Revealing the nonlinear response of a tunneling two-level system ensemble using coupled modes

Atomic sized two-level systems (TLSs) in amorphous dielectrics are known as a major source of loss in superconducting devices. In addition, individual TLSs are known to induce large frequency shifts due to strong coupling to the devices. However, in the presence of a broad ensemble of TLSs these shifts are symmetrically canceled out and not observed in a typical single-tone spectroscopy experiment. We introduce a two-tone spectroscopy on the normal modes of a pair of coupled superconducting coplanar waveguide resonators to reveal this effect. Together with an appropriate saturation model this enables us to extract the average single-photon Rabi frequency of dominant TLSs to be ${\mathrm{\Omega }}_0/2\pi \approx 79$ kHz. At high photon numbers we observe an enhanced frequency shift due to nonlinear kinetic inductance when using the two-tone method and estimate the value of the nonlinear coefficient as $K/2\pi \approx -1\times {10}^{-4}$ Hz/photon. Furthermore, the lifetime of each resonance can be controlled (increased) by pumping of the other mode as demonstrated both experimentally and theoretically.

Figure 1

(a) Design of the coupled resonators device [15]. Empty area represents Si substrate, gray area represents metal (Al). Feedline and resonators are black for clarity. Dark-red dashed box marks the coupling region. (b) Cartoon of the experimental idea. Magenta bars represent the resonators, the $x$ axis is the position along their lengths. In analogy to coupled pendula the coupling results in the forming of a symmetric and an antisymmetric modes which differ only by the phase between the electric fields in both resonators (blue and red curves) [16]. TLSs (represented by X) experience the same total field intensity (black curves) in both modes, except for the small coupling area in which the resonators' fields interfere. This allows ignoring other details than the frequency detuning between the modes.

Figure 2

(a) Resonance frequency shift and (b) internal loss (shown in a log-log scale in the inset) vs $\langle N\rangle$ for the probe-only experiment for the resonances at 5626 MHz (green) and 5689 MHz (blue). Dashed lines in (a) are linear fits of the high power shifts to a Kerr nonlinearity model as detailed in the text. Dashed lines in (b) are fits to the standard TLS model ([23], Eq. (29)). Inset in (a): A typical fit of the transmission data.

Figure 3

(a) Resonance frequency shift and (b) internal loss vs the average number of photons $\langle N\rangle$ in the pumped mode for the pump-probe experiment. Green (blue) points are for probe at 5626 MHz (5689 MHz) and pump at 5689 MHz (5626 MHz). The dashed lines in (a) are linear fits of the high power shifts to a Kerr nonlinearity model as detailed in the text. Solid lines are sums of the fits to the prediction of the generalized model of frequency shift due to TLSs [Eq. (4)] and the linear fits. In (a) constant shifts were added for clarity.

Figure 4

Qualitative picture of the generalized TLS saturation model for the resonance shift. A black X represents a TLS. A red X represents a saturated TLS. Green arrows represent the level repulsion (the length indicating the repulsion strength). Red arrows represent the pump tone. (a) In the absence of pumping the TLSs are symmetrically distributed around a resonance, resulting in a broadening without a frequency shift. (b) A symmetric pump around one resonance narrows the linewidth without a frequency shift. (c) When two resonances are present, strongly pumping symmetrically around the lower resonance results in an asymmetric repulsion for the higher resonance, causing a frequency shift. (d) When the pump power increases, the repulsion of the higher resonance becomes more and more symmetric, causing a decrease in the frequency shift.