Dipole elements for superconducting circuits

Abstract

To provide a superconducting dipole element for use in a superconducting microwave quantum circuit.SOLUTION: The present invention relates to an inductive dipole element for a superconducting microwave quantum circuit. The dipole element comprises a DC-SQUID formed by a pair of Josephson junctions shunted by an inductance. The Josephson junctions have equal energy, and the Josephson junctions and the inductance are arranged such that each of the junctions forms a loop with the inductance. The two loops are asymmetrically threaded with external magnetic DC fluxes φext1 and φext2, respectively, such that φext1=π and φext2=0. Parametric pumping is enabled by modulating the total flux φΣ=φext,1+φext,2 threading the dipole element, thereby allowing even-wave mixing between modes that participate in the dipole element with no Kerr-like interactions.SELECTED DRAWING: Figure 1e

Description

[Technical Field]

[0001] The present invention relates generally to nonlinear mixing elements for superconducting microwave quantum circuits, and more particularly to superconducting dipole elements for use in superconducting microwave quantum circuits.

[0002] The field of the invention is more particularly, but not exclusively, that of superconducting circuit-based quantum technologies. [Background technology]

[0003] Electromagnetic radiation waves do not normally interact with each other as they travel through simple media such as vacuum or air. For two or more waves to interact, they must encounter some kind of nonlinearity.

[0004] For example, in the field of nonlinear optics, certain crystals are used to double the frequency of incident radiation through a process called second-harmonic generation. This conversion results from the slightly nonlinear potential energy experienced by electrons in the crystal when influenced by the large electromagnetic field of a laser.

[0005] In the microwave range, such interactions are possible. For example, frequency mixers take two input frequencies and output their sum and difference. To perform this conversion, they rely on carefully designed circuits containing nonlinear electronic components: diodes and transistors.

[0006] In the quantum world, electromagnetic radiation is carried by individual photons. Wave mixing is the process of destroying many incoming photons and generating several new photons, subject to the constraint that the total energy must be conserved. For example, in second harmonic generation, the conversion mechanism destroys two photons of frequency ω and generates one photon of frequency 2ω. Because this conversion involves three photons, it is called a three-wave mixing process. Also, if the input energy is

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[0007] When the desired dynamics does not conserve energy (these interactions are also called non-resonant), the technique of parametric pumping is used. For example, a destroying photons of frequency ω b In the process of generating photons of two frequencies ω a and ω b Conversion between (ω a <ω b ) can be considered. This is a two-wave mixing process, but it is not resonant. To conserve the energy of this dynamics, we instead consider the frequency ω that contributes the lost energy. p =ω b -ω a By adding a third electromagnetic tone, called the pump, we need to use a three-wave mixing process. These types of relationships are also called frequency matching conditions. This new process occurs at a frequency ω a and one pump photon to generate a photon at frequency ω b This results in the generation of a single photon, leading to the overall desired conversion dynamics. Furthermore, increasing the amplitude of the pump tone improves the conversion rate.

[0008] Superconducting circuits are an excellent physical platform for manipulating microwave photons and experimenting with such mixing dynamics at the quantum level. Superconductivity introduces very small dissipation into the circuit, resulting in long-lived electromagnetic modes that can be used to trap microwave photons and enhance their interaction. The mixing capability is provided by a lossless nonlinear element, the Josephson junction. It consists of a thin insulating layer separating two superconducting leads.

[0009] Like conventional electrical circuits, superconducting circuits are designed by arranging capacitors, inductors, and Josephson junctions to tune specific dynamics such as signal amplification or frequency conversion. Another focus of superconducting circuits is realizing highly anharmonic modes whose energy level structure resembles that of atoms. Thus, quantum information can be encoded into two specific addressable energy levels that form a qubit.

[0010] To be more quantitative, the mixing ability of the system is given by the resonant nonlinear term in the Hamiltonian. For example, the aforementioned conversion process is given by

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[0011] In superconducting circuits, these terms are expressed as the Josephson junction energy E J cos(φ), where E J is the Josephson energy of the junction, which can be adjusted during fabrication, and φ is the phase difference across the junction, which is the time integral of that voltage difference. The Josephson junction can be viewed as an inductive element to the first order of φ, and its energy is the inductor E L φ 2 / 2, where E L is the energy of the inductor.

[0012] Annihilation operators a1, ,a n For a circuit hosting n modes, the phase φ is JPEG0007758720000005.jpg5170, where the coefficient φ i is related to how much mode i affects the phase difference across the junction. i It is said that the larger the value, the more modes i are involved in the junction. J cos(φ) causes only even wave mixing.

