Patent Publication Number: US-2023163737-A1

Title: Traveling wave parametric amplifier

Description:
FIELD 
     The present invention relates to superconducting traveling wave parametric amplifiers, TWPAs. 
     BACKGROUND 
     Parametric amplifiers are in effect mixers, wherein a weaker input signal may be amplified by mixing it with stronger pump signal, producing a stronger output signal as a result. Parametric amplifiers rely on a nonlinear response of a physical system to generate amplification. Such amplifiers may comprise standing wave parametric amplifiers or traveling wave parametric amplifiers, wherein a traveling wave parametric amplifier uses a series of nonlinear elements distributed along a transmission line, such as a coplanar waveguide, for example. In case the nonlinear elements comprise Josephson junctions, the amplifier may be referred to as a Josephson traveling wave parametric amplifier, JTWPA. In a JTWPA, the Josephson junctions are maintained in superconducting condition and carry a supercurrent. 
     In use, a signal is added to the strong oscillator signal, resulting in a sum signal wherein an amplitude envelope exhibits variance at a frequency which is a difference between the signal and oscillator frequencies. Since in the waveguide transmission line, a phase velocity is dependent on amplitude, a phase of the summed signal at the end of the line will vary in accordance with a difference in the two frequencies. In effect, the nonlinear waveguide transmission line converts amplitude modulation into phase modulation. In case the non-linearity is strong enough, this will result in a gain at the signal frequency. 
     SUMMARY OF THE INVENTION 
     According to some aspects, there is provided the subject-matter of the independent claims. Some embodiments are defined in the dependent claims. 
     According to a first aspect of the present invention, there is provided a travelling wave parametric amplifier comprising a transmission line comprising therein a plurality of Josephson elements and a plurality of shunt capacitors, and wherein at least some of the shunt capacitors are dispersive capacitors comprising an open-ended, distributed transmission line. 
     According to a second aspect of the present invention, there is provided a method, comprising providing a transmission line comprising therein a plurality of Josephson elements and a plurality of shunt capacitors, and wherein at least some of the shunt capacitors are dispersive capacitors comprising an open-ended, distributed transmission line. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1 A  illustrates a section of an example amplifier in accordance with at least some embodiments of the present invention; 
         FIG.  1 B  illustrates a section of an example amplifier in accordance with at least some embodiments of the present invention; 
         FIG.  2 A  illustrates simulated dispersion in a TWPA in accordance with at least some embodiments of the present invention; 
         FIG.  2 B  illustrates phase shifting in a TWPA in accordance with at least some embodiments of the present invention; 
         FIG.  3 A  illustrates three-wave mixing simulated gain in accordance with at least some embodiments of the present invention; 
         FIG.  3 B  illustrates pump and second harmonic powers in accordance with at least some embodiments of the present invention; 
         FIG.  3 C  illustrates an example TWPA in accordance with at least some embodiments of the present invention, and 
         FIG.  4    is a flow graph of a method in accordance with at least some embodiments of the present invention. 
     
    
    
