Optomechanical force sensing is an established measurement technique that can reach remarkable precision. In most applications, the system exerting the force on the mechanical oscillator is treated classically and we are not interested in any coherence between states of the system that give rise to different forces. A full quantum treatment, however, enables richer physics since measuring more such systems can lead to interference effects. In this talk, I will show that the coherence can survive the measurement and can be used for quantum-technological applications. I will consider a model example of spin readout in superconducting qubits. Coupling two transmon qubits to mechanical oscillators and reading out the mechanical positions using a single beam of light provides information on the total spin of the qubits. It is thus possible to conditionally generate entanglement between the two qubits. The system represents a basic quantum network with superconducting circuits. The scheme has modest requirements on the system parameters; it does not require ground-state cooling or resolved-sideband regime and can work with quantum cooperativity moderately larger than unity. Afterwards, I will consider another scheme, namely nondestructive detection of a single photon using an optomechanical transducer. The basic idea is similar to spin readout; the photon exerts a force on a mechanical oscillator and the the force is measured optically. I will argue that such a measurement is subject to a quantum limit due to backaction of the transducer on the dynamics of the photon and that this result also applies to other techniques of nondestructive photon detection, such as methods using Kerr interaction between the single photon and a meter beam. Finally, I will show numerically that measurