bimodal microwave system

Abstract

The technique targets a multimode, nonlinear, superconducting microwave transmission line 10 forced 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 half that of the other mode(s). This allows the transmission line 10 to act as a circulator, amplifier, and / or reciprocally adjustable coupler (depending on the set of pumps forcing the line) for a signal to be transmitted S, while exhibiting robust phase matching and performance that increases exponentially with the length of the transmission line 10. (See Figure 1 for abbreviations.)

Description

Title of the invention: Bimodal microwave system 1. Scope of the invention

[0001] The invention falls within the field of microwaves, particularly in the field of quantum technologies.

[0002] Microwaves are a frequency range widely used in multiple fields, such as telecommunications or quantum computing. The generation, processing and detection of these signals are carried out using a wide range of active and passive components (mixers, amplifiers, circulators, filters, switches, diplexers...). 2. Prior art

[0003] In the field of quantum technologies, superconducting circuits resonate in the microwave range, and in fact, their control is achieved via microwave signals. In microwave devices potentially suitable for the quantum domain, three functionalities are of interest: amplification, circulation, and reciprocal tunable coupling.

[0004] 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 the measurement fidelity.

[0005] To overcome this limitation, Josephson junction-based amplifiers have been developed over the past twenty years. The performance of such amplifiers approaches the quantum limit; that is, the added noise is as low as is permitted by the limits of quantum physics. However, these amplifiers operate over a narrow frequency band.

[0006] For the past ten years, traveling-wave parametric amplifiers, or TWPAs, have further improved these performance levels, 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 driven by a high-amplitude microwave called a pump. The main obstacle to the realization and performance of TWPAs is ensuring phase matching of the pump with the signal to be amplified and its mirror signal (called the 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, noise backpropagates towards the system emitting the signal to be amplified.

[0007] Circulators are non-reciprocal components that separate signals according to their direction of propagation. The dominant technology for circulators is based on strong magnetic fields, which are incompatible with superconducting circuits. Several implementations compatible with superconducting circuits have been demonstrated, but they suffer from low bandwidth (less than 100 MHz), poor tunability (less than 1 GHz), and limited isolation (< 20 dB).

[0008] In conclusion, an adjustable coupler is a two-port reciprocal element that allows adjustment of the fraction of power incident on one port that is transmitted to the other port or reflected. Adjustable couplers integrated into superconducting circuits now exhibit low bandwidth and limited noise robustness.

[0009] Thus, there is a need for a component capable of providing these functionalities, operating over a wide bandwidth, with high performance and robust phase matching. The invention improves this situation. 3. Description of the invention

[0010] To this end, the invention relates to a low-level microwave system that does not have the aforementioned drawbacks. The invention proposes a microwave system comprising a superconducting microwave transmission line, a first microwave source capable of generating at least one first signal, called the pump, a second microwave source capable of generating at least one second signal, called the transmission signal, said sources each being connected to one end of the transmission line so as to inject their respective signal into the transmission line,

[0011] the transmission line supporting a plurality of propagation modes,

[0012] a first mode, called the slow mode, of said plurality of propagation modes having a phase velocity less than half the phase velocity of the other mode(s), called the fast mode(s), of the plurality of propagation modes,

[0013] said slow mode being forced by the pump, said or said fast modes supporting at least one signal to be carried, the amplitude of at least one signal to be carried being at least ten times less than the amplitude of at least one pump,

[0014] the transmission line comprising a plurality of non-linear inductive elements, each exhibiting a kinetic inductance,

[0015] the transmission line being configured so that, when the frequency of at least one pump is equal to a function of the frequency of at least one signal to be transported and the propagation speeds of the propagation modes, the set of modes participates within the nonlinear inductive elements in a resonant mixing of the waves of each propagation mode.

[0016] This new type of microwave system makes it possible to obtain a transmission of a signal to be carried that can be unidirectional, amplified or both at the same time time, depending on the nature of the pump signal, all while ensuring robust phase matching.

[0017] The unidirectionality of the transmission means that the signal to be carried at the second frequency can circulate 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 direction.

[0018] This unidirectionality is a consequence of the resonant mixing of the waves, itself resulting from the large difference in speeds between the slow and fast modes. The speed (or velocity) of these modes is an intrinsic consequence of the physical characteristics of the transmission line, in particular its inductance and capacitance per unit length.

[0019] This unidirectionality arises from the interactions between microwaves passing through the line and interacting at the level of nonlinear inductive elements, which allow the pump signal to convert the signal to be transmitted into a mirror signal ("idler") with a different frequency than the signal to be transmitted and propagating in the opposite direction. These exchanges occur at the level of these nonlinear inductive elements. The unidirectionality thus blocks the backpropagation towards the source of noise produced by a device to which the transmission line is connected.

[0020] Thus, the signal to be transmitted (at its own frequency) can only travel in the desired direction. Furthermore, noise at the frequency of the signal to be transmitted, propagating in the opposite direction to the signal to be transmitted, is converted into a mirror signal, preventing backpropagation of noise. In effect, the signal to be transmitted has its backpropagation 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 transmitted in the absence of this unidirectionality.

[0021] Thus, when it is desired to transmit a signal of a given frequency via the microwave system, it is sufficient to set the pump to the correct frequency. Furthermore, the signal frequency adjustment band is very wide, typically between 2 and 12 GHz.

