^{1}, M. H. Devoret

^{2}and I. Siddiqi

^{1}

### Abstract

We review the theory, fabrication, and implementation of the Josephsonbifurcationamplifier (JBA). At the core of the JBA is a nonlinear oscillator based on a reactively shunted Josephson junction. A weak input signal to the amplifier couples to the junctioncritical current and results in a dispersive shift in the resonator plasma frequency . This shift is enhanced by biasing the junction with a sufficiently strong microwave current to access the nonlinear regime where varies with . For a drive frequency such that , the oscillator enters the bistable regime where two nondissipative dynamical states and , which differ in amplitude and phase, can exist. The sharp dependent transition from to forms the basis for a sensitive digital threshold amplifier. In the vicinity of the bistable regime , analog amplification of continuous signals is also possible. We present experimental data characterizing amplifier performance and discuss two specific applications—the readout of superconducting qubits (digital mode) and dispersive microwave magnetometry (analog mode).

We would like to thank M. Metcalfe, C. Rigetti, E. Boaknin, and V. Manucharyan for their contribution to the JBA and CBA experiments. We would like to acknowledge M. Hatridge for his contribution to the magnetometry experiments. Funding from the following sources is acknowledged: AFOSR under Grant No. FA9550-08-1-0104 (R.V. and I.S.), ONR under Grant No. N00014-07-1-0774 (I.S.), NSA under ARO Contract No. W911NF-05-1-0365, NSF under Grant No. DMR-0653377 (M.H.D.), the Keck Foundation (M.H.D.), the Agence Nationale de la Recherche (M.H.D.) and College de France (M.H.D.).

I. INTRODUCTION

II. THEORY

III. DESIGN AND FABRICATION

IV. EXPERIMENTAL SETUP

V. RESULTS

A. Frequency domain measurements

B. Time domain measurements

VI. APPLICATIONS

A. Qubit readout

B. High speed magnetometry

VII. CONCLUSIONS

### Key Topics

- Qubits
- 47.0
- Bifurcations
- 45.0
- Josephson junctions
- 35.0
- Critical currents
- 34.0
- Oscillators
- 33.0

## Figures

Dissipative (a) and dispersive (b) techniques to determine . In (a), we directly sample the barrier height by biasing with a current . By measuring the escape rate out of the zero voltage state, one can infer the height of the barrier. In (b), we drive the system with an ac excitation and measure the response at the drive frequency to determine the plasma frequency .

Dissipative (a) and dispersive (b) techniques to determine . In (a), we directly sample the barrier height by biasing with a current . By measuring the escape rate out of the zero voltage state, one can infer the height of the barrier. In (b), we drive the system with an ac excitation and measure the response at the drive frequency to determine the plasma frequency .

Computed steady state junction (a) voltage amplitude and (b) phase of the JBA when driven with an ac excitation . The different traces correspond to different drive amplitudes . Solutions are plotted as a function of normalized drive frequency for an oscillator with . and denote the low amplitude phase lagging and the high amplitude phase leading oscillation states, respectively. The black dot indicates a typical bias point where the system is highly sensitive to changes in .

Computed steady state junction (a) voltage amplitude and (b) phase of the JBA when driven with an ac excitation . The different traces correspond to different drive amplitudes . Solutions are plotted as a function of normalized drive frequency for an oscillator with . and denote the low amplitude phase lagging and the high amplitude phase leading oscillation states, respectively. The black dot indicates a typical bias point where the system is highly sensitive to changes in .

Schematic diagram of the JBA. A junction with critical current and a parametrically coupled input is driven by a rf pump signal, which provides the power for amplification. When biased in the vicinity of the dynamical bifurcation point, the phase of the reflected signal depends sensitively on the input signal. The circulator (C) is used to couple signals in and out of the resonator. It also suppresses the noise of the following amplifier reaching the oscillator and helps ensure that the fluctuations felt by the oscillator correspond to the noise of a resistor at the bath temperature . Inset: examples of parametric input coupling circuits using a SQUID and S-SET.