[0013] Therefore,

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[0014] The challenge with wave mixing in superconducting circuits is that while one mixing term is usually desired for a given application, the expansion of the joint cosine potential results in excess terms. When using parametric pumping, the frequency matching condition can be used to select some of these terms by adjusting the pump frequency. Nevertheless, there are always terms for which the frequency matching condition cannot be verified and which cannot be made non-resonant. These are of the form a i +2 a i 2 and a i + a i a j + a jThese are the Kerr or cross-Kerr terms, or all higher even-order terms generated in the same way. The primary effect of these terms is to shift the frequency of the mode when it is introduced. However, they also have detrimental effects such as limiting the dynamic range of the engineering process and deforming the quantum state. Also, increasing the pump power to enhance the nonlinear interactions can destabilize the system.

[0015] Improving the properties of wave mixing using superconducting circuits is a topic of ongoing research. For example, Sivak et al., Phys. Rev. Applied 11, 054060 (2019) proposed a new dipole called SNAIL (Superconducting Nonlinear Asymmetric Inductive Element) that allows three-wave mixing (and more generally odd-wave mixing) without the Kerr term.

[0016] SUMMARY OF THE INVENTION It is an object of the present invention to provide a dipole element for a superconducting circuit that overcomes the drawbacks of the prior art. [Prior art documents] [Non-patent literature]

[0017] [Non-Patent Document 1] Sivak et al.,Phys.Rev.Applied 11,054060(2019) Summary of the Invention [Means for solving the problem]

[0018] According to one aspect of the present invention, there is provided an inductive dipole element for a superconducting microwave quantum circuit, the dipole element including a DC-SQUID (Superconducting Quantum Interference Device) formed by a pair of Josephson junctions shunted by an inductance, the Josephson junctions having equal energy, and the Josephson junctions and the inductance arranged such that each of the junctions forms a loop with the inductance. ext1 -φ ext2 =φ ext1 +φext2 =π, respectively, and the external static (DC) magnetic flux φ ext1 and φ ext2 is asymmetrically penetrated by φ ext1 = π and φ ext2 = 0, hence the name asymmetrically threaded SQUID (ATS). Parametric pumping is possible by modulating the global magnetic flux that threads the ATS. This pumping allows even-wave mixing between modes involving dipoles, without Kerr-like terms. These latter terms are parasitic in parametric pumping applications, but they cancel out thanks to the ATS symmetry.

[0019] In one embodiment, the inductance comprises a superconducting wire made of flat or granular superconducting material. Thus, the embodiment provides an alternative configuration of the dipole element.

[0020] In another embodiment, the inductance comprises a set of Josephson junctions connected in series with one another, thus providing another alternative configuration of dipole elements.

[0021] In one embodiment, the external DC magnetic flux is applied by a superconducting wire through which both DC and AC currents circulate, and I is φ ext1 and I2 induces φ ext2 , and the superconducting wire is adjacent to the superconducting loop.

[0022] In another embodiment, the superconducting wire is directly inductively connected to the wire of the loop, so that the superconducting bias wire not only shares a mutual inductance with the loop, but also shares a real inductance with the loop wire by being directly connected.

[0023] In another embodiment, the superconducting wire is connected to one input current I Σ is the total magnetic flux φ of the dipole element Σ =φ ext,1 +φ ext,2 and biasing it to a different input current, I Δ is the differential magnetic flux φ of the dipole elementΔ =φ ext,1 -φ ext,2 are positioned to bias the

[0024] In one embodiment, the parametric pumping is performed by generating an oscillating current I Σ Thus, embodiments provide a practical way to couple a pump to the global magnetic flux of a dipole element.

[0025] In another embodiment, φ is modulated with an appropriate modulation phase and amplitude selected to cancel higher order effects. Δ =φ ext,1 -φ ext,2 By modulating the .lambda. .lambda., the parametric pumping capability can be improved.

[0026] In one embodiment, one of the junctions is replaced by a DC-SQUID to account for the natural asymmetry in the junction energy.

