Bi-SQUID Versus dc SQUID in Flux-Driven Traveling-Wave Parametric Ampli ﬁ er

: Characteristics of arti ﬁ cial waveguide lines composed of ﬁ nite-size arti ﬁ cial cells are analyzed and discussed with relation to designing traveling-wave parametric ampli ﬁ ers. From this point of view, the ﬂ ux-driven ampli ﬁ er suggested recently can be considered as the most successful ampli ﬁ er design. To increase dynamic range of the ampli ﬁ er, dc SQUID cells of the arti ﬁ cial line in use can be substituted for bi-SQUID cells. The 1 dB compression points of the ampli ﬁ er gain are calculated and compared for both types of cells.


Introduction
Parametric amplifiers (PAs) are known as ultimate low-noise amplifiers; however, such conventional cavity-based amplifiers are burdened with a gain-bandwidth trade-off.To overcome the drawback, traveling-wave parametric amplifiers (TWPAs) were suggested.TWPAs are based on using waveguide lines with nonlinear reactive parameters (of capacitive or inductive types) such as a kinetic inductance microwave line [1] or artificial lines composed of lumped cells with nonlinear capacitors or inductances.As the superconducting kinetic inductance of Josephson junctions  =  /cos ( ) is strongly nonlinear (here,  = Φ /(2 ), Φ = ℎ/(2), and  is the Josephson junction critical current and  is the Josephson junction phase), the artificial lines composed of superconducting cells containing Josephson junctions are generally used in the designs of Josephson traveling-wave parametric amplifiers (JTWPAs) [2][3][4][5][6][7][8][9][10][11][12].The amplifiers capable of working at low and very low temperatures can provide the extremely high sensitivity approaching the quantum limit level.Therefore, JTWPAs are currently considered as promising readout devices for use in the field of precision quantum measurements (including single-photon detectors), quantum communications and quantum computing (see [13,14] and review [15]).
However, when the pump and signal waves propagate in the common artificial Josephson line of JTWPAs, the attainable gain is limited by depletion of the pump wave [8,9,15] because the initial amplitude of the pump wave is restricted by critical current values of the used Josephson junctions.To overcome this problem, a two-line JTWPA design with a magnetic flux drive was suggested [16].In the design, a separate non-Josephson (linear) transmission line is used for the pump wave propagation.This line is inductively coupled with SQUID-like cells of the signal artificial line and, therefore, the pump wave produces a traveling wave of magnetic flux applied to the signal line cells.This magnetic flux wave provides modulation of the cell inductances in the traveling-wave fashion.
In this paper, we discuss the characteristics of the artificial waveguide lines composed of lumped element cells in connection with designing the Josephson traveling wave parametric amplifiers and consider the increase in dynamic range of the two-line fluxdriven amplifier by substitution of dc SQUID cells for bi-SQUID cells.