[0022] Another advantage of this system is that it produces exponential isolation as a function of the optical length of the transmission line. Transmission line isolation is a way of quantifying its unidirectionality. Indeed, isolation is defined as the ratio between the signal injected at the line input and the signal actually transmitted at the line end, typically expressed in dB. The isolation is described as "robust," meaning that it remains effective even in the event of minor disturbances.

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

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[0031] 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. In one particular aspect, non-linear inductive elements are configured to provide four-wave or three-wave mixing, and in which the signal to be carried is injected by the signal source with a frequency and speed The propagation speed of the slow mode is rP, the propagation speed of the fast mode is rs, the pump being injected by the pump source into the transmission line in the same direction as the signal to be carried, the pump frequency a'P / 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 to 2 for three-wave mixing: k = D' In one particular aspect, non-linear inductive elements are configured to provide four-wave or three-wave mixing, and in which the signal to be carried is injected by the signal source with a frequency and speed The propagation speed of the slow mode is vp, the propagation speed of the fast mode is Ks the pump being injected by the pump source into the transmission line in the opposite direction to the signal to be carried, the pump frequency œp / 2tt 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 to 2 for three-wave mixing: k This pump source tuning allows the system to produce a circulation effect with robust phase matching and exponential isolation with the length of the transmission line. Here, this frequency selection enables a conversion effect at the level of the nonlinear inductive elements. More precisely, a "signal" photon is converted into a "mirror" photon by using two "pump" photons (for the four-wave mixing case) or one "pump" photon (for the three-wave mixing case). This conversion achieves the desired unidirectional effect. This effect can be obtained with a pump and a signal propagating in the same direction or in opposite directions. According to one particular aspect, the non-linear inductive elements are configured to provide four-wave or three-wave mixing, and in which the signal The signal to be transported is injected by the signal source with a frequency Ijt; the propagation speed of the slow mode is vp, and the propagation speed of the fast mode is y

[0032] the pump being injected simultaneously by the pump source on both sides of the transmission line, the pump frequency œP!2TT 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 to 2 for three-wave mixing:

[0033] =

[0034] Here, the pump source is adjusted so that the microwave system acts as a reciprocally adjustable coupler. A conversion process occurs at the level of the nonlinear inductive elements in which a pump is applied in both directions of the transmission line. A "pump" photon in one direction 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.

[0035] According to a particular aspect, the non-linear inductive elements are configured to ensure four-wave mixing, the system comprising two microwave sources each injecting a pump signal (PI, P2) at two frequencies a)Pj2TT and satisfying the following equation, where is the speed of the fast mode, and vp the speed of the slow mode supporting the two pump signals:

[0036] £-1

[0037] With such a pump configuration, the system here fulfills the role of an amplifier, and a broadband one at that, since the frequencies of the two pumps are independent of that of the signal to be transmitted. At the level of the nonlinear inductive elements, a two-mode "squeezing" process occurs whereby a "signal" photon and a "mirror" photon are created by destroying two "pump" photons.

[0038] According to a particular aspect, 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 a part of the non-linear inductive elements.

[0039] In this configuration, known as discrete, the electrodes can form a plurality of cells, each cell comprising one or more nonlinear inductive elements. These Josephson junction-based connections can be simple Josephson junctions, SQUIDs, or a dipole of the type configured to allow three-way mixing waves such as a SNAIL (“Superconducting Nonlinear Asymmetric Inductive eElement”) reduced to one degree of freedom.

[0040] 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) passing through the transmission line.

[0041] According to a particular aspect, the transmission line comprises a plurality of electrodes, at least one of the electrodes being made of a material with high kinetic inductance, thus forming at least a part of the non-linear inductive elements.

[0042] This configuration makes it possible to obtain a continuous transmission line. By high kinetic inductance, we can understand an inductance per square typically equal to or greater than 0.01 nH.

[0043] 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 both electrodes to allow wave mixing.

[0044] In such a system, the transmission line is described as symmetrical. This symmetrical transmission line 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 greater capacitance per unit length of the second capacitance over the first capacitance per unit length.

[0045] According to a particular aspect, the transmission line comprises two electrodes and a ground,

[0046] the electrodes each being connected to ground by a first capacitance per unit length respective, 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 first capacitances per unit length respective,

[0047] 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 level of the first electrode.

[0048] In such a configuration, the second electrode is a conventional electrode, and wave mixing occurs in the first electrode. Such an architecture of An asymmetric transmission line simplifies the integration of the transmission line into an external microwave circuit. Indeed, since the propagation modes are 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.

[0049] According to 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 folding.