[0027] According to another aspect of the present invention, there is provided a superconducting microwave quantum circuit including a dipole element according to embodiments disclosed herein, the dipole element being capacitively shunted and configured to operate at a frequency ω b This nonlinear mode, called the buffer, forms a resonant electromagnetic mode at a frequency ω a is capacitively coupled to a linear resonator at frequency ω p =2ω a -ω b When pumped with, the ATS will

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[0028] Other advantages and features of the present invention will become apparent from the following detailed description of illustrative embodiments thereof and from the accompanying drawings. [Figure 1a] FIG. 1 is a diagram illustrating an example of an inductive dipole including a Josephson junction. [Figure 1b] FIG. 1 is a diagram illustrating an example of an inductive dipole including a Josephson junction. [Figure 1c] FIG. 1 is a diagram illustrating an example of an inductive dipole including a Josephson junction. [Figure 1d] FIG. 1 is a diagram illustrating an example of an inductive dipole including a Josephson junction. [Figure 1e] FIG. 2 is a diagram illustrating a schematic diagram of a dipole element according to an exemplary embodiment of the present invention. [Figure 2] FIG. 10 shows an optical image of an asymmetrically threaded DC-SQUID according to an exemplary embodiment of the present invention. [Figure 3] FIG. 1 is a circuit diagram of an ATS dipole in a practical implementation according to an exemplary embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0029] It is fully understood that the embodiments described below are in no way limiting. Variations of the present invention may be considered in isolation from other features, including only a selection of features described below, if this selection of features confers a technical advantage or is sufficient to differentiate the present invention over the state of the art. This selection includes at least one, preferably functional, feature without structural details, or including only some of the structural details, if only these structural details are sufficient to confer a technical advantage or to differentiate the present invention over the state of the art.

[0030] In particular, all variants and all embodiments described can be combined together if this combination is not objectionable from a technical point of view.

[0031] In the figures, elements common to several figures retain the same reference numerals.

[0032] Figures 1a to 1e show several nonlinear inductive dipoles using Josephson junctions. The phase difference of the dipoles is denoted by φ and corresponds to the time integral of the voltage.

[0033] Figure 1a shows the energy E J cos(φ), where E J is the Josephson junction energy, and φ is the phase difference between the junctions.

[0034] Figure 1b shows the energy 2E J cos(φ ext )cos(φ), and the external magnetic flux φ ext is provided by a coil powered by a current source. DC-SQUIDs produce a tunable energy E J ⇔2E J cos(φ ext ) can be viewed as a single Josephson junction.

[0035] Figure 1c corresponds to an RF-SQUID, and Figure 1d shows a SNAIL element. Both elements require a flux bias circuit (current source and coil) to pass an external magnetic flux through the loop.

[0036] FIG. 1e is a schematic diagram of a dipole element configuration according to one embodiment of the present invention.

[0037] A dipole element, or asymmetrically threaded SQUID (ATS) shown in Figure 1e, comprises a pair of Josephson junctions 2, 3 arranged in parallel to form a loop. In the example shown, the dipole is symmetric, which means that the Josephson junctions 2, 3 have the same Josephson energy.

[0038] The loop 5 of the dipole element is shunted in the middle (electrically coupled in parallel) by the inductance 4. The inductance 4 therefore defines two loops, each including a Josephson junction 2, 3. A dipole according to embodiments disclosed herein therefore includes two loops 6, 8 formed by the Josephson junctions 2, 3 and the inductance 4.

[0039] According to alternative embodiments, the inductance 4 may be constituted by a chain of Josephson junctions, a flat superconducting wire or one made of granular aluminum, or any other inductive device suitable for implementation with the dipole of the present invention.

[0040] The dipole element is a magnetic flux φ penetrating the first loop 6. ext,1 and the magnetic flux φ penetrating the second loop 8 ext,2 The bias is applied by a DC magnetic field of φ ext,1 = π and φ ext,2 In the case of E = 0, the dipole L φ 2 / 2+2E J sinφ Σ It has an energy of the form sinφ, where φ Σ is a small deviation of the total magnetic flux penetrating the ATS. Σ =φ p (p+p + ) so that φ Σ When coupled to , the dipole energy expansion has only even terms, but no Kerr-form terms. Finally, by tuning the pump frequency, the desired mixing terms can be selected. With this dipole, any even-wave mixing process can be designed, with one of the waves being the pump. Another advantage of the ATS is that it is well suited to parametric pumping, due to the infinite possibilities.

[0041] Central inductance 4, E L φ 2The unlimited potential provided by / 2 prevents the system from escaping to higher energy states when it is strongly pumped. This property enables ATSs according to embodiments disclosed herein to be used for sensitive parametric pumping tasks, such as stabilizing quantum states of coupled electromagnetic modes to form long-lived qubits.

[0042] Therefore, ATS dipoles according to embodiments of the present invention avoid problems that can arise in the presence of Kerr nonlinearities by canceling these terms through symmetry.

[0043] Importantly, according to the present invention, the dipole elements and parametric pumping maintain ATS symmetry. In particular, small asymmetries in the junction energy resulting from unavoidable fabrication imprecision result in parasitic Kerr nonlinearities.

[0044] To compensate for this asymmetry, at least one of the junctions 2, 3 can be made flux tunable by replacing it with a SQUID itself.