Effect of Finite Size of Artificial Cells
Usually, the JTWPA designs are based on implementation of the artificial righthanded lines containing Josephson junctions, although the left-handed artificial lines can also be used as recently reported in [17].The right-handed artificial lines correspond to the existing natural distributed waveguide lines.Therefore, the natural distributed lines are often modeled by the right-handed artificial LC lines under the stipulation that the length of the distributed line pieces described by the lumped LC cells goes to zero.The limit processing (when ,  → 0) leads to telegrapher's equations and hence to the differential wave equation [18].
However, when the lumped cells of the implemented artificial lines model the continuous line pieces of finite length, one should describe the discrete system using discretevalued equations as it is realized in [19] (versus continuum approximation in [12]) and partly in [16].Nevertheless, to simplify analysis of the artificial network, many authors consider the structure as a really distributed continuous waveguide line to describe propagation of the pumping, signal and idler waves in the artificial structure using a wavetype equation for a continuous medium.In the approach, wave impedance of the artificial line is considered as a real-value impedance Z0 of the corresponding really distributed waveguide line.However, this is generally incorrect.Figure 1 illustrates simple reasoning which can be used to calculate wave impedance of the long artificial line, consisting of Z1, Z2 Γ-cells.If Z0 is the wave impedance, then it does not change with connection of the same additional cell.This requirement yields in the following equation: In case of a conventional dissipativeless cell line with the cell element impedances  =  and  = () , the wave impedance of the artificial line is complex: where the angle  changes from  = 0 at  → 0 to  = /2 at cutoff frequency Thus, the wave impedance changes from the real value  = / at a low frequency to the imaginary value i at  →  .This fact means that the artificial cell line cannot be matched with any constant real-valued impedance over the total frequency band up to the cutoff frequency.For example, when the cell line is connected to the real-valued impedance  = /, the reflection factor is imaginary and frequency-dependent: The absolute magnitude of the factor changes from || = 0 at  → 0 to || = 1 at  →  .Of course, the problem vanishes with decreasing the cells' size (at  → 0 and  → 0), i.e., with limit processing of the distributed waveguide line when  →  = /.
To mitigate the reflection problem, one could increase the cutoff frequency of the artificial line as compared with frequencies of the pumping, signal and idler signals; however, it additionally leads to the intense depletion of the pump wave due to the unwanted leak of the pump power to the higher harmonics and intermodulation components occurring in the frequency band ranging up to the cutoff frequency.Moreover, the described dependence of the wave impedance on frequency automatically leads to the frequencydependent phase velocity decreasing with frequency.

One-Line Amplifier Design
In the three-wave operation mode when  +  =  , finding the reasonable tradeoff seems very problematic because both the wave reflection and the phase mismatch of the pumping, signal and idler waves enforce setting the cutoff frequency much higher than the wave frequencies, but it does not allow for keeping the higher harmonics and the intermodulation components out of the frequency band.The restrictions can be mitigated by using the four-wave operation mode when  +  = 2 .In this case, all three frequencies  ,  ,  can be located nearer to each other in the vicinity of ~ /3 so that both the harmonics and the intermodulation components produced by the Kerr nonlinearity are above the cutoff frequency, and the reflection factor (at  /3) is quite moderate: || ≃ 0.15.At the same time, the frequency closeness makes the efficient filtering out of the pump frequency from the amplified signal more difficult.

Two-Line Amplifier Design
The flux-driven JTWPA with two-line amplifier design suggested in [16] consists of two coupled separate artificial waveguide lines.The first artificial LC line, where both the signal and idler waves propagate, is based on using a serial array of dc SQUIDs coupled inductively to the second superconducting LC transmission line carrying the pumping wave.The latter is linear (non-Josephson) and therefore is free from any limitation on the pump wave amplitude and from energy leakage to higher harmonics as well.The pump signal applies rf magnetic flux Φ to the SQUID cells and, due to the cells' nonlinear properties, provides modulation of the dipole cell inductances  (with a depth m) in the wave manner through influence of the maximum superconducting current | | of the dc SQUID cell.The inductance modulation can be provided either on the pump frequency when a dc flux bias |Φ | < Φ /2 is additionally applied: or on the double frequency when no dc flux is applied (Φ = 0): However, in the latter case, the modulation depth m is considerably less than m due to the approximately ||-like dependence [20] of the maximum superconducting current | | of dc SQUID on the applied magnetic flux Φ .At a low loop inductance (as assumed in [16]), | | ≈ 2 |cos ( /2)| , where  = 2Φ /Φ , and hence at where  = 2Φ /Φ and  = 2Φ /Φ are the normalized values of the applied dc and rf magnetic fluxes.This expression shows that dc flux changes the maximum superconducting current (from 2 to 2 cos ( /2)) and hence the dipole inductance of dc SQUID and modulation of the inductance from ( 7) to ( 6) with a modulation depth about four times higher.
The inductance modulation waves ( 6) and ( 7) yield in the signal frequency mixing either to  ±  or to 2 ±  corresponding to the three-wave and four-wave operation modes, when  is the signal frequency and the idler frequency is  =  −  or  = 2 −  , respectively.In both cases, the signal and idler wave frequencies can be optimally located in the vicinity of ~ /3, where the reflection factor is moderate, || ≃ 0.15, and the phase velocities of the waves are quite close to each other:  ≈  ≈ 1/√ (with using dimensionless coordinate x normalized on the cell length "a"), and therefore they can be tuned closely to the phase velocity of the pump wave propagating in the second line.For example, when one sets the pump frequency as  ≃ (3/4) •  in threewave mode or 2 ≃ (3/4) •  in four-wave mode, the signal and idler frequency band can range from  ≃  /4 to  ≃  /2, that enables eliminating wave components at frequencies  +  , >  and 2 +  , >  , respectively, and the reflection factor at signal frequency  and idler frequency  decreases to || ≲ 0.2. Figure 2 shows the reflection factor |Γ()| for the artificial waveguide line of lumped L and C elements which is connected to resistor  =  ≡ /, as well as optimal values of both the pump wave frequency   in the three-wave mode (or double frequency 2  in the four-wave mode) and the frequency range for signal and idler waves.

