Optimization of Adiabatic Superconducting Logic Cells by Using π Josephson Junctions

Adiabatic superconducting logic circuits can ensure the practical implementation of operations with the energy dissipation below the Landauer limit. However, applications of the existing solutions are limited because of two contradictory requirements of a high energy efficiency and a sufficiently fast response of devices. Josephson junctions with a negative critical current (π junctions) allow one to obtain a certain form of the potential energy of superconducting circuits and, as a result, a practically required degree of control of dynamic processes in the proposed reversible logic cells. The features of the current transport and balance of Josephson phases in circuits with π junctions make it possible to improve the coupling between the parts of a reversible computer by a factor more than 2. At the same time, the continuous evolution of the state is ensured at higher critical currents and higher characteristic voltages of the main Josephson junctions of adiabatic superconducting logic cells, which allows an increase in the response rate.

INTRODUCTION

The processing of large amounts of data requires the search for most energy efficient methods for calculations. In this respect, the application of superconducting materials seems promising [1–4]. The possibility of achieving the energy dissipation of zettajoule energy per logic operator with a duration of picoseconds is demonstrated with adiabatic superconducting logic devices with physically reversible switching processes between stable states (logic 0 and 1) [5–10]. Devices based on adiabatic superconducting logic circuits are considered as processors for next-generation supercomputers [11], cryogenic memory [12], and specialized cryptographic [13] and neuromorphic [14] processors. Advantages of adiabatic superconducting logic devices are particularly relevant in interfaces for quantum computers [15–20].

Adiabatic superconducting logic circuits are based on a parametric quantron (PQ, see the left panel of Fig. 1), i.e., a single-junction superconducting interferometer, where the critical current I_c of the Josephson junction can be varied under an external action.

Fig. 1.

(Color online) (Left panel) Circuit of the parametric quantron. The state of the cell is determined by the external magnetic flux Φ_e, I_c is the critical current of the Josephson junction, and L is the inductance of the circuit. (Right panel) Circuit of the parametric quantron with the SQUID instead of the Josephson junction. The phases of the junctions in the SQUID are controlled by the independent magnetic fluxes Φ_x and Φ_t.

We describe the “energy picture” of the operation of the PQ. The potential energy (potential) is the sum of the Josephson energy and the magnetic energy stored in an inductor:

$$\frac{{2\pi U}}{{{{E}_{{\text{J}}}}}} = - \cos (\varphi ) + \frac{{{{{(\varphi - {{\varphi }_{{\text{e}}}})}}^{2}}}}{{2l}}.$$

(1)

Here, φ is the Josephson phase of the junction; φ_e = $\frac{{2\pi {{\Phi }_{{\text{e}}}}}}{{{{\Phi }_{0}}}}$ and $l = 2\pi {{I}_{{\text{c}}}}L{\text{/}}{{\Phi }_{0}}$ are the normalized external magnetic flux and inductance, respectively; and E_J = I_cΦ₀ is the Josephson energy of the junction, where Φ₀ is the magnetic flux quantum. The external magnetic flux Φ_e determines the current circulating in a superconducting loop and, thereby, the phase of the Josephson junction φ. The screening of the external magnetic flux by the circulating current depends on the critical current and the inductance of the circuit. Expression (1) illustrates the relation between the external magnetic flux and the Josephson phase: the energy minimum for a given normalized external magnetic flux φ_e is reached at a certain Josephson phase φ. An increase in the critical current I_c leads to the increase in the Josephson energy and the normalized inductance l, which reduces the curvature of the parabola of the inductive term in the potential energy, and the single-well potential is transformed to the double-well form. This change is called the activation of a cell, when its logic state, 0 or 1, is specified corresponding to one of the two potential wells. Two states correspond to the directions of current circulation in the PQ (the ground state in the single-well potential corresponds to zero circulating current). Information transfer upon the successive activation of cells is implemented due to the magnetic coupling between their circuits. Cells in logic circuits are activated periodically, whereas the external action inducing activation is called the clock signal because it determines the frequency of information processing operations. To design adiabatic superconducting logic cells, it is necessary to ensure the adiabatic evolution of the potential. For this, the coordinate of a cell in the phase space should be a single-valued and desirably linear function of the activating signal intensity. The linearity of this dependence reflects the controllability of the cell. It is also necessary to reach the maximum phase difference between logic states for their reliable distinguishability and information transfer between cells. The deviation of the phase of a logic state of the cell in the activated state from the initial phase induced from the neighboring phase determines the enhancement of the information- transmitting signal. This work demonstrates how the use of Josephson heterostructures with a negative critical current (π junctions) opens new ways to the “balance” between the Josephson and inductive energies in the electric circuit. New possibilities of the adjustment of the potential energy of the system allow the practical implementation of its controlled dynamics and thus advancement in the fabrication of promising adiabatic superconducting logic cells.

