arXiv:1508.04317 · Physical review. B./Physical review. B

Effective model for a short Josephson junction with a phase discontinuity

E. Goldobin, Sergey V. Mironov, A. I. Buzdin, R. G. Mint︠s︡, D. Koelle, R. Kleiner

Published Apr 25, 2016• 11 citations

Abstract

We consider a short Josephson junction with a phase discontinuity $\ensuremath{\kappa}$ created, e.g., by a pair of tiny current injectors, at some point ${x}_{0}$ along the width of the junction. We derive the effective current-phase relation (CPR) for the system as a whole, i.e., reduce it to an effective pointlike junction. From the effective CPR we obtain the ground state of the system and predict the dependence of its critical current on $\ensuremath{\kappa}$. We show that in a large range of $\ensuremath{\kappa}$ values the effective junction behaves as a ${\ensuremath{\varphi}}_{0}$ Josephson junction, i.e., has a unique ground state phase ${\ensuremath{\varphi}}_{0}$ within each $2\ensuremath{\pi}$ interval. For $\ensuremath{\kappa}\ensuremath{\approx}\ensuremath{\pi}$ and ${x}_{0}$ near the middle of the junction one obtains a ${\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$ junction, i.e., a Josephson junction with degenerate ground state phase ${\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$ within each $2\ensuremath{\pi}$ interval. Further, in view of possible escape experiments especially in the quantum domain, we investigate the scaling of the energy barrier and eigenfrequency close to the critical currents and predict the behavior of the escape histogram width $\ensuremath{\sigma}(\ensuremath{\kappa})$ in the regime of the macroscopic quantum tunneling.

DOI: 10.1103/physrevb.93.134514

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Weak coupling Josephson junction as a current probe: effect of dissipation on escape dynamics

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arXiv:1508.04317 · Physical review. B./Physical review. B

Effective model for a short Josephson junction with a phase discontinuity

E. Goldobin, Sergey V. Mironov, A. I. Buzdin, R. G. Mint︠s︡, D. Koelle, R. Kleiner

Published Apr 25, 2016• 11 citations

arXiv HTML PDF

Abstract

We consider a short Josephson junction with a phase discontinuity

\ensuremath{\kappa}

created, e.g., by a pair of tiny current injectors, at some point

{x}_{0}

along the width of the junction. We derive the effective current-phase relation (CPR) for the system as a whole, i.e., reduce it to an effective pointlike junction. From the effective CPR we obtain the ground state of the system and predict the dependence of its critical current on

\ensuremath{\kappa}

. We show that in a large range of

\ensuremath{\kappa}

values the effective junction behaves as a

{\ensuremath{\varphi}}_{0}

Josephson junction, i.e., has a unique ground state phase

{\ensuremath{\varphi}}_{0}

within each

2\ensuremath{\pi}

interval. For

\ensuremath{\kappa}\ensuremath{\approx}\ensuremath{\pi}

and

{x}_{0}

near the middle of the junction one obtains a

{\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}

junction, i.e., a Josephson junction with degenerate ground state phase

{\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}

within each

2\ensuremath{\pi}

interval. Further, in view of possible escape experiments especially in the quantum domain, we investigate the scaling of the energy barrier and eigenfrequency close to the critical currents and predict the behavior of the escape histogram width

\ensuremath{\sigma}(\ensuremath{\kappa})

in the regime of the macroscopic quantum tunneling.

