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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
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
Chalmers University of Technology
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)
Chalmers University of Technology
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
Chalmers University of Technology
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
Chalmers University of Technology
Examples of superconducting qubits
NEC, Chalmers, Yale, JPL
CEA Saclay TU Delft, MIT, IPHT Jena
NIST, UCSB
charge charge/phase phase flux
Chalmers University of Technology
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|
Chalmers University of Technology
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
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Josephson inductance
Any change in Josephson current will result in a finite voltage across the Josephson junction
The junction acts like a (nonlinear) inductor!!
Chalmers University of Technology
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.
Chalmers University of Technology
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.
Chalmers University of Technology
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
Chalmers University of Technology
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)
Chalmers University of Technology
Superconducting QUantum Interference Device (SQUID) Ib
See T. van Duzer, “Principles of Superconductive Devices and Circuits”, 2nd edition, Prentice Hall
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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
Chalmers University of Technology
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
Chalmers University of Technology
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
Chalmers University of Technology
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) =
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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!!
Chalmers University of Technology
0
1
Macroscopic quantum tunneling rates
Possibility to distinguish between 0 and 1 state!!
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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
Chalmers University of Technology
J. Claudon Phys. Rev. Lett. 93, 187003, (2004)
0 1
Spectroscopy, relaxation time and Rabi oscillations using magnetic flux read out pulse
Chalmers University of Technology
pulse
Schematics of switching event measurement
Chalmers University of Technology
3He-4He dilution refrigerator
Base temperature T=15 mK
Gas handling panel Qubit frequency: GHz (275 mK)
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Qubit: Nb/AlOx/Nb SQUID 5
mm
Magnetic flux line
SQUID
DC bias current
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DC current pulse DC flux pulse + MW
shaping pulse
mixer
MW source
Counter (events+ time difference)
Power combiner/
divider
comparator
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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.