Bimodal microwave system

Abstract

The invention relates to a multimode non-linear superconducting microwave transmission line 10 boosted by at least one pump P. The pump P is supported by one of the propagation modes of the transmission line 10, the velocity of which is at least two times lower than that of the one or more other modes. This allows the transmission line 10 to act as a circulator, amplifier and / or reciprocal adjustable coupler (according to the set of pumps boosting the line) for a signal to be transported S, while exhibiting robust phase matching and exponential performance with the length of the transmission line 10.

Description

[0001] DESCRIPTION

[0002] TITLE OF THE INVENTION: Bimodal microwave system

[0003] 1. Field of the invention

[0004] The invention relates to the field of microwaves, particularly in the field of quantum technologies.

[0005] Microwaves are a frequency range widely used in multiple fields, such as telecommunications and quantum computing. The generation, processing, and detection of these signals are achieved using a wide range of active and passive components (mixers, amplifiers, circulators, filters, switches, diplexers, etc.).

[0006] 2. Prior art

[0007] In quantum technologies, superconducting circuits resonate in the microwave range, and thus are controlled via microwave signals. In microwave devices potentially suitable for the quantum domain, three features are of interest: amplification, circulation, and adjustable reciprocal coupling.

[0008] Amplifiers are used to amplify an input signal, for example to measure a signal. However, most amplifiers add noise to the measured signals. In the case of weak signals (typically on the order of a single photon) used by quantum technologies, this noise limits measurement fidelity.

[0009] To overcome this limitation, Josephson junction amplifiers have been developed over the past twenty years. The performance of such amplifiers approaches the quantum limit, i.e., the added noise is as low as what is permitted by the limits of quantum physics. However, these amplifiers operate over a narrow frequency band.

[0010] Over the past ten years, traveling-wave parametric amplifiers (TWPAs) have further improved these performances, as they allow operation over a wide frequency band. TWPAs are based on a single-mode, non-linear transmission line doped with Josephson junctions. The transmission line is forced by a high-amplitude microwave called a pump. The main obstacle to the realization of TWPAs and their performances is to ensure the "phase-matching" of the pump with the signal to be amplified and its mirror signal (called "idler") propagating in the same direction as the signal to be amplified. Another obstacle to their use is that they are not completely directional, that is to say, noise is back-propagated towards the system emitting the signal to be amplified.

[0011] Circulators are non-reciprocal components that separate signals based on their direction of propagation. The dominant technology for circulators relies on strong magnetic fields, which are incompatible with superconducting circuits. Several implementations compatible with superconducting circuits have been demonstrated but suffer from low bandwidth (less than 100 MHz), poor tunability (less than 1 GHz), and limited isolation (< 20 dB).

[0012] Finally, an adjustable coupler is a 2-port reciprocal element that allows adjusting the fraction of the power incident on one port that is transmitted to the other port or reflected. Adjustable couplers integrated into superconducting circuits today have low bandwidth and limited noise robustness.

[0013] Thus, there is a need for a component capable of providing these functionalities, operating over a wide bandwidth, performing well and with robust phase matching. The invention improves the situation.

[0014] 3. Statement of the invention

[0015] To this end, the invention aims at a microwave system of a level type not having the drawbacks stated above. The invention proposes a microwave system comprising a superconducting microwave transmission line, a first microwave source capable of generating at least a first signal, called pump, a second microwave source capable of generating at least a second signal, called transport, said sources each being connected to one end of the transmission line so as to inject their respective signal into the transmission line, the transmission line supporting a plurality of propagation modes, a first mode, called slow mode, of said plurality of propagation modes having a phase velocity less than half the phase velocity of the other mode(s), called fast mode(s), of the plurality of propagation modes, said slow mode being forced by the pump,said so-called fast mode(s) supporting the at least one signal to be transported, the amplitude of the at least one signal to be transported being at least ten times lower than the amplitude of the at least one pump, the transmission line comprising a plurality of non-linear inductive elements, each having a kinetic inductance, the transmission line being configured so that, when the frequency of the at least one pump is equal to a function of the frequency of the at least one signal to be transported and the propagation speeds of the propagation modes, all of the modes participate within the non-linear inductive elements in a resonant mixture of the waves of each propagation mode.,

[0016] This new type of microwave system allows for transmission of a signal to be transported that can be unidirectional, amplified or both at the same time, depending on the nature of the pump signal, while ensuring robust phase matching. By transmission line configuration is meant a judicious choice of physical characteristics of the transmission line. These physical characteristics may, among other things, relate to the inductance per unit length and / or the capacitance per unit length of the line.

[0017] By function is meant a function defined by the physical characteristics of the line, and chosen so that when the pump signal is at the appropriate frequency, dependent on the frequency of the signal to be transported, then the resonant mixing described above occurs. In other words, the transmission line is configured — or has an inductance per unit length and a capacitance per unit length — so that, for a given signal frequency to be transported, when the frequency of the at least one pump is equal to a function of the frequency of the at least one signal to be transported and the propagation speeds of the propagation modes, then all of the modes participate within the non-linear inductive elements in a resonant mixing of the waves of each propagation mode.

[0018] Unidirectionality of transmission means that the signal to be carried at the second frequency can travel in one direction of the line, from the end of the transmission line where the source is connected to the other end, but not in the other.

[0019] This unidirectionality is a consequence of the resonant mixing of waves, which itself results from the large difference in the speeds of the slow and fast modes. The speed (or velocity) of these modes is an intrinsic consequence of the physical characteristics of the transmission line, notably its inductance and capacitance per unit length.

[0020] This unidirectionality results from the interactions between the microwaves crossing the line and interacting at the level of the non-linear inductive elements, which allow the pump signal to convert the signal to be transported into a mirror signal ("idler") of a different frequency from the signal to be transported and propagating in the opposite direction to it. These exchanges occur at the level of these non-linear inductive elements. Unidirectionality thus blocks backpropagation towards the source of noise produced by a device to which the transmission line is connected.

[0021] Thus, the signal to be transported (at its own frequency) can only travel in the desired direction. In addition, noise at the frequency of the signal to be transported propagating in the opposite direction to the signal to be transported is converted into a mirror signal, preventing noise from backpropagating. In fact, the signal to be transported sees its propagation in the indirect direction blocked, inducing a reflection at another frequency corresponding to the mirror signal, hence its name. In many applications, this mirror signal is harmless, unlike the reflection of the signal to be transported in the absence of this unidirectionality.

