Physical Review B

Spectroscopic resonance broadening in a Josephson junction qubit due to current noise

Haitan Xu, A. J. Berkley, Roberto Ramos, M. A. Gubrud, P. R. Johnson, Frederick W. Strauch, Alex J. Dragt, J. R. Anderson, Christopher J. Lobb, F. C. Wellstood

Published Feb 23, 2005• 25 citations

Publisher

Abstract

We analyze the effect of dissipation and low-frequency current noise on quantum coherence in a current-biased Josephson junction that has low damping. Developing a stochastic Bloch equation for the reduced density matrix of the system, we determine the resonance response of the system to a weak external microwave current drive. We characterize the response by finding the spectroscopic coherence time ${T}_{2}^{*}$ as a function of the energy relaxation time ${T}_{1}$ and the current noise power spectral density. For current noise with a constant spectral density up to a frequency ${f}_{c}⪢{(2\ensuremath{\pi}{T}_{1})}^{\ensuremath{-}1}$, we find that the resonance broadening is proportional to the noise spectral density, while for ${f}_{c}⪡{(2\ensuremath{\pi}{T}_{1})}^{\ensuremath{-}1}$, the broadening is proportional to the rms current noise. We compare our results to microwave spectroscopy data on $100\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}{\mathrm{m}}^{2}$ Nb and Al tunnel junctions at milli-Kelvin temperatures and find good agreement between the measured response spectra and our model simulations.

DOI: 10.1103/physrevb.71.064512

Related papers

Measurements of quantum noise in resistively shunted Josephson junctions

R. H. Koch, Dale J. Van Harlingen, John L. Clarke

Physical review. B, Condensed matterJul 1, 1982117 citations

Decoherence in a Josephson-junction qubit

A. J. Berkley, H. Xu, M. A. Gubrud +4 more

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Observation of quantum stochastic synchronization in a dissipative quantum system

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Decoherence in Josephson vortex quantum bits: Long-Josephson-junction approach to a two-state system

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Physical Review BJun 6, 200621 citations

A method of determining microwave dissipation of Josephson junctions with non-linear frequency response

Heng-Jie Chen, Hang Xue, Shaoxiong Li +1 more

Acta Physica SinicaJan 1, 20191 citation

Physical Review B

Spectroscopic resonance broadening in a Josephson junction qubit due to current noise

Haitan Xu, A. J. Berkley, Roberto Ramos, M. A. Gubrud, P. R. Johnson, Frederick W. Strauch, Alex J. Dragt, J. R. Anderson, Christopher J. Lobb, F. C. Wellstood

Published Feb 23, 2005• 25 citations

Publisher

Abstract

{T}_{2}^{*}

as a function of the energy relaxation time

{T}_{1}

and the current noise power spectral density. For current noise with a constant spectral density up to a frequency

{f}_{c}⪢{(2\ensuremath{\pi}{T}_{1})}^{\ensuremath{-}1}

, we find that the resonance broadening is proportional to the noise spectral density, while for

{f}_{c}⪡{(2\ensuremath{\pi}{T}_{1})}^{\ensuremath{-}1}

, the broadening is proportional to the rms current noise. We compare our results to microwave spectroscopy data on

100\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}{\mathrm{m}}^{2}

Nb and Al tunnel junctions at milli-Kelvin temperatures and find good agreement between the measured response spectra and our model simulations.

