Heat measurement of quantum interference

Abstract

Coherence is a key property of quantum systems, and it plays a central role in the operation and performance of quantum heat engines and refrigerators. Despite its importance for the fundamental understanding in quantum thermodynamics and its technological implications, coherence effects in heat transport have not been observed previously. Here, we measure quantum features in the heat transfer between a qubit and a thermal bath. The system is formed of a driven flux qubit galvanically coupled to a $λ/4$ coplanar-waveguide resonator that is coupled to a heat reservoir. This thermal bath is a normal-metal mesoscopic resistor, whose temperature can be measured and controlled. We detect interference patterns in the heat current due to driving-induced coherence. In particular, resonance peaks in the heat transferred to the bath are found at driving frequencies which are integer fractions of the resonator frequency. A selection rule on the even/odd parity of the peaks holds at the qubit symmetry point. We present a theoretical model based on Floquet theory that captures the experimental results. The studied system provides a platform for studying the role of coherence in quantum thermodynamics. Our work opens the possibility to demonstrate a true quantum thermal machine where heat is measured directly.

Figures (3)

• Figure 1: The studied system and device. a, Conceptual description of the studied system consisting of a driven qubit embedded in a cavity. The coupling $g$ is the energy exchange rate between qubit and cavity, while the coupling $\gamma$ is the energy exchange rate between the cavity and its environment bath. b, Schematic of a driven qubit-resonator coupled weakly to a thermal bath. The driven system transfers heat $P_{\text{b}}$ to the thermal bath raising its temperature $T_{\text{b}}$. The heat transferred to the bath is measured by observing the steady-state temperature difference between the bath and the substrate (phonon bath). c, Optical micrograph of the fabricated device. A $\lambda/4$ resonator is shunted by a flux qubit (inside blue rectangle). The open side of the resonator is weakly capacitively attached to an on-chip thermal bath made of a Cu film (inside red rectangle). The resonator is weakly attached to a feedline for RF scattering measurement. d, SEM image of the flux qubit made of Al film (blue) whose frequency $f_{\text{q}}$ can be tuned by an external magnetic flux $\Phi_{\text{ext}}$. The resonator made of Nb film (orange) is clean-contacted to the Al film. e, SEM image of the resonator (orange) capacitively coupled to the Cu film (inside yellow rectangle). f, SEM image of the Cu film (red) that is connected to four superconducting Al leads (green) with an insulating layer, forming NIS junctions for probing the electronic temperature $T_{\text{b}}$ in the Cu resistor. g, Two-tone spectroscopy showing the qubit-resonator spectrum. h, One-tone spectroscopy showing Rabi splittings where the qubit and resonator are at resonance.
• Figure 2: Heat transfer at fractional frequencies. a, At the symmetry point, $\Phi_{\text{ext}}/\Phi_0 = 0.5$, heat transfer occurs at driving frequencies $f_{\text{d}} = f_{\text{r}}/l$, where $l$ is an odd integer number due to selection rules. Here $f_{\text{r}}=7.025~\text{GHz}$ is the bare resonator frequency. b, At flux bias $\Phi_{\text{ext}}/\Phi_0 \neq 0.5$, the bias symmetry is broken, and heat transfer occurs at both even and odd integer fractions of $f_{\text{r}}$. The drive amplitude is fixed at $V_{\text{d}}=100~\text{mV}$. c-d, Zoomed-in data from inside the dashed rectangles of (a) and (b) respectively. e, 2D heat maps of measured heat current as a function of drive amplitude $V_{\text{d}}$ and external flux $\Phi_{\text{ext}}$ at driving frequencies $f_{\text{d}} = f_{\text{r}}/l$, top row for $l$ odd and bottom row for $l$ even.
• Figure 3: Quantum interference in heat transport. Heat current $P_{\text{b}}$ at fixed driving frequency with varying driving amplitude and external flux at (a)$f_{\text{d}}=2.3~\text{GHz}$ and (b)$f_{\text{d}}=1.735~\text{GHz}$. These frequencies roughly correspond to $f_{\text{r}}/3$ and $f_{\text{r}}/4$, respectively. The left panels show the measurement results with varying voltage amplitude $V_{\text{d}}$ and the right panels show the results from simulations using the quantum Rabi model (harmonic oscillator truncated as a $5$-level system) with varying drive amplitude $A_{\text{d}}$. Above the green dashed lines of $W$-shaped "event horizon", the heat current can be detected (above regions I and II). The vertical white dashed lines numbered 1 and 2 in the theory panels mark the conditions $\epsilon_0=f_{\text{d}}$ and $\epsilon_0=2f_{\text{d}}$, respectively. c, Comparison between measured (blue dots) and simulated (orange lines) heat current at fixed $f_{\text{d}}=2.3~\text{GHz}$ for three values of the drive amplitude.