Abstract
Quantum information can be stored in micromechanical resonators, encoded as quanta of vibration known as phonons. The vibrational motion is then restricted to the stationary eigenmodes of the resonator, which thus serves as local storage for phonons. In contrast, we couple propagating phonons to an artificial atom in the quantum regime and reproduce findings from quantum optics, with sound taking over the role of light. Our results highlight the similarities between phonons and photons but also point to new opportunities arising from the characteristic features of quantum mechanical sound. The low propagation speed of phonons should enable new dynamic schemes for processing quantum information, and the short wavelength allows regimes of atomic physics to be explored that cannot be reached in photonic systems.
A sound proposition for quantum communication
Quantum computers exploit the quantum-mechanical properties of materials to store and manipulate information stored in the quantum states of atoms or artificial atoms. Although there are a number of quantum platforms under investigation already, Gustafsson et al. present another, based on the propagation of sound waves on the surface of a crystal (see the Perspective by Ruskov and Tahan). The ability to tune the system and the slow propagation speeds of the acoustic waves offer new opportunities to control and process quantum information.
The quantum nature of light is revealed and explored in its interaction with atoms, which can be either elemental or artificial. Artificial atoms typically have transition frequencies in the microwave range and can be designed on a microchip with parameters tailored to fit specific requirements. This makes them well suited as tools to investigate fundamental phenomena of atomic physics and quantum optics. In the form of superconducting qubits, they have seen extensive use in closed spaces (electromagnetic cavities), where they have ample time to interact with confined microwave radiation (1–3). These experiments have recently been extended to quantum optics in open one-dimensional (1D) transmission lines, where the atom interacts with itinerant microwave photons (4–7). We present an acoustic equivalent of such a system, where the quantum properties of sound are explored, rather than those of light.
At the intersection between quantum informatics and micromechanics, recent milestones include the coupling between a superconducting qubit and a vibrational mode (8, 9), hybrids of mechanical resonators and electrical microwave cavities (10), and the use of mechanics to interface between microwaves and optical photons (11, 12). The system we present here is another manifestation of mechanics in the quantum regime, but one that differs fundamentally from the suspended resonators mentioned above. In our case, the phonons are not bound to the eigenmodes of any structure but consist of surface acoustic waves (SAWs) that propagate freely over long distances, before and after interacting with an atom in their path.
In the domain of quantum information, SAWs with high power have been used to transport electrons and holes in semiconductors (13–15). This stands in contrast with our use of SAWs, where the power is much too low to transport charge carriers, and we instead focus on the quantum nature of the phonons themselves.
We do this by coupling an artificial atom directly to the SAWs through piezoelectricity, so that this mode of interaction becomes the dominant one for the atom. This means that we can communicate with the atom bidirectionally through the SAW channel, exciting it acoustically as well as listening to its emission of propagating surface phonons.
The idea that this might be feasible was put forward in previous work (16), where the use of a single-electron transistor as a sensitive probe for SAWs was demonstrated. Earlier work on the interaction between phonons and two-level systems includes the demonstration of SAW absorption by quantum dots (17) and theoretical treatments of phonon quantum networks (18) and phononic coupling to dopants in silicon (19, 20).
The acoustically coupled atom
Although there are several types of SAWs, we use the term to denote Rayleigh waves (21–23), which propagate elastically on the surface of a solid within a depth of approximately one wavelength. At and above radio frequencies (RF), the SAW wavelength is short enough that the surface of a microchip can serve as a medium of propagation. By use of a piezoelectric substrate, SAWs can be generated efficiently from electrical signals and converted back to the electrical domain after propagating acoustically over a long distance on the chip. This is used extensively in commercial applications such as microwave delay lines and filters (22–24).
The primary component in a microelectronic SAW device is the interdigital transducer (IDT), which converts power from the electrical to the acoustic domain and vice versa. In its simplest form, it consists of two electrodes, each made of many long fingers deposited as thin films on a piezoelectric substrate. The fingers of the two electrodes are interdigitated so that an ac voltage applied between the electrodes produces an oscillating strain wave in the surface of the substrate. This wave is periodic in both space and time and radiates as a SAW away from each finger. The periodicity 
of the fingers defines the acoustic resonance of the IDT, with the frequency given by 
where 
is the SAW propagation speed. When the IDT is driven electrically at 
, the SAWs emanating from all fingers interfere constructively, resulting in strong acoustic beams launching from the IDT in the two directions perpendicular to the fingers.