[0050] In such a case, referred to as a limiting case, it is still possible to obtain three- or four-wave mixing despite the wavelengths of the pump signals and the signal to be transported no longer satisfying the so-called continuous limit assumption during mixing. This makes it possible to obtain the above functionalities, in particular circulation, despite signals with short wavelengths. 4. List of figures

[0051] The proposed technique, as well as its various advantages, will be more easily understood in light of the following description of illustrative and non-limiting embodiments thereof, and the accompanying drawings, among which:

[0052] - [Fig. 1] represents an example of a microwave system comprising a source pump, signal source and transmission line as detailed in this disclosure;

[0053] - [Fig.2] represents an example of an electrical diagram of an elementary cell of the transmission line of the [Fig.1];

[0054] - [Fig.3] represents an energy-momentum diagram of a four-photon exchange at the level of a Josephson junction of the electrical diagram of [Fig.2];

[0055] - [Fig.4] represents the simulated spectral decompositions of signals according to their direction propagation and the side on which they are injected into the transmission line of the [Fig.1];

[0056] - [Fig.5] represents a microscopic view of the transmission line of the [Fig.1];

[0057] - [Fig.6] represents the view of [Fig.5] on which the diagram is superimposed electrical of the [Fig.2];

[0058] - [Fig.7] represents a photograph of a portion of the transmission line of the [Fig.l];

[0059] - [Fig.8] represents an asymmetric SQUID cell, an alternative to the cell of the [Fig.2];

[0060] - [Fig.9] represents a four-wave exchange in a limiting case where the hypothesis of the The continuous limit is not satisfied; and

[0061] - [Fig. 10] represents an example of a cell in a three-way transmission line electrodes. 5. Detailed description

[0062] 5.1. General principle of the invention

[0063] The general principle of the invention consists of a non-linear multimode superconducting microwave transmission line forced by at least one pump signal, hereinafter referred to as a pump, supported by its slowest propagation mode, whereby the transmission line can act as a circulator, amplifier and / or reciprocally 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.

[0064] The transmission line, for this purpose, comprises a plurality of non-linear inductive elements, each exhibiting a kinetic inductance.

[0065] This can, for example, be achieved using electrodes comprising a plurality of high kinetic inductance links, hence the non-linear nature of the transmission line. Several examples for obtaining this high kinetic inductance will be described below.

[0066] By propagation mode, for a structure invariant in a direction Z (the longitudinal direction of the transmission line) we mean a linear combination of voltages (or currents) between the electrodes such that this combination propagates without deforming.

[0067] Each propagation mode of the transmission line has its own specific propagation speed. 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) traversing the transmission line and carried by that given mode.

[0068] The transmission line comprises a plurality of electrodes, and in fact exhibits as many propagation modes as 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.

[0069] A first of these propagation modes has a propagation speed at least twice as low as the propagation speed of at least a second propagation mode. The first mode is referred to hereafter as the slow mode, and the at least one other second mode is referred to as the fast mode(s).

[0070] Pump forcing in the slow mode means that a signal to be transmitted, supported by the fast mode, propagates through the transmission line and can benefit from one or more features (circulation, amplification, adjustable reciprocal coupling), provided that the frequency of the signal to be transmitted is a function of the pump frequency and the propagation speeds of the modes. The constraints relating the propagation speeds of the modes, the frequency of the pump signal, and the frequency of the signal to be transmitted will be detailed below.

[0071] In other words, by adjusting the pump frequency, it is possible to precisely control the frequency at which the signal to be transmitted benefits from the desired functionality. In fact, this microwave system exhibits robust phase matching. Furthermore, the system's performance increases exponentially with the length of the transmission line, as will be seen below.

[0072] 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 transmitted in both directions. Furthermore, the pump and the signal to be transmitted are not necessarily injected at the same end of the transmission line. In other words, the pump source and the signal source are two distinct elements.

[0073] . For the remainder of the description and for the sake of brevity, we will describe the case where the The pump and the signal to be transmitted propagate in the same direction, from the source to the other end of the transmission line. Other cases will be discussed at the end of the description; see sections 5.5 and 5.6 in particular.

[0074] In general terms, the term "source" means any device that emits microwaves. In particular, the source of the signal to be transmitted can be any device, including quantum devices, that emits a signal.

[0075] For the sake of brevity, these two sources are shown directly connected to the ends of the transmission line, but this is not a necessity. In particular, the pump source can be located away from the transmission line and connected to it via various cables and components. The pump source can thus be at room temperature (around 300 K), while the transmission line is at a low temperature (for example, around 10 mK) to achieve the superconducting effect.

[0076] For the remainder of the description, the direction from the input (i.e. the end of the transmission line into which the source of the signal to be transported is connected) to the output is called the direct direction (waves propagating in this direction), and the reverse direction is called the indirect direction (waves backpropagating, i.e. propagating in the other direction).

[0077] 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 backpropagation of the signal to be transmitted), and losses as the attenuation of the signal to be transmitted in the forward direction (which we wish to limit, in order to transmit the signal as effectively as possible in the desired direction). The stronger the isolation, the more directional the microwave system.

[0078] 5.2. Two-mode transmission line

[0079] Reference is made to figures 1 to 6.

[0080] Fig. 1 represents a microwave system 1 comprising a transmission line 10 and a microwave source 12.

[0081] 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 end 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 two-electrode transmission line is described here, but it is possible to construct a transmission line with three or more electrodes, as will be described below.

[0082] 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 dashed lines in [Fig. 1], and is shown in more detail in [Fig. 2], to which reference is now made.

[0083] The cell 20 here comprises two Josephson junction lines 200 and 210. These Josephson junction lines 200 and 210 act as the high kinetic inductance link mentioned above. Each Josephson junction line 200 and 210 is connected to a common ground 30 via a first capacitance 220, 230 of capacitance Cg. The Josephson junction lines 200 and 210 are connected to each other by a second capacitance 240 of capacitance C1.

