Interacting Qubit-Photon Bound States with Superconducting Circuits.

Qubits strongly coupled to a photonic crystal give rise to qubit-photon dressed bound states. These bound states, comprising the qubits and spatially localized photonic modes induced around the qubits, are the basis for many exotic physical scenarios. The localization of these states changes with qubit detuning from the photonic crystal band edge, offering an avenue of in situ control of bound-state interaction. Here, we present experimental results from a device with two transmon qubits coupled to a superconducting microwave photonic crystal and realize tunable on-site and interbound state interactions. We observe a fourth-order two-photon virtual process between bound states indicating strong coupling between the photonic crystal and transmon qubits. Because of their localization-dependent interaction, these states offer the ability to realize one-dimensional chains of bound states with tunable and potentially long-range interactions that preserve the qubits' spatial organization. The widely tunable, strong, and robust interactions demonstrated with this system are promising benchmarks towards realizing larger, more complex systems that use bound states to build and study quantum spin models.

In the strong-coupling domain, a qubit coupled to a photonic band edge forms an exponentially localized photonic mode at the qubit position, which together with the qubit forms a qubit-photon dressed bound state [1-7]. Photonic crystals are natural avenues to realize these bound states due to their intrinsically tailorable band structure, and characteristic Bloch mode electric field distribution [8] which enables access to strong coupling with qubits [9-13]. Bound states in multiqubit photonic crystal devices are an ideal platform to study many-body quantum optics in one-dimensional systems [6,7,14-19]. Unlike many qubits coupled to a common cavity mode but similar to the case of some optical multimode cavities [20,21], coupling to a band edge creates bound states that intrinsically preserve the spatial organization of qubits, offering the ability to create one-dimensional chains of bound states with tunable and potentially long-range interactions. The promise of engineering interaction profiles beyond the intrinsic flip-flop with additional microwave drive tones further opens the possibility of simulating a wide range of quantum spin models in future devices [17]. In this paper, we demonstrate and characterize the underlying, fundamental tunable on-site and interbound-state interactions in a superconducting microwave photonic crystal device coupled to two transmon qubits.

A single dressed bound state, seeded by a single qubit in a crystal, is itself a unique avenue of study. Liu et al. first directly detected such a bound state in a stepped-impedance microwave crystal coupled to a single transmon qubit [13]. That work characterized the dependence of localization length on detuning between the transmon qubit and the band edge and further confirmed the existence of the localized state in the band gap when the bare transmon qubit is in the passband—an unmistakable signature of non-Markovianity, as seen in Figs. 1(e), 1(f), and Appendix D. We note that the transmission dip observed in Figs. 1(e) and 1(f) is due to the reflection from the transmon qubit [22-24]. State localization is tunable in situ with frequency through a range determined by device parameters, including transmon qubit-waveguide coupling and band curvature. Compared with previous work, we attain increased localization in this device [Fig. 1(b)] due mainly to a flatter band dispersion, realized by tailoring the unit cell of the photonic crystal (see Appendix A for a detailed discussion of the experimental parameters of our system). The bound-state localization length in this device is still widely tunable, which is critical for realizing strong, tunable interaction between spatially separated bound states. As the different coupling regimes translate to dramatically altered system behavior [7], it is important to determine which domain our system falls under. In systems such as the one presented here, qubit emission into the waveguide being larger than the other decay rates (coherent atom-photon interaction rates larger than decay rates) is the minimal coupling criterion, upon which the dressed bound state within the gap can be spectrally resolved [7]. The strong coupling criterion corresponds to the situation where a bare qubit resonant with the band edge gives rise to a bound state that is shifted from the band edge by more than the bound state’s linewidth [7,13]. In our finite system, we observe an approximately 250-MHz separation between the bound state and the band edge with bound-state linewidth of 4 MHz when a qubit is resonant with the band edge, thus firmly reaching the strong coupling condition [see Figs. 1(b), 1(e), and 1(f)]. By fabricating two transmon qubits in the photonic crystal [see Fig. 1(a) and Appendix B for a discussion on coupling transmons to photonic crystals], we realize multiple, spectrally resolvable bound states and can study interbound-state interaction.

FIG. 1.

