Compact resonator with shifted higher modes for quantum buses

Abstract

A resonator is based on a coplanar waveguide (CPW) structure that includes a first end portion having a first width and configured to couple to a first qubit, a middle portion having a second width narrower than the first width, and a second end portion having a third width that is wider than the second width and configured to couple to a second qubit.

Description

Background Art Technical Field

[0001] The present disclosure relates generally to superconducting devices and, more particularly, to coupling qubits together.

[0002] Description of Related Technology

[0003] Superconducting quantum computing is an implementation of a quantum computer using superconducting electronic circuits. Quantum computing exploits quantum phenomena for information processing and communication. Various models of quantum computing and quantum simulation exist. The fundamental building block of a gate-based quantum computer is the quantum bit (qubit). A qubit is a generalization of a bit, which has two possible states, but due to its quantum nature, can be in a superposition of the two states. A quantum gate is a generalization of a logic gate, however, it describes the transformation that one or more qubits will undergo after a gate is applied to them, given their initial state.

[0004] There are many different implementations of qubits using superconducting circuits. A common element is the Josephson junction. A Josephson junction is a weak connection between two superconducting electrodes that allows Cooper pairs to tunnel dissipatively between the electrodes. Due to the relationship between the voltage across the junction and the tunneling current, the Josephson junction can in some cases be considered a nonlinear lossless inductor. One of the most successful implementations of a qubit is the so-called Transmon qubit. A Transmon qubit consists of a Josephson junction connected in parallel with a parallel capacitor, thus forming an anharmonic oscillator. The two states of the qubit are then taken to be the two lowest energy levels of the anharmonic oscillator (called the ground state and the first excited state). The energy difference between the two states corresponds to a frequency in the range of several GHz (typically around 5 GHz). To read out the qubit state, a microwave signal is applied to a microwave readout cavity coupled to the qubit (e.g., via a coupling capacitor). The emitted (or reflected) microwave signal passes through multiple thermal isolation stages and low-noise amplifiers, which are used to block or reduce noise and improve the signal-to-noise ratio. Information about the qubit state can be inferred from the amplitude and / or phase of the return / output microwave signal. The microwave signal that carries quantum information about the state of a qubit is typically weak (e.g., on the order of a few microwave photons). To measure this weak signal, a low-noise quantum confinement amplifier (QLA), such as a Josephson amplifier and a traveling-wave parametric amplifier (TWPA), can be used as a preamplifier (i.e., the first amplification stage) at the output of the quantum system to boost the quantum signal while adding the minimum amount of noise dictated by quantum mechanics to improve the signal-to-noise ratio of the output chain. In addition to Josephson amplifiers, certain Josephson microwave components can be used in scalable quantum processors. These components use Josephson amplifiers or Josephson mixers, such as Josephson circulators, Josephson isolators, and Josephson mixers.

[0005] The ability to include more qubits is significant for enabling the potential of quantum computers. Applicants have recognized that improvements can be made along two main dimensions to increase the computational power and reliability of quantum computers. The first is the qubit count itself. The more qubits there are in a quantum processor, the more states that can, in principle, be manipulated and stored. The second is low error rate, which is related to precisely manipulating qubit states and performing sequential operations that provide consistent results rather than just unreliable data. Therefore, in order to improve the fault tolerance of quantum computers, a large number of physical qubits should be used to store logical qubits. In this way, local information is delocalized, making the quantum computer less susceptible to local errors, and measurements are performed in the eigenvalue basis of the qubits, similar to parity checks in classical computers, leading to more fault-tolerant qubits. Summary of the Invention

[0006] According to one embodiment, a resonator includes a coplanar waveguide (CPW) structure including a first end portion having a first width and configured to couple to a first qubit, a middle portion having a second width narrower than the first width, and a second end portion having a third width that is wider than the second width and configured to couple to a second qubit.

[0007] In one embodiment, the first width is substantially equal to the third width.

[0008] In one embodiment, the middle portion of the CPW structure is folded.

[0009] In one embodiment, at least one of the first end portion or the second end portion has an S-structure. The S-structure may surround one or more bonding structures. At least one of the one or more bonding structures may be an under-bump metallurgy (UBM).

