Control of the Dynamic Regimes of a Superconducting Nanointerferometer

An analytical solution has been found to describe dynamic processes in a superconducting nanointerferometer with negligibly low inductances that is included in a high-Q factor resonator. The effect of nonlinearity in the system, as well as the effect of external parametric pumping, has been analyzed. The screening of detected dynamic modes in the studied nanostructure in the resonator is performed in a wide range and their positions on the plane of the parameters have been determined. A significant influence of phase effects on the evolution of the system has been demonstrated and phase relations allowing one to control the output signal intensity have been evaluated. The detected effects open new possibilities for developing and testing basic elements of modern quantum computing systems.

INTRODUCTION

Systems with Josephson junctions integrated in waveguides (cavities) constitute an important part of modern quantum computing systems and breakthrough fundamental experiments. The simplest example is a nanoSQUID (plan view of the thin-film structure, the calculation of current spreading, the equivalent circuit, and the typical micrograph are shown in Fig. 1). Such systems are used, e.g., to adjust the coupling between artificial atoms through a resonator and to analyze their states, to study the dynamic Casimir effect, etc. [1–6]. In such applications, even an episodic excess of the current flowing through an element above the critical value leads to the formation of voltage pulses and provokes the “quasiparticle poisoning” of superconducting quantum registers. These practical requirements stimulate the analysis of low-dissipative dynamic processes in an interferometer [7, 8] and possibilities of controlling them, in particular, by the application of quasimonochromatic radiation (“pump”) [9]. The search for the solution of the “equation of motion” for the nanointerferometer in the resonator under the action of the ac pump signal is also of interest because the consideration of the system behavior in the quantum mode is sometimes necessary (both the nanostructure and the field in the resonator can be described quantum mechanically) [12]. In this work, an analytical solution is found to describe dynamic processes in the superconducting nanointerferometer with negligibly low inductances and the screening of detected dynamic modes is performed in a wide region of the parameters of the studied nanosystem.

Fig. 1.

(Color online) (a) Topology of the two-contact SQUID included in the coplanar waveguide with the control line to specify the pump magnetic flux in the interferometer ring. (b) Simulation of current spreading in the SQUID in the 3D-MLSI software package [10, 11]. (c) Circuit of the two-contact SQUID with the negligibly low geometric inductance under the action of the external magnetic flux ${{\Phi }_{{\text{e}}}}$. (d) Scanning electron microscopy image of the nanoSQUID.

MODEL

We study a two-contact interferometer in the form of a superconducting thin-film ring with two “weak points” in the form of constrictions. The sketch of a SQUID integrated in a superconducting resonator with the possibility of applying a magnetic pump signal is presented in Fig. 1a (superconducting thin-film regions are shown in blue). The transport current I_S flowing through the interferometer is induced by the source of the input signal V_in. In addition, a pump magnetic field is applied to the interferometer from a control line. The described design is widely used in recent experiments because of its simplicity and compatibility with technological solutions used in superconducting quantum computing (see, e.g., [13]).

To analyze the current spreading in considered superconducting thin-film structures (t $\ll$ l, t ~ λ_L, where t is the thickness of the superconductor, l is the characteristic planar dimension of the film, and λ_L is the London penetration depth), we numerically solved the London equation using the finite element method [10]. According to the results shown in Fig. 1b, two weak points—Josephson junctions—are formed in the superconducting contour because the current density in constrictions marked by blue ovals in Fig. 1b exceeds the critical value [14, 15]. Furthermore, using the 3D-MLSI software package [10, 11], we calculated the inductance matrix from the found spatial distribution; the inductance of the interferometer loop appeared to be about 0.65 pH. Since the critical current of the Josephson junctions is I_c = I_c1 = I_c2 ≈ 25 μA (Fig. 1c), the inductance of the interferometer loop is β_l = $\frac{{2\pi L{{I}_{{{\text{c}}1}}}}}{{{{\Phi }_{0}}}}$ = 0.05 $\ll$ 1. Therefore, the SQUID can be considered in the approximation of a small dimensionless inductance β_l and its total phase $\varphi (t)$ can be introduced [16, 17]. This quantity, which is the difference of the phases of the complex order parameter on the opposite “ends” of the interferometer (1 and 0 in Fig. 1c), serves as a generalized coordinate. In this case, the behavior of the system is similar to the behavior of a single Josephson junction (i.e., a nonlinear oscillator), where the total critical current depends on the external magnetic flux Φ_e. The total supercurrent through such an interferometer can be represented in the form

$${{I}_{{\text{s}}}}(t) = {{I}_{{\text{c}}}}\cos ({{\varphi }_{{\text{e}}}})\sin (\varphi (t)),$$