A platform for interacting dressed bound states.—

(a) A 16-site microwave photonic crystal is realized by alternating sections of high and low impedance coplanar waveguide. Two transmon qubits (multilevel, anharmonic energy ladder) are in neighboring unit cells in the middle of the device, centered in the high impedance sections for maximal coupling to the band edge at 7.8 GHz [all values presented in units of (2π) Hz, i.e., ωBE = 7.8 (2π) GHz]. For this experiment, the passband (band gap) refers to states above (below) the band-edge frequency. Each transmon is individually tunable in frequency via a local flux bias line. (b) Bound-state linewidth, an indirect measure of localization, varies with bare transmon qubit frequency. The wide range over which photon localization can be tuned indicates the feasibility of realizing a chain of strongly interacting bound states. Experimentally measured and simulated linewidths are shown in red and black, respectively. Inset: Overlay of simulated S21 from the transfer matrix method (blue) and measured high-power S21 (black) shows good agreement in bare crystal characteristics. (c) The interaction between bound states will be determined by overlap of their localized photonic envelopes with the qubits. (d) One can couple more qubits to the band edge by adding them to other cells of the photonic crystal. In such a system, the localization-length-dependent interaction of the bound states would preserve the spatial organization of qubits across the crystal, and determine the many-body structure of the interactions. (e) Experimental data and (f) hopping model simulation for S21 vs single-qubit frequency and probe frequency. The bare band edge is at 7.797 GHz. The bright peak in the band gap is the dressed qubit-photon bound state. The bound state always exists within the band gap for qubit frequencies (the other qubit is far detuned and has negligible effect) both above and below the band edge—a clear signature of non-Markovianity.

The nature of interbound-state interaction makes this platform intrinsically well suited for investigating one-dimensional chains of bound states [see Fig. 1(c)]. Realizing sizable chains is possible by increasing the number of unit cells—a property that does not impact the Bloch mode distribution or band dispersion. Thus, qubits can be in separate unit cells but realize nearly identical coupling to the band edge. As the strength of interbound-state interaction depends on the spatial overlap of the photonic wave functions with the qubits, the distance separating qubits (set by device design) is directly mapped into the interactions of the system, maintaining the chainlike interaction pattern. Furthermore, in the investigation of bound states, the finite size of the crystal is a practical advantage: the overlaps of bound states with ends of the crystal lead to only quasi-bound states, allowing for detection through transmission measurements (see Appendix C 2 for a proof). Direct detection of such a state in this way was first demonstrated in Liu et al. [13].

Controlling photon-mediated interaction between superconducting qubits has been demonstrated in other one-dimensional systems—such as two qubits in a cavity [25,26] or in a linear waveguide [27]. However, in these cases the distance between the qubits was effectively eliminated (i.e., standing-wave interaction in a cavity) or otherwise reduced (modulo wavelength in a linear dispersion waveguide). Thus, photonic crystals and tunable bound states offer a fundamentally distinct form of interaction.

In addition to determining localization length, the frequency of the bound state also determines on-site interaction strength. In Figs. 2(a) and 2(b), we characterize the dependence of the transition frequencies between the three lowest levels of the bound state on bare transmon qubit frequency, and observe the steady reduction in bound-state anharmonicity from over 350 MHz to 0 MHz as the transmon qubit is tuned from deep in the band gap to the passband. Here, we have defined the bound-state anharmonicity as , where and are the dressed bound-state frequencies [see level diagram in Fig. 2(a)]. This is dramatically more than the approximately 10% modification of transmon qubit anharmonicity with frequency expected when a transmon qubit is strongly coupled to a cavity mode [28].

FIG. 2.

Probing the bound-state energy levels.—

(a) The anharmonicity of the bound state is dependent on bare transmon qubit frequency ω01, demonstrating a tunable on-site interaction strength. In blue (red), the first (second) transition of the bound state is measured across a range of bare qubit frequencies (inset: simulation). Upper left corner: level diagram of the bare transmon and dressed transmon. (b) Decreasing anharmonicity with increasing bound-state frequency shown in red for experimental data and black for simulation. (c) Power spectrum of a resonantly driven bound state for increasing drive amplitude. Sidebands are linearly displaced from the central peak with increasing drive amplitude, characteristic of the Mollow triplet. Inset: second-order autocorrelation measurement for drive amplitude = 0.2 is consistent with single photon, antibunched transport. (d) Emission spectrum of a resonantly driven (≈7.59 GHz) bound state (induced by a qubit at 7.9 GHz, which is above the band edge located at 7.8 GHz) as a function of drive power. At low drive power, only the Mollow triplet is observed. With increasing power we see four additional sidebands, two on either side of the original Rabi sidebands, which together are the transitions between the three lowest levels of the anharmonic bound state (|0〉, |1〉, |2〉). The white crosses are from numerical simulations (see Appendix E). We have included five transmon qubit levels in our simulation. See text for the discussion of the seventh sideband around 7.25 GHz.

