Propagation of microwave photons along a synthetic dimension

The evenly spaced modes of an electromagnetic resonator are coupled to each other by appropriate time modulation, leading to dynamics analogous to those of particles hopping between different sites of a lattice. This substitution of a real spatial dimension of a lattice with a “synthetic” dimension in frequency space greatly reduces the hardware complexity of an analog quantum simulator. Complex control and readout of a highly multimoded structure can thus be accomplished with very few physical control lines. We demonstrate this concept with microwave photons in a superconducting transmission line resonator by modulating the system parameters at frequencies near the resonator's free spectral range and observing propagation of photon wave packets in time domain. The linear propagation dynamics are equivalent to a tight-binding model, which we probe by measuring scattering parameters between frequency sites. We extract an approximate tight-binding dispersion relation for the synthetic lattice and initialize photon wave packets with well-defined quasimomenta and group velocities. As an example application of this platform in simulating a physical system, we demonstrate Bloch oscillations associated with a particle in a periodic potential and subject to a constant external field. The simulated field strongly affects the photon dynamics despite photons having zero charge. Our observation of photon dynamics along a synthetic frequency dimension generalizes immediately to topological photonics and single-photon power levels, and expands the range of physical systems addressable by quantum simulation.

Figure 1

Coplanar waveguide resonator with modulated SQUID termination. (a) Schematic of resonator including external coupling and flux line. Standing waves depict the local amplitude of five adjacent modes; the lowest five modes are displayed for visual clarity though not directly accessed in this work. (b) Schematic of lattice sites along the synthetic dimension when the flux bias is modulated at frequency $\mathrm{\Omega }$ near the average free spectral range. Coupling to the next higher (lower) frequency site can be envisioned as absorbing (emitting) a photon from (to) the modulating field. (c) Three lowest-energy eigenfunctions of a five-site tight-binding Hamiltonian with uniform nearest-neighbor coupling. The narrow-band input signal shown excites only eigenfunctions with nonzero amplitude at site 0 (solid red and evenly dashed blue lines), and subsequent output signals at other sites are emitted only from these two eigenfunctions. (d) Optical micrograph of the resonator, fabricated with aluminum on a $14\times 14$ mm chip of high-resistivity silicon. The device layout is similar to that in Refs. [46, 47]. Key features are shown in scanning electron micrographs: (e) external-coupling capacitor, (f) superconducting air bridges to suppress parasitic slotline resonances, and (g) SQUID array termination and flux line. The scale bars are, respectively, 50, 20, and $50\mu \mathrm{m}$. The inward crimping of the airbridge span in (f) is a systematic fabrication defect and is discussed in Appendix pp1

Figure 2

Experimental spectra demonstrating flux tunability and tight-binding coupling. (a) Parallel tuning of modes 31, 32, and $33\left(n=-1,0,1\right)$ over 17 MHz as the flux line voltage bias is swept (the frequency axis is collapsed between each mode). The thin vertical line denotes the DC bias used for modulation experiments; the blue rectangular overlay displays a typical flux modulation amplitude equivalent to $0.062{\mathrm{\Phi }}_0$. (b) Continuous-wave (CW) reflection spectra at site $n=0$, measured as a VNA reflection coefficient as the modulation frequency is swept from 153.9 to $156\text{MHz}$. VNA spectra correspond to Eq. (9). The dashed line and corresponding line trace at $155.1\text{MHz}$ correspond to “resonant modulation” at $\mathrm{\Omega }/2\pi =155.1\text{MHz}$ used in subsequent experiments. (c) Transient spectra for scattering between input site $n=0$ and output sites $(-1,0,1)$, corresponding to Eq. (10). Traces for ${\mathit{S}}_{00}$ (green, middle) and ${\mathit{S}}_{10}$ (blue, upper) are vertically shifted by 0.1 and 0.2, respectively. The (green, middle) trace may be compared to the VNA trace in (b) as they represent the same scattering parameter with (b) and without (c) direct reflection.

Figure 3

Transient propagation from a single-site initial state. (a) Pulse schematic for exciting a single mode to steady state, applying flux modulation and measuring the output voltage while the excitation propagates and rings down. All voltage traces in (c)–(h) are displayed from the arrival of the modulation pulse to an intermediate time before the end of the modulation pulse and smoothed with a 16-point moving average. The numerical values are the voltage amplitudes $|V_{\text{ADC}}|$, normalized so that each colormap has a maximum value of 1. (b) Schematic of on-site energies in the rotating frame, for modulation resonant with the free spectral range (top) and detuned by an amount $\mathrm{\Delta }$ (bottom). We show the ideal case with no on-site disorder. (c), (f) Experimental magnitude of slowly varying voltage envelopes with flux modulation at (155.1, 152.1) MHz, respectively, compare the theory calculations in (d) and (g) and the overlaid traces at $n=0$ in (e) and (h).

