CN113239548A - Mixing modeling method based on high-temperature superconducting Josephson junction - Google Patents

Mixing modeling method based on high-temperature superconducting Josephson junction Download PDF

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CN113239548A
CN113239548A CN202110538509.1A CN202110538509A CN113239548A CN 113239548 A CN113239548 A CN 113239548A CN 202110538509 A CN202110538509 A CN 202110538509A CN 113239548 A CN113239548 A CN 113239548A
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高翔
李焕新
卜祥元
安建平
刘珩
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Beijing Institute of Technology BIT
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Abstract

The invention relates to a mixing modeling method based on a high-temperature superconducting Josephson junction, and belongs to the technical field of microwave and terahertz communication and high-temperature superconduction. The method comprises modeling based on parasitic signal selection, mixed solution of impedance matrix, noise matrix solution and solution of noise temperature and conversion gain expression: 1) establishing a six-port model based on parasitic signal selection; 2) hybrid solving an impedance matrix, including single-tone current excitation, perturbation of current or voltage; 3) solving a noise matrix; 4) the conversion gain and noise temperature are solved. The superconducting mixer designed by the frequency mixing modeling method has the advantages of low noise, extremely wide intermediate frequency band, high frequency upper limit, low power requirement and the like; the method overcomes the defects that the existing method has very low calculation speed, can not calculate the noise temperature of the mixer and can not provide conversion gain. The fundamental wave mixing can be analyzed, harmonic mixing conversion gain and noise temperature expression are also given, and the application prospect is wide; the established model is proposed for the first time and has universality.

Description

Mixing modeling method based on high-temperature superconducting Josephson junction
Technical Field
The invention relates to a mixing modeling method based on a high-temperature superconducting Josephson junction, and belongs to the technical field of microwave and terahertz communication and high-temperature superconduction.
Background
Compared with a conventional semiconductor terahertz mixer, the superconducting mixer has the advantages of low noise, extremely wide middle-frequency band, high frequency upper limit, low power requirement and the like; compared with a low-temperature superconducting mixer, a low-temperature facility required by the high-temperature superconducting mixer is more miniaturized and cheaper, and the high-temperature superconducting mixer has a good application prospect as a front-end device of a terahertz communication system.
However, many terahertz frequency mixers based on high-temperature superconducting technology adopt harmonic frequency mixing, which has the advantages of reducing the frequency of local oscillation signals and reducing the requirements on the local oscillation source of the frequency mixer. However, due to the particularity of the high-temperature superconducting Josephson junction, the difficulty of theoretical derivation and model establishment of frequency mixing analysis is very large, and no accurate and feasible modeling and analyzing method based on harmonic frequency mixing of the high-temperature superconducting Josephson junction is provided. Therefore, it is urgently needed to develop a mixing modeling analysis method based on the high-temperature superconducting josephson junction, which can analyze and predict the mixing performance, especially the harmonic mixing performance, of the high-temperature superconducting josephson junction, and the performance mainly comprises noise temperature, conversion gain and the like.
Disclosure of Invention
The invention aims to provide a truly feasible mixing modeling analysis method based on a high-temperature superconducting Josephson junction aiming at the blank of the harmonic mixing modeling analysis of the high-temperature superconducting Josephson junction.
In order to achieve the above object, the specific method of the present invention is as follows.
The high-temperature superconducting Josephson junction-based frequency mixing modeling method comprises modeling based on parasitic signal selection, mixing to solve an impedance matrix, solving a noise matrix, and solving a noise temperature and conversion gain expression, and specifically comprises the following steps:
step 1, selecting modeling based on parasitic signals, and specifically comprising the following sub-steps:
step 1.1, establishing an equivalent circuit based on an RSJ model establishing model;
wherein the equivalent circuit comprises a middle part, an upper end and other parts;
wherein the middle part is an equivalent circuit of a Josephson junction, and the equivalent circuit is formed by an independent thermal noise current delta InnDriving, and connecting the resistor R with the circuit;
the upper end of the equivalent circuit is a direct current bias circuit of a Josephson junction and a loaded local oscillator circuit, and the direct current bias circuit comprises direct current IdcDC source voltage Vs_dcD.c. voltage VdcDC impedance RdcLocal oscillation source voltage Vs_loLocal oscillator voltage VloLocal oscillator impedance ZloLocal oscillator current Ilo；
The other part of the equivalent circuit is a mirror image omegal1＝ωlo-ωoIntermediate frequency omegaoRadio frequency omegaun＝n*ωlo+ωoFrequency omegau1＝ωlo+ωo、ωu(n-1)＝(n-1)*ωlo+ωo、ωu(n+1)＝(n+1)*ωlo+ωoThe ports formed by the circuits are all small signal ports;
wherein the mirror image ωlo-ωoCircuit of (2), called mirror circuit, intermediate frequency ωoSo-called intermediate frequency circuit, radio frequency n x ωlo+ωoReferred to as radio frequency circuitry;
wherein the radio frequency circuit comprises a radio frequency source voltage VsigRadio frequency voltage VunRadio frequency impedance ZunAnd a radio frequency current Iun(ii) a Intermediate frequency circuit including intermediate frequency voltage VoIntermediate frequency impedance ZoAnd inFrequency current Io(ii) a Mirror circuit including a mirror voltage Vl1Mirror impedance Zl1And a mirror current Il1，(n-1)*ωlo+ωoA circuit at a frequency including a voltage V at the corresponding frequencyu(n-1)Impedance Zu(n-1)And current Iu(n-1)，(n+1)*ωlo+ωoA circuit at a frequency including a voltage V at the corresponding frequencyu(n+1)Impedance Zu(n+1)And current Iu(n+1)。
Step 1.2, accepting or rejecting by analyzing the conversion efficiency of the parasitic signal corresponding to each port, namely, selecting the parasitic signal, specifically: suppose the frequency is i x ωlo+ωoConversion efficiency eta of the signal to the intermediate frequency ∈ iαAiming at the high-temperature superconducting Josephson junctions with fixed k and alpha, the conversion efficiency corresponding to signals with different frequencies in the table 1 is obtained, and other parasitic signals are ignored according to the size of the conversion efficiency to obtain a six-port model;
wherein, six ports are: image signal omegalo-ωoRadio frequency n x omegalo+ωoAnd frequency omegalo+ωo、(n-1)*ωlo+ωo、(n+1)*ωlo+ωoAnd an intermediate frequency signal port;
TABLE 1 conversion efficiency contribution of radio frequency and other spurious signals
Figure BDA0003070809230000021
Wherein the scale factors k and alpha are constants;
step 2, solving an impedance matrix in a mixing mode, wherein the impedance matrix comprises single-tone current excitation and current or voltage perturbation, and the method specifically comprises the following substeps:
step 2.1 the specific process of single-tone current excitation is as follows:
step 2.1.1, establishing a linear expression of the normalized voltage vector of the high-temperature superconducting Josephson junction, which specifically comprises the following steps: high temperature superconducting Josephson junction normalized voltage vector vsHigh temperature superconducting Josephson junction at equal impedance matrix Z and corresponding frequencyNormalized current vector isThe product of (a);