(1)

where φ_e = π(Φ_e/Φ₀) is the “external phase,” Φ₀ = h/(2e) is the magnetic flux quantum, and Φ_e is the given external magnetic flux. If the external flux is described by a harmonic function with a certain frequency, the system is a parametric element with a flux pump with respect to the external signal [18, 19]. In this work, the dynamics of the system exposed to harmonic signals and pump is analytically studied within the resistively and capacitively shunted junction model [8, 9].

The total equation of the dynamics of the considered system exposed to the harmonic signal ${{I}_{{\text{s}}}}(\tau ){\text{/}}{{I}_{{\text{c}}}}$ = i_scosΩτ and the external magnetic flux pump φ_e(τ) = ${{\varphi }_{0}} + \varepsilon \cos ({{\Omega }_{{\text{p}}}}\tau + \Delta )$ is conveniently represented in the dimensionless form

$$\ddot {\varphi } + \alpha \dot {\varphi } + {\text{cos}}(\varphi + \varepsilon {\text{cos}}({{\Omega }_{{\text{p}}}}\tau + \Delta ))\sin \varphi = {{i}_{{\text{s}}}}{\text{cos}}\Omega \tau ,$$

(2)

where an overdot means the derivative with respect to the dimensionless time $\tau = t{{\omega }_{{{\text{plasma}}}}}$; $\Omega = \omega {\text{/}}{{\omega }_{{{\text{plasma}}}}}$ is the dimensionless frequency in units of the plasma frequency ${{\omega }_{{{\text{plasma}}}}} = \sqrt {2e{{I}_{{\text{c}}}}{\text{/}}(\hbar C)}$; $\alpha = (1{\text{/}}R)\sqrt {\hbar {{I}_{{\text{c}}}}{\text{/}}(2e)}$, where C and R are the capacitance and resistance of the SQUID, respectively; $\varepsilon$ is the normalized amplitude of the variable component φ_e(τ) with the frequency Φ_p, and Δ is the shift of the pump phase with respect to the signal. At a small pump amplitude ($\varepsilon \ll 1$),

$$\begin{gathered} \cos {\kern 1pt} {\kern 1pt} \text{[}{{\varphi }_{0}} + \varepsilon \cos ({{\Omega }_{{\text{p}}}}\tau + \Delta )] \\ \cong \cos {{\varphi }_{0}} - \sin {{\varphi }_{0}}\varepsilon \cos ({{\Omega }_{{\text{p}}}}\tau + \Delta ). \\ \end{gathered} $$

Since the constant component of the external phase ${{\varphi }_{0}}$ can be chosen arbitrarily, the dynamic equation (2) can be represented in the form

$$\ddot {\varphi } + \alpha \dot {\varphi } + [1 + \varepsilon \cos ({{\Omega }_{{\text{p}}}}\tau + \Delta )]\sin \varphi = {{i}_{{\text{s}}}}{\kern 1pt} \cos \Omega \tau .$$

(3)

ANALYSIS OF VARIOUS DYNAMIC MODES

We first consider the effect of the external parametric pump under the condition of low intrinsic nonlinearity in the system. Retaining only terms linear in φ and introducing the eigenfrequency of the oscillator ω₀ = 1, we arrive at the following equation describing the dynamics of the damped Josephson oscillator exposed to the external parametric pump:

$$\ddot {\varphi } + \alpha \dot {\varphi } + [1 + \varepsilon \cos ({{\Omega }_{{\text{p}}}}\tau + \Delta )]\omega _{0}^{2}\varphi = {{i}_{{\text{s}}}}{\kern 1pt} \cos \Omega \tau .$$