[0010] In one embodiment, the first width and the third width are based on widths that provide capacitance and inductance of the first and second ends, respectively, which increase the frequency of modes higher than a fundamental frequency of the CPW structure.

[0011] In one embodiment, the coupling between the first qubit and the first end is capacitive.

[0012] In one embodiment, the length of the first end portion and the length of the second end portion are both shorter than the length of the middle portion.

[0013] In one embodiment, the inductance of the middle portion is higher than the inductance of each of the first end portion and the second end portion.The capacitance of the middle portion may be lower than the capacitance of each of the first end portion and the second end portion.

[0014] In one embodiment, the first mode of the resonator is a factor of 2 or more greater than the fundamental frequency of the resonator.

[0015] According to one embodiment, a quantum bus system includes a first qubit and a second qubit. A CPW structure includes a first end portion having a first width and coupled to the first qubit; a middle portion having a second width narrower than the first width; and a second end portion having a third width wider than the second width and coupled to the second qubit.

[0016] In one embodiment, the middle portion of the CPW structure is folded, and at least one of the first end portion or the second end portion has an S-structure. The S-structure may surround one or more bonding structures.

[0017] In one embodiment, the first wide bandwidth and the third width are based on providing widths of capacitance and inductance of the first end and the second end, respectively, which increase the frequency of modes higher than a fundamental frequency of the CPW structure.

[0018] In one embodiment, the coupling between the first qubit and the first end is capacitive.

[0019] In one embodiment, the length of the first end portion and the length of the second end portion are both shorter than the length of the middle portion.

[0020] In one embodiment, the inductance of the middle portion is higher than the inductance of each of the first end portion and the second end portion, and the capacitance of the middle portion is lower than the capacitance of each of the first end portion and the second end portion.

[0021] In one embodiment, the first mode of the resonator is a factor of 2 or more greater than the fundamental frequency of the resonator.

[0022] According to one embodiment, a method for coupling qubits includes coupling a first end of a CPW structure to a first qubit, coupling a second end of the CPW structure to a second qubit, and configuring a middle portion of the CPW structure to have a width smaller than both the first end and the second end.

[0023] In one embodiment, the frequency of modes higher than the fundamental frequency of the CPW structure is increased by adjusting the geometry of the first end and the second end of the CPW structure.

[0024] In one embodiment, the coupling between the first qubit and the first end is capacitive.

[0025] In one embodiment, the first mode of the CPW structure is configured to be a factor of at least 2 times the fundamental frequency of the resonator.

[0026] These and other features will become apparent from the following detailed description of illustrative embodiments of the invention, which is to be read in connection with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] The accompanying drawings are illustrative embodiments. They do not show all embodiments. In addition or alternatively, other embodiments may be used. Details that may be obvious or unnecessary may be omitted to save space or for more effective description. Some embodiments may be implemented with additional components or steps and / or without all components or steps shown. When the same number appears in different drawings, it refers to the same or similar components or steps.

[0028] Figure 1A An exemplary architecture of a coplanar waveguide transmission line resonator consistent with exemplary embodiments is shown.

[0029] Figure 1B A symbolic representation of a coplanar waveguide transmission line coupled between two load impedances is provided.

[0030] Figure 2 Example resonators carrying half-wavelength and full-wavelength are shown.

[0031] Figure 3A The resonance of a conventional coplanar waveguide transmission line resonator is shown.

[0032] Figure 3B Resonance of a coplanar waveguide transmission line resonator using a paddle structure is shown, consistent with an exemplary embodiment.

[0033] Figure 4A A graph showing selected fundamental frequencies of the resonators is shown.

[0034] Figure 4B Shown Figure 4A A graph of the first harmonic of the resonator.

[0035] Figure 5 An exemplary resonator structure having a middle portion and two end portions having a width wider than the width of the middle portion is shown according to an exemplary embodiment.

[0036] Figure 6 An exemplary coplanar waveguide transmission line resonator having ends surrounding under-bump metallization that may be used for flip-chip packaging is shown, consistent with exemplary embodiments.

[0037] Figure 7 An exemplary plan view of a multi-qubit computing system is shown, consistent with the illustrative embodiments. DETAILED DESCRIPTION

[0038] In the following detailed description, numerous specific details are set forth by way of example to provide a thorough understanding of the relevant teachings. However, it should be understood that the present teachings can be practiced without these details. In other instances, well-known methods, procedures, components, and / or circuits have been described at a relatively high level, without detail, to avoid unnecessarily obscuring aspects of the present teachings.