the principle of step 2.1.1 is: the current and the voltage corresponding to a six-port in a circuit model of the high-temperature superconducting Josephson junction are small signals, and the current and the voltage of the small signal circuit model are in a linear relation;
step 2.1.2, a time domain nonlinear Josephson equation set is established, single-tone current corresponding to small signal frequency is added for excitation, and superconducting current of a high-temperature superconducting Josephson junction and junction resistance R flowing through one end of the equation setjThe other end is the sum of direct current, local oscillator current, single-tone current excitation current vector and noise current flowing into the high-temperature superconducting Josephson junction; normalizing the Josephson equation set, solving to obtain a normalized superconducting phase difference vector of the Josephson junction, and deriving a normalized time variable by the normalized superconducting phase difference vector to obtain a time domain normalized junction voltage vector of the Josephson junction;
wherein the superconducting current of the high-temperature superconducting Josephson junction is the critical current I of the junctionjThe product of the sine function sin phi (t) of the superconducting phase difference of the junction;
step 2.1.3, Fourier transform is carried out on the normalized voltage vector to obtain Fourier transform coefficients at the frequency positions corresponding to the 6 ports, namely the normalized voltage vector of the corresponding port;
step 2.1.4 repeating step 2.1.1-2.1.3, and obtaining the mean value of the normalized voltage vectors of each port by carrying out weighted average on all the normalized voltage vectors of each port;
step 2.1.5, taking the ratio of the mean value of the normalized voltage vector of each port obtained in the step 2.1.4 to the normalized single-tone current excitation current as the value of each impedance element in the impedance matrix Z;
wherein, the normalized single-tone current excitation current is the normalization of the single-tone current excitation current vector;
step 2.2, performing voltage or current perturbation to obtain a sub-process specifically comprising the following steps:
step 2.2.1 similar to step 2.1.2, the non-linear josephson equation is re-established, except that it is no longer the equationAdding single-tone excitation current, and solving to obtain the time domain normalization junction voltage of the Josephson junction, which specifically comprises the following steps: establishing a time domain nonlinear Josephson equation set, wherein one end of the equation set is a superconducting current of a high-temperature superconducting Josephson junction and a junction resistance R flowing through the superconducting currentjThe other end is the sum of direct current, local oscillator current and noise current flowing into the high-temperature superconducting Josephson junction; carrying out normalization on the Josephson equation set and then solving to obtain a normalized superconducting phase difference of the Josephson junction, and carrying out derivation on the normalized superconducting phase difference to a normalized time variable tau to obtain a time domain normalized junction voltage of the Josephson junction;
wherein the superconducting current of the high-temperature superconducting Josephson junction is the critical current I of the junctionjThe product of the sine function sin phi (t) of the superconducting phase difference of the junction;
step 2.2.2, Fourier transform is carried out on the time domain normalization junction voltage of the Josephson junction, the frequencies of the Fourier transform are direct current, local oscillation and n-1, n and n +1 harmonics, and the Fourier transform coefficients of the frequencies are obtained and are the normalization voltage at the corresponding frequency;
step 2.2.3 repeat step 2.2.1-2.2.2, and the port normalized voltage mean values corresponding to the direct current, local oscillator and other harmonic wave positions are obtained by performing weighted average on all normalized voltages at each corresponding frequency position<vdc>、<vlo>、<vq>(q＝n-1，n，n+1)；
Step 2.2.4 normalizing the voltage mean value of the corresponding port of the direct current, the local oscillator and other harmonic wave positions<vdc>、<vlo>、<vq>Normalized current i written as direct current and local oscillatordc、iloA function of (a);
the principle of step 2.2.4 is: port normalized voltage mean value corresponding to direct current, local oscillator and other harmonic wave positions<vdc>、<vlo>、<vq>Normalized current i along with direct current and local oscillatordc、ilo(ii) a change;
step 2.2.5, the function obtained in the step 2.2.4 is solved for differentiation to obtain a differential equation;
step 2.2.6The frequency q omega is measuredlo+ωoAnd q ωlo-ωoAt a voltage of approximately d<vq>Will frequency ωlo+ωoAnd ωlo-ωoAt a voltage and current of approximately d, respectively<vlo>And d<ilo>The voltage and current at the intermediate frequency and its conjugate are approximated to d, respectively<vdc>And d<idc>Obtaining a differential equation after said approximation;
step 2.2.7 substitutes the differential equation obtained in step 2.2.6 into the differential equation of step 2.2.5 and solves to obtain the frequency of single-tone current excitation as mirror image, intermediate frequency and frequency omega respectivelyu1The value of the transformed impedance vector of (a);
step 3, solving the noise matrix, specifically comprising the following substeps:
step 3.1, a time domain nonlinear Josephson equation set is established, one end of the equation set is superconducting current of the high-temperature superconducting Josephson junction and junction resistance R flowing throughjThe other end is the sum of direct current, local oscillator current and noise current flowing into the high-temperature superconducting Josephson junction; carrying out normalization on the Josephson equation set and then solving to obtain a normalized superconducting phase difference of the Josephson junction, and carrying out derivation on the normalized superconducting phase difference to obtain a time domain normalized junction voltage of the Josephson junction;
step 3.2, carrying out Fourier transform on the time domain normalization junction voltage of the Josephson junction, and obtaining Fourier transform coefficients of all items, namely the normalization voltage of each port;
each frequency of Fourier transform is the frequency corresponding to each port;
step 3.3 repeat steps 3.1-3.2, and obtain the mean value of the normalized voltages of each port by performing weighted average on all the normalized voltages of each port
Figure BDA0003070809230000041
<v0_o>、<v0_uq>(q ═ n-1, n, n + 1); subtracting the corresponding normalized voltage mean value from the normalized voltage of each port to obtain the voltage noise of each port;
step 3.4, Fourier series expansion is carried out on the voltage noise obtained in the step 3.3, and the frequency is solved to be omegaoFourier coefficients of 6 ports are formed, and the Fourier coefficients of the 6 ports form a normalized voltage noise vector;
wherein, the period of Fourier series expansion is marked as T', omegaoIs an intermediate frequency omegaoA corresponding analog angular frequency;
step 3.5, multiplying the normalized voltage noise vector obtained in the step 3.4 by the conjugate transpose of the normalized voltage noise vector, and then multiplying by a time coefficient 2T' to obtain a noise matrix;
step 4, solving a noise temperature and conversion gain expression, wherein the specific process is as follows:
step 4.1 obtaining normalized voltage vector v of high-temperature superconducting Josephson junction according to each port circuit in circuit modelsAnd a high temperature superconducting Josephson junction normalized current vector isAnother equation of (1);
wherein, each port circuit corresponds to the other part of the equivalent circuit in the step 1.1;
the principle of the step 4.1 is as follows: kirchhoff's voltage law;
step 4.2 the equations of step 2.1.1 and step 4.1 are combined to obtain isTo obtain a normalized intermediate frequency current ioAnd the expression of the mean square thereof;
and 4.3, obtaining a semi-analytical expression corresponding to the conversion gain and the noise temperature based on the definition of the conversion gain and the noise temperature.