Figure 4

Calibration and propagation of an initial wave packet. Color online. (a) Approximate dispersion along the frequency dimension, extracted from peaks in the 2D Fourier transform of transient propagation data as in Fig. 3. Data are arbitrarily normalized to a maximum of 1. Dashed vertical lines indicate target $k$ values for the initial states in (d) (yellow, leftmost) and (e) (cyan, rightmost). (b) Schematic of the nearest-neighbor interference experiment used to calibrate the on-chip amplitudes $\beta \left(0\right)$. Interference occurs in the triangular overlap between light cones emanating from neighboring sites. (c) Representative calibration data, with $r\approx 1$ for maximum interference amplitude. Sites $-1$ and 0 are excited by a two-frequency pulse with phase offset ${\theta }_{\text{AWG}}$ defined at the start of the pulse. Output voltage amplitude from site $-1$ is plotted for short modulation times as the phase offset is swept. (d) Propagation of a five-site wave packet with initial $k_{\text{eff}}\approx 0.5\pi$, showing negative group velocity and coherent reflection of the wave packet at the $n=-6$ barrier site. (e) Directional Bloch oscillation of a five-site wave packet with initial $k_{\text{eff}}\approx 0.78\pi$.

Figure 5

Experimental setup. The sample is located at the mixing-chamber plate of a dilution refrigerator, packaged in a microwave PCB and copper enclosure, and surrounded by cryogenic magnetic shielding. All instruments are phase-locked by a 10 MHz rubidium frequency standard (SRS SIM940); the output of both local oscillator sources is high-pass filtered to remove feedthrough from the 10 MHz clock. Notation follows Refs. [61, 62].

Figure 6

On-site Hamiltonian parameters extracted by fitting measured reflection coefficients of 29 modes. (a) Frequency disorder in two different rotating frames corresponding to flux modulation at $155.1\text{MHz}$ (black circles) and $155.0\text{MHz}$ (green squares). $155.1\text{MHz}$ was used to approximate “resonant” modulation, where in the ideal case all points would fall on a horizontal line at $\mathrm{\Delta }=0$. After carefully refitting the reflection coefficients, this “resonant” description applies more accurately to modulation at $155.0\text{MHz}$; however, a systematic detuning of about $100\text{kHz}$ per site has a minimal effect for modulation times $t\lesssim 1\mu \text{s}$. The range of frequency sites accessed in the experiments of this paper is highlighted with a light-blue background between two vertical dashed lines, representing “barrier sites” with high loss rates. (b) Loss rates to external coupling $({\kappa }_e$, blue circles) and internal degrees of freedom $({\kappa }_i$, red squares). Internal loss rates at relative mode numbers $n=\{-6,7,10\}$ fall in the range ${\kappa }_i/2\pi \sim 0.52\text{MHz}$, outside the vertical bounds of the plot. In particular, the high loss rates at $n=\{-6,7\}$ determine the “barrier sites” reported in this paper.

Figure 7

Zero-point phase distribution and related quantities. (a) Predicted phase magnitudes for the first 60 modes of the device using Eq. (E4), with a maximum at $n=9$ followed by decay at higher $n$. All modes satisfy $|{\varphi }_n^{\text{zp}}|\ll 1$. (b) Variation in free spectral range (FSR) in the absence of fabrication disorder, plotted as a discrete second derivative of the mode frequencies. The FSR increases monotonically with $n$, stabilizing asymptotically for $n\gg 30$. (c) Predicted single-hopping rates for single-tone modulation at 155.1 MHz, increasing linearly with modulation amplitude. (d) Predicted double-hopping rates for single-tone modulation at $155.1\text{MHz}$, increasing quadratically with modulation amplitude. At the experimental value $\mathrm{\Phi }=0.062{\mathrm{\Phi }}_0,|J_{n,n+2}|<0.05|J_{n,n+1}|$; however, the double-hopping rates are still used for theory comparisons as in Fig. 3. For plots (b)–(d), the relative magnitude of the traces in each plot increases with modulation amplitude, though in (b) this effect is negligible.