Our sample is fabricated on the (100) surface of a semi-insulating GaAs substrate, chosen for its piezoelectric and mechanical properties (Fig. 1A). The IDT is visible on the left side of the sample, with an enlargement in Fig. 1B. It has 
finger pairs, with an overlapping width of W = 25 μm. The fingers, made of aluminum capped with palladium, are aligned so that the SAW propagates in the [011] direction of the crystal, at a speed 
.
(A) Electron micrograph of the sample (top view in false color). The IDT, shown to the left, converts electrical signals to SAWs and vice versa. It has 
periods of fingers, each consisting of one finger from each electrode, with a periodicity of 600 nm. The width of the SAW beam is given by the overlap of the fingers, W = 25 μm. SAWs from the IDT propagate a distance of 100 μm before reaching the qubit, which is shown to the right. All electrodes without explicit connections are grounded. (B and C) Enlargements of the IDT. (D and E) Enlargements of the transmon qubit. The qubit has 
finger periods, with double fingers to suppress internal mechanical reflections (22, 23, 26). (F) Semiclassical circuit model for the qubit. Ctr is the geometric capacitance of the finger structure. It is shunted by a SQUID, which acts as a nonlinear inductance LJ that can be adjusted with a magnetic flux 
. 
is the acoustic admittance element that can pick up a SAW from the IDT (red arrow) to produce electrical excitation in the qubit and regenerate it as a SAW with a phase shift (blue arrow) [(27), semiclassical and full quantum model]. In addition to SAWs from the IDT, radio-frequency signals applied through the gate capacitance 
can be used to address transitions in the qubit.
In our case, the IDT makes use of internal reflections to achieve strong electro-acoustic power conversion that would otherwise be infeasible (22, 23). This results in a narrow bandwidth of ~1 MHz around 
. The IDT is coupled to a low-noise cryogenic high-electron-mobility transistor amplifier via a circulator and an isolator. Through the circulator and the IDT, we can launch a SAW beam toward the artificial atom, which is shown to the right in Fig. 1A with enlargements in Fig. 1, D and E. Conversely, the IDT can pick up leftward-propagating SAW phonons emitted or reflected by the atom. All experiments were done in a dilution refrigerator with a base temperature of 20 mK, that is, with 
. At this low temperature, the charge carriers in the substrate are fully frozen out and the Bose-Einstein distribution gives a population of less than 10−4 thermal phonons per mode around 
.
The artificial atom in our setup is a superconducting qubit of the transmon type (25), positioned 100 μm away from the IDT (Fig. 1, A, D, and E). A transmon consists of a superconducting quantum interference device (SQUID) shunted by a large geometric capacitance Ctr with charging energy 
. The Josephson energy EJ of the SQUID can be tuned with a magnetic flux 
. The Josephson inductance 
forms a resonant circuit together with Ctr, and the nonlinearity of LJ gives rise to the anharmonic energy spectrum that is characteristic for an atom. The transmon is ideally suited for coupling to SAWs because the shunt capacitance can be designed as a finger structure, like an IDT. The charge on Ctr then relates directly to the mechanical strain of Rayleigh waves in the underlying substrate surface.
The periodicity of the capacitor fingers defines the resonance frequency where the acoustic coupling of the qubit is strongest. By design, the qubit and the IDT have the same acoustic resonance frequency, 
. The finger structure of the qubit has 
periods. In contrast with the IDT, each period of the qubit consists of four fingers, in a configuration that greatly diminishes internal mechanical reflections (22, 23, 26). This, along with the lower Ntr compared with 
, means that the qubit has a much wider acoustic bandwidth than the IDT (~250 MHz). From the geometry and materials of the device, we estimate 
fF. In addition to the strong coupling to SAWs, the qubit couples weakly to an RF gate through a capacitance 
. The gate (in contrast with the IDT) has a high bandwidth, so we can use it to excite qubit transitions away from 
as well as to apply RF pulses.
The acoustic coupling rate of the qubit, 
, is an important characteristic of the system, with 
representing the average time it takes the qubit to relax from the first excited state 
to the ground state 
by emitting an acoustic phonon at the transition frequency 
. The acousto-electric conversion of a finger structure is commonly represented as a complex and frequency-dependent acoustic admittance element 
, where electrical dissipation represents conversion to SAWs (22, 23). By inserting this element into a semiclassical model of a transmon, we get the circuit shown in Fig. 1F. Here, the qubit is approximated as a harmonic oscillator, and the model is thus not valid for qubit states beyond the first excited one, 
. When LJ is adjusted so that the transition frequency 
between 
and 
resonates with 
, 
is real-valued and we get the coupling strength as the power loss rate of the parallel resonant circuit. Using this model, we find 
MHz, where 
is a geometry factor and 
is a material parameter that defines the strength of the piezoelectric coupling [(27), semiclassical model].