Advantageous effects
The invention provides a high-temperature superconducting Josephson junction-based mixing modeling method, which comprises the following steps of modeling based on parasitic signal selection, mixing and solving an impedance matrix, solving a noise temperature and converting a gain expression, and has the following beneficial effects:
1. compared with a conventional semiconductor terahertz mixer, the superconducting mixer has the advantages of low noise, extremely wide middle-frequency band, high frequency upper limit, low power requirement and the like; compared with a low-temperature superconducting mixer, a low-temperature facility required by the high-temperature superconducting mixer is more miniaturized and cheaper, and the high-temperature superconducting mixer has a good application prospect as a front-end device of a terahertz communication system; this advantage becomes very important since harmonic mixing has very low frequency requirements on the local oscillator signal, especially for terahertz mixing; however, the existing modeling analysis method for the high-temperature superconducting mixer can only analyze fundamental wave mixing, and although an article indicates that (Advanced Design System) ADS software is adopted to perform modeling simulation analysis on harmonic mixing of the high-temperature superconducting Josephson junction, the method is very slow in calculation speed, cannot calculate the noise temperature of the mixer, and cannot provide an expression of conversion gain; the frequency mixing modeling analysis method based on the high-temperature superconducting Josephson junction establishes a six-port current model by considering the contribution of different parasitic signal conversion efficiencies, omits most parasitic signals, greatly improves the calculation speed, can analyze fundamental wave frequency mixing, also analyzes conversion gain and noise temperature of the harmonic wave frequency mixing, and provides a semi-analytic expression of the two, so the application prospect is very wide;
2. the modeling method based on parasitic signal selection is characterized in that other parasitic signals are omitted through analysis of contribution to parasitic signal conversion efficiency, and the calculation speed of the method is greatly improved; the model is proposed for the first time, has universality and is significant;
3. the hybrid solving impedance matrix comprises single-tone current excitation and current or voltage perturbation, wherein the single-tone current excitation is to add single-tone current excitation with different frequencies at different ports into a nonlinear Josephson equation set to obtain the junction voltage of a corresponding time domain normalized high-temperature superconducting Josephson junction, then carry out Fourier transform, obtain the voltage mean value at different frequencies by repeating the process, and divide the voltage mean value or the conjugate of the voltage mean value by the corresponding single-tone current excitation or the conjugate of the voltage mean value, namely the conversion impedance at the corresponding single-tone current excitation and the frequency, thereby obtaining the solution of the impedance matrix, and the method is also proposed for the first time and has innovation; of current or voltageThe perturbation is to solve the nonlinear Josephson equation without small signal exciting current to obtain the time domain normalized junction voltage of the high temperature superconductive Josephson junction, and to perform Fourier transform, and to repeat the above steps to obtain the port voltage mean value corresponding to the DC, local oscillator and other harmonic parts, and to write it as the function of the normalized current of the DC and local oscillator, then to differentiate the function, and to regard the voltage after the harmonic and intermediate frequency are mixed for one time as the differential of the voltage at the harmonic part, and to regard the voltage or current after the local oscillator and intermediate frequency are mixed for one time as the differential of the voltage or current at the local oscillator part, and to regard the voltage or current at the intermediate frequency and its conjugation as the differential of the voltage or current at the DC part, and to link the above differential equation or equation to obtain the frequency of single-tone current excitation as the mirror image, the intermediate frequency and the ωu1Transformed impedance vector z of timel1、zo、zu1Taking the value of (A); the method is also put forward for the first time, and the z is further acceleratedl1、zo、zu1Solving the speed has great significance;
4. solving a noise matrix, namely solving a nonlinear Josephson equation without small-signal excitation current to obtain a time domain normalized junction voltage of the high-temperature superconducting Josephson junction, performing Fourier transform, repeating the steps to obtain a mean value of the voltage of each port, and subtracting the corresponding mean value from the voltage of each port to obtain the voltage noise of each port; performing Fourier series expansion on the noise, wherein Fourier coefficients at the intermediate frequency are values of elements in corresponding normalized voltage noise vectors; multiplying the vector by the product of the conjugate transpose of the vector and multiplying by twice the time coefficient to obtain a matrix which is a noise matrix; because the frequency of the intermediate frequency is very small relative to the frequency of other ports, the approximate value of the noise matrix is obtained through the autocorrelation or cross-correlation calculation of the voltage mean value of each port, and the calculation speed is accelerated; the solving method of the noise matrix provides a noise calculation method and a fast approximate calculation method of the noise calculation method at different frequencies of six ports, is also proposed for the first time, and has pioneering significance for noise solving of a similar model in the future;
5. solving a noise temperature and conversion gain expression, and obtaining another equation set of the normalized voltage vector and the current vector of the high-temperature superconducting Josephson junction through each port circuit in the circuit model; obtaining a solution of a current vector by simultaneous equations, and further obtaining an intermediate frequency current and a square mean value thereof; obtaining a corresponding semi-analytical expression based on the definition of conversion gain and noise temperature; the method gives a semi-analytic expression of conversion gain and noise temperature, is not only suitable for fundamental wave mixing analysis of the high-temperature superconducting Josephson junction, but also suitable for harmonic mixing analysis of the high-temperature superconducting Josephson junction, and fills a plurality of blanks in the field.