(4)

Of a separate interest is the resonance case where the frequency of the applied signal coincides with the eigenfrequency of the oscillator Ω = ω₀ and the pump frequency is twice as high (Ω_p = 2ω₀), so that the parametric resonance occurs. The analytical solution of this problem was found in the form

$$\begin{gathered} \varphi (\tau ) = {\text{cos(}}{{\omega }_{0}}\tau )(A + {{\mu }_{1}}{{e}^{{{{k}_{1}}\tau }}} + {{\mu }_{2}}{{e}^{{{{k}_{2}}\tau }}}) \\ + {\text{ sin(}}{{\omega }_{0}}\tau )(B + {{\eta }_{1}}{{\mu }_{1}}{{e}^{{{{k}_{1}}\tau }}} + {{\eta }_{2}}{{\mu }_{2}}{{e}^{{{{k}_{2}}\tau }}}), \\ \end{gathered} $$

(5)

where

$${{k}_{{1,2}}} = \frac{{ - 4\omega _{0}^{2}\alpha \pm \sqrt { - 4\omega _{0}^{2}{{\alpha }^{4}} + {{\varepsilon }^{2}}{{\alpha }^{2}}\omega _{0}^{4} + 4{{\varepsilon }^{2}}\omega _{0}^{6}} }}{{2({{\alpha }^{2}} + 4\omega _{0}^{2})}},$$

(6)

and the coefficients ${{\mu }_{1}}$, ${{\mu }_{2}}$, ${{\eta }_{1}}$, ${{\eta }_{2}}$, A, and B can be found from initial conditions (the detailed derivation is presented in the supplementary material).

The analysis of the evaluated equation makes it possible to reveal various dynamic modes of the s-ystem depending on the relation between the parameters α and $\frac{{{{\omega }_{0}}\varepsilon }}{2}$ (see Fig. 2). Interplay between the p-rocesses of parametric amplification and damping appears in the system. In particular, in the case of strong attenuation, a uniform solution attenuates completely; as a result, only oscillations with a constant amplitude maintained by the applied input signal “survive” at long times. Examples of dynamic processes in such a stationary mode are presented in Fig. 2a for several phases Δ. At a stronger pump, the system itself generates exponentially growing oscillations with the amplitude depending on the input signal, and the parametric generation occurs in any case even in the absence of the input signal (see Fig. 2b). A specific case is the compensation of attenuation by pump-induced amplification under the condition α = $\frac{{{{\omega }_{0}}\varepsilon }}{2}$. This mode presented in Fig. 2c is characterized by the unlimited linear increase in the amplitude of oscillations (the mode is similar to the resonance swinging of the amplitude of oscillations of an oscillator by an external force in the absence of the attenuation). It is noteworthy that the amplitude in this case increases linearly at all phases except for Δ = $\frac{{3\pi }}{2}$, at only damping solutions for amplitudes of oscillations in Eq. (5) and a particular solution with a constant amplitude are possible. In this mode, the absolute minimum of the output signal amplitude is reached at this pump phase.

Fig. 2.

(Color online) Josephson phase $\phi (\tau )$ versus the normalized time for (a) the stationary mode (α = 0.2, $\varepsilon = 0.25$); (b) the exponential growth mode of oscillations (α = 0.1, $\varepsilon = 0.5$); and (c) the linear growth mode (α = 0.25, $\varepsilon = 0.5$); ${{i}_{{\text{s}}}} = 1$.