[0022] Thus, when we want to transport a signal of a given frequency via the microwave system, we simply need to set the pump to the right frequency to do so. The signal frequency adjustment band is also very wide, typically between 2 and 12 GHz. Another advantage of this system is that it produces exponential isolation depending on the optical length of the transmission line. The isolation of the transmission line is a way of quantifying its unidirectionality. In fact, isolation is the ratio between the signal injected at the line input and the signal actually transmitted at the end of the line, typically expressed in dB. The isolation is described as "robust", meaning that it remains effective even in the event of low disturbances.

[0023] The microwave system is further capable of acting as an adjustable reciprocal coupler and / or a broadband amplifier, in both cases exhibiting robust phase matching.

[0024] Thus, this new type of microwave system not only features robust phase matching, but is also multifunctional simply by changing the nature of the pump forcing the line, all with performance increasing exponentially with the length of the transmission line.

[0025] In a particular aspect, the non-linear inductive elements are configured to provide four-wave mixing or three-wave mixing, and wherein the signal to be transported is injected by the signal source with a frequency > s / 2n, the propagation speed of the slow mode is v P , the propagation speed of the fast mode is v s, the pump being injected by the pump source into the transmission line in the same direction as the signal to be transported, the pump frequency pl2n being set to satisfy the following equation, where k is an integer equal to 1 when the non-linear inductive elements are configured to allow four-wave mixing and 2 for three-wave mixing:

[0026] In a particular aspect, the non-linear inductive elements are configured to provide four-wave mixing or three-wave mixing, and wherein the signal to be transported is injected by the signal source with a frequency > s / 2n, the propagation speed of the slow mode is v P , the propagation speed of the fast mode is v s, the pump being injected by the pump source into the transmission line in the opposite direction to the signal to be transported, the pump frequency <j pl n étant réglée pour satisfaire l'équation suivante, où k est un entier égal à 1 lorsque les éléments inductifs non-linéaire sont configurés pour permettre un mélange à quatre ondes et à 2 pour un mélange à trois ondes : Such a pump source setting allows the system to produce a circulation effect with robust phase matching and exponential isolation with the length of the transmission line. Here, this choice of frequencies makes it possible to obtain, at the level of the non-linear inductive elements, a conversion effect. More precisely, a "signal" photon is converted into a "mirror" photon by mobilizing two "pump" photons (for the four-wave mixing case) or one "pump" photon (case of a three-wave mixing). This conversion makes it possible to obtain the desired unidirectionality effect. This effect can be obtained with a pump and a signal propagating in the same direction or in opposite directions.

[0027] In a particular aspect, the non-linear inductive elements are configured to provide four-wave mixing or three-wave mixing, and wherein the signal to be transported is injected by the signal source with a frequency > s / 2n, the propagation speed of the slow mode is v P , the propagation speed of the fast mode is v s , the pump being injected simultaneously by the pump source on both sides of the transmission line, the pump frequency being set to satisfy the following equation, where k is an integer equal to 1 when the nonlinear inductive elements are configured to allow four-wave mixing and 2 for three-wave mixing:

[0028] Here, the pump source is adjusted so that the microwave system acts as a reciprocal adjustable coupler. A conversion process takes place at the nonlinear inductive elements in which a pump is applied in both directions of the transmission line. A "pump" photon in one of the two directions is destroyed, while a "pump" photon in the other direction is created. This allows a "signal" photon to be reflected without changing its frequency. The resulting reflection coefficient is then proportional to the product of the amplitudes of the pumps applied at each end.

[0029] In a particular aspect, the non-linear inductive elements are configured to provide four-wave mixing, the system comprising two microwave sources each injecting a pump signal (Pl, P2) at two frequencies p l2n and (j Pï l ii satisfying the following equation, where v s is the speed of fast mode, and v Pthe slow mode speed supporting both pump signals:

[0030] With such a pump setting, the system here fulfills the role of an amplifier, which is also broadband since the frequencies of the two pumps are independent of that of the signal to be transported. At the level of the non-linear inductive elements, a two-mode "squeezing" process occurs by which a "signal" photon and a "mirror" photon are created by destroying two "pump" photons.

[0031] In a particular aspect, the transmission line comprises a plurality of electrodes, at least one of the electrodes being interrupted by Josephson junction-based connections so as to form at least a portion of the non-linear inductive elements.

[0032] In this so-called discrete configuration, the electrodes can form a plurality of cells, each cell comprising one or more non-linear inductive elements. These Josephson junction-based connections can be simple Josephson junctions, SQUIDs or a dipole of the type configured to allow three-wave mixing such as a SNAIL (Superconducting Nonlinear Asymmetric Inductive element) reduced to one degree of freedom.

[0033] In the case of Josephson junctions as non-linear inductive elements, these can interrupt the electrode(s) by forming a short gap in front of the wavelength of the microwaves (pump, signal to be transported) crossing the transmission line.

[0034] In a particular aspect, the transmission line comprises a plurality of electrodes, at least one of the electrodes being made of a high kinetic inductance material, thereby forming at least a portion of the non-linear inductive elements.

[0035] This configuration allows for a continuous transmission line. High kinetic inductance means a square inductance typically equal to or greater than 0.01 nH.

[0036] According to a particular aspect, the transmission line comprises two electrodes and a ground, the two electrodes having an equal inductance per unit length, the electrodes being connected to the ground by a first capacitance per unit length of the same value, and the electrodes being connected to each other by a second capacitance per unit length, the second capacitance per unit length being greater than the first capacitance per unit length, the non-linear inductive elements being present on the two electrodes to allow wave mixing.

[0037] In such a system, the transmission line is called symmetrical. The transmission line thus symmetrical defines two propagation modes, symmetrical and antisymmetrical. The slow mode corresponds to the antisymmetrical mode and the fast mode corresponds to the symmetrical mode. The fact that the antisymmetrical mode is slower than the symmetrical mode is a consequence of the superiority of the second capacitances per unit length over the first capacitances per unit length.

[0038] In a particular aspect, the transmission line comprises two electrodes and a ground, the electrodes each being connected to ground by a respective first capacitance per unit length, and the electrodes being connected to each other by a second capacitance per unit length, the second capacitance per unit length being less than the respective first capacitances per unit length, the non-linear inductive elements being present on a first electrode, and a second electrode having linear inductances, whereby a microwave traveling in the second electrode and supported by the slow mode induces a magnetic field biasing the non-linear inductive elements within the first electrode, inducing wave mixing at the first electrode.

[0039] In such a configuration, the second electrode is a conventional electrode, and the wave mixing occurs in the first electrode. Such an asymmetric transmission line architecture simplifies the integration of the transmission line into an external microwave circuit. Indeed, the propagation modes being very close to those of decoupled lines (the slow mode being mainly supported by the second electrode and the fast mode by the first electrode), it is possible to directly connect the first electrode to a line of an external circuit, one of which seeks to process the signal, and to inject the pump into the first electrode.