Drawings
FIG. 1 is a current model diagram of a high temperature superconducting Josephson junction;
FIG. 2 is a graph of conversion efficiency contributions for various spurious signals;
FIG. 3 is a spectral plot of the mean value of port voltages for single-tone current excitation;
FIG. 4 is a simulation of the transformed impedance versus normalized DC voltage for each port for single-tone current excitation at RF;
FIG. 5 is a comparison graph of simulation and actual measurement of the relationship between conversion gain and normalized DC bias current at different temperatures or local oscillator powers;
FIG. 6 is a comparison graph of simulation and actual measurement of the relationship between noise temperature and DC voltage at different temperatures;
fig. 7 is a graph of conversion gain or noise temperature versus intermediate frequency at different temperatures.
Detailed Description
The invention relates to a frequency mixing modeling analysis method based on a high-temperature superconducting Josephson junction, which comprises modeling based on parasitic signal selection, hybrid solving of an impedance matrix, solving of a noise matrix, and solving of a noise temperature and conversion gain expression. The invention is further illustrated and described in detail below with reference to the figures and examples.
Example 1
The invention provides a powerful tool for frequency mixing analysis and performance prediction of a high-temperature superconducting Josephson junction based on a frequency mixing modeling analysis method of the high-temperature superconducting Josephson junction, which comprises fundamental frequency mixing and harmonic frequency mixing, and the specific derivation process is as follows:
corresponding to the step 1, establishing a circuit model based on a parasitic signal selection method;
corresponding to steps 1.1 to 1.3, as shown in fig. 1, the middle part of the equivalent circuit of the model is the equivalent circuit of the josephson junction, the equivalent circuit is an RSJ model, i.e. a parallel resistor R circuit, and an independent thermal noise current δ I is usednnDriving; the upper end of the equivalent circuit is a direct current bias circuit of a Josephson junction and a loaded local oscillator circuit, and the direct current bias circuit comprises direct current IdcDC source voltage Vs_dcD.c. voltage VdcDC impedance RdcLocal oscillation source voltage Vs_loLocal oscillator voltage VloLocal oscillator impedance ZloLocal oscillator current Ilo(ii) a The other part of the equivalent circuit is a frequency mirror image omegalo-ωoIntermediate frequency omegaoRadio frequency n x omegalo+ωoAnd frequency omegalo+ωo、(n-1)*ωlo+ωo、(n+1)*ωlo+ωoThe ports formed by the circuits are small signal ports, and the radio frequency circuit comprises a radio frequency source voltage VsigRadio frequency voltage VunRadio frequency impedance ZunRadio frequency current Iun(ii) a For intermediate frequency circuits, including intermediate frequency voltage VoIntermediate frequency impedance ZoIntermediate frequency current Io(ii) a For mirror circuits, including the mirror voltage Vl1Intermediate frequency impedance Zl1Mirror current Il1(ii) a For a ωLO+ωOThe circuit at a frequency comprises a voltage V at the corresponding frequencyuaImpedance at frequency ZuaCurrent at frequency Iua；
Wherein a is n-1, n + 1;
corresponding to step 1.4, as shown in fig. 2, according to table 1, a certain high temperature superconducting josephson junction mixer is subjected to simulation analysis for 4 th harmonic mixing, 10 th harmonic mixing and 20 th harmonic mixing, and a mirror image ω is found through a simulation curvelo-ωoRadio frequency n x omegalo+ωoAnd frequency omegalo+ωo、(n-1)*ωlo+ωo、(n+1)*ωlo+ωoThe mixing efficiency of the parasitic signals is high, and other parasitic signals are ignored; but for fourth harmonic mixing, the image ωlo-ωoAnd parasitic signal (n +1) × ωlo+ωoAlso left off; in consideration of the universality of the patent, six ports are adopted for theoretical modeling by adding the intermediate frequency port.
Corresponding to step 2, the derivation process of the hybrid solution impedance matrix is as follows:
the hybrid solution impedance matrix consists primarily of single-tone current excitations, perturbations in current or voltage.
Corresponding to step 2.1, the derivation of the single-tone current excitation is as follows:
corresponding to step 2.1.1, a linear relationship between the current and the voltage at each frequency corresponding to each small signal is established as follows:
vs＝Z·is+δvs (1)
equation (1) is written as
Figure BDA0003070809230000071
Wherein v iss＝Vs/RIcIs a normalized voltage vector, is＝Is/RIcNormalized current vector, δ vsIs the noise vector, R is the resistance explained by the quasi-particle current, IcThe maximum superconducting current of the junction is,
Figure BDA0003070809230000081
is a vector of the voltage that is,
Figure BDA0003070809230000082
is a current vector. The subscripts of the parameters in the vector correspond to the frequencies of the radio frequency and other parasitic ports.
Establishing a time domain nonlinear Josephson equation set corresponding to the step 2.1.2, and adding single-tone current excitation corresponding to the small signal frequency:
Figure BDA0003070809230000083
where t is time, r ═ 1,1 …]TH is the Planck constant, e is the charge of the electron, phi (t) ═ phil1(t),φo(t),φu1(t),φu(n-1)(t),φun(t),φu(n+1)(t)]TIs the difference in superconducting phase across the junction, IdcIs a DC-biased current, IloIs the current of the local oscillator, ω ═ ωl1,ωo,ωu1,ωu(n-1),ωun,ωu(n+1)]TIs a signal frequency vector, I ═ Il1',Io',Iu1',Iu(n-1)',Iun',Iu(n+1)']TIs a single-tone current excitation vector that includes spurious signals at different frequencies. Josephson junction noise is represented by the following autocorrelation function:
Figure BDA0003070809230000084
where k is the boltzmann constant and T is the ambient temperature.
Normalizing the system of josephson equations to obtain the following formula:
dφ(τ)/dτ+sinφ(τ)＝[idc+2ilocosΩloτ]r+2i′cosΩτ+rδinn(τ) (5)
wherein idc＝Idc/Ic、ilo＝Ilo/IcAnd δ inn＝δInn/IcIs a normalized current, i ═ il1',io',iu1',iu(n-1)',iun',iu(n+1)']T＝I'/IcIs a normalized current vector, τ ═ 2eRIcH) t is normalized time, Ωlo＝h/ωlo2eRIcIs the normalized frequency, Ω ═ Ωl1,Ωo,Ωu1,Ωu(n-1),Ωun,Ωu(n+1)]T＝h/ω2eRIcIs made ofThe frequency vector is normalized. Likewise, equation (4) becomes
<δinn(t)δinn(t′)>＝Γδ(τ-τ′) (6)
Wherein r 2ekT/hIcIs a noise parameter.
