Harmonic Suppression in a Distributed Resonator for a Traveling-Wave Parametric Amplifier

Abstract

A traveling-wave parametric amplifier (TWPA) transmission line having an improved bandwidth includes one or more unit cells. Each unit cell includes a capacitor and a Josephson junction connected in series and leading to ground, configured to provide inductance and non-linearity. One or more distributed resonators are coupled to the transmission line. The distributed resonators are configured to (i) use a first harmonic mode of the resonator for resonance phase matching in the transmission line and (ii) prevent the second mode of the resonator from interacting with the intermodulation product of the transmission line.

Description

Technical Field

[0001] The present disclosure generally relates to superconducting signal amplifiers.

Background Art

[0002] A traveling wave parametric amplifier (TWPA) is a commonly used superconducting amplifier. The TWPA includes a non-linear LC transmission line, and Josephson junctions provide inductance and non-linearity. Since the TWPA has high gain, low noise, and sufficient bandwidth and saturation power for the readout of multiple qubits, it is a main amplifier in some quantum computing systems.

[0003] When amplifying a signal tone using a TWPA having a pump tone, the TWPA amplifies the signal tone by taking in power from the pump tone. This is a four-wave mixing process in which two pump photons are converted into a new photon at the signal frequency and a photon at the idler frequency. In this process, both energy (frequency) and momentum (wave vector) should be conserved. For small signals, these two criteria can be satisfied simultaneously. However, for large pump tones, a phase shift depending on power may be formed, and thus the criterion of the wave vector is no longer satisfied. To satisfy both criteria simultaneously, resonant phase matching is introduced. Specifically, by periodically adding microwave resonators throughout the transmission line, the dispersion of the transmission line is changed so that the two criteria are satisfied simultaneously at the pump tone near the frequency of the microwave resonator. These microwave resonators are called dispersion resonators or dispersion features, and they are components that are added to the TWPA transmission line to add dispersion when amplifying a signal using the TWPA transmission line.

[0004] Unfortunately, there may be an additional dip in gain that reduces the available bandwidth of the TWPA due to the addition of the distributed resonator. This dip occurs where the frequency of the second mode of the distributed resonator is phase - matched with a four - wave mixing process that generates higher - order intermodulation products. SUMMARY OF THE INVENTION

[0005] Some embodiments of the present disclosure provide a traveling - wave parametric amplifier (TWPA) transmission line having an improved bandwidth. The transmission line includes one or more unit cells. Each unit cell includes a capacitor and a Josephson junction connected in series that provide inductance and non - linearity and lead to ground. One or more distributed resonators are coupled to the transmission line. Each distributed resonator is configured to (i) use the first harmonic mode of the resonator for resonant phase matching in the transmission line and (ii) prevent the second mode of the resonator from interacting with the intermodulation products of the transmission line. Preventing the second mode from interacting with the intermodulation products of the device removes a dip in signal gain, thereby expanding the available bandwidth of the TWPA. In other words, the resonator is configured to remove or suppress coupling to the second mode in the frequency range of the intermodulation products of the TWPA.

[0006] In some embodiments, the node - coupling method is used to suppress coupling to the second mode. Specifically, the resonator is a quarter - wavelength resonator coupled to the transmission line at the voltage node of the second mode of the resonator via a coupling capacitor. The resonator includes a first transmission line leading from the voltage node of the second mode to ground and a second transmission line leading to an open circuit. The first mode is the fundamental frequency of the resonator, and the second mode is the third harmonic of the fundamental frequency of the resonator.

[0007] In some embodiments, the conversion resonator method is used, and thus the resonator is configured to detune or decouple the second mode of the resonator by several gigahertz so that the second mode of the resonator does not match the intermodulation product of the transmission line. The resonator is an eighth-wavelength transmission line resonator including an inductor leading to the ground and is in parallel with an open-circuit transmission line. In some embodiments, the open-circuit transmission line is a conversion equivalent capacitor functioning as a capacitor shunted to the ground. In some embodiments, the conversion equivalent capacitor has an effective capacitance determined based on (i) the characteristic impedance of the open-circuit transmission line and (ii) the target dispersion frequency of the resonator, while the inductor leading to the ground has an inductance set according to the target dispersion frequency of the resonator and the effective capacitance of the conversion equivalent capacitor.

