature before the superposition has been created. For a quantum eraser, in fact, the results should remain unchanged even if the acquisition and erasure of the which-path information occurs after the registration of the interferometric effect.
Realizations of quantum erasers have so far used photons, in both the optical and microwave bands. The first experimental realization used optical photons and marked the photon's propagation through a specific path by creating a path-specific polarization [6]. The first delayed-choice eraser test [7] triggered the emission of entangled photon pairs on each path of the interferometer, using one set of photons to complete the propagation through the interferometer, and the other set to mark and erase the which-path information. Further tests used setups where the marking of the which-path information and the interference detection took place at spatially distant locations, making the test robust to locality loopholes [8]. More recently, a quantum eraser test using superconducting qubits and microwave photons was realized using a Ramsey interferometer, where the which-path information was acquired by coupling to an ancillary cavity [9].
Here, we propose and implement a quantum eraser scheme using surface acoustic wave (SAW) phonons [7]. Building on a previously demonstrated interferometer [1], we implement the quantum erasure process by constructing a two-phonon entangled state, with the second phonon marking the which-path information. The slow propagation of this 'herald' phonon is exploited to delay the which-path information detection after detection of the result of the interferometric process, allowing for a delayedchoice quantum erasure.
The interferometry scheme we use for the quantum eraser protocol is described in Ref. [1]. The experimental layout of the device is shown in Fig. 1a. Two nominally identical superconducting qubits [5,31], Q 1 and Q 2 , are coupled via two tunable inductive couplers [8] to a phonon channel comprising a central interdigitated transducer (IDT) located between two reflective mirror gratings. Each qubit can relax into this channel at a rate κ(t), controlled by its tunable coupler, emitting counterpropagating itinerant surface acoustic wave phonons via the IDT when the qubit is tuned near the IDT operating frequency of ∼ 4 GHz. The two SAW mirrors, made of thin metallic gratings on either side of the IDT, ensure reflection of the phonons back towards the IDT when the phonons are in the mirrors' 125 MHz-wide operating bandwidth. Either qubit can efficiently re-absorb the itinerant phonons after the phonons complete a ∼ 500 ns-long round-trip: The tunable couplers' dynamic tuning is used to shape each emitted phonon wavepacket as well as to control their absorption [2], enabling in theory their complete re-capture by either of the qubits [34]. Experimentally, the qubit-to-qubit transfer efficiency is measured to be η ∼ 65%, limited by acoustic losses in the SAW device [1].
Here, we make use of the three lowest-energy qubit states, |g , |e and |f . The qubits' anharmonicities χ/2π = (ω ef − ω ge )/2π are respectively −179 and −188 MHz. The qubit intrinsic lifetimes are T 1 = 18 µs for both qubits, while the g-e  Figure 1. Experimental set-up and quantum eraser scheme. (a) Two transmon superconducting qubits (blue) are coupled to a surface acoustic wave phononic channel (grey) via a central interdigitated transducer (IDT, green), using which both qubits can emit and capture itinerant phonons. The IDT is placed between two reflective mirror gratings (orange) that define a Fabry-Pérot cavity and reflect phonons within the mirrors' bandwidth back towards the IDT. Two tunable couplers (red) are used to dynamically control the coupling between the qubits and the IDT, allowing shaping the wavepackets of emitted phonons, and ensuring their efficient re-absorption after completing the 500 ns acoustic round-trip. The couplers also enable the controlled partial release of phonons. (b) Optical micrograph of the device, showing (top) the acoustic Fabry-Pérot structure on a lithium niobate chip, (bottom) the two superconducting qubits and associated superconducting wiring on a separate sapphire chip, and (middle) a side-view of the flip-chip assembled device. (c) When one of the qubits (Q1) swaps a half-phonon (A) into the acoustic channel, an interferometer can be implemented (green box): once A completes a round-trip within the acoustic cavity, its re-absorption probability by Q1 depends on the relative phase accumulated by Q1 and A, and leads to interference in Q1's final excitation probability. To implement a quantum eraser, we generate an entangled phonon herald marking the which-path information by generating a second, entangled phonon (B) conditionally on Q1 being in |e (blue box): this suppresses the interference. Capture and detection of the entangled herald B by Q2 acquires the which-path information after the interference of Q1 and A is complete, making this a time-delayed herald. Subsequently applying a π/2 pulse to Q2 equalizes its |g and |e populations, erasing the which-path information and restoring the interference, thereby completing a delayed quantum eraser measurement. transition has a Ramsey T 2,ge,R = 1.2 µs (0.8 µs) for Q 1 (Q 2 ), and T 2,ef,R = 0.4 µs for both qubits' e-f transition. More details on the device and the phonon emission-capture protocol are available in [1].