Schematic diagram of the JBA. A junction with critical current and a parametrically coupled input is driven by a rf pump signal, which provides the power for amplification. When biased in the vicinity of the dynamical bifurcation point, the phase of the reflected signal depends sensitively on the input signal. The circulator (C) is used to couple signals in and out of the resonator. It also suppresses the noise of the following amplifier reaching the oscillator and helps ensure that the fluctuations felt by the oscillator correspond to the noise of a resistor at the bath temperature . Inset: examples of parametric input coupling circuits using a SQUID and S-SET.

(a) Circuit diagram of the JBA. (b) Circuit diagram of the CBA with a half-wave transmission line cavity embedded with a Josephson junction in the middle. (c) The equivalent LCR circuit for the CBA.

(a) Circuit diagram of the JBA. (b) Circuit diagram of the CBA with a half-wave transmission line cavity embedded with a Josephson junction in the middle. (c) The equivalent LCR circuit for the CBA.

Principle of operation of the JBA. The curves in all three panels are the reflected signal phase response for three different drive powers. The red (right) and blue (left) curves correspond to two different values of the small oscillation plasma frequency . The value of for the blue curve is indicated by the vertical solid line. Panel (a) is for the smallest drive power and corresponds to linear regime. For a larger drive power in (b), the JBA oscillator is in the single-valued nonlinear regime. When biased with sufficient power to access the bistable regime (c), the phase response exhibits a sharp transition. Thus, the maximum phase shift for a given change in increases with drive power [(a)–(c)] and is achieved at the bias points with increasing detuning as indicated by the vertical dashed lines.

Principle of operation of the JBA. The curves in all three panels are the reflected signal phase response for three different drive powers. The red (right) and blue (left) curves correspond to two different values of the small oscillation plasma frequency . The value of for the blue curve is indicated by the vertical solid line. Panel (a) is for the smallest drive power and corresponds to linear regime. For a larger drive power in (b), the JBA oscillator is in the single-valued nonlinear regime. When biased with sufficient power to access the bistable regime (c), the phase response exhibits a sharp transition. Thus, the maximum phase shift for a given change in increases with drive power [(a)–(c)] and is achieved at the bias points with increasing detuning as indicated by the vertical dashed lines.

Steady state solutions of Eq. (4) for increasing (left to right) values of reduced detuning . The magnitude squared of reduced oscillation amplitude is plotted as a function of reduced drive power . The and axes have been scaled to clearly depict the variation with . We observe multivalued solutions for . The turning points of the curves for are the bifurcation points and are indicated for the curve with .

Steady state solutions of Eq. (4) for increasing (left to right) values of reduced detuning . The magnitude squared of reduced oscillation amplitude is plotted as a function of reduced drive power . The and axes have been scaled to clearly depict the variation with . We observe multivalued solutions for . The turning points of the curves for are the bifurcation points and are indicated for the curve with .

Switching probability of the JBA when energized with a pulse of duration and amplitude . The curves indicate the probability of switching, as a function of from the low amplitude to the high amplitude state when biased near the upper bifurcation point . The red (left) and blue (right) curves correspond to smaller and larger junction critical currents, respectively. The shift depends on the change in and the relative detuning . The width depends on the intensity of fluctuations in the oscillator.

Switching probability of the JBA when energized with a pulse of duration and amplitude . The curves indicate the probability of switching, as a function of from the low amplitude to the high amplitude state when biased near the upper bifurcation point . The red (left) and blue (right) curves correspond to smaller and larger junction critical currents, respectively. The shift depends on the change in and the relative detuning . The width depends on the intensity of fluctuations in the oscillator.

