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In quantum optics, light–matter interaction has conventionally been studied using small atoms interacting with electromagnetic fields with wavelength several orders of magnitude larger than the atomic dimensions1,2. In contrast, here we experimentally demonstrate the vastly different ‘giant atom’ regime, where an artificial atom interacts with acoustic fields with wavelength several orders of magnitude smaller than the atomic dimensions. This is achieved by coupling a superconducting qubit3 to surface acoustic waves at two points with separation on the order of 100 wavelengths. This approach is comparable to controlling the radiation of an atom by attaching it to an antenna. The slow velocity of sound leads to a significant internal time-delay for the field to propagate across the giant atom, giving rise to non-Markovian dynamics4. We demonstrate the non-Markovian character of the giant atom in the frequency spectrum as well as non-exponential relaxation in the time domain. By coupling a superconducting qubit to surface acoustic waves the ‘giant atom’ regime is realized, where an atom is coupled to a field with wavelength orders of magnitude smaller than the atomic size. This leads to non-Markovian qubit dynamics.

We benchmark the decoherence of superconducting qubits to examine the temporal stability of energy-relaxation and dephasing. By collecting statistics during measurements spanning multiple days, we find the mean parameters $\overline{T_{1}}$ = 49 $\mu$s and $\overline{T_{2}^{*}}$ = 95 $\mu$s, however, both of these quantities fluctuate explaining the need for frequent re-calibration in qubit setups. Our main finding is that fluctuations in qubit relaxation are local to the qubit and are caused by instabilities of near-resonant two-level-systems (TLS). Through statistical analysis, we determine switching rates of these TLS and observe the coherent coupling between an individual TLS and a transmon qubit. Finally, we find evidence that the qubit's frequency stability is limited by capacitance noise. Importantly, this produces a 0.8 ms limit on the pure dephasing which we also observe. Collectively, these findings raise the need for performing qubit metrology to examine the reproducibility of qubit parameters, where these fluctuations could affect qubit gate fidelity., Comment: 15 pages ArXiv version rev2

We investigate three types of amplification processes for light fields coupling to an atom near the end of a one-dimensional (1D) semi-infinite waveguide. We consider two setups where a drive creates population inversion in the bare or dressed basis of a three-level atom and one setup where the amplification is due to higher-order processes in a driven two-level atom. In all cases, the end of the waveguide acts as a mirror for the light. We find that this enhances the amplification in two ways compared to the same setups in an open waveguide. Firstly, the mirror forces all output from the atom to travel in one direction instead of being split up into two output channels. Secondly, interference due to the mirror enables tuning of the ratio of relaxation rates for different transitions in the atom to increase population inversion. We quantify the enhancement in amplification due to these factors and show that it can be demonstrated for standard parameters in experiments with superconducting quantum circuits.

We report the observation of Electromagnetically Induced Transparency (EIT) of a mechanical field, where a superconducting artificial atom is coupled to a 1D-transmission line for surface acoustic waves. An electromagnetic microwave drive is used as the control field, rendering the superconducting transmon qubit transparent to the acoustic probe beam. The strong frequency dependence of the acoustic coupling enables EIT in a ladder configuration due to the suppressed relaxation of the upper level. Our results show that superconducting circuits can be engineered to interact with acoustic fields in parameter regimes not readily accessible to purely electromagnetic systems.

Microscopic two-level system (TLS) defects at dielectric surfaces and interfaces are among the dominant sources of loss in superconducting quantum circuits, and their properties have been extensively probed using superconducting resonators and qubits. We report on spectroscopy of TLSs coupling to the strain field in a surface acoustic wave (SAW) resonator. The narrow free spectral range of the resonator allows for two-tone spectroscopy where a strong pump is applied at one resonance, while a weak signal is used to probe a different mode. We map the spectral hole burnt by the pump tone as a function of frequency and extract parameters of the TLS ensemble. Our results suggest that detuned acoustic pumping can be used to enhance the coherence of superconducting devices by saturating TLSs.

