Large Collective Lamb Shift of Two Distant Superconducting Artificial Atoms

Virtual photons can mediate interaction between atoms, resulting in an energy shift known as a collective Lamb shift. Observing the collective Lamb shift is challenging, since it can be obscured by radiative decay and direct atom-atom interactions. Here, we place two superconducting qubits in a transmission line terminated by a mirror, which suppresses decay. We measure a collective Lamb shift reaching 0.8% of the qubit transition frequency and twice the transition linewidth. We also show that the qubits can interact via the transmission line even if one of them does not decay into it.

Figure 1

Experimental setup. (a) A conceptual sketch of the setup. Two atoms are placed in front of a mirror and interact via virtual photons of different frequencies. (b) A photo of the device. Qubit 1 is placed $L\simeq 33\mathrm{mm}$ from Qubit 2 (enlargement on the right; the two bright parts form the qubit capacitance and the gap between them is bridged by two Josephson junctions forming a superconducting quantum interference device loop), which sits at the end of the transmission line (TL), i.e., at the mirror. The characteristic impedance of the TL is $Z_0\simeq 50\mathrm{\Omega }$. The relatively long distance $L$ makes it easier to tune Qubit 1 between nodes and antinodes of the electromagnetic (EM) field in the TL by tuning the qubit transition frequency. This tuning can be used to calibrate the velocity $v$ of the EM field in the TL [72]. (c) Signal routing for the experiment. Each qubit frequency can be tuned by local magnetic fields via local voltage biases ($V_1$, $V_2$) and both frequencies can be tuned by a global magnetic field via $V_3$. For measurements, a coherent signal at frequency ${\omega }_p$ is generated by a vector network analyzer (VNA) at room temperature and fed through attenuators (red squares) to the sample, which sits in a cryostat cooled to 20 mK. At this low temperature, the number of thermal photons is negligible and the thermal excitation of the qubits should be less than a few percent [73]. The reflected signal passes a bandpass filter (BPF) and amplifiers, and is then measured with the VNA.

Figure 2

Single-tone spectroscopy of the individual qubits. (a), (b) Electromagnetic mode structure (red curve) in the TL seen by Qubit 1 (Q1) and Qubit 2 (Q2), respectively. (c), (d) Amplitude reflection coefficient $|r|$ for a weak coherent probe as a function of probe frequency ${\omega }_p$ and qubit transition frequency (controlled by the voltages $V_1$ and $V_3$ for Qubit 1 and Qubit 2, respectively). Panel (c) shows how the response disappears when Qubit 1 ends up at a node for the EM field around 4.75 GHz. During these measurements, the frequency of the other qubit is tuned far from resonance with the probe. (e), (f) Linecuts from panels (c) and (d) as indicated. Dots are experimental data, and solid curves are fits following Ref. [37]. The extracted parameters are given in Table S1 in the Supplemental Material [72]. The linewidth of the dip, which occurs at the resonance ${\omega }_p={\omega }_{10}$, is set by the qubit decoherence rate $\gamma =\mathrm{\Gamma }/2+{\gamma }_{\varphi }$, where $\mathrm{\Gamma }$ is the relaxation rate and ${\gamma }_{\varphi }$ is the pure dephasing rate. Relaxation into other channels than the TL will affect the extracted value of ${\gamma }_{\varphi }$. The depth of the dip is set by the ratio $\mathrm{\Gamma }/{\gamma }_{\varphi }$; since $\mathrm{\Gamma }$ decreases close to the node of the field, the dip in linecut $A$ is shallower than that in $B$.

Figure 3

Collective Lamb shift. (a) The amplitude reflection coefficient for a weak probe as a function of the probe frequency ${\omega }_p$ and the transition frequency of Qubit 1 (controlled through $V_1$). The frequency of Qubit 2 is fixed at ${\omega }_{10}/2\pi =4.75\mathrm{GHz}$ and the frequency of Qubit 1 is tuned through resonance with this frequency. (b) Theory simulation [72] of the single-tone spectroscopy data in panel (a). The simulation is done with previously fitted parameters from Table 1, with the exceptions of the free parameters ${\beta }_1=0.323$, ${\beta }_2=0.306$, and ${\gamma }_{\varphi }/2\pi =2.3\mathrm{MHz}$ for Qubit 1, which all are close to the values in Table 1. The agreement between the data in (a) and the simulation in (b) is excellent. We note that the extinction of the signal around ${\omega }_p/2\pi =4.65\mathrm{GHz}$ and ${\omega }_p/2\pi =4.85\mathrm{GHz}$ can be explained by an interplay between detuning and decoherence [72]. (c) A linecut of the data and theory [dashed line in panels (a) and (b)], at the point where the two qubits are on resonance and the CLS $2\mathrm{\Delta }$ is most clearly visible. From this figure, we extract a CLS of $2\mathrm{\Delta }\simeq 2\pi \times 38\mathrm{MHz}$. The inset shows the level structure of the eigenstates of the qubits that are coupled through the CLS. The two dips in the reflection correspond to the symmetric and antisymmetric eigenstates $|s⟩$ and $|a⟩$. Each of these two states have the same decay rate, giving a linewidth ${\gamma }_c$ smaller than the CLS due to the presence of the mirror, as explained in the text.

Figure 4

Energy shift as a function of input power. (a) Amplitude reflection coefficient $|r|$ as a function of probe frequency ${\omega }_p$ and probe power $P$ for both qubit frequencies fixed at ${\omega }_{10}/2\pi =4.75\mathrm{GHz}$. The data agree very well with theoretical simulations [72]. At high probe powers, the qubits are saturated and most photons are simply reflected from the mirror, resulting in $|r|\approx 1$. (b) The extracted splitting $2\mathrm{\Delta }$ (red points) from panel (a) as a function of $P$, and the splitting extracted from theoretical simulations (red curve) [72]. The splitting is clearly independent of the input power in this wide range.