Therefore, while we may treat the one-excitation and two-excitation bound states as first and second excited states of a new effective anharmonic transmon qubit [13], it is important to note that this effective transmon qubit differs in frequency and anharmonicity from the bare multilevel transmon qubit. Defining the three lowest bare transmon qubit levels as |0〉 |1〉, and |2〉, here the two-excitation bound is largely due to the coupling of the second transmon qubit transition (|1〉 ↔ |2〉 with the band edge rather other the multiphoton effects [7,15] (see Appendix F 1).

Numerical simulations, modeling the photonic crystal as a coupled cavity array with free parameters fit to match the band curvature from the dispersion relation [7,29] (see Appendix C for details), show similar dependence of anharmonicity on detuning [see Fig. 2(a) inset and Fig. 2(b)]. Unlike the transfer matrix method [30-32], this approach can extend beyond the single-excitation manifold to capture the higher levels of the bound state, as well as the Lamb shift of the qubit frequency, observed when including next-nearest-neighbor hopping between coupled cavities. Each transmon qubit is modeled as a three-level ladder (unless otherwise mentioned) with negative anharmonicity, and with the |0〉 ↔ |1〉 and |1〉 ↔ |2〉 transitions coupled with amplitudes g and , respectively, to its local cavity site. It is critical to include level |2〉 to accurately reproduce the two-excitation manifold observed in experiment.

The tunable level structure also emerges in the emission spectrum of a continuously driven bound state [Figs. 2(c) and 2(d)], induced by a single qubit with bare frequency above the band edge. At low drive amplitude or Rabi frequency, transmission across the crystal via the bound state exhibits antibunching [see Fig. 2(c), inset] [33], consistent with single-photon transport of a two-level system and resonant pump (see Appendix D 4) [34-36]. As Rabi frequency is increased, we see a Mollow triplet emission spectrum [Fig. 2(c)], characteristic of a driven two-level system. When the Rabi frequency is on the same order as the anharmonicity, the bound state can no longer be approximated as a two-level system. In this domain, the steady state will be a mixture of the three eigenstates obtained by diagonalizing the drive Hamiltonian in the Hilbert space spanned by |0〉 |1〉, and |2〉. Transitions between all eigenstates result in six sidebands [37]. These six sidebands are visible in Fig. 2(d), though emission intensity varies greatly among them due to eigenstate population. A seventh transition is evident in the data [7.25 GHz in Fig. 2(d)]. This additional transition is due to the fourth effective transmon qubit level while its curvature is reproduced by including a fifth effective transmon qubit level in our numerical simulations (see Appendix E). Crucial to reproducing this transition in our theoretical simulations is taking into account that the bound-state level structure cannot be defined by a single anharmonicity, i.e., given the anharmonicity of the bound state, the frequency of the fourth level of the bound state is not simply given by as is expected for a transmon [38] [see transmon level diagram in Fig. 2(a)].

We observe the flip-flop interaction (by a flip-flop interaction between two qubits with basis states |0〉, |1〉, we mean a Hamiltonian proportional to |01〉〈10| + |10〉〈01|) between the two the spatially bound states measuring the avoided crossing in transmission when the bound states are tuned into resonance. As these qubits (while we really have a multilevel systems coupled to the photonic crystal, we refer to them as qubits when one can ignore higher levels) are a fixed distance apart (9 mm) and there is negligible direct capacitive coupling, the strength of the flip-flop interaction will be entirely determined by the overlap of the localized photonic mode of one qubit with the other qubit, controllable here via the qubit frequencies.

In Fig. 3(a), the frequency of one qubit is held constant while the other is tuned through resonance. Measuring transmission at the single-photon level reveals an avoided crossing between the and levels of the coupled dressed bound states. Transmission amplitude of a bound state dims when the bound state and bare qubit are near resonance [see Fig. 3(a) inset]. From this plot, we can extract a resonant bound-state–bound-state interaction of 120 MHz for a 7.73-GHz bare qubit frequency. In comparison, Figs. 3(b)–3(d) show reduced interaction strength when both qubits are further detuned from the band edge, 6.125 GHz, 6.75 GHz, and 7.625 GHz, respectively, for the fixed qubit frequency.

FIG. 3.