An analytical solution of Eq. (4) was also obtained for the case where the applied signal frequency Ω differs from the eigenfrequency of the system, but the parametric resonance condition Ω_p = 2Ω holds. In this case, the general solution was evaluated in the form

$$\begin{gathered} \varphi [\tau ] = ({{\delta }_{1}}{\text{exp}}\left( {{{\gamma }_{1}}\tau } \right) + {{\delta }_{2}}{\text{exp}}\left( {{{\gamma }_{2}}\tau } \right) + {{A}_{0}}){\text{cos}}\left( {\Omega \tau } \right) \\ + \,({{\delta }_{1}}{{\chi }_{1}}{\text{exp}}\left( {{{\gamma }_{1}}\tau } \right) + {{\delta }_{2}}{{\chi }_{2}}{\text{exp}}\left( {{{\gamma }_{2}}\tau } \right) + {{B}_{0}}){\text{sin}}\left( {\Omega \tau } \right))], \\ \end{gathered} $$

(7)

where the parameters γ₁ and γ₂ have a rather complex form and the coefficients δ₁, δ₂, χ₁, χ₂, A₀, and B₀ can be found from initial conditions (the detailed derivation is presented in the supplementary material). Similar to the resonance case, the general solution demonstrates three dynamic modes for various parameter values:

$${{K}_{0}} = 4{{\alpha }^{2}}{{\left( {\frac{\Omega }{{{{\omega }_{0}}}}} \right)}^{2}} + 4{{\left( {{{\omega }_{0}} - \frac{{{{\Omega }^{2}}}}{{{{\omega }_{0}}}}} \right)}^{2}} - {{\varepsilon }^{2}}\omega _{0}^{2}.$$

(8)

In particular, if ${{K}_{0}} > 0$, the real parts of the exponential coefficients ${{\gamma }_{1}}$ and ${{\gamma }_{2}}$ are negative, and the solution with the time approaches a constant amplitude of oscillations, which corresponds to the stationary mode. If ${{K}_{0}} < 0$, ${{\gamma }_{1}}$ and ${{\gamma }_{2}}$ have opposite signs, and the exponential growth mode occurs. If ${{K}_{0}} = 0$, ${{\gamma }_{1}} = 0$, ${{\gamma }_{2}} < 0$, and the linear growth mode occurs. The expression obtained for the for parameter K₀ allows one to plot the (pump parameter, applied signal frequency) map in order to illustrate the dynamics of the system at a fixed damping factor (see Fig. 3).

Fig. 3.

(Color online) (Ω, ε) Map of the parameter ${{K}_{0}}$ at the fixed damping factor α = 0.2. Here, Ω is the applied signal frequency and ε is the pump amplitude.

It is seen in Fig. 3 that four changes in dynamic mode occurs with increasing applied signal frequency at any parameter $\alpha$ if the pump intensity is not too low. Indeed, at large detuning values, the structure reaches the stationary mode, which changes to the linear growth mode when approaching the resonant frequency. The exponential growth mode is observed near the resonant frequency. This mode is not reached in the case of a low pump intensity because of a large role of damping in the system. The stationary mode occurs at frequencies far from the resonant one.

EFFECT OF THE PUMP PHASE Δ ON THE DYNAMICS OF THE SYSTEM

The pump phase Δ is an important parameter that determines the dynamics of the system and allows one to control it. Let us analyze the influence of this phase in the resonant case. To illustrate the found features of the system, we plotted the maps ${\text{|}}\varphi (\Delta ,\tau ){\text{|}}$ for the exponential growth and stationary modes in Figs. 4a and 4b, respectively.

Fig. 4.

(Color online) (Δ, τ) Map of |φ| at the fixed parameters ${{i}_{{\text{s}}}} = 1$ an ${{\omega }_{0}} = 1$ in (a) the exponential growth mode (α = 0.1, $\varepsilon = 0.5$) and (b) the stationary mode (α = 0.2, $\varepsilon = 0.25$). Here, Δ is the pump phase and τ is the time.

It is seen that a local extremum of the output signal φ(τ) is reached at certain phase values Δ existing in each of the considered modes. In particular, it is seen in Fig. 4 that the maximum of the function φ(τ) is reached primarily at the phase Δ = $\frac{\pi }{2}$ for both modes, whereas the minimum is reached near the phase Δ = $\frac{{3\pi }}{2}$. This result is also in good agreement with the dependences φ(τ) presented in Fig. 2 for various Δ va-lues.

For a more detailed analysis of the influence of the pump phase, the (Δ, ε) maps of the amplitudes of the exponentially growing and stationary solutions at the damping factor $\alpha = 0.3$ are plotted in Fig. 5, where Δ is the pump phase and ε is the pump amplitude.