Obtaining phi (tau) by solving equation (5); deriving τ by φ (τ) is the time domain normalized junction voltage vector v' (τ) of the Josephson junction.
Corresponding to step 2.1.3, fourier transform is performed on the time domain normalization junction voltage vector v' (τ) to obtain the following formula:
Figure BDA0003070809230000085
wherein v isl1'＝[vl1_l1',vl1_o',vl1_u1',vl1_u(n-1)',vl1_un',vl1_u(n+1)']T、vo'＝[vo_l1',vo_o',vo_u1',vo_u(n-1)',vo_un',vo_u(n+1)']T、vu1'＝[vu1_l1',vu1_o',vu1_u1',vu1_u(n-1)',vu1_un',vu1_u(n+1)']T、vu(n-1)'＝[vu(n-1)_l1',vu(n-1)_o',vu(n-1)_u1',vu(n-1)_u(n-1)',vu(n-1)_un',vu(n-1)_u(n+1)']T、vun'＝[vun_l1',vun_o',vun_u1',vun_u(n-1)',vun_un',vun_u(n+1)']TAnd vu(n+1)'＝[vu(n+1)_l1',vu(n+1)_o',vu(n+1)_u1',vu(n+1)_u(n-1)',vu(n+1)_un',vu(n+1)_u(n+1)']TAre vectors of the transformed voltage under excitation by different single-tone currents, each vector containing the transformed voltage at a different frequency.
Corresponding to step 2.1.4, the port voltages vary with normalized time due to the presence of noise, written as:
Figure BDA0003070809230000091
repeating the above process K times to obtain the average value of the voltage vector of each port<vl1'>、<vo'>、<vu1'>、<vu(n-1)'>、<vun'>、<vu(n+1)'>The following formula:
Figure BDA0003070809230000092
the impedance matrix Z is solved for step 2.1.5 using the following equation:
Figure BDA0003070809230000093
wherein Z is [ Z ]l1 zo zu1 zu(n-1) zun zu(n+1)]Comprising transformed impedance vectors at different single-tone current excitations, each vector comprising transformed impedances at different spurious frequencies.
Corresponding to step 2.2, the derivation of the perturbation of current or voltage is as follows:
corresponding to step 2.2.1, the single-tone excitation current is not added any more, and the nonlinear josephson equation is reestablished as follows:
Figure BDA0003070809230000094
where phi (t) is also the superconducting phase difference of the junction.
The time domain normalization junction voltage v of the Josephson junction is obtained by normalizing equation (11) and then solving to obtain a numerical solution of phi (tau)0(τ)＝dφ(τ)/dτ。
Corresponding to step 2.2.2, for v0(tau) carrying out Fourier transform, wherein the frequencies of the Fourier transform are direct current, local oscillation and n-1, n +1 subharmonics:
Figure BDA0003070809230000095
wherein v isdcIs a DC offset, vloIs the local oscillator voltage, vqIs the q-order harmonic signal omega of the local oscillatorq＝q*ΩloThe voltage of (c).
Corresponding to the step 2.2.3, repeating the steps 2.2.1-2.2.2 to obtain the port voltage mean values corresponding to the direct current, the local oscillator and other harmonic wave positions<vdc>、<vlo>、<vq>(q ═ n-1, n, n +1) is as follows:
Figure BDA0003070809230000096
Figure BDA0003070809230000097
Figure BDA0003070809230000098
corresponding to step 2.2.4, the voltage mean value of each port changes with the normalized current of the direct current and the local oscillator, and therefore, the voltage mean values are written into the following function form:
<vdc>＝f0(idc,|ilo|) (16)
<vlo>＝ilof1(idc,|ilo|) (17)
Figure BDA00030708092300001013
corresponding to step 2.2.5, differentiating the functions (16), (17), (18) is:
Figure BDA0003070809230000101
Figure BDA0003070809230000102
Figure BDA0003070809230000103
corresponding to step 2.2.6, the frequency q ω is measuredlo+ωoAnd q ωlo-ωoThe voltage at is regarded as d<vq>Will frequency ωlo+ωoAnd ωlo-ωoThe voltage and current at are respectively considered as d<vlo>And d<ilo>The voltage and current at the intermediate frequency and its conjugate are respectively considered as d<vdc>And d<idc>Then, there are:
Figure BDA0003070809230000104
Figure BDA0003070809230000105
Figure BDA0003070809230000106
Figure BDA0003070809230000107
Figure BDA0003070809230000108
corresponding to step 2.2.7, equations (22) to (26) are substituted into (19) to (21) to obtain the following results:
Figure BDA0003070809230000109
Figure BDA00030708092300001010
Figure BDA00030708092300001011
Figure BDA00030708092300001012
thus, the frequencies of the single-tone current excitation are mirror, intermediate, and ω, respectivelyu1Transformed impedance vector z of timel1、zo、zu1The expression of (a) is as follows:
Figure BDA0003070809230000111
Figure BDA0003070809230000112
Figure BDA0003070809230000113
Figure BDA0003070809230000114
Figure BDA0003070809230000115
Figure BDA0003070809230000116
Figure BDA0003070809230000117
wherein
Figure BDA0003070809230000118
Corresponding to step 3, the derivation process for solving the noise matrix is as follows:
obtaining the time domain normalization junction voltage v of formula (11) and the Josephson junction corresponding to step 3.10(τ)。
Corresponding to step 3.2, for v0(τ) performing a fourier transform, except that each frequency is a frequency corresponding to each port:
Figure BDA0003070809230000119
corresponding to the step 3.3, repeating the step 3.1-3.2 to obtain the average value of the voltage of each port
Figure BDA00030708092300001110
<v0_o>、<v0_uq>(q＝n-1，n，n+1)：
Figure BDA00030708092300001111
v0_o＝<v0_o>+δv0_o (40)
v0_uq＝<v0_uq>+δv0_uq (41)
Figure BDA00030708092300001112
Figure BDA00030708092300001113
Figure BDA00030708092300001114
Subtracting the corresponding voltage mean value from the voltage of each port to obtain the voltage noise of each port
Figure BDA00030708092300001115
δv0_o、δv0_uq。
In correspondence with the step 3.4,
Figure BDA0003070809230000121
δv0_o、δv0_uall curves are varying, so the fourier series is expanded as follows:
Figure BDA0003070809230000122
Figure BDA0003070809230000123
Figure BDA0003070809230000124
if Δ f is 1/T' the normalized bandwidth frequency of the mixer noise, then there are:
δvs＝[cl1_N,co_N,cu1_N,cu(n-1)_N,cun_N,cu(n+1)_N]T (48)
wherein N satisfies 2 pi N/T' ═ omegao。
Corresponding to step 3.5, the expression of the six-dimensional noise matrix S is as follows:
Figure BDA0003070809230000125
in specific implementation, when the precision requirement of the noise matrix is not high, the method of the following step 3.6 is adopted to obtain an approximate noise matrix:
the intermediate frequency is very small relative to the frequencies of the other ports, so that the mean value of the normalized voltages obtained for the ports is obtained
Figure BDA0003070809230000126
<v0_o>、<v0_uq>And performing autocorrelation or cross-correlation calculation to obtain an approximate noise matrix so as to accelerate the calculation speed.