     EMBODIMENTS 
     In accordance with solutions disclosed herein, a traveling wave parametric amplifier, TWPA, may be enabled to use higher critical current densities and higher plasma frequencies, which results in the benefit that a device footprint of the TWPA may be reduced. As the plasma frequency need not be as precisely defined as in prior solutions, Josephson junctions need not be manufactured to as high a precision as earlier. Yet still, in TWPAs manufactured according to principles disclosed herein, dispersion characteristics are highly reproducible. A high TWPA gain is Obtained due to phase-matching of three-wave mixing by purposefully inefficient generation of the 2 nd  harmonic of the pump tone. In addition, due to a lower dielectric loss in shunt capacitors, a higher gain and lower added noise in three- and four-wave mixing in TWPAs is obtained 
     These benefits are obtained in TWPAs as will be described herein below. In brief, dispersion is introduced into the TWPA using shunt capacitive elements, such as shunt capacitive elements of TWPA unit cells. These dispersive shunt capacitors each comprise an open-ended, distributed transmission line. In comparison to the typical realization of the shunt capacitors as parallel plates separated by a dielectric layer, this reduces dielectric loss in the TWPA. In flux-tuneable TWPAs, generating the dispersion mainly with the shunt capacitors rather than with the plasma resonance of Josephson elements is desirable: this decouples the dispersion from the magnetic flux tuning. In some embodiments, the open-ended, distributed transmission line is arranged in the form of a spiral, which reduces a physical footprint of the TWPA when manufactured on a chip. Another option is a polygonal shape. 
     Overall, for example in quantum computation, signals may be attenuated for transmission even to a single-photon or a near-single-photon regime. Detecting such signals presents challenges owing to their low amplitude. Therefore, suitable amplifiers may be employed to increase the amplitudes of received signals prior to their provision to detector elements, where the information encoded into these received signals may be recovered. As another example, a single-photon regime communication may be employed in communicating encryption keys in a secure manner using quantum communication, such that eavesdropping without detection is made very difficult. 
     The present disclosure focuses on a superconductive realization of the TWPA, where the center trace of a transmission line is an array of Josephson junction based elements, known as Josephson elements, which constitute a non-linear inductance. The non-linearity enables a mixing process which provides power gain for a weak signal that propagates along the same direction as a strong radio frequency, rf, pump tone. The strength of the pump tone is measured with the ratio between the pump current amplitude Ip and a critical current Ic of the Josephson element. The nature of the non-linearity depends on the arrangement of Josephson junctions within the element. The simplest realization is the use of a single Josephson junction as the non-linear element: the associated Taylor expansion of the inductance is a constant plus a term proportional to (Ip/Ic)∧2, that is, a Kerr non-linearity. While the Kerr term results in a desired four-wave mixing process, it also changes the wavevectors of the rf tones as a function of the pump power, an effect that may be compensated with dispersion engineering. The balancing of the wavevectors, also called phase matching, allows an exponential increase of the TWPA gain as a function of the device length. Due to the typically narrowband dispersive features embedded into the transmission line, the center frequency of gain is a fixed quantity in this example of the TWPA. 
     In general, TWPAs exist in two categories, namely operating based on three-wave mixing,  3 WM, and devices operating based on four-wave mixing,  4 WM. These mixing concepts are originally from the field of non-linear optics. In  3 WM, the pump tone f_p is at twice the signal frequency f_s which is to be amplified, and the device tends to produce an unwanted second harmonic of the pump tone. The generation of this second harmonic may be referred to as second harmonic generation, SHG. In  4 WM, the pump tone is in the same frequency band as the signal to be amplified, and an unwanted third harmonic of the pump tends to be generated. Regardless of whether a TWPA is based on  3 WM or  4 WM, the TWPA device comprises a transmission line which has a frequency-dependent wavenumber. 
     Furthermore, in a device producing gain, an idler tone f_i is generated at a frequency of f_p−f_s ( 3 WM) or 2f_p−f_s ( 4 WM). Achieving a high gain requires phase matching, which means that the sum of the signal and idler wavenumbers (k_s+k_i) must be close to k_p ( 3 WM) or 2k_p ( 4 WM), with k_p the pump wavenumber. At the same time, the device must be dispersive in order to have phase mismatching of the unwanted harmonic generation of the pump Dispersiveness in this context means that the wavenumber as a function of frequency deviates from a linear trend at frequencies above f_p. This behavior is illustrated in  FIG.  2 A  One way to control the dispersion is by adjusting the Josephson plasma frequency, f j  The plasma frequency describes the effect of the tunnel junction&#39;s capacitance, which connects in parallel to non-linear Josephson inductance. In some embodiments, the TWPA consists of cascaded unit cells, each of which features a series impedance and a shunt impedance. In the limit of low frequency where the dispersion has no practical effect, the series impedance is the reactance of an inductor L and the shunt impedance is the reactance of a capacitor C. These quantities determine the unit cell LC eigenmode f 0 =(2η√{square root over (LC)}) −1 . Together with the plasma frequency, the eigenmode sets a cut-off of the transmission line. At the cut-off, the real part of the wavenumber goes to zero, implying that wave propagation is not supported along the transmission line. Just below the cut-off the wavenumber diverges, showing a superlinear trend. One way of suppressing unwanted harmonic generation in TWPAs involves lowering of the plasma frequency towards the frequency of the pump harmonic generation. In more detail, f j  may be comparable in magnitude to 2f_p in a  3 WM TWPA, and 3f_p for a  4 WN TWPA. 
     A TWPA design which relies on a low and/or precise value of the plasma frequency involves challenges, however. The plasma frequency is essentially a Josephson junction fabrication process parameter that is controlled by oxidation pressure and time Changing the plasma frequency means that the junction critical current density is changed as well. This implies that designing for a low plasma frequency means that the physical size of the junctions will be large, which may in turn affect the overall footprint of the TWPA on the a chip. 