[0039] The present disclosure relates generally to superconducting devices, and more specifically to efficient readout of interconnected qubits in a quantum processor. The connections between qubits, sometimes referred to herein as qubits, are typically mediated by a bus resonator. A resonator is provided herein that has relatively low losses and mitigates the effects of unwanted extra modes, thereby improving the quality of qubit readout. Furthermore, the bus resonator has a relatively small footprint, which reduces the overall area of ​​the entire quantum processor. Due to this minimization in size, the quantum processor can operate at higher frequencies.

[0040] Example Architecture

[0041] Figure 1A An exemplary architecture 100 of a coplanar waveguide (CPW) transmission line resonator is shown, consistent with the illustrative embodiments. Figure 1B A symbolic representation of a CPW transmission line coupled between two load impedances ZL is provided. The load impedance ZL represents the combined load of the coupling capacitor and the wide portion (L1) of the transmission line. Resonator 100 can be used to connect qubits together. This connection is necessary to generate entanglement between the qubits. For example, there can be a qubit grid connected together by a CPW transmission line resonator between each pair of qubits. Resonator 100 allows these qubits to interact with each other by providing the necessary coupling for qubit entanglement.

[0042] Architecture 100 represents a coplanar waveguide structure that includes an intermediate portion 104, sometimes referred to herein as a main portion, between a first end 102 and a second end 106. First end 102 can be coupled (e.g., capacitively or inductively) to a first qubit, while second end 106 can be coupled (e.g., capacitively or inductively) to a second qubit. As shown, main portion 104 has a suitable length L2, while first end 102 and second end 106 can each have different lengths. In one embodiment, first end 102 and second end 106 have similar lengths L1. In one embodiment, the ratio of length L2 to L1 is at least 0.2. The width W1 of first end 102 and the width W3 of second end 106 are each wider than the width W2 of intermediate portion 104. In one embodiment, the widths of first end 102 and second end 106 are equal (i.e., W1=W3).

[0043] In one embodiment, the actual dimensions of the middle and end portions are based on the middle portion 104 having an inductance greater than the end portions 102, 106 by a predetermined factor (e.g., 2). Additionally, the dimensions of the middle and end portions are also based on the middle portion having a capacitance less than the end portions 102, 106 by a predetermined factor (e.g., 2). Thus, the inductance and capacitance of the resonator are different in different portions of the resonator. The resonator 100 carries a standing microwave mode. The effective (i.e., lumped) circuit parameters of the resonator 100 are obtained by multiplying the waveform per unit length by the inductance L or capacitance C. By optimizing the inductance and capacitance per unit length along the length of the transmission line 170, a bus resonator is obtained that exhibits less interference with higher modes. The following equation can be used to obtain an expression for the resonance condition:

[0044]

[0045] in,

[0046]

[0047]

[0048] The resonance condition of the fundamental mode can be written as:

[0049]

[0050] The resonance condition of the first harmonic can be written as:

[0051] Z2tan(β1l1)=Z1tan(β2l2 / 2) (Equation 5)

[0052] In the above equations, we use the fact that the imaginary part of the impedance is zero for the fundamental mode and the imaginary part of the admittance is zero for the first excited mode, as seen from the center of the resonator structure. The solutions to Equations 4 and 5 can be easily found numerically.

[0053] To better understand the features of the novel resonator 100, it may be helpful to discuss the general operation of the resonator. The resonant frequencies of a resonator (referred to herein as normal modes or simply modes) are generally equally spaced multiples (harmonics) of a lowest frequency, called a fundamental frequency. For example, for a typical λ / 2 resonator, there are multiple modes on the transmission line that implements the resonator, including a fundamental mode and additional frequencies based on multiples of 2 from the fundamental frequency. Unlike traditional resonators where the modes are simple multiples of the fundamental frequency, by using the "paddle" structure of the resonator 100, the higher modes are shifted further out (e.g., by a factor of 2.3 or more), thereby substantially reducing the impact of the higher modes on qubit coupling. For example, for a half-wave resonator, the second mode is pushed higher in frequency than 2X the fundamental frequency. By shifting the higher modes further away from (i.e., higher than) the fundamental frequency in frequency, the qubits experience less undesirable Purcell losses and reduce parasitic coupling between qubits mediated by the higher resonant frequencies.