[0040] In a particular aspect, the wavelength of the combination of the signal wave to be transported and the pump is less than half the size of the non-linear inductive elements, whereby the resonant wave mixing induces spectral aliasing.

[0041] In such a case, referred to as a limiting case, it is still possible to obtain a three- or four-wave mixing despite the wavelengths of the pump signals and the signal to be transported which, during mixing, no longer satisfy the so-called continuous limit hypothesis. This makes it possible to obtain the above functionalities, in particular circulation, despite signals with short wavelengths.

[0042] 4. List of figures

[0043] The proposed technique, as well as the various advantages it presents, will be more easily understood, in the light of the following description of illustrative and non-limiting embodiments thereof, and the appended drawings among which:

[0044] [Fig. 1] depicts an exemplary microwave system comprising a pump source, a signal source, and a transmission line according to the present disclosure;

[0045] [Fig. 2] represents an example of an electrical diagram of an elementary cell of the transmission line of figure 1;

[0046] [Fig. 3] represents an energy-momentum diagram of a four-photon exchange at a Josephson junction in the electrical diagram of Figure 2;

[0047] [Fig- 4] represents the simulated spectral decompositions of signals according to their direction of propagation and the side from which they are injected into the transmission line of Figure 1; [Fig. 5] represents a microscope view of the transmission line of Figure 1;

[0048] [Fig. 6] represents the view of figure 5 on which the electrical diagram of figure 2 is superimposed;

[0049] [Fig- 7] represents a photograph of a portion of the transmission line of Figure 1;

[0050] [Fig. 8] represents an asymmetric SQUID cell, an alternative to the cell of Figure 2;

[0051] [Fig. 9] represents a four-wave exchange in a limiting case where the continuous limit assumption is not satisfied; and

[0052] [Fig. 10] shows an example of a cell of a three-electrode transmission line.

[0053] 5. Detailed description

[0054] 5.1. General principle of the invention

[0055] The general principle of the invention consists of a multi-mode non-linear superconducting microwave transmission line forced by at least one pump signal, hereinafter referred to as pump, supported by the slowest of its propagation modes, whereby the transmission line can act as a circulator, amplifier and / or reciprocal adjustable coupler (depending on the set of pumps forcing the line) for a signal to be carried, while exhibiting robust phase matching and exponential performance with the length of the transmission line.

[0056] The transmission line, to do this, comprises a plurality of non-linear inductive elements, each having a kinetic inductance.

[0057] This can be achieved, for example, by using electrodes comprising a plurality of links with high kinetic inductance, hence the non-linear nature of the transmission line. Several examples for achieving this high kinetic inductance will be described below.

[0058] By propagation mode, we mean for a structure invariant in a Z direction (the longitudinal direction of the transmission line) a linear combination of voltages (or currents) between the electrodes such that this combination propagates without deforming.

[0059] Each of the transmission line propagation modes has its own propagation speed through the transmission line. The propagation speed of each mode is intrinsically derived from the physical characteristics of the line (inductance and capacitance per unit length). The propagation speed of a given mode also depends on the characteristics (voltage and current) of the microwave signal(s) traveling through the transmission line and being supported by the given mode.

[0060] The transmission line comprises a plurality of electrodes, and in fact has as many propagation modes as there are electrodes, as will be described below. The line may further comprise an additional electrode, called a ground electrode, taken as a potential reference (or ground plane), not shown (as an electrode) in the figures. A first of these propagation modes has a propagation speed at least twice lower than the propagation speed of at least one second propagation mode. The first mode is hereinafter called the slow mode, and the at least one other second mode is called the fast mode(s).

[0061] Forcing by the pump on the slow mode induces that a signal to be transported supported by the fast mode propagates through the transmission line and can benefit from one or more functionalities (circulation, amplification, adjustable reciprocal coupling), provided that the signal to be transported has a frequency that is a function of that of the pump and the propagation speeds of the modes. The constraints linking the propagation speeds of the modes, the frequency of the pump signal and the frequency of the signal to be transported will be detailed below.

[0062] In other words, by adjusting the pump frequency, it is possible to precisely adjust the frequency at which the signal to be transported benefits from the desired functionality. In fact, this microwave system exhibits robust phase matching. The system's performance is also exponential with the length of the transmission line, as will be seen below.

[0063] To generate the pump, the system includes a microwave source connected to one end of the transmission line. It is possible to place a source at each end of the transmission line, for example to block the flow of the signal to be transported in both directions. The pump and the signal to be transported are not necessarily injected at the same end of the transmission line. In other words, the source of the pump and the source of the signal to be transported are two distinct elements.

[0064] . For the rest of the description and for the sake of brevity, we will describe the case where the pump and the signal to be transported propagate in the same direction, from the source to the other end of the transmission line. The other cases will be discussed at the end of the description, see sections 5.5 and 5.6 in particular.

[0065] A source is generally understood to mean any device that emits microwaves. In particular, the source of the signal to be transported can be any device, including a quantum device, that emits a signal.

[0066] For brevity, these two sources are shown directly connected to the ends of the transmission line, but this is not necessary. In particular, the pump source can be remote from the transmission line and transported to the transmission line through various cables and components. The pump source can thus be at room temperature (around 300 K), while the transmission line is at low temperature (e.g., around 10 mK) to achieve the superconducting effect.

[0067] For the remainder of the description, we call the direction from the input (i.e. the end of the transmission line to which the source of the signal to be transported is connected) to the output the forward direction (the waves propagate in this direction), and the reverse direction the indirect direction (the waves backpropagate, i.e. propagate in the other direction).

[0068] It is possible to characterize the performance of a circulator or an amplifier. For an amplifier, this is simply the gain (output / input amplitude ratio). For a circulator, isolation is defined as the attenuation (for example expressed in dB) in the indirect direction (which is the desired effect: blocking the backpropagation of the signal to be transported) and losses as the attenuation of the signal to be transported in the forward direction (which is what we want to limit, to transmit the signal to be transported as well as possible in the desired direction). The stronger the isolation, the more directional the microwave system.

[0069] 5.2. Dual-mode transmission line

[0070] Reference is made to Figures 1 to 6.

[0071] Figure 1 shows a microwave system 1 comprising a transmission line 10 and a microwave source 12.

[0072] The transmission line 10 here comprises two electrodes 14 and 16 in parallel. The source 12 is connected to the electrodes 14 and 16 at one of the ends 110 of the transmission line. For the sake of brevity, this end will be referred to as the input 110 of the line. It should be noted that a transmission line with two electrodes is described here, but that it is possible to produce a transmission line with three or more electrodes, as will be described below.