Fig. 5.

(Color online) (Δ, ε) Map of the amplitude of the solution given by Eq. (5) at α = 0.3 and the time τ = 13 in (a) the exponential growth mode and (b) the stationary mode. Here, Δ is the pump phase and ε is the pump amplitude.

In the absence of damping in the system, only the exponential growth of the solution is observed because of the parametric resonance. In this case, the pump phase Δ_min at which the amplitude of the output signal φ(τ) reaches minimum. In the case of small pump amplitudes, it can be determined from the equation

$$2 = \varepsilon \cos ({{\Delta }_{{\min }}})\left( {\frac{1}{2} + \frac{1}{{1 + \sin (\Delta )}}} \right).$$

(9)

The analysis of this equation showed that Δ_min is close to $\frac{{3\pi }}{2}$. As a result, $\frac{1}{{1 + \sin ({{\Delta }_{{\min }}})}} \gg \frac{1}{2}$, and Eq. (9) with Δ = Δ_min is reduced to the form

$$2 = \varepsilon \cos ({{\Delta }_{{\min }}}) - 2\sin ({{\Delta }_{{\min }}}).$$

(10)

The solution of this equation gives the following expression for the pump phase ensuring the minimum amplitude of the solution in the exponential growth mode:

$${{\Delta }_{{\min }}} = {\text{arcsin}}\left( {\frac{\varepsilon }{{\sqrt {{{\varepsilon }^{2}} + 4} }}} \right) - {\text{arcsin}}\left( {\frac{2}{{\sqrt {{{\varepsilon }^{2}} + 4} }}} \right).$$

(11)

For comparison with the case of a nonzero damping factor, this dependence is plotted on the map in Fig. 5a.

In the presence of damping in the system in the exponential growth mode α < $\left( {\frac{{{{\omega }_{0}}\varepsilon }}{2}} \right)$, the dependence Δ_min(ε) is similar to that in the absence of damping (see Fig. 5a). The effect of the damping factor α on the threshold pump amplitude ensuring the occurrence of this mode is as follows: the increase in the damping factor results in the increase in the pump amplitude at which the critical phase Δ = $\frac{{3\pi }}{2}$ is reached.

At smaller pump amplitudes, the stationary mode occurs, corresponding to the solution limited in absolute value with the minimum reached at Δ = $\frac{{3\pi }}{2}$, as seen in Fig. 5b. in this case, the maximum amplitude of the solution is reached at phases in a wide vicinity of $\frac{\pi }{2}$ because the amplitude decreases very smoothly. The existence of two local extrema at Δ = $\frac{\pi }{2}$ (maximum) and at Δ = $\frac{{3\pi }}{2}$ (minimum) is easily seen from the eq-uation

$$ - \frac{\varepsilon }{2}\sin (\Delta )\cos (\Delta ) + \cos (\Delta )\left( {\frac{\alpha }{{{{\omega }_{0}}}} + \frac{\varepsilon }{2}\sin (\Delta )} \right) = 0.$$

(12)

It was also found that, unlike the resonant case, the conditions for the minimum output signal significantly depend on both the magnitude and sign of a nonzero frequency detuning. Maps presented in Figs. 6 and 7 are similar to those shown in Fig. 5 but are obtained for various input signal frequencies.

Fig. 6.

(Color online) (Δ, ε) Map of the amplitude of the solution given by Eq. (7) at the parameters τ = 13, $\alpha = 0.1$, $\Omega = 1.05$, ${{i}_{s}} = 1$, and ${{\omega }_{0}} = 1$ in (a) the exponential growth mode and (b) the stationary mode. Here, Δ is the pump phase and ε is the pump amplitude.

Fig. 7.

(Color online) (Δ, ε) Map of the amplitude of the solution given by Eq. (7) at the parameters τ = 13, $\alpha = 0.1$, $\Omega = 0.95$, ${{i}_{s}} = 1$, and ${{\omega }_{0}} = 1$ in (a) the exponential growth mode and (b) the stationary mode. Here, Δ is the pump phase and ε is the pump amplitude.