A two-path interferometer can be realized in this device, shown in Fig. 1b, by initializing one of the qubits (here Q 1 ) in its excited state and using its coupler to emit a half-phonon (A) with a symmetric wavepacket into the SAW channel. This results in the superposition state writing Q 1 's state first and the phonon state second. Applying a detuning pulse on Q 1 of varying length introduces a relative phase ϕ between the states |e0 and |g1 (defined here to be the phase accumulated by the phonon with respect to the qubit), yielding oscillations in the qubit occupancy after Q 1 re-captures the phonon [1]. The origin of the interference can be understood by considering the outgoing acoustic field. This field has two contributions: the reflection of the incoming field combined with the field emitted by the qubit, whose population is also affected by the incoming field. External control of the qubit coupling rate κ(t) ensures that the two contributions are equal in amplitude. The energy in the outgoing acoustic field thus only depends on the relative phase factor e iϕ . When ϕ = 0, absorption is the time-reversed emission process so that the qubit goes back to |e . The interference can be seen as destructive since the acoustic field reflected from the qubit acquires a π phase shift and cancel out the acoustic field re-emitted by the qubit and thus no phonon is re-emitted. When ϕ = π, the interference is constructive, and the qubit energy is transferred to the acoustic channel, leaving the qubit in |g ; the re-emitted phonon eventually decays in the acoustic channel. The final state of the system can thus be written as a function of ϕ, resulting in the observation of an interference pattern in Q 1 's final excited state probability P e (t f ) when sweeping the phase ϕ, with a period of 2π. Two steps are required to realize a quantum eraser in this interferometer configuration. The first is to create which-path information, i.e. a herald indicating whether the qubit remained excited or instead phonon A was emitted in the acoustic channel. Obtaining this information should result in the disappearance of the interference pattern, because this entangles the system under observation -the qubit and traveling phonon A -with the measurement apparatus. The second step is to erase this knowledge, and look for a recovery of the interferometric pattern. Here, we use a protocol similar to that used in the original quantum eraser proposal [4] as shown in Fig. 1b. This protocol requires the on-demand generation of a second, entangled phonon to serve as a herald of the first, signalling phonon. Following the signaling half-phonon emission, we apply a transition-selective π pulse on the e-f transition of the qubit Q 1 , then turn on the coupler, inducing Q 1 to emit a second phonon B if initially in |e . This phonon thereby heralds that the qubit is in its excited state (and that there is no A phonon in the channel). Including the herald, the system state before re-absorption of phonon A is then displaying the entanglement of phonons A and B. The entanglement of Q 1 with phonon B makes the two states of the interferometer orthogonal, even after re-capture of phonon A, and prevents any interference. Phonon B is then captured by qubit Q 2 , putting Q 2 in |e if Q 1 was in |e , transferring phonon B's entanglement to Q 2 and thus placing the which-path information in Q 2 (this occurs after the interference has taken place, due to phonon B's long (0.5 µs) transit time). The which-path information can be erased by subsequently applying a π/2 pulse to Q 2 , mapping Q 2 's state to a superposition of |e and |g . For a particular phase choice for this π/2 erasure pulse, the final state of the system can be written as where qubit Q 2 's state is written last. This expression shows that Q 2 's state remains entangled with the interferometer, but a measurement along its quantization axis no longer yields which-path information. The interference is therefore not directly recoverable by only measuring Q 1 , but can be restored with a joint measurement of Q 1 and Q 2 . This is similar to photon-based realizations of quantum eraser tests [6][7][8][9], and the original quantum eraser proposal [4].