[0073] The transmission line 10 comprises a plurality of cells 20 connected in series. This is referred to as a discrete transmission line. The cells 20 are identical in the example described here, although this is not essential. One of these cells 20 is outlined in dotted lines in Figure 1, and is shown in more detail in Figure 2, which is now referred to.

[0074] The cell 20 here comprises two lines 200 and 210 based on Josephson junctions. These lines 200 and 210 based on Josephson junctions play the role of high kinetic inductance connection mentioned above. Each line 200 and 210 based on Josephson junctions is connected to a common ground 30 via a first capacitance 220, 230 of capacity C g . The Josephson junction-based lines 200 and 210 are connected together by a second capacitance 240 of capacity.

[0075] The Josephson junctions of lines 200, 210 each have a so-called Josephson energy, denoted Ej which is specific to each of the Josephson junctions. Here, the energy Ej of the Josephson junctions is inversely proportional to their inductance, i.e. Ej = <Po / Lj où (p Q = <p0 / 2n le quantum de flux réduit.

[0076] The Josephson junction-based lines 200 and 210 of the cells 20 (connected in series) are connected together one after the other, so as to form the electrodes 14 and 16. Formulated differently, the electrodes 14 and 16 are electrodes doped by Josephson junctions facing each other in pairs, and are connected between the electrodes by a series of capacitances 240 of capacity between each pair of facing Josephson junctions.

[0077] As explained above, the electrodes together define as many modes of the transmission line as the number of electrodes, here two modes. Each mode has its own propagation speed, which results from the characteristics of the transmission line. The slowest mode has a velocity through the transmission line at least twice lower than the velocity of the other mode(s). Here, the line defines a slow mode and a fast mode.

[0078] The source 12, connected to the electrodes 14 and 16 at the input 110 of the transmission line 10, is configured to generate through the electrodes 14 and 16 the combination of a pump and a signal to be transported. Each of these two signals has a frequency, which is adjustable. The source can be in two distinct parts, a pump source (denoted P in FIG. 1) and a source of the signal to be transported (denoted S in FIG. 1).

[0079] As mentioned above, the source is not necessarily connected directly to the ends of the transmission line. In practice, the microwave source in particular may be at a constant temperature, while the transmission line is at a low temperature, typically a few milliKelvin. The pump source can then be connected to the end of the line via various cables and components to carry the produced pump signal.

[0080] The pump propagates in slow mode, and the signal to be transported in fast mode. The electrodes are thus crossed by an electromagnetic signal in the microwave frequency.

[0081] 5.3. Symmetrical configuration and four-wave mixing

[0082] Reference is always made to Figure 2.

[0083] Cell 20 is described as symmetrical, that is to say that the energies Ej of the Josephson junctions of lines 200 and 210 are substantially identical (see below).

[0084] At cell 20 (identified by the index n for the following), we define a voltage C> a n at the input of the first electrode 14 and a voltage ^a.n+i at the output of the first electrode 14. Similarly, we define a voltage <P b n at the input of the second electrode 16 and a voltage b,n+i at the output of the second electrode 16.

[0085] We consider the case of a signal to be transported that is weak compared to the pump signal, i.e. e s " e P where e denotes a normalized amplitude.

[0086] The transmission line thus formed, symmetrical due to the equality of the energies Ej of the Josephson junctions, defines two modes of propagation. More precisely, by neglecting the non-linearity of the Josephson junctions, each equivalent to an induction Lj = <Po / Ej (où (p Q = h / (2e) is the reduced quantum flux), the symmetry of the transmission line imposes that the two propagation modes are respectively symmetric (noted A) and antisymmetric (noted Z). We define <P a and C> / , as the generalized phases, i.e. the integral of the voltage with respect to time. Formulated differently, mode A is equal to the average of <P a And <P b , and the Z mode is equal to half the difference between the signals <t> a And <1> & . We actually define the A signals at the input of the cell n and Z n following:

[0087] In the continuous limit (id is a "X where a is the dimension of the cell 20 in the direction of propagation of the signal and is the wavelength of the propagation signal), the capacitance c per unit length and the inductance l per unit length of each of the symmetrical and antisymmetrical modes are:

[0088] The following quantities are defined:

[0089] This results in the following values ​​for the propagation speeds of the symmetric and antisymmetric modes:

[0090] Thus, the factor / z is the square root of the ratio of the propagation speeds of the slow mode and the fast mode, since v-^ / v^ = / z 1 / 2 . Since this factor / z is greater than 1, it follows that mode A is the slow mode, and mode Z is the fast mode. We also deduce that this factor / z is a function of the first C g and second capacitances.

[0091] It is possible to physically represent what happens at the level of a cell in the following way: a voltage of the signal Z n (respectively A n ) charges the capacitors C g arranged in parallel (respectively the capacitor in parallel and the capacitors C g in series) while the signal current Z n (respectively A n ) flows through the two Josephson junctions arranged in parallel (respectively arranged in series). The velocity of mode A (which acts as a pump) can be lowered by fixing C g » . The inventors estimated that this ratio v^ / v^ must be at least equal to 2. In practice, this ratio can be between 2 and 10, and the inventors produced a transmission line with a ratio of 3. The non-linearity of the Josephson junctions allows, at their level, a multi-photon interaction between all the signals propagating through the propagation line, since the Josephson junctions support the currents of all the modes. If the currents are small compared to the critical currents of the Josephson junctions, the dominant effect of the junctions is to allow a four-wave exchange ("four-wave mixing"), shown schematically in Figure 3.

[0092] In operation of system 1, a transfer of an excitation of the signal to be transported (denoted S, propagating in the forward direction) to a mirror signal (idler, denoted I and propagating in the indirect direction, i.e. from the output to the input) occurs at a Josephson junction by depositing two excitations in the pump signal (denoted P) propagating in the forward direction. The signal to be transported S and the mirror signal I propagate according to the Z symmetry (i.e. the Z mode), while the pump signal P propagates according to the A symmetry (i.e. the A mode).

[0093] Conservation of energy and conservation of momentum (and proportional to the wave vector) impose the vector equality of Figure 3: (k being the momentum and > being the pulsation of a photon S, I or P). This implies: kj 4- 2k p — k s

[0094] The pulsation (ro) and frequency (ÛJ / 27T) of a wave are proportional to its energy, hence these terms can be used interchangeably depending on the context.