Optical image of the microfabricated on-chip capacitor is shown in the left panel. The top Al layer forms two capacitors in series with the Cu underlayer as the common electrode. Top right panel indicates the profile of the different layers in the ground plane. The additional layers (Ti and Cr) sandwiching the Cu layer are used to protect the Cu layer during the deposition of SiN. These additional layers are not required for Nb ground planes. The bottom right panel shows a SEM image of a typical Josephson junction, which is shunted by the capacitor. (b) Circuit schematic of the JBA chip including stray elements. The junction inductance is shunted with a capacitor including the stray inductance and stray resistance . is the intrinsic junction capacitance. The stray inductance and resistance arise due to imperfect screening currents and the finite conductivity of the ground planes.

Optical image of the microfabricated on-chip capacitor is shown in the left panel. The top Al layer forms two capacitors in series with the Cu underlayer as the common electrode. Top right panel indicates the profile of the different layers in the ground plane. The additional layers (Ti and Cr) sandwiching the Cu layer are used to protect the Cu layer during the deposition of SiN. These additional layers are not required for Nb ground planes. The bottom right panel shows a SEM image of a typical Josephson junction, which is shunted by the capacitor. (b) Circuit schematic of the JBA chip including stray elements. The junction inductance is shunted with a capacitor including the stray inductance and stray resistance . is the intrinsic junction capacitance. The stray inductance and resistance arise due to imperfect screening currents and the finite conductivity of the ground planes.

Cryogenic microwave measurement setup. The rf excitation line is attenuated and filtered with reactive components. The signal return line has isolators and lossy filters. Typical values of components along with the type of coaxial cables used are indicated.

Cryogenic microwave measurement setup. The rf excitation line is attenuated and filtered with reactive components. The signal return line has isolators and lossy filters. Typical values of components along with the type of coaxial cables used are indicated.

Schematic (a) and optical image (b) of the combined differential rf and dc biasing scheme. Shielded twisted-pair wires implement a four wire dc measurement. A 180° hybrid coupler is used to split the rf signal into two out of phase components, creating the differential rf drive. The rf and dc signals are combined using a bias tee before they reach the device (inside blue rectangle). A superconducting Al box (green rectangle) is used to shield the device from low frequency magnetic fields since the twisted pairs have to be separated before they can be combined with the rf signals. This differential biasing scheme rejects noise over a wide frequency range.

Schematic (a) and optical image (b) of the combined differential rf and dc biasing scheme. Shielded twisted-pair wires implement a four wire dc measurement. A 180° hybrid coupler is used to split the rf signal into two out of phase components, creating the differential rf drive. The rf and dc signals are combined using a bias tee before they reach the device (inside blue rectangle). A superconducting Al box (green rectangle) is used to shield the device from low frequency magnetic fields since the twisted pairs have to be separated before they can be combined with the rf signals. This differential biasing scheme rejects noise over a wide frequency range.

Normalized reflected signal phase as a function of excitation frequency for sample 5. The open circles are measured data for . The solid line is calculated from the equivalent circuit model shown in the inset. The magnitude of the reflected signal is unity within experimental uncertainty.

Normalized reflected signal phase as a function of excitation frequency for sample 5. The open circles are measured data for . The solid line is calculated from the equivalent circuit model shown in the inset. The magnitude of the reflected signal is unity within experimental uncertainty.

Inverse squared of the plasma frequency as a function of the inverse critical current for samples 1, 2, 4, and 5. Solid lines are linear fits to the data corresponding to the model in Fig. 11 with a single best fit line drawn for samples 1 and 2, which nominally differ only in .

Inverse squared of the plasma frequency as a function of the inverse critical current for samples 1, 2, 4, and 5. Solid lines are linear fits to the data corresponding to the model in Fig. 11 with a single best fit line drawn for samples 1 and 2, which nominally differ only in .

Normalized reflected signal phase as a function of excitation frequency and excitation power is shown for sample 5. Experimental data are shown in the bottom panel, while the top panel is the result of numerical simulations. A vertical slice taken at (dashed line) shows the abrupt transition between two oscillation states of the system.

Normalized reflected signal phase as a function of excitation frequency and excitation power is shown for sample 5. Experimental data are shown in the bottom panel, while the top panel is the result of numerical simulations. A vertical slice taken at (dashed line) shows the abrupt transition between two oscillation states of the system.

