Josephson Circuits I.JJ RCSJ Model as Circuit ElementDC Current DriveDC Voltage DriveAC Voltage DriveAC Voltage DriveAC Voltage vs Current DrivesVoltage StandardHYPRES: Voltage Standard ChipParametric InductorInductance along V=0 branchThe DC SQUID (damped)DC SQUID Equivalent CircuitDC SQUID Equivalent CircuitDC SQUID Voltage ModulationDC SQUID Voltage ModulationDC SQUID SensitivityDC SQUID and Thermal NoiseEquivalent Tunable JunctionTunable Inductance along V=0 branchThree-Junction Loop MeasurementsThermal Activation of Nb Persistent Current QubitCoupling between qubitsMassachusetts Institute of Technology 6.763 2003 Lecture 14Josephson Circuits I.Outline1. RCSJ Model Review2. Response to DC and AC Drives• Voltage standard3. The DC SQUID4. Tunable Josephson JunctionOctober 21, 2003Massachusetts Institute of Technology 6.763 2003 Lecture 14JJ RCSJ Model as Circuit ElementiandTherefore,Massachusetts Institute of Technology 6.763 2003 Lecture 14DC Current DriveA. Static Solution: Return CurrentB. Dynamical SolutionMassachusetts Institute of Technology 6.763 2003 Lecture 14DC Voltage DriveThe voltage source is DC with v=V0, so thatThe resulting current across the JJ is ac The current across the resistor is dcThe total current is then <i>V0Massachusetts Institute of Technology 6.763 2003 Lecture 14AC Voltage DriveThe voltage sourceThen the guage-invariant phase is The current across the resistor isThe resulting current across the JJ isThe current across the capacitor isThe total current is thenMassachusetts Institute of Technology 6.763 2003 Lecture 14AC Voltage DriveMassachusetts Institute of Technology 6.763 2003 Lecture 14AC Voltage vs Current DrivesMassachusetts Institute of Technology 6.763 2003 Lecture 14Voltage Standard10 V conventional Josephson voltage standard chip. The chip is 1 cm x 2 cm and contains 20,208 series connected Nb-AlOx -Nb junctions.http://www.boulder.nist.gov/div814/div814/whatwedo/volt/dc/JVS.htmlMassachusetts Institute of Technology 6.763 2003 Lecture 14HYPRES: Voltage Standard ChipHYPRES is the only commercial manufacturer of the superconducting integrated circuit used in Primary Voltage Standard Systems. HYPRES chips are used in the primary voltage standards in national laboratories around the world including Italy, France, United Kingdom, Australia, China, Malaysia, Japan, England, Canada, Norway, United States, Netherlands and Mexico.The HYPRES Josephson Junction Array Voltage Standard circuits provide the ultimate accuracy for realizing and maintaining the SI Volt.Features/Specifcations• Niobium/Aluminum Oxide/Niobium, SiO2 dielectric, Niobium wiring technology.• All Niobium technology. Refractory. Impervious to moisture and thermal cycling.• 20,208 Josephson junctions (10 V chip) 3,660 Josephson junctions (1 V chip)• 18 x 38 micrometers junction area.• Installed in a FR-4 epoxy glass mount.• RF input WR-12 waveguide flange.• RF Distribution - 16 way parallel x 1263 cells in series (10 V) - 4 way parallel x 915 cells in series (1 V)• Designed for a frequency range of 72-78 GHz• Operating temperature of 4.2 K• Common DC terminal resistance is < 1 Ohm - typical• Approximately 10 mW operating power at the input flange for 10 V chip (2 mW for 1 V)• -11V to +11 V range for 10 V chip.• -2.5 V to + 2.5 V range for 1 V chip.• Stability time is typically 1 hour for the 10 V chip, 5 hours for the 1V chip• 0.005 PPM accuracy at 10 V (10 V chip)• 0.05 PPM accuracy at 1V (1V chip)• Calibration certificate supplied with each chip.• Two (2) year warrantyhttp://www.hypres.com/Massachusetts Institute of Technology 6.763 2003 Lecture 14Parametric InductorTake the time derivative of the currents, and for the Josephson term:The parameteric (time-dependent) inductance can be defined as*On the zero-voltage branch, for Is<< I0and I0+ Is<< Ic, then I0~ Icsin φ, so thatMassachusetts Institute of Technology 6.763 2003 Lecture 14Inductance along V=0 branch]2[COILπΦ|I/IcMassachusetts Institute of Technology 6.763 2003 Lecture 14The DC SQUID (damped)Φ0Our goal is to show that the DC SQUID circuit in (a) is equivalent to the circuit for a single junction with an effective Icand an effective resistance. The inductance of the SQUID loop in (a) is considered negligible.In fact, we will also show later that there is an equivalent statement even for the underdamped cass.Massachusetts Institute of Technology 6.763 2003 Lecture 14DC SQUID Equivalent CircuitΦ0and the fact that the junctions are identical1/R+Use flux quantization, sinI =IcMassachusetts Institute of Technology 6.763 2003 Lecture 14DC SQUID Equivalent CircuitΦ0withTherefore, for this overdamped equivalent circuit, for i > ICMassachusetts Institute of Technology 6.763 2003 Lecture 14DC SQUID Voltage ModulationFrom Van Duzer and Turner,Figure 5.11a, page 272Massachusetts Institute of Technology 6.763 2003 Lecture 14IBiasDC SQUID Voltage ModulationIBias002ΦoΦovoltageIB1IB2ΦBiasRC-Shunted SQUIDIbiaskept slightly above ICΦbias= 0.75Φofor sensitivityGoal is to maximize:LRddV≈Φ+ V__IB2IB1Massachusetts Institute of Technology 6.763 2003 Lecture 14DC SQUID SensitivityLsΦ0LsAssume that Ls> LJ = Φ0/2 π Ic(Φext), then the range of modulation of the current is aboutThe sensitivity of the output voltage to the input flux isWith RD ~R = 1 Ohm and L ~ 1 nH, then the sensistivity is about one microvolt per flux quantum, so that small fractions of a flux quantum can be measured.Massachusetts Institute of Technology 6.763 2003 Lecture 14DC SQUID and Thermal NoiseLsΦ0LsAssume again that Ls> LJ = Φ0/2 π Ic(Φext), then to prevent thermal noise from effecting the SQUID’s performance, one needs Energy stored in the SQUID >> Thermal Energy Therefore, Ls< 1 nH at 4 K, and 0.1 nH at 100 K.Massachusetts Institute of Technology 6.763 2003 Lecture 14Equivalent Tunable Junctioni+R1C1-++with Lloop<< LJIcsinϕ2Icsinϕ1R2C2v--i++CIcsinϕRv--Massachusetts Institute of Technology 6.763 2003 Lecture 14Tunable Inductance along V=0 branch]2[COILπΦ|Φ [in Φο]Massachusetts Institute of Technology 6.763 2003 Lecture 14Three-Junction Loop Measurements1pF0.45µm1.1µm0.55µm1.1µmI-V-I+V+Three-junction LoopJct. Size ~ 0.45µm, 0.55µmLoop size ~16x16µm2L3-junction~ 30pHIc~1 & 2µAEJ/Ec~ 350 & 550 0≡DC SQUIDShunt capacitors ~ 1pFJct. Size ~ 1.1µm Loop size ~20x20µm2LSQUID~ 50pHIc~10 & 20µAM~35pHJc~350 & 730A/cm21≡20 µm• Measure switching current of DC