As seen, the maximum and minimum amplitudes of the solution are significantly shifted with a decrease in the input frequency compared to the resonant value. The dependence of these conditions on the pump amplitude also changes strongly. The dependence Δ_min(ε) in the exponential growth mode begins not at the point Δ = $\frac{{3\pi }}{2}$ but at the point shifted towards π or 2π depending on the sign of the frequency detuning, as clearly seen in Figs. 6 and 7. Thus, the output signal can be controlled in practice by varying the frequency detuning. An additional analysis showed that an increase in damping in the system somehow inhibits the shift of the beginning of the dependence Δ_min(ε) with an increase in the frequency detuning from the resonance.

The stationary mode occurs at a low pump intensity (see Figs. 6b and 7b). We note that the threshold pump amplitude ε₀ separating the stationary mode from the exponential growth mode in the case of a nonzero frequency detuning, in contrast to the resonant case, is nonzero even in the absence of damping in the system:

$${{\varepsilon }_{0}} = \frac{{2\sqrt {{{\alpha }^{2}}{{\Omega }^{2}} + {{\Omega }^{4}} - 2{{\Omega }^{2}}\omega _{0}^{2} + \omega _{0}^{4}} }}{{\omega _{0}^{2}}}.$$

(13)

In the stationary mode, the phase at which the output signal is minimal or maximal no longer depends on the pump amplitude (see Figs. 6b and 7b). For this case, we evaluated the following expression for the phase Δ_max at which the maximum the output signal is ma-ximal:

$${{\Delta }_{{\max }}} = \left\{ \begin{gathered} \pi - \arctan \left[ {\frac{{\alpha \Omega }}{{1 - {{\Omega }^{2}}}}} \right],\quad \Omega < 1, \hfill \\ - \arctan \left[ {\frac{{\alpha \Omega }}{{1 - {{\Omega }^{2}}}}} \right],\quad \Omega > 1. \hfill \\ \end{gathered} \right.$$

(14)

According to Eq. (14), the phase ensuring the maximum output amplitude significantly depends on the input signal frequency and the damping factor but is not determined by the pump amplitude. In the resonance limit, Δ_max → π/2, whereas Δ_max → π or 2π in the presence of a significant positive or negative frequency detuning, respectively. In this case, the lower the damping factor, the sharper the transition from one phase value to the other at a certain frequency detuning. As in the resonant case, the phase at which the output amplitude is maximal differs by π from the phase at which it is maximal.

DETECTION OF HYSTERESIS IN THE SYSTEM WITHOUT PARAMETRIC PUMPING

Nonlinearity in the system was first taken into account in the absence of parametric pumping. It was shown that, at low input currents, it is sufficient to retain all terms up to the third order in the Taylor expansion of the nonlinear term sinφ in Eq. (3), where the coefficient of the φ³ term is denoted as –β in order to keep the generality. As a result, Eq. (3) is reduced to the form

$$\ddot {\varphi } + \alpha \dot {\varphi } + \varphi \omega _{0}^{2} - \beta {{\varphi }^{3}} = {{i}_{{\text{s}}}}{\kern 1pt} \cos \Omega \tau .$$

(15)

To obtain an analytical solution, we substitute φ in the form $\varphi [\tau ] = \frac{{\tilde {\varphi }}}{2}{{e}^{{i\Omega \tau }}} + \frac{{\tilde {\varphi }{\text{*}}}}{2}{{e}^{{ - i\Omega \tau }}}$ into Eq. (15) and arrive at the following equation for ${\text{|}}\tilde {\varphi }{{{\text{|}}}^{2}}$:

$${\text{|}}\tilde {\varphi }{{{\text{|}}}^{2}}{{\alpha }^{2}}{{\Omega }^{2}} + {{\left( {\omega _{0}^{2} - {{\Omega }^{2}} - \frac{{3\beta }}{4}{\text{|}}\tilde {\varphi }{{{\text{|}}}^{2}}} \right)}^{2}}{\text{|}}\tilde {\varphi }{{{\text{|}}}^{2}} = i_{s}^{2}.$$

(16)

