chalmers university of technology superconducting qubits...

Chalmers University of Technology

Superconducting qubits (Phase qubit)

Quantum informatics (FKA 172)

Thilo Bauch ([email protected])

Quantum Device Physics Laboratory, MC2, Chalmers University of Technology


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


Basic features of quantum integrated circuits

Low dissipation: superconductivity

Zero resistance is necessary condition for the

preservation of quantum coherence

Cooper pair condensate described by a single wave

function

Superconducting qubits

Resi

stan

ce (Ω

)

Temperature (K)


Superconducting materials

Al 1.2 K 170 µeV Nb 9.2 K 1.5 meV

transition temperature

Energy gap

Cooper pair

Ψ(r)=nS1/2eiθ(r)

Cooper pair condensate is descirbed by a single

wave function

where nS : Cooper pair density


Energy scale of quantum integrated circuits

kBT<<ħω01<<Δ |0>

|1> ħω01

Typical energies (see later): ω01/2π ≈ 5-20 GHz

No thermal excitations! We are able to prepare system in ground state |0>

Two level system protected from quasi-particles (low intrinsic dissipation)

From kBT=hν 1 GHz corresponds to 50 mK => Experiments in dilution refrigerator


Examples of superconducting qubits

NEC, Chalmers, Yale, JPL

CEA Saclay TU Delft, MIT, IPHT Jena

NIST, UCSB

charge charge/phase phase flux


Key ingredient: non-linear, non-dissipative element.

Tunnel (Josephson) junction

Superconductor 1 Superconductor 2 Tunnel barrier

Ψ1=nS1/2eiθ1 Ψ2=nS

1/2eiθ2

|Ψ| |Ψ2| x

x

|Ψ1|


Tunnel (Josephson) junction Josephson equations

Superconductor 1 Superconductor 2 Tunnel barrier

Ψ1=nS1/2eiθ1 Ψ2=nS

1/2eiθ2

Josephson 1

Josephson 2

dissipationless (Josephson) current

finite voltage state


Josephson inductance

Any change in Josephson current will result in a finite voltage across the Josephson junction

The junction acts like a (nonlinear) inductor!!


Equation of motion for the current biased Josephson junction

Ib R C Ic or LJ

The bias current splits into three currents. From the currents through the resistor, capacitor, and

Josephson element we get:

By replacing the voltage across the parallel RCLJ 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.


Dynamics of the current biased Josephson junction

plasma frequency

ωP

ΔU

barrier height

quality factor, only the real part of the admittance causes dissipation

Josephson inductance

junction capacitance The shunting admittance= (impedance)-1 is frequency dependent and accounts for all processes causing dissipation.


Curr

ent

0

IC

Voltage 2Δ/e 0

Current Voltage Characteristic of a Josephson junction WITHOUT thermal

or quantum fluctuations

1

2

3

4

6

Slope of phase particle trajectory is determined by the quality factor Q. The lower Q the steeper the trajectory (more energy loss).

Tilt of the washboard potential is determined by the bias current

Slope 1/RN

Ir

5


Properties of the current voltage characteristics of a Josephson junction

Curr

ent

0 Ir

Voltage 2Δ/e 0

1

2 3

4

6

Slope 1/RN

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. 200-210 in “Introduction to Superconductivity”,

Second Edition, by M. Tinkham, McGraw-Hill, Inc.)

5

Ic

(π/4)(2Δ/e)


Superconducting QUantum Interference Device (SQUID) Ib

See T. van Duzer, “Principles of Superconductive Devices and Circuits”, 2nd edition, Prentice Hall


Escape mechanisms from V=0 to V=0

cross over temperature

Thermal Activation

H.A. Kramers, Pysica 7, 284 (1940)

Macroscopic Quantum Tunneling

A.O. Caldeira, A.J. Leggett, PRL 46, 211 (1981)

ΔU


Curr

ent

0

IC

Voltage 2Δ/e 0

Current Voltage Characteristic of a Josephson junction WITH thermal

or quantum fluctuations

1

2

3

4

6

IS

switching current IS < critical current IC

5

stochastic process

IS


Switching probability from V=0 to V=0

Probability to switch in time interval between t and t+dt

Probability NOT to switch up to time t

Probability to switch after (small) time interval dt

Ib

IC

t

Bias junction at a fixed current < IC and wait until junction switches and measure time difference

t0

V

t

tS

t0=0 repeat 10000 times+

histogram

Here and stands for thermal or quantum escape rate


t

Fix current pulse length and height (<IC)

V

.....

Ib

IC

t

Probability to switch during a current pulse of height I and width

number of current pulses number of switching events

Switching probability P(I) =


Energy levels in tilted washboard potential

Ic=3.9 µA C=3.1 pF Ib/IC=0.977

For bias currents close to the critical current we can approximate the system by a two level system!!


0

1

Macroscopic quantum tunneling rates

Possibility to distinguish between 0 and 1 state!!


Measurement of Rabi oscillations Bi

as c

urre

nt I

b IC

0 time

time

Har

mon

ic m

w si

gnal

0

For a fixed microwave pulse length repeat sequence 5000 times to accumulate statistics

Read out pulse 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.

At the working point the barrier height is large enough to prevent the phase particle to escape the matastable well

Working point current pulse


J. Claudon Phys. Rev. Lett. 93, 187003, (2004)

0 1

Spectroscopy, relaxation time and Rabi oscillations using magnetic flux read out pulse


pulse

Schematics of switching event measurement


3He-4He dilution refrigerator

Base temperature T=15 mK

Gas handling panel Qubit frequency: GHz (275 mK)


Qubit: Nb/AlOx/Nb SQUID 5

mm

Magnetic flux line

SQUID

DC bias current


DC current pulse DC flux pulse + MW

shaping pulse

mixer

MW source

Counter (events+ time difference)

Power combiner/

divider

comparator


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 T1 of the first excited state for a fixed bias current

Measure Rabi oscillations for a fixed bias current; determine roughly the driven coherence time.

chalmers university of technology superconducting qubits...

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