Increase in Dynamic Range Using bi-SQUID Cells
The dynamic range (DR) of an amplifier can be defined as the maximum-to-minimum ratio of the output signal power within the range of the linear relation between the output signal and input signal, starting from the minimum value of the output signal amplitude restricted by the root-mean-square (rms) value of the output noise.
Although the gain of the flux-driven JTWPA is not restricted by the pump wave depletion, the output signal power is inevitably limited by nonlinear distortions as the rf current amplitude increases and nears the maximum superconducting current of the cells (as inductances) in use, namely dc SQUIDs suggested in [16].In fact, a linear wave-type equation with a periodically modulated reactive parameter, which was derived in [16] for the magnetic flux Φ(, ) in the frame of distributed line approximation (using dimensionless coordinate x normalized on the cell spacing "a"), gives an exponential increase in amplitudes with x of both the signal and idle traveling flux waves Φ (, ) and Φ (, ).However, to obtain an output power, one needs to know the amplitude of the current traveling waves  (, ) and  (, ) .The relation between the flux and current amplitudes, Φ and  , is linear at low values: Φ =  , where k is the wave number (reduced to the reciprocal of the cell spacing),  is the running inductance (dipole inductance of SQUID-cell) and Φ is amplitude of the flux per unit length.But this ratio becomes nonlinear with an increase in the wave amplitudes and then the current wave amplitude can be written as  = Φ / −  (Φ ).This leads to the power gain reduction by factor  =  /(Φ /) rising the current amplitude nearing the maximum superconducting current value of the SQUID cells in use.The linear domain limit determines the dynamic range of the amplifier.The end point of the linear range can be characterized by the signal amplitude corresponding to the power gain reduction by 1 dB; this is called the 1 dB compression point.
Increasing the critical current  of the used Josephson junctions to increase the maximum superconducting current of the SQUID cells cannot be an acceptable solution for extending the linear domain, because the critical current magnitude determines some other normalized parameters answering for the amplifier's performance.For example, one has to increase geometric loop inductance in order to increase the flux applied by the pumping wave, but one has to decrease the normalized value of the loop inductance  = 2 /Φ (8) by decreasing the junction critical current to increase both the modulation depth of the maximum superconducting current of the cell and the characteristic Josephson junction inductance  = Φ /(2 ) (9) and hence the modulation depth of the dipole cell inductance.
To increase the linear domain of the flux-driven JTWPA at the tradeoff-assigned magnitude of the junction critical current, one needs to modify the cells in use in order to increase the linear domain of the cell characteristics describing the relation between transport current via the cell  produced by the signal and idler waves and the corresponding phase drop Δ on the cell.The phase drop answers the flux (Φ /2π)Δ per length of the waveguide line composed of the cells.
For this purpose, one can substitute the dc SQUID cells for bi-SQUID cells as suggested recently in [21] and shown in Figure 3.In the cell, the required improvement may be achieved through redirection of the currents flowing inside the cell with the current via the added third Josephson junction.The mechanism can be explained in the following way.