[0084] The Josephson junctions of lines 200, 210 each exhibit a Josephson energy, denoted Ej, which is specific to each Josephson junction. Here, the energy Ej of the Josephson junctions is inversely proportional to their inductance, i.e. Ej = (p / Lj where = $J2jt the reduced flux quantum.

[0085] The Josephson junction-based lines 200 and 210 of the cells 20 (connected in series) are linked together one after the other, so as to form electrodes 14 and 16. In other words, 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 capacitance Ct between each pair of facing Josephson junctions.

[0086] As explained above, the electrodes together define as many modes of the transmission line as there are electrodes, here two modes. Each mode has its own propagation speed, which is determined by the characteristics of the transmission line. The slower mode has a velocity through the transmission line at least half that of the other mode(s). Here, the line defines a slow mode and a fast mode.

[0087] The source 12, connected to electrodes 14 and 16 at input 110 of the transmission line 10, is configured to generate, through electrodes 14 and 16, the combination of a pump and a signal to be transmitted. 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 transmitted (denoted S in [Fig. 1]).

[0088] As mentioned above, the source is not necessarily directly connected to the ends of the transmission line. In practice, the microwave source in particular can 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 pump signal it produces.

[0089] The pump propagates in slow mode, and the signal to be transmitted in fast mode. The electrodes are thus traversed by an electromagnetic signal in the microwave frequency.

[0090] 5.3. Symmetrical configuration and four-wave mixing

[0091] Reference is always made to [Fig.2].

[0092] Cell 20 is described as symmetric, that is to say that the energies E j of the Josephson junctions of lines 200 and 210 are substantially identical (see below).

[0093] At the level of cell 20 (identified by the index n for the following), a voltage is defined at the input of the first electrode 14 and a voltage at the output of the first electrode 14. In a similar way, a voltage is defined at the input of the second electrode 16 and a voltage ^n+| at the output of the second electrode 16.

[0094] We consider the case of a signal to be transported that is weak compared to the pump signal, i.e. ^P where f denotes a normalized amplitude.

[0095] The transmission line thus formed, symmetrical due to the equality of the energies Ej of the Josephson junctions, defines two propagation modes. More precisely, neglecting the non-linearity of the Josephson junctions, each equivalent to a

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[0107] induction Lj — (pj Ej (where is 'C "UX 9quantique reduced), the symmetry of the The transmission line dictates that the two propagation modes are respectively symmetric (denoted A) and antisymmetric (denoted S). We define 0 and 4 as the generalized phases, i.e., the integral of the voltage with respect to time. Put another way, mode A is equal to the average of 0 and 4, and mode E is equal to half the difference between the signals. We therefore define the following signals at the cell input: Art = 2' ~ 2 In continuous limit (id esta where a is the dimension of cell 20 in the direction of signal propagation and 2 is the wavelength of the propagating signal), the capacitance c per unit length and the inductance 1 per unit length of each of the symmetric and antisymmetric modes are: _ 2Q _ L l a 1^- 2a c^— ua The following quantities are defined: / i=l + 2CiICg <vg= l / ^LjCg) The following values ​​for the propagation speeds of the symmetric and antisymmetric modes result from this: vE = 0.00 Thus, the factor is the square root of the ratio of the propagation velocities of the mode slow and fast mode, since 1 / 2. Since this factor fi is greater than 1, It follows that mode A is the slow mode, and mode E is the fast mode. It also follows that this factor is a function of the first Cg and second C. capacitances. It is possible to physically represent what happens at the cell level as follows: a signal voltage (respectively A„) charges the capacitors Cg arranged in parallel (respectively the capacitor Cs in parallel and the capacitors Cg in series) while the signal current E„ (respectively A„) 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 setting Cg > C„. The inventors estimated that this ratio vE / vA should be at least equal to 2. In practice, this ratio can be between 2 and 10, and the inventors have created a transmission line with a ratio of 3.

[0108] The non-linearity of Josephson junctions allows, at their level, a multi-photon interaction between all signals propagating through the propagation line, since Josephson junctions support the currents of all modes. If the currents are small compared to the critical currents of the Josephson junctions, the dominant effect of the junctions is to allow four-wave mixing, as shown schematically in [Fig. 3].

[0109] During the operation of system 1, at a Josephson junction, a transfer occurs from 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 reverse direction, i.e., from the output to the input) by depositing two excitations in the pump signal (denoted F) propagating in the forward direction. The signal to be transported $ and the mirror signal Z propagate with E symmetry (i.e., mode S), while the pump signal P propagates with A symmetry (i.e., mode A).

[0110] The conservation of energy and the conservation of momentum (proportional to the wave vector) impose the vector equality in Figure 3: (where & is the momentum and w is the angular frequency of a 5,I or P photon). This implies: [YES] c>)j + 2a>P= œs

[0112] k{ + 2kP = ks

[0113] The angular frequency (t<;!) and the frequency (oj / 2n) of a wave are proportional to its energy, hence the fact that these terms can be used interchangeably depending on the context.

[0114] Since the signal to be carried $ and the mirror signal I propagate in mode E, we deduce ks = k;—- uf / yf- Similarly, the pump signal P se Propagating in mode A, we deduce kp — ü'p / v^-. In this diagram, the propagation speed of mode E can be visualized as the absolute value of the slope of lines 300 and 310 (carrying the vectors of the photon of the signal to be transported, 5, and the mirror signal, Z, respectively). Similarly, the propagation speed of mode A can be visualized (in this diagram) as the absolute value of the slope of 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, the direction of the pulse. Line 300 carrying the vector P of the photon of the signal to be transported and line 310 carrying the vector Z of the photon of the mirror signal are thus symmetrical with respect to the y-axis.