To analyze our system quantitatively, we have developed an extended model that takes the anharmonic nature of the qubit into account [(27), full quantum model] (28–33). A fully quantum mechanical model for the transmon (25) gives its energy levels as
In our extended model, we also need to account for the spatial extension of the qubit, because it interacts with SAWs over a distance of Ntr wavelengths. We do this by considering one interaction point per finger and accounting for the SAW phase shifts between the different points. However, we assume that the propagation time along the qubit is short compared with the inverse coupling frequency, 
. This means that each emitted phonon leaves the qubit entirely before the next one is emitted, an assumption that is valid in our experiments. It is interesting to note that the opposite limit can be reached, where dressed states should form between excitations in the qubit and phonons localized within its finger structure. Our model predicts the same nonlinear reflection for phonons that has previously been observed for photons (4, 5), and we compare it with experimental data in Figs. 2, 3, and 5.
(A) Reflection coefficient 
of the IDT versus frequency 
, with the qubit far detuned (
). The dip in the curve represents conversion of applied electrical power into SAWs. The complex background reflection acquired with the qubit far detuned has been subtracted from panels (B), (D), (E), and (F), which thus show the change 
in total reflection coefficient caused by the qubit. (B) 
versus frequency and qubit flux, 
. As 
increases, the first transition frequency of the qubit, 
, periodically tunes in and out of the slim band where the IDT can address it acoustically. On resonance, the qubit reflects the incoming SAW beam. (C) Periodic modulation of 
with magnetic flux, fitted to the reflection peaks in (B). (D) Enlargement of one reflection peak from (B). (E) Cross section through (D) at 
(blue). A fit to the theoretical expression for the qubit reflection (red) gives the acoustic coupling rate, 
MHz. (F) Flux modulation of 
versus power applied to the IDT (blue; theoretical fit in red). The rate at which the qubit reflects phonons from the incoming coherent beam is limited by the coupling rate 
. Thus, the acoustic reflection coefficient of the qubit goes to zero for 
, and the flux modulation vanishes. This saturation at the single-phonon level is evidence for the two-level nature of the qubit.
(A) Transduction from the gate to the IDT versus driving power and qubit detuning. We apply a continuous RF signal to the gate at fixed frequency, 
, and adjust the detuning of the first qubit transition from this frequency, 
. The color scale shows the normalized coherent SAW amplitude detected by the IDT. No electrical signal is applied to the IDT. At low applied gate power, the qubit can only be excited at zero detuning, where it emits SAWs in proportion to the applied RF amplitude (constant 
). At higher power, two photons from the gate at 
can excite the 
state of the qubit, which subsequently emits two phonons. For still higher power, photon-to-phonon conversions of increasingly higher orders are visible. (B) Theoretical simulation [(27), full quantum model]. The markup shows the values of 
, where n photons at frequency 
can excite the nth state of the qubit and give rise to the emission of n SAW phonons.
Acoustic reflection measurements
An atom in an open 1D geometry reflects weak resonant coherent radiation perfectly (in the absence of pure dephasing). This is also predicted both by our semiclassical and quantum mechanical models. However, at irradiation powers comparable to or larger than one photon per relaxation time, the 
state of the qubit becomes populated to an appreciable degree, which reduces its ability to reflect phonons of frequency 
and leads to an increase in transmission (4, 5, 7, 34). We measure the nonlinear acoustic reflection of the qubit by applying a coherent microwave tone of frequency 
to the IDT, which transmits part of the power in the form of a SAW propagating toward the qubit.
On IDT resonance (
), ~25% of the applied electrical power reflects against the IDT without converting to acoustic power (Fig. 2A). Here, the qubit is tuned off resonance so that 
. Of the power that leaves the IDT in the form of SAWs, half is emitted in the rightward direction. With all losses accounted for, ~8% of the applied electrical power reaches the qubit in the form of a SAW [(27), extracted sample parameters]. Of the acoustic power that reflects against the qubit, an appreciable part converts back to electrical power in the IDT and can be detected.
The qubit resonance frequency 
is modulated by the magnetic flux 
applied through the SQUID loop, with a periodicity 
. As we tune the flux, we observe an increase in the reflected SAW when 
coincides with 
. As shown in Fig. 2, B and C, we can fit the flux modulation 
to these resonance points. The absence of reflection outside the frequency band of the IDT shows that acoustic coupling strongly dominates over any electrical cross-talk between the IDT and the qubit.