[0045] In one embodiment, two adjacent superconducting wires represented by coils 7, 9 allow the circuit to be flux biased. When currents I1 and I2 flow through these wires, an external magnetic flux φ ext,1 and φ ext,2 respectively run through the two loops of the ATS. The current source has both a DC component that sets the operating point of the ATS and an AC component at the pump frequency that resonates the desired mixing process. Modulation symmetry is achieved by controlling the relative amplitudes and phases of the two AC components of currents I1 and I2. Considering the dipole energy described in the previous paragraph, the ideal modulation is Σ =φ ext,1 +φ ext,2 In reality, we only deal with the differential flux φ Δ =φ ext,1 -φ ext,2Some corrections must be made by modulating the .pi. tones as well. These corrections are necessary to compensate for the direct driving of the dipole by the pump tones.

[0046] In an alternative embodiment, the superconducting bias line shares a real inductance with the loop instead of the mutual inductance of the standard mutual inductance equivalent circuit.

[0047] In another embodiment, for practicality, the bias line is connected to a current source I Σ and I Δ is the total magnetic flux φ Σ and differential flux φ Δ (See Figure 2.) A superconducting wire coupled to the total magnetic flux is used to transmit the microwave pump to the ATS, generating an oscillating total magnetic flux (φ needed only for correction) that serves as the pumping parameter. Δ (as opposed to

[0048] According to this embodiment, the symmetry of the modulation is such that the pump is φ Σ This is achieved by having an on-chip hybrid (equal split transmission line) that allows the system to address only

[0049] Figure 2 shows an optical image of an asymmetrically threaded DC-SQUID according to an exemplary embodiment of the invention. The electromagnetic dipole is fabricated on-chip and placed within the superconducting circuit (partially shown). In the embodiment of Figure 2, the shunt inductance 4 separating two loops, each containing Josephson junctions 2, 3, is formed by a chain or array of five Josephson junctions (represented by five crosses centered between pairs of identical Josephson junctions 2, 3 that form the SQUID). The left and right magnetic flux lines 10, 11 are connected to the same input via an on-chip hybrid (not shown). The lines are fed by a microwave frequency pump and a DC current I Σ flows, and the total magnetic flux φ Σ The magnetic flux line 12 at the bottom of the image is the current I Δ and each loop carries a magnetic flux of ±φ ΔCombining these two controls, the ATS is biased at the π / 0 asymmetric DC operating point, which is ext1 -φ ext2 =φ ext1 +φ ext2 This means that φ = π ext1,2 are the external magnetic fluxes passing through the two loops respectively. ATS is the frequency ω p The oscillating magnetic flux is the magnetic flux φ Σ It works by modulating the

[0050] An example configuration of a superconducting quantum circuit implementing an ATS according to embodiments disclosed herein is shown in Figure 3. The ATS may be implemented in a circuit that realizes a four-wave mixing interaction, where one of the waves is a pump tone, the frequency of which is selected to make the interaction resonant. In particular, the ATS can be implemented to engineer a two-to-one photon exchange interaction between two microwave resonators.

[0051] Referring to Figure 3, the ATS dipole is shunted with a capacitive device 13 to form a resonant mode 20. This nonlinear resonator 20, called a buffer, is capacitively coupled 14 to a linear resonator 30, called a storage mode, modeled by an LC circuit (inductance 15, capacitor 16).

[0052] ATS is a function of the frequency ω p =2ω a -ω b is pumped by, where ω a , b are the frequencies of the storage cavity 30 and the buffer cavity 20, respectively. This parametric pumping mediates a two-to-one photon exchange interaction between the two modes. The storage mode and the buffer mode resonate in the GHz range.

[0053] "Such exchange interactions are of great importance in applications towards quantum computing and quantum error correction. For example, the interactions can be used to stabilize a new type of qubit called a cat qubit. Cat qubits are promising candidates for hardware-efficient quantum error correction, enabling the protection and stabilization of quantum information."

[0054] Dipoles according to embodiments disclosed herein may be implemented for microwave photodetection applications or for realizing logical operations between qubits.