Figure BDA0003070809230000127
Figure BDA0003070809230000128
Figure BDA0003070809230000129
Figure BDA00030708092300001210
Figure BDA00030708092300001211
Figure BDA00030708092300001212
In which the period of the Fourier transform of the P equation (38), q1,q2＝1,n-1,n,n+1。
Corresponding to step 4, the derivation process of the expression of the conversion gain and the noise temperature is as follows:
corresponding to the step 4.1, v is obtained according to the kirchhoff voltage law and each port circuit in the circuit modelsAnd isAnother equation of (a):
vs+Zs·is＝vsig (56)
corresponding to step 4.2, equations (1) and (56) of the simultaneous steps are obtained to obtain isThe solution of (a):
is＝(Z+Zs)-1·(vsig-δvs) (57)
if order
Figure BDA0003070809230000131
yo＝[yl1_o yo_o yu1_o yu(n-1)_o yun_o yu(n+1)_o] (59)
Then the intermediate frequency current ioComprises the following steps:
io＝yo·(vsig-δvs)＝yo_unvSig-yo·δvs (60)
the expression of the mean square value is:
Figure BDA0003070809230000132
corresponding to step 4.3, gain G is convertedmixTemperature T of noisemixAre respectively:
Figure BDA0003070809230000133
Figure BDA0003070809230000134
example 2
For the mixing modeling method based on the high-temperature superconducting Josephson junction proposed in example 1, experimental verification is carried out by adopting a certain fourth harmonic high-temperature superconducting Josephson junction mixer.
The mixer has very good radiation coupling and isolation effects at frequencies around 160GHz and 640GHz, and is therefore a fourth harmonic mixer. The temperature of the experiment was adjusted by a small refrigerator. The impedances at the different frequencies used in the experiment are shown in table 2.
TABLE 2 test results of impedance at different frequencies
Figure BDA0003070809230000135
The impedance of the Josephson junction is only 3 ohms, much less than the frequency ω in Table 3u3The impedance of (c) is ignored, so only ω in fig. 1 needs to be consideredo、ωu1And ωunThree ports. Then testing and calculating the critical current I of the junction to be used for the gain and noise temperature of the swivelcThe results are shown in Table 3.
TABLE 3 Critical Current test results of Josephson junctions at different temperatures
Figure BDA0003070809230000141
In the simulation calculation of the case, the periods of the fourier transforms of equations (7) and (12) are both the periods of the intermediate frequency 2.4GHz, but the integration step lengths are 1/50 of the periods of the radio frequency 642.24GHz and the intermediate frequency 2.4GHz respectively, so that the calculation speed is increased on the premise of not influencing the calculation accuracy. Fig. 3 is a spectrum diagram of the mean value of the port voltage when excited by single-tone current, in which voltages of direct current, intermediate frequency, local oscillator, radio frequency and other harmonic frequencies are seen, which corresponds to equation (7) or (12). As the frequency of the harmonic increases, the voltages corresponding to odd and even harmonics decrease, which is consistent with the mixing phenomenon of semiconductors. However, the voltage of the even harmonics is always smaller than the voltage of the odd harmonics, especially the voltage of the second harmonics is smaller than the voltage of the third harmonics, which may be due to the peculiarities of the nonlinear mixing of josephson junctions. Albeit at a direct current and local oscillatorMuch higher than the intermediate frequency and omegau1But this does not affect the solution of the transformed impedance.
The mean of the voltages in fig. 3 divided by the corresponding single-tone current excitation in equation (5) is the transformed impedance at the corresponding frequency given by equation (10). Fig. 4 is a simulation diagram showing the relationship between the transformed impedance of each port and the normalized dc voltage when excited by a single-tone current at radio frequency. It is clear that the switching impedance varies periodically with the normalized voltage of the dc current with a period that coincides with the sharp step of the josephson junction, and that the impedance between the two periods is discontinuous, which may be due to the sharp step phenomenon of the high temperature superconducting josephson junction. Impedance zo_unThe imaginary part of (a) is close to 0, so that only the simulation result of the real part thereof is given.
Fig. 5 is a comparison graph of simulation and actual measurement of the relationship between the conversion gain and the normalized dc bias current at different temperatures or local oscillator powers.
The experimental temperatures of the graphs (5a), (5b) and (5c) are respectively 20K, 40K and 60K, the local oscillator powers are respectively-43.45 dBm, -42.7dBm and-42.3 dBm, the experimental temperatures of the graph (5d) are both 40K, and the local oscillator powers are respectively-45.7 dBm. Both experimental and simulation results show that the conversion gain is reduced along with the reduction of the temperature, and the optimal direct current bias is reduced along with the increase of the true power. Under different experimental temperatures or local oscillator powers, the simulation result of the conversion gain is well matched with the actual measurement result, which proves the correctness of the calculation method of the conversion gain deduced in case 1. In fig. 5, the experimental results are always lower than the simulation results, which may be due to environmental changes or human operations.
FIG. 6 is a comparison graph of simulation and measurement of noise temperature versus DC voltage at different temperatures. The graph shows that the noise temperature also varies periodically with the dc voltage and the period is the same as the sharp steps. The noise temperature between the two periods is also discontinuous and may be caused by the sharp steps of the high temperature superconducting josephson junction. In addition, fig. 6 also shows that the measured result of the noise temperature is substantially consistent with the simulation result, which proves the correctness of the noise temperature calculation method in case 1. In addition, the fact that the measured value in fig. 6 is always lower than the theoretical calculation result may also be caused by the change of the environment or the manual operation in the test process.
