Formation of robust bound states of interacting microwave photons

Abstract

Systems of correlated particles appear in many fields of modern science and represent some of the most intractable computational problems in nature. The computational challenge in these systems arises when interactions become comparable to other energy scales, which makes the state of each particle depend on all other particles¹. The lack of general solutions for the three-body problem and acceptable theory for strongly correlated electrons shows that our understanding of correlated systems fades when the particle number or the interaction strength increases. One of the hallmarks of interacting systems is the formation of multiparticle bound states^2-9. Here we develop a high-fidelity parameterizable fSim gate and implement the periodic quantum circuit of the spin-½ XXZ model in a ring of 24 superconducting qubits. We study the propagation of these excitations and observe their bound nature for up to five photons. We devise a phase-sensitive method for constructing the few-body spectrum of the bound states and extract their pseudo-charge by introducing a synthetic flux. By introducing interactions between the ring and additional qubits, we observe an unexpected resilience of the bound states to integrability breaking. This finding goes against the idea that bound states in non-integrable systems are unstable when their energies overlap with the continuum spectrum. Our work provides experimental evidence for bound states of interacting photons and discovers their stability beyond the integrability limit.

Fig. 1. Bound states of photons.

a, In a 1D chain of qubits hosting bound states, an initial state with adjacent photons evolves into a superposition of states in which the photons remain bound together. b, Interactions between photons can lead to destructive interference for paths in which photons do not stay together, thus suppressing separation. c, Schematic of the gate sequence used in this work. Each cycle of evolution contains two layers of fSim gates that connect the even and odd pairs, respectively. The fSim gate has three controllable parameters, which set the kinetic energy (θ), the interaction strength (ϕ) and a synthetic magnetic flux (β). The median gate infidelity, measured with cross-entropy benchmarking, is 1.1% (see Supplementary Information).

Fig. 2. Trajectory of bound photons.

a, Time- and site-resolved photon occupancy on a 24-qubit ring for photon numbers n_ph = 1–5. To measure a n_ph photon bound state, n_ph adjacent qubits are prepared in the $∣1⟩$ state. b, Schematic and example of bitstrings in $\mathcal{T}$ and $\mathcal{S}$. Centre of mass is defined as the centre of n_ph adjacent occupied sites. c, Evolution of the centre of mass of n_ph bound states. Each trajectory is similar to the single-photon case, highlighting the composite nature of the bound states. d, Extracted maximum (Max.) group velocity from the trajectory of the centre of mass. Black line, theoretical prediction. Exp., experimental. e, Decay of the bound state into the single excitations continuum due to dephasing. For all panels, θ = π/6 and ϕ = 2π/3, and the trajectories are averaged over all possible initial states. Data are postselected for number of excitations equal to n_ph.

Fig. 3. Band structure of multiphoton bound states.

a, Schematic of circuit used for many-body spectroscopy. n_ph adjacent qubits are prepared in the $\mid +⟩$ state, before evolving the state with a variable number of fSim gates. The phase of the bound state is probed by measuring the correlator $⟨{\sigma }_i^{+}..{\sigma }_{i+n_{\mathrm{ph}}-1}^{+}⟩$ for all sets of n_ph adjacent qubits. b, Real (top, Re) and imaginary (bottom, Im) parts of the n_ph = 2 correlator. c, Band structure for n_ph = 1–5 (top to bottom), obtained via a 2D Fourier transform in space and time of the n_ph correlators. Colour scale: absolute square of the Fourier transform, ∣A_k,ω∣². Dashed curves: theoretical prediction in equation (3). d, Band structure for n_ph = 2 in the weakly interacting (ϕ < 2θ) regime, displaying the emergence of a bound state only at momenta near k = ±π. Dashed black lines: theoretically predicted momentum threshold for the existence of the bound state (see Supplementary Information). e, Flux dependence of the n_ph = 2 band structure, displaying a gradual momentum shift as the flux increases (Φ₀ = 2πN_Q). Orange circles and dashed line indicate the peak position of the band. f, Extracted momentum shifts as a function of flux for n_ph = 1–5 (top to bottom), indicating that the rate of shifting scales linearly with the photon number of the bound states, that is, the pseudo-charge q of each bound state is proportional to its number of photons. Coloured lines: theoretical prediction.

Fig. 4. Resilience to integrability breaking.

a, Schematic of the 14-qubit chain with seven extra sites in red to break the integrability. b, Integrability is broken via an extra layer of fSim gates (red) between the chain and the extra qubits, with ${\varphi }^{\prime }=\varphi$ and a gradually varied ${\theta }^{\prime }$. c, Decaying probability of remaining bound for different swap angles ${\theta }^{\prime }$. Similar to Fig. 2e, the bound state decays into the continuum due to the dephasing. d, Probability of remaining bound after 20 and 40 cycles as ${\theta }^{\prime }$ is swept. e, Spectroscopy of the n_ph = 3 bound state for different ${\theta }^{\prime }$. Note that the bound state survives even for ${\theta }^{\prime }=\theta$. f, Half-width of the momentum-averaged spectra (from g) as a function of ${\theta }^{\prime }$. The grey line indicates the result for the chain without the extra qubits. g, Momentum-averaged quasi-energy spectra for varying ${\theta }^{\prime }$ fitted with Lorentzian. The bound state peak slowly disappears with the increase of ${\theta }^{\prime }$. AU, arbitrary units.