III. WHICH-PATH HERALD
Implementing the quantum eraser scheme hinges on our ability to emit a heralding phonon (phonon B) on Q 1 's e-f transition, while preserving Q 1 's excited and ground-state populations. For a superconducting qubit coupled to a microwave environment, this can be achieved by either engineering the qubit's environment [35][36][37], or manipulating the qubit's coupling to the environment [38]. In our experiment, we make use of the former and harness the frequency-dependent response of the IDT [3,6,7,26,29]. For a non-reflective uniform IDT of the type used here, the power conversion between microwave electrical and acoustic signals is proportional to the IDT conductance G a (ω): where X = πN (ω − ω c )/ω c , N = 20 is the number of IDT finger pairs, ω c = 2πv/p the IDT central radial frequency, p = 0.985 µm the IDT pitch and v the SAW velocity within the IDT. The uniform profile of the IDT implies that G a = 0 for X = ±π. At the corresponding frequencies ω ±π , the qubit relaxation by phonon emission should be suppressed. For this device, the qubit anharmonicity α is quite close to the difference between the IDT conductance minima at ω ±π and the IDT central frequency ω c . By tuning the qubit's g-e emission frequency to ω ge ∼ ω π , the e-f transition is brought close to the IDT main emission peak, ω ef ∼ ω c . Phonon emission on the e-f transition is thus close to its maximum, while emission on the g-e transition is heavily suppressed, making the proposed quantum eraser scheme possible. This is shown in Fig. 2. Qubit Q1 frequency ω ge /2π Figure 2. Single-qubit frequency-and state-dependent energy decay. (a) We monitor the decay of Q1's state after excitation respectively to |e and |f (pulse sequence is in inset of panel b), dominated by emission of phonons into the IDT. Q1's coupler is set to maximum coupling and Q2's coupler is turned off. (b) Fitting the population evolution (see [40]) enables us to extract the transition rate κge of transition g-e (blue) and the transition rate κ ef of transition e-f (red) as a function of Q1 frequency. The frequency dependence of each transition rate is seen to follow the frequency-dependence of the IDT conductance (c). We identify two operating points ωA and ωB. At frequency ωge = ωA, phonon emission on the g-e transition dominates, resulting in phonon emission at ωge/2π = 3.95 GHz within the mirror bandwidth (orange), while decay on the e-f transition is suppressed. Similarly, at frequency ωge = ωB = 2π × 4.15 GHz, phonon emission on the e-f transition dominates, resulting in phonon emission at ω ef = ωge − |α|= 2π × 3.97 GHz (grey dashed line), also within the mirror bandwidth, while decay on the g-e transition is suppressed.
The κ ge (ω ge ) emission rate displays close to the expected behavior, as shown in Fig. 2. Similarly, κ ef (ω ge ) also displays roughly the expected behavior: a shift in frequency by α compared to κ ge (ω ge ) and a factor of two increase in the rate (×2.1 comparing the κ ge to the κ ef maxima), as expected for a weakly anharmonic qubit. The expected behavior, plotted as solid gray lines for both emissions in Fig. 2b, is calculated from the qubit coupling to the IDT and the internal IDT frequency reflections, using an electrical model for the circuit and a coupling-of-modes model for the IDT [7]. These account for the nonlinearity of the qubit using "black-box quantization" [8,41]. The resulting modeled rates only account partially for the experimental results: while the agreement is satisfactory for the e-g decay rates, we find a 50 MHz misalignment in the modeled maximum of the f -e decay compared to measurements. The modeling is explained in detail in [40].