PARAMETRIC QUANTRON BASED ON THE SQUID

Since it is difficult to rapidly control the critical current of a single Josephson junction in practice, a double-junction SQUID whose effective critical current depends on the magnetic flux inside its contour is used instead of it (see the right circuit in Fig. 1 and a more detailed circuit of such a PQ in Fig. 2). The Josephson energy in the system is controlled by the magnetic flux Φ_t specified in the SQUID (see Fig. 1); it corresponds to the phase 2φ_t (see Fig. 2). The magnetic flux specified in the main contour Φ_x corresponds to the phase φ_x. Thus, the state of the cell is now determined by the magnetic fluxes Φ_t and Φ_x. They correspond to the dimensionless phases 2φ_t and φ_x, respectively, which can be controlled independently by varying these magnetic fluxes.

Fig. 2.

Circuit of the parametric quantron with the SQUID instead of one Josephson junction. Inductances l and ${{l}_{q}}$ are normalized to the same critical current I_c, and the total inductance of the SQUID is $2l$.

We note that the PQ circuit, being symmetric, can be activated both by varying the phase φ_t and transmitting information in the phase φ_x and by varying the phase φ_x and transmitting information in the phase φ_t. We first consider the latter variant. The equations for the sum and difference phases of Josephson junctions 1 and 2 in the PQ in the superconducting regime have the form

$${{\varphi }_{ + }} + (l + 2{{l}_{q}})\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) = {{\varphi }_{x}},$$

(2a)

$${{\varphi }_{ - }} + l\sin ({{\varphi }_{ - }})\cos ({{\varphi }_{ + }}) = {{\varphi }_{t}},$$

(2b)

where ${{\varphi }_{ + }} = \frac{{{{\varphi }_{1}} + {{\varphi }_{2}}}}{2}$ and ${{\varphi }_{ - }} = \frac{{{{\varphi }_{1}} - {{\varphi }_{2}}}}{2}$.

A solution of this system of equations for φ₊ and φ_– can be obtained through the search for the root ${{x}_{0}}$ of an arbitrary continuous function $f(x)$ having only one root in a given interval $[a,b]$. Thus, the dependences φ_–(φ₊) and φ_x(φ₊), as well as the implicit dependence φ_–(φ_x) corresponding to the variation of φ_– from 0 to 2π, can be determined.

The degree of control of the dynamics of the PQ by the external clock signal φ_x can be estimated by the difference φ₊ – φ_x, where the phase (φ₊ in this case) closest to the clock phase is called the “main” phase of the cell. At a small phase of the logic state ${{\varphi }_{t}} \approx 0$ induced due to the coupling to the neighboring cell and low inductances $l < 1$, we have

$${{\varphi }_{ + }} - {{\varphi }_{x}} \approx (l + 2{{l}_{q}})\sin ({{\varphi }_{ + }}).$$

(3)

Thus, to exactly determine the dynamics by the external signal, the minimum possible inductances ${{l}_{q}} \to 0$ are necessary. For this reason, this inductance was disregarded in [21–25].

Another method to improve the control of the dynamics of the PQ by the clock signal is the introduction of a negative mutual inductance between the arms of the SQUID (see Fig. 3). Then, the mutual inductance coefficient m becomes negative, and Eqs. (2a) and (2b) take the form

$${{\varphi }_{ + }} + (l + 2{{l}_{q}} - m)\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) = {{\varphi }_{x}},$$

(4a)

$${{\varphi }_{ - }} + (l + m)\sin ({{\varphi }_{ - }})\cos ({{\varphi }_{ + }}) = {{\varphi }_{t}}.$$

(4b)

Fig. 3.