Histograms of the reflected signal phase at . The upper histogram contains counts with a measurement time . The lower panel, taken with the latching technique, has counts with a measurement time . The dashed line represents the discrimination threshold between the and states.

Histograms of the reflected signal phase at . The upper histogram contains counts with a measurement time . The lower panel, taken with the latching technique, has counts with a measurement time . The dashed line represents the discrimination threshold between the and states.

Switching probability curves at as a function of the normalized drive current . The discrimination power is the maximum difference between the two curves. The two curves differ by approximately 1% in , and the curve corresponding to the higher critical current lies at higher values of .

Switching probability curves at as a function of the normalized drive current . The discrimination power is the maximum difference between the two curves. The two curves differ by approximately 1% in , and the curve corresponding to the higher critical current lies at higher values of .

The reduced escape rate as a function of at different bath temperatures (decreasing from left to right). Data are shown for two samples with (a) and and (b) and . The solid lines are straight line fits to the data. The slope and intercept of these fits are used to extract the escape temperature and the bifurcation current .

The reduced escape rate as a function of at different bath temperatures (decreasing from left to right). Data are shown for two samples with (a) and and (b) and . The solid lines are straight line fits to the data. The slope and intercept of these fits are used to extract the escape temperature and the bifurcation current .

vs obtained from the data in Fig. 16. The solid lines are a plot of Eq. (16) for and , respectively. Data show excellent agreement with the theoretical prediction. The dashed line is the classical dependence . The arrow indicates the lowest escape temperature measured in the dc escape measurements for the corresponding sample.

vs obtained from the data in Fig. 16. The solid lines are a plot of Eq. (16) for and , respectively. Data show excellent agreement with the theoretical prediction. The dashed line is the classical dependence . The arrow indicates the lowest escape temperature measured in the dc escape measurements for the corresponding sample.

Schematic of the qubit measurement setup. The quantronium qubit is comprised of two small Josephson junctions and a large junction in a loop. This latter junction is shunted by a capacitor and forms the nonlinear oscillator of the JBA. The qubit state is manipulated by sending pulses to the gate (write port), while readout operation is performed by sending a pulse to the nonlinear resonator via the read port. The circulator (C) is used to separate the incident and reflected signals. The phase of the reflected signal carries information about the qubit state.

Schematic of the qubit measurement setup. The quantronium qubit is comprised of two small Josephson junctions and a large junction in a loop. This latter junction is shunted by a capacitor and forms the nonlinear oscillator of the JBA. The qubit state is manipulated by sending pulses to the gate (write port), while readout operation is performed by sending a pulse to the nonlinear resonator via the read port. The circulator (C) is used to separate the incident and reflected signals. The phase of the reflected signal carries information about the qubit state.

Summary of qubit coherence measurements. The Larmor frequency of the qubit is 9.513 GHz with . (a) Rabi oscillations as a function of the duration of a square pulse applied on the gate. Solid green curve is an exponentially decaying sinusoidal fit with . (b) Decay of the excited state, prepared by applying a pulse, as a function of the waiting time between the preparation and readout pulses. Solid green curve is an exponential fit with a decay constant . The dashed black line indicates the value of in the absence of a pulse. (c) Ramsey fringes obtained with two pulses separated by the time interval . The pulse frequency was detuned from the Larmor frequency by 20 MHz. The green curve is an exponentially decaying sinusoidal fit. The decay time is 320 ns. (d) Switching probability as a function of drive current amplitude for ground and excited qubit states. The vertical dotted line is the value of the drive current at which the maximal difference in is observed. The solid line connects the observed data points in the state, and the dashed line is a copy of the solid line horizontally shifted to overlap the state data at low values of .

