Although superconducting systems provide a promising platform for quantum computing, their networking poses a challenge as they cannot be interfaced to light---the medium used to send quantum signals through channels at room temperature. We show that mechanical oscillators can mediated such coupling and light can be used to measure the joint state of two distant qubits. The measurement provides information on the total spin of the two qubits such that entangled qubit states can be postselected. Entanglement generation is possible without ground-state cooling of the mechanical oscillators for systems with optomechanical cooperativity moderately larger than unity; in addition, our setup tolerates a substantial transmission loss. The approach is scalable to generation of multipartite entanglement and represents a crucial step towards quantum networks with superconducting circuits.

1. Measurement-Induced Long-
Distance Entanglement with
Optomechanical Transducers
Ondřej Černotík and Klemens Hammerer
Leibniz Universität Hannover
Palacký University Olomouc, 23 September 2015

2. Quantum information
2
Processing
Superconducting qubits,
trapped ions, …
Schoelkopf
Blatt
Transfer
Light
Zeilinger
Storage
Solid-state spins, atomic
ensembles, mechanical
oscillators, …
Lehnert Polzik
Superconducting qubits

3. SC qubits can be interfaced with light
using spin ensembles.
3
C. O’Brien et al., PRL 113, 063603 (2014)
K. Xia & J. Twamley, PRA 91, 042307 (2015)
C. O’Brien et al.

4. Mechanical oscillators can also mediate
the coupling.
4
T. Bagci et al., Nature 507, 81 (2014)R. Andrews et al., Nature Phys. 10, 321 (2014)
Z. Yin et al., PRA 91, 012333 (2015)

5. Mechanical oscillators can also mediate
the coupling.
5
K. Xia et al., Sci. Rep. 4, 5571 (2014)
K. Stannigel et al., PRL 105, 220501 (2010)

6. Vg
Mechanical coupling
Single-qubit readout
Two-qubit readout Implementations
This talk will be about:

7. Coupling light and SC qubits to
mechanical oscillators
Vg

8. 8
a x
!,  ⌦, ¯n
!(x) ⇡ !(0) +
d!
dx
x
Cavity frequency:
g0 =
d!
dx
xzpf =
!
L
xzpfCoupling strength:
xzpf =
r
~
2m⌦
x = xzpf (b + b†
),
M . A s p e l m e y e r, T.
Kippenberg, F. Marquardt,
RMP 86, 1391 (2014)
Hamiltonian:
H = ~!(x)a†
a + ~⌦b†
b
H = ~!a†
a + ~⌦b†
b + ~g0a†
a(b + b†
)
Optomechanical interaction arises due to
radiation pressure.

9. ⌦
Strong coupling can be achieved using
laser driving.
9
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: strong optical drive a ! ↵ + a
Interaction Hamiltonian Hint = ~g0↵(a + a†
)(b + b†
)
M . A s p e l m e y e r, T.
Kippenberg, F. Marquardt,
RMP 86, 1391 (2014)
Red-detuned drive:
Hint ⇡ ~g(a†
b + b†
a)
Optomechanical cooling
!L = ! ⌦

10. Strong coupling can be achieved using
laser driving.
10
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: strong optical drive a ! ↵ + a
⌦
Interaction Hamiltonian Hint = ~g0↵(a + a†
)(b + b†
)
M . A s p e l m e y e r, T.
Kippenberg, F. Marquardt,
RMP 86, 1391 (2014)
Blue-detuned drive:
Hint ⇡ ~g(ab + a†
b†
)
Two-mode squeezing
!L = ! + ⌦

11. ⌦
Strong coupling can be achieved using
laser driving.
11
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: strong optical drive a ! ↵ + a
⌦
Interaction Hamiltonian Hint = ~g0↵(a + a†
)(b + b†
)
M . A s p e l m e y e r, T.
Kippenberg, F. Marquardt,
RMP 86, 1391 (2014)
Resonant drive:
Hint ⇡ ~g(a + a†
)(b + b†
)
Position readout
! = !L

12. ' = '1 '2
Josephson junction is a basic building
block of SC circuits.
12
Superconductor
Insulator (∼ 1 nm)
Superconductor
Junction parameters:
• critical current ,
• capacitance ,
• phase
I0
C
EJ =
~I0
2e
EC =
(2e)2
2C
Josephson energy
charging energy
Energy scale:
K. Bennemann & J. Ketterson, Superconductivity (Springer)
p
⇢ei'1
p
⇢ei'2

13. ' = '1 '2
Josephson junction is a basic building
block of SC circuits.
13
Superconductor
Insulator (∼ 1 nm)
Superconductor
Junction parameters:
• critical current ,
• capacitance ,
• phase
I0
C
V =
~
2e
˙', I = I0 sin '
Josephson relations
˙I = I0 cos(') ˙'
V =
~
2e
1
I0 cos '
˙I = L(') ˙I
K. Bennemann & J. Ketterson, Superconductivity (Springer)
p
⇢ei'1
p
⇢ei'2

14. Charge qubit is a voltage-biased JJ.
14
Electrostatic energy:
ECoulomb = 4EC(N Ng)2
Ng =
CgVg
2e
, EC =
e2
2C
Cg
Vg
Ng
Energy
K. Bennemann & J. Ketterson, Superconductivity (Springer)