Circuit of the parametric quantron with the mutual negative magnetic coupling between the inductive arms of the SQUID.

The introduced inductive coupling reduces the difference φ₊ – φ_x and increases the amplitude of the nonlinear term in Eq. (4b), ensuring the possibility of a larger deviation of φ_– from φ_t, and, therefore, a greater enhancement of the information-carrying si-gnal.

When the PQ is clocked through the phase φ_t and information is transferred in the phase φ_x, the dependences φ₊(φ_–) and ${{\varphi }_{t}}({{\varphi }_{ - }})$, as well as the implicit dependence φ₊(φ_t), can be obtained by analogy with the solution of the system of Eqs. (2a) and (2b). This clocking method does not require the additional mutual inductance because the control of the dynamics of the PQ by the external signal (smallness of the difference φ_– – φ_t) directly depends on the smallness of the inductances of the arms of the SQUID l and the enhancement of the information-carrying signal (the difference φ₊ – φ_x) becomes significant with an increase in the amplitude of the nonlinear term in Eq. (2a).

A great attention has paid recently to this method of using the PQ [5–13, 26–31]. The corresponding adiabatic cell is called the adiabatic quantum flux parametron (AQFP). The circuit variant with the negative mutual inductance of the arms, where the phases φ_x and φ_t are used to clock the signal and to transfer information, respectively, is traditionally called the n‑SQUID [21–25] (n stands for negative).

PARAMETRIC QUANTRON WITH π JUNCTIONS

The potential energy of the PQ has the form

$$\begin{gathered} \frac{{2\pi U}}{{{{E}_{{\text{J}}}}}} = - 2\cos ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) \\ + \;\frac{{{{{({{\varphi }_{ + }} - {{\varphi }_{x}})}}^{2}}}}{{l + 2{{l}_{q}}}} + \frac{{{{{({{\varphi }_{ - }} - {{\varphi }_{t}})}}^{2}}}}{l}. \\ \end{gathered} $$

(5)

Figure 4 shows the Josephson term in Eq. (5). When the PQ is used as the n-SQUID to implement the adiabatic switching process, we propose to modify this term as shown in Fig. 5. This modification requires a “negative addition” to the Josephson energy, which is possible in the n-bi-SQUID with the π junction ${{J}_{3}}$ (see Fig. 6).

Fig. 4.

(Color online) Map of the Josephson term in the potential energy of the parametric quantron U_J; arrows illustrate the evolution of the state upon the activation of the cell. The inset shows the potential energy profiles U for the main phase φ_x = (from bottom to top) 0, $0.25\pi$, $0.5\pi$, $0.75\pi$, and $\pi$ at ${{\varphi }_{t}} = 1.5$, $l = 1.5$, and ${{l}_{q}} = 0.08$, where colored circles indicate the ground states for different φ_x values. Here and below, energies are given in units of the Josephson energy E_J.

Fig. 5.

(Color online) Josephson term in Eq. (7) for the potential energy of the n-bi-SQUID with the π junction and profiles of the potential energy at ${{\varphi }_{t}} = 1.5$, $l = 3$, ${{l}_{q}} = 0.2$, $m = 2$, and i_c3 = 0.5. The notation and normalization as in Fig. 4.

Fig. 6.

(Color online) Circuit of the parametric quantron with the mutual negative magnetic coupling between the inductive arms of the bi-SQUID with the π junction ${{J}_{3}}$.

In this case, a new term corresponding to the added junction appears in Eq. (4b) for the difference phase and the system of Eqs. (4a) and (4b) takes the form

$${{\varphi }_{ + }} + (l - m + 2{{l}_{q}})\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) = {{\varphi }_{x}},$$

(6a)

$${{\varphi }_{ - }} + (l + m)\sin ({{\varphi }_{ - }})\cos ({{\varphi }_{ + }}) - {{i}_{{c3}}}\sin (2{{\varphi }_{ - }}) = {{\varphi }_{t}}.$$

(6b)

The potential energy of the n-bi-SQUID is given by the expression

$$\begin{gathered} \frac{{2\pi U}}{{{{E}_{{\text{J}}}}}} = - 2\cos ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) + \frac{{{{{({{\varphi }_{ + }} - {{\varphi }_{x}})}}^{2}}}}{{l - m + 2{{l}_{q}}}} \\ + \;\frac{{{{{({{\varphi }_{ - }} - {{\varphi }_{t}})}}^{2}}}}{l} + {{i}_{{{\text{c}}3}}}\cos (2{{\varphi }_{ - }}). \\ \end{gathered} $$