[0008] The TWPA is useful for scaling up quantum computers. By using either the node coupling method or the conversion resonator method, the gain dip can be removed or pushed outside the range of the intermodulation (IM) product, thereby expanding the usable readout bandwidth of the TWPA. The expanded usable readout bandwidth of the TWPA can be used to either increase the number of multiplexed qubits measured by a single TWPA or to further space out the readout resonators to reduce crosstalk. Removing the gain dip facilitates detuning between the readout resonator and the qubit, reducing parcel loss in the quantum system.

[0009] The foregoing summary is intended to function as a concise introduction to some embodiments of the present disclosure. It is not meant to be an introduction or overview of all inventive subject matter disclosed in this document. The following detailed description of the invention, and the drawings referenced in the detailed description of the invention, further describe the embodiments described in the summary and other embodiments. Accordingly, the summary, the detailed description of the invention, and the drawings are provided to understand all embodiments described in this document. Further, the claimed subject matter is not limited by the illustrative details in the summary, the detailed description of the invention, and the drawings, but rather, the claimed subject matter is defined by the appended claims since the claimed subject matter can be embodied in other specific forms without departing from the spirit of the subject matter.

Brief Description of the Drawings

[0010] The drawings are of exemplary embodiments. They do not illustrate all embodiments. Other embodiments may be used additionally or alternatively. Details that may be apparent or unnecessary for space conservation or more effective illustration may be omitted. Some embodiments may be practiced using additional components or steps and / or without using all of the illustrated components or steps. When the same numerals are shown in different drawings, they refer to the same or similar components or steps.

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DETAILED DESCRIPTION OF THE INVENTION

[0021] In the following embodiments of the invention, many specific details are set forth by way of example in order to provide a thorough understanding of the relevant teachings. However, it should be apparent that the present teachings may be practiced without these details. In other instances, well-known methods, procedures, components, and / or circuits are described relatively summarily, without detail, in order to avoid unnecessarily obscuring aspects of the present teachings.

[0022] The backbone of a Josephson TWPA is a non - linear transmission line. A unit cell of the transmission line includes a capacitor leading to ground and a Josephson junction connected in series, thereby providing both inductance and non - linearity. For a signal tone of frequency f s propagating through a TWPA driven by a pump tone of frequency f p , the TWPA amplifies the signal tone by taking power from the pump tone. This is a four - wave mixing process in which two pump photons are converted into a new photon at the signal frequency and a photon at the idler frequency. (The idler is a microwave tone that is a result of the non - linear amplification process; in the four - wave mixing process, the idler frequency is f i = 2f p - f s .)

[0023] In this process, both energy (frequency) and momentum (wave - number vector) are conserved, i.e., 2ω p = ω s + ω i and 2k p = k s + k i should hold (ω p , ω s , and ω i are the angular frequencies of the pump, signal, and idler respectively, while kp , k s , and k j is the wave number vector of the pump, signal, and idler, respectively). For small signals, these two criteria can be satisfied simultaneously. However, for large pump tones, a phase shift that depends on power may be formed, and thus the criterion of the wave number vector (2k p = k s + k i ) is no longer satisfied.

[0024] To satisfy both criteria simultaneously, resonant phase matching is introduced. Specifically, by periodically adding microwave resonators throughout the transmission line, the dispersion k(ω) of the transmission line is changed so that the two criteria (conservation of energy and momentum) are satisfied simultaneously at the pump tone near the frequency of the microwave resonator. These microwave resonators are called dispersion resonators or dispersion features, and they are components that are added to the TWPA transmission line to add dispersion when using the TWPA transmission line to amplify a signal.