We extract two operating points, both within the IDT mirror bandwidth (3.91 GHz-4.03 GHz). At ω ge = ω A = 2π×3.95 GHz, the g-e emission time is 1/κ ge = 9.3 ± 0.1 ns while the e-f decay is suppressed by a relative factor κ ge /κ ef = 5.9 ± 0.1, strongly favoring the emission of phonons on the g-e transition. When ω ge = ω B = 2π × 4.15 GHz, the emission time is 1/κ ef = 4.8 ± 0.1 ns, with the phonon emitted at ω ef = ω B − |α|= 2π × 3.97 GHz while the decay on g-e is suppressed by a factor κ ef /κ ge = 84 ± 3: this is the operating point for emitting the which-path herald.
Operating at frequency ω A , we use the tunable couplers to efficiently shape the emitted and absorbed wave-packets, see [1,2,34,40,[42][43][44]. The couplers are controlled so the emitted wavepackets have a cosecant shape with characteristic time 1/κ c = 15 ns [40]. In Fig. 3a, we measure the transfer efficiency by emitting one phonon using Q 1 's g-e transition and capturing it later using Q 2 's g-e transition, with an efficiency η A = P 2e (t f )/P 1e (0) = 0.66 ± 0.01, limited by acoustic losses [1,40]. The same operation realized using Q 1 's e-f transition (Fig. 3b) while operating at frequency ω B yields the same efficiency, η B = P 2e (t f )/P 1f (0) = 0.64 ± 0.02. Due to the imperfectly-suppressed 1/κ ge = 0.4 µs decay, a small population is transferred from |e to |g during this process, leading to P 1g (t f ) = 0.06 ± 0.02. As a consequence, exciting and then emitting a phonon on the e-f transition to herald the which-path information will have at most a η h = 94 ± 2% efficiency due to this spurious decay. The probability of actually detecting this information is limited to η B .

IV. QUANTUM ERASURE IMPLEMENTATION
We implement the full quantum eraser scheme as shown in Fig. 4. First, we demonstrate single-phonon interferometry without heralding: Qubit Q 1 , initialized in |e , emits, and later re-captures, a half-phonon on its g-e transition at ω A . Following release, a detuning pulse applied to Q 1 accumulates a phase ϕ between the traveling half-phonon and Q 1 (pulse sequence in panel a; intermediate measurements in panel b). This results in an interference pattern in the final excitation probability P e1 (t 1 = 650 ns) of Q 1 as a function of ϕ (panel c). The oscillations have an average occupation of 0.41 with peak-to-peak amplitude 0.49. These are reduced from the ideal values of 1/2 and 1 due to acoustic losses, Q 1 decoherence, and the finite readout visibility. Taking these effects into account, a numerical model (see [40]) provides similar results (panel c).
A which-path herald is generated by inserting an intermediate π pulse on Q 1 's e-f transition followed by emission of a phonon at ω B on Q 1 's e-f transition, returning Q 1 to |e (see panels Fig. 4a and b). Generating the herald destroys the interference pattern, as expected. The amplitude in the heralded P e1 displays small fluctuations with amplitude ∼ 0.01. This could be  attributed to the imperfect information acquisition discussed in Fig. 3, with our model shown by the dashed line, but is below the noise threshold. We note that even if the heralding phonon is not captured and detected via Q 2 , the interference is not recovered, as Q 1 's state is now irremediably coupled to the herald and thus to the environment. The final step of the quantum eraser test is to erase the heralded information, and thereby recover the interference pattern. As the heralding phonon marks whether Q 1 was in |e , its capture using Q 2 followed by a π/2 pulse on Q 2 's g-e transition erases the information that could distinguish the two paths. This erasure can be performed in a time-delayed manner by capturing the herald and measuring Q 2 after the measurement of Q 1 . We thus implement the measurement of Q 1 immediately following its interaction with the returning half-phonon, completing the interferometry, and before absorbing and detecting the herald using Q 2 . This requirement limits Q 1 's readout time to 200 ns, decreasing its readout visibility from 96% to 81%.