DOI: 10.1103/physrevb.71.064512

Related papers

Measurements of quantum noise in resistively shunted Josephson junctions

R. H. Koch, Dale J. Van Harlingen, John L. Clarke

Measurements have been made of the low-frequency spectral density of the voltage noise in current-biased resistively shunted Josephson tunnel junctions under conditions in which the noise mixed-down from frequencies near the Josephson frequency (${\ensuremath{\nu}}_{J}$) to the measurement frequency ($\ensuremath{\ll}{\ensuremath{\nu}}_{J}$) is in the regime $h{\ensuremath{\nu}}_{J}>{k}_{B}T$. In this limit, quantum corrections to the mixed-down noise are important. The spectral densities measured on junctions with current-voltage characteristics close to the Stewart-McCumber model were in excellent agreement with the predicted values, with no fitted parameters. The mixed-down noise for a wide range of bias voltages was used to infer the spectral density of the current noise in the shunt resistor at frequency $\ensuremath{\nu}$. With no fitted parameters, this spectral density at frequencies up to 500 GHz was in excellent agreement with the prediction $(\frac{2h\ensuremath{\nu}}{R})coth(\frac{h\ensuremath{\nu}}{2{k}_{B}T})$. The presence of the zero-point term, $\frac{2h\ensuremath{\nu}}{R}$, at frequencies $h\ensuremath{\nu}>{k}_{B}T$ was clearly demonstrated. The current-voltage characteristics of a junction with ${\ensuremath{\beta}}_{L}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{L}_{s}{I}_{0}}{{\ensuremath{\Phi}}_{0}}\ensuremath{\sim}1$ and ${\ensuremath{\beta}}_{C}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{I}_{0}{R}^{2}C}{{\ensuremath{\Phi}}_{0}}\ensuremath{\ll}1$, where ${I}_{0}$ is the critical current, $C$ is the junction capacitance, and ${L}_{s}$ is the shunt inductance, showed structure at voltages where the Josephson frequency was near a subharmonic of the ${L}_{s}C$ resonant frequency. The additional nonlinearity of the $I\ensuremath{-}V$ characteristic caused mixing down of noise near higher harmonics of the Josephson frequency, thereby greatly enhancing the voltage noise. The measured noise was in good agreement with that predicted by computer simulations.Measurements have been made of the low-frequency spectral density of the voltage noise in current-biased resistively shunted Josephson tunnel junctions under conditions in which the noise mixed-down from frequencies near the Josephson frequency (${\ensuremath{\nu}}_{J}$) to the measurement frequency ($\ensuremath{\ll}{\ensuremath{\nu}}_{J}$) is in the regime $h{\ensuremath{\nu}}_{J}>{k}_{B}T$. In this limit, quantum corrections to the mixed-down noise are important. The spectral densities measured on junctions with current-voltage characteristics close to the Stewart-McCumber model were in excellent agreement with the predicted values, with no fitted parameters. The mixed-down noise for a wide range of bias voltages was used to infer the spectral density of the current noise in the shunt resistor at frequency $\ensuremath{\nu}$. With no fitted parameters, this spectral density at frequencies up to 500 GHz was in excellent agreement with the prediction $(\frac{2h\ensuremath{\nu}}{R})coth(\frac{h\ensuremath{\nu}}{2{k}_{B}T})$. The presence of the zero-point term, $\frac{2h\ensuremath{\nu}}{R}$, at frequencies $h\ensuremath{\nu}>{k}_{B}T$ was clearly demonstrated. The current-voltage characteristics of a junction with ${\ensuremath{\beta}}_{L}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{L}_{s}{I}_{0}}{{\ensuremath{\Phi}}_{0}}\ensuremath{\sim}1$ and ${\ensuremath{\beta}}_{C}\ensuremath{\equiv}\frac{2\ensuremath{\pi}{I}_{0}{R}^{2}C}{{\ensuremath{\Phi}}_{0}}\ensuremath{\ll}1$, where ${I}_{0}$ is the critical current, $C$ is the junction capacitance, and ${L}_{s}$ is the shunt inductance, showed structure at voltages where the Josephson frequency was near a subharmonic of the ${L}_{s}C$ resonant frequency. The additional nonlinearity of the $I\ensuremath{-}V$ characteristic caused mixing down of noise near higher harmonics of the Josephson frequency, thereby greatly enhancing the voltage noise. The measured noise was in good agreement with that predicted by computer simulations.