[0025] FIG. 1 shows a TWPA100 with dispersion resonators inserted to maintain the constraints regarding conservation of energy and momentum, consistent with an exemplary embodiment. As shown, the TWPA transmission line 100 is characterized by a chain of Josephson junction inductive elements (labeled "J") connected in series to a capacitive element (labeled "C") leading to ground. The TWPA transmission line 100 also has periodic dispersion resonators inserted, including resonators 111 and 112 coupled to the transmission line 100 by coupling capacitors.

[0026] There are multiple different techniques for implementing dispersion in a TWPA. FIGS. 2A - 2B show an exemplary quarter - wavelength resonator 210 used as a dispersion feature in a TWPA transmission line 200. As shown, the TWPA transmission line 200 is characterized by a chain of Josephson inductive elements (labeled "J") and capacitive elements (labeled "C"). A periodic dispersion resonator including the quarter - wavelength resonator 210 can also be inserted into the TWPA transmission line 200. The advantage of this type of dispersion feature is that it is easy to make all the resonators uniform. The problem associated with this method is that the quarter - wavelength resonator has multiple (harmonic) modes.

[0027] FIG. 2A shows the first mode (dotted line) and the second mode (dashed line) of the quarter - wavelength resonator 210. The first or lowest mode (fundamental frequency or first harmonic) of the quarter - wavelength resonator is used for resonance phase matching in the TWPA. The next - lowest mode (second mode) of the quarter - wavelength resonator is a harmonic at three times the fundamental frequency, or the third harmonic. (If a half - wavelength resonator were used, the second mode should be twice the fundamental frequency, or the second harmonic.) In this implementation of the TWPA, there is a dip in the gain that reduces the usable bandwidth of the TWPA. This dip occurs where the second mode of the dispersion resonator 210 phase - matches with the generation of higher - order intermodulation products.

[0028] Figures 3 and 4 show plots of the performance of the TWPA200 based on the parameter values shown in FIG. 2B. FIG. 3 shows the dip in transmission in the second mode of the dispersion resonator for an unpumped TWPA. This figure includes a plot of the simulation of the scattering parameters of an unpumped TWPA with a dispersion resonator of 8.75 GHz. This plot also shows the interaction with the second mode at 26 GHz. FIG. 4 shows the dip in gain in the second mode of the dispersion resonator for a pumped TWPA. This figure includes a simulation of the scattering parameters of the TWPA when pumped at 8.55 GHz. As shown in FIGS. 3 and 4, two normal dips in gain can be observed around 8.5 GHz due to the stop band generated by the dispersion resonator around the pump frequency. However, due to the generation of the IM product around the second mode of the dispersion resonator, there is an additional gain dip around 7.9 GHz.

[0029] During the parametric amplification process, unwanted tones are also generated in the intermodulation products of the signal and pump frequencies. Unfortunately, the higher harmonics of the dispersion resonator have parasitic interactions with the intermodulation (IM) products, thereby increasing the coupling to the IM products such as ω IM3 =2ω p +ω s and ω IM5 =4ω p -ω s etc. (ω IM3 and ω IM5 are the angular frequencies of the third and fifth order intermodulation products). This occurs under the phase matching conditions (i) k IM3 =2k p +k s and (ii) k IM5 =2k p +k i (where k IM3 and k IM5 are the wave vectors of the third and fifth order intermodulation products). This is the signal frequency f sappears in the device as a dip in gain (shown by dip 410 in FIG. 4 labeled "parasitic dip"). This dip causes the available signal bandwidth of the device to decrease by up to 300 MHz or more, thereby limiting the range of readout frequencies that are compatible with the TWPA. Also, since the dip frequency depends on the pump frequency f p this limits the range of available pump frequencies that are compatible with the desired bandwidth.

[0030] Some embodiments of the present disclosure provide a method for eliminating this dip in the signal gain of a TWPA. In some embodiments, the dip in gain is eliminated by removing or suppressing coupling to the second (or higher order) harmonic. Specifically, at the voltage node of the second mode, the resonator is coupled to the transmission line. This prevents the transmission line from interacting with the second mode of the distributed resonator, thereby avoiding the dip in gain. This method is also referred to as the node coupling method.