15. Charge qubit is a voltage-biased JJ.
15
Cg
Vg
Ng
Energy
Two-level approximation:
H = 2EC(2Ng 1) z
EJ
2
x
Total Hamiltonian:
H = 4EC(N Ng)2
+ EJ cos '
EC EJ
K. Bennemann & J. Ketterson, Superconductivity (Springer)

16. Mechanical coupling is achieved using a
mechanically compliant capacitor.
16
Charge qubit with a movable gate
H = 4EC[N Ng(x)]2
+ EJ cos ' + ~⌦b†
b
Vg
x
Gate charge: Ng(x) ⇡
CgVg
2e
+
Vg
2e
dCg
dx
x
Hint = 2EC
Vg
e
dCg
dx
xzpf (b + b†
) z
Interaction Hamiltonian:
T. Heikkilä et al., PRL 112, 203603 (2014)

17. Optical readout of a
superconducting qubit

18. We can use an optomechanical system to
read out the state of a qubit.
18
d⇢ = i[Hint, ⇢]dt + LT ⇢dt +
p
H[aei
]⇢dW
LT ⇢ = i[HT , ⇢] + {(¯n + 1)D[b] + ¯nD[b†
]}⇢ + D[a]⇢
D[O]⇢ = O⇢O† 1
2 (O†
O⇢ + ⇢O†
O)
H[O]⇢ = (O hOi)⇢ + ⇢(O†
hO†
i)
H = z(b + b†
) + !b†
b + g(a + a†
)(b + b†
) = Hint + HT
H. Wiseman & G. Milburn, Quantum
measurement and control (Cambridge)

19. We adiabatically eliminate the transducer
degrees of freedom.
19
d⇢q = ( meas + mech)D[ z]⇢qdt +
p
measH[ z]⇢qdW
meas = 16
2
g2
!2
, mech =
2
!2
(2¯n + 1)
Efficient readout:
meas mech
Optomechanical cooperativity C =
4g2
 ¯n
1

20. Intermezzo: Adiabatic elimination of
Gaussian quantum systems
System Transducer
System

21. Gaussian systems are described by
quadratic Hamiltonians, linear jumps, and
homodyne measurements.
21
d⇢ = i[H, ⇢]dt +
X
n
D[jn]⇢dt +
X
m
H[ m]⇢dWm
H =
1
2
rT
Rr, jn = ⇠T
n r, m = (cm + imm)T
r
r = (q1, p1, . . . , qN , pN )T

22. Dynamics can be described using
statistical moments of canonical operators.
22
Mean values:
Covariance matrix:
d⇢ = i[H, ⇢]dt +
X
n
D[jn]⇢dt +
X
m
H[ m]⇢dW
dx = Axdt +
X
m
( cm mm)dW
˙ = A + AT
+ 2N 2
X
m
( cm mm)( cm mm)T
x = tr{r⇢}
ij = tr{[ri, rj]+⇢} 2xixj

23. Dynamics can be described using
statistical moments of canonical operators.
23
Mean values:
Covariance matrix:
d⇢ = i[H, ⇢]dt +
X
n
D[jn]⇢dt +
X
m
H[ m]⇢dW
dx = Axdt +
X
m
( cm mm)dW
˙ = A + AT
+ 2N 2
X
m
( cm mm)( cm mm)T
x = tr{r⇢}
ij = tr{[ri, rj]+⇢} 2xixj

24. Description using statistical moments
enables adiabatic elimination.
24
OC et al., PRA 92, 012124 (2015)ˇ
System Transducer
⇢ x,

25. Two-qubit readout using light

26. Joint measurement on two qubits can
generate entanglement between them.
26
1
z + 2
z
| 0i = (|0i + |1i)(|0i + |1i)
!
8
<
:
|00i
|11i
|01i + |10i
C. Hutchison et al., Canadian J. Phys. 87, 225 (2009)
D. Ristè et al., Nature 502, 350 (2013)
N. Roch et al., PRL 112, 170501 (2014)

27. The system is treated similarly to a single
qubit.
27
meas = 16
2
g2
!2
, mech =
2
!2
(2¯n + 1)
d⇢q =
1
T1
D[ j
]⇢qdt +
✓
1
T2
+ mech
◆
D[ j
z]⇢qdt+
+ measD[ 1
z
2
z]⇢qdt +
p
measH[ 1
z
2
z]⇢qdW

28. We can postselect entangled two-qubit
states.
28

29. We can postselect entangled two-qubit
states.
29

30. We can also include optical losses.
30

31. We can also include optical losses.
31

32. Experimental implementations of
half-parity measurements

33. The mechanical system can be formed by
a nanobeam.
33
G. Anetsberger et al., Nature Phys. 5, 909 (2009)
J. Pirkkalainen et al., Nat. Commun. 6, 6981 (2015)

34. The mechanical oscillator can be a
membrane.
34
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)

35. Mechanical oscillators can also couple to
flux qubits.
35
F. Xue et al., NJP 9, 35 (2007)

36. Other kinds of qubits can be used as
well.
36
P. Rabl et al., PRB 79, 041302 (2009)
P. Rabl et al., Nature Phys. 6, 602 (2010)
S. Kolkowitz et al., Science 335, 1603 (2012)

37. Summary
37
Vg

38. Mechanical oscillators can mediate
interaction between light and SC qubits.
38
• Strong optomechanical cooperativity,
• Sufficient qubit lifetime
C =
4g2
 ¯n
1

39. More complex schemes can be designed.
39
• Measurement feedback
• Two-mode optomechanical driving
• More experimental implementations