As Q 2 is still entangled with the interferometer, simply tracing out Q 2 's state (equivalently, not measuring Q 2 ) will not recover the interference pattern; instead, we must condition the measurements of Q 1 on measurements of Q 2 , even though measuring Q 2 does not yield any heralded information (see Eq. (4)). In Fig. 4d, we plot all joint qubit probabilities as a function of ϕ: all have an oscillation pattern of amplitude 0.12, while the excitation probabilities P e1 , P e2 for each qubit evaluated separately only display very weak oscillations, below 1%. To make a fair comparison with the original interference pattern, we next consider the conditional measurement P e1|e2 = P ee /(P ge + P ee ), the probability of measuring Q 1 in |e conditioned on Q 2 being measured in |e . This probability has a mean identical to that measured without a herald, but the amplitude of the oscillations is reduced by 48%, due to the inefficient capture of the second phonon and thus an incomplete erasure of information, as well as the additional decoherence in Q 2 .
A model taking into account these losses and Q 2 's finite coherence time partially accounts for the amplitude reduction, as shown by the dashed line. We attribute the remaining discrepancy to decoherence occurring during the measurement, which we have not taken into account.
In conclusion, we have successfully completed a quantum eraser protocol, using an acoustic Fabry-Pérot interferometer. We realized three distinct steps in this process, first observing an interferogram; next, marking the which-path information which makes the interference fringes disappear, and third, erasing the which-path information which leads to the recovery of an interference signal. The erasure of the which-path information occurs after registering the result of the interference, making this a delayed-choice quantum eraser. The which-path detection was implemented by signaling using a heralding phonon.
This construct enabled us to demonstrate and exploit a two-phonon entanglement, opening the door to two-phonon interferometry, acoustic Bell tests [45] and phonon coherence length measurements [46]. Phonon heralding as demonstrated here could also be used to mitigate propagation losses in future acoustic experiments and implement for example high-fidelity acoustic quantum state transfer and remote entanglement, using schemes analogous to Refs. [47,48].   With Q1 in |e , its coupler is used to half-release a phonon at ωA (blue). Q1's frequency is then detuned, accumulating a phase ϕ between the half-phonon and Q1's |e state. An optional π pulse on Q1's e-f transition (red) is followed by the coupler-controlled emission of a phonon at ωB (orange), heralding that Q1 is in |e and returning Q1 to |e . Following the optional heralding, Q1 catches the half-phonon (blue) and is measured, completing the interferometry. Following Q1's measurement, Q2 catches the optional heralding phonon (orange) and is measured at time t2 = 1.1 µs. (b) Left: Q1's |e and |f state populations as a function of time t, showing the unheralded Pe(t) for ϕ = 0 and ϕ = π, which at time t1 displays the interference maximum and minimum, and for the heralded Pe, which at time t1 does not have a ϕ dependence. Also shown is Q1's |f state population when the herald is generated. Right: Q2's excited state Pe(t) when the herald is generated, which ideally would reach the value 1/2 but is limited by acoustic losses to η b × 0.5 ≈ 0.32. (c) Interference fringes Pe(ϕ) are visible when the herald is absent, but disappear when the herald reports which-path information (Q1 in |e ). If the herald is generated but the information in Q2 is erased, by applying a π/2 pulse on Q2's g-e transition, the fringes reappear when Q1's measurement is conditioned on measuring Q2 in |e . This occurs even though Q1's measurement was already complete by the time the information in Q2 is erased. (d) Probability of measuring Q2 in |e , showing lack of dependence on ϕ. Also shown are variations in two-qubit probabilities Pgg, Pge, Peg and Pee. Inset shows the lack of variation of both qubits' |e -state probabilities Pe1 and Pe2 as a function of ϕ when applying the π ef pulse. Dashed lines in all panels correspond to a numerical model taking into account the qubits' decoherence and phonon losses, see [40].
Supplementary Materials for Quantum erasure using entangled surface acoustic phonons

I. DEVICE, EXPERIMENTAL SETUP AND TECHNIQUES
The flip-chip device, setup and techniques used for this experiment are strictly identical to [1], except that the data shown in this paper were acquired in a separate cool-down of the cryostat used for the experiment (base temperature < 7 mK). A full wiring diagram and a description of the room-temperature set-up may be found in Ref. [2]. The fabrication description is given in Ref. [3]. The circuit is shown in Fig. S1. Compared to Ref. [1], we note a 5% shift in the nominal values of the Josephson junctions of the two tunable couplers, as well as an overall reduction of the coherence times of the qubits.