(7)

A new term with a halved period in the phase ${{\varphi }_{ - }}$ reduces the potential barrier between potential wells $\{ {{\varphi }_{ + }},{{\varphi }_{ - }}\} = \{ 0,0\}$ and $\{ {{\varphi }_{ + }},{{\varphi }_{ - }}\} = \{ \pi ,\pi \}$. Comparing the insets of Figs. 4 and 5, where the potential energy profiles in the process of evolution are shown, one can see the effect of the modification of the circuit: the barrier for transitions between stable states (at logic operations) in the n-bi-SQUID with the π junction becomes lower by a factor of almost 2.

The described modification of the Josephson term in the potential energy is appropriate in the case of the activation of the PQ by varying the phase φ_x. In the opposite case, where clocking is ensured by the phase φ_t, it is more convenient to add a similar term but periodic in the total phase. This variant can be implemented by adding the π Josephson junction ${{J}_{4}}$ in parallel to the main inductor of the circuit ${{l}_{q}}$ (see Fig. 7) to the initial PQ circuit (see Fig. 2). Then, Eqs. (2a) and (2b) take the form

$$\begin{gathered} {{\varphi }_{ + }} + (l + 2{{l}_{q}})\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) \\ - \;{{l}_{q}}{{i}_{{{\text{c}}4}}}\sin ({{\varphi }_{ + }} + l\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }})) = {{\varphi }_{x}}, \\ \end{gathered} $$

(8a)

$${{\varphi }_{ - }} + l\sin ({{\varphi }_{ - }})\cos ({{\varphi }_{ + }}) = {{\varphi }_{t}},$$

(8b)

where i_c4 is the normalized critical current of the junction added to the circuit. The potential energy (5) is modified to the form

$$\frac{{2\pi U}}{{{{E}_{{\text{J}}}}}} = - 2\cos ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) + \frac{{{{{({{\varphi }_{ + }} - {{\varphi }_{x}})}}^{2}}}}{{l + 2{{l}_{q}}}}$$

$$ + \;\frac{{{{{({{\varphi }_{ - }} - {{\varphi }_{t}})}}^{2}}}}{l} + {{i}_{{{\text{c}}4}}}\cos ({{\varphi }_{ + }} + l\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}))$$

(9)

$$ - \;\frac{{{{l}_{q}}}}{4}\left( {1 - \frac{{2{{l}_{q}}}}{{l + 2{{l}_{q}}}}} \right)i_{{{\text{c}}4}}^{2}\cos (2[{{\varphi }_{ + }} + l\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }})]).$$

Fig. 7.

(Color online) Circuit of the parametric quantron with the π junction ${{J}_{4}}$ connected in parallel to the main inductor in the circuit ${{l}_{q}}$.

New terms in Eqs. (8а) and (9) in the variant with clocking of the PQ through the phase φ_t do not have the strict doubled period in the phase ${{\varphi }_{ + }}$. However, these terms raise the potential well of the Josephson relief at $\{ {{\varphi }_{ + }},{{\varphi }_{ - }}\} = \{ 0,0\}$ to the level at $\{ {{\varphi }_{ + }},{{\varphi }_{ - }}\}$ = $\{ \pm \pi {\text{/}}2, \pm \pi {\text{/}}2\}$ (see Fig. 8), which again allows the adiabatic switching of the PQ. In this case, the critical current of the additional junction should be i_c4 ≈ 2.

Fig. 8.

(Color online) Josephson term U_J in Eq. (9) for the potential energy of the parametric quantron with the π junction, which is connected in parallel to the main inductor in the circuit ${{l}_{q}}$, and the parameters $l = 0.2$, ${{l}_{q}} = 3$, and i_c4 = 2. The inset shows the potential energy profiles U for the main phase φ_t = (from bottom to top) 0, $0.25\pi$, $0.5\pi$, $0.75\pi$, and $\pi$.