Summary of qubit coherence measurements. The Larmor frequency of the qubit is 9.513 GHz with . (a) Rabi oscillations as a function of the duration of a square pulse applied on the gate. Solid green curve is an exponentially decaying sinusoidal fit with . (b) Decay of the excited state, prepared by applying a pulse, as a function of the waiting time between the preparation and readout pulses. Solid green curve is an exponential fit with a decay constant . The dashed black line indicates the value of in the absence of a pulse. (c) Ramsey fringes obtained with two pulses separated by the time interval . The pulse frequency was detuned from the Larmor frequency by 20 MHz. The green curve is an exponentially decaying sinusoidal fit. The decay time is 320 ns. (d) Switching probability as a function of drive current amplitude for ground and excited qubit states. The vertical dotted line is the value of the drive current at which the maximal difference in is observed. The solid line connects the observed data points in the state, and the dashed line is a copy of the solid line horizontally shifted to overlap the state data at low values of .

(a) Schematic of a resonator terminated with an unshunted two junction SQUID. The input flux applied via a coil modulates the critical current of the SQUID, resulting in a shift in the resonant frequency. (b) Optical image of Al coplanar stripline resonator terminated with an unshunted SQUID, shown in the SEM image on the right.

(a) Schematic of a resonator terminated with an unshunted two junction SQUID. The input flux applied via a coil modulates the critical current of the SQUID, resulting in a shift in the resonant frequency. (b) Optical image of Al coplanar stripline resonator terminated with an unshunted SQUID, shown in the SEM image on the right.

Reflected signal phase (color) as a function of drive power and frequency. This plot is similar to the one in Fig. 13, but here the data were obtained by sweeping power in both directions at a given frequency. Alternate sweep directions are interlaced, allowing one to see the hysteresis in the driven response (striped region in the top left corner). The upper and lower bifurcation boundaries are clearly visible. Inset: reflected signal phase (color) as a function of drive frequency and flux bias for a drive power of −140 dBm. The linear resonant frequency shows the expected periodic dependence on flux bias.

Reflected signal phase (color) as a function of drive power and frequency. This plot is similar to the one in Fig. 13, but here the data were obtained by sweeping power in both directions at a given frequency. Alternate sweep directions are interlaced, allowing one to see the hysteresis in the driven response (striped region in the top left corner). The upper and lower bifurcation boundaries are clearly visible. Inset: reflected signal phase (color) as a function of drive frequency and flux bias for a drive power of −140 dBm. The linear resonant frequency shows the expected periodic dependence on flux bias.

Effective flux noise of the magnetometer as a function of drive power. This quantity is the smallest change in the flux one can detect in a 1 Hz bandwidth. Data are shown for a different sample than the one in Fig. 21. The dotted curves (from top to bottom) correspond to data for three increasing values of dc flux bias points. Lowest effective flux noise obtained is . The dashed lines indicate the expected reduction in flux noise with power for a comparable linear resonator. Biasing in the nonlinear regime yields an order of magnitude reduction in flux noise.

Effective flux noise of the magnetometer as a function of drive power. This quantity is the smallest change in the flux one can detect in a 1 Hz bandwidth. Data are shown for a different sample than the one in Fig. 21. The dotted curves (from top to bottom) correspond to data for three increasing values of dc flux bias points. Lowest effective flux noise obtained is . The dashed lines indicate the expected reduction in flux noise with power for a comparable linear resonator. Biasing in the nonlinear regime yields an order of magnitude reduction in flux noise.

## Tables

Variables used in Eqs. (2)–(4) expressed in terms of circuit parameters.

Variables used in Eqs. (2)–(4) expressed in terms of circuit parameters.

Sample parameters from Ref. 11. and are measured values. and are fit values to the data. Samples 1 and 2 have a 100 nm thick Au underlayer, sample 3 has a 50 nm thick Nb underlayer, samples 4 and 6 have a thick Cu underlayer, and sample 5 has a 200 nm thick Nb underlayer.

Sample parameters from Ref. 11. and are measured values. and are fit values to the data. Samples 1 and 2 have a 100 nm thick Au underlayer, sample 3 has a 50 nm thick Nb underlayer, samples 4 and 6 have a thick Cu underlayer, and sample 5 has a 200 nm thick Nb underlayer.

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