Josephson-junction-embedded transmission-line resonators: From Kerr medium to in-line transmon

We provide a general method to find the Hamiltonian of a linear circuit in the presence of a nonlinearity. Focusing on the case of a Josephson junction embedded in a transmission-line resonator, we solve for the normal modes of the system by taking into account exactly the effect of the quadratic (i.e., inductive) part of the Josephson potential. The nonlinearity is then found to lead to self and cross-Kerr effects, as well as beam-splitter-type interactions between modes. By adjusting the parameters of the circuit, the Kerr coefficient $K$ can be made to reach values that are weak ($K<\kappa$), strong ($K>\kappa$), or even very strong ($K\gg \kappa$) with respect to the photon-loss rate $\kappa$. In the latter case, the $\mathrm{resonator}+\mathrm{junction}$ circuit corresponds to an in-line version of the transmon. By replacing the single junction by a SQUID, the Kerr coefficient can be tuned in situ, allowing, for example, the fast generation of Schrödinger cat states of microwave light. Finally, we explore the maximal strength of qubit-resonator coupling that can be reached in this setting.

Figure 1

A $LC$ circuit in parallel with a Josephson junction can be equivalently represented by a $LC$, whose parameters are renormalized by the junction capacitance $C_J$ and the linear Josephson inductance $L_J$, in parallel with a purely nonlinear element represented by the spiderlike symbol.

Figure 2

(a) Discretized representation of a transmission-line resonator capacitively coupled to input and output ports and with a Josephson junction interrupting the center conductor at position $x_J$. (b) Lumped element representation of the transmission $\mathrm{line}+\mathrm{junction}$. The normal modes of the resonator, dressed by the junction's capacitance and linear inductance, are represented by $LC$ circuits biasing the junction's nonlinear inductance (spiderlike symbol).

Figure 3

Typical example of the first three normal mode envelopes of a resonator with a Josephson junction at $x_J=0$.

Figure 4

(a) Frequency ${\omega }_1$ and (b) Kerr nonlinearity $K_{11}$ of the first mode as a function of $x_J$ and $E_J$. The black lines indicate the contour of constant $K_{11}/\kappa =1$ and 10, with ${\kappa }_{\lambda /2}/2\pi =1.5$ MHz. The dashed white lines indicate $x_J/\ell =0.5$, 0.75, and $0.95$. These positions are used below. The resonator has a total length $2\ell =1.2$ cm, characteristic impedance $Z^0=50\Omega$ and input and output capacitors $C_i=C_o=10$ fF. In the absence of the junction, it has a fundamental $\lambda /2$ mode of oscillation at ${\omega }_{\lambda /2}/2\pi =4.95$ GHz. Unless otherwise stated, the numerical calculations in this section have been realized with this set of parameters.

Figure 5

(a) Frequencies, (b) detunings from two-photon processes, and (c) self-Kerr coefficients $K_{mm}$ of the first four normal modes of a nonlinear resonator as a function of the Josephson junction position, starting from the center ($x_J=0$) to the right edge ($x_J=\ell$) of the resonator. Because of the shape of the mode envelopes $u_m\left(x\right)$, moving the junction away from the center increases the nonlinearity on the higher resonator modes and thus the inharmonicity. The dashed vertical line indicates at $x_J=\ell /2$ corresponds to the junction location chosen in Fig. 6. The parameters are given in the text.

Figure 6

(a) Frequencies, (b) detunings from high-order photon processes, and (c) self-Kerr effects $K_{mm}$ and Rabi frequencies $g_{mn}$ for ${\Phi }_{\mathrm{rf}}=0.02{\Phi }_0$ of the first three normal modes of a nonlinear resonator with a large Josephson junction, as a function of the external flux ${\Phi }_x$. The SQUID is placed at position $x_J=\ell /2$. Parameters are given in the text. The vertical dashed line present at ${\Phi }_x=0.37{\Phi }_0$ corresponds to the operating point of the JPC used in Ref. [20].

Figure 7

(a) Frequency ${\omega }_r$, radiative relaxation rate $\kappa$, and self-Kerr coefficient $K$ of the first mode as a function of external flux. (b) Pulse shape of the external flux and Kerr coefficient as a function of time for Schrödinger cat state generation. (c) Density plot of the numerically obtained resonator Wigner function immediately after the flux pulse acting on an initially coherent state $|\alpha =2\rangle$. The device parameters are given in text.

Figure 8

(a) Density plot of the average number of photons as a function of time and external flux in the presence of an external drive of amplitude $\epsilon /2\pi =2$ MHz and of frequency ${\omega }_d={\omega }_r-K/2$. (b) Line cuts for the values of flux indicated by I and II in panel (a) and corresponding, respectively, to $K/\kappa \sim 1/25$ and $\sim 200$.

Figure 9

${\omega }_1/{\omega }_p$ (full red line) and $K_{11}/E_{C,T}$ (green dashed line) vs total resonator length $2\ell$. For very short lengths, the circuit essentially behaves as a transmon of frequency $\hslash {\omega }_p=\sqrt{8E_{C,T}{E}_J}$ and anharmonicity $E_{C,\mathrm{T}}=e^2/\left[2(2\ell C^0/4+C_J)\right]$.

Figure 10

In-line transmon (right) capacitively coupled to a low-$Q$ resonator (left). The in-line transmon can be coherently controlled, for example, using the right (green) control port and the low-$Q$ resonator measured in reflection using the left (red) readout port.

Figure 11

(a) Effective circuit of a junction coupled to the current fluctuations of an oscillator as suggested by Devoret et al.[23]. Close to the resonance and with $C_J\gtrsim C$ such that the $\lambda /2$ mode approximation is valid, this lumped circuit is a good representation of the circuit shown in panel (b). (c) Effective circuit for a junction coupled to the voltage fluctuations of an oscillator. (d) For the physical implementation with a transmission-line resonator to be equivalent to (c) around resonance, it is required that if ${\ell }_q\ll 2\ell$ for the $\lambda /2$ mode approximation to be valid. As stated in the text, the effective circuits of panels (a) and (c) are a valid representation of the circuits of panel (b) and (d) only around their resonance frequency.

Figure 12

(a) Frequencies, (b) nonlinear Kerr coefficient, and (c) participation ratios of the first two normal modes of the resonator for a junction located near the end of the resonator. We have assumed a SQUID of total Josephson energy $E_{J\Sigma }/2\pi =19$ GHz and asymmetry $d=5\%$ placed at a distance ${\ell }_q=260$ $\mu$m from the right extremity of a resonator with bare fundamental frequency ${\omega }_r/2\pi =4.95$ GHz. This corresponds to a charging energy of the island of $E_C/2\pi \approx 400$ MHz [blue dotted line in panel b].