Fig. 7 is a graph of conversion gain or noise temperature versus intermediate frequency at different temperatures. Simulation results of conversion gain or noise temperature change with the change of the intermediate frequency, because the impedance of the harmonic mixer is different at different frequencies. At approximately the intermediate frequency of 14Ghz, the conversion gain and noise temperature both reach optimum values. The solid line in the figure is simulation; scattering points are actually measured; simulations and measurements were performed for 40K, 60K and 20K. The results show that: the variation trend of the simulation curves of conversion gain and noise temperature is consistent for different experimental temperatures. At intermediate frequencies, the conversion gain or noise temperature is either lower or higher than the simulation curve at different experimental temperatures, for the same reasons as the environmental changes or human manipulation mentioned above.
In summary, case 2 analyzes many characteristics of the conversion gain and the noise temperature of the harmonic mixer through comparison of various simulation and experiment results, and provides a spectrogram of the junction voltage of equations (7) and (12) in case 1 and simulation curves of some conversion impedances changing along with the normalized direct current voltage in equation (10), and finally verifies the correctness of the mixing modeling analysis method based on the high-temperature superconducting josephson junction provided by case 1 through comparison of simulation results and actual measurement results.
While the preferred embodiments of the present invention have been described above, the present invention should not be limited to the embodiments and the drawings disclosed. Equivalents and modifications may be made without departing from the spirit of the disclosure, which is to be considered as within the scope of the invention.

Claims (9)

1. A mixing modeling method based on a high-temperature superconducting Josephson junction is characterized in that: the method comprises modeling based on parasitic signal selection, mixed solution of an impedance matrix, solution of a noise matrix, and solution of a noise temperature and conversion gain expression, and specifically comprises the following steps:
step 1, selecting modeling based on parasitic signals, and specifically comprising the following sub-steps:
step 1.1, establishing an equivalent circuit based on an RSJ model establishing model;
wherein the equivalent circuit comprises a middle part, an upper end and other parts;
wherein the middle part is an equivalent circuit of a Josephson junction, and the equivalent circuit is formed by an independent thermal noise current delta InnDriving, and connecting the resistor R with the circuit;
the other part of the equivalent circuit is a mirror image omegal1＝ωlo-ωoIntermediate frequency omegaoRadio frequency omegaun＝n*ωlo+ωoFrequency omegau1＝ωlo+ωo、ωu(n-1)＝(n-1)*ωlo+ωo、ωu(n+1)＝(n+1)*ωlo+ωoThe ports formed by the circuits are all small signal ports;
wherein the mirror image ωlo-ωoCircuit of (2), called mirror circuit, intermediate frequency ωoSo-called intermediate frequency circuit, radio frequency n x ωlo+ωoReferred to as radio frequency circuitry;
step 1.2, accepting or rejecting by analyzing the conversion efficiency of the parasitic signal corresponding to each port, namely, selecting the parasitic signal, specifically: suppose the frequency is i x ωlo+ωoConversion efficiency eta of the signal to the intermediate frequency ∈ iαAiming at the high-temperature superconducting Josephson junctions with fixed k and alpha, the conversion efficiency corresponding to signals with different frequencies in the table 1 is obtained, and other parasitic signals are ignored according to the size of the conversion efficiency to obtain a six-port model;
wherein, six ports are: image signal omegalo-ωoRadio frequency n x omegalo+ωoAnd frequency omegalo+ωo、(n-1)*ωlo+ωo、(n+1)*ωlo+ωoAnd an intermediate frequency signal port;
step 2, solving an impedance matrix in a mixing mode, wherein the impedance matrix comprises single-tone current excitation and current or voltage perturbation, and the method specifically comprises the following substeps:
step 2.1 the specific process of single-tone current excitation is as follows:
step 2.1.1, establishing a linear expression of the normalized voltage vector of the high-temperature superconducting Josephson junction, which specifically comprises the following steps: high temperature superconducting Josephson junction normalized voltage vector vsNormalized current vector i equal to impedance matrix Z and corresponding frequency of high-temperature superconducting Josephson junctionsThe product of (a);
step 2.1.2, a time domain nonlinear Josephson equation set is established, single-tone current corresponding to small signal frequency is added for excitation, and superconducting current of a high-temperature superconducting Josephson junction and junction resistance R flowing through one end of the equation setjThe other end is the sum of direct current, local oscillator current, single-tone current excitation current vector and noise current flowing into the high-temperature superconducting Josephson junction; normalizing the Josephson equation set, solving to obtain a normalized superconducting phase difference vector of the Josephson junction, and deriving a normalized time variable by the normalized superconducting phase difference vector to obtain a time domain normalized junction voltage vector of the Josephson junction;
step 2.1.3, Fourier transform is carried out on the normalized voltage vector to obtain Fourier transform coefficients at the frequency positions corresponding to the 6 ports, namely the normalized voltage vector of the corresponding port;
step 2.1.4 repeating step 2.1.1-2.1.3, and obtaining the mean value of the normalized voltage vectors of each port by carrying out weighted average on all the normalized voltage vectors of each port;
step 2.1.5, taking the ratio of the mean value of the normalized voltage vector of each port obtained in the step 2.1.4 to the normalized single-tone current excitation current as the value of each impedance element in the impedance matrix Z;
step 2.2, performing voltage or current perturbation to obtain a sub-process specifically comprising the following steps:
step 2.2.1 is similar to step 2.1.2, reestablishes the nonlinear josephson equation, except that the equation does not add single-tone excitation current any more, obtains the time domain normalization junction voltage of the josephson junction by solving, and specifically comprises the following steps: establishing a time domain nonlinear Josephson equation set, wherein one end of the equation set is a superconducting current of a high-temperature superconducting Josephson junction and a junction resistance R flowing through the superconducting currentjThe other end is direct current flowing into the high-temperature superconducting Josephson junctionThe sum of the current, the local oscillator current and the noise current; carrying out normalization on the Josephson equation set and then solving to obtain a normalized superconducting phase difference of the Josephson junction, and carrying out derivation on the normalized superconducting phase difference to a normalized time variable tau to obtain a time domain normalized junction voltage of the Josephson junction;