Physical review. B, Condensed matterJul 1, 1982117 citations

Decoherence in a Josephson-junction qubit

A. J. Berkley, H. Xu, M. A. Gubrud +4 more

The zero-voltage state of a Josephson junction biased with constant current consists of a set of metastable quantum energy levels. We probe the spacings of these levels by using microwaves to induce transitions and thereby enhance the escape rate to the voltage state. The widths of the resonances give a measurement of the spectroscopic coherence time of the two metastable states involved in the transitions. We observe a decoherence time shorter than that expected from dissipation alone in resonantly isolated $20\ifmmode\times\else\texttimes\fi{}5\ensuremath{\mu}{\mathrm{m}}^{2}$ area $\mathrm{Al}/\mathrm{AlO}x/\mathrm{Al}$ junctions at 60 mK. The data are well fit by a model that includes the dephasing effects of both low-frequency current noise and the escape rate to the voltage state. We discuss implications for quantum computation using current-biased Josephson-junction qubits, including limits on the minimum number of levels needed in the well.The zero-voltage state of a Josephson junction biased with constant current consists of a set of metastable quantum energy levels. We probe the spacings of these levels by using microwaves to induce transitions and thereby enhance the escape rate to the voltage state. The widths of the resonances give a measurement of the spectroscopic coherence time of the two metastable states involved in the transitions. We observe a decoherence time shorter than that expected from dissipation alone in resonantly isolated $20\ifmmode\times\else\texttimes\fi{}5\ensuremath{\mu}{\mathrm{m}}^{2}$ area $\mathrm{Al}/\mathrm{AlO}x/\mathrm{Al}$ junctions at 60 mK. The data are well fit by a model that includes the dephasing effects of both low-frequency current noise and the escape rate to the voltage state. We discuss implications for quantum computation using current-biased Josephson-junction qubits, including limits on the minimum number of levels needed in the well.

arXiv:quant-ph/0303001Physical review. B, Condensed matterAug 18, 200350 citations

Observation of quantum stochastic synchronization in a dissipative quantum system

G. M. Xue, Ming Gong, Huikai Xu +8 more

We report the observation of quantum stochastic synchronization (QSS) in a dissipative quantum system---a moderately damped current-biased Josephson junction. We show that when the junction is biased subcritically by a combination of a constant current and a small sinusoidal modulation signal, forced phase synchronization between the otherwise random, incoherent quantum tunneling and the signal may arise as a consequence of the interplay between quantum fluctuation and the weak periodic drive. It is found that optimal synchronization occurs at a signal frequency ${f}_{\text{QSS}}$ that is comparable to the quantum tunneling rate. We also find a phenomenon called ``modulation induced suppression of quantum noise'': for signal frequency well below ${f}_{\text{QSS}},$ the power spectral density of escape time distribution of the junction could be reduced substantially from the floor level of the unmodulated system. Furthermore, we demonstrate that the application of a small signal with proper phase can suppress the average tunneling rate and enhance the stability of the metastable system. Our experimental results agree well with the numerical calculations without the use of adjustable parameters.We report the observation of quantum stochastic synchronization (QSS) in a dissipative quantum system---a moderately damped current-biased Josephson junction. We show that when the junction is biased subcritically by a combination of a constant current and a small sinusoidal modulation signal, forced phase synchronization between the otherwise random, incoherent quantum tunneling and the signal may arise as a consequence of the interplay between quantum fluctuation and the weak periodic drive. It is found that optimal synchronization occurs at a signal frequency ${f}_{\text{QSS}}$ that is comparable to the quantum tunneling rate. We also find a phenomenon called ``modulation induced suppression of quantum noise'': for signal frequency well below ${f}_{\text{QSS}},$ the power spectral density of escape time distribution of the junction could be reduced substantially from the floor level of the unmodulated system. Furthermore, we demonstrate that the application of a small signal with proper phase can suppress the average tunneling rate and enhance the stability of the metastable system. Our experimental results agree well with the numerical calculations without the use of adjustable parameters.

Physical Review BDec 3, 201411 citations

Suppression of macroscopic quantum tunneling in a large Josephson junction coupled to a resonator