[0095] Since the signal to be transported S and the mirror signal Z propagate according to the mode Z, we deduce k s = a) s / v-£, k! = —(JÔJ / VZ. In the same way, the pump signal P propagates according to mode A, we deduce k P = a) P / v & . We can visualize in this diagram the speed of propagation of the Z mode as the absolute value of the slope of the lines 300 and 310 (carrying the vectors of the photon respectively of the signal to be transported S and of the mirror signal I). In the same way, the propagation speed of the A mode is visualized (in this diagram) as the absolute value of the slope of the line 320 carrying the vector P of the two photons of the pump signal. The sign of the slope depends on the direction of propagation (direct or indirect), or in other words on the direction of the pulse. The line 300 carrying the vector S of the photon of the signal to be transported and the line 310 carrying the vector I of the photon of the mirror signal are thus symmetrical with respect to the ordinate axis.

[0096] It follows from the above equations and assumptions that this four-wave exchange process is resonant for the following values:

[0097] When these conditions are met, the signal to be transported is reflected and the desired isolation is obtained. Thus, the signal to be transported (at its own frequency) can only circulate in the forward direction, at the cost of the backpropagation of the mirror signal in the indirect direction at another frequency. In other words, the signal to be transported sees its propagation in the indirect direction blocked, which causes a reflection at another frequency corresponding to the mirror signal, hence its name. In many applications, this mirror signal is harmless, unlike the reflection of the signal to be transported if there were no this circulation effect. This also makes it possible to block the backpropagation of noise at the frequency of the signal to be transported.

[0098] Since the slopes of the lines 300, 310 and 320 of figure 3 (i.e. the speeds And v s) are intrinsically linked to the physical characteristics of the transmission line, it is easy to understand that the adjustment of the pump, i.e. the norm of the vector P, imposes the norms and therefore the values ​​of the vectors S and I. In fact, the adjustment of the pump to a given frequency imposes the frequency of the signal to be transported and its mirror signal. Conversely, if we are targeting a particular frequency for the signal to be transported, it is sufficient to adjust the pump frequency to the appropriate value to do so. This results in an easily adjustable microwave isolator with robust phase matching. By reversing the direction of the pump (i.e. the direction in which the pump signal propagates), the direction of the isolation is also reversed.

[0099] As explained above, it is important that the Ej energies of the Josephson junctions are as identical as possible. By as identical as possible, we mean that they are equal to ± 5%, and preferably ± 1%. Thus, the inventors estimate that a difference ("mismatch") of approximately 10% in the Ej energies induces a leakage of approximately -20 dB between the slow mode and the fast mode, i.e. the pump (which is the strongest signal in amplitude) pollutes the signal to be transported (weaker) and vice versa. The inventors have however found that the higher the C Cg ratio, the lower the leakage. More precisely, the higher this C Cg ratio, the greater the spatial overlap of the A and Z modes. It is also possible to reduce this leakage at certain specific frequencies by adapting the energy dispersion of each mode, but at the cost of reduced adjustability of the system.

[0100] It has been seen above that the / z factor and the speeds of the slow and fast modes arise from the physical characteristics of the transmission line, in particular the inductances and capacitances per unit length. However, for a / z triplet, given, several combinations of inductances and capacitances are possible. It is thus possible to choose such a combination which ensures the transmission line a given equivalent impedance, this being in the symmetrical bimodal case above equal to ~ (L / C) for the effective L and C of each mode.

[0101] For example, it may be desirable to obtain an equivalent impedance of 50 fl, to make the transmission line a standard line, like a coaxial cable. Other values ​​are possible, and this relative freedom in the choice of equivalent impedance makes this transmission line architecture suitable for becoming a standard in microwave devices.

[0102] Additionally, one can aim for a matched impedance at the ends of the transmission line, so as to avoid reflections that could harm the unidirectionality of the device. This can be achieved by having the line impedance equal to that of the input and / or output of the line.

[0103] It has been seen that the transmission line, thanks to the two electrodes, defines two "coupled" modes, each dependent on the signal passing through each of the electrodes. This follows, as seen above, from the fact that the electrodes are, at regular intervals, connected by the capacitances. Thus, without these capacitances, each electrode would define its own mode (independently of each other), and the pump signal from one of the electrodes would have no influence on the signal to be transported circulating in the other electrode.

[0104] The symmetrical transmission line thus described can act as a four-port "logical" circulator, with two physical ports (the ends of the line) and two frequencies. If the signal at the second frequency (that of the signal to be transported) is not detected, this is equivalent to dissipating this signal to be transported in a load, whereby system 1 fulfills the function of an isolator.

[0105] The transmission line thus described also allows it to act as an adjustable reciprocal coupler. In fact, by connecting a pump at the same time in both directions of the transmission line, this results in isolation in both directions - hence the term reciprocal - adjusted by the amplitude of each of the pumps.

[0106] To summarize in other words, in this example of a transmission line, it comprises two electrodes and a ground, the two electrodes having an inductance per unit of equal length, the electrodes being connected to the ground by a first capacitance per unit of length of the same value, and the electrodes being connected to each other by a second capacitance per unit of length, the second capacitance per unit of length being greater than the first capacitance per unit of length, the non-linear inductive elements being present on both electrodes to allow wave mixing. This makes it possible to obtain the desired circulation, reciprocal coupling or amplifier effects.

[0107] 5.4. Comparison with TWPAs

[0108] The major challenge in the design of known TWPAs (see part 2: prior art) is to ensure that all waves propagate at exactly the same speed (i.e., obtain phase matching). Indeed, in such TWPAs, if the pump propagates faster than the signal to be transported and the mirror signal, energy / momentum conservation is never achieved, and isolation is not obtained. This often happens in practice, typically because of a weak self-phase modulation of the pump compared to the crossed phase modulation of the signal to be transported and its mirror signal.

[0109] In contrast, the microwave system 1 described here is robust to small disturbances in signal velocity.

[0110] Furthermore, for a TWPA of known type, the conversion is reversible and the amplitude of the signal to be transported and the mirror signal oscillate as a function of the position in the TWPA transmission line. This implies that the conversion is perfect provided that the pump amplitude is adjusted to a precise value allowing a complete conversion at the end of the device, not allowing robust isolation.

[0111] Conversely, the microwave system 1 as described has an isolation that increases exponentially with the optical length of the device, or in other words the length of the line divided by the wavelength. Thus, if we increase the inductance and capacitance values ​​of the line, we increase the length of the line. However, this implies moving away from the continuous limit hypothesis stated above.

[0112] 5.5. Simulation of the symmetric configuration

[0113] Reference is made to Figure 4, which represents the simulated spectral decompositions of signals. The simulated transmission line comprises 400 cells, with 140 f F, C g worth 116 f F, and Lj worth 600 pH. The pump is injected to the left of the transmission line (as in Figure 1 described above), in the direct propagation direction (left to right) at a frequency Pl2n = 3.2 GHz. The phase across the Josephson junctions is of the order of a few tenths of a flux quantum in the simulated implementation. For each scheme, the frequency is on the abscissa and the scale is in GHz, and the number of cells is on the ordinate (from 0 to 400). The signal to be transported is injected at a frequency < s of 8.9 GHz.

