The document discusses the integration of superconducting circuits and optomechanical transducers in quantum networks, emphasizing the potential of superconducting qubits for quantum computing. It outlines various quantum mechanics concepts such as quantum gates, teleportation, and error correction, while detailing how mechanical oscillators facilitate interaction between microwaves and light. Additionally, it highlights the challenges due to differences in parameters between superconducting and optical systems and the impact of thermal mechanical baths on qubit performance.

1. Quantum networks
with superconducting circuits
and optomechanical transducers
Ondřej Černotík
Leibniz Universität Hannover
IST Austria, 10 November 2016
-

2. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Superconducting systems are among the
best candidates for quantum computers.
2
• Quantum gates and processors
L. DiCarlo et al., Nature 460, 240 (2009); ibid. 467, 574 (2010); A. Fedorov et
al., Nature 481, 170 (2011)
• Quantum teleportation
L. Steffen et al., Nature 500, 319 (2013)
• Quantum simulations
A. Houck et al., Nature Physics 8, 292 (2012)
• Quantum error correction
A. Córcoles et al., Nature Commun. 6, 6979
(2015); J. Kelly et al., Nature 519, 66 (2015);
D. Ristè et al., Nature Commun. 6, 6983 (2015)
R. Schoelkopf, Yale

3. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Light is ideal for quantum communication
due to low losses and noise.
3
• Quantum key distribution
F. Grosshans et al., Nature 421, 238 (2003); T. Schmitt-Manderbach et al., PRL
98, 010504 (2007); H. Yin et al., PRL 117, 190501 (2016)
• Quantum teleportation
D. Bouwmeester et al., Nature 390, 575 (1997); A. Furusawa et al., Science 282,
706 (1998); H. Yonezawa et al., Nature 431, 430 (2004); T. Herbst et al., PNAS
112, 14202 (2015)
• Loophole-free Bell test
B. Hensen et al., Nature 526, 682 (2015); M. Giustina
et al., PRL 115, 250401 (2015); L. Shalm et al., ibid.,
250402 (2015)
A. Zeilinger

4. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
There is a large gap between
superconducting and optical systems.
4
Superconducting circuits Optical communication
10 GHz 200 THzfrequency
625 0.03thermal occupation
(300 K)
0.5 K 10,000 K
ground state
temperature

5. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Mechanical oscillators can mediate
coupling between microwaves and light.
5
R. Andrews et al., Nature Phys. 10, 321 (2014)
K. Stannigel et al., PRL 105, 220501 (2010)

6. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ 6
Parity measurements
in circuit QED
Optomechanical
force sensing
Long-distance entanglement
of superconducting qubits
Fext

7. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Full control of a qubit is possible using an
electromagnetic field.
7
Hint = g(a + + a†
)
A. Blais et al., PRA 69, 062920 (2004)
Jaynes–Cummings interaction

8. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Full control of a qubit is possible using an
electromagnetic field.
8
Hint =
g2
a†
a z
A. Blais et al., PRA 69, 062920 (2004)
dispersive interaction

9. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Dispersive coupling can be used to read
out the qubit state.
9
|0i
|1i
R. Vijay et al., PRL 106, 110502 (2011)
K. Murch et al., Nature 502, 211 (2013)
Hint =
g2
a†
a z

10. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Spin measurement can be used to
generate entanglement of two qubits.
10
C. Hutchison et al., Canadian J. Phys. 87, 225 (2009)
N. Roch et al., PRL 112, 170501 (2014)
|11i
|00i
|01i + |10i
| 0i = (|0i + |1i)(|0i + |1i)

11. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Optomechanical interaction arises due to
radiation pressure.
11
a x
!,  ⌦, ¯n
!(x) ⇡ !(0) +
d!
dx
x
Cavity frequency:
g0 =
d!
dx
xzpf =
!
L
xzpfCoupling strength:
xzpf =
r
~
2m⌦
x = xzpf (b + b†
),
Hamiltonian:
H = ~!(x)a†
a + ~⌦b†
b
H = ~!a†
a + ~⌦b†
b + ~g0a†
a(b + b†
)
M. Aspelmeyer, et al.,
RMP 86, 1391 (2014)

12. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
⌦
Strong coupling can be achieved using
laser driving.
12
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: strong optical drive a ! ↵ + a
Interaction Hamiltonian Hint = ~g0↵(a + a†
)(b + b†
)
M. Aspelmeyer, et al.,
RMP 86, 1391 (2014)
Red-detuned drive:
Hint ⇡ ~g(a†
b + b†
a)
Optomechanical cooling
!L = ! ⌦

13. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
⌦
Strong coupling can be achieved using
laser driving.
13
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: strong optical drive a ! ↵ + a
⌦
Interaction Hamiltonian Hint = ~g0↵(a + a†
)(b + b†
)
M. Aspelmeyer, et al.,
RMP 86, 1391 (2014)
Blue-detuned drive:
Hint ⇡ ~g(ab + a†
b†
)
Two-mode squeezing
!L = ! + ⌦

14. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
⌦
Strong coupling can be achieved using
laser driving.
14
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: strong optical drive a ! ↵ + a
Interaction Hamiltonian Hint = ~g0↵(a + a†
)(b + b†
)
M. Aspelmeyer, et al.,
RMP 86, 1391 (2014)
Resonant drive:
Hint ⇡ ~g(a + a†
)(b + b†
)
Position readout
! = !L

15. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Standard quantum limit bounds the
sensitivity of displacement measurements.
15
A. Clerk et al., RMP 82, 1155 (2010)
M. Aspelmeyer et al., RMP 86, 1391 (2014)
˙x = !mp
˙p = !mx p g(a + a†
) + ⇠ + Fext
˙a =

2
a igx +
p
ain
Fext

16. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Standard quantum limit bounds the
sensitivity of displacement measurements.
16
A. Clerk et al., RMP 82, 1155 (2010)
M. Aspelmeyer et al., RMP 86, 1391 (2014)
Fext
pout = i(aout a†
out)
=
4g!m
p

(!2
m !2 + i !)( + 2i!)
✓
Fext + ⇠
2g
p

 + 2i!
xin
◆
+
 2i!
 + 2i!
pin

17. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Optomechanical transducer acts as a
force sensor.
17
F = ~ /(
p
2xzpf )
S2
F (!) = x2
zpf /[8g2 2
m(!)]Sensitivity:
⌧meas =
S2
F (!)
F2
=
!2
m
16 2g2
⌧ T1,2Measurement time:
H = z(b + b†
) + !mb†
b + g(a + a†
)(b + b†
)

18. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
The thermal mechanical bath affects the
qubit.
18
mech = S2
f (!) =
2 2
!2
m
¯nDephasing rate:
⌧meas <
1
mech
! C =
4g2
 ¯n
>
1
2

19. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
The system can be modelled using a
conditional master equation.
19
D[O]⇢ = O⇢O† 1
2 (O†
O⇢ + ⇢O†
O)
H[O]⇢ = (O hOi)⇢ + ⇢(O†
hO†
i)
H. Wiseman & G. Milburn, Quantum
measurement and control (Cambridge)
d⇢ = i[H, ⇢]dt + Lq⇢dt +
2X
j=1
{(¯n + 1)D[bj] + ¯nD[b†
j]}⇢dt
+ D[a1 a2]⇢dt +
p
H[i(a1 a2)]⇢dW
H =
2X
j=1
j
z(bj + b†
j) + !mb†
jbj
+ g(aj + a†
j)(bj + b†
j) + i

2
(a1a†
2 a2a†
1)

20. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
The transducer is Gaussian and can be
adiabatically eliminated.
20
OC et al., PRA 92, 012124 (2015)ˇ
2 qubits
Mechanics,
light

21. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
We obtain an effective equation for the
qubits.
21
d⇢q =
2X
j=1

1
T1
D[ j
] +
✓
1
T2
+ mech
◆
D[ j
z] ⇢qdt
+ measD[ 1
z + 2
z]⇢qdt +
p
measH[ 1
z + 2
z]⇢qdW
meas = 16
2
g2
!2
m
, mech =
2
!2
m
(2¯n + 1)

22. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Optical losses introduce additional
dephasing.
22
p
⌘ measH[ 1
z + 2
z]⇢q
(1 ⌧) measD[ 1
z]⇢q

23. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
A transmon qubit can capacitively couple
to a nanobeam oscillator.
23
G. Anetsberger et al., Nature Phys. 5, 909 (2009)
J. Pirkkalainen et al., Nat. Commun. 6, 6981 (2015)
= 2⇡ ⇥ 5.8 MHz
g = 2⇡ ⇥ 900 kHz
 = 2⇡ ⇥ 39MHz
!m = 2⇡ ⇥ 8.7 MHz
Qm = 5 ⇥ 104
T = 20 mK
¯n = 48
T1,2 = 20 µs
C = 10

24. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
A transmon qubit can capacitively couple
to a nanobeam oscillator.
24
= 2⇡ ⇥ 5.8 MHz
g = 2⇡ ⇥ 900 kHz
 = 2⇡ ⇥ 39MHz
!m = 2⇡ ⇥ 8.7 MHz
Qm = 5 ⇥ 104
T = 20 mK
¯n = 48
T1,2 = 20 µs
C = 10
⌘
Psucc
Psucc
OC and K. Hammerer, PRA 94, 012340 (2016)ˇ

25. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
With high-frequency mechanical
oscillators, modulated interaction can be
used.
25
H = z(b + b†
) ig(a + a†
)(b b†
)
meas = 16
2
g2
 2
, mech =
2
(2¯n + 1)

26. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Microwave cavity can improve the
lifetime of the qubit.
26
meas = 256
2
g2
ag2
c
2
a!2
mc
,
deph = 4
2
a
+ 256
2
g4
a
3
a!2
m
+ 16
2
g2
a
2
a!2
m
(2¯n + 1)
H = z(a + a†
)
iga(a a†
)(b + b†
)
+ !mb†
b + gc(c + c†
)(b + b†
)
a
b
c

27. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Both techniques can also be combined in
one system.
27
a
b
c
H = z(a + a†
)
iga(a a†
)(b + b†
)
igc(c + c†
)(b b†
)
meas = 1024
2
g2
ag2
c
2
a
2c
,
deph = 4
2
a
+ 64
2
g2
a
2
a
(2¯n + 1)

28. Ondrej Cernotík (Hannover): Quantum networks with SC qubits and OM transducersˇˇ
Mechanical oscillators can mediate
interaction between light and SC qubits.
28
OC and K. Hammerer, PRA 94, 012340 (2016)ˇ
-
C =
4g2
 ¯n
>
1
2
• Strong optomechanical cooperativity,
• Sufficient qubit lifetime