The universal PQ circuit that allows the most free selection of the parameters to implement the adiabatic (and even physically reversible) switching process upon clocking through both phases ${{\varphi }_{x}}$ and ${{\varphi }_{t}}$ is the PQ that contains two Josephson junctions ${{J}_{3}}$ and ${{J}_{4}}$ and has a negative mutual inductance of the arms of the SQUID (see Fig. 9).

Fig. 9.

(Color online) Circuit of the parametric quantron that is based on the n-bi-SQUID with the π junction ${{J}_{3}}$ and is supplemented with the π junction ${{J}_{4}}$ connected in parallel to the main inductor in the circuit ${{l}_{q}}$.

The equations for the phases of such a PQ have the form

$$\begin{gathered} {{\varphi }_{ + }} + (l - m + 2{{l}_{q}})\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) \\ - \;{{l}_{q}}{{i}_{{{\text{c}}4}}}\sin ({{\varphi }_{ + }} + (l - m)\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }})) = {{\varphi }_{x}}, \\ \end{gathered} $$

(10a)

$$\begin{gathered} {{\varphi }_{ - }} + (l + m) \\ \times \;(\sin ({{\varphi }_{ - }})\cos ({{\varphi }_{ + }}) - {{i}_{{{\text{c}}3}}}\sin (2{{\varphi }_{ - }})) = {{\varphi }_{t}}. \\ \end{gathered} $$

(10b)

The potential energy has the form

$$\frac{{2\pi U}}{{{{E}_{{\text{J}}}}}} = - 2\cos ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) + \frac{{{{{({{\varphi }_{ + }} - {{\varphi }_{x}})}}^{2}}}}{{l - m + 2{{l}_{q}}}}$$

$$\begin{gathered} + \;\frac{{{{{({{\varphi }_{ - }} - {{\varphi }_{t}})}}^{2}}}}{{l + m}} + {{i}_{{{\text{c}}4}}}\cos ({{\varphi }_{ + }} + l\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }})) \\ + \;{{i}_{{{\text{c}}3}}}\cos (2{{\varphi }_{ - }}) - \frac{{{{l}_{q}}}}{4}\left( {1 - \frac{{2{{l}_{q}}}}{{l - m + 2{{l}_{q}}}}} \right) \\ \end{gathered} $$

(11)

$$ \times \;i_{{{\text{c}}4}}^{2}\cos (2[{{\varphi }_{ + }} + l\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }})]).$$

Here, the last term is small both at $l \to 0$, which is the case of the most frequent use of the PQ with the main phase φ_t, and at ${{l}_{q}} \to 0$, which is a necessary condition for the use of the PQ as the n-SQUID. In the case of the clocking of the PQ through the phase φ_t, a positive inductive feedback between the arms of the SQUID can be used to increase the deviation of ${{\varphi }_{ + }}$ from ${{\varphi }_{x}}$ (in this case, the sign of m in Eq. (11) should be changed).

ESTIMATE OF THE EFFECT OF π JUNCTIONS

The introduction of π junctions in the PQ reduces the requirements to the design of the logic cell. Below, we consider the examples of the n-SQUID and AQFP with the parameters taken from [6, 25].

The detailed analysis of the practically implemented adiabatic cells is presented in the supplementary material. We emphasize that experimentalists used Josephson junctions with a low critical current (and low characteristic frequencies) for the normalized inductances to ensure a certain adiabatic evolution of the state of the PQ. To this end, intermediate states with three wells should not appear at the transformation of the single-well potential to the double-well one. Otherwise, an error is possible in information transfer: thermal fluctuations can induce a change in the sign of the phase associated with the logic state with respect to the phase specified at the input of the cell.

This problem can also be solved by correcting the Josephson term in the potential energy. Using the PQ with two π junctions (see Fig. 9) with relatively high critical currents, we transform the initial Josephson relief (which does not ensure the certain evolution of the state) to the “required” relief, which allows the operation at a standard temperature of 4.2 K. Details of the calculations and necessary illustrations are given in the auxiliary material.

The introduction of π Josephson junctions in the n-SQUID changes the evolution trajectory of the PQ state of the n-SQUID (see Figs. 10а, 10b): the gain of the information transmission signal, i.e., the difference ${{\varphi }_{ - }}$ – ${{\varphi }_{t}}$, as well as the “controllability” of the state by the external signal, i.e., the difference φ₊ – φ_x, changes. The solid line in Fig. 10a shows the dependence ${{\varphi }_{ - }}({{\varphi }_{x}})$ of the difference phase of the n-SQUID with the parameters from the experiment [30] on the main phase. The corresponding dependence of the difference between the sum phase of the n-SQUID φ₊ and the main phase φ_x is shown in Fig. 10c.

