Controlling Atom-Photon Bound States in an Array of Josephson-Junction Resonators

Engineering the electromagnetic environment of a quantum emitter gives rise to a plethora of exotic light-matter interactions. In particular, photonic lattices can seed long-lived atom-photon bound states inside photonic band gaps. Here, we report on the concept and implementation of a novel microwave architecture consisting of an array of compact superconducting resonators in which we have embedded two frequency-tunable artificial atoms. We study the atom-field interaction and access previously unexplored coupling regimes, in both the single- and double-excitation subspace. In addition, we demonstrate coherent interactions between two atom-photon bound states, in both resonant and dispersive regimes, that are suitable for the implementation of swap and cz two-qubit gates. The presented architecture holds promise for quantum simulation with tunable-range interactions and photon transport experiments in the nonlinear regime.

Popular Summary

Engineering the electromagnetic environment of a quantum emitter makes it possible to observe many exotic phenomena involving atom-light interactions. In particular, coupling quantum emitters to a finite-band waveguide can lead to the formation of long-lived atom-photon bound states with energies outside the photonic band. These bound states, characterized by a photonic cloud localized around the atom, can mediate long-range coherent qubit-qubit interactions, opening opportunities for quantum simulations of interacting spin models. In our experiment, we induce the formation of these bound states by engineering an array of microwave resonators coupled to superconducting qubits.

Our implementation of the slow-light waveguide combines a small footprint with strong interresonator and emitter-resonator couplings. We demonstrate full individual control of the bound states, extensively studying the bound-states-mediated interaction between two qubits. In particular, we realize for the first time an excitation swap and a ZZ interaction between two atom-photon bound states, demonstrating that our system contains the building blocks to implement quantum gates.

We expect this architecture to enable quantum simulation of spin models with tunable long-range interactions, possibly also in two dimensions. Moreover, the intrinsic nonlinearity of the emitters and that of the waveguide, tunable by design, open exciting possibilities to study correlated photon transport and collective effects in quantum optics.

Figure 1

Two artificial atoms interacting with a coupled-cavity array. (a) Concept: qubit 1, ${\mathrm{Q}}_1$ (blue), and qubit 2, ${\mathrm{Q}}_2$ (red), are locally coupled to an array of 21 cavities (green). Each atom can be independently controlled through an individual driving line (same color as the atom) and its state measured via a dedicated readout resonator (yellow). Each qubit is dressed by a photon, giving rise to a corresponding atom-photon bound state (see shaded cloud localized around each atom). (b) Energy diagram of dressed states as a function of the bare qubit frequency ${\omega }_q$. For a fixed ${\omega }_q$, the bound state emerges as a discrete energy $E$ separated from the photonic band (green shaded region). The color scale of the bound state (from green to red) indicates whether the excitation is mostly photonic or atomic in nature. (c) Micrograph of the sample and (d) simplified experimental setup: An array of 21 Josephson-junction (JJ) resonators (green) is capacitively coupled at its edges to coplanar waveguides directly connected to the input (${\mathrm{WG}}_{\mathrm{in}}$) and output (${\mathrm{WG}}_{\mathrm{out}}$) of two cryostat lines. The transmon qubits Q1 and Q2 are coupled to resonator (cavity) 10 and 12, respectively, and driven through dedicated charge lines XY-${\mathrm{Q}}_1$ and XY-${\mathrm{Q}}_2$, respectively. Each qubit state is detected through 2 quarter-wavelength coplanar resonators (yellow) inductively multiplexed on a transmission line measured through ${\mathrm{RO}}_{\mathrm{in}}$ and ${\mathrm{RO}}_{\mathrm{out}}$. The qubit frequencies are tuned by injecting a magnetic flux in the transmon SQUID loop with individual flux lines (Z-${\mathrm{Q}}_1$ and Z-${\mathrm{Q}}_2$). Details of the JJ resonator and $Q_2$, the JJ array, and the ${\mathrm{Q}}_2$ SQUID and flux line are shown in panels (c1)–(c3), respectively.

Figure 2

Single coherent tone spectroscopy of the coupled cavity array. (a) Frequency landscape of the resonant modes of the systems: ${\mathrm{Q}}_1$ (blue), ${\mathrm{Q}}_2$ (red) (tuned away from the band in this measurement), readout resonators (yellow), and array modes (green). (b) Transmitted amplitude, $|S_{21}|$ vs frequency, $\omega$, for different input powers, and $P_{\mathrm{in}}$ at room temperature. The traces are vertically offset in steps of 1.1 and filled to their own baseline for clarity.