[0114] The signal represented on each graph injected to the left (figs. 4A and 4B) or to the right (figs. 4C and 4D) of the transmission line, and its attenuation is represented (visible in gray level) in the direct direction (noted figs. 4A and 4C) or indirect direction (noted figs. 4B and 4D).

[0115] In the case of left injection (ends. 4A, 4B), we measure an isolation of about 20 dB and a negligible loss (on the scale shown). In the case of right injection (ends. 4C, 4D), we measure an isolation of 20 dB and a loss that is also negligible (on the scale shown). We also observe that the total attenuation is exponential with the number of cells (i.e. with the length of the transmission line crossed by the signal).

[0116] 5.6. Pump sets and functionalities As described above, the microwave system described can implement three functionalities.

[0117] The first sought-after functionality is that of circulator (also called isolator), and has already been described extensively above. Circulation is a conversion effect, that is to say that a "signal" photon is converted into a "mirror" photon by mobilizing two "pump" photons (for the case of four-wave mixing) or one "pump" photon (case of three-wave mixing, described below).

[0118] In this circulator, the signal to be transported at a given frequency can only circulate in one direction when the circulation is implemented. We are under the assumptions of a signal to be transported and its mirror signal moving in the opposite direction (i.e. k,. k s < 0, which ensures isolation of the exponential line with the number of cells) and |v = |v s | > 21 v P |, and that the cell size is negligible compared to the wavelengths of the signals (this hypothesis is called "quasi-continuous", see above). By distinguishing the case where the pump and the signal to be transported propagate (i) in the same direction or (ii) in opposite directions, we arrive at the following two constraints on the frequency of the pump signal depending on the case (i) or (ii):

[0119] In these equations (i) and (ii), ro denotes the pulsation (i.e. the frequency to within a factor of 2n) and v the speed of the signal to be transported S or of the pump P, respectively. We thus note that the form of these constraints allows the circulator to have great tunability, i.e. it is easy to adjust the pump frequency for a signal frequency to be transported target OJ S given.

[0120] It is possible to obtain the circulation functionality when | v, | and | v s | differ slightly, provided they are both a factor of 2 greater than the pump velocity. The equations are slightly more complex, but the general idea remains the same.

[0121] It is also possible to obtain circulation without the "quasi-continuous" assumption, although this induces frequency aliasing phenomena.

[0122] The second desired functionality is that of a reciprocal adjustable coupler. This is also a conversion process, in which a pump is applied in both directions of the transmission line. A "pump" photon in one of the two directions is destroyed, while a "pump" photon in the other direction is created. This allows a "signal" photon to be reflected without changing its frequency, i.e. OJ S = ) I and k s = k,. Formulated differently, the signal and its mirror have the same frequency. The laws of conservation of energy and momentum are then written OJ S + = OJ S + and

[0123] — k s + k P = —kp + k s . This induces the following constraint (iii) to obtain an adjustable reciprocal coupling:

[0124] The product of the pump amplitudes in each direction of the reciprocal adjustable coupler sets the coupling value of dipoles connected to both ends of the line.

[0125] The third desired functionality is amplification. This corresponds to a two-mode "squeezing" process by which a "signal" photon and a "mirror" photon are created by destroying two "pump" photons. The inventors have notably obtained amplification using the following process: two pumps PI and P2 are applied at two different frequencies and (i)p2 whose speeds are approximately equal, in opposite directions and at least half lower (in absolute value) than the signal S and its mirror I

[0126] |fpil ~ |vp21 < l^sl / 2 = \vj | / 2

[0127] We then arrive at constraint (iv) on the pump set allowing us to obtain an amplifier:

[0128] It should be noted here that constraint (iv) on the set of pumps to obtain an amplifier links the frequencies of the two pumps PI and P2, but does not link them to the frequency of the signal S to be amplified. This implies that the amplification is obtained whatever the value of > s In other words, the amplifier is broadband.

[0129] We have just seen three functionalities that the microwave system according to the invention makes it possible to obtain. These three functionalities, which can be summarized in the sets of constraints (i), (ii), (iii) and (iv) all have in common that for a given signal frequency to be processed (for circulation, coupling, amplification), it is easy to adjust the pump frequency(ies) to obtain the desired functionality. In other words, phase matching is in all cases easy to obtain, and robust.

[0130] 5.7. Transmission line structure

[0131] Reference is made to Figures 5 to 7. Figures 5 and 6 are microscope views of a prototype transmission line, a photograph of which is shown in Figure 7.

[0132] The prototype transmission line 700 visible in Figure 7 is arranged on a substrate 705. In parallel with the transmission line 700, the substrate supports test contacts 710 (“test pads”), wires 720 connecting the grounds together so as to form a common ground and, on each side of the line, a standard bimodal line 730 (without coupling of the electrodes) which is optional. The test contacts 710 have the function of verifying that there are no bugs in the nanofabrication process of the transmission line. To do this, their resistance is measured when hot, supposedly proportional to the inductance of the junctions when cold. These test contacts 710 are obviously optional in practice.

[0133] This prototype line is obtained here by lithography, which allows for great miniaturization in it, and industrialization of its manufacture. All processes capable of manufacturing small circuits are obviously possible to manufacture such a line.

[0134] We now move to the microscopic level, with particular reference to Figure 5. For the sake of brevity, the electrode at the top of the figure is called the upper electrode, and the electrode at the bottom of the figure is called the lower electrode.

[0135] The line is formed on a substrate 205. The substrate may be made of silicon, or alternatively of sapphire or any other material suitable for this function.

[0136] The prototype electrodes are made of aluminum. The Josephson junctions can be made of superconducting granular aluminum. For superconducting granular aluminum junctions, the transmission line forced by a pump produces the desired functionalities at cryogenic temperatures well below the Kelvin, for example 10 mK.

[0137] It is also possible, as mentioned above, to manufacture these junctions in another superconducting material with high kinetic inductance, such as niobium titanium nitride NbTiN.

[0138] These capacitors of the transmission line thus etched are planar, parallel and alternately arranged, as described below.

[0139] The capacitors coupling the electrodes here comprise, in order, a first aluminum electrode, an alumina (Al2O3) insulator and a second niobium electrode. In order to symmetrize the line, as can be seen in Figure 5, the order can be alternated between two adjacent capacitors (for example aluminum / alumina / niobium then niobium / alumina / aluminum).