     If the TWPA has an adjustable magnetic flux degree of freedom which affects the Josephson element critical current, the plasma frequency will be adjustable as well There arc several reasons of why magnetic flux adjustability would be desired. Firstly, it may be desired for setting the characteristic impedance level of the TWPA transmission line to a correct value, for example to 50 ohm Secondly, it may be desired for enabling three-wave mixing. If the dispersiveness of the TWPA depends primarily on the plasma frequency, then the dispersion could be coupled in a potentially undesired way to the impedance matching —three-wave mixing condition. 
     In prior literature, the theory of phase matching in  3 WM Josephson TWPAs has been addressed only a few times. In the literature, second harmonic generation, SHG, is often either not discussed in sufficient detail or the device design attempts to block it altogether by stopband engineering. The phase mismatching of the SHG is quantified with the parameter Δk SHG =k 2p −2k p ≠0. However, the phase mismatching of the SHG via the unit cell and plasma cut-off inevitably leaves some residual phase mismatch for the mixing process which gives gain. This residual mismatch is quantified with the parameter Δk TWPA =k p −k s −k i ≠0. This may, in turn, limit the gain available from the  3 WM TWPA. Furthermore, Josephson TWPAs in general have the drawback of dielectric loss in shunt capacitors of the transmission line. The shunt capacitors are typically implemented as parallel plate capacitors comprising a dielectric material that is lossy and non-linear at millikelvin temperatures, due to the presence of two-level fluctuators. The dielectric loss has a negative effect on the gain and noise temperature of the TWPA. 
     TWPAs in accordance with the present disclosure implement dispersion control. Instead of relying solely on the plasma frequency of the series inductive element of the TWPA unit cell, dispersion is introduced in the shunt capacitors of the transmission line as well. To be specific, dispersive capacitors comprising an open-ended, distributed transmission line are used to introduce dispersion into the transmission line. This produces several benefits. Firstly, this reduces dielectric loss in the TWPAs. Secondly, in TWPAs with the adjustable magnetic flux degree of freedom, this decouples the dispersion from the flux adjustments of the gain and of the characteristic impedance. The physical footprint on a chip of the open-ended distributed transmission line may be small if, for example, a doubly-wound spiral geometry is chosen for it. By open-ended, distributed transmission line it is meant a structure that lacks a galvanic connection to the TWPA ground and in which capacitance and inductance are distributed continuously throughout the material. 
     Concerning the dispersive capacitors, an open-ended stub of transmission line behaves as a lumped capacitor at low-frequencies, but shows resonances at certain high frequencies, the first of which corresponds to the stub being a quarter of wavelength long The first resonance frequency is selected in such a way that the capacitance value has only minimal frequency dependence at the signal, pump and idler frequencies. This paves the way for a good phase matching of the gain process. However, the effective value of the capacitance starts to increase at frequencies higher than f_p, which provides phase mismatching of the harmonic generation. To be specific, the harmonic generation means transferring energy from the pump signal to a frequency equal to 2f_p, corresponding to SHG in  3 WM or to 3f_p, corresponding to  4 WM. In a  4 WM TWPA, the precise value of the phase mismatch is not important as long as the third harmonic generation efficiency remains low, as described in [5]. By contrast, in a  3 WM TWPA there is an optimal value for the phase mismatching of the SHG. 
     In the field of non-linear optics of light, it has been discovered that cascaded multiwave mixing processes may occur. In the  3 WM TWPA, a cascaded process occurs where second harmonic generation is followed by the transfer of the pump energy from the second harmonic, 2f_p, back to the pump frequency f_p. This cascaded process occurs whenever phase matching of second harmonic generation, SHG, is not perfect. When a dispersive capacitor is responsible for the frequency dependence of the TWPA wavenumber, that is, responsible for the dispersion, having a finite phase mismatch of the second harmonic generation, SI-R implies that there is some residual phase mismatch in the gain process as well. Examples of finite phase mismatch include Δk SHG ; being 15% and 20% of k_p. Examples of residual phase mismatch include Δk TWPA  being 1,5% and 2% of k_p. It has been found, that there is an optimal combination of 1) the SHG wavenumber mismatch Δk SHG , 2) the amplification wavenumber mismatch Δk TWPA , and 3) the TWPA pump power. At this optimum, the pump tone receives phase kicks from inefficient SHG, in a way that exactly compensates for the phase mismatch between the pump, signal and idler accumulating in the gain process. Furthermore, using a set of dispersive capacitors to generate dispersion in the transmission line allows several adjustments for optimizing TWPA operation. Firstly, the pump frequency may be shifted and new values may be obtained for the wavenumber mismatches. Secondly, the pump power can be shifted, and/or thirdly, if the TWPA is at the magnetic flux bias point that nulls  4 WM while retaining finite  3 WM, some magnetic flux offset may be applied to obtain positive or negative phase shifts from weak  4 WM. An operating point of the TWPA is in a three-dimensional space defined by these axes of adjustments. The magnetic flux may be offset from its tentative optimum value, which corresponds to zero Kerr, to obtain a positive or negative Kerr value which may improve the phase matching. A potential drawback of the non-zero Kerr is potential activation of generation of a third harmonic of the pump frequency. 
     Insight into the multiwave mixing processes can be gained by studying coupled mode equations, CMEs, of  3 WM in the absence of  4 WM and in the absence of losses. The CMEs describe the evolution of the wave amplitudes as a function of position x along the transmission line of the TWPA. The CMEs of the SHG in the absence of parametric amplification read 
       ∂ x   A   p   =−ikA*   p   A   2p   e   −iΔk     SHG     x  
 