Finally, for the desired function $\chi = {\text{|}}\tilde {\varphi }{{{\text{|}}}^{2}}$, we obtain the cubic equation

$$\begin{gathered} {{\chi }^{3}} - \frac{8}{{3\beta }}(1 - {{\Omega }^{2}}){{\chi }^{2}} \\ + \frac{{16}}{{9{{\beta }^{2}}}}({{\alpha }^{2}}{{\Omega }^{2}} + {{(1 - {{\Omega }^{2}})}^{2}})\chi - \frac{{16i_{s}^{2}}}{{9{{\beta }^{2}}}} = 0. \\ \end{gathered} $$

(17)

The analysis of solutions reveals various cases. If the discriminant of the cubic equation is D > 0, the equation has a single real root, whereas the equation in the case of a negative discriminant has three real solutions, which correspond to the presence of hysteresis in the system. Thus, knowing the dependence of the discriminant of Eq. (17) on the parameters i_s, Ω, $\alpha$, and $\beta$, one can easily determine when hysteresis in the system can be observed. To plot the so-called “map” of the determinant, we found a solution of the equation $D({{i}_{{\text{s}}}},\Omega ,\alpha ,\beta ) = 0$, which determines the relation of the input signal i_s and the frequency Ω and has two branches:

$${{i}_{{{{{\text{s}}}_{{D{{{ = 0}}_{{1,2}}}}}}}}} = \frac{{2\sqrt 2 }}{{9\sqrt \beta }}\sqrt {(1 - {{\Omega }^{2}})(9{{\alpha }^{2}}{{\Omega }^{2}} + {{{(1 - {{\Omega }^{2}})}}^{2}}) \mp \tilde {\Omega }} ,$$

(18)

where

$$\tilde {\Omega } = \sqrt {{{{( - 3{{\alpha }^{2}}{{\Omega }^{2}} + {{{(1 - {{\Omega }^{2}})}}^{2}})}}^{3}}} .$$

The both branches of the function ${{i}_{{{{{\text{s}}}_{{D = 0}}}}}}$ of the frequency Ω given by Eq. (18) are real under the condition

$$\Omega \leqslant {{\Omega }_{{\text{h}}}} = \frac{{\sqrt {3{{\alpha }^{2}} + 2 - \sqrt 3 \alpha \sqrt {(3{{\alpha }^{2}} + 4)} } }}{{\sqrt 2 }}{\kern 1pt} .$$

(19)

At frequencies specified by this inequality, the radicand in the expression below Eq. (18) is nonnegative. Thus, the applied signal frequency range where hysteresis is possible is determined. Figure 8a presents both branches of Eq. (18) separating the regions of the negative and positive discriminants (solid line) with allowance for the found frequency constraint. Hysteresis is possible only at D < 0. It is noteworthy that the second branch of the solution given by Eq. (18) implies an upper bound on the frequency lower than Ω_h for large input signal amplitudes, so that hysteresis is no longer observed for signals with the amplitude slightly higher than unity (see the inset of Fig. 8a).

Fig. 8.

(Color online) (a) ${{i}_{{{{{\text{s}}}_{{D = 0}}}}}}(\Omega )$ plot at α = 0.1 and β = 1/6, where the red and blue dashed horizontal lines mark the values ${{i}_{{{{{\text{s}}}_{{D = 0}}}}}}({{\Omega }_{h}}) = 0.09758$ and ${{i}_{{{{{\text{s}}}_{{\max }}}}}} = 0.14118$. (b) Frequency response at α = 0.1, β = 1/6, and various applied signal amplitudes.

It is clearly seen in Fig. 8a that there are three characteristic regions of the current amplitude where hysteresis is observed in one range of the possible frequencies, in two frequency ranges, and at all possible frequencies satisfying condition (19).

In particular, at small amplitudes ${{i}_{{\text{s}}}} < {{i}_{{{{{\text{s}}}_{{D = 0}}}}}}$ $({{\Omega }_{{\text{h}}}})$, hysteresis occurs in a low frequency range expanding with an increase in the current amplitude i_s, which is clearly confirmed by the frequency response presented in Fig. 8b for ${{i}_{{\text{s}}}} = 0.05$. It is seen that three points on the frequency response correspond to each frequency in the range $\Omega \in [0;{\kern 1pt} 0.17971)$. However, the corresponding branches of solutions in Fig. 8b are very close to each other.

