Quantum Optics with Propagating Microwaves in Superconducting Circuits I
2015 AMO Summer School
Io-Chun, Hoi
Outline
1. Introduction to quantum electrical circuits
2. Introduction to superconducting artificial atom
3. Quantum optics with superconducting circuits
4. Single atom scattering
Io-Chun Hoi
Introduction to quantum electrical circuits
Io-Chun Hoi
Io-Chun Hoi
Quantum electrical circuits
Coherent superposition states:
Q
Φ
ChargeFlux
Charge on a capacitor: Current or magnetic flux in an inductor:
+1
2( )
+1
2( )
Probabilistic character.
The superposition states collapse when measure.
Properties:
Macrosopic system
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Conventional electrical circuits First transistor 1947
Introduced 2007Clock speed >3GHzNumber of transistors820millionManufacturing technology 45nm
Dual-core Intel processor
Basic elements:
Fig. from Intel
Fig. from Intel
Introduction to superconducting artificial atom
Io-Chun Hoi
Io-Chun Hoi
Superconducting circuits are like LEGOS
Basic Elements of Superconducting Circuits
Capacitance Inductance
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Josephson Junction:Non-disspative nonlinear inductance
LJ L C
Tunnel barrier between two superconductors
Al
Al
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Fabrication of Josephson Junction
Φ
+Q
−Q
U
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Constructing linear quantum electrical circuits
H = ω(a†a +1
2)
H =Q2
2C+Φ2
2L
H =Q̂2
2C+Φ̂2
2L
ω =1
LC
Classical physics:Quantum mechanics:
ω
Φ̂,Q̂⎡⎣ ⎤⎦ = i
Analogy with a moving particle in a harmonic potential H =
p2
2m+1
2kx2
M. H. Devoret, A. Wallraff, and J. M. Martinis. Superconducting qubits: A short review. http://arxiv.org/abs/cond-mat/0411174v1, 2004.
Quantization
∼ GHz
LCΦ
-1.0
-0.5
0.0
0.5
1.0
Ener
gy(E
J)
-4 -2 0 2 4Phi (rad)
0
1
2
3U
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Constructing nonlinear Quantum circuit: Artificial atom
U = −EJ cosφ
φ
LJ =
4eIc cos πΦext
Φ0
⎛
⎝⎜⎞
⎠⎟
Replace linear inductance by Josephson junction(Nonlinear inductance)
Transition become addressable!
Emission spectrum
Frequencyω01ω12
α =ω01 −ω12
Anharmonicity:
C LJ
kBT << ω << Δ s
How to operate electrical circuits quantum mechanically?
Avoid dissipation
Work at low temperaturesProvide reset of the circuit(Ground state)
Avoid broaden energy levels
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Superconducting gap energy
T@mKω / 2π ∼ 4 − 8GHz
Family of superconducting artificial atom
Focus on Cooper Pair Box and Transmon! G. Wendin and V. S. ShumeikoLow Temp. Phys., 33(9):724-744, 2007.
J. Clarke and F. K. Wilhelm. Nature, 453:1031–1042, 2008.
Fig. fromMichel Devoret. Linneaus summer school in quantum engineering. 2010.
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4
3
2
1
0
Energy
(Ec)
1.00.80.60.40.20.0ng
EJ/Ec=0.5
|0>
|0>|1>
|1>
1/√2(|0>-|1>)
1/√2(|0>+|1>)
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Artificial atom I: The Single-Cooper Pair Box
EQ = 4EC =(2e)2
2CΣ
ng = CgVg / (2e)
CΣ = Cg +CJ
But the coherence time is short (few ns)due to charge noise! Y. Nakamura et al. Nature, 398:786–788, 1999.
H = −1
2Echσ z −
1
2EJσ x
Ech = EQ (1− 2ng )
Map to a spin 1/2 particle inmagnetic field.
Depends on external flux
σ z ,σ x :Pauli matrix
EJ / Ec < 1
Coherent oscillations between ground state and excited state in time domain, demonstrated by
Decoherence of artificial atom
Relaxation rate Pure dephasing rate
Random switching
Enviroment Enviroment
Phase randomization ω01→ω01 + δω01(t)
e− iω01t1 → 0
Γ01 Γϕ
(Effect from the environment)
ω01
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CS
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Artificial atom II: The transmon
25
20
15
10
5
0
Energy
(Ec)
1.00.80.60.40.20.0ng
EJ/Ec=30
Jens Koch et al.
Insensitive to the charge noise
Long coherence time.Physical Review A, 76(4):042319, 2007.
20 < EJ / Ec < 100
8
6
4
2
0
Energy
(Ec)
1.00.80.60.40.20.0ng
EJ/Ec=0.5
8
6
4
2
0
Energ
y(Ec
)
1.00.80.60.40.20.0ng
EJ/Ec=1
8
6
4
2
0
Energ
y(Ec
)
1.00.80.60.40.20.0ng
EJ/Ec=5
25
20
15
10
5
0
Energy
(Ec)
1.00.80.60.40.20.0ng
EJ/Ec=30
-1.0
-0.5
0.0
0.5
1.0
Ener
gy(E
J)
-4 -2 0 2 4Phi (rad)
Natural atomOptical photons
Superconducting artificial atom Microwave photons
0
1
2
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Compare with optical photon, the frequency of microwave photon is 106 less.