Interacting bound states.—

Interaction between bound states is characterized by the avoided crossing (observed in S21 measurement) that arises while tuning one qubit (y axis) through resonance with the other (fixed). (a) An avoided crossing of 240 MHz is observed when the fixed qubit is at 7.73 GHz. The two points where transmission amplitude of a bound state dims are understood as the bound-state peak being resonant with the qubit frequency. (a), inset—Hopping model simulation of the one-excitation manifold is consistent with experimental observation. The lamb shift in the hopping model originates from next-nearest-neighbor interaction between coupled cavities. (b),(c),(d) Tunable bound-state interaction strength is illustrated in example bound-state avoided level crossings for a fixed qubit whose bare frequency is circa 6.125, 6.75, and 7.625 GHz. As qubits are detuned further from the band edge, bound states are more tightly localized, reducing overlap and thus reducing interaction. (e),(f) Transmission when the qubits are on resonance across a range of qubit frequencies in the experiment and the simulation, respectively. The uneven linewidths of the two bound states when they occur at the same frequency suggest they are symmetric (higher-frequency bound state) and antisymmetric (lower-frequency bound state) states (see main text). (g) Bound-state avoided crossing and qubit population (from simulation) as a function of average bound-state frequency. A steady reduction in interaction strength occurs with increasing detuning from the band edge (moving deeper into the band gap) due to increasing localization of the bound states. Hopping model simulation (black) captures this detuning-dependent behavior observed in experiment (red). Near the band edge, both bound states (blue and cyan) have a significant photonic contribution.

To characterize this aspect of the two bound-state interaction, we map the magnitude of the avoided crossing as a function of detuning. In Fig. 3(g), the qubits are maintained on resonance with one another while being simultaneously tuned through the band gap. Theoretical modeling [Figs. 3(g) and 8(e)] shows experimental data to be consistent with localized photonic states and with interaction via wave-function overlap. In the limit where the qubit is deep in the gap, the Markovian approximation holds as is evident in the probability that the bare qubits are in the excited state (obtained from the hopping model) in Fig. 3(g) [see Fig. 8(c) for the photonic probability of the bound states]. Here, the localized mode and flip-flop interaction both have the same distance dependence e− where , is the localization δ is the detuning between the bare qubit and the band edge, a is the unit cell size, and the band-edge dispersion is ω = ω0 + αa2(k − k0)2 (see Appendix D 2). The corresponding flip-flop interaction Hamiltonian is . When the bare qubit frequencies are near the band edge the probability that the bare qubits are in the excited state decreases [Fig. 3(g)] and we have non-Markovian behavior (see Appendix D 3 for a detailed on the breakdown of Markovian behavior). While our experiment studies steady states, in other settings, where an initial state evolves in the absence of an input field, non-Markovianity can lead to the preservation of entanglement during time evolution [6,39-41]. It would be interesting to study in future work whether there is also entanglement in steady state and how to best measure it in our system.

FIG. 8.

Exact solution versus Born-Markov approximation. Here, the red lines are the exact solution while the blue lines are the Born-Markov approximation. The dashed black lines mark the band edge and the thick black line is the bare qubit frequency. Here, we take ω0 = 7.8 GHz, α = 1.155 GHz, and g = 0.5275 GHz. (a) Single qubit. (b) Two qubits for |z1 − z2| = 1. (c) Theoretical photon population of the bound states. (d) Linewidth ratio of the two bound states as a function of bound-state energy. (e) Avoided crossing on resonance versus bare qubit frequency. We see that the experimental data are well described by the Markovian solution only deep in the gap.

We now turn to the qubit nature of the bound states when the bare qubit frequencies are resonant. The qubit part of the wave function of these two bound states (obtained from the effective Hamiltonian of the system) is (approximately) either symmetric (|0〉|1〉 + |1〉|0〉) or antisymmetric (|0〉|1〉 − |1〉|0〉) We theoretically predict that the higher (lower) frequency bound state at resonance in Fig. 3(a) is symmetric (antisymmetric) (see Appendix D 2). This turns out to affect transmission through the system. Intuitively, the antisymmetric state is expected to be dimmed as the exponential parts of the localized photonic states cancel each other out; and hence the linewidth, which is proportional to the wave function at the end of the photonic crystal, is smaller. However, because the band edge is not at zero momentum in our system, it turns out the symmetric state is actually dimmed and has a smaller linewidth, as we prove in Appendix D 2. In Fig. 3(e), we see that the bound states at the same transmission frequency (with different bare qubit frequencies) have drastically different linewidths with the higher-frequency bound state having a smaller linewidth, consistent with our numerical simulations [Figs. 3(f) and 8(d)]. This provides some indirect experimental evidence that the qubit part of the higher (lower) frequency bound-state wave function is indeed symmetric (antisymmetric).