The ability to control and measure the temperature of propagating microwave modes down to very low temperatures is indispensable for quantum information processing, and may open opportunities for studies of heat transport at the nanoscale, also in the quantum regime. Here we propose and experimentally demonstrate primary thermometry of propagating microwaves using a transmon-type superconducting circuit. Our device operates continuously, with a sensitivity down to $4\times 10^{-4}$ photons/$\sqrt{\mbox{Hz}}$ and a bandwidth of 40 MHz. We measure the thermal occupation of the modes of a highly attenuated coaxial cable in a range of 0.001 to 0.4 thermal photons, corresponding to a temperature range from 35 mK to 210 mK at a frequency around 5 GHz. To increase the radiation temperature in a controlled fashion, we either inject calibrated, wideband digital noise, or heat the device and its environment. This thermometry scheme can find applications in benchmarking and characterization of cryogenic microwave setups, temperature measurements in hybrid quantum systems, and quantum thermodynamics.

We numerically and experimentally investigate the phononic loss for superconducting resonators fabricated on a piezoelectric substrate. With the help of finite element method simulations, we calculate the energy loss due to electromechanical conversion into bulk and surface acoustic waves. This sets an upper limit for the resonator internal quality factor Q i. To validate the simulation, we fabricate quarter wavelength coplanar waveguide resonators on GaAs and measure Q i as function of frequency, power and temperature. We observe a linear increase of Q i with frequency, as predicted by the simulations for a constant electromechanical coupling. Additionally, Q i shows a weak power dependence and a negligible temperature dependence around 10 mK, excluding two level systems and non-equilibrium quasiparticles as the main source of losses at that temperature.

We explore routing of propagating phonons in analogy with previous experiments on photons. Surface acoustic waves (SAWs) in the microwave regime are scattered by a superconducting transmon qubit. The transmon can be tuned on or off resonance with the incident SAW field using an external magnetic field or the Autler-Townes effect, and thus the reflection and transmission of the SAW field can be controlled in time. We observe 80% extinction in the transmission of the low power continuous signal and a 40 ns rise time of the router. The slow propagation speed of SAWs on solid surfaces allows for in-flight manipulations of the propagating phonons. The ability to route short, 100 ns, pulses enables new functionality, for instance to catch an acoustic phonon between two qubits and then release it in a controlled direction., 10 pages, 11 figures put into 4 figures, 1 table

Virtual photons can mediate interaction between atoms, resulting in an energy shift known as a collective Lamb shift. Observing the collective Lamb shift is challenging, since it can be obscured by radiative decay and direct atom-atom interactions. Here, we place two superconducting qubits in a transmission line terminated by a mirror, which suppresses decay. We measure a collective Lamb shift reaching 0.8% of the qubit transition frequency and exceeding the transition linewidth. We also show that the qubits can interact via the transmission line even if one of them does not decay into it., Comment: 7+5 pages, 4+2 figures

Artificial atoms coupled to surface acoustic waves (SAWs) have played a crucial role in the recent development of circuit quantum acoustodynamics (cQAD). In this paper, we have investigated the interaction of an artificial atom and SAWs beyond the weak coupling regime, focusing on the role of the interdigital transducer (IDT) that enables the coupling. We find a parameter regime in which the IDT acts as a cavity for the atom, rather than an antenna. In other words, the atom forms its own cavity. Similar to an atom coupled to an explicit cavity, this regime is characterized by vacuum-Rabi splitting, as the atom hybridizes with the phononic vacuum inside the IDT. This hybridization is possible because of the interdigitated coupling, which has a large spatial extension, and the slow propagation speed of SAWs. We work out a criterion for entering this regime from a model based on standard circuit-quantization techniques, taking only material parameters as inputs. Most notably, we find this regime hard to avoid for an atom on top of a strong piezoelectric material, such as LiNbO$_3$. The SAW-coupled atom on top of LiNbO$_3$ can thus be regarded as an atom-cavity-bath system. On weaker piezoelectric materials, the number of IDT electrodes need to be large in order to reach this regime., 11 pages, 5 figures