In the absence of both the signal and idler waves (i.e., at  = 0) when only the dc magnetic flux Φ is applied to an assigned operation point, Josephson phases  and  of the junctions J1 and J2 in symmetric cells of both types, dc SQUID and bi-SQUID, are equal in magnitude and opposite in their signs in regard to the direction of transport current, i.e.,  = − , due to the induced counterclockwise circular screening supercurrent.Figure 4a schematically shows the phases and the currents flowing through the junctions J1 and J2 when only the dc magnetic flux is applied to the cells.At a negligibly small geometric inductance of the device loops, as assumed in [16],  = − =  /2 = Φ /Φ .
for dc SQUID (in detail, see the macroscopic quantum interference effect in [20] and  −  =  − (sin  − sin  )/2 (12) for bi-SQUID in accordance with the schematic in Figure 4, where the junctions J1 and J2 are on the left and right sides, respectively, and J3 is above them.In both equations, l is the normalized loop inductance ( 8),  = 2 Φ Φ ⁄ is the normalized value of the applied magnetic flux Φ = Φ + Φ , where Φ is the applied dc magnetic flux setting operation point, and Φ is the rf magnetic flux which is applied by the pump wave to modulate the cell dipole inductance; correspondingly, φ = φ + φ .
In dc SQUID, transport current ib causes displacement of both the points 1 and 2 shown in Figure 4a in the sinusoid in the direction corresponding to either the increase in the junction phases at  > 0 or the decrease in the ones at  < 0, and some drop in the Josephson phase Δ appears on the cell: Δ > 0 or Δ < 0 , respectively.The relation between the phase drop and the transport current is evidently nonlinear.As seen from the sketch (Figure 4a), the nonlinearity results from either the first junction at the positive transport current (corresponding to red color of the shown arrows) or the second junction (green arrows) at the negative transport current.
In bi-SQUID, the nonlinearity problem is considerably mitigated through the current flowing via the added Josephson junction J3.As follows from the numerical simulation (see [21]), this current always flows through the third junction in the same direction independently of the transport current direction; however, on different paths through the other circuit elements, there is a dependence on the transport current direction and therefore it changes distribution of the transport current between the main junctions J1 and J2.These two possible paths of the current  are shown in Figure 4b by the red lines and red arrows for the positive transport current and by the green lines and arrows for the negative transport current.In both cases, the current  provides a decrease in the current flowing through that junction which is responsible for nonlinearity in the relation between the phase drop Δ and the transport current  (i.e., for nonlinearity of the dipole inductance of the cell).
As an example, Figure 5 shows phase drops Δ on both dc SQUID (green line) and bi-SQUID (blue line) having the same critical current  of the basic junctions, J1 and J2, and the same normalized geometric inductance  = 0.5 versus the transport current  =  / created by both the signal and idler waves.The same dc magnetic flux  =  /3 is applied to both devices to the set operation point.In bi-SQUID, the normalized critical current of the third junction  ≡  / = 2.The presented dependences evidence the appreciable increase in the linearity range when using bi-SQUID cells and hence the increase in the dynamic range of the flux-driven JTWPA.Quantitative characterization of the linearity domain increase can be conducted by considering the 1 dB compression points of the power gain.Figures 6 and 7 present the calculated 1 dB compression points as a function of the applied dc bias flux  = 2Φ /Φ , setting the operation point for both the dc SQUID cell and the bi-SQUID cell at the same values, l = 1 and l = 0.75, of the normalized cell inductances.It is evidenced that dynamic range of the flux-driven JTWPA can be increased by two to three times when using bi-SQUID cells.