Figure 3

Single coherent tone spectroscopy as a function of ${\mathrm{Q}}_2$ frequency. (a) Frequency landscape: ${\mathrm{Q}}_1$ is tuned to its lowest frequency. The ${\mathrm{Q}}_2$ frequency is tuned from its largest frequency at 6.6 GHz to 4.1 GHz. (b) Low-power coherent tone spectroscopy through the resonator array as a function of the ${\mathrm{Q}}_2$ frequency (dashed red line). (c) Calculated transmission spectrum via input-output theory using the single-excitation Hamiltonian Eq. (3). (d) Detail of the measurement in panel (a) showing how the atom-photon bound state approaches the band and eventually becoming the last array modes. The dashed black line shows data fit via Eq. (4). (e) Total decay rate of the bound state as a function of its frequency. The red data are obtained from the transmission spectroscopy, while the blue are from the atom spectroscopy (see Sec. 2c). The dashed black line represents the theoretically expected decay rate. Insets: calculated distribution of the excitation over the array (green dots) and between the two qubits (blue and red, respectively) for two distinct bound-state frequencies: ${\omega }_{\mathrm{BS}2}=6.226\mathrm{GHz}$ (e1) and ${\omega }_{\mathrm{BS}2}=6.510\mathrm{GHz}$ (e2).

Figure 4

Isolated bound state in single excitation subspace. (a) Pulse scheme. A Gaussian pulse (red line) with frequency ${\omega }_p$ drives ${\mathrm{Q}}_2$, while ${\mathrm{Q}}_1$ is far detuned by the flux pulse (blue shaded area). The readout pulse (in yellow) reads the population in both qubits. On the side, ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$ (blue and red dots, respectively), the photonic band (green dots), and the readout resonator (yellow dots) frequencies are depicted in relation to the flux pulses. (b) Population of ${\mathrm{Q}}_2$ as a function of flux pulse amplitude ${\mathrm{\Phi }}_{q2}$ and driving pulse frequency ${\omega }_p$. The black line shows the fit of the bound-state frequency given in Eq. (4) as a function of the expected bare ${\mathrm{Q}}_2$ frequency (red dashed line). (c) Bound-state frequency extracted from panel (b). (d) Atomic fraction of the excitation. The black line shows the expected value [Eq. (b24)] using the parameters extracted from the fit in panels (b) and (c).

Figure 5

Isolated bound state in the two-excitation subspace. (a) Pulse scheme. A $\pi$ pulse (red line) excites ${\mathrm{Q}}_2$, and a second Gaussian pulse brings it to the third level when ${\omega }_{p2}$ is resonant with the ef transition. When ${\omega }_{q2}$ is not resonant, a $\pi$ pulse brings the qubit back to the ground state. (b) Absolute value of the dressed anharmonicity ${\beta }_{\text{dress}}$. The arrow indicates the value of the bare anharmonicity. The black solid line shows the calculated ${\beta }_{\text{dress}}$, with the parameters extracted from the fit in Fig. 4. Finally, the black dashed line indicates the dressed anharmonicity of an ideal two-level atom.

Figure 6

Avoided crossing and interaction strength between two bound states. (a) Pulse scheme. A Gaussian pulse (red line) with frequency ${\omega }_p$ drives the two bound states tuned in resonance with two flux pulses (red and blue shaded areas). The populations of the qubits are read out (yellow line) at their highest frequencies. (b)–(g) Individual bound-state population on ${\mathrm{Q}}_1$ [$P_1$ (b)–(d)] and on ${\mathrm{Q}}_2$ [$P_2$ (e)–(g)] as a function of the bare frequency of ${\mathrm{Q}}_2$ for different frequencies of ${\mathrm{Q}}_1$. The dashed black lines are the expected bound-state frequencies calculated from Hamiltonian (c1). The green dashed line indicates the photonic band edge. (h) Bound-state coupling $U$, as defined in the main text, extracted from the measurements reported in panels (b)–(g). The black dashed line represents the expected value calculated from the Hamiltonian Eq. (3), while the solid line is obtained by the extended model given in Eq. (c1), which takes into account the next-nearest-neighbor interaction.

Figure 7

Time-resolved energy swap. (a) Pulse sequence. A $\pi$ pulse (red line) excites ${\mathrm{Q}}_2$, and after 30 ns, the two bound states are tuned in resonance with two flux pulses (blue and red shaded area) for a variable duration $\tau$. The qubits are tuned back to their unbiased frequencies, and their populations are read out (yellow line). (b)–(e) Chevron pattern. Measured bound-state population of ${\mathrm{Q}}_1$ (left column) and ${\mathrm{Q}}_2$ (right column) as a function of the interaction time and of the bare frequency of ${\mathrm{Q}}_2$. The two rows correspond to two different values of ${\omega }_{q1}$, at its largest frequency [(b) and (c)] and 6.02 GHz [(d) and (e)], respectively. (f,g) Horizontal line cuts in panels (b)–(e) (blue and red dashed lines) displaying the population of ${\mathrm{Q}}_1$ (blue) and ${\mathrm{Q}}_2$ (red) as a function of the interaction time. In both plots, the relaxation time is fitted to an exponential decay (black dashed line).

Figure 8

Bound-state interaction in two-excitation subspace. (a) Pulse sequence. After tuning the bound-state frequency with flux pulses (red and blue shaded area), they are excited (red and blue lines) and their population is read out. (b) Cross-Kerr interaction and avoided crossing between the states $|02⟩$ and $|11⟩$ for the frequency of the ${\mathrm{Q}}_2$ bound state ${\omega }_{\mathrm{BS}/2}/2\pi =6.653\mathrm{GHz}$. The dashed line represents the expected value calculated with the Hamiltonian Eq. (c1). (c) Energy-level scheme for the bare (left) and dressed (right) states.