[0140] This particular geometry of parallel and alternating plane capacitors makes it possible to obtain large capacities for a reduced footprint on the substrate, which allows for great miniaturization of the transmission line.

[0141] Indeed, in order to reduce the size of the transmission line, it is desirable to obtain relatively low velocities for the propagation modes. As explained above (section 5.3), the velocities of these modes are proportional to l / ^ LjCg). It is therefore desirable that the product LjC g is high. This implies that the transmission line has either a large inductance Lj (that of Josephson junctions, more generally of the superconductor line with high kinetic inductance), or a large capacitance C g .

[0142] The inventors found that a large inductance Lj limits the usable pump power, and therefore the power of the signal to be transported, the latter always having to remain weak compared to the pump signal.

[0143] An example of a transmission line with discrete elements ("lumped") has just been described here. Alternatively, it is possible to produce a transmission line whose geometry and capacitances are defined not by discrete elements but by continuous equivalents, for example two metal tracks side by side, each having a capacitance per unit length.

[0144] 5.8. Asymmetrical line

[0145] Reference is made to Figure 8, which depicts an example of an unbalanced transmission line architecture.

[0146] In this example, a cell 400 comprises a first electrode 410 called a linear electrode “Z” and a second electrode 420 called a SQUID electrode “s”. The coupling between electrodes is here considered sufficiently weak so that the two propagation modes are very close to the decoupled modes, that is to say that the voltage-current couple V h i L in the first electrode 410 at the input of the cell 400 defines a mode, and the other voltage-current pair V s ., i s in the second electrode 420 at the input of the cell 400 defines the other mode. Here, only an inductive coupling is represented, symbolized by the double arrow M.

[0147] The linear electrode 410 and the SQUID electrode 420 have a different structure, hence the name asymmetric.

[0148] More precisely, the linear electrode 410 comprises at the level of the cell 400 a simple inductance denoted L t . The “SQUID” electrode 420 comprises at the level of the cell 400 a pair of identical Josephson junctions 422 mounted in parallel. This pair of identical electrodes in parallel is called a SQUID, referenced 424. The first electrode 410 is connected to ground via a capacitor, and the second electrode 420 is connected to ground via a second capacitor C s .

[0149] The SQUID 424 has the particularity of being sensitive to the flow of the magnetic field inside the loop delimited by the two Josephson junctions 422. This magnetic field can be biased using a direct current, in which case the current i s is composed of a direct current added to the current of the propagating mode s. It is also possible to bias all or part of the transmission line as a whole with an external magnetic field. In this case, it is <p qui est composé d'une valeur de courant continu additionné au flux dû au couplage avec le mode propagatif l dans la première électrode. La combinaison du couple de jonctions, du courant continu et du flux à l'intérieur de la boucle peut être vu comme équivalente à une seule jonction de Josephson, de paramètre E s dynamically dependent on all these factors.

[0150] Depending on the exact DC / external flux configuration and inductive coupling, a 3- or 4-wave mixing is obtained in the s-mode equations, via the "effective" junction parameter E s .

[0151] Thus, apart from the abstraction £ s and A -> Z, we find a system which is very similar to the symmetrical transmission line described above. This involves choosing the physical parameters of the line in such a way as to respect the conditions on the propagation speeds of the two modes, i.e. the latter formula depending on the choice of direct current / external flux.

[0152] The major advantage of this asymmetric architecture is that integration into an external microwave circuit is simplified. Indeed, since the propagation modes are very close to those of decoupled lines, it is possible to directly connect the second electrode 420 to a line of an external circuit, one of which is seeking to process the signal, without having to convert to the asymmetric / symmetric multiconductor modes £ and A of the symmetric line described above. The pump is directly injected into the first electrode 410 of the mode l.

[0153] Formulated in other words and to generalize to the continuous case, the transmission line comprises two electrodes and a ground. The electrodes are each connected to ground by a respective first capacitance per unit length, and the electrodes are connected to each other by a second capacitance per unit length, the second capacitance per unit length being less than the respective first capacitances per unit length. The non-linear inductive elements are present on a first electrode (here electrode 420) of the electrodes and a second electrode (here electrode 410) having linear inductances, whereby a microwave circulating in the second electrode and supported by the slow mode induces a magnetic field biasing the non-linear inductive elements within the first electrode, inducing wave mixing at the first electrode.

[0154] 5.9. Limiting case We have seen so far a transmission line operating under the hypothesis of continuous approximation, that is to say that the length of a cell is much less than the wavelength of the microwaves. However, it is possible with a transmission line as described above (for example, discrete symmetric, see parts 5.2 and 5.3 above), to obtain a four-wave mixture without this hypothesis. Such a mixture is represented in figure 9. The ordinate corresponds to the energy (or the frequency / pulse), and the abscissa to the phase. The curves of the two modes 900 and 910 each represent a dispersion relation, flattened at a frequency noted f cu t-off- We observe that they are approximately linear in the vicinity of the origin of the frequency-phase diagram, which corresponds to the continuous approximation above

[0155] However, for certain frequencies of the signal S and the pump P (two photons) very far from the continuous approximation, a four-wave mixing can occur thanks to spectral folding, because the phases (i.e. the abscissa) are defined modulo 2n, i.e. the values ​​-l-7r and — n coincide. Here we can make an analogy with the stroboscopic effect.

[0156] A similar three-wave mixing can be obtained. A signal S to mirror I conversion is also shown here (pump arrows in a given direction), but the reverse is also possible (it is sufficient to reverse the direction of the arrows of the P vectors of the pump photons).

[0157] From a quantitative point of view, for a line composed of discrete elements (Josephson junctions) Lj and capacities C, we have the formula f cu toff = 2 / (2^^).

[0158] And taking into account the plasma frequency of the junctions, which is represented by an additional capacitance, in parallel with the junction on the electrical diagram, we obtain the following value:

[0159] 5.10. Generalizations

[0160] A symmetrical transmission line was seen, in fact defining a symmetrical mode A and an antisymmetrical mode Z corresponding to the slow mode and the fast mode. However, a non-symmetrical line architecture is perfectly conceivable, and in such a case the slow mode and the fast mode are no longer the symmetrical and antisymmetrical modes. In such a more general configuration, the transmission line forced by the pump nevertheless retains its directional properties described above.

[0161] Reference is made to Figure 10, which represents a case of a three-electrode line. The pump is supported on the slow mode, and the signal to be transported on one of the other modes of the line, the third mode possibly being unused. The central inductance is linear, and has the function of slowing down the slow mode. This configuration with a central inductance and Josephson junctions in parallel makes it possible to obtain a device called RF-SQUID, making it possible to obtain a three-wave mixing in the presence of a direct current.