Fig. 10.

(Color online) (а, b) Difference phase ${{\varphi }_{ - }}$ and (c, d) the phase difference φ₊ – φ_x at the phase φ_t = 0.1 corresponding to the input logic signal versus the main phase φ_x for the (а, c) n-SQUID and (b, d) parametric quantron with π junctions. The parameters correspond either to (solid lines in panels (а, c)) experimental implementations [30] or to (dashed lines in panels (а, c)) a triple increase in the critical currents of the Josephson junctions compared to the experiment, as well to the parameters taken to plot Fig. S4 in the supplementary material with the introduction of π junctions in the circuit (see Figs. 9b, 9d).

The calculation was performed with the system of Eqs. (4a) and (4b). The same dependences for the n-SQUID with the parameters corresponding to the triple increase in the critical currents of the Josephson junctions are shown by the dashed lines in Figs. 10a and 10c. Since segments of the dependences with negative derivatives $\frac{{d{{\varphi }_{ - }}}}{{d{{\varphi }_{x}}}}$ and $\frac{{d({{\varphi }_{ + }} - {{\varphi }_{x}})}}{{d{{\varphi }_{x}}}}$ are not implemented in experiments, the dynamics of such a cell cannot be adiabatic and involves jumps of the phase.

The considered dependences for the PQ corresponding to the introduction of π junctions (the parameters are similar to those used in Fig. S4) that are obtained with the system of Eqs. (10a) and (10b) are shown in Figs. 10b and 10d. The presence of π junctions allows one to implement the adiabatic switching of the PQ. The magnetic flux at the logic state transfer for the cell with π junctions increases in a wide range of the main phase. This circumstance reduces the requirements to the transformer of the magnetic flux coupling the cells in the circuit.

However, the accuracy of controlling the dynamics of the cell by the external signal decreases: the area under the plot of |φ₊ – φ_x|/π increases by 50%, which reduces the maximum response rate of the cell.

DEVELOPMENT OF THE AQFP CONCEPT

The AQFP circuit in the representation of the clock signal of the PQ through the φ_t phase (see Fig. 2) can also be improved by using π junctions (see Fig. 9). For the sake of convenience, the authors of [5, 7, 8, 10, 26–29] divided the inductor of the main circuit in the AQFP into two parts (see Fig. 11a), which allowed them to create separate magnetic couplings to specify/read the signal. In this case, the effective inductance of the entire circuit remained relatively low. This variant of the AQFP is described by the equations

$$\begin{gathered} {{\varphi }_{ + }} + \left( {l + 2\frac{{{{l}_{{{\text{in}}}}}{{l}_{{{\text{out}}}}}}}{{{{l}_{{{\text{in}}}}} + {{l}_{{{\text{out}}}}}}}} \right) \\ \times \;\sin ({{\varphi }_{ + }})\cos ({{\varphi }_{ - }}) = {{\varphi }_{x}}\frac{{{{l}_{{{\text{out}}}}}}}{{{{l}_{{{\text{in}}}}} + {{l}_{{{\text{out}}}}}}}, \\ \end{gathered} $$

(12a)

$${{\varphi }_{ - }} + l\sin ({{\varphi }_{ - }})\cos ({{\varphi }_{ + }}) = {{\varphi }_{t}}.$$

(12b)

Fig. 11.

(Color online) Circuit of the adiabatic quantum flux parametron (а) with the separation of the main inductor in the circuit ${{l}_{q}}$ into two parts l_in and l_out and (b) with the addition of two π junctions.

Introducing the notation ${{l}_{q}} = \frac{{{{l}_{{{\text{in}}}}}{{l}_{{{\text{out}}}}}}}{{{{l}_{{{\text{in}}}}} + {{l}_{{{\text{out}}}}}}}$ and ${{\varphi }_{x}} = {{\varphi }_{x}}\frac{{{{l}_{{{\text{out}}}}}}}{{{{l}_{{{\text{in}}}}} + {{l}_{{{\text{out}}}}}}}$, one can obviously reduce Eqs. (12a) and (12b) to Eqs. (2a) and (2b).