[0031] FIGS. 5A-5B conceptually illustrate harmonic suppression by node coupling. These figures show a TWPA transmission line 500 characterized by a chain of Josephson inductive elements (labeled "J") and capacitive elements (labeled "C"). The TWPA transmission line 500 also has periodic distributed resonators such as a quarter-wavelength resonator 520 inserted therein. As shown, the quarter-wavelength resonator 520 is coupled to the TWPA transmission line 500 at the voltage node 505 of the second mode by a coupling capacitor 510. This node coupling prevents the intermodulation product of the transmission line 500 from interacting with the second mode of the resonator, thus removing the parasitic gain dip. (Alternatively, the node coupling may be achieved by inductively coupling the resonator to the transmission line at the current node of the second mode.)

[0032] Figures 6 and 7 respectively show the corresponding plots of FIGS. 3 and 4 in which the gain dips in the TWPA are removed by node coupling. These plots relate to simulations performed based on the parameter values shown in FIG. 5B. FIG. 6 shows the removal of the dip in the transmission in the TWPA by node coupling. This figure includes a plot of the scattering parameters of the unpumped TWPA (TWPA500) where the distributed resonator is at 8.75 GHz and the second mode node (e.g., voltage node 505) is coupled. As shown, the first harmonic of the distributed resonator is still at 8.75 GHz. However, the dip at 26 GHz due to the second mode is significantly reduced. FIG. 7 shows the removal of the gain dip in the pumped TWPA (TWPA500) by node coupling. This figure includes a simulation of the scattering parameters of the TWPA when the TWPA is pumped at 8.55 GHz for gain and the second mode node is coupled. As shown, the gain dip at 7.9 GHz due to the second mode of the distributed resonator has disappeared.

[0033] In some embodiments, by configuring the distributed resonator to detune or decouple the second or higher harmonics by several gigahertz, the gain dip is eliminated while maintaining the ease and uniformity of fabrication of the quarter-wavelength resonator. Specifically, the quarter-wavelength resonator is replaced with an inductor that is in parallel with the Richard's transformation equivalent of the capacitor to form an eighth-wavelength transmission line resonator. Thus, the eighth-wavelength transmission line resonator is also referred to as a Richard's transformation resonator.

[0034] Figures 8A - 8B show an exemplary quarter - wavelength transmission line resonator. These figures show a TWPA transmission line 800 characterized by a chain of Josephson inductive elements (labeled "J") and capacitive elements (labeled "C"). The TWPA transmission line 800 also has periodic distributed resonators such as the conversion resonator 810 inserted therein. As shown, the TWPA 800 is coupled to the conversion resonator 810 via a coupling capacitor 815. The conversion resonator 810 is formed by an inductor 820 (going to ground) in parallel with a quarter - wavelength transmission line 830 connected to an open circuit. The open - circuit transmission line 830 effectively functions as a capacitor shunted to ground and thus also as a transformation equivalent capacitor. Thus, this method is also referred to as the Richard conversion resonator method. The higher - order modes of this resonator structure no longer match the IM product of the parametric process in the TWPA transmission line 800, thereby avoiding gain dips. This method has the additional benefit that the length of the transmission line section is shortened, and thus the overall footprint of the resonator is reduced.

[0035] The inductor 820 and the conversion equivalent capacitor 830 are parameterized to detune or decouple second - order (or higher - order) harmonics. Specifically, when combined in parallel with the inductor 820 to form the resonant circuit 810, the first (harmonic) mode of the open - circuit transmission line 830 is at the desired or target dispersion frequency. The effective capacitance C0 of the open - circuit transmission line 830 is set by the characteristic impedance Z0 of the line and the desired dispersion frequency ω of the resonator. Specifically: res as follows:

Number

[0036] The inductance in the resonator (e.g., inductor 820) is set as follows:

Number

[0037] These values may be corrected by the value of the coupling capacitance C c (e.g., the capacitance of coupling capacitor 815).