       ∂ x   A   2p   =−ikA   p   2   e   iΔk     SHG     x  
 
     where A_p is the slowly varying dimensionless amplitude of the pump, A_2p the slowly varying dimensionless amplitude of the second harmonic of the pump, and κ is a  3 WM non-linear coefficient in units of 1/m. In the limit of phase matched SHG (Δk SHG =0) 
       | A   2p ( x )/ A   p (0)| 2 =[tanh(Γ x )] 2 ,
 
     which implies potential for a 100% unidirectional conversion of the pump into the second harmonic Here, the TWPA pump strength equals Γ=κ|A p (0)|. Conversely, in the limit of weak SHG efficiency the pump experiences a phase distortion 
     
       
         
           
             
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                         1 
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     The CMEs of near-degenerate parametric amplification in the absence of the SHG and in the limit of small phase mismatch Δk TWPA &lt;&lt;k p  read 
       ∂ x   A   p ≈0
 
       ∂ x   A   s ≈−0.25 iκΔ*   i   A   p   e   −iΔk     TWPA     x  
 
       ∂ x   A   i ≈−0.25 iκA*   s   A   p   e   −iΔk     TWPA     x  
 
     where A_s and A_i are the slowly varying dimensionless amplitudes of the signal and idler, respectively Being near-degenerate means in this context that the signal and idler frequencies are almost identical. The pre-factor 0.25 arises from the frequency difference of 0.25 between the signal and the second harmonic of the purr p. In the limit of phase matched parametric amplification (Δk TWPA =0) 
       | A   s ( x )/ A   s (0)| 2 =1+[sinh(0.25Γ x )| 2 ,
 
     which describes the power gain experienced by the signal. Considering the SHG and the parametric amplification simultaneously, we observe the optimum that achieves phase matching of the parametric amplification via inefficient SHG: 
       ΔΦ NL   /x+Δk   TWPA =0.
 
     In the limit of weak SHG efficiency a condition for the optimal TWPA pump strength is obtained: 
       Γ 2   =Δk   TWPA   2   +Δk   TWPA   Δk   SHG .
 