[0162] Doped electrodes have been seen at Josephson junctions. However, it is possible to replace Josephson junctions with superconductors with high kinetic inductance. Since inductance is nonlinear, this allows the aforementioned wave mixing to be obtained.

[0163] Among the superconductors with high kinetic inductance, one can consider using one of the following materials for an electrode: Gral, NbTin, granular Indium... "High kinetic inductance" can be quantified by "inductance per square greater than approximately 10pH".

[0164] A material with high kinetic inductance is equivalent to a chain containing many high-energy (i.e. low-inductance) junctions. Choosing an electrode made of such a high-kinetic inductance superconductor allows for greater robustness, at the cost of less nonlinearity.

[0165] In other words, doping electrodes with Josephson junctions to obtain kinetic inductance is not the only way to achieve such a result, and some materials natively exhibit kinetic inductance. It is thus possible to manufacture the entire electrode in such a material with high kinetic inductance. This is called a continuous electrode, as opposed to a discrete electrode like the one shown in Figure 2, organized into cells.

[0166] Four-wave mixing has been seen at Josephson junctions, allowing directionality to be obtained. However, it is possible to manufacture a transmission line with certain circuits (SNAIL, RF SQUID, etc.) whose forcing by a pump allows three-wave mixing to be obtained (a signal photon to be transported, a mirror signal photon, and a pump photon). The publication https: / / pubs.aip.org / aip / apl / article / 110 / 22 / 222603 / 33972 describes a dipole with three-wave mixing without four-wave mixing whose implementation is compatible with the transmission line described above. This dipole is called SNAIL, for "Superconducting Nonlinear Asymmetric Inductive element." In such a case, the Josephson junction in Figure 2 can be replaced by such a dipole, and three-wave mixing is obtained under pump forcing and injection of a signal to be transported.The resulting three-wave mixing is similar to the four-wave mixing shown in Figure 5, except that the two "P" arrows (two pump photons are exchanged) are replaced by a single "P" arrow (a single pump photon is exchanged). The equations for the pump sets described in Section 5.6 are similar, except that all occurrences of < must be replaced. P by <D P / 2 since the energy of the pump photon in three-wave mixing is double compared to its four-wave equivalent.< / t>

Claims

CLAIMS 1. Microwave system comprising a superconducting microwave transmission line, a first microwave source capable of generating at least a first signal, called a pump, a second microwave source capable of generating at least a second signal, called a transport signal, said sources each being connected to one end of the transmission line so as to inject their respective signal into the transmission line, the transmission line supporting a plurality of propagation modes, a first mode, called a slow mode, of said plurality of propagation modes having a phase velocity less than half the phase velocity of the other mode(s), called a fast mode(s), of the plurality of propagation modes, said slow mode being forced by the pump, said fast mode(s) supporting the at least one signal to be transported, the amplitude of the at least one signal to be transported being at least ten times lower than the amplitude of the at least one pump,the transmission line comprising a plurality of non-linear inductive elements, each having a kinetic inductance, the transmission line being configured so that, when the frequency of the at least one pump is equal to a function of the frequency of the at least one signal to be transported and the propagation speeds of the propagation modes, all of the modes participate within the non-linear inductive elements in a resonant mixture of the waves of each propagation mode., 2. System according to claim 1, wherein the non-linear inductive elements are configured to provide four-wave mixing or three-wave mixing, and wherein the signal to be transported is injected by the signal source with a frequency > s / 2n, the propagation speed of the slow mode is v P , the propagation speed of the fast mode is v s, the pump being injected by the pump source into the transmission line in the same direction as the signal to be transported, the pump frequency being set to satisfy the following equation, where k is an integer equal to 1 when the nonlinear inductive elements are configured to allow four-wave mixing and 2 for three-wave mixing:

3. The system of claim 1, wherein the non-linear inductive elements are configured to provide four-wave mixing or three-wave mixing, and wherein the signal to be transported is injected by the signal source with a frequency ) S / 2TI, the propagation speed of the slow mode is v P , the propagation speed of the fast mode is v s , the pump being injected by the pump source into the transmission line in the opposite direction to the signal to be transported, the pump frequency <jP ln being set to satisfy the following equation, where k is an integer equal to 1 when the nonlinear inductive elements are configured to allow four-wave mixing and 2 for three-wave mixing:

4. The system of claim 1, wherein the non-linear inductive elements are configured to provide four-wave mixing or three-wave mixing, and wherein the signal to be transported is injected by the signal source with a frequency > s / 2n, the propagation speed of the slow mode is v P , the propagation speed of the fast mode is v s , the pump being injected simultaneously by the pump source on both sides of the transmission line, the pump frequency <j Pln being set to satisfy the following equation, where k is an integer equal to 1 when the nonlinear inductive elements are configured to allow four-wave mixing and 2 for three-wave mixing:

5. System according to claim 1, in which the non-linear inductive elements are configured to provide four-wave mixing, the system comprising two microwave sources each injecting a pump signal (P1, P2) at two frequencies (j P ln and <j Pï ln satisfying the following equation, where v s is the speed of fast mode, and v P the slow mode speed supporting both pump signals: ÛJ P2 6. System according to one of the preceding claims, in which the transmission line comprises a plurality of electrodes, at least one of the electrodes being interrupted by connections based on Josephson junctions so as to form at least part of the non-linear inductive elements.

7. System according to one of the preceding claims, in which the transmission line comprises a plurality of electrodes, at least one of the electrodes being made of a high kinetic inductance material, thus forming at least part of the non-linear inductive elements.

8. System according to one of claims 1 to 7, in which the transmission line comprises two electrodes and a ground, the two electrodes having an equal inductance per unit length, each of the electrodes being connected to the ground by a respective first capacitance having the same first capacitance value per unit length, and the electrodes being connected to each other by a second capacitance having the same second capacitance value per unit length, the second capacitance value per unit length being greater than the first capacitance value per unit length, the non-linear inductive elements being present on the two electrodes to allow wave mixing.

9. A system according to one of claims 1 to 7, wherein the transmission line comprises two electrodes and a ground, the electrodes each being connected to ground by a respective first capacitance per unit length, and the electrodes being connected to each other by a second capacitance per unit length, the second capacitance per unit length being less than the respective first capacitances per unit length, the non-linear inductive elements being present on a first electrode, and a second electrode having linear inductances, whereby a microwave circulating in the second electrode and supported by the slow mode induces a magnetic field biasing the non-linear inductive elements within the first electrode, inducing wave mixing at the first electrode.

10. System according to one of the preceding claims, in which the wavelength of the combination of the signal wave to be transported and the pump is less than half the size of the non-linear inductive elements, whereby the resonant wave mixing induces spectral folding.

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