Quantum model for rf-SQUID-based metamaterials enabling three-wave mixing and four-wave mixing traveling-wave parametric amplification

A quantum model for Josephson-based metamaterials working in the three-wave mixing (3WM) and four-wave mixing (4WM) regimes at the single-photon level is presented. The transmission line taken into account, namely Josephson traveling wave parametric amplifier (JTWPA), is a bipole composed of a chain of rf-SQUIDs, which can be biased by a DC current or a magnetic field to activate the 3WM or 4WM nonlinearities. The model exploits a Hamiltonian approach to analytically determine the time evolution of the system both in the Heisenberg and interaction pictures. The former returns the analytic form of the gain of the amplifier, while the latter allows recovering the probability distributions vs time of the photonic populations, for multimodal Fock and coherent input states. The dependence of the metamaterial's nonlinearities is presented in terms of circuit parameters in a lumped model framework while evaluating the effects of the experimental conditions on the model validity.

Figure 1

Electrical equivalent of a repetition of three elementary cells (periodicity $a$) of the rf-SQUIDs based JTWPA. Each cell consists of a superconducting loop containing a geometrical inductance $L_{\mathrm{g}}$, a Josephson junction, with an associated capacitance $C_{\mathrm{J}}$ and inductance $L_{\mathrm{J}}$, and a ground capacitor $C_{\mathrm{g}}$. The series can be biased both through an external DC magnetic field $B$ and a flowing current $I_{\mathrm{DC}}$. $\mathrm{\Delta }\widehat{\mathrm{\Phi }}$ is the magnetic flux difference across the nodes of a cell, while ${\widehat{V}}_{C_{\mathrm{g}}}$ is the voltage drop across the ground capacitor.

Figure 2

Hamiltonian coupling parameters ${\chi }_i$ characterizing the Hamiltonian (11) vs the normalized DC flux bias ($\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{DC}}/{\mathrm{\Phi }}_0$). The Hamiltonian coefficient related to the constant flux bias (${\chi }_0$) has been scaled by a factor of ${10}^{-12}$, the noninteracting-modes Hamiltonian coupling constants (${\chi }_1^{\left(n\right)}$) have been shifted by the frequency of the corresponding photon (${\omega }_{\mathrm{i}}$) and scaled by a factor ${10}^{-5}$ while ${\chi }_3^{(p,s,i)}$ has been scaled by a factor of ${10}^{-2}$. The indices in the superscripts vary with the considered mode and can take the values p (pump), s (signal), and i (idler). The blue vertical lines indicate the flux biases at which the amplifier works as a three-wave mixer, while the red vertical lines indicate the flux biases at which the amplifier works as a four-wave mixer (see Sec. 2c for a detailed description). The coupling parameters ${\xi }_{n,n}$ and ${\xi }_{n,l}$ refer to the self-phase and cross-phase modulation due to the 4WM interaction. The circuit parameters used to perform the numerical evaluations and plots are summarized in Table 1.

Figure 3

(a) Photon number gain $G$ of the JTWPA under the undepleted pump approximation expressed by equation (24) in the 4WM (light blue/orange) and 3WM (blue/red) regimes. For each pump power one can see the cases with and without zero phase mismatch, respectively $\mathrm{\Psi }=0$ and $\mathrm{\Psi }\ne 0$. (b) Squeezing spectrum $S$ [Eq. (33)] as a function of the signal frequency calculated for a vacuum input state in the 4WM and 3WM regimes. Different colors express different pump currents ($I_{\mathrm{p}}$) for which $G$ and $S$ are calculated: blue/light blue $I_{\mathrm{p}}/I_{\mathrm{c}}=0.99$ ($P_{\mathrm{p}}=-64.6\mathrm{dB}$), red/orange $I_{\mathrm{p}}/I_{\mathrm{c}}=0.81$ ($P_{\mathrm{p}}=-66.3\mathrm{dB}$). The working points in the 3WM and 4WM cases are respectively $\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{DC},3\mathrm{WM}}/{\mathrm{\Phi }}_0=0.25$ and $\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{DC},4\mathrm{WM}}/{\mathrm{\Phi }}_0=0$. Refer to Table 1 for the experimental parameters used in the computations

Figure 4

Added-noise number ($\mathcal{A}$) as a function of the normalized pump current, for a pump frequency of $12\mathrm{GHz}$ and a signal frequency of $7\mathrm{GHz}$ in 3WM working point. The dashed curves are calculated in case of negligible phase mismatch while the solid curves in the case of non-negligible phase mismatch. The blue curves are considered for an input Fock state where the idler is in the vacuum state ($|N_{\mathrm{in}}^{\mathrm{S}},0\rangle$), while the red curves for an input Fock state with one idler photon ($|N_{\mathrm{in}}^{\mathrm{S}},1\rangle$).

Figure 5

Time evolution inside the medium, from its input port ($t=0$) to the output port ($t=t_T$) of the probability distribution $P_{\mathrm{F}}$ to find $N_{\mathrm{fin}}^{\mathrm{S}}$ signal photons for three different initial Fock states (a)${|3\rangle }_{\mathrm{s}}{|0\rangle }_{\mathrm{i}}$, (b)${|3\rangle }_{\mathrm{s}}{|2\rangle }_{\mathrm{i}}$, and (c) ${|6\rangle }_{\mathrm{s}}{|6\rangle }_{\mathrm{i}}$ calculated for $I_{\mathrm{p}}=0.5I_{\mathrm{c}}$. The dashed white lines represent the time evolution of the expectation value $\langle {\widehat{n}}_{\mathrm{s}}\rangle$.

Figure 6

Time evolution inside the medium, from its input port ($t=0$) to the output port ($t=t_T$) of the probability distribution $P_{\mathrm{C}}$ to find $N_{\mathrm{fin}}^{\mathrm{S}}$ signal photons for three different initial bimodal coherent states ${|\alpha \rangle }_{\mathrm{s}}{|\beta \rangle }_{\mathrm{i}}$, (a) ${|1\rangle }_{\mathrm{s}}{|0\rangle }_{\mathrm{i}}$, (b) ${|0\rangle }_{\mathrm{s}}{|1\rangle }_{\mathrm{i}}$, and (c) ${|1\rangle }_{\mathrm{s}}{|1\rangle }_{\mathrm{i}}$ calculated for $I_{\mathrm{p}}=0.2I_{\mathrm{c}}$. The dashed purple lines represent the time evolution of the expectation value $\langle {\widehat{n}}_{\mathrm{s}}\rangle$.

Figure 7

(a) The plot shows three sets of curves calculated for different values of the screening parameter $\beta$ representing the cell parameters for a $50\mathrm{\Omega }$ matching of the signal mode, at $7\mathrm{GHz}$, as a function of the critical current $I_{\mathrm{c}}$. The solid lines refer to the left axis and report the geometrical inductance $L_{\mathrm{g}}$ vs $I_{\mathrm{c}}$. On the contrary, the dashed curves refer to the right axis and represent the ground capacitance $C_{\mathrm{g}}$ vs $I_{\mathrm{c}}$.

(b) Gain $G$ of the JTWPA in 3WM mode (solid line) as a function of the critical current $I_{\mathrm{c}}$ for different values of the screening parameter $\beta$. The dashed curves represent the limit above which the undepleted pump approximation cannot be considered valid considering the input state $|1,0\rangle$ while the dotted curves considering the input state $|100,0\rangle$ (see Eq. S2 within the Supplemental Material [24]).