[0038] Figures 9 and 10 each show corresponding plots of FIGS. 3 and 4 in which the gain dip in the TWPA800 is removed by the Richard conversion resonator. These plots relate to simulations performed based on the parameter values shown in FIG. 8B. FIG. 9 shows the removal of the dip in the transmission in the unpumped TWPA by the Richard conversion resonator. The figure includes a plot of the scattering parameters of the unpumped TWPA where the dispersion resonator is at 8.75 GHz, but here the dip due to the second mode is moved from 27 GHz within the IM product range to beyond 30 GHz outside the IM product range. FIG. 10 shows the removal of the gain dip in the pumped TWPA by the Richard conversion resonator. As shown, when the TWPA is pumped (e.g., at 8.55 GHz) for gain, the dip at 7.9 GHz has disappeared.

[0039] The TWPA can be useful for scaling up quantum computers. By using either the node coupling method (described above by referring to FIGS. 5 - 7) or the conversion resonator method (described by referring to FIGS. 8 - 10), the gain dip can be removed or pushed outside the IM product range, thereby expanding the usable readout bandwidth of the TWPA. The expansion of the usable readout bandwidth of the TWPA can be used to either increase the number of multiplexed qubits measured by a single TWPA or to further space out the readout resonators to reduce crosstalk. The removal of the gain dip facilitates detuning between the readout resonator and the qubit, reducing parcel loss in the quantum system. (The parcel effect is the enhancement or loss of the natural emission rate of the quantum system due to its environment.)

[0040] The descriptions of the various embodiments of the present teachings have been presented for illustrative purposes, but are not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terms used herein are chosen to best explain the principles of the embodiments, the practical application, or the technological improvements found in the marketplace, or to enable others skilled in the art to understand the embodiments disclosed herein.

[0041] In the foregoing, the best mode and / or what is considered to be other examples have been described, but it is to be understood that various modifications may be made thereto, that the subject matter disclosed herein may be implemented in various forms and examples, and that the teachings may be applied to many uses and only a portion thereof is described herein. The following claims are intended to claim any and all uses, modifications, and variations within the true scope of the present teachings.

[0042] The components, steps, features, objects, benefits, and advantages described herein are merely illustrative. None of these, nor the descriptions associated therewith, are intended to limit the scope of protection. Although various advantages have been described herein, it will be understood that not all embodiments necessarily include all advantages. Unless otherwise stated, all measurements, values, ratings, positions, sizes, dimensions, and other specifications described herein, including those in the following claims, are approximate and not exact. They are intended to have a reasonable range consistent with the functions to which they relate and what is customary in the technical fields to which they relate.

[0043] Many other embodiments are contemplated. These include embodiments having fewer, additional, and / or different components, steps, features, objectives, benefits, and advantages. These also include embodiments in which the components and / or steps are arranged and / or ordered in a different manner.

[0044] Although the foregoing has been described in conjunction with exemplary embodiments, it is understood that the term "exemplary" means merely an example and not the best or optimal. Except as specifically described immediately above, nothing described or illustrated is intended to or should be construed to make public any component, step, feature, objective, benefit, advantage, or equivalent, whether or not it is recited in the claims.

[0045] It is to be understood that the terms and expressions used herein have the ordinary meaning ascribed to such terms and expressions in each respective field of investigation and study, unless otherwise specifically defined herein. Relative terms such as first, second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. The terms "comprises," "comprising," or any other variation thereof are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. An element preceded by "a" or "an" does not, without further constraints, preclude the existence of additional identical elements in the process, method, article, or apparatus that includes that element.

[0046] A summary of the disclosure is provided to enable a reader to quickly ascertain the nature of the technical disclosure. The summary is presented with the understanding that it is not to be used to interpret or limit the scope or meaning of the claims. Also, in the detailed description of the invention that follows, it is to be understood that various features are grouped together in various embodiments for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed embodiments require more features than are expressly recited in each claim. Rather, as reflected in the following claims, the inventive subject matter lies in less than all of the features of a single disclosed embodiment. Accordingly, the following claims are hereby incorporated into the detailed description of the invention, with each claim standing on its own as a separately claimed subject matter.