     If the plasma frequency and the first resonance mode of the shunt capacitors coincide, the characteristic TWPA line impedance will be frequency independent at low frequencies, to the first order. This may be beneficial from an impedance matching point of view at the signal, pump and idler frequencies. 
     Concerning dielectric losses of the proposed dispersive capacitor, as the transmission line of the capacitor is, for example, a planar structure made from a superconducting thin film on a low-loss substrate such as high-resistivity silicon, most of the electric field energy will reside in the substrate. This is to be contrasted against parallel plate capacitors in which almost all of the field energy is in the lossy dielectric layer. 
     Benefits of using the dispersive capacitors include an ability to use high critical current densities and high plasma frequencies in the TWPA, which may help reduce the device footprint. Further, relaxed requirements of Josephson junction fabrication are obtained, because some error in the plasma frequency can be tolerated. A further benefit is high reproducibility of the dispersion in TWPA fabrication. Yet further, a high TWPA gain may be obtained, due to phase matching of  3 WM parametric amplification by inefficient SHG. A higher gain and lower added noise in  3 WM or  4 WM TWPAs is possible due to the lower dielectric loss in the shunt capacitors. 
       FIG.  1 A  illustrates a section of an example amplifier in accordance with at least some embodiments of the present invention. The JTWPA of  FIG.  1    comprises a coplanar waveguide transmission line, which comprises Josephson elements  110  and shunt capacitors  120 . Each one of the Josephson elements comprises a loop, with at least one Josephson junction of a first size arranged in one side of the loop. At least one Josephson junction of a second size may further be arranged in another side of the loop, Examples of the loop include SQUIDs and SNAILs. As a specific example, the loops may comprise two large Josephson junctions on one side and one smaller Josephson junction on the other side, the smaller junction being 23% of the larger ones. This is an implementation of a SNAIL. In some embodiments, the Josephson elements  110  each comprise one and only one loop. The Josephson elements  110  are connected with each other with waveguides capable of conveying electromagnetic waves, as is known in the art. The overall coplanar waveguide transmission line, a section of which is illustrated in  FIG.  1   , has an input at the left, arranged to receive the signal to be amplified and a strong oscillator signal, which are mixed in the waveguide in the non-linear Josephson elements  110 . In some embodiments, a unit comprises two consecutive Josephson elements  110 . At an output is at the right, such that a phase-modulated amplified signal is eventually obtained as output at the end of the coplanar waveguide transmission line, a segment of which is illustrated in  FIG.  1 A . Two wiring layer elements  101 , which are the ground planes of the coplanar waveguide, may each comprise a superconductor covered with an insulator, for example. 
     In general, a Josephson element, such as a single junction, a superconducting quantum interference device (SQUID), an asymmetric SQUID, or a more complex Josephson element such as a flux-qubit-like circuit, can be described using an effective potential energy: 
         U   eff (φ)/ E   j   =c   2 φ 2   +c   3 φ 3   +c   4 φ 4 + . . .
 