For this run, we implemented in addition a three-state dispersive readout. Each qubit readout resonator is a λ/4 resonator inductively coupled to a λ/2 Purcell filter. A 500-ns microwave tone is applied at resonance with each qubit readout resonator and the transmitted signal is successively amplified by a traveling-wave parametric amplifier [4], a high-electron mobility transistor amplifier, and a room-temperature amplifier, before homodyne mixing and recording the integrated value of the quadrature amplitudes I and Q. To estimate the fidelity of the preparation and readout of each state, we successively prepare each qubit in |g , |e or |f and repeat each measurement 4000 times. The state-dependent dispersive shift of the readout resonator allows us to attribute a sector of the IQ plane to each state, enabling us to identify the qubit state from any single-shot readout based on its recorded I and Q values. These calibrations also determine the fidelity of each state readout, which are all above 90%, see Table S1. Data shown in Fig. 3 and Fig. 4b in the main text are corrected for readout errors using this calibration.
To perform a delayed-choice quantum eraser test, we modified the Q 1 measurement procedure to fit within a phonon round-trip time, by shortening its readout pulse from 500 ns to 200 ns. When performing a two-state readout, this decreases the visibility of Q 1 's |e and |g states to 81%.

II. RELAXATION RATES AND CIRCUIT MODELING
In this section, we describe the modeling and the measurements of the relaxation rates when one qubit (Q 1 ) is maximally coupled to the IDT, and the other qubit is disconnected (coupler 2 turned off), see Fig. 2 in the main text. For a given qubit frequency, the operating points are determined by (1) maximizing the coupler-induced frequency shift on the qubit, (2) maximizing the other qubit relaxation time. We prepare the qubit in |e or |f by the successive application of resonant π pulses and measure the qubit state populations after a varying amount of time t during which coupler 1 is open. The measurements realized on Q 1 for the two operating points described in the main text are shown in Fig. S3.
A weakly anharmonic transmon-or xmon-style qubit is expected to have decay rates very similar to a harmonic oscillator [5], with the population of the |e and |f excited states evolving aṡ where κ ef = 2κ ge and κ gf = 0. Here, due to the IDT response, we measure a very different behavior. We first make the assumption that κ gf = 0, as this two-phonon relaxation is expected to be exponentially suppressed for a transmon [5]. To determine the rates κ ge and κ ef , we start by fitting the decay from |e after excitation to |e with a single Figure S1. Electrical circuit. Elements in blue are the qubit equivalent circuits, in red the variable couplers and the inductive couplers between the qubit sapphire chip and the acoustic lithium niobate chip, and green the interdigitated transducer (IDT) for phonon emission and capture. Figure S2. Qubits Q1 (a) and Q2 (b) single-shot readout using a 500-ns readout pulse. Dots indicate the coordinates in the IQ-plane of each integrated integrated readout pulses for the 4000 measurements realized after preparing each qubit either in |g (blue), |e (red) or |f (green). This calibration allows us to assign any given measurement to the ground, excited or second excited state, as separated by the black lines in the IQ plane. Corresponding fidelities are given in inset.
decaying exponential for all qubit frequencies. We only consider times past the transient on-set of the coupler (t ≥ 3 ns) and prior to any re-excitation of the qubit by the phonons reflected off the mirrors (t < 500 ns) when within the mirrors' bandwidth.
We also fix the steady-state populations by measuring the qubit population without any microwave excitation. This fit determines κ ge . We repeat the same single decaying exponential fit for the decay from |f after excitation to |f , determining κ ef . The |e population evolution after excitation to |f is modeled by Eq. S7 using the two fitted rates. The resulting fits are shown in Fig. S3 for the operating frequencies ω A and ω B defined in the main text for Q 1 , and agree very well with the data. The frequency dependence of κ ge and κ ef is shown in Fig. 2 of the main text.