DOI: 10.1103/physrevb.93.134514

Related papers

Evidence of macroscopic quantum tunneling from both wells in a φ Josephson junction

R. Menditto, Michael Merker, M. Siegel +3 more

We study $\mathrm{Nb}\ensuremath{-}{\mathrm{AlO}}_{x}$-Nb Josephson junctions (JJs) with a phase-discontinuity $\ensuremath{\kappa}$ created by a pair of current injectors attached to one of the Nb electrodes. For $\ensuremath{\kappa}\ensuremath{\approx}\ensuremath{\pi}$ the Josephson potential energy $U$ as a function of the average phase $\ensuremath{\psi}$ across the JJ has the form of a $2\ensuremath{\pi}$-periodic double-well potential. Thus, the device behaves as a $\ensuremath{\varphi}$ JJ with degenerate ground state phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$ (the value of $\ensuremath{\varphi}$ depends on the system parameters). Experimentally, the existence of two wells of the potential is confirmed by the observation of two different critical currents ${I}_{\mathrm{c}\ifmmode\pm\else\textpm\fi{}}$, corresponding to the escape from different wells. We investigate the escape of the Josephson phase from both wells by collecting statistics of the switching currents. The histogram of switching current exhibits two peaks corresponding to ${I}_{\mathrm{c}\ifmmode\pm\else\textpm\fi{}}$. The dependence of the width ${\ensuremath{\sigma}}_{+}$ and ${\ensuremath{\sigma}}_{\ensuremath{-}}$ of each peak on the bath temperature $T$ indicates the transition from thermal activation to macroscopic quantum tunneling (MQT) at ${T}^{★}\ensuremath{\approx}260\phantom{\rule{0.28em}{0ex}}\mathrm{mK}$ as $T$ decreases. We argue that the observed saturation value of ${\ensuremath{\sigma}}_{+}$ and ${\ensuremath{\sigma}}_{\ensuremath{-}}$ below ${T}^{★}$ is indeed related to quantum tunneling rather than to parasitic noise in the system, as the histogram width can be reduced by tuning the value of $\ensuremath{\kappa}$ away from $\ensuremath{\pi}$. The comparison of the experimental escape rate $\mathrm{\ensuremath{\Gamma}}$ with theoretical predictions further confirms MQT of the phase $\ensuremath{\psi}$ from both wells.We study $\mathrm{Nb}\ensuremath{-}{\mathrm{AlO}}_{x}$-Nb Josephson junctions (JJs) with a phase-discontinuity $\ensuremath{\kappa}$ created by a pair of current injectors attached to one of the Nb electrodes. For $\ensuremath{\kappa}\ensuremath{\approx}\ensuremath{\pi}$ the Josephson potential energy $U$ as a function of the average phase $\ensuremath{\psi}$ across the JJ has the form of a $2\ensuremath{\pi}$-periodic double-well potential. Thus, the device behaves as a $\ensuremath{\varphi}$ JJ with degenerate ground state phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$ (the value of $\ensuremath{\varphi}$ depends on the system parameters). Experimentally, the existence of two wells of the potential is confirmed by the observation of two different critical currents ${I}_{\mathrm{c}\ifmmode\pm\else\textpm\fi{}}$, corresponding to the escape from different wells. We investigate the escape of the Josephson phase from both wells by collecting statistics of the switching currents. The histogram of switching current exhibits two peaks corresponding to ${I}_{\mathrm{c}\ifmmode\pm\else\textpm\fi{}}$. The dependence of the width ${\ensuremath{\sigma}}_{+}$ and ${\ensuremath{\sigma}}_{\ensuremath{-}}$ of each peak on the bath temperature $T$ indicates the transition from thermal activation to macroscopic quantum tunneling (MQT) at ${T}^{★}\ensuremath{\approx}260\phantom{\rule{0.28em}{0ex}}\mathrm{mK}$ as $T$ decreases. We argue that the observed saturation value of ${\ensuremath{\sigma}}_{+}$ and ${\ensuremath{\sigma}}_{\ensuremath{-}}$ below ${T}^{★}$ is indeed related to quantum tunneling rather than to parasitic noise in the system, as the histogram width can be reduced by tuning the value of $\ensuremath{\kappa}$ away from $\ensuremath{\pi}$. The comparison of the experimental escape rate $\mathrm{\ensuremath{\Gamma}}$ with theoretical predictions further confirms MQT of the phase $\ensuremath{\psi}$ from both wells.

