Exploiting Kerr cross nonlinearity in circuit quantum electrodynamics for nondemolition measurements

We propose a scheme for dispersive readout of stored energy in one mode of a nonlinear-superconducting microwave ring resonator by detection of the frequency shift of a second mode coupled to the first via a Kerr nonlinearity. Symmetry is used to enhance the device responsivity while minimizing self-nonlinearity of each mode. Assessment of the signal to noise ratio indicates that the scheme will function at the single-photon level, allowing quantum nondemolition measurement of the photon-number state of one mode. Experimental data from a simplified version of the device demonstrating the principle of operation are presented.

Figure 1

(Color online) Proposed circuit for implementation of nondemolition measurement using the Kerr effect. The superconducting ring resonator (red) is made from two transmission lines $T_{a,b}$ of unequal lengths (as indicated) joined at the ends by nonlinear elements realized by nominally identical dc SQUIDs $S_{1,2}$. The $180°$ hybrid causes the pump to couple only to ring modes with an odd-voltage profile with respect to the horizontal midline of the device; the power splitter causes the probe to couple only to even modes.

Figure 2

(Color online) Simulated device response near $\alpha =3/2$, where $\alpha$ fixes the relative lengths of $T_a$ and $T_b$ (see Fig. 1). Inset: phase response of $M_1$ as pump power $P_{pump}$ is swept from 0 to 12 fW in steps of 2 fW for $\alpha =3/2$. $f_{r0,probe}=7.1322\text{GHz}$ is the probe-resonance frequency in the absence of any pump excitation and $\delta f_{r,probe}$ is the shift of the probe-resonant response under probe excitation.

Figure 3

(Color online) Predicted histogram of phase-measurement outcomes $\theta$ (measured relative to the zero-pump power phase) for different measurement SNR for the device in Fig. 1. Measurement is simulated for a coherent state with average photon number $n_{av}=1$. The quantum nature of the radiation field is not evident for $\text{SNR}=1$ but for the predicted values of $\text{SNR}=7.1$ clearly resolved photon-number peaks will be seen.

Figure 4

(Color online) Experimentally measured cross-Kerr response of simple SQUID-tuned resonator. The circuit schematic is much simpler than the optimized one above (Fig. 1). A CPW resonator with $3\mu \text{m}$ center strip and $2\mu \text{m}$ slots was used. The dc SQUID loop had an area of $5\times 5\mu {\text{m}}^2$ with each junction having $I_c=1.2\mu \text{A}$. The SQUID was not flux biased for data in this figure. Readout line was used for pump excitation as well as measuring the forward scattering parameter $S_{21}$ for the probe signal. Inset shows resonant phase response of the probe for different values of the pump power. Probe-resonance frequencies are plotted in the main plot; the blue line is guide to the eye. The initial linear behavior of this curve at low-pump power (slope indicated by black dashed line) is a manifestation of the cross-Kerr effect.