We now consider the possibility of a two-phonon relaxation process, under the hypothesis that it could be strongly enhanced due to the frequency dependence of the IDT [6]. The two-phonon relaxation is expected to be maximal when ω ge − |α|/2 matches the IDT central frequency. We fit the population evolution from |f using a two-parameter fit and keeping κ ge as given by fitting the decay from |e after excitation to |e . The result is shown in Fig. S3c. The extracted κ gf reaches a maximum of 1/30 ns. At the operating frequencies of the main text, ω A and ω B , the ratio κ gf /κ ef is below 10%. The uncertainty of this determination is also quite large when κ ef is large -more than 50% whenever κ ef /2π 20 MHz. We thus conclude that even if this process occurs, it is negligible in our experiment. We attempted to model the frequency-dependent relaxation rates using a circuit model for the qubit-coupler-IDT system. The IDT is modeled using a coupling-of-modes model [7], taking into account the internal reflections occurring between the electrodes of the IDT, thus allowing us to infer the IDT admittance as a function of frequency, see Fig. S4a. For reference, we also study the ideal case of an uniform transducer with no internal reflections, where the IDT admittance is given by where C 0 is the IDT electrical capacitance, G a (ω) the IDT conductance given in the main text, and B a the IDT susceptance related to its conductance by an Hilbert transformation B a (ω) = G a (ω) * [−1/πω]. We then derive the equivalent impedance Z(ω) for the circuit shown in Fig. S1 looking into terminals A-B for qubit Q 1 (or C-D for qubit Q 2 ). We extract the circuit resonant frequencies by looking for the zeros of Z(ω) in the complex plane. Ignoring losses in the IDT (by setting Re[Y (ω)] = 0), we identify three modes: the qubit, the IDT series resonance, and the mode created by the IDT capacitor and the couplers' inductance networks. We then re-evaluate the frequencies of these modes in the presence of IDT loss. To extract the qubit relaxation rate and its anharmonicity, we approximate the circuit near the qubit resonance ω q as an RLC series circuit, with effective parameters (S11) The qubit relaxation rate κ ge = 1/T 1 is then given by T 1 ω q = Q, where the qubit quality factor is given by Q = To evaluate the anharmonicity, we split the effective qubit inductance L eff into its non-linear part, arising from the qubit SQUID inductance L q (φ q ), and its linear part L eff − L q (φ q ). The anharmonicity is then given by [8]: (S12) Finally, the relaxation rate from state |f is given by κ ef = 2/T 1,ef where T 1,ef × (ω q + α) = Q ef and Q ef is evaluated considering the following updated circuit parameters: To obtain the model relaxation rates shown in Fig. 2 of the main text, we use the parameters listed in Table S1 as input parameters. The non-design parameters were calibrated as follow: The qubit capacitance was adjusted to reproduce the qubit anharmonicity and the qubit SQUID inductance was adjusted to reproduce the measured qubit bare frequency. The couplers' Josephson junction inductances were calibrated using the qubit frequency shift induced by the coupler, with the qubit tuned at a non-zero emission point for the IDT. The mutual inductive coupling between the two chips was calibrated using the qubit-qubit direct electrical coupling (∼ g/2π = 1.1 MHz) at a non-zero emission point for the IDT. The SAW velocity for the IDT was adjusted to match the frequencies of the two zero emission points. Finally the IDT reflectivity r is an imaginary free parameter, whose value is expected to be small (|r| 1%) for a 30-nm-thick aluminum transducer fabricated on a 128 • Y − X lithium niobate wafer [7]. (c) Induced qubit anharmonicity for both types of transducers. The error in κ ef seems to arise from an overestimate of the anharmonicity.

III. NUMERICAL MODELING
In this last section, we address the numerical modeling of the quantum eraser experiment, as well as the transfers from Fig. 3 in the main text. In the quantum eraser experiment, the system comprises two qubits (Q 1 and Q 2 and the itinerant wavepackets corresponding to the phonons A and B.