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arXiv:1207.3013Physical Review LettersSep 7, 2012142 citations

Weak coupling Josephson junction as a current probe: effect of dissipation on escape dynamics

Jani M. Kivioja, T E Nieminen, Julien Claudon +3 more

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Tunable ±φ,φ0 , and φ0±φ Josephson junction

E. Goldobin, D. Koelle, R. Kleiner

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Quantum Mechanics of a Macroscopic Variable: The Phase Difference of a Josephson Junction

John L. Clarke, Andrew N. Cleland, Michel Devoret +2 more

Experiments to investigate the quantum behavior of a macroscopic degree of freedom, namely the phase difference across a Josephson tunnel junction, are described. The experiments involve measurements of the escape rate of the junction from its zero voltage state. Low temperature measurements of the escape rate for junctions that are either nearly undamped or moderately damped agree very closely with predictions for macroscopic quantum tunneling, with no adjustable parameters. Microwave spectroscopy reveals quantized energy levels in the potential well of the junction in excellent agreement with quantum-mechanical calculations. The system can be regarded as a "macroscopic nucleus with wires."Experiments to investigate the quantum behavior of a macroscopic degree of freedom, namely the phase difference across a Josephson tunnel junction, are described. The experiments involve measurements of the escape rate of the junction from its zero voltage state. Low temperature measurements of the escape rate for junctions that are either nearly undamped or moderately damped agree very closely with predictions for macroscopic quantum tunneling, with no adjustable parameters. Microwave spectroscopy reveals quantized energy levels in the potential well of the junction in excellent agreement with quantum-mechanical calculations. The system can be regarded as a "macroscopic nucleus with wires."

ScienceFeb 26, 1988359 citations

Resonant enhancement of macroscopic quantum tunneling in Josephson junctions: Influence of coherent two-level systems

M. V. Fistul

We report a theoretical study of the macroscopic quantum tunneling (MQT) in small Josephson junctions containing randomly distributed two-level systems. We focus on a Josephson phase escape for switching from the superconducting (the zero-voltage) state to a resistive one. Above the crossover temperature ${T}_{cr}$ the thermal fluctuations of the Josephson phase induce such a switching, and as $T<{T}_{cr}$ the regime of the MQT occurs. In the absence of two-level systems (TLSs) a magnetic field applied parallel to the junction plane results in a smooth reduction of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$, where $\mathrm{\ensuremath{\Phi}}$ is an applied magnetic flux. As the TLSs are present in Josephson junctions we obtain a resonant enhancement of the MQT. This phenomenon manifests itself by a narrow peak in the dependence of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$ occurring in the intermediate range of $\mathrm{\ensuremath{\Phi}}$, i.e., $0<\mathrm{\ensuremath{\Phi}}<{\ensuremath{\phi}}_{0}$ (${\ensuremath{\phi}}_{0}$ is the magnetic flux quantum). We explain this effect quantitatively by a strong resonant suppression of the potential barrier for the Josephson phase escape that is due to the coherent quantum Rabi oscillations in two-level systems present in the junction.We report a theoretical study of the macroscopic quantum tunneling (MQT) in small Josephson junctions containing randomly distributed two-level systems. We focus on a Josephson phase escape for switching from the superconducting (the zero-voltage) state to a resistive one. Above the crossover temperature ${T}_{cr}$ the thermal fluctuations of the Josephson phase induce such a switching, and as $T<{T}_{cr}$ the regime of the MQT occurs. In the absence of two-level systems (TLSs) a magnetic field applied parallel to the junction plane results in a smooth reduction of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$, where $\mathrm{\ensuremath{\Phi}}$ is an applied magnetic flux. As the TLSs are present in Josephson junctions we obtain a resonant enhancement of the MQT. This phenomenon manifests itself by a narrow peak in the dependence of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$ occurring in the intermediate range of $\mathrm{\ensuremath{\Phi}}$, i.e., $0<\mathrm{\ensuremath{\Phi}}<{\ensuremath{\phi}}_{0}$ (${\ensuremath{\phi}}_{0}$ is the magnetic flux quantum). We explain this effect quantitatively by a strong resonant suppression of the potential barrier for the Josephson phase escape that is due to the coherent quantum Rabi oscillations in two-level systems present in the junction.