We present an experimental demonstration as well as a theoretical model of an integrated circuit designed for the manipulation of a microwave field down to the single-photon level. The device is made of a superconducting resonator coupled to a transmission line via a second frequency-tunable resonator. The tunable resonator can be used as a tunable coupler between the fixed resonator and the transmission line. Moreover, the manipulation of the microwave field between the two resonators is possible. In particular, we demonstrate the swapping of the field from one resonator to the other by pulsing the frequency detuning between the two resonators. The behavior of the system, which determines how the device can be operated, is analyzed as a function of one key parameter of the system, the damping ratio of the coupled resonators. We show a good agreement between experiments and simulations, realized by solving a set of coupled differential equations., 10 pages, 6 figures

We investigate nondegenerate parametric oscillations in a multimode superconducting microwave resonator that is terminated by a SQUID. The parametric effect is achieved by modulating magnetic flux through the SQUID at a frequency close to the sum of two resonator-mode frequencies. For modulation amplitudes exceeding an instability threshold, self-sustained oscillations are observed in both modes. The amplitudes of these oscillations show good quantitative agreement with a theoretical model. The oscillation phases are found to be correlated and exhibit strong fluctuations which broaden the oscillation spectral linewidths. These linewidths are significantly reduced by applying a weak on-resonance tone, which also suppresses the phase fluctuations. When the weak tone is detuned, we observe synchronization of the oscillation frequency with the frequency of the input. For the detuned input, we also observe an emergence of three idlers in the output. This observation is in agreement with theory indicating four-mode amplification and squeezing of a coherent input., 7 pages, 8 figures

We report on the experimental observation of period multiplication in parametrically driven tunable superconducting resonators. We modulate the magnetic flux through a superconducting quantum interference device, attached to a quarter-wavelength resonator, with frequencies $n\omega$ close to multiples, $n=2,\,3,\,4,\,5$, of the resonator fundamental mode and observe intense output radiation at $\omega$. The output field manifests $n$-fold degeneracy with respect to the phase, the $n$ states are phase shifted by $2\pi/n$ with respect to each other. Our demonstration verifies the theoretical prediction by Guo et al. in PRL 111, 205303 (2013), and paves the way for engineering complex macroscopic quantum cat states with microwave photons.

Entangled pairs of microwave photons are commonly produced in the narrow frequency band of a resonator, which represents a modified vacuum density of states. We generate and investigate the entanglement of a stream of photon pairs, generated in a semi-infinite broadband transmission line, terminated by a superconducting quantum interference device (SQUID). A weak pump signal modulates the SQUID inductance, resulting in a single time-varying boundary condition, and we detect all four quadratures of the microwave radiation emitted at two different frequencies separated by 0.7 GHz. Power calibration is done in situ, and we find positive logarithmic negativity and two-mode squeezing below the vacuum in the observed radiation, indicating entanglement.

We have observed period-tripling subharmonic oscillations, in a superconducting coplanar waveguide resonator operated in the quantum regime, $k_B T \ll \hbar\omega$. The resonator is terminated by a tunable inductance that provides a Kerr-type nonlinearity. We detected the output field quadratures at frequencies near the fundamental mode, $\omega/2\pi \sim 5\,$GHz, when the resonator was driven by a current at $3\omega$ with an amplitude exceeding an instability threshold. The output radiation was red-detuned from the fundamental mode. We observed three stable radiative states with equal amplitudes and phase-shifted by $120^\circ$. The downconversion from $3\omega$ to $\omega$ is strongly enhanced by resonant excitation of the second mode of the resonator, and the cross-Kerr effect. Our experimental results are in quantitative agreement with a model for the driven dynamics of two coupled modes.