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Superconducting circuitsQuantum optics
Microwave photonsOptical photons
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Comparison of the toolboxes
Detect I, Q
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1. Photons and “atom” interaction can be engineered 2. The photons can be guided by waveguides; beam alignment is not needed.
3. Large vacuum field E0,rms 0.2V / m due to small mode volume 4. Standard on-chip fabrication technique 5. Tunable transition energy of the “atom” 6. Mechanical stable
Atom-light interaction on single photon level
Advantages of quantum circuit
dE0
Quantum optics with superconducting circuits
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Io-Chun Hoi
Io-Chun Hoi
Fig: O. Astafiev, et al. 327, 840 Science (2010)
Resonant scattering
Incoming light Atom/dipole emits light
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Resonant scattering in 3D space
G. Wrigge et al. Nature Phys. 4, 60 (2008). M. Tey et al. Nature Phys. 4, 924 (2008).
The extinction signal is due to interference
Incoming light
Sum
Resonant scattering in 3D spaceAtom/dipole emits light
U. Håkanson
Fig. from
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Fully coherent: no transmission, perfect reflection.
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D.E. Chang et al. Nature Physics 3, 807(2007)
Resonant scattering in 1D waveguide
Fully coherent: no transmission, perfect reflection.
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Relaxation dominated by transmission line.O. Astafiev, et al. 327, 840 Science (2010) IoChun, Hoi et al. PRL 107, 073601 (2011)
λ >> d λ ∼ cm d ∼ μm Size of “atom”
Wavelength of EM field
Point like atom/dipole!
Al
Al
D.E. Chang et al. Nature Physics 3, 807(2007)
Resonant scattering in 1D waveguide
Fully coherent: no transmission, perfect reflection.
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Relaxation dominated by transmission line.O. Astafiev, et al. 327, 840 Science (2010) IoChun, Hoi et al. PRL 107, 073601 (2011)
λ >> d λ ∼ cm d ∼ μm Size of “atom”
Wavelength of EM field
Point like atom/dipole!
Al
Al
2nm
Fig. from E. Olsson & S. M. Nik
JJ
D.E. Chang et al. Nature Physics 3, 807(2007)
Resonant scattering in 1D waveguide
CC
CJS
φJ
φ0φ1Lφ2L φ2Rφ1R2LLL0 L0 L0 L0
C0 C0 C0 C0
Quantum circuit model
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Relaxation rate into 1D transmission line, indicates the strength of coupling!
Γ10ω012 Cc
2Z
4CΣ
CΣ = Cc + CJSZ =
L0C0
Strong interaction limit:
Fully coherent.
Transmission and reflection
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Saturation of transmission
Nonlinear nature of the atom!
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Transmission comparing to theory
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12
10
8
6
4
2
-140 -135 -130 -125 -120[dBm]
5 6 7 8 910
2 3 4 5 6 7 8
[MHz]
Total scattered BW=10MHz BW=100MHz
Elastic scattered Input field
Output Power(nW)
/ 2πΩ p
Pp
VR
2
VR
2
Ω p/2π
BW
30 MHz 83 MHz 250 MHz
δω p / 2π
ω10
Ω p
Coherent vs Incoherent scattering
I.-C. Hoi et al.
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Tunable artificial atom
Φext /Φ0Φext /Φ0
f01
f12
( f12 + f01) / 2
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Two-Photon Transition
Low powerHigh power
Extract: EJ ,Max = 13GHzEc = 590MHzEJ / Ec = 23
Only two-photon transition occurs!Only 0-1 transition occurs!
Fully coherent: perfect reflected by the atom.
measure the phase coherent signal.
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Two-Tone Spectroscopy
Two-Tone Spectroscopy
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ω p =ω12
T
1.0
0.8
0.6
0.4
0.2
0.07.47.27.06.86.66.46.2
GHz
Pump @ 7.1GHz Pump off-135dBm -131dBm -127dBm -123dBm -119dBm -115dBm
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ω p / 2π
Higher level effect
(Low Power)
Anharmonicity: ω12 / 2π = 6.38GHzω10 / 2π = 7.1GHz α =ω01 −ω12 720MHz
5.2
5.0
4.8
4.6
-140 -130 -120 -110
1.0
0.9
0.8
ω10
Ω p
P01 [dBm]
[GHz]
ωp /2π
Tp,1
Mollow triplet
O. Astafiev, et al. 327, 840 Science (2010)
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B.R. Mollow, Phys.Rev. 188, 1969 (1969)
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T
[dBm] Pc
ωp /2π[GHz]
A. A. Abdumalikov, Jr et al. PRL 104, 193601 (2010)
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To be continued…