[0054] Now refer to Figure 2 , which shows an example resonator carrying half wavelength 220 and full wavelength 230. The standing waves on the resonator as a transmission line are only multiples of half wavelength. Figure 2 As shown, the width of the resonator is smaller in the middle 204 than at the ends 202 and 206, creating a "paddle-like" structure for the resonator. In this way, the speed of the traveling wave traveling through the transmission line changes. The boundary conditions change depending on how the standing wave looks, thus affecting the fundamental mode from the secondary excited mode.

[0055] Figure 3A and 3B The resonance of a conventional CPW transmission line resonator 300A and the resonance of a CPW transmission line resonator 300B using a paddle structure are shown, respectively. For the purpose of discussion, the graph 300A is based on a capacitance of 165e-12F / m and an inductance of 2*pi*1e-7H / m. The graph 300A shows that the fundamental frequency (e.g., fundamental mode) is at approximately 7GHz and the frequency of the first harmonic is 2X, i.e., 1.5GHz. For comparison, Figure 3B A graph based on a resonator having a paddle-like structure is shown, wherein the first and second ends of the resonator each have a higher capacitance but a lower inductance than the middle portion. For purposes of discussion and not limitation, C1 = 165e-12 / m, L1 = 2*pi*1e-7 / m, C2 = 105e-12 / m, and L2 = 4*pi*1e-7 / m. The graph shows that there is a significant increase in the spacing between the fundamental frequency and the first harmonic (i.e., in Figure 3B2.4X the frequency in the example shown). It should be noted that although the fundamental frequency is illustrated as being shifted to a lower frequency, it can be set (i.e., controlled) by appropriate selection of the capacitance and inductance of each portion of the resonator. For example, the fundamental frequency can be maintained at 7 GHz, reduced below 7 GHz, or even increased based on design choices.

[0056] A notable feature of this architecture is that the higher modes are shifted by a factor of more than 2X (in Figure 3B The further apart the modes are in frequency, the lower the stray coupling between the fundamental and the corresponding harmonics, providing a substantially better quality readout of the subject qubit.

[0057] Figures 4A to 4B An example transmission line is shown in the simulator for selecting its fundamental frequency and first harmonic, respectively. Figures 4A to 4B In the example shown, the total length of the resonator including the wide portion and the narrow portion is 7 mm. Figure 4A A selected fundamental frequency of 6.4 GHz is shown. Compared to a conventional resonator with a first harmonic at 12.8 GHz, high frequency finite element simulation software (HFSS) indicates that by using the paddle structure discussed herein, the first harmonic is significantly shifted out to 17.5 GHz, providing an additional separation of nearly 5 GHz from the fundamental frequency.

[0058] Now refer to Figure 5 , which shows an exemplary resonator structure having a middle portion 510 and two end portions 502 and 530 having a width wider than the width of the middle portion 510 according to an exemplary embodiment. Figure 5 In the example, the resonator 500 is folded into a compact design. More specifically, the middle section 510 is folded, for example with a meandering structure, thereby saving real estate. In one embodiment, the structure of the narrow portion of the center bar is selected to be scientifically less than 10μm (for example 2μm), while the wide portion of the center bar is selected to be significantly wider than 10μm (for example 30μm). The narrow portion of the resonator is close to half of the total length, while the wider end portions are each close to a quarter of the length. The width of the narrow portion is taken to be within the range that can be reliably produced using optical lithography. Using other manufacturing techniques, such as electron beam lithography, even narrower center lines (sub-micron) can be used. By leaving open areas at the ends, better mode suppression of the slotline mode is provided, with fewer grounding points. In one embodiment, the folded middle section representing the sensing segment can be tightly bent to a pitch of 60μm to minimize the footprint.