[0055] While the present invention has been described in conjunction with numerous embodiments, it is evident that many alternatives, modifications, and variations will be apparent to those skilled in the applicable arts. Accordingly, it is intended to embrace all such alternatives, modifications, equivalents, and variations that fall within the spirit and scope of the present invention. [Explanation of symbols]

[0056] 2. Josephson junction 3 Josephson junction 4 Inductance 5 Loop 6 First Loop 7 coils 8 Second Loop 9 coils 10 Magnetic flux lines 11 Magnetic flux lines 12 Magnetic flux lines 13 Capacitive Devices 14 Capacitive coupling 15 Inductance 16 Capacitors 20 Buffer resonator, nonlinear resonator 30 Storage resonators, linear resonators

Claims

1. 1. A superconducting microwave quantum circuit including an inductive dipole element, the dipole element including a DC-SQUID formed by a pair of Josephson junctions shunted by an inductance, the Josephson junctions and the inductance being arranged such that each of the Josephson junctions forms a loop with the inductance, whereby the inductance defines two loops; The superconducting microwave quantum circuit, wherein the circuit is configured to stabilize a cat qubit.

2. The circuit a superconducting wire adjacent to the loop; to circulate a current through the superconducting wire, and thereby an external magnetic flux φ ｅｘｔ，１ through one of the two loops, and the external magnetic flux φ ｅｘｔ，２ a current source configured to pass through the other of the two loops; The circuit of claim 1 , comprising:

3. The circuit of claim 2 , wherein the current source is configured to circulate a current in the superconducting wire having both a DC and an AC component.

4. 4. The circuit of claim 3, wherein the current source and the superconducting wire are each configured to pass different magnetic fluxes through the loop, thereby setting an operating point of the inductive dipole element with a DC component.

5. 4. The circuit of claim 3, wherein the circuit includes a linear resonator, the dipole element is capacitively shunted to form a nonlinear resonator coupled to the linear resonator, and the current source is configured to parametrically pump the dipole element with the AC component at a pump frequency to create a nonlinear interaction between the linear resonator and the nonlinear resonator.

6. The linear resonator has a first resonant frequency ω ａ and the nonlinear resonator has a second resonant frequency ω ｂ and the pump frequency ω ｐ is the difference ω obtained by subtracting the second resonance frequency from twice the first resonance frequency ｐ = 2ω ａ -ω ｂ and the nonlinear interaction is a two-to-one photon exchange.

7. The current source and the superconducting wire are Δ passes through one loop, and -φ Δ One input current I passes through another loop. Σ is the total magnetic flux φ in both loops Σ and another input current I Δ 3. The circuit of claim 2, wherein the is arranged to bias both loops with a differential magnetic flux.

8. The superconducting wire is A pair of superconducting wires connected to the same current source via an on-chip hybrid or split transmission line, the pair of superconducting wires carrying an input current I Σ and the total magnetic flux φ Σ a pair of superconducting wires configured to bias both loops at φ Δ passes through one loop and -φ Δ The input current I Δ and biasing both loops with a differential magnetic flux.

8. The circuit of claim 7, comprising:

9. 8. The circuit of claim 7, wherein the current source and superconducting line are configured to set an operating point of the inductive dipole element by combining a total magnetic flux and a differential magnetic flux such that each magnetic flux is threaded through the loop.

10. The circuit further comprises a linear resonator, the dipole elements being capacitively shunted to form a nonlinear resonator coupled to the linear resonator, and the current source is configured to generate a total magnetic flux φ Σ The input current I Σ 8. The circuit of claim 7, configured to parametrically pump the dipole elements at a pump frequency by oscillating a frequency of .pi.

11. The linear resonator has a first resonant frequency ω ａ and the nonlinear resonator has a second resonant frequency ω ｂ and the pump frequency ω ｐ is the difference ω obtained by subtracting the second resonance frequency from twice the first resonance frequency ｐ = 2ω ａ -ω ｂ and the nonlinear interaction is a two-to-one photon exchange.

12. 11. The circuit of claim 10, wherein the current source is configured to parametrically pump the dipole elements to enable even-wave mixing between modes involving the dipole elements without Kerr-like interactions.

13. 11. The circuit of claim 10, wherein the current source is further configured to parametrically pump the dipole elements by modulating the differential magnetic flux with an appropriate modulation phase and amplitude selected to cancel higher order effects.

14. 10. The circuit of claim 1, further comprising a linear resonator, wherein the dipole elements are capacitively shunted to form a nonlinear resonator coupled to the linear resonator.

15. The circuit of claim 14 , wherein the linear resonator and the nonlinear resonator both resonate in the GHz band.

16. The circuit of claim 1 , wherein the inductance comprises a superconducting wire made of flat or granular superconducting material.

17. 10. The circuit of claim 1, wherein the inductance comprises a chain of Josephson junctions connected in series with one another.

18. 18. The circuit of claim 17, wherein the chain of Josephson junctions comprises five Josephson junctions.

19. 2. The circuit of claim 1, wherein the circuit further comprises a SQUID, and wherein one of the Josephson junctions is comprised by the SQUID.

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