Brad R. Trees, Yaser H. Helal, Joshua Schiffrin +1 more

We employ a functional integral technique to calculate the macroscopic quantum tunneling rate at zero temperature of a current-biased Josephson junction weakly coupled to a resonator. We allow for the effects of environmental dissipation on both the junction and the resonator via baths of harmonic oscillators, and we consider both cases of weak and strong junction damping. The resonator can be, in principle, any quantum harmonic oscillator that couples to the junction bilinearly in either the coordinate or the velocity. The Josephson phase difference $\ensuremath{\phi}$ is the tunneling variable in this system, and the low-lying bound states in the junction's potential energy well are weakly metastable due to tunneling through a finite-width energy barrier. There has been some interest in using such biased junctions as qubits with the resonator providing multiqubit coupling. The main result of this work is that coupling to the resonator has a suppressive effect on the junction's tunneling: the stronger the coupling strength between the junction and resonator, the greater the reduction of the tunneling rate. Including damping to the junction also suppresses tunneling, a well-known result, but damping of the resonator actually reduces the magnitude of suppression; i.e., damping the resonator partially counteracts the suppression provided directly by the junction-resonator coupling. Details of the junction-resonator coupling yield interesting variations on this theme that may be useful for qubit design. For example, the coupling ${U}_{\mathit{int}}$ between a current-biased junction and an AlN dilatational resonator should depend on the angular frequency of the resonator ${\ensuremath{\omega}}_{R}$ in a power-law fashion [A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)], specifically ${U}_{\mathit{int}}\ensuremath{\propto}{({\ensuremath{\omega}}_{R})}^{n}$, where $n=1∕2$ . We find that for any value of $n$ in the range $0<n<1$ the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for $n=1∕2$ , the tunneling rate is maximally suppressed for ${\ensuremath{\omega}}_{R}∕{\ensuremath{\omega}}_{J}\ensuremath{\approx}1$, where ${\ensuremath{\omega}}_{J}$ is the bias current-dependent plasma frequency of the junction.We employ a functional integral technique to calculate the macroscopic quantum tunneling rate at zero temperature of a current-biased Josephson junction weakly coupled to a resonator. We allow for the effects of environmental dissipation on both the junction and the resonator via baths of harmonic oscillators, and we consider both cases of weak and strong junction damping. The resonator can be, in principle, any quantum harmonic oscillator that couples to the junction bilinearly in either the coordinate or the velocity. The Josephson phase difference $\ensuremath{\phi}$ is the tunneling variable in this system, and the low-lying bound states in the junction's potential energy well are weakly metastable due to tunneling through a finite-width energy barrier. There has been some interest in using such biased junctions as qubits with the resonator providing multiqubit coupling. The main result of this work is that coupling to the resonator has a suppressive effect on the junction's tunneling: the stronger the coupling strength between the junction and resonator, the greater the reduction of the tunneling rate. Including damping to the junction also suppresses tunneling, a well-known result, but damping of the resonator actually reduces the magnitude of suppression; i.e., damping the resonator partially counteracts the suppression provided directly by the junction-resonator coupling. Details of the junction-resonator coupling yield interesting variations on this theme that may be useful for qubit design. For example, the coupling ${U}_{\mathit{int}}$ between a current-biased junction and an AlN dilatational resonator should depend on the angular frequency of the resonator ${\ensuremath{\omega}}_{R}$ in a power-law fashion [A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)], specifically ${U}_{\mathit{int}}\ensuremath{\propto}{({\ensuremath{\omega}}_{R})}^{n}$, where $n=1∕2$. We find that for any value of $n$ in the range $0<n<1$ the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for $n=1∕2$, the tunneling rate is maximally suppressed for ${\ensuremath{\omega}}_{R}∕{\ensuremath{\omega}}_{J}\ensuremath{\approx}1$, where ${\ensuremath{\omega}}_{J}$ is the bias current-dependent plasma frequency of the junction.

Physical Review BDec 13, 20076 citations

Decoherence in Josephson vortex quantum bits: Long-Josephson-junction approach to a two-state system