Physical Review BJul 9, 20154 citations

We consider a short Josephson junction with a phase discontinuity

\ensuremath{\kappa}

created, e.g., by a pair of tiny current injectors, at some point

{x}_{0}

\ensuremath{\kappa}

. We show that in a large range of

\ensuremath{\kappa}

values the effective junction behaves as a

{\ensuremath{\varphi}}_{0}

Josephson junction, i.e., has a unique ground state phase

{\ensuremath{\varphi}}_{0}

within each

2\ensuremath{\pi}

interval. For

\ensuremath{\kappa}\ensuremath{\approx}\ensuremath{\pi}

and

{x}_{0}

near the middle of the junction one obtains a

{\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}

junction, i.e., a Josephson junction with degenerate ground state phase

{\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}

within each

2\ensuremath{\pi}

\ensuremath{\sigma}(\ensuremath{\kappa})

in the regime of the macroscopic quantum tunneling.

U

as a function of the average phase $\ensuremath{\psi}$ across the JJ has the form of a $2\ensuremath{\pi}$-periodic double-well potential. Thus, the device behaves as a $\ensuremath{\varphi}$ JJ with degenerate ground state phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$ (the value of $\ensuremath{\varphi}$ depends on the system parameters). Experimentally, the existence of two wells of the potential is confirmed by the observation of two different critical currents ${I}_{\mathrm{c}\ifmmode\pm\else\textpm\fi{}}$, corresponding to the escape from different wells. We investigate the escape of the Josephson phase from both wells by collecting statistics of the switching currents. The histogram of switching current exhibits two peaks corresponding to ${I}_{\mathrm{c}\ifmmode\pm\else\textpm\fi{}}$. The dependence of the width ${\ensuremath{\sigma}}_{+}$ and ${\ensuremath{\sigma}}_{\ensuremath{-}}$ of each peak on the bath temperature

T

indicates the transition from thermal activation to macroscopic quantum tunneling (MQT) at ${T}^{★}\ensuremath{\approx}260\phantom{\rule{0.28em}{0ex}}\mathrm{mK}$ as

T

decreases. We argue that the observed saturation value of ${\ensuremath{\sigma}}_{+}$ and ${\ensuremath{\sigma}}_{\ensuremath{-}}$ below

{T}^{★}

is indeed related to quantum tunneling rather than to parasitic noise in the system, as the histogram width can be reduced by tuning the value of $\ensuremath{\kappa}$ away from $\ensuremath{\pi}$. The comparison of the experimental escape rate $\mathrm{\ensuremath{\Gamma}}$ with theoretical predictions further confirms MQT of the phase $\ensuremath{\psi}$ from both wells.

{T}_{cr}

the thermal fluctuations of the Josephson phase induce such a switching, and as

T<{T}_{cr}

the regime of the MQT occurs. In the absence of two-level systems (TLSs) a magnetic field applied parallel to the junction plane results in a smooth reduction of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$, where $\mathrm{\ensuremath{\Phi}}$ is an applied magnetic flux. As the TLSs are present in Josephson junctions we obtain a resonant enhancement of the MQT. This phenomenon manifests itself by a narrow peak in the dependence of ${T}_{cr}(\mathrm{\ensuremath{\Phi}})$ occurring in the intermediate range of $\mathrm{\ensuremath{\Phi}}$, i.e., $0<\mathrm{\ensuremath{\Phi}}<{\ensuremath{\phi}}_{0}$ (${\ensuremath{\phi}}_{0}$ is the magnetic flux quantum). We explain this effect quantitatively by a strong resonant suppression of the potential barrier for the Josephson phase escape that is due to the coherent quantum Rabi oscillations in two-level systems present in the junction.