The optimal parameters $l = 0.2$ and ${{l}_{q}} = 1.6$ of the AQFP were determined in [6]. The switching dynamics at these parameters will be certain. However, the inductance of the SQUID corresponding to such a small normalized value $l = 0.2$ can hardly be ensured at the minimum allowable critical current I_c = 50 μA. The minimum normalized inductance reached in experiments [5, 29] is $l = 0.4$, which is twice as high. The dynamics of the AQFP at this inductance is no longer certain (see Fig. 12).

Fig. 12.

(Color online) Potential energy of the adiabatic quantum flux parametron calculated by Eq. (5) with the normalized inductances $l = 0.4$ and ${{l}_{q}} = 1.6$ for the phases φ_x = 0 and φ_t = (а) 2.1, (b) 2.22, and (c) 2.3 corresponding to external sources of the magnetic flux (see Fig. 11a).

The introduction of two π junctions in the circuit (see Fig. 11b) can ensure the certain and more controllable evolution of the state of this cell. The equations describing this cell are similar to Eqs. (10) and (11) because the magnetic coupling between the inductive arms of the SQUID is absent: $m = 0$. The Josephson term of the potential energy at the critical currents of the π junctions i_c3 = 0.6 and i_c4 = 1.4 changes as shown in Fig. 13. In this case, the dynamics of the AQFP is certain even at high inductances $l = 0.4$ and ${{l}_{q}} = 2.4$ (see Fig. 14). The gain of the information-carrying signal and the quality of controlling the state of the AQFP are shown in Fig. 15. The magnetic flux transfer and the controllability quality increase by 125 and 40%, respectively.

Fig. 13.

(Color online) (а) Josephson relief of the potential energy of the adiabatic quantum flux parametron, (b) the relief corresponding to additional terms in Eq. (11), and (c) the Josephson relief of the adiabatic quantum flux parametron with π junctions (see Fig. 11b) at the parameters $l = 0.4$, ${{l}_{q}} = 2.4$, i_c3 = 0.6, and i_c4 = 1.4.

Fig. 14.

(Color online) Potential energy of the adiabatic quantum flux parametron calculated by Eq. (11) with the normalized inductances $l = 0.4$ and ${{l}_{q}} = 2.4$ and the critical currents of the π junctions i_c3 = 0.6, and i_c4 = 1.4 for the phases φ_x = 0 and φ_t = (а) 1, (b) 1.68, and (c) 1.9 corresponding to external sources of the magnetic flux (see Fig. 11b).

Fig. 15.

(Color online) (а, b) Total phase ${{\varphi }_{ + }}$ and (c, d) the phase difference φ_– – φ_t at the phase φ_x = 0.1 corresponding to the input logic signal versus the main phase φ_t for the (а, c) conventional adiabatic quantum flux parametron and (b, d) adiabatic quantum flux parametron with π junctions. The calculations for the adiabatic quantum flux parametron presented in panels (а, c) were performed with the parameters $l = 0.4$ and ${{l}_{q}} = 1.6$ from known experimental works [23, 31]; the parameters $l = 0.4$, ${{l}_{q}} = 2.4$, i_c3 = 0.6, and i_c4 = 1.4 were taken for the cell proposed in panels (b, c).

CONCLUSIONS

The results obtained in this work have indicated that the addition of π junctions in the parametric quantron circuit allows one not only to ensure the uniqueness of the dynamics of cells of the n-SQUID and the adiabatic quantum flux parametron in the range of practically possible parameters but also to improve both the enhancement of the transmitted signal and, in the case of the adiabatic quantum flux parametron, the quality of the control of the circuit state.

At the same time, the range of the clock signal, where the enhancement in magnetic flux occurs, expands. This expansion allows the reduction of the number of phases of the clock signal upon the multiphase clocking by the alternating current.

In addition, an increase in the critical currents (and characteristic voltages) of main Josephson junctions in circuits makes it possible to increase their response speed. Thus, the application of π junctions seems promising for the improvement of the parameters of multicomponent circuits such as adiabatic microprocessors [11, 13, 14] and interfaces for superconducting quantum processors [20, 32, 33] based on adiabatic superconducting logic.