[0059] In some embodiments, the first end 502 and / or the second end 530 may have an "S" structure. For example, an S structure may be used to surround the connection points 540(A) to 540(D). For example, each of the connection points 540(A) to 540(D) ​​may receive a wire bond via one or more pads, bumps, wire bonds, etc. It should be noted that the resonator 500 may also be used in the context of a bump bond package. In this regard, Figure 6 An exemplary CPW transmission line resonator is shown with an end portion surrounded by under-bump metallization (UBM) that can be used for flip-chip packaging, consistent with exemplary embodiments. Thus, the larger open area at the end portion of resonator 600 can accommodate relatively large bump bonds 602(A) to 602(D), respectively. The UBM can be evaporated into the region of resonator structure 600 and appropriately stripped.

[0060] Back to Figure 5 In one embodiment, the inductance of middle portion 510 is a factor of 2 or greater than the inductance of end portions 502 and 530. Alternatively or additionally, the capacitance of middle portion 510 is a factor of at least 2 less than the inductance of each of end portions 502 and 530.

[0061] Figure 7 An example plan view of a multi-qubit computing system 700 consistent with an illustrative embodiment is shown. Plan view 700 includes qubits in each corner, where each pair of qubits is coupled together by a corresponding resonator. For example, each qubit (704(A) through 704(D)) can be a transport qubit, which is a superconducting charge qubit that has reduced sensitivity to charge noise. In some embodiments, the two superconductors are capacitively shunted to reduce sensitivity to charge noise. The transport achieves its reduced sensitivity to charge noise by increasing the ratio of the Josephson energy to the charge energy, which can be achieved by using a larger shunt capacitor.

[0062] like Figure 7 As shown, each pair of qubits is coupled together through a CPW transmission line resonator having the structure discussed herein. For example, qubits 704(A) and 704(B) are coupled together through resonator 706; qubits 704(B) and 704(D) are coupled together through resonator 708; qubits 704(D) and 704(C) are coupled together through resonator 710; and qubits 704(A) and 704(C) are coupled together through resonator 712. In various embodiments, each pair of qubits is capacitively or inductively coupled to adjacent resonators. By using Figure 7 The architecture provides a very area-efficient and low-interference multi-qubit computing system 700.

[0063] in conclusion

[0064] The description of various embodiments of the present teachings has been presented for illustrative purposes, but is not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, practical applications, or improvements over existing technologies in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments disclosed herein.

[0065] While the above describes what is believed to be the best state and / or other examples, it should be understood that various modifications may be made therein, and that the subject matter disclosed herein may be implemented in a variety of forms and embodiments, and that the teachings may be applied to many applications, only some of which have been described herein. It is intended that the appended claims claim any and all applications, modifications, and variations that fall within the true scope of the present teachings.

[0066] The components, steps, features, objects, benefits, and advantages discussed herein are illustrative only. None of them, or the discussion thereof, is intended to limit the scope of protection. Although various advantages have been discussed herein, it will be understood that not all embodiments necessarily include all advantages. Unless otherwise indicated, all measurements, values, ratings, positions, amplitudes, sizes, and other specifications in this specification, including those set forth in the appended claims, are approximate and not exact. They are intended to be within a reasonable range consistent with their associated functions and customary practices in the art to which they pertain.

[0067] Many other embodiments are also contemplated. These include embodiments with fewer, additional, and / or different components, steps, features, objects, benefits, and advantages. These also include embodiments in which components and / or steps are arranged and / or ordered differently.

[0068] Although the foregoing has been described in conjunction with exemplary embodiments, it should be understood that the term "exemplary" is merely meant to be an example, rather than the best or optimal. Except as immediately above, nothing stated or illustrated is intended or should be construed as conferring upon the public any component, step, feature, object, benefit, advantage, or equivalent, whether or not recited in the claims.

[0069] Should be understood that, unless specific meaning is elaborated in addition herein, the terms and expressions used herein have the common meaning consistent with these terms and expressions about their corresponding respective investigation and research fields. Relational terms such as first and second etc. can be used only to distinguish one entity or action from another, without requiring or implying any actual such relationship or order between these entities or actions. The term "comprises", "comprising" or any other variation thereof is intended to cover non-exclusive inclusion, so that the process, method, article or device comprising a series of elements not only include those elements, but also can include other elements that are not clearly listed or are inherent to such process, method, article or device. In the absence of further constraints, the element with "a" or "an" in front does not exclude the presence of additional identical elements in the process, method, article or device comprising the element.