A key characteristic of reflection against an atom in one dimension is the nonlinear dependence of the reflection coefficient on the power of the applied coherent tone: Only when the flux of incoming phonons is much lower than one per interaction time, 
, does the qubit produce full reflection. As the power increases, the 
state of the qubit becomes partly populated, which reduces its ability to reflect phonons of frequency 
(4, 5, 7, 34). This power-dependent saturation is shown in Fig. 2F. By fitting our theoretical model to these data, we find good agreement for 
MHz with negligible dephasing, which implies full reflection from the qubit in the low-power limit. All theoretical plots use the same values for the fitted parameters [(27), extracted sample parameters].
Electrical driving in the steady state
In addition to the acoustic excitation discussed above, we can address transitions in the qubit with the electrical gate and use the IDT to pick up the SAW phonons that the qubit emits. We expect the qubit to selectively emit one-phonon states when driven electrically on resonance, in a process similar to the nonlinear reflection observed under acoustic driving (Fig. 3A). With the frequency of the RF signal applied to the gate fixed at 
, we observe the dependence of the emitted phonon flux on the qubit detuning 
as well as the applied RF power 
. As 
increases from zero, we first see acoustic emission from the qubit at 
. At higher power, additional peaks show up for 
, which correspond to the excitation of higher states of the qubit by multiple photons, and subsequent relaxation into multiple phonons. The offsets between the peaks reflect the anharmonicity of the qubit. Fitting the positions of these multiphonon peaks allows us to determine the Josephson energy 
and 
of the qubit to good accuracy. We find 
GHz and 
GHz. This charging energy agrees well with our geometry-based estimate of the transmon capacitance.
Time domain experiments
The slow propagation of SAWs compared with electromagnetic waves allows us to clearly establish that the qubit couples to the IDT via phonons rather than photons. We do this by applying microwave pulses to the gate and studying the signal that reaches the IDT in the time domain. Figure 4 shows the results of such experiments using 1 μs long pulses with frequency 
. When the qubit is tuned far away from resonance (
), we see a cross-talk signal reaching the IDT from the gate. This is virtually independent of frequency, and we attribute it to stray capacitance between the electrical transmission lines.
(A) By applying microwave pulses to the gate at 
and listening to the qubit in the time domain with the IDT, we can clearly determine that the qubit predominantly emits acoustically. When the qubit is strongly detuned (
), a certain cross-talk reaches the transmission line of the IDT from the gate (red). This cross-talk is independent of power and frequency, and we attribute it to capacitive coupling between the transmission lines of the gate and the IDT. With the qubit near resonance, 
, the transduction coefficient increases substantially above the cross-talk level when the acoustic signal adds in phase with the cross-talk (blue) and decreases when the SAW and the cross-talk have opposite phases (black). (B) Enlargement of (A). The emission from the qubit is observed ~40 ns after the onset of the cross-talk, which corresponds to the acoustic propagation time from the qubit to the IDT.
Also, when the qubit is tuned close to its resonance, 
, the cross-talk signal is the first to rise above the noise floor. This leading edge serves as a time reference, showing the arrival of the electrical pulse to the gate. After the cross-talk edge, it takes another ~40 ns before we observe the emission from the qubit, which corresponds to the acoustic propagation time from the qubit to the IDT. This shows unequivocally that the signal from the qubit is phononic. The phase of the SAW phonons emitted coherently by the qubit is sensitive to 
, and with small variations around 
, we can go from the case where the qubit emission and the cross-talk interfere constructively (blue) to the case where they interfere destructively (black).
Because the IDT partly reflects phonons impinging on it from the qubit, additional features can be seen that correspond to acoustic round trips from the IDT to the qubit and back. These acoustic reflections add up gradually with time, superimposed on the transient response of the IDT, until the steady-state signal is established after ~1 μs. Just as in the steady-state experiments, the acoustic emission relative to the applied gate power increases with decreasing power 
, until it saturates at 
.
Hybrid two-tone spectroscopy
To characterize the dynamics of the phonon-qubit interaction in greater detail, we perform hybrid two-tone spectroscopy, where a continuous acoustic probe tone with low power is launched from the IDT toward the qubit at fixed frequency 
, and its reflection is measured. At the same time, we vary the qubit detuning 
and apply a continuous electrical control tone to the gate, with varying frequency 
and power 
. When the qubit absorbs photons from the gate, we observe an impact on its reflection of phonons from the IDT (Fig. 5).