To further study tunable on-site interaction, we probe the interacting bound states beyond the one-excitation manifold using spectroscopic measurements [see Fig. 4(a)]. Similar to spectroscopy of qubits in cavities, we can use transmission at the band edge to help detect bound-state transitions, a technique that provides sharper contrast compared to transmission measurement for the more highly localized bound states and allows detection of higher-dressed transitions, such as the transition between |0〉 and |2〉 driven by two photons of frequency ω02=2.

FIG. 4.

Interaction between two-excitation levels of two bound states.—

(a) Spectroscopic measurement while tuning one bound state through the other (qubit fixed at 7.2 GHz), reveals survival of strong interaction into the two-excitation manifold. Crossed and dotted lines are guides to the eye to discern the levels belonging to the first (|01〉 and |10〉) and second (|02〉, |20〉, and |11〉) excitation manifolds, respectively. Here, we have labeled our states as |transmon one; transmon two〉. In addition to the |02〉(|20〉) and |11〉 avoided level crossing, we also detect a two-photon virtual interaction between |02〉 and |20〉 white box). This interaction—fourth order in coupling g—manifests itself in avoided level crossings up to and exceeding 20 MHz. For comparison, the |02〉(|20〉)-|11〉 and |01〉-|10〉 interactions are second order in g and thus both are significantly stronger. Inset: two-photon avoided crossing strength versus average bound-state frequency. (b) Numerical simulation for fixed bare qubit frequency of 7.27 GHz, with the one (two) excitation manifold in black (red).

With this technique we detect interaction between |02〉, |20〉, and |11〉 of the coupled bound states, observed as avoided level crossings. In addition to the single-photon exchange interaction between |02〉 |20〉 |11〉 and [26], remarkably we measure the two-photon virtual interaction between |20〉 and |02〉, despite the fact that this process is fourth order in coupling g (see Appendix F 2). This two-photon interaction shows consistent dependence on detuning: increasing in strength (from 0 MHz to over 10 MHz) as the bound states shift towards the band edge and the states become more delocalized [see inset of Fig. 4(a)]. Numerical simulations [Fig. 4(b)] are consistent with experimental data and capture the relative magnitudes of interaction between levels as well as frequency dependence on coupling strengths. Observation of this small interaction highlights the overall strength of interbound-state coupling possible via overlap alone.

The widely tunable on-site and interbound-state interactions demonstrated with this device and consistent theoretical simulations are promising benchmarks towards realizing larger, more complex systems of bound states. Examples of these systems include spin models with both local and long-range interactions, which arise when the bare qubit frequencies are deep in the band gap [17], and complicated multiqubit-photon bound states, which arise when the bare qubit frequencies are in the passband (see, e.g., Ref. [7]). Beyond stepped impedance coplanar waveguides, there are undoubtedly numerous ways to realize superconducting microwave photonic crystals, including lumped element or Josephson junction-based designs, that are equally compatible with superconducting qubits. Regardless of the platform, behavior of bound states due to qubit-band edge coupling will mirror the behavior characterized in this work—elevating this platform above any single experimental design choice.

While the bound states were centered in neighboring unit cells in this device, this is not a limitation or requirement for future experiments as the range of localization can be accordingly set via the basic crystal parameters, as seen by comparing bound-state linewidths measured here with those reported previously [13]. Therefore, one can realize a one-dimensional chain of bound states in a moderately sized photonic crystal, where individual control over the qubits would allow dialing up or down long-distance interaction between sets of qubits.

In this work the interactions were in situ tunable via qubit frequency (DC flux), a static quantity on the timescale of the bound-state lifetime. Dynamically controllable interactions would introduce an additional tool for designing and manipulating spin Hamiltonians [17]. One method for realizing this type of fast timescale control is flux pumping, a technique involving microwave frequency modulation of the qubit frequency along the flux bias line [42-45]. Another potential pathway would use an auxiliary microwave field through the crystal itself. Here, the qubits could be maintained on resonance deep in the band gap such that there is minimal interaction via bound-state overlap. A single rf control tone can be turned on to induce a transition close to the passband, thus redressing the bound states into new, effective bound states with interaction strength depending on properties of the microwave drive. The addition of several drives or precisely shaped microwave pulses (made possible by commercial high-speed arbitrary microwave waveform generators [46]) promise not only changing interaction strength but also modifying the shape of the interaction itself—from an exponential to a sum of exponentials—leading to a wide range of possibilities including power-law-decaying interactions [17]. These supplementary forms of tunable control would expand the ability of the qubit-photonic crystal platform to realize a broader class of tunable spin models.