[0115] It follows from the above equations and assumptions that this four-wave exchange process is resonant for the following values:

[0116] 0,,= ((1-^) / (1 + ¾)¼ it)s = ( 1 + ) œP = (vA + v^kp

[0118] When these conditions are met, the signal to be transmitted is reflected, and the desired isolation is achieved. Thus, the signal to be transmitted (at its own frequency) can only propagate in the forward direction, at the cost of backpropagation of the mirror signal in the reverse direction at another frequency. In other words, the signal to be transmitted has its backpropagation blocked, which causes 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 transmitted if this circulation effect did not occur. This also makes it possible to block the backpropagation of noise at the frequency of the signal to be transmitted.

[0119] Since the slopes of lines 300, 310, and 320 in Figure 3 (i.e., the velocities and ve) are intrinsically linked to the physical characteristics of the transmission line, it is readily understood that the pump setting, i.e., the magnitude of the vector P, dictates the magnitudes and therefore the values ​​of the vectors S and I. In fact, setting the pump to a given frequency dictates the frequency of the signal to be transmitted and its mirror signal. Conversely, if a particular frequency is targeted for the signal to be transmitted, it suffices to set the pump frequency to the appropriate value. This results in a microwave isolator that is easy to tune and has 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.

[0120] As explained above, it is important that the energies Ej of the Josephson junctions be 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 about 10% in the energies Ej induces a leakage of about -20 dB between the slow and fast modes, i.e., the pump (which is the signal with the higher amplitude) pollutes the signal to be carried (which is weaker), and vice versa. The inventors have, however, observed that the higher the Cj / Cg ratio, the lower the leakage. More precisely, the higher this Cj / Cg ratio, the greater the spatial overlap of the A and E 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 system adjustability.

[0121] It has been seen above that the factor fl and the speeds of the slow and fast modes derive from the physical characteristics of the transmission line, in particular the inductances and capacitances per unit length. However, for a given triplet rA' %, Several combinations of inductances and capacitances are possible. It is therefore possible to choose such a combination which ensures a given equivalent impedance for the transmission line, this being in the symmetric bimodal case above equal to ^(l / c) for the effective L and C of each mode.

[0122] For example, it may be desirable to obtain an equivalent impedance of 50 Ω, 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.

[0123] Furthermore, an impedance matched to the ends of the transmission line can be targeted to avoid reflections that could impair the unidirectionality of the device. This can be achieved by having a line impedance equal to that of the line's input and / or output.

[0124] 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 electrode. This follows, as seen above, from the fact that the electrodes are connected at regular intervals by capacitances. Thus, without these capacitances, each electrode would define its own mode (independently of the other), and the pump signal from one of the electrodes would have no influence on the signal to be transmitted circulating in the other electrode.

[0125] The symmetrical transmission line thus described can act as a circulator with four "logical ports", with two physical ports (the ends of the line) and two frequencies. If the signal is not detected at the 2nd frequency (that of the signal to be transmitted), this is equivalent to dissipating the signal to be transmitted into a load, thereby fulfilling the function of an isolator.

[0126] The transmission line thus described also allows it to act as an adjustable reciprocal coupler. Indeed, by connecting a pump simultaneously in both directions of the transmission line, isolation results in both directions — hence the term reciprocal — adjusted by the amplitude of each of the pumps.

[0127] To summarize in other words, in this example of a transmission line, it comprises two electrodes and a ground, the two electrodes having equal inductance per unit length, the electrodes being connected to 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 both electrodes to to allow wave mixing. This makes it possible to obtain the desired circulation, reciprocal coupling or amplification effects.

[0128] 5.4. Comparison with TWPAs

[0129] The major challenge in the design of known TWPAs (see Part 2: Prior Art) is to ensure that all waves propagate at precisely the same speed (i.e., achieve phase matching). Indeed, in such TWPAs, if the pump propagates faster than the signal to be carried and the mirror signal, energy / momentum conservation is never achieved, and isolation is not obtained. This often occurs in practice, typically due to weak self-phase modulation of the pump compared to the crossed-phase modulation of the signal to be carried and its mirror signal.

[0130] On the contrary, the microwave system 1 described herein is robust to small disturbances in signal velocity.

[0131] Furthermore, for a known type of TWPA, the conversion is reversible, and the amplitude of the signal to be transmitted and the mirror signal oscillate as a function of the position in the transmission line of the TWPA. This implies that the conversion is perfect provided that the pump amplitude is adjusted to a precise value allowing complete conversion at the end of the device, but not allowing robust isolation.

[0132] Conversely, the microwave system 1 as described exhibits insulation that increases exponentially with the optical length of the device, or in other words, with the length of the line divided by the wavelength. Thus, if the inductance and capacitance values ​​of the line are increased, the length of the line is also increased. However, this implies a departure from the continuous limit assumption stated above.