Bi-SQUID Cell Parameters
The observed increase in dynamic range when using bi-SQUID cells depends first of all on the magnitudes of the main parameters, which are the normalized cell inductance l and the normalized critical current of the third Josephson junction  =  / .Both the parameters influence the current redirection inside the cell.Increase in the inductive parameter l, on the one hand, improves the redirection, but on the other hand, a high value of l decreases the fractional Josephson junction contribution to the cell inductance and hence essentially restricts the obtainable modulation of the reactive parameter.Therefore, the magnitude  ~ 1 can be considered as a tradeoff value of this parameter.
Next, Figure 8 shows the 1 dB compression point of the gain of the flux-driven JTWPA with bi-SQUID cells versus the normalized value of the third Josephson junction  .An increase in the critical current of this junction results in the monotonic rise of the point, i.e., improvement of the linearity.However, most of the improvement can be achieved with  ~2 to 3. Except the discussed main parameters, one has to consider satellite parameters which can also influence the linear domain of the bi-SQUID cell and hence the gain reduction.These are the second loop inductance and the magnetic flux which can be applied to the loop.Figure 9 schematically shows all inductances in the bi-SQUID cell, where L is the main part of the principal loop inductance of the cell, and  and  are parts of the second loop inductance with the normalized values: Unfortunately, inductance of the second loop cannot be decreased exactly down to zero, and some flux can occur in the loop as well.In a rough way, the latter can be considered as the flux Φ of a locally homogeneous magnetic field applied to the bi-SQUID in order to apply to its principal loop dc magnetic flux Φ needed to set the operation point.In this case, the ratio of the fluxes,  = Φ /Φ , can be estimated by the ratio of the loop areas and hence roughly by the ratio of the loop geometric inductances in a square (believing roughly that the loop inductance is proportional to linear size r, while the loop area S~r 2 ).
Interestingly, the moderate normalized inductance component  even slightly rises the 1 dB gain compression points (up to ≲ 10%) as shown in Figure 10, but the applied dc flux has an inverse influence and causes a reduction in the overpowering positive impact of the inductance component  as seen in Figure 10.
As for the other inductance part  , its increase reduces the 1 dB gain compression points (up to ≲ 10%) as seen in Figure 11.These effects can be explained as follows.Inductance  is connected in series with the third junction J3 and therefore increases inductive impedance of the branch as compared to the parallel branch containing junctions J1 and J2.On the contrary, the second inductance part  changes the inductive impedance ratio in the reverse direction by increasing the inductive impedance of the J1-J2 branch.Therefore, the inductance parts oppositely influence the current redirection mechanism inside the bi-SQUID cell and hence the 1 dB gain compression points.The minimum value of the output signal of the flux-driven JTWPA actually does not change when substituting the dc SQUID cells for bi-SQUID cells.In fact, fluctuations arising in Josephson junctions can be described using the Langevin method [20,22,23], i.e., by adding noise current sources connected in parallel to the junctions.The in-phase current fluctuations (in regard to the direction of the transport current) arising in the basic junctions J1 and J2 yield fluctuations in both the transport current and the phase drop Δ on the cell.These are the fluctuations forming the output noise of the flux-driven JTWPA.The antiphase current fluctuations arising in the junctions J1 and J2 cause a circular noise current in the cell loop and hence some small fluctuations in the dc flux setting operation point of the cell.The added Josephson junction J3 can contribute only to the circular noise current; however, this contribution is about compensated by the shunting impact of the junction on the antiphase current fluctuations produced by the junctions J1 and J2.In such a way, the influence of the added junction J3 on the noise characteristics of the JTWPA can be considered negligibly small.