[0070] The Abstract of the Disclosure is provided to allow the reader to quickly ascertain the nature of the technical disclosure. It should be understood that it is not intended to interpret or limit the scope or meaning of the claims. In addition, in the foregoing detailed description, it can be seen that various features are grouped together in various embodiments in order to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that the claimed embodiments have more features than those expressly recited in each claim. On the contrary, as reflected in the appended claims, the inventive subject matter lies in less than all the features of a single disclosed embodiment. Therefore, the following claims are hereby incorporated into the Detailed Description, with each claim independently serving as a separately claimed subject matter.

Claims

1. A resonator, comprising: Coplanar waveguide (CPW) structures, including; a first end portion having a first width and configured to couple to a first qubit; a middle portion having a second width narrower than the first width; and a second end portion having a third width wider than the second width and configured to couple to a second qubit, The inductance of the middle portion is higher than the inductance of each of the first end portion and the second end portion. The resonator of claim 1 , wherein the first width is substantially equal to the third width. The resonator according to claim 1 , wherein the middle portion of the CPW structure is folded. The resonator according to claim 1 , wherein at least one of the first end portion or the second end portion has an S-structure. The resonator of claim 4 , wherein the S-structure surrounds one or more bonding structures. The resonator of claim 5 , wherein at least one of the one or more bonding structures is an under bump metallurgy (UBM).

7. The resonator of claim 1 or 2, wherein the first width and the third width are based on widths providing capacitance and inductance of the first end and the second end, respectively, the widths increasing the frequency of modes above a fundamental frequency of the CPW structure.

8. A resonator according to claim 1 or 2, wherein the coupling between the first qubit and the first end is capacitive.

9. The resonator according to claim 1 or 2, wherein the length of the first end portion and the length of the second end portion are both shorter than the length of the middle portion. 10 . The resonator of claim 9 , wherein a capacitance of the middle portion is lower than a capacitance of each of the first end portion and the second end portion.

11. A resonator according to claim 1 or 2, wherein the first mode of the resonator is higher than a factor of 2 of the fundamental frequency of the resonator.

12. A quantum bus system comprising: The first qubit; a second qubit; as well as Coplanar waveguide (CPW) structure, including: a first end portion having a first width and coupled to a first qubit; a middle portion having a second width narrower than the first width; and a second end portion having a third width wider than the second width and coupled to a second qubit, The inductance of the middle portion is higher than the inductance of each of the first end portion and the second end portion.

13. The system of claim 12, wherein: The middle part of the CPW structure is folded; and At least one of the first end portion or the second end portion has an S-structure. The system of claim 13 , wherein the S-structure surrounds one or more engagement structures.

15. The system of any one of preceding claims 12 to 14, wherein the first and third widths are based on widths providing capacitance and inductance of the first and second ends, respectively, the widths increasing the frequency of modes above a fundamental frequency of the CPW structure.

16. The system of any preceding claim 12 to 14, wherein the coupling between the first qubit and the first end is capacitive.

17. The system of any one of preceding claims 12 to 14, wherein the length of the first end portion and the length of the second end portion are both shorter than the length of the middle portion.

18. The system according to any one of the preceding claims 12 to 14, wherein: The capacitance of the middle portion is lower than the capacitance of each of the first end portion and the second end portion.

19. The system of any one of the preceding claims 12 to 14, wherein the first mode of the resonator is above a factor of 2 times the fundamental frequency of the resonator.

20. A method of coupling quantum bits, comprising: coupling a first end of a coplanar waveguide (CPW) structure to a first qubit; coupling a second end of the CPW structure to a second qubit; as well as providing a middle portion of the CPW structure to have a width smaller than a width of the first end portion and a width of the second end portion, The inductance of the middle portion is higher than the inductance of each of the first end portion and the second end portion.

21. The method of claim 20, further comprising increasing the frequency of modes higher than a fundamental frequency of the CPW structure by adjusting the geometry of the first and second ends of the CPW structure.

22. A method according to claim 20 or 21, wherein the coupling between the first qubit and the first end is capacitive.

23. The method of claim 20 or 21, further comprising providing the first mode of the CPW structure to be a factor of at least 2 times the fundamental frequency of the resonator.

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