We have developed and tested a doubly tunable resonator, with the intention to simulate fast motion of the resonator boundaries in real space. Our device is a superconducting coplanar-waveguide half-wavelength microwave resonator, with fundamental resonant frequency ~5 GHz. Both of its ends are terminated by dc-SQUIDs, which serve as magnetic-flux-controlled inductances. Applying a flux to either SQUID allows tuning of the resonant frequency by approximately 700 MHz. By using two separate on-chip magnetic-flux lines, we modulate the SQUIDs with two tones of equal frequency, close to twice that of the resonator's fundamental mode. We observe photon generation, at the fundamental frequency, above a certain pump amplitude threshold. By varying the relative phase of the two pumps we are able to control the photon generation threshold, in good agreement with a theoretical model for the modulation of the boundary conditions. At the same time, some of our observations deviate from the theoretical predictions, which we attribute to parasitic couplings, resulting in current driving of the SQUIDs., Comment: 7 pages, 4 figures

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 unique 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 which cannot be reached in photonic systems., Comment: 30 pages, 6 figures, 1 table

The conversion efficiency of electric microwave signals into surface acoustic waves in different types of superconducting transducers is studied with the aim of quantum applications. We compare delay lines containing either conventional symmetric transducers (IDTs) or unidirectional transducers (UDTs) at 2.3 GHz and 10 mK. The UDT delay lines improve the insertion loss with 4.7 dB and a directivity of 22 dB is found for each UDT, indicating that 99.4 % of the acoustic power goes in the desired direction. The power lost in the undesired direction accounts for more than 90 % of the total loss in IDT delay lines, but only ~3 % percent of the total loss in the FEUDT delay lines., 4 pages (5 including references), 3 figures

In the dynamical Casimir effect (DCE), a boundary condition for the electromagnetic field is changed rapidly, resulting in generation of pairs of photons [1]. This was experimentally demonstrated in 2011 [2]. Both generation of photons and two-mode squeezing were observed. Quantum theory predicts that the photon pairs are entangled and that the resulting electromagnetic field is in a nonclassical state at zero temperature. Johansson et al. [3] recently suggested how to test entanglement of the DCE radiation by evaluating the log negativity. Here we present such an experimental test. We have measured a sample consisting of an open transmission line terminated by a superconducting quantum interference device (SQUID). The electrical length of the transmission line can be modulated, by changing the magnetic flux through the SQUID loop. Broadband radiation of photon pairs is observed along with strong two mode squeezing. The system is characterized at a temperature of < 20 mK, and the signal level is calibrated using the SQUID as a shot-noise source. We investigate the cross-correlations for the field at two different frequencies and we recover the covariance matrix. From this matrix we determine the log negativity, which shows weakly positive values indicating that the two modes are entangled. A source of entangled photons can be useful in quantum information applications, for instance to entangle qubits [4].

We propose two different setups to generate single photons on demand using an atom in front of a mirror, along with either a beam-splitter or a tunable coupling. We show that photon generation efficiency ~99% is straightforward to achieve. The proposed schemes are simple and easily tunable in frequency. The operation is relatively insensitive to dephasing and can be easily extended to generate correlated pairs of photons. They can also in principle be used to generate any photonic qubit of the form $\mu |0 \rangle + \nu |1\rangle$ in arbitrary wave-packets, making them very attractive for quantum communication applications., Comment: 10 pages, Added appendix

It has recently been demonstrated that surface acoustic waves (SAWs) can interact with superconducting qubits at the quantum level. SAW resonators in the GHz frequency range have also been found to have low loss at temperatures compatible with superconducting quantum circuits. These advances open up new possibilities to use the phonon degree of freedom to carry quantum information. In this chapter, we give a description of the basic SAW components needed to develop quantum circuits, where propagating or localized SAW-phonons are used both to study basic physics and to manipulate quantum information. Using phonons instead of photons offers new possibilities which make these quantum acoustic circuits very interesting. We discuss general considerations for SAW experiments at the quantum level and describe experiments both with SAW resonators and with interaction between SAWs and a qubit. We also discuss several potential future developments.

14 págs.; 4 figs.; Open Access funded by Creative Commons Atribution Licence 4.0, We show how to use relativistic motion to generate continuous variable Gaussian cluster states within cavity modes. Our results can be demonstrated experimentally using superconducting circuits where tuneable boundary conditions correspond to mirrors moving with velocities close to the speed of light. In particular, we propose the generation of a quadripartite square cluster state as a first example that can be readily implemented in the laboratory. Since cluster states are universal resources for universal one-way quantum computation, our results pave the way for relativistic quantum computation schemes., D.E.B. was in part supported by the UK Engineering and Physical Science Research Council grant number EP/J005762/1. I.F. and C.S acknowledge support from EPSRC (CAF Grant No. EP/G00496X/2 to I.F.). G.J. and P.D. would like to acknowledge funding from the Swedish Research Council and from the EU through the ERC and the projects PROMISCE and SQALEQIT. Financial support by Fundación General CSIC (Programa ComFuturo) is acknowledged by C.S.

