Optomechanical systems offer a promising route towards frequency conversion between microwaves and light. Current theoretical and experimental efforts focus on approaches based on either optomechanically induced transparency (suffering from limited conversion bandwidth) or adiabatic passage (requiring time-dependent control). In my talk, I will present two alternative strategies for optomechanical transduction that avoid these limitations. In the first one, entanglement between two superconducting qubits is generated by using transducers as force sensors; jointly measuring the force with which the qubits act on the transducers leads to conditional generation of entanglement between the qubits. The other device uses spatially adiabatic frequency conversion in an array of optomechanical transducers, allowing for large conversion bandwidth with time-independent control.

1. Novel approaches to
optomechanical transduction
Ondřej Černotík
Max Planck Institute for the Science of Light, Erlangen, Germany
Leibniz University Hannover, Germany
Aarhus University, 18 April 2018

2. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ 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 simulations
A. Houck et al., Nature Physics 8, 292 (2012); R. Barends et al., Nature Commun.
6, 7654 (2015); Y. Salathé et al., PRX 5, 021027 (2015)
• 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)
• Quantum networks
Quantum communication, distributed
quantum computing, …
Schoelkopf group, Yale
Superconducting circuits are among the
best candidates for quantum computers.

3. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Various systems can serve as an interface
between microwaves and light.
3
J. Bochmann et al., Nature Phys. 9, 712 (2013)
R. Andrews et al., Nature Phys. 10, 321 (2014)
T. Bagci et al., Nature 507, 81 (2014)
K.C. Balram et al., Nature Photon. 10, 346 (2016)
A. Okada et al., arXiv:1705.04593
K. Takeda et al., arXiv:1706.00532
Mechanical oscillators
Electrooptics
A. Rueda et al., Optica 3, 597 (2016)
Magnons
R. Hisatomi et al., PRB 93, 174427 (2016)
C. O’Brien et al., PRL 113, 063603 (2014)
Spins/spin ensembles

4. Novel approaches to optomechanical
transduction
Entanglement generation
-
Optomechanical transduction
Spatially adiabatic conversion

5. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Optomechanical interaction is due to
radiation pressure.
5
Cavity frequency:
!c(ˆx) ⇡ !c(0) +
d!c
dx
ˆx
Coupling strength: g0 =
d!c
dx
xzpf =
!c
L
xzpf
Hamiltonian:
ˆH = ~!c(ˆx)ˆa†
ˆa + ~!m
ˆb†ˆb
ˆH = ~!cˆa†
ˆa + ~!m
ˆb†ˆb + ~g0ˆa†
ˆa(ˆb + ˆb†
)
ˆx = xzpf (ˆb + ˆb†
), xzpf =
r
~
2m!m
ˆa
ˆx
!m
!c
M. Aspelmeyer, T. Kippenberg, and
F. Marquardt, RMP 86, 1391 (2014)

6. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Strong coupling can be achieved by
suitable driving.
6
Optomechanical coupling is weak
Solution: strong optical drive ˆa ! ↵ + ˆa, g0 ! g = g0↵
Interaction Hamiltonian ˆHint ⇡ ~g(ˆa + ˆa†
)(ˆb + ˆb†
)
g0 ⌧ 
M. Aspelmeyer, T. Kippenberg, and
F. Marquardt, RMP 86, 1391 (2014)

7. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Driving frequency is an important control
parameter.
7
M. Aspelmeyer, T. Kippenberg, and
F. Marquardt, RMP 86, 1391 (2014)
!m
Red-detuned drive:
State swapˆHint = ~g(ˆa†ˆb + ˆb†
ˆa)
!L = !c !m
!
!c!L
 < !m
!m
Blue-detuned drive:
Squeezing
!L = !c + !m
ˆHint = ~g(ˆaˆb + ˆa†ˆb†
)
!!c !L
 < !m
Resonant drive:
Position readout
!L = !c
ˆHint = ~g(ˆa + ˆa†
)(ˆb + ˆb†
)
!
!c = !L


8. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Optomechanical interaction can be
observed in various systems.
8

9. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Mechanical oscillators are good
transducers for microwaves and light.
9
R. Andrews et al., Nature Phys. 10, 321 (2014)
g2
g1
ˆa1
ˆa2
ˆb
ˆa2,in ˆa2,out
ˆa1,outˆa1,in
ˆH = g1(ˆa†
1
ˆb + ˆb†
ˆa1) + g2(ˆa†
2
ˆb + ˆb†
ˆa2)
ˆa1
ˆa2 ˆb
ˆa1,in/out
ˆa2,in/out

10. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Mechanical oscillators are good
transducers for microwaves and light.
10
R. Andrews et al., Nature Phys. 10, 321 (2014)
g2
g1
ˆa1
ˆa2
ˆb
ˆa2,in ˆa2,out
ˆa1,outˆa1,in
ˆH = g1(ˆa†
1
ˆb + ˆb†
ˆa1) + g2(ˆa†
2
ˆb + ˆb†
ˆa2)
dtˆa(t) = Aˆa(t) + Bˆain(t)
ˆaout(t) = Cˆa(t) + Dˆain(t)
ˆaout(!) = S(!)ˆain(!) = [D C(A + i!1)B]ˆain(!)

11. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Mechanical oscillators are good
transducers for microwaves and light.
11
R. Andrews et al., Nature Phys. 10, 321 (2014)
g2
g1
ˆa1
ˆa2
ˆb
ˆa2,in ˆa2,out
ˆa1,outˆa1,in
ˆH = g1(ˆa†
1
ˆb + ˆb†
ˆa1) + g2(ˆa†
2
ˆb + ˆb†
ˆa2)
g2
1
1
=
g2
2
2
Impedance matching
i ⌧ !mResolved-sideband regime
Strong cooperativity Ci =
4g2
i
i ¯n
1
/
g2
1
1
=
g2
2
2
⌧ i

12. OC and K. Hammerer, PRA 94, 012340 (2016)ˇ
Measurement-induced entanglement of
superconducting qubits

13. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
We can measure qubits using dispersive
interaction with fields.
13
Dispersive coupling ˆHint = ˆzˆa†
ˆa
|0i
|1i
D. Ristè et al., PRL 109, 050507 (2012)
amplitude
phase

14. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
We can measure qubits using dispersive
interaction with fields.
14
C. Hutchison et al., Canadian J. Phys. 87, 225 (2009)
N. Roch et al., PRL 112, 170501 (2014)
Dispersive coupling ˆHint = ˆzˆa†
ˆa
|11i
|00i
|01i + |10i
| 0i = (|0i + |1i)(|0i + |1i)
amplitude
phase

15. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
D[ ˆO]ˆ⇢ = ˆOˆ⇢ ˆO† 1
2
( ˆO† ˆOˆ⇢ + ˆ⇢ ˆO† ˆO)
H[ ˆO]ˆ⇢ = ( ˆO h ˆOi)ˆ⇢ + ˆ⇢( ˆO†
h ˆO†
i)
ˆH =
2X
j=1
ˆj
z(ˆbj + ˆb†
j) + !m
ˆb†
j
ˆbj + g(ˆaj + ˆa†
j)(ˆbj + ˆb†
j)
+ i

2
(ˆa1ˆa†
2 ˆa2ˆa†
1)
dˆ⇢ = i[ ˆH, ˆ⇢]dt + Lq ˆ⇢dt +
2X
j=1
{(¯n + 1)D[ˆbj] + ¯nD[ˆb†
j]}ˆ⇢dt
+ D[ˆa1 ˆa2]ˆ⇢dt +
p
H[i(ˆa1 ˆa2)]ˆ⇢dW
Extension to room temperature is possible
with optomechanical transducers.
15

16. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
ˆa
ˆb
g
Optomechanical transducer can act as a
force sensor.
16
F = ~ /(
p
2xzpf )
Sensitivity: S2
F (!) = x2
zpf /8g2 2
m(!)
Measurement time: ⌧meas =
S2
F (!)
F2
=
!2
m
16 2g2
⌧ T1,2
ˆH = ˆz(ˆb + ˆb†
) + !m
ˆb†ˆb + g(ˆa + ˆa†
)(ˆb + ˆb†
)

17. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
The thermal mechanical bath affects the
qubit.
17
Dephasing rate: mech = S2
f (!) =
2 2
!2
m
¯n
⌧meas <
1
mech
! C =
4g2
 ¯n
>
1
2
ˆa
ˆb
g
¯n

18. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Gaussian systems can be adiabatically
eliminated from conditional dynamics.
18
OC, D.V. Vasilyev, and K. Hammerer,
PRA 92, 012124 (2015)
ˇ
ˆHint = ˆsT
ˆr
operatorsˆrˆs
state descriptionx,ˆ⇢
System Transducer
...

19. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
meas = 16
2
g2
!2
m
, mech =
2
!2
m
(2¯n + 1)
dˆ⇢q =
2X
j=1

1
T1
D[ˆj
] +
✓
1
T2
+ mech
◆
D[ˆj
z] ˆ⇢qdt
+ measD[ˆ1
z + ˆ2
z]ˆ⇢qdt +
p
measH[ˆ1
z + ˆ2
z]ˆ⇢qdW
We can obtain an effective equation for
the qubits.
19

20. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Optical losses introduce additional
dephasing.
20
(1 ⌧) measD[ˆ1
z]ˆ⇢q
p
⌘ measH[ˆ1
z + ˆ2
z]ˆ⇢q

21. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Entanglement generation is possible with
existing technology.
21
G. Anetsberger et al., Nature Phys. 5, 909 (2009)
J. Pirkkalainen et al., Nat. Commun. 6, 6981 (2015)
OC and K. Hammerer, PRA 94, 012340 (2016)ˇ
= 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
!m
Qm
⌘
0.2
1.0

22. OC, S. Mahmoodian, and K. Hammerer, arXiv:1707.03339ˇ
Spatially adiabatic frequency conversion
in optomechanical arrays

23. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ 23
g1
g2
L. Tian, PRL 108, 153604 (2012)
Y.-D. Wang and A.A. Clerk, PRL 108,
153603 (2012)
Efficient transduction is possible using
adiabatic state transfer.
ˆ
ˆ
ˆ ˆH(t) = g1(t)(ˆc†
1
ˆb + ˆb†
ˆc1) + g2(t)(ˆc†
2
ˆb + ˆb†
ˆc2)
=
q
g2
1(t) + g2
2(t)[ ˆd†
1(t)ˆb + ˆb† ˆd1(t)]
ˆd1(t) =
1
p
g2
1 + g2
2
[g1(t)ˆc1 + g2(t)ˆc2]
ˆd2(t) =
1
p
g2
1 + g2
2
[g2(t)ˆc1 g1(t)ˆc2]

24. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ 24
Efficient transduction is possible using
adiabatic state transfer.
G1
G2
ˆ
ˆ
ˆ
ˆH = vi
Z
dzˆa†
i (z)@zˆai(z) +
Z
dzGi(z)ˆa†
i (z)ˆb(z) + H.c.
ˆd1(z) =
1
p
G2
1 + G2
2
[G1(z)ˆa1 + G2(z)ˆa2]
ˆd2(z) =
1
p
G2
1 + G2
2
[G2(z)ˆa1 G1(z)ˆa2]

25. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
ˆa(z+
j , !) = Sj(!)ˆa(zj , !)
ˆa(L, !) =
NY
j=1
Sj(!)ˆa(0, !)
25
Adiabatic conversion can be
approximated in a transducer array.
ˆc1
ˆc2
ˆb
ˆ
ˆ
dˆci
dt
=
i
2
ˆci igi
ˆb +
p
iˆai(zj )
dˆb
dt
=
2
ˆb ig1ˆc1 ig2ˆc2 +
p ˆbin
ˆai(z+
j ) =
p
iˆci ˆai(zj )

26. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Conversion bandwidth can be increased
by increasing the array size.
26
Conversion probability off resonance
p1 / g1g2/!3
⌧ 1
pN = Np1 / g1g2N/!3
1 transducer
transducersN
! =
4
p
2
3
g1g2N
!1/3

27. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Conversion bandwidth can be increased
by increasing the array size.
27
p1 / g1g2/!3
⌧ 1
pN = Np1 / g1g2N/!3
1 transducer
transducersN
! =
4
p
2
3
g1g2N
!1/3
10
50
200
N = 1

28. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Added noise is reduced by increasing the
array size.
28
Single transducer ˆa(z+
j , !) = Sj(!)ˆa(zj , !) + Vj(!) ˆfj
Output spectrum S2
out(!) / S2
in(!) +
1
C
, C =
4g2
 ¯n
Many transducers S2
out(!) / S2
in(!) +
N
C
Dark mode ˆd2(z+
j , !) / ˆd2(zj , !) +
1
N
Vj(!) ˆfj
Total noise S2
add(!) /
1
CN
Collective cooperativity Ccoll =
4g2
N
 ¯n
> 1

29. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Efficient conversion is possible in
presence of optical losses.
29
Propagation loss Backscattering
RL
⌘
⌘ = 0.05
0.01
0.2
0.001
L/R = 0
0.1
0.5
0.9

30. Ondrej Cernotík (MPL Erlangen): Novel approaches to optomechanical transductionˇˇ
Optomechanical transducers present a
promising route to frequency conversion.
30
Arbitrary input
Small bandwidth
OC, S. Mahmoodian, and K. Hammerer, arXiv:1707.03339ˇ
Large bandwidth, noise resilient
Arbitrary signals
OC and K. Hammerer, PRA 94, 012340 (2016)ˇ
Entanglement generation
Builds on existing experiments
System Transducer
...
OC, D.V. Vasilyev, and K. Hammerer, PRA 92, 012124 (2015)ˇ
Gaussian transducers under continuous measurement
General recipe for adiabatic elimination