[0133] 5.5. Simulation of the symmetrical configuration

[0134] Reference is made to Figure 4, which represents the simulated spectral decompositions of signals. The simulated transmission line comprises 400 cells, with values ​​of 140 fF, Cg of 116 fF, and Lj of 600 pH. The pump is injected to the left of the transmission line (as in Figure 1 described above), in the forward propagation direction (left to right) at a frequency of a)P / 2Φ = 3.2 GHz. The phase across the Josephson junctions is on the order of a few tenths of a flux quantum in the simulated implementation. For each scheme, the frequency is on the x-axis and the scale is in GHz, and the number of cells is on the y-axis (from 0 to 400). The signal to be transmitted is injected at a frequency of 8.9 GHz.

[0135] The signal represented on each graph is injected to the left (Figs. 4A and 4B) or to the right (Figs. 4C and 4D) of the transmission line, and its attenuation is shown (visible in greyscale) in the direct direction (noted , figs. 4A and 4C) or indirect direction (noted , figs. 4B and 4D).

[0136] In the case of left-hand injection (fins 4A, 4B), an isolation of approximately 20 dB and negligible loss are measured (at the scale shown). In the case of right-hand injection (fins 4C, 4D), an isolation of 20 dB and also negligible loss are measured (at the scale shown). It is also observed that the total attenuation is exponential with the number of cells (i.e., with the length of the transmission line traversed by the signal).

[0137] 5.6. Pump sets and features

[0138] As described above, the described microwave system can implement three functionalities.

[0139] The first desired functionality is that of a circulator (also called an 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 a four-wave mixture) or one "pump" photon (case of a three-wave mixture, described below).

[0140] In this circulator, the signal to be transported at a given frequency can only circulate in one direction when circulation is activated. We assume that the signal to be transported and its mirror signal travel in the opposite direction (i.e., kt.ks < 0, which ensures isolation of the exponential line with the number of cells) and |V / | = || > 2|vP|, and that the cell size is negligible compared to the wavelengths of the signals (this assumption 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): [0M1) (<) [01«l (ii) =

[0143] In these equations (i) and (ii), w denotes the angular frequency (i.e., the frequency to within a factor of 2) and t the velocity of the signal to be transported S or of the pump P, respectively. It can thus be seen that the form of these constraints allows the circulator to exhibit a high degree of tunability, i.e., it is easy to adjust the pump frequency P for a given target signal frequency.

[0144] It is possible to obtain the circulation functionality when |V / | and |^j differ slightly, provided that they are both greater than the pump velocity by a factor of 2. The equations are slightly more complex, but the general idea remains the same.

[0145] It is also possible to obtain circulation without the "quasi-continuous" assumption, although this induces frequency folding phenomena.

[0146] The second desired feature 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 direction is destroyed, while a "pump" photon in the other direction is created. This makes it possible to reflect a "signal" photon without changing its frequency, i.e., ~ εᵢ and kᵢ - kᵢ. Put another way, the signal and its mirror have the same frequency.

[0147] The laws of conservation of energy and momentum can then be written as ü-s+mp = + ù)p and -ks + kP- -kP + ks. This induces the following constraint (iii) for obtaining an adjustable reciprocal coupling:

[0148] M

[0149] The product of the pump amplitudes in each direction of the reciprocal adjustable coupler fixes the coupling value of dipoles connected to the two ends of the line.

[0150] The third desired feature 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 according to the following process: two pumps PI and P2 are applied at two different frequencies and with approximately equal velocities, in opposite directions, and at least half as high (in absolute value) as the signal S and its mirror 1:

[0151]

[0152] This leads to constraint (iv) on the pump set, which allows for obtaining an amplifier:

[0153] £-1 (iv) Ü)PI- ^(1^2

[0154] It should be noted here that the 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 amplification is obtained regardless of the value of (,\ In other words, the amplifier is broadband.

[0155] We have just seen three functionalities that the microwave system according to the invention makes possible. These three functionalities, which can be summarized by 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 easy to achieve in all cases and robust.

[0156] 5.7. Transmission line structure

[0157] Reference is made to Figures 5 to 7. Figures 5 and 6 are microscopic views of a prototype transmission line, a photograph of which is shown [Fig.7].

[0158] The prototype transmission line 700 visible [Fig.7] is arranged on a substrate 705. In parallel with the transmission line 700, the substrate supports test contacts 710 (“testpads”), 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 electrode coupling) which is optional.

[0159] The test contacts 710 are used to verify that there are no defects in the nanofabrication process of the transmission line. This is done by measuring their hot resistance, which is supposedly proportional to the inductance of the cold junctions. These test contacts 710 are, of course, optional in practice.

[0160] This prototype line is obtained here by lithography, which allows for significant miniaturization and industrialization of its manufacture. All processes suitable for manufacturing small circuits are obviously conceivable for manufacturing such a line.

[0161] We now consider the microscopic level, with reference in particular to [Fig.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.

[0162] The line is formed on a substrate 205. The substrate can be made of silicon, or alternatively of sapphire or any other material suitable for this function.

[0163] The electrodes of the prototype are here made of aluminum. The Josephson junctions can be made of superconducting granular aluminum. For junctions made of superconducting granular aluminum, the pump-forced transmission line produces the desired functionalities at cryogenic temperatures much below 1 Kelvin, for example 10 mK.

[0164] It is also conceivable, as mentioned above, to manufacture these junctions in another superconducting material with high kinetic inductance, such as niobium titanium nitride NbTiN.

[0165] These capacitors of the transmission line thus engraved are planar, parallel and in alternating arrangement, as described below.

