dressed state amplification by a superconducting qubit e. il‘ichev, outline introduction:...
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Dressed state amplification by a superconducting qubit
E. Il‘ichev,
Outline
Introduction: Qubit-resonator system
Parametric amplification
Quantum amplifier
Lasing (charge qubit)
Dressed state lasing (flux qubit)
Conclusion
• Superconducting Ring + one or more Josephson junctions
• Contact sizes + process parameters
• Operation: external magnetic flux near half flux quantum Persistent current Ip Energy of external field
)2/(2 0 xIpE
20
Flux qubit
Qubits energy- Eigenbasis
• Tunnel-coupling produces splitting -> D – energy gap
xz hhH 22
• Eigenstates of the qubit |g> and |e>Splitting adjustable via the external field
• Hamiltonian in eigenbasis
22,2
qzq
q hHBias
Ene
rgy
• coplanar waveguide resonator 50 ohm characteristic impedance– Length determines the resonance
frequencyl / 2 fundamental mode 2.5 GHzHigh Q ~ 106
Thermal population at 20 mK
• Hamiltonian
• Relaxation with rate k
0025.0
1
10
kT
hth
e
n
2
10 aahH R
Superconductor
CcCc
Resonator
• In flux basis (g= M Ip Ir)
• Or in eigenbasis of Qubits
zaahgH int
xq
zq
aahgH
int
Simulation:
Due to tapering M is 4 times lager
Coupling between resonator and qubit
Parametric amplification
M. Rehak, Appl. Phys. Lett., 104, 162604, (2014).
f f
pump
signal
pump
signalidler
IN OUT
Superconducting coplanar waveguide resonator with a pair of flux qubits
Resonator design with its nonlinear element –
pair of flux qubits
Gain
Quantum amplifier
O. Astafiev, et. al., Phys. Rev. Lett 104, 183603 (2010).
11
Spontaneous emission
Noise level of the 4K amplifier is 10-22 W/Hz!
3
2
1
32
21
13
-100 -50 0 50 100-5
0
5
10
15
/2 = 40 MHz
31/2 (MHz)
31/2 = 24 MHz
31/2 (MHz)
S (10
-25 W
/Hz)S
(10
-25 W
/Hz)
/2
(MH
z)
22
21
2121
2)(
fS
Noise spectral density (weak driving limit)
31
12
0.9
1.0
1.1
-200 -100 0 100 200
-0.1
0.0
0.1
0.2
21
/2
(MH
z)
21/2
(MH
z)
31/2 (MHz)
|t|A
rg t
Amplification
Stimulated emission
f13
21/2 (MHz)
|t|A
rg t
Amplification
13
2132231
1122
/11
t
Transmission at resonance without pure dephasing Maximum transmission
2132231 3
8
11max t
4
322
4
111 32
21
3
2
1
31
32 >> 21
21/2 = 11 MHz
32/2 = 35 MHz
1 10 100
1.00
1.04
1.08
Tra
nsm
issi
on
am
plitu
de
Probing amplitude 21/2 (MHz)
Linear amplificationregime
Quantum amplifier gain
14
Single qubit lasing
O. Astafiev, et. al., Nature 449, 588-590 (2007)
3
2
1
nD
G32
Geff
Lasing principle
16Josephson quasiparticle current
0
2
2 0
2e
e+e
Far away from degeneracy, 0 state is decoupled from 2
11
JQP cycle: 2 1 0
2
0
1
IJQP
Population inversion
2
E
EI JJQP
Vb >2e
17
CC
C
rg
V b
J E J
g a te
res o n ato r
p ro b e
g ro u n d
is lan d
N + 1 g02
0
1
The three level atom in the resonator
island
Josephson junctionsgate
probeelectrode
to resonator
18
-2 -1 00
1
S (
10-2
1 W
/Hz)
amplifiernoise
Emission spectrum
f (MHz)
Nphoton >2P p
= 30
19
0.0
0.5
1.0
Nor
mal
ised
tran
smis
sion
ampl
itude
9.895 9.900 9.905-4-3-2-10123
f (GHz)
Ph
ase
Laser is OFF
Laser is ON
Amplification
0.5
1.0
1.5
20
Dress-state lasing
21
2
1Geff
Coupling
Φi
VLT
L
CT
Ib
M
Tank-qubit arrangement
TeffCL
10
Φi
VLT
L
CT
Ib
M
M2=k2LLT;
22
;1
,1;
)(1
2
222
2
22
.