One of the most surprising predictions of modern quantum theory is that the vacuum of space is not empty. In fact, quantum theory predicts that it teems with virtual particles flitting in and out of existence. While initially a curiosity, it was quickly realized that these vacuum fluctuations had measurable consequences, for instance producing the Lamb shift of atomic spectra and modifying the magnetic moment for the electron. This type of renormalization due to vacuum fluctuations is now central to our understanding of nature. However, these effects provide indirect evidence for the existence of vacuum fluctuations. From early on, it was discussed if it might instead be possible to more directly observe the virtual particles that compose the quantum vacuum. 40 years ago, Moore suggested that a mirror undergoing relativistic motion could convert virtual photons into directly observable real photons. This effect was later named the dynamical Casimir effect (DCE). Using a superconducting circuit, we have observed the DCE for the first time. The circuit consists of a coplanar transmission line with an electrical length that can be changed at a few percent of the speed of light. The length is changed by modulating the inductance of a superconducting quantum interference device (SQUID) at high frequencies (~11 GHz). In addition to observing the creation of real photons, we observe two-mode squeezing of the emitted radiation, which is a signature of the quantum character of the generation process., Comment: 12 pages, 3 figures

The design, simulation and measurements of an on-chip Mach-Zehnder interferometer operating in the microwave regime are described. Using microwave signals in microfabricated superconducting Al waveguides, the concept of an interferometer is transferred from optics to on-chip. Tuning of the path length of one of the interferometer arms is executed through the tunable inductance of a SQUID. By placing one or more SQUIDs in the waveguide structure and by varying the magnetic flux through the SQUID loop, the total SQUID inductance can be tuned. In this way, a phase difference leading to destructive or constructive interference at the interferometer output can be achieved. Thorough software simulations were performed to determine the different design parameters, assign a desired working frequency and provide a reference for comparison with experimental results. Measurements at 300 mK show an effective working frequency close to the simulations, with a deviation smaller than 0.05 GHz. The behavior of the interferometer is very similar to the simulations as well. The on-off ratio exceeds 40 dB.

We describe in this review progress in techniques for measuring the quantum states of the single Cooper-pair box (SCB) at the Jet Propulsion Laboratory, and the use of these techniques to characterize their performance as quantum bits. Initially, the Radio-Frequency-Single-Electron Transistor was used to measure conventional and differential versions of the SCB. To overcome some of the difficulties associated with the radio-frequency single-electron transistor, a transition was made to readout based on the quantum capacitance technique. Using this method, entanglement between two SCBs was demonstrated spectroscopically and with ground state characterization.

We have observed energy level quantization in an all high critical temperature superconductor d-wave Josephson junction. From the measurements we have also extracted the quality factor Q of the junction which is of the order of 40. These results indicate that the role of dissipation mechanisms in high temperature superconductors has to be revised, and may also have consequences for the class of solid state “quiet” quantum bit with longer coherence time.

We demonstrate the operation of a differential single Cooper-pair box, a charge qubit consisting of two aluminum islands, isolated from ground, coupled by a pair of small-area Josephson junctions. We have tested four devices, all of which show evidence of quasiparticle poisoning. The devices are characterized with microwave spectroscopy and temperature dependence studies.