[0166] The capacitors coupling the electrodes here comprise, in order, a first aluminum electrode, an alumina insulator (AZ2Q0) and a second niobium electrode. In order to symmetrize the line, as can be seen in [Fig. 5], the order can be alternated between two adjacent capacitors (for example aluminum / alumina / niobium then niobium / alumina / alumina).

[0167] This particular geometry of parallel and alternating planar capacitors makes it possible to obtain large capacities for a reduced footprint on the substrate, which allows for a large miniaturization of the transmission line.

[0168] Indeed, in order to reduce the congestion of the transmission line, it is desirable to obtain relatively low velocities for the propagation modes. As explained above (part 5.3), the velocities of these modes are proportional to 1 / H, so it is desirable that the product LjCg be high. This implies that the transmission line exhibits either a large inductance Lj (that of the Josephson junctions, more generally of the superconducting line with high kinetic inductance), or a large capacitance Cg.

[0169] The inventors have observed that a large inductance Lj limits the usable pump power, and therefore the power of the signal to be carried, the latter always having to remain low compared to the pump signal.

[0170] We have just described here an example of a transmission line with discrete ("lumped") elements. It is alternatively possible to make a transmission line whose geometry and capacitances are defined not by discrete elements but by continuous equivalents, for example two side-by-side metallic tracks, each having a capacitance per unit length.

[0171] 5.8. Asymmetric line

[0172] Reference is made to [Fig.8], which describes an example of a non-symmetric transmission line architecture.

[0173] In this example, a cell 400 comprises a first electrode 410 called the linear electrode "1" and a second electrode 420 called the SQUID electrode "s". The coupling between the electrodes is considered here to be sufficiently weak that the two propagation modes are very close to decoupled modes; that is, the voltage-current couple q in the first electrode 410 at the input of the cell 400 defines one mode, and the other voltage-current couple V# is in the second electrode 420 at the input of the cell 400 defines the other mode. Here, only inductive coupling is represented, symbolized by the double arrow M.

[0174] The linear electrode 410 and the SQUID electrode 420 have a different structure, hence the name asymmetric.

[0175] More specifically, the linear electrode 410 comprises, at cell 400, a simple inductance denoted Lt. The "SQUID" electrode 420, for its part, comprises, at cell 400, a pair of identical Josephson junctions 422 connected 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 capacitance Ch and the second electrode 420 is connected to ground via a second capacitance Cs.

[0176] The SQUID 424 has the particularity of being sensitive to the flux 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 is composed of a direct current added to the current of the propagating mode s.

[0177] 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 $ which consists of a DC current value added to the flux due to coupling with the propagation mode1 in the first electrode. The combination of the junction pair, the DC current, and the flux inside the loop can be viewed as equivalent to a single Josephson junction, with parameter Es dynamically dependent on all these factors.

[0178] Depending on the exact configuration of direct current / external flux and inductive coupling, a 3 or 4 wave mixture is obtained in the equations of the s mode, through the parameter of the "effective" junction Es.

[0179] Thus, apart from the abstraction of E s and A, we find a system that is very similar to the symmetrical transmission line described above. This implies choosing the physical parameters of the line so as to respect the conditions on the propagation speeds of the two modes, that is to say

[0181] this last formula depending on the choice of direct current / external flux.

[0182] The major advantage of this asymmetric architecture is that its 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 that seeks to process the signal, without having to convert to the asymmetric / symmetric E and A multiconductor modes of the symmetric line described above. The pump, for its part, is directly injected into the first electrode 410 of the modez.

[0183] Put another way, and to generalize to the continuous case, the transmission line comprises two electrodes and a ground. Each electrode is connected to ground by a 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 first respective capacitances per unit length. The nonlinear inductive elements are present on a first electrode (here electrode 420) and a second electrode (here electrode 410) exhibiting 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.

[0184] 5.9. Limiting case

[0185] So far, we have seen a transmission line operating under the assumption of the continuous approximation, that is, the length of a cell is much smaller than the wavelength of microwaves. However, it is possible, with a transmission line such as the one described above (for example, discrete symmetric, see parts 5.2 and 5.3 above), to obtain a four-wave mixture without this assumption. Such a mixture is shown in Figure 9. The ordinate corresponds to the energy (or 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 denoted f cutoff. 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.

[0186] However, for certain frequencies of the signal S and the pump P (two photons) far removed from the continuous approximation, a four-wave mixture can occur due to spectral folding, since the phases (i.e., the abscissa) are defined modulo 2tf, i.e., the values ​​+Æ and coincide. An analogy can be drawn here with the stroboscopic effect.

[0187] An analogous three-wave mixture 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 suffices to reverse the direction of the arrows in the P vectors of the pump photons).

[0188] From a quantitative point of view, for a line composed of discrete elements (Josephson junctions) L; and capacities C, we have the formula r lr J cutoff ~ 2 / ^^6)

[0189] And taking into account the plasma frequency of the junctions, which is represented by an additional capacitance C j, in parallel with the junction on the electrical diagram, we obtain the following value:

[0190] ft + eutojj t \ VJ '■ J /

[0191] 5.10. Generalizations

[0192] A symmetric transmission line has been observed, effectively defining a symmetric mode A and an antisymmetric mode E corresponding to the slow and fast modes. However, a non-symmetric line architecture is perfectly conceivable, and in such a case the slow and fast modes are no longer the symmetric and antisymmetric modes. In such a more general configuration, the line forced transmission by the pump however retains its directional properties described above.