     here E j  is the Josephson energy, and p is the superconducting phase. The c 2  term relates to critical current and linear part of Josephson inductance, the c 3  term relates to 3-wave mixing and the c 4  term relates to 4-wave mixing, which is also known as the Kerr nonlinearity. 
     Normally single junctions and SQUIDs, including asymmetric SQUIDs, have c 3 =0, whereby 3-wave mixing does not occur, and non-linearity is provided by the Kerr term. 3-wave mixing means the ability to pump at twice the input frequency, which is desirable. 3-wave mixing could be activated by injecting a de current, but however, the Kerr term would remain non-zero. 
     Nonlinearity provided by the Kerr term is associated with the need for resonant phase matching, in practice the pump signal is given a small phase increment at regular intervals along the transmission line. This is due to the pump having a different phase velocity from the signal (at the frequency f_p) and the idler (at the frequency f_i). This phase mismatch increases with the pump power. Conservation of energy implies the existence of an idler frequency at the output, the frequency of which is located at the “mirror image” of the signal frequency with respect to the pump, f_i=2f_p−f_s. In detail, in the Kerr mode, phase mismatch and gain depend on the same parameter, the Kerr nonlinearity. The three frequencies are related by f_s+f_i=f_p in the case of 3-wave mixing. To minimize the amount of reflections, both ends of the TWPA further need to have good impedance match at each of the frequencies f_i, f_s and f_p. 
     The JTWPA of  FIG.  1    has, in the waveguide, shunt capacitors  120 , interspersed between the Josephson elements  110 . Two Josephson elements  110  between every two shunt capacitors  120  is one example, to which the invention is not limited, indeed, in various embodiments there may be three or more Josephson elements  110  between every two shunt capacitors  120 . The shunt capacitors  120  form the majority of the shunt capacitance of the transmission line. The JTWPA of  FIG.  1    is a coplanar waveguide. The shunt capacitors  120  may be parallel-plate capacitors. Overall the JTWPA may comprise a large number of Josephson elements  110 , such as 200, 800 or 1600 elements. For example, there may be more than fifteen Josephson elements  110 . 
     The JTWPA of  FIG.  1    is further furnished with an integrated flux bias line, FBL,  130 . Flux bias line  130  is a two-port circuit that takes a serpentine path, ranging from one side of the coplanar waveguide to the other. In other words, the flux bias line meanders and proceeds alternatingly, repeatedly on either side of the coplanar waveguide transmission line. The flux bias line  130  forms an upper electrode of the shunt capacitors  120  in places where it crosses over to another side of the waveguide, as illustrated in  FIG.  1 A , The flux bias line  130  connects to the ground planes  101  of the transmission line through resistors  150 , the value of which is much smaller than the reactive impedance of the shunt capacitors  120 , at the relevant frequencies f_i, f_s and f_p. The purpose of the resistors is to provide an rf path to ground from the shunt capacitors  120 . At the same time, the resistors and the flux bias line  130  enforce a similar electric potential of the ground planes at the frequencies f_i, f_s, and f_p. 
     As illustrated, flux bias line  130  extends on one side of the waveguide, parallel. to the waveguide, before ranging over to another side of the waveguide at a place corresponding to one of the shunt capacitors  120 , to again extend parallel to the waveguide on said another side of the waveguide. Where flux bias line  130  extends parallel to the waveguide, it may be connected to the ground planes  101  with the resistors. A dc current in the flux bias line  130  generates the magnetic field gradient for the Josephson elements  110 . The flux bias line is configured to generate a magnetic flux threading each of the loops of the Josephson elements  110 . The flux bias line may be connected to equalize electrical potentials of ground planes of the coplanar waveguide transmission line. This connection may be through resistors, for example. 
     In the TWPA of  FIG.  1 A , some of the shunt capacitors are dispersive capacitors  140 , A dispersive capacitor  140  is provided with an open-ended, distributed transmission line which causes a dispersive effect in the transmission line of the TWPA, as described herein above. The open-ended, distributed transmission line is in the example of  FIG.  1 A  arranged in a spiral arrangement to save space. 
     In some embodiments, the spiral arrangement is a double Archimedean spiral arrangement. In a double Archimedean spiral arrangement, a first conductor of the transmission line is arranged in a spiral and a second conductor of the transmission line is disposed in the same area in a shape that is a mirror image of the shape of the first conductor of the transmission line, the first and second conductors of the transmission lines not touching. The first conductor is connected with ground plane  101 , and the other one with the central transmission line of the TWPA. Although illustrated as having only a few turns in  FIG.  1 A  for the sake of clarity of the illustration, the spiral arrangement in practical implementations may have a much larger number of turns, such as several dozens, for example. In some embodiments with two concentric spirals, the spirals are not mirror images of each other, but concentrically arranged and winding in the same direction. 
     A low spiral eigenfrequency leads to large capacitor values, whereas a high spiral eigenfrequency may mean that dispersion is not sufficiently generated. One example is an approximately 40 GHz eigenmode of 4,6 turns of the spiral made of a thin superconducting film on silicon substrate. The film has a trace width of 1.9 micrometers and a gap of 1.1 micrometers. The diameter of the resulting geometry is approximately 75 micrometers. 
     In principle, all the shunt capacitors might be dispersive capacitors. On the other hand, in various embodiments only a part of the shunt capacitors are dispersive capacitors. For example, only every nth shunt capacitor is a dispersive capacitor, with n selected from the group {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. A benefit of not making every shunt capacitor dispersive is that cross-talk is avoided between the dispersive capacitors. 300 micrometers of separation for 75 micrometer spiral capacitors is a sufficient separation to avoid cross-talk, for example. In general, a separation equal to or greater than the diameter of a spirally wound capacitor may be sufficient to avoid cross-talk. The dispersive capacitors need not have exactly the same capacitance as the other shunt capacitors, although ideally the difference is small. 
     A second benefit of not making every shunt capacitor dispersive is the ability to do stopband engineering. Whenever the electrical separation between two spirals, as measured along the TWPA transmission line, equals an integer multiple of one half of the wavelength, a stopband will occur. It may be beneficial to place this kind of a stopband at or near the frequency of the higher harmonic generation of the pump. 
       FIG.  1 B  illustrates a section of an example amplifier in accordance with at least some embodiments of the present invention. The TWPA of  FIG.  1 B  is the same one as in  FIG.  1 A , the difference between these figures being that a longer section of the transmission line is shown in  FIG.  1 B , 