Figure 8

Model for SQUID array. (a) Circuit diagram of a SQUID array as a ladder of Josephson junctions and capacitors, in which the twin junctions of each SQUID have been lumped into a single flux-dependent $E_{Jn}$. (b) Distribution of phases for a randomly generated array of eight SQUIDs as a function of flux bias, calculated in the linear voltage-divider approximation in Eqs. (F4, F5, F6, F7). Large disorder is used to emphasize that larger phase drops occur across larger inductances, which are designated by smaller $E_{Jn}$ near zero flux, and smaller $d_n$ near half-integer flux. (c) Geometric interpretation of the dynamics in Eq. (F3) for a two-SQUID array. The phase configuration lives on the intersection between the plane and cosine surface, where the plane oscillates along the direction of the dashed line. We approximate the intersection as a parabola confined to the center well and assume the configuration adiabatically follows the bottom of the parabola. (d) Comparison of $E_J$ calculations for four random eight-SQUID arrays. (Red, middle) curves represent the full sum in Eq. (F12) (green, upper) curves represent the single-SQUID approximation in Eq. (F13), and (black, lower) curves represent the zero-disorder limit where both models coincide. Where these three curves differ, the effective $E_J$ is smallest for the zero-disorder limit and largest for the single-SQUID approximation.

Figure 9

Diagrams and extended-time behavior for nearest-neighbor interference. (a) Example off-diagonal diagram satisfying Eq. (I3), in which the product of all propagation phases depends only on the distance between the initial and final sites. (b) Example diagram connecting $n=(0,1)$ with only single hops such that the total number of transitions is odd; this type of diagram is included in deriving Eq. (I7). (c) Example diagram connecting $n=(0,0)$ with an odd number of hops by using a diagonal matrix element; we ignore such diagrams at short times. (d), (e) Experimental and calculated voltage magnitudes over $1\mu \text{s}$ for initial sites $n=(-1,0)$, with output measured at $n=-1$. The first $0.25\mu \text{s}$ of (d) are reproduced in Fig. 4. The calculation in (e) assumes a uniform coupling rate $J/2\pi =1.25\text{MHz}$, uniform loss $\kappa /2\pi =90\text{kHz}$, and a calibration phase ${\theta }_{n=-1}^{\text{calib}}=-0.20\pi$. This calibration phase absorbs the modulation phase for simpler plot labeling.

Figure 10

Fourier transforms for second-nearest-neighbor coupling. Color online. (a) Single-tone modulation at 310.2 MHz showing two symmetric dispersion periods. White (solid) and blue (dashed) curves are fits to models incorporating the lowest and two lowest modulation harmonics, respectively. (b) Two-tone modulation at 155.1 and 310.2 MHz showing asymmetric dispersion. The white (solid) curve is a fit incorporating only the lowest harmonic of each tone; the blue (dashed) curve includes the second harmonic of the $2\mathrm{\Omega }$ tone as in (a). Dots (cyan) overlaid on both plots show the maximum amplitude for each value of $k_f\mathrm{\Omega }$.

Figure 11

Experimental data for time-reversed coupling and parasitic oscillations. Color online. (a), (b) Multisite output traces for modulation pulses at amplitudes of $0.062{\mathrm{\Phi }}_0$ and $0.031{\mathrm{\Phi }}_0$ respectively; all traces are smoothed by a 16-point moving average. Pulse durations are chosen to minimize the amplitude remaining in site 0 during the delay between pulses (here $0.5\mu \text{s})$. (c) Single-site output traces at $n=0$ as the delay is swept from 0.05 to $1\mu \text{s}$, with flux amplitude $0.031{\mathrm{\Phi }}_0$. (d) Single-site output traces at $n=0$ for varying resonator excitation amplitude and flux amplitude $0.062{\mathrm{\Phi }}_0$, showing four periods of parasitic oscillation within the $1\mu \text{s}$ pulse delay. (e) Single-site reflection spectra at $n=0$, sweeping modulation frequency at larger detuning. Dashed lines indicate approximate modulation frequencies that induce Bloch oscillations at 4 MHz. (f) Single-site reflection spectra at $n=0$, sweeping modulation frequency near the expected lowest harmonic of the device. A large redshift is seen for modulation at 75.5 MHz, which we attribute to a cross-Kerr interaction when the lowest harmonic is directly excited by the flux line.

Figure 12

1D scattering model. (a) Schematic for reflected and transmitted waves when an incident wave approaches a single defect from the left. Uniform nearest-neighbor coupling and zero on-site energy outside the defect are assumed. (b) Amplitudes of reflected and transmitted waves as a function of defect size and effective wave vector $k_{\text{eff}}\equiv k_{\text{i}}+{\theta }_J$. Overlaid plots display $|A_{\text{r}}{|}^2$ (blue-to-red gradient corresponding to widening downward peaks) and $|A_{\text{t}}{|}^2$ (green-to-purple gradient corresponding to widening upward peaks) as $|k_{\text{eff}}|$ varies from $0.01\pi$ to $0.5\pi$.