[0193] Reference is made to [Fig. 10], which represents a three-electrode transmission line. The pump is supported in the slow mode, and the signal to be transmitted is carried in one of the other modes of the line, the third mode being optional. The central inductor is linear and serves to slow down the slow mode. This configuration, with a central inductor and parallel Josephson junctions, makes it possible to obtain a device called RF-SQUID, which allows for three-wave mixing in the presence of a direct current.

[0194] Doped electrodes at Josephson junctions have been observed. However, it is conceivable to replace the Josephson junctions with superconductors of high kinetic inductance. Since the inductance is non-linear, this makes it possible to obtain the aforementioned wave mixing.

[0195] Among the high kinetic inductance superconductors, 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 lOpH”.

[0196] 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 reduced nonlinearity.

[0197] Put another way, 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 fabricate the entire electrode from such a high-kinetic-inductance material. This is then referred to as a continuous electrode, as opposed to a discrete electrode like the one shown [Fig. 2], which is organized into cells.

[0198] Four-wave mixing has been observed at Josephson junctions, enabling directionality. However, it is possible to fabricate a transmission line incorporating certain circuits (SNAIL, RF SQUID, etc.) whose pumping forces produce a three-wave mixture (a photon of the signal to be transmitted, 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 but without four-wave mixing, the implementation of which 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 by pumping and injecting a signal to be transmitted. The resulting three-wave mixture is similar to the four-wave mixture shown in Figure 5, except that the two arrows "P" (two pump photons are exchanged) are replaced by a single arrow "P" (only one pump photon is exchanged). The pump set equations described in Section 5.6 are similar, except that all occurrences of must be replaced by φP / 2 since the energy of the pump photon in the three-wave mixture is double that of its four-wave equivalent.

Claims

1.

2. Demands A microwave system comprising a superconducting microwave transmission line, a first microwave source capable of generating at least one first signal, called the pump, a second microwave source capable of generating at least one second signal, called the signal to be carried, 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 the slow mode, of said plurality of propagation modes having a phase velocity less than half the phase velocity of the other mode(s), called the 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 carried, the amplitude of the at least one signal to be carried being at least ten times less than the amplitude of the at least one pump,the transmission line comprising a plurality of nonlinear inductive elements, each exhibiting a kinetic inductance, the transmission line being configured such that, when the frequency of at least one pump is equal to a function of the frequency of at least one signal to be carried and the propagation speeds of the propagation modes, the set of modes participates within the nonlinear inductive elements in a resonant mixing of the waves of each propagation mode. A system according to claim 1, wherein the nonlinear inductive elements are configured to provide four-wave or three-wave mixing, and wherein the signal to be carried is injected by the signal source with a frequency of / 2tt, the propagation speed of the slow mode is vp, the propagation speed of the fast mode is of the pump being injected by the pump source into the transmission line in the same direction as the signal to be carried, the pump frequency (jpI2it) being set to satisfy the following equation, where k is an integer equal to 1 when the inductive elements Non-linear systems are configured to allow four-wave mixing and two-wave mixing: k = T--me

3. A system according to claim 1, wherein the nonlinear inductive elements are configured to provide four-wave or three-wave mixing, and wherein the signal to be carried is injected by the signal source with a frequency (ωp / 2π), the propagation speed of the slow mode is vp, the propagation speed of the fast mode is vp, and the pump is injected by the pump source into the transmission line in the opposite direction to the signal to be carried, the pump frequency Wpp 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 to 2 for three-wave mixing: k Wp = 1 / (1π)

4. A system according to claim 1, wherein the nonlinear inductive elements are configured to provide four-wave or three-wave mixing, and wherein the signal to be carried is injected by the signal source with a frequency of λiT, the propagation speed of the slow mode is νp, the propagation speed of the fast mode is νs, 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 to 2 for three-wave mixing: λiPikiMs

5. A system according to claim 1, wherein the nonlinear 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 Wp1 / 2tt and Wp2 / 2tt satisfying the following equation, where is the velocity of the fast mode, and vp the speed of the slow mode supporting both pump signals: §4

6. A system according to any one of the preceding claims, wherein the transmission line comprises a plurality of electrodes, at least one of the electrodes being interrupted by links based on Josephson junctions so as to form at least a portion of the nonlinear inductive elements.

7. A system according to any one of the preceding claims, wherein the transmission line comprises a plurality of electrodes, at least one of the electrodes being made of a material with high kinetic inductance, thus forming at least a part of the nonlinear inductive elements.

8. A system according to any one of claims 1 to 7, wherein 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, nonlinear inductive elements being present on both electrodes to permit wave mixing.

9. A system according to any one of claims 1 to 7, wherein the transmission line comprises two electrodes and a ground, the electrodes each being connected to the ground by a first capacitance per unit length respective, 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 first capacitances per unit length respective, the nonlinear 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 nonlinear inductive elements within the first electrode, inducing wave mixing at the first electrode. 28

10. A system according to any one of the preceding claims, wherein the wavelength of the combination of the signal to be carried and the pump is less than half the size of the nonlinear inductive elements, whereby the resonant wave mixing induces spectral folding.