Superconducting qubits (Phase qubit) Quantum informatics (FKA 172)

Transcription

1 Superconducting qubits (Phase qubit) Quantum informatics (FKA 172) Thilo Bauch Quantum Device Physics Laboratory, MC2, Chalmers University of Technology

2 Qubit proposals for implementing a quantum computer Microscopic degree of freedom Spin of electrons or nuclei Transition dipoles of atoms or ions in vacuum +Very well isolated from environmet -Hard to couple (qubit-qubit, qubit-control/readout) Quantum integrated circuits Collective electrodynamic modes of macroscopic electrical elements +-Intrinsically large electromagnetic cross-section

3 Basic features of quantum integrated circuits Resistance (Ω) Low dissipation: superconductivity Zero resistance is necessary condition for the preservation of quantum coherence Cooper pair condensate described by a single wave function Temperature (K) Superconducting qubits

4 Superconducting materials transition temperature Energy gap Al 1.2 K 170 µev Nb 9.2 K 1.5 mev Cooper pair condensate is descirbed by a single wave function Cooper pair Ψ(r)=n S 1/2 e iθ(r) where n S : Cooper pair density

5 Energy scale of quantum integrated circuits ħω 01 1> 0> k B T<<ħω 01 <<Δ No thermal excitations! We are able to prepare system in ground state 0> Two level system protected from quasi-particles (low intrinsic dissipation) Typical energies (see later): ω 01 /2π 5-20 GHz From k B T=hν 1 GHz corresponds to 50 mk => Experiments in dilution refrigerator

6 Examples of superconducting qubits charge charge/phase phase flux NEC, Chalmers, Yale, JPL CEA Saclay TU Delft, MIT, IPHT Jena NIST, UCSB

7 Key ingredient: non-linear, non-dissipative element. Tunnel (Josephson) junction Tunnel Superconductor 1 barrier Superconductor 2 Ψ 1 =n 1/2 S e iθ 1 Ψ 2 =n 1/2 S e iθ 2 x Ψ Ψ 1 Ψ 2 x

8 Tunnel (Josephson) junction Josephson equations Chalmers University of Technology Tunnel Superconductor 1 barrier Superconductor 2 Ψ 1 =n S 1/2 e iθ 1 Ψ 2 =n S 1/2 e iθ 2 Josephson 1 dissipationless (Josephson) current Josephson 2 finite voltage state

9 Josephson inductance Any change in Josephson current will result in a finite voltage across the Josephson junction The junction acts like a (nonlinear) inductor!!

10 Equation of motion for the current biased Josephson junction Chalmers University of Technology The bias current splits into three currents. From the currents through the resistor, capacitor, and Josephson element we get: I b R C I c or L J By replacing the voltage across the parallel RCL J circuit using the second Josephson equation (see 2 slides before) we get the equation of motion for the fictitious phase particle with mass proportional to the capacitance: From this equation we can directly determine the potential of the system U.

11 Dynamics of the current biased Josephson junction junction capacitance Josephson inductance The shunting admittance= (impedance) -1 is frequency dependent and accounts for all processes causing dissipation. ω P ΔU barrier height plasma frequency quality factor, only the real part of the admittance causes dissipation

12 Tilt of the washboard potential is determined by the bias current Current Voltage Characteristic of a Josephson junction WITHOUT thermal or quantum fluctuations 4 Slope 1/R N I C 3 Current I r Voltage 2Δ/e Slope of phase particle trajectory is determined by the quality factor Q. The lower Q the steeper the trajectory (more energy loss).

13 Properties of the current voltage characteristics of a Josephson junction Chalmers University of Technology Ambegaokar-Baratoff: In the tunnel limit (barrier transparency << 1) where is the superconducting gap and is the absolute value of the elementary charge. where is the retrapping current. (see pp in Introduction to Superconductivity, Second Edition, by M. Tinkham, McGraw-Hill, Inc.) I c Current I r (π/4)(2δ/e) Slope 1/R N 5 4 2Δ/e Voltage

14 Superconducting QUantum Interference Device I b (SQUID) See T. van Duzer, Principles of Superconductive Devices and Circuits, 2nd edition, Prentice Hall

15 Escape mechanisms from V=0 to V=0 Thermal Activation Chalmers University of Technology ΔU H.A. Kramers, Pysica 7, 284 (1940) Macroscopic Quantum Tunneling cross over temperature A.O. Caldeira, A.J. Leggett, PRL 46, 211 (1981)

16 Current Voltage Characteristic of a Josephson junction WITH thermal or quantum fluctuations Chalmers University of Technology switching current I S < critical current I C stochastic process 4 Current I C I S 3 2 I S Voltage 2Δ/e

17 Switching probability from V=0 to V=0 Probability to switch after (small) time interval dt Here and stands for thermal or quantum escape rate Probability to switch in time interval between t and t+dt Probability NOT to switch up to time t I b Bias junction at a fixed current < I C and wait until junction switches and measure time difference I C V t 0 =0 t S t t repeat times+ histogram t 0

18 Probability to switch during a current pulse of height I and width I b Fix current pulse length and height (<I C ) I C V t... Switching probability P(I) = number of switching events number of current pulses t

19 Energy levels in tilted washboard potential Ic=3.9 µa C=3.1 pf I b /I C =0.977 Chalmers University of Technology For bias currents close to the critical current we can approximate the system by a two level system!!

20 Macroscopic quantum tunneling rates Chalmers University of Technology 1 0 Possibility to distinguish between 0 and 1 state!!

21 At the working point the barrier height is large enough to prevent the phase particle to escape the matastable well Measurement of Rabi oscillations Harmonic mw signal Bias current I b I C 0 0 Working point current pulse Read out pulse time time At the measuring point the barrier is lowered by applying a short dc current pulse on top of the long bias current pulse. If qubit is in state 1: switching (escape of phase particle) probability is very high. If qubit is in state 0: Switching probability is very low. For a fixed microwave pulse length repeat sequence 5000 times to accumulate statistics

22 Spectroscopy, relaxation time and Rabi oscillations using magnetic flux read out pulse 1 0 J. Claudon Phys. Rev. Lett. 93, , (2004)

23 Schematics of switching event measurement pulse

24 3 He- 4 He dilution refrigerator Gas handling panel Base temperature T=15 mk Qubit frequency: GHz (275 mk)

25 Qubit: Nb/AlO x /Nb SQUID Chalmers University of Technology SQUID 5 mm Magnetic flux line DC bias current

26 comparator DC current pulse DC flux pulse + MW shaping pulse MW source Counter (events+ time difference) mixer Power combiner/ divider

27 Measure probability for the SQUID/junction to switch in time interval t and t+dt as a function of bias current, when the SQUID/junction is in the ground state Measure the relaxation time T 1 of the first excited state for a fixed bias current Measure Rabi oscillations for a fixed bias current; determine roughly the driven coherence time.