1. The document proposes using optomechanical transducers to entangle superconducting qubits over long distances. 2. An optomechanical transducer can act as a force sensor to measure superconducting qubits coupled to a mechanical oscillator. 3. The mechanical oscillator is modeled using a conditional master equation and can be adiabatically eliminated to obtain an effective equation describing the qubits.

1. Measurement-induced long-distance
entanglement of superconducting qubits
using optomechanical transducers
Ondřej Černotík and Klemens Hammerer
Leibniz Universität Hannover
Erice, 1 August 2016
PRA 94, 012340

2. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
Superconducting systems are among the
best candidates for quantum computers.
2
• Controlling microwave fields with qubits
Hofheinz et al., Nature 454, 310 (2008); Nature 459, 546 (2009)
• Feedback control of qubits
Ristè et al., PRL 109, 240502 (2012); Vijay et al., Nature 490, 77 (2012);
de Lange et al., PRL 112, 080501 (2014)
• Entanglement generation
Ristè et al., Nature 502, 350 (2013); Roch et al., PRL 112, 170501 (2014);
Saira et al., PRL 112, 070502 (2014)
• Quantum error correction
Córcoles et al., Nature Commun. 6, 6979
(2015), Kelly et al., Nature 519, 66 (2015),
Ristè et al., Nature Commun. 6, 6983 (2015)
R. Schoelkopf

3. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
Entanglement between two qubits can be
generated by measurement and
postselection.
3
C. Hutchison et al., Canadian J. Phys. 87, 225 (2009)
N. Roch et al., PRL 112, 170501 (2014)
Hint = za†
aDispersive coupling
|11i
|00i
|01i + |10i

4. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
We want to extend the distance over
which the qubits become entangled.
4
-
Other proposals:
K. Stannigel et al., PRL 105, 220501 (2010)
B. Clader, PRA 90, 012324 (2014)
Z. Yin et al., PRA 91, 012333 (2015)
Experiments:
J. Bochmann et al., Nat. Physics 9, 712 (2013)
R. Andrews et al., Nat. Physics 10, 321 (2014)
T. Bagci et al., Nature 507, 81 (2014)
K. Balram et al., Nat. Photon. 10, 346 (2016)
----
1. Force sensing
2. Equation of motion
3. Feasibility

5. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
Optomechanical transducer acts as a
force sensor.
5
F = ~ /(
p
2xzpf )
S2
F (!) = x2
zpf /[8g2 2
m(!)]Sensitivity:
! ⌧ !m
⌧meas =
S2
F (!)
F2
=
!2
m
16 2g2
⌧ T1,2Measurement time:
H = z(b + b†
) + !mb†
b + g(a + a†
)(b + b†
)

6. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
The thermal mechanical bath affects the
qubit.
6
mech = S2
f (!) =
2 2
!2
m
¯nDephasing rate:
⌧meas <
1
mech
! C =
4g2
 ¯n
>
1
2

7. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
The system can be modelled using a
conditional master equation.
7
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)

8. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
The transducer is Gaussian and can be
adiabatically eliminated.
8
OC et al., PRA 92, 012124 (2015)ˇ
2 qubits
Mechanics,
light

9. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
We obtain an effective equation for the
qubits.
9
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)

10. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
Optical losses introduce additional
dephasing.
10
p
⌘ measH[ 1
z + 2
z]⇢q
(1 ⌧) measD[ 1
z]⇢q

11. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
A transmon qubit can capacitively couple
to a nanobeam oscillator.
11
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
⌘
Psucc
Psucc

12. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
The mechanical oscillator can be a
membrane.
12
R. Andrews et al., Nature Phys. 10, 312 (2014)
T. Bagci et al., Nature 507, 81 (2014)
J. Pirkkalainen et al., Nature 494, 211 (2013)

13. Ondrej Cernotík (Hannover): Entanglement of superconducting qubits, PRA 94, 012340ˇˇ
Mechanical oscillators can mediate
interaction between light and SC qubits.
13
OC & K. Hammerer, PRA 94, 012340ˇ
-
C =
4g2
 ¯n
>
1
2
• Strong optomechanical cooperativity,
• Sufficient qubit lifetime