We apply a continuous acoustic probe tone to the IDT with constant frequency 
and low power. At the same time, a continuous control tone with varying power 
and detuning from the probe, 
, is applied to the gate. The color scale shows the electrical reflection coefficient 
of the IDT versus 
and the qubit detuning 
. (A to C) Experimental data for increasing 
. (D to F) Theoretical simulations corresponding to the experiments in (A) to (C). Selected features are highlighted and the corresponding qubit transitions illustrated in (G) to (J). (G) In the absence of a control signal on the gate, 
is modulated by the qubit flux, which is seen in (A) and (D) as a vertical feature at 
. (H) For 
, a strong control tone contributes to the saturation of the first qubit transition. (I) Provided that the 
state of the qubit is appreciably populated by the control tone, nonlinear acoustic reflection can occur also for 
. We observe this when a moderately strong control tone is applied at 
. (J) For moderate to strong electrical driving of the 
transition, Rabi splitting of the qubit energy levels gives rise to an Autler-Townes doublet (35, 36). For very high control tone power [as shown in (F)], higher-order qubit states are populated and subject to Rabi splitting. This gives rise to a complex set of features that is well captured by the theoretical model [(27), full quantum model].
For low values of 
, the acoustic reflection is modulated only by the qubit detuning, as demonstrated in Figs. 2, B to E. For higher 
and when 
coincides with 
, the control tone contributes to the population of the 
state of the qubit. When the frequencies of the probe and control tones coincide, this results in saturation of the 
transition, as seen in Fig. 2F. If the control tone is tuned to populate the 
state of the qubit and the condition 
GHz is fulfilled, the 
transition exhibits nonlinear acoustic reflection.
For still higher control powers, we observe a rich set of spectral features, which agrees well with our model [(27), full quantum model]. Of particular interest is the Autler-Townes doublet, which is caused by Rabi splitting of the 
state at strong electrical driving of the 
transition (35, 36).
Outlook and summary
Our SAW device occupies a middle ground between fixed mechanical resonators and transmission lines for free photons, and it is relevant to compare its features with both of these related systems.
Although their itinerant nature is an essential property of SAW phonons, they can also be confined into cavities by on-chip Bragg mirrors. Such cavities compare well with suspended resonators also in other respects than their natural integration with propagating waves: They can be fabricated for mode energies well above 
for temperatures attainable with standard cryogenic equipment, and because the motion takes place directly in the cooled substrate, thermalization is excellent. The few available studies also indicate that high-frequency SAW cavities can have comparatively high quality factors at low temperature (37, 38).
In comparison to photons, SAW phonons have several striking features. Their speed of propagation is lower by a factor of ~105, and their wavelength at a given frequency is correspondingly shorter. The slow speed means that qubits can be tuned much faster than SAWs traverse interqubit distances on a chip. This enables new dynamic schemes for trapping and processing quanta.
SAW phonons furthermore give access to a regime where the size of an atom substantially exceeds the wavelength of the quanta it interacts with. This is the opposite of the pointlike interaction realized so far in photonics, cavity quantum electrodynamics (QED), and circuit QED (28). In our device, the qubit is a modest factor 
times longer than the SAW wavelength, but this can be extended substantially in a device designed for such investigations.
In the device presented here, the coupling strength between SAWs and the qubit is also moderate. This is necessary to discriminate between the different qubit transition energies due to the low anharmonicity of the transmon design. However, in a strongly piezoelectric material such as LiNbO3, the many coupling points of an IDT-shaped qubit should make it possible to reach the regimes of “ultrastrong coupling” (31, 39, 40), 
and even “deep strong coupling” (41), 
, which are difficult to access with the standard electrical dipole coupling of photonic systems.
In conclusion, we have demonstrated nonclassical interaction between surface acoustic waves and an artificial atom in the regime of strong coupling. Our experiments suggest that phonons can serve as propagating carriers of quantum information, in analogy with itinerant photons in quantum optics. The data are in good quantitative agreement with a theoretical quantum model, which captures the nonlinear acoustic reflection and relaxation of the atom. These results open up new possibilities in quantum experiments, because SAWs can reach regimes of coupling strength and time domain control that are not feasible with photons.
Supplementary Materials
References and Notes
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵Materials and methods are available as supplementary materials on Science Online.
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Acknowledgments: We acknowledge financial support from the Swedish Research Council (VR), the Wallenberg Foundation, and the European Union through the European Research Council and the Scalable Superconducting Processors for Entangled Quantum Information Technology project. The samples were made at the Nanofabrication Laboratory at Chalmers. We acknowledge fruitful discussions with P. Leek, R. Manenti, and D. Niesner. M.V.G. acknowledges funding from the Wenner-Gren Foundations and support from X. Zhu and P. Kim during the manuscript preparation.