Conclusions
In such a way, with the substitution of dc SQUID cells for bi-SQUID cells, the dynamic range of the flux-driven JTWPA increases by a factor of two to three due to the current redirection mechanism inside the cells.To achieve this, it suffices to set the normalized value of the cell loop l~1 and set two to three times as much critical current of the third junction as compared to the basic junctions to provide sufficiency of the mechanism.The negative contribution of satellite parameters, inductance of the second low-inductance loop and the magnetic flux occurring in the loop can be reduced down to several percent when the inductance is three to four times less than the main inductance of the cell.
Moreover, the two-line design of the flux-driven JTWPA allows for maximal mitigation of the restrictions following from the complex-valued frequency-dependent wave impedance of the used artificial waveguide lines composed of lumped elements.

Figure 1 .
Figure 1.If Z0 is the wave impedance of a long artificial line of Γ-cells, the value Z0 does not change with cell addition.

Figure 2 .
Figure 2. Reflection factor |Γ()| for the artificial waveguide of lumped L and C elements which is connected to resistor  =  ≡ / ; optimal values of both the pump wave frequency  in 3wave mode (or double frequency 2 in 4-wave mode) and the frequency range for signal and idler waves.

Figure 3 .
Figure 3.After[21].The inductively coupled elementary cells of the pump and signal transmission lines of JTWPA based on using bi-SQUIDs.C and C1 are ground capacitances of the two lines:   line for the pump wave and  line for the signal and idler waves, where L is inductance of the bi-SQUID cell.When the waves propagate in an artificial line and produce a relatively slow-varying transport current  flowing via a symmetric dc SQUID and a symmetric bi-SQUID both in the superconducting state, this current (when the wave frequencies are much less than both the Josephson characteristic and plasma frequencies  = 2  /Φ ,  = 2 (Φ  ) ⁄ ;  and  are the junction capacitance and normal resistance) is carried practically by only the superconducting currents through Josephson junctions J1 and J2; therefore, the Josephson phases  and  of the junctions obey the current relation  ≡   ⁄ =  + = sin + sin

Figure 4 .
Figure 4.After [21].(a) Josephson phases and currents through the junctions J1 and J2 described by two points in sinusoid on the current-phase plane when some dc magnetic flux is applied to the cell, dc SQUID or bi-SQUID, inducing counterclockwise circular screening supercurrent, and bias current is absent.The red and green arrows show directions of displacement of the points at positive (red color) and negative (green color) bias current.(b) Paths of the current flowing through the additional junction in bi-SQUID at positive (red lines and red arrows) and negative (green lines and arrows) bias currents.The junctions J1 and J2 are on the left and right sides, respectively, and J3 is above them.

Figure 5 .
Figure 5.After [21].Phase drops on both dc SQUID (green line) and bi-SQUID (blue line) with  = 2 and same value  = 0.5 of the normalized geometric inductance versus transport current.The same dc magnetic flux  =  /3 is applied to both devices.

Figure 6 .
Figure 6.The current wave amplitude  / corresponding to the 1 dB compression point of the gain of JTWPA vs. normalized value of the applied dc bias flux  = 2Φ /Φ with dc SQUID cells (red line) and bi-SQUID cells (black line) at the same value l = 1 of the normalized cell inductances.

Figure 7 .
Figure 7.The current wave amplitude  / corresponding to the 1 dB compression point of the gain of JTWPA vs. normalized value of the applied dc bias flux  = 2Φ /Φ with dc SQUID cells (red line) and bi-SQUID cells (black line) at the same value l = 0.75 of the normalized cell inductances.

Figure 8 .
Figure 8.The current wave amplitude  / corresponding to the 1 dB compression point of the gain of JTWPA with bi-SQUID cells versus normalized value of the third Josephson junction  =  / at  = 0.75 and  = 2Φ /Φ .

Figure 9 .
Figure 9. Bi-SQUID cell schematic, where L is the main part of the principal loop inductance of the cell, and  and  are parts of the second loop inductance.Pink line on the top is the inductance of the cell of the line, where pump wave propagate.

Figure 10 .
Figure 10.The 1 dB gain compression points versus the normalized inductance component  of the second loop inductance of the bi-SQUID cell when no magnetic flux is applied to the loop (red line) and when magnetic flux Φ = Φ is applied to the loop (black line).In both cases, the normalized value of the main loop inductance is  = 0.75, the normalized dc flux applied to the main loop is  = 2πΦ /Φ = 1.6 and  = 0.01 (normalized value of  ).Factor  = ( ) / corresponds approximately to the loop square ratio.

Figure 11 .
Figure 11.The 1 dB gain compression points versus the normalized inductance component  of the second loop inductance of the bi-SQUID cell when no magnetic flux is applied to the loop.Normalized value of the main inductance  = 0.75, normalized dc flux applied.