We propose and demonstrate a new read-out technique for a superconducting qubit by dispersively coupling it to a Josephson parametric oscillator. We employ a tunable quarter-wavelength superconducting resonator and modulate its resonant frequency at twice its value with an amplitude surpassing the threshold for parametric instability. We map the qubit states onto two distinct states of classical parametric oscillation: one oscillating state, with $185\pm15$ photons in the resonator, and one with zero oscillation amplitude. This high contrast obviates a following quantum-limited amplifier. We demonstrate proof-of-principle, single-shot readout performance, and present an error budget indicating that this method can surpass the fidelity threshold required for quantum computing., Comment: 11 pages, 5 figures

We investigate the backaction of superconducting single-electron transistor (SSET) continuously measuring a Cooper-pair box. Due to the minimized backaction of the SSET, we observe a 2e periodic Coulomb staircase according to the two-level system Hamiltonian of the Cooper-pair box. We demonstrate that we can control the quantum broadening of the ground state in-situ. We perform spectroscopy measurements and demonstrate that we have full control over the Cooper-pair box Hamiltonian. The ability to reduce the backaction is a necessary condition to use the SSET as a quantum state readout for the CPB as a qubit.

We study decoherence of a Josephson charge qubit using fast pulses to perform qubit rotations. The gate charge dependence of the decoherence rate indicates that the low frequency fluctuations affecting the qubit are from charges distributed with a 1/f-type spectrum. Assuming the form Sq(ω) = α/|ω|, we find $$\sqrt \alpha$$ = 4 × 10−3e, which is slightly higher than the value found from very low frequency noise measurements. PACS numbers: 03.67.-a, 75.40.+r, 85.25.Cp.

We have analyzed a radio-frequency single-electron-transistor (RF-SET) circuit that includes a high-electron-mobility-transistor (HEMT)amplifier, coupled to the single-electron-transistor (SET) via an impedance transformer. We consider how power is transferred between different components of the circuit, model noise components, and analyze the operating conditions of practical importance. The results are compared with experimental data on SETs. Good agreement is obtained between our noise model and the experimental results. Our analysis shows, also, that the biggest improvement to the present RF-SETs will be achieved by increasing the charging energy and by lowering the HEMT amplifier noise contribution.

We present numerical results on the current mirror effect in a single-electron turnstile capacitively coupled with a one-dimensional (1D) array of small tunnel junctions. Contrary to conventional operation with a sinusoidal signal source, the single-electron turnstile is driven by charge solitons (and anti-solitons) transferred through the 1D array. Then, the current through the turnstile can be locked to the current through the 1D array; this is the current mirror effect. We numerically demonstrate the current mirror effect in our circuit, where a 4-junction turnstile and a 20-junction 1D array are coupled at their middle nodes via a coupling capacitor. We introduce a coupling parameter, Qc, to define the strength of coupling between the 1D array and the turnstile. We also introduce a current mirror index, CMI, to evaluate the quality of the current mirror effect. The dependence of CMI upon the soliton length, the current through the 1D array, and Qc is discussed.

There are several dualities between Josephson and single charge tunneling, being conjugate phenomena, where either the phase (flux) or the number (charge) states are well determined while the complementary entity (number, phase) is undetermined. A zero voltage, phase-dependent current flows in the Josephson junction up to a critical current value while no current flows in the Coulomb blockade region, up to a threshold voltage, in a single charge element having high resistance and small Josephson coupling. The dual of the extremely flux sensitive DC superconducting quantum interference device (SQUID) is the single charge transistor (SET). It has an outstanding charge sensitivity but its high impedance severely limits its bandwidth. A superconducting radio frequency version, RF-SET, is related to the common RF-SQUID and its bandwidth is improved orders of magnitude compared to the usual SET. A charge sensitivity of the order of 3×10 −6 e/√Hz has been demonstrated. It can be used, e.g., as a read out of the charge of a quantum “qu-bit”. The latter is, in this case, based upon the charge state of a small island, being a superposition of a charge or not a charge, which can be compared with a flux based qu-bit where currents go clock- or anti-clockwise in a SQUID loop.

We show that the physics underlying the dynamical Casimir effect may generate multipartite quantum correlations. To achieve it, we propose a circuit quantum electrodynamics (cQED) scenario involving superconducting quantum interference devices (SQUIDs), cavities, and superconducting qubits, also called artificial atoms. Our results predict the generation of highly entangled states for two and three superconducting qubits in different geometric configurations with realistic parameters. This proposal paves the way for a scalable method of multipartite entanglement generation in cavity networks through dynamical Casimir physics., Improved version and references added. Accepted for publication in Physical Review Letters