Figure 9

Complete wiring diagram and room-temperature setup. The coherent tone to the waveguide is generated and measured with a vector network analyzer (VNA). The ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2$ driving (blue and red, respectively) are made by analog up-conversion of the pulse generated by an arbitrary waveform generator (AWG). The flux lines are controlled by two AWG channels. Finally, the qubit readout (yellow) is controlled by the up- and down-conversion of a pulsed generated by an AWG and logged by an analog-to-digital converter (ADC).

Figure 10

State preparation of the isolated ${\mathrm{Q}}_1$. (a) Population of ${\mathrm{Q}}_1$ as a function of flux pulse amplitude ${\mathrm{\Phi }}_{q1}$ and driving pulse frequency ${\omega }_p$. The black line shows the fit of the bound-state energy given in Eq. (4) to the measured data. The white dashed line shows the bare ${\mathrm{Q}}_1$ frequency calculated with the parameters extracted from the fit. (b) Bound-state frequency extracted from panel (a) as a function of the bare ${\mathrm{Q}}_1$ frequency. (c) Atomic fraction of the excitation, extracted from the relative population measured in the qubit. The black line shows the expected value of the mixing angles as predicted from the ideal case theory given in Eq. (b24) using the parameters extracted from the fit in (a) and (b).

Figure 11

Single coherent tone spectroscopy of a single mode. (a) Magnitude and (b) phase of the transmission of the mode of the coupled-cavity array at 5.552 GHz as a function of power. The self-Kerr from the JJ array resonator is inherited by the modes. The coherent tone addresses only one cavity mode, so the global fit of a Kerr nonlinear resonator (solid line) is used to extract the self-Kerr $K$ and the photon number $n$ of the array mode. The shade areas indicate the deviation from zero transmission (deviation from the zero phase).

Figure 12

Circuit model for $N=21$ capacitively coupled LC resonators (in green). The resonators at the edges of the array are capacitively coupled to a $50\text{−}\mathrm{\Omega }$ transmission line. Two transmon qubits are coupled at sites $x=10$ (blue) and $x=12$ (red).

Figure 13

(a) Sketch of the two-excitation spectrum (upper part) for a single qubit as a function of the qubit frequency. The bound states in the case of a linear resonator (black solid line) and an ideal two-level qubit (black dashed lines) are also shown for comparison (see main text for more details). (b) Excitation distribution of the two-photon bound state as a function of the single-excitation bound-state frequency. Specifically, we plot the probability of finding one or two excitations on the transmon, respectively given by $P_q={\sum }_{x=1}^N|c_2(x){|}^2$ and $P_{qq}=|c_{22}{|}^2$, along with the two-photon population $P_{ph}={\sum }_{x=1}^N{\sum }_{y=1}^N|u(x,y){|}^2$. (c,d) Spatial profile of the normalized two-photon wave function for the frequencies highlighted by the dashed vertical line in panel (b). The considered parameters are those corresponding to ${\mathrm{Q}}_2$ (see Table 1), with ${\omega }_r/2\pi =5.7\mathrm{GHz}$, $J/2\pi =249\mathrm{MHz}$, $g_2/2\pi =311\mathrm{MHz}$, and ${\beta }_2/2\pi =-257\mathrm{MHz}$ for the transmon qubit.

Figure 14

Sketch of the two-excitation spectrum (upper part) as a function of the ${\mathrm{Q}}_1$ frequency in the case of two qubits. We set the frequency of ${\mathrm{Q}}_2$ to ${\omega }_{q2}/2\pi =6.45\mathrm{GHz}$ with the remaining parameters fixed to ${\omega }_r/2\pi =5.7\mathrm{GHz}$, $J/2\pi =249\mathrm{MHz}$, $g_2/2\pi =311\mathrm{MHz}$, $g_1/2\pi =338\mathrm{MHz}$, ${\beta }_2/2\pi =-257\mathrm{MHz}$, and ${\beta }_1/2\pi =-266\mathrm{MHz}$. In this way, the bound states correspond to the one shown in Fig. 8.

Figure 15

Interaction strength between two bound states. The four lines represent the calculated coupling between two bound states, for qubits coupled to different array sites($x_i$) and with different qubit-cavity coupling strengths $g$.

Figure 16

Transmission cross-talk. (a) Micrograph of the sample bonded in the copper sample box. The 1.5-mm aluminium bond wires connecting the inner conductor of a coaxial cable to the bond pad on the chips have direct electromagnetic cross-talk. (b) Model of the cross-talk consisting in two power splitters/combiners: A small portion of the signal bypasses the waveguide and is recollected at the output, constructively or destructively interfering with the signal through the waveguide, and producing a nonzero transmission outside the band. (c) Measured transmission through the sample (green dots), which is not well reproduced considering only the ideal case (input-output theory at low power, red dashed line). The sum on the signal through the array and the cross-talk (black solid line) explain the nonzero transmission and the “pairing” effect.