       FIG.  2 A  illustrates simulated dispersion in a TWPA in accordance with at least some embodiments of the present invention. The figure illustrates the relationship between frequency of a signal propagating through the TWPA, on the horizontal axis, and a total phase change incurred by the signal while traversing the length of the TWPA on the vertical axis. The phase change of the vertical axis is known as the S 21  (transmission) phase. Dispersion here means, that the wavenumber vs. frequency behaves almost linearly until frequency f_p, after which it behaves superlinearly, as illustrated. 
       FIG.  2 B  illustrates phase shifting in a TWPA in accordance with at least some embodiments of the present invention. When the wave vectors k_pump=k_signal+k_idler, the situation is optimal. In case energy is transferred from the pump frequency to the second harmonic, the pump is weakened. However, where energy is transferred from the second harmonic back to the pump frequency, the problem of energy bleed from pump to second harmonic is solved. The pump phase shift arises in connection with the transfer of energy from the second harmonic back to the pump. The phase shill is therefore a good sign. In some embodiments, this pump phase shift cancels a TWPA phase mismatch caused by low-frequency dispersion. 
       FIG.  3 A  illustrates three-wave mixing simulated gain in accordance with at least some embodiments of the present invention. Signal and idler powers are shown in dimensionless units. In these simulations, a 10 GHz pump frequency is used, with Δk TWPA /k p ≈0,019 and Δk SHG /k p  ≈0,24. As f_p=f_s+f_i always, an optimum is where k_p=k_s+k_i. Δk TWPA =k_p−k_s−k_i, which is nonzero, and Δk SHG =k_2p−2k_p. In the figure, Γ=0,09 indicates pump strength, defined as I_p/I_c times a factor which is dependent on the magnetic flux. The dashed line is a theoretical upper bound of the signal power. 
       FIG.  3 B  illustrates pump and second harmonic powers in dimensionless units in accordance with at least some embodiments of the present invention. Amplitude losses visible in the figure are incurred from the dielectric materials and the resistors, which are discussed herein above. 
       FIG.  3 C  illustrates the top view of an example TWPA in accordance with at least some embodiments of the present invention. The illustrated TWPA is manufactured on a chip. The example TWPA illustrated in the figure is 9-periodic in the sense that it has 8 unit cells with two Josephson elements in each cell, followed by one cell with a dispersive capacitor instead of any Josephson elements, repeated over and over along the length of the transmission line. 24 cells are in bends and 93 in the long, straight sections resulting in 24 93=117 cells, which is divisible by 9. The spiral-shaped distributed transmission lines of the dispersive capacitors are visible as dark circles in the Figure, disposed on either side along the length of the overall transmission line. 
       FIG.  4    is a flow graph of a method in accordance with at least some embodiments of the present invention. The phases of the illustrated method may be performed in a factory apparatus, an auxiliary device or a personal computer, for example, or in a control device configured to control the functioning thereof, when installed therein. 
     Phase  410  comprises providing a transmission line comprising therein a plurality of Josephson elements and a plurality of shunt capacitors Phase  420  specifies, that at least some of the shunt capacitors are dispersive capacitors comprising an open-ended, distributed transmission line. For example, the open-ended, distributed transmission line may be arranged in a spiral shape or a polygonal shape. In some embodiments there is further provided a flux bias line configured to generate a magnetic flux threading each of loops of the Josephson elements by meandering alternatingly, repeatedly on either side of the transmission line, which is a coplanar waveguide transmission line, such that the flux bias line crosses over the coplanar waveguide transmission line where a shunt capacitor is disposed on the coplanar waveguide transmission line. 
     It is to be understood that the embodiments of the invention disclosed are not limited to the particular structures, process steps; or materials disclosed herein; but are extended to equivalents thereof as would be recognized by those ordinarily skilled in the relevant arts. It should also be understood that terminology employed herein is used for the purpose of describing particular embodiments only and is not intended to be limiting. 
     Reference throughout this specification to one embodiment or an embodiment means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention, Thus, appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same embodiment, Where reference is made to a numerical value using a term such as, for example, about or substantially, the exact numerical value is also disclosed. 
     As used herein, a plurality of items, structural elements, compositional elements, and/or materials may be presented in a common list for convenience. However, these lists should be construed as though each member of the list is individually identified as a separate and unique member. Thus; no individual member of such list should be construed as a de facto equivalent of any other member of the same list solely based on their presentation in a common group without indications to the contrary. In addition, various embodiments and example of the present invention may be referred to herein along with alternatives for the various components thereof. It is understood that such embodiments, examples, and alternatives are not to be construed as de facto equivalents of one another, but are to be considered as separate and autonomous representations of the present invention. 
     Furthermore, the described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the preceding description, numerous specific details are provided, such as examples of lengths, widths, shapes, etc., to provide a thorough understanding of embodiments of the invention. One skilled in the relevant art will recognize, however, that the invention can be practiced without one or more of the specific details, or with other methods, components, materials, etc. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of the invention. 
     While the forgoing examples are illustrative of the principles of the present invention in one or more particular applications, it will be apparent to those of ordinary skill in the art that numerous modifications in form, usage and details of implementation can be made without the exercise of inventive faculty, and without departing from the principles and concepts of the invention. Accordingly, it is not intended that the invention be limited, except as by the claims set forth below. 
     The verbs “to comprise” and “to include” are used in this document as open limitations that neither exclude nor require the existence of also un-recited features. The features recited in depending claims are mutually freely combinable unless otherwise explicitly stated. Furthermore, it is to be understood that the use of “a” or “an”, that is, a singular form, throughout this document does not exclude a plurality. 
     INDUSTRIAL APPLICABILITY 
     At least some embodiments of the present invention find industrial application in amplification of low-amplitude signals. 
     ACRONYMS LIST 
     