step 2.2.2, Fourier transform is carried out on the time domain normalization junction voltage of the Josephson junction, the frequencies of the Fourier transform are direct current, local oscillation and n-1, n and n +1 harmonics, and the Fourier transform coefficients of the frequencies are obtained and are the normalization voltage at the corresponding frequency;
step 2.2.3 repeat step 2.2.1-2.2.2, and the port normalized voltage mean values corresponding to the direct current, local oscillator and other harmonic wave positions are obtained by performing weighted average on all normalized voltages at each corresponding frequency position<vdc>、<vlo>、<vq>(q＝n-1，n，n+1)；
Step 2.2.4 normalizing the voltage mean value of the corresponding port of the direct current, the local oscillator and other harmonic wave positions<vdc>、<vlo>、<vq>Normalized current i written as direct current and local oscillatordc、iloA function of (a);
the principle of step 2.2.4 is: port normalized voltage mean value corresponding to direct current, local oscillator and other harmonic wave positions<vdc>、<vlo>、<vq>Normalized current i along with direct current and local oscillatordc、ilo(ii) a change;
step 2.2.5, the function obtained in the step 2.2.4 is solved for differentiation to obtain a differential equation;
step 2.2.6 frequency q ωlo+ωoAnd q ωlo-ωoAt a voltage of approximately d<vq>Will frequency ωlo+ωoAnd ωlo-ωoAt a voltage and current of approximately d, respectively<vlo>And d<ilo>The voltage and current at the intermediate frequency and its conjugate are approximated to d, respectively<vdc>And d<idc>Obtaining a differential equation after said approximation;
step 2.2.7 differentiation of the values obtained in step 2.2.6Substituting the equation into the differential equation in step 2.2.5 and solving to obtain the mirror image, the intermediate frequency and the frequency omega of the single-tone current excitation respectivelyu1The value of the transformed impedance vector of (a);
step 3, solving the noise matrix, specifically comprising the following substeps:
step 3.1, a time domain nonlinear Josephson equation set is established, one end of the equation set is superconducting current of the high-temperature superconducting Josephson junction and junction resistance R flowing throughjThe other end is the sum of direct current, local oscillator current and noise current flowing into the high-temperature superconducting Josephson junction; carrying out normalization on the Josephson equation set and then solving to obtain a normalized superconducting phase difference of the Josephson junction, and carrying out derivation on the normalized superconducting phase difference to obtain a time domain normalized junction voltage of the Josephson junction;
step 3.2, carrying out Fourier transform on the time domain normalization junction voltage of the Josephson junction, and obtaining Fourier transform coefficients of all items, namely the normalization voltage of each port;
step 3.3 repeat steps 3.1-3.2, and obtain the mean value of the normalized voltages of each port by performing weighted average on all the normalized voltages of each port
Figure FDA0003070809220000031
<v0_o>、<v0_uq>(q ═ n-1, n, n + 1); subtracting the corresponding normalized voltage mean value from the normalized voltage of each port to obtain the voltage noise of each port;
step 3.4, Fourier series expansion is carried out on the voltage noise obtained in the step 3.3, and the frequency is solved to be omegaoFourier coefficients of 6 ports are formed, and the Fourier coefficients of the 6 ports form a normalized voltage noise vector;
wherein, the period of Fourier series expansion is marked as T', omegaoIs an intermediate frequency omegaoA corresponding analog angular frequency;
step 3.5, multiplying the normalized voltage noise vector obtained in the step 3.4 by the conjugate transpose of the normalized voltage noise vector, and then multiplying by a time coefficient 2T' to obtain a noise matrix;
step 4, solving a noise temperature and conversion gain expression, wherein the specific process is as follows:
step 4.1 obtaining normalized voltage vector v of high-temperature superconducting Josephson junction according to each port circuit in circuit modelsAnd a high temperature superconducting Josephson junction normalized current vector isAnother equation of (1);
step 4.2 the equations of step 2.1.1 and step 4.1 are combined to obtain isTo obtain a normalized intermediate frequency current ioAnd the expression of the mean square thereof;
and 4.3, obtaining a semi-analytical expression corresponding to the conversion gain and the noise temperature based on the definition of the conversion gain and the noise temperature.
2. The method of claim 1, wherein the method comprises: step 1.1 the middle part of the equivalent circuit is formed by an independent noise current deltaInnDriving, and connecting the circuit for resistor R.
3. The method of claim 2, wherein the method comprises: step 1.1 the upper end of the equivalent circuit includes a direct current IdcDC source voltage Vs_dcD.c. voltage VdcDC impedance RdcLocal oscillation source voltage Vs_loLocal oscillator voltage VloLocal oscillator impedance ZloAnd local oscillator current Ilo。
4. The method of claim 3, wherein the method comprises: in step 1.1, the RF circuit includes an RF source voltage VsigRadio frequency voltage VunRadio frequency impedance ZunAnd a radio frequency current Iun(ii) a Intermediate frequency circuit including intermediate frequency voltage VoIntermediate frequency impedance ZoAnd intermediate frequency current Io(ii) a Mirror circuit including a mirror voltage Vl1Mirror impedance Zl1And a mirror current Il1，(n-1)*ωlo+ωoCircuits at frequency, including corresponding frequencyVoltage V at rateu(n-1)Impedance Zu(n-1)And current Iu(n-1)，(n+1)*ωlo+ωoA circuit at a frequency including a voltage V at the corresponding frequencyu(n+1)Impedance Zu(n+1)And current Iu(n+1)。
5. The method of claim 4, wherein the method comprises: the conversion efficiency corresponding to the different frequency signals in step 1.2 is shown in table 1:
TABLE 1 conversion efficiency contribution of radio frequency and other spurious signals
Figure FDA0003070809220000041
Wherein the scale factors k and alpha are constants.
6. The method of claim 5, wherein the method comprises: in step 2.1.5, the normalized single-tone current excitation current is the normalization of the single-tone current excitation current vector.
7. The method of claim 6, wherein the method comprises: in step 2.1.2 and step 2.2.1, the superconducting current of the high temperature superconducting Josephson junction is the critical current I of the junctionjAnd the product of the sine function sin phi (t) of the superconducting phase difference of the junction.
8. The method of claim 7, wherein the method comprises: in step 3.2, each frequency of the fourier transform is a frequency corresponding to each port.
9. The method of claim 8, wherein the method comprises: in step 4.1, each port circuit corresponds to the other part of the equivalent circuit in step 1.1; the principle of the step 4.1 is as follows: kirchhoff's voltage law.
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CN115115054A (en) * 2022-06-28 2022-09-27 合肥本源量子计算科技有限责任公司 Circuit structure, quantum chip and quantum computer

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