Doppler-induced tunable and shape-preserving frequency conversion of microwave wave packets

Abstract

In superconducting electronics, the ability to control the frequency of microwave wave packets is crucial for several applications, such as the operation of superconducting quantum processors and the readout of superconducting sensors. We introduce a new approach to microwave frequency conversion harnessing a dynamic Doppler effect induced by a propagating front separating regions of different phase velocities. Employing a high-kinetic-inductance superconducting transmission line in a travelling-wave geometry, we were able to implement frequency shifts of microwave wave packets at 500 MHz and 4 GHz of up to 3.7 % while fully preserving their temporal shape. In contrast to conventional methods based on frequency-mixing, our Doppler-induced frequency-conversion method avoids spurious mixing products, is continuously tunable by a quasi-dc current amplitude, and allows to imprint arbitrary patterns on the instantaneous frequency profile of temporally long wave packets. By engineering transmission lines that allow for larger phase-velocity changes and/or by cascading multiple Doppler-induced frequency conversions, an unlimited amount of frequency shifting is in principle attainable. These features demonstrate the potential of our frequency-conversion technique as a promising tool for advanced control of microwave wave packets for different quantum applications.

Figures (9)

• Figure 1: Experimental setup and concept.a, schematic outline of the experimental setup. Two DACs (digital-to-analogue converters) are employed to create a microwave wave packet (WP) and a control pulse (CP), which are routed to the opposite ports of a superconducting high-kinetic-inductance transmission line and counter-propagate through it. The outgoing wave packet is routed through a splitter to an ADC (analogue-to-digital converter). b, cryogenic microwave setup: the superconducting device is packaged in a copper holder equipped with printed circuit boards and microwave connectors. A scanning electron microscopy (SEM) image shows the microscopic structure of the superconducting transmission line, implemented as a coplanar waveguide loaded with finger-shaped capacitor elements, patterned in a thin NbTiN film on a Si substrate DW2023. c, measured output wave packets in absence (gold line) and presence (blue line) of a control pulse for identical input wave packets. The carrier frequency of the first wave packet coincides with that of the input wave packet, i.e. 500 MHz, while the carrier frequency of the second wave packet is visibly increased due to the Doppler-induced frequency-conversion effect induced by the falling front of the control pulse.
• Figure 2: Frequency shifting dynamics. The dynamics of the frequency-conversion experiment is shown for four configurations a, b, c and d, corresponding to different delays between the wave packet (WP) and the control pulse (CP): a, WP and CP do not meet inside the device. b, WP meets the rising front of CP inside the device and undergoes a redshift. c, WP meets both rising front and falling front of CP inside the device. The respective red- and blueshift of the WP cancel out. d, WP meets the falling front of CP inside the device and undergoes a blueshift. Column I: spacetime diagrams showing the evolution of the WP and the CP for the four configurations. The red and blue colour of the originally gold WP indicate the spacetime regions in which it has been red- or blue-shifted, respectively. Column II: snap-shots of the WP and CP relative positions in space at a given instant in time. The arrows indicate the respective direction of movement. Column III: experimental data of the wave packet at the output of the device for the four configurations sketched in I and II, demonstrating the expected frequency shift with respect to the input frequency. $\tau_\mathrm{wp}$ is the pulse durations of the WP. The white curve in the inset of b,III shows the data along a fixed-time cut indicated as a dotted white line in the main plot. The red solid line is a quadratic fit to the data, while the red dashed line marks the centre of the parabola, i.e. the output frequency $\omega_\mathrm{out}/(2\pi)$. Details on the measurement are discussed in the main text.
• Figure 3: Instantaneous frequency shift. Merged data obtained from 70 individual $\tau_\mathrm{WP}=15\,$ns long wave packets with an initial carrier frequency $\omega_\mathrm{in}/(2\pi)=4\,\mathrm{GHz}$ which have encountered a $\tau_\mathrm{CP}=40\,$ns long control pulse with an amplitude $1.58\,\mathrm{mA}$ for different delays between $0-128\,$ns. The output wave packets have been digitally down-converted at 191 different frequencies between 3.86 GHz and 4.24 GHz and the resulting magnitude is plotted as colourmap. For each delay an instantaneous frequency shift at a single point inside the wave packet has been determined from the phase evolution. The combined data from all wave packets has been superimposed on the colourmap as a white line. The inset shows the phase evolution in the $I$--$Q$ plane for two output wave packet digitally down-converted at $\omega_\mathrm{in}$, one of which underwent no frequency shift (yellow markers) and one of which was redshifted (red markers). As a guide to the eye we have overlaid red arrows in the latter case.
• Figure 4: Pulse shape preservation. Measured envelopes of wave packets with initial frequency $\omega_\mathrm{in}/(2\pi)=4$ GHz, pulse length $\tau_\mathrm{WP}=30$ ns and non-trivial temporal envelopes. The wave packets indicated in gold (red) have passed through the device in absence (presence) of a rising front of height $0.52\,\mathrm{mA}$. Digital down-conversion was performed at the expected output frequencies, respectively. In the lower panels the relative difference between the fixed-frequency and the shifted-frequency pulses is plotted.
• Figure 5: Amplitude-controlled frequency shift. The inset shows the unwrapped measured phase $\varphi$ of wave packets with an initial frequency of $\omega_\mathrm{in}/(2\pi)=4.0\,\mathrm{GHz}$ that have interacted with the rising front of a control pulse and that have consecutively been digitally down-converted at $\omega_\mathrm{d}=\omega_\mathrm{in}$. Data are plotted for control-pulse amplitudes between $I_\mathrm{CP}=0.08\,\mathrm{mA}$ (lightest orange line) and $I_\mathrm{CP}=2.03\,\mathrm{mA}$ (darkest red line). Each line represents the phase data for a fixed $I_\mathrm{CP}$ and has been constructed by merging the data obtained from 24 wave packets with different delays $\Delta t$ with respect to the control pulse. The main plot shows the frequency shift obtained from the slope of the phase data in the inset. The frequency shift was fitted with Eq. \ref{['eq:shift_from_current']} taking into account both quadratic and quartic contributions in $I_\mathrm{CP}$. The fit is shown as light grey line.