Claims

1. A transmission line having one or more unit cells, each unit cell including a capacitor and a Josephson junction connected in series and leading to ground, configured to provide inductance and non-linearity, a transmission line, and One or more resonators coupled to the transmission line, each resonator being configured to (i) use a first mode of the resonator for resonance phase matching in the transmission line, and (ii) prevent a second mode of the resonator from interacting with an intermodulation product of the transmission line, a resonator, and An apparatus comprising.

2. The apparatus according to claim 1, wherein the resonator is configured to remove or suppress coupling of the second mode between the transmission line and the resonator in a frequency range of the intermodulation product of the transmission line.

3. The apparatus according to claim 1 or 2, wherein the resonator is coupled to the transmission line at a voltage node of the second mode of the resonator via a coupling capacitor.

4. The apparatus according to claim 3, wherein the resonator is a quarter-wavelength resonator, the first mode is a fundamental frequency of the resonator, and the second mode is a third harmonic of the fundamental frequency of the resonator.

5. The apparatus according to any one of claims 1 to 4, wherein the resonator is coupled to the transmission line at a current node of the second mode of the resonator via an inductive coupling.

6. The apparatus according to any one of claims 1 to 5, wherein the resonator is configured to detune or decouple the second mode by several GHz so that the second mode of the resonator does not match the intermodulation product of the transmission line.

7. The apparatus according to any one of claims 1 to 6, wherein the resonator has an inductor leading to the ground and is in parallel with an open-circuit transmission line.

8. The apparatus according to claim 7, wherein the resonator is an eighth-wavelength transmission line resonator.

9. The apparatus according to claim 7 or 8, wherein the open-circuit transmission line is a conversion equivalent capacitor operating as a capacitor shunted to the ground.

10. The apparatus according to claim 9, wherein the conversion equivalent capacitor has an effective capacitance determined based on (i) a characteristic impedance of the open-circuit transmission line and (ii) a target dispersion frequency of the resonator.

11. The inductor leading to the ground has an inductance set according to the target dispersion frequency of the resonator and the effective capacitance of the conversion equivalent capacitor, the apparatus according to claim 10.

12. Providing a transmission line including one or more unit cells, each unit cell including a capacitor and a Josephson junction connected in series and configured to provide inductance and non-linearity and leading to the ground, providing a transmission line; Coupling one or more resonators to the transmission line; Configuring the resonator such that (i) a first mode of the resonator is used for resonance phase matching in the transmission line and (ii) a second mode of the resonator does not interact with an intermodulation product of the transmission line; A method comprising:

13. The method according to claim 12, wherein the resonator is configured to remove or suppress coupling of the second mode between the transmission line and the resonator in a frequency range of the intermodulation product of the transmission line.

14. The method according to claim 12 or 13, wherein the resonator is a quarter-wavelength resonator coupled to the transmission line at a voltage node of the second mode of the resonator via a coupling capacitor.

15. The method according to any one of claims 12 to 14, wherein the resonator includes a first transmission line leading from the voltage node of the second mode to the ground and a second transmission line leading to an open circuit.

16. The method according to any one of claims 12 to 15, wherein the resonator is configured to detune or decouple the second mode by several GHz so that the second mode of the resonator does not match the intermodulation product of the transmission line.

17. The method according to any one of claims 12 to 16, wherein the resonator is an eighth-wavelength transmission line resonator including an inductor leading to the ground and being in parallel with an open circuit transmission line.

18. The method according to claim 17, wherein the open circuit transmission line is a conversion equivalent capacitor operating as a capacitor shunted to the ground.

19. The method according to claim 18, wherein the conversion equivalent capacitor has an effective capacitance determined based on (i) a characteristic impedance of the open circuit transmission line and (ii) a target dispersion frequency of the resonator.

20. The method according to claim 19, wherein the inductor leading to the ground has an inductance set according to the target dispersion frequency of the resonator and the effective capacitance of the conversion equivalent capacitor.

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