.....
d
EdLk
d
dILk
d
dILk
QVd
dI
d
dII
tIC
IMVVQ
V
Tq
T
Tq
TTq
qqq
b
T
qTTT
> kT > h
Phenomenological approach
We found quantum-mechanical correction, but at low temperature kT<< it is negligible:
Ya. S. Greenberg and E. Il’ichev PRB 77, 094513 (2008)
Ya. S. Greenberg et al., PRB 66, 214525, 2002
M. Grajcar et al., PRB 69, 060501, 2004
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
(fx) (GHz)
Tank cooling
M. Grajcar et al., Nature Phys., 4, 612, (2008)
Spectroscopy with oscillator as a detector
Rabi resonances
• Atom + photon field• Energy states split on
Allowed transitions by dipoles matrix elementfluorescence triplet
C. Coen-Tannoudji, J. Dupont-Rock, and G. Grynberg, Atom-Photon Interactions. Basic Principles and Applications (JohnWiley, New York, 1998)
Dressed systems in quantum optics
• Population depends on detuning
• Use additional signal with a tunable frequency-> Gain or attenuation
• Dressed state laser
C. Coen-Tannoudji, J. Dupont-Rock, and G. Grynberg, Atom-Photon Interactions. Basic Principles and Applications (JohnWiley, New York, 1998)
F. Y. Wu , S. Ezekiel,M. Ducloy, and B. R. Mollow, Phys. Rev. Lett. 38 1077, (1977)
Dressed systems in quantum optics
• Splitting of the levels in the resonance point is proportional to
• Hamiltonian
• Neglecting constant offset the energy is proportional to N
• For small g and large N a variable:
Ng
aahvh
H zR
D 02
|20>
|10>
.constR
2
2 2
qR
gN
g0
g1
g2e1
e0
Ng~
Ene
rgy
detuning
Dressed levels
• Reasonable N=105 and g=1MHz
• Therefore effective two-level system: quasi equilibrium levels |1> and |2>
kHzNR
NR 5.1)0()1()( 0
)1(
)(
NR
NR
N+1
N
N-1
|2>
|1>
Detuning changes the role of relaxation
Effective inversion of population
GG
G
G
Dressed levels
1122221111 ,12
11
2
1LLL
RRRR
|2>
|1>
d0
1122
1111
L
L
2222
2211
L
L
Inversion population
)(],[0 LHih
RxRzR gaahgaahhH )()
2
1(
2 0
• System resonator – qubit • Qubit: = 3.6 GHz, Ip = 12 nA, g~0.8
MHz• + Gold resistance• Fundamental mode below the qubit gap:
resonant interaction is absent• Additional microwave field generates an
effective two-level system• Good qubit-resonator coupling • High photon numbers in the resonator
possible
Rq
HR
q
HHqHR
Ngg
Ng
22222
2,
2
|21>
|20>
|10>
Lasing
Damping
Lasing: experimental realization
• Input signal 2.5 GHz
Lasing: experimental realization
• Fitting parameter /2G p = 60 MHz and G /2f p = 20MHz
)(],[0 LHih
Lasing: fitting
Emmision
G.Oelsner et. al. Phys. Rev. Lett. 110, 053602 (2013) • P. Neilingeret. al. Phys. Rev. B 91, 104516 (2015).
Conclusion
• Level inversion of a driven qubits is used to produce lasing at the Rabi frequency
• The qubit is adjusted for stable resonance conditions and rapid relaxation.
• Harmonics of the resonator determine the driving field for good coupling and high photon number
• The experimental results are described by a full quantum theory - on the base of the dressed states.