Ju H. Kim, Ramesh P. Dhungana, Kee-Su Park

We investigated decoherence of a Josephson vortex quantum bit (qubit) in dissipative and noisy environment. As the Josephson vortex qubit (JVQ) is fabricated by using a long Josephson junction (LJJ), we use the perturbed sine-Gordon equation to describe the phase dynamics representing a two-state system and estimate the effects of quasiparticle dissipation and weakly fluctuating critical and bias currents on the relaxation time ${T}_{1}$ and on the dephasing time ${T}_{\ensuremath{\phi}}$. We show that the critical current fluctuation does not contribute to dephasing of the qubit in the lowest order approximation. Modeling the weak current variation from magnetic field fluctuations in the LJJ by using the Gaussian colored noise with long correlation time, we show that the coherence time ${T}_{2}$ is limited by the low frequency current noise at very low temperatures. Also, we show that a ultra-long coherence time may be obtained from the JVQ by using experimentally accessible value of physical parameters.We investigated decoherence of a Josephson vortex quantum bit (qubit) in dissipative and noisy environment. As the Josephson vortex qubit (JVQ) is fabricated by using a long Josephson junction (LJJ), we use the perturbed sine-Gordon equation to describe the phase dynamics representing a two-state system and estimate the effects of quasiparticle dissipation and weakly fluctuating critical and bias currents on the relaxation time ${T}_{1}$ and on the dephasing time ${T}_{\ensuremath{\phi}}$. We show that the critical current fluctuation does not contribute to dephasing of the qubit in the lowest order approximation. Modeling the weak current variation from magnetic field fluctuations in the LJJ by using the Gaussian colored noise with long correlation time, we show that the coherence time ${T}_{2}$ is limited by the low frequency current noise at very low temperatures. Also, we show that a ultra-long coherence time may be obtained from the JVQ by using experimentally accessible value of physical parameters.

Physical Review BJun 6, 200621 citations

A method of determining microwave dissipation of Josephson junctions with non-linear frequency response

Heng-Jie Chen, Hang Xue, Shaoxiong Li +1 more

Based on Josephson junction (JJ), superconducting quantum bit (qubit) is operated at frequencies of several GHz. Dissipation of JJs in this frequency range can cause energy relaxation in qubits, and limit coherence time, therefore it is highly concerned and needs to be determined quantitatively. The dissipation of JJs can be quantified by microwave quality factor. It is usually done at very low temperature (~mK) to determine whether a JJ is suitable for qubit devices by measuring the quality factor. In this paper, a method based on nonlinear frequency response of JJs is proposed to determine the quality factor. This method can be used in thermal activation regime, which may bring great conveniences to experiments. To analyze high frequency properties of JJs, the dynamic equation of a current-biased JJ, which describes high frequency oscillation of the JJ, is introduced. A fourth-order potential approximation is used to obtain the analytical equation of non-linear response. The dependence on quality factor, as well as on amplitude, of difference between JJ’s plasma frequency and resonant frequency, is derived from the equation. The approximate treatment is quantitatively validated by our numerical simulations with practical JJ parameters including different environment influences. Thus, based on nonlinear frequency response of JJs, a reliable and simple method to determine quality factor of JJ is proposed, which is desirable for exploring JJ based microwave devices such as parametric amplifier, superconducting qubit. Being driven well into the nonlinear microwave response regime, due to frequency-amplitude interaction, the resonant frequency of a current bias JJ deviates from the JJ’s plasma frequency. The deviation is directly related to the microwave quality factor. Hence, the quality factor can be deducted from the experimental measurement of the resonant frequency deviation, with different microwave power values applied. In comparison with linear resonance experiment, the nonlinear resonance used by the proposed method produces stronger signal. Therefore it is more robust against external noise. When being conducted at high temperature, the proposed method is more reliable. The accuracy of the measured quality factor primarily depends on those of the JJ’s parameters such as critical current and capacitance, while those parameters can be experimentally determined with high precision.Based on Josephson junction (JJ), superconducting quantum bit (qubit) is operated at frequencies of several GHz. Dissipation of JJs in this frequency range can cause energy relaxation in qubits, and limit coherence time, therefore it is highly concerned and needs to be determined quantitatively. The dissipation of JJs can be quantified by microwave quality factor. It is usually done at very low temperature (~mK) to determine whether a JJ is suitable for qubit devices by measuring the quality factor. In this paper, a method based on nonlinear frequency response of JJs is proposed to determine the quality factor. This method can be used in thermal activation regime, which may bring great conveniences to experiments. To analyze high frequency properties of JJs, the dynamic equation of a current-biased JJ, which describes high frequency oscillation of the JJ, is introduced. A fourth-order potential approximation is used to obtain the analytical equation of non-linear response. The dependence on quality factor, as well as on amplitude, of difference between JJ’s plasma frequency and resonant frequency, is derived from the equation. The approximate treatment is quantitatively validated by our numerical simulations with practical JJ parameters including different environment influences. Thus, based on nonlinear frequency response of JJs, a reliable and simple method to determine quality factor of JJ is proposed, which is desirable for exploring JJ based microwave devices such as parametric amplifier, superconducting qubit. Being driven well into the nonlinear microwave response regime, due to frequency-amplitude interaction, the resonant frequency of a current bias JJ deviates from the JJ’s plasma frequency. The deviation is directly related to the microwave quality factor. Hence, the quality factor can be deducted from the experimental measurement of the resonant frequency deviation, with different microwave power values applied. In comparison with linear resonance experiment, the nonlinear resonance used by the proposed method produces stronger signal. Therefore it is more robust against external noise. When being conducted at high temperature, the proposed method is more reliable. The accuracy of the measured quality factor primarily depends on those of the JJ’s parameters such as critical current and capacitance, while those parameters can be experimentally determined with high precision.