         
           3 WM three wave mixing 
           4 WM four wave mixing 
         CME coupled mode equation 
         fl Idler frequency 
         fP Oscillator/pump frequency 
         fS Signal frequency 
         Ic Critical current of Josephson junction 
         Ip Pump current amplitude 
         JTWPA Josephson traveling wave parametric amplifier 
         SHG second harmonic generation 
         SQUID superconducting quantum interference device 
         SNAIL superconducting nonlinear asymmetric inductive element 
         TWPA traveling wave parametric amplifier 
       
    
     REFERENCE SIGNS LIST 
       
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 110 
                 Josephson element 
               
               
                   
                 120 
                 Shunt capacitor (parallel-plate capacitor) 
               
               
                   
                 130 
                 Flux bias line 
               
               
                   
                 140 
                 Capacitor 
               
               
                   
                 150 
                 Resistor 
               
               
                   
                 101 
                 Wiring layer element 
               
               
                   
                 301 
                 Superconducting part 
               
               
                   
                 302 
                 Tunnel Junction 
               
               
                   
                 410-420 
                 Phases of the method of FIG. 4 
               
               
                   
                   
               
            
           
         
       
     
     CITATION LIST 
     
         
         [1] httpsilarxiv.orgiabs/1907,10158 “A photonic crystal Josephson traveling wave parametric amplifier”, preprint 
         [2] https://arxiv, org/abs/1804.09109 “Flux-driven Josephson traveling-wave parametric amplifier”, preprint and journal article Phys. Rev. Applied 12, 044051 (2019). 
         [3] https://doi.org/10.11031PhysRevApplied. 6.034006 “Josephson Traveling-Wave Parametric Amplifier with Three-Wave Mixing”, journal article 
         [4] https://escholarship.orglucliteml4nw2m1sj “Nonlinear Light-Matter Interactions in Metamaterials”, doctoral dissertation 
         [5] littps://ieeexploreleee.org/abstractIdocument/8666769 “Symmetric Traveling Wave Parametric Amplifier”, journal article 
         [6] https://arxiv.org/abs/1912.05349 “Capturing complex behaviour in Josephson Travelling Wave Parametric Amplifiers”, preprint 
         [7] https://doi.org/10.1364/01117.000028 “Self-focusing and self-defocusing, by cascaded second-order effects in KTP”, Optics Letters 17, 28 (1992), journal article 
         [8] haps://doi.org/10.1.364/01-27.000043 “Observation of 99% pump depletion in single-pass second-harmonic generation in a periodically poled lithium riii.Dbate waveguide”, Optics Letters 27, 43 (2002), journal article 
         [9] http://dx.doi.org/10.2528/P1ER14011901 “Travelling wave mechanism and novel analysis of the planar Archimedean spiral antenna in free space”, Progress in Elearomagnetics Research 145. 287 (2014), journal article