When ${{i}_{{{{{\text{s}}}_{{D = 0}}}}}}({{\Omega }_{{\text{h}}}}) < {{i}_{{\text{s}}}} < {{i}_{{{{{\text{s}}}_{{\max }}}}}}$, hysteresis occurs at low frequencies and at frequencies close to Ω_h. However, the second branch of Eq. (18) yields an additional more stringent bound on the input signal frequency. In this case, we revealed an interesting effect of cutting of a frequency range where hysteresis is absent. The larger the current amplitude, the narrower the cut range. Moreover, this cut range is absent at ${{i}_{{\text{s}}}} = {{i}_{{{{{\text{s}}}_{{{\text{max}}}}}}}}$. This case is very interesting because the current amplitude ${{i}_{{{{{\text{s}}}_{{\max }}}}}}$ is threshold: for ${{i}_{{\text{s}}}} > {{i}_{{{{{\text{s}}}_{{\max }}}}}}$, hysteresis is observed in the entire region of possible applied signal frequencies allowed for hysteresis. In this mode, solution branches presented in Fig. 8b diverge significantly and hysteresis is manifested noticeably. At high frequencies, D > 0 and hysteresis no longer occurs. It is remarkable that the frequency response curve without discontinuities has the maximum height just at ${{i}_{{\text{s}}}} = {{i}_{{{{{\text{s}}}_{{\max }}}}}}$. The condition of zero derivative of Eq. (18) with respect to Ω gives the following formula for ${{i}_{{{{{\text{s}}}_{{\max }}}}}}$:

$$\begin{gathered} {{i}_{{{{{\text{s}}}_{{\max }}}}}} \\ = \,\frac{{\sqrt { - 621{{\alpha }^{6}}\, + \,108{{\alpha }^{4}}\, + \,1296{{\alpha }^{2}}\, + \,64\, - \,\sqrt {{{{(9{{\alpha }^{2}}\, - \,4)}}^{6}}} } }}{{72\sqrt \beta }}. \\ \end{gathered} $$

(20)

Let us discuss the influence of damping on the revealed effects. An increase in damping in the system significantly narrows the allowed frequency range and strongly increases the critical current ${{i}_{{{{{\text{s}}}_{{\max }}}}}}$; the regime with the “cut” frequency range is observed in a very narrow region of the input current amplitudes. As a result, for weak input signals, hysteresis can occur only at low frequencies but is manifested very weakly. At large input signal amplitudes, a stringent upper bound on the frequency occurs. The region of parameters ensuring a noticeable manifestation of hysteresis is significantly narrowed.

Thus, hysteresis has been revealed on the frequency response curve of the system with cubic nonlinearity in the absence of parametric pumping. It has been shown that its appearance and related effects strongly depend on the initial parameters of the system. Regions of the parameters where hysteresis occurs depending on the frequency and amplitude of the applied signal have been determined.

We also note that the simultaneous effect of nonlinearity and parametric pumping in the system can lead to very strong amplification and “swinging” of the output signal. The study of this mode is beyond the scope of this work. However, damping in the system at a rather low pump intensity will limit the output current amplitude, and a mode with hysteresis will occur but with other parameters.

CONCLUSIONS

To summarize, an analytical solution has been found to describe dynamic processes in a superconducting nanointerferometer included in a high-Q factor resonator. The effect of cubic nonlinearity in the system, as well as the effect of external parametric pumping, has been analyzed in detail. Regions of the system parameters for the implementation of the stationary mode, as well as of exponential and linear growth modes, have been determined. A significant influence of the pump phase on the dynamics of the system has been demonstrated. The phase relations ensuring the maximum amplification and suppression of the output signal have been evaluated. It has been shown that the output signal can be controlled by varying the frequency detuning, which is very important for the design of elements of microwave quantum systems. Hysteresis has been detected in frequency responses and regions of the parameters for its implementation have been determined.