Acta Physica SinicaJan 1, 20191 citation

{T}_{2}^{*}

as a function of the energy relaxation time

{T}_{1}

and the current noise power spectral density. For current noise with a constant spectral density up to a frequency

{f}_{c}⪢{(2\ensuremath{\pi}{T}_{1})}^{\ensuremath{-}1}

, we find that the resonance broadening is proportional to the noise spectral density, while for

{f}_{c}⪡{(2\ensuremath{\pi}{T}_{1})}^{\ensuremath{-}1}

, the broadening is proportional to the rms current noise. We compare our results to microwave spectroscopy data on

100\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}{\mathrm{m}}^{2}

Nb and Al tunnel junctions at milli-Kelvin temperatures and find good agreement between the measured response spectra and our model simulations.

{I}_{0}

is the critical current,

C

is the junction capacitance, and

{L}_{s}

is the shunt inductance, showed structure at voltages where the Josephson frequency was near a subharmonic of the

{L}_{s}C

resonant frequency. The additional nonlinearity of the $I\ensuremath{-}V$ characteristic caused mixing down of noise near higher harmonics of the Josephson frequency, thereby greatly enhancing the voltage noise. The measured noise was in good agreement with that predicted by computer simulations.

\mathrm{Al}/\mathrm{AlO}x/\mathrm{Al}

junctions at 60 mK. The data are well fit by a model that includes the dephasing effects of both low-frequency current noise and the escape rate to the voltage state. We discuss implications for quantum computation using current-biased Josephson-junction qubits, including limits on the minimum number of levels needed in the well.

{f}_{\text{QSS}}

that is comparable to the quantum tunneling rate. We also find a phenomenon called ``modulation induced suppression of quantum noise'': for signal frequency well below

{f}_{\text{QSS}},

the power spectral density of escape time distribution of the junction could be reduced substantially from the floor level of the unmodulated system. Furthermore, we demonstrate that the application of a small signal with proper phase can suppress the average tunneling rate and enhance the stability of the metastable system. Our experimental results agree well with the numerical calculations without the use of adjustable parameters.

{U}_{\mathit{int}}

between a current-biased junction and an AlN dilatational resonator should depend on the angular frequency of the resonator ${\ensuremath{\omega}}_{R}$ in a power-law fashion [A. N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)], specifically ${U}_{\mathit{int}}\ensuremath{\propto}{({\ensuremath{\omega}}_{R})}^{n}$, where

n=1∕2

. We find that for any value of

n

in the range

0<n<1

the junction's tunneling rate exhibits a nonmonotonic dependence on ${\ensuremath{\omega}}_{R}$, and for

n=1∕2

, the tunneling rate is maximally suppressed for ${\ensuremath{\omega}}_{R}∕{\ensuremath{\omega}}_{J}\ensuremath{\approx}1$, where ${\ensuremath{\omega}}_{J}$ is the bias current-dependent plasma frequency of the junction.

{T}_{1}

and on the dephasing time ${T}_{\ensuremath{\phi}}$. We show that the critical current fluctuation does not contribute to dephasing of the qubit in the lowest order approximation. Modeling the weak current variation from magnetic field fluctuations in the LJJ by using the Gaussian colored noise with long correlation time, we show that the coherence time

{T}_{2}

is limited by the low frequency current noise at very low temperatures. Also, we show that a ultra-long coherence time may be obtained from the JVQ by using experimentally accessible value of physical parameters.