We model the qubits as anharmonic oscillators with bosonic creation operatorsŝ i . Their non-interacting Hamiltonians in the frame rotating at the frequency of phonon A is given by where ∆ i is the detuning of qubit i with respect to phonon A and α i its anharmonicity. We also define the qubit matrix element operatorsŝ ge,i = |g e| andŝ ef,i = |e f | to take into account transition-dependent effects.
The itinerant wavepackets are modeled as bosonic modes. To accurately describe the emission and capture of the phonons, and model the evolution of the populations in these itinerant bosonic modes, we use the theory derived in [9]. Each interaction of the qubits with the acoustic channel requires the use of two wavepackets: an input wavepacket and an output wavepacket.
As we wish to model four interactions (the half-emission (or the herald emission) and the half-phonon capture (or the herald capture) for phonon mode A (or B)), we only need to consider six wavepackets: First, a in (t) (b in (t)), the input acoustic field that interacts with qubit Q 1 during the phonon A (B) emission at time t e,A (t e,B ). Second, a rt (t) (b rt (t)) is the acoustic output field describing the result from the interaction of a in (t) (b in (t)) with the qubit and contains the emitted phonon A (B). This field will then be used as input for the second interaction with the qubit after it completes one round-trip during the capture process. Third, the field c out (t) (b out (t)), the acoustic output field containing the phonon resulting from this second interaction. According to [9], bosonic annihilation operators can be used to describe the quantum state contained in these wavepackets, defined aŝ where the functions u w (t) describe the wavepacket envelopes and are normalized such that |u w (t)| 2 dt = 1.
To describe the interactions of the qubits with the acoustic channel during the quantum eraser experiment, we thus use six bosonic creation operators, three (â in ,â rt , andâ out ) for the half-phonon and three (b in ,b rt , andb out ) for the heralding phonon. We note thatâ rt andb rt correspond to what we call phonons A and B in the main text.
In the frame rotating at the phonon A emission frequency, the stationary Hamiltonian of the system is where ∆ b,i is the detuning of phonon B with respect to the frequency of phonon A.
The interaction of the sub-system comprising one incoming bosonic modeâ x and one outgoing bosonic modeâ y interacting with one of the qubits at time t, either on its g-e or e-f transition,ĉ =ŝ ge,i orĉ =ŝ ef,i , at the coupling rate κ i (t) set by the coupler, is described by this master equation: where the HamiltonianĤ(t) is given bŷ and the Lindblad operator is given byL 0 (t) = κ i (t)ĉ + g in (t)â x + g out (t)â y .
In the above equations, the coupling coefficients are given by g in (t ) = √ κ c √ 1 + e −κct , and (S23) using the cosecant wavepackets from the experiment. We simulate the total evolution of the system using four consecutive integrations of Eq. S20. In addition, we include the action of qubit decoherence and acoustic losses by including the following Lindblad dissipation operators: 1/T 1ŝi for the intrinsic qubit relaxation, 1/T φŝ † geŝge and 1/T φŝ † efŝ ef for the qubit decoherence for the g-e and e-f transitions, with 1/T φ,ge|ef = 1/T 2,R,ge|ef −1/(2T 1 ), and √ κ aâ , √ κ bb for the acoustic losses, with κ a and κ b defined to match the round-trip transfer efficiency η a = e −κaτ and η b = e −κ b τ . The qubits' XY drives are modeled using H D / = β(ŝ i e iω d t +ŝ † i e −iω d t ), where β is adjusted to give the measured rotation.
We perform these master equations simulations using QuTip [10], with the control sequences defined in Fig. 3 and 4 of the main text as inputs. The model input parameters are given in Table S1. The extracted populations are shown in Fig. 3 and 4 of the main text, corrected for readout errors only in Fig. 4b, and giving good agreement with the measured data.  Table S1. Device parameters for the two qubits, parameters related to the interdigitated acoustic transducer (IDT), the tunable couplers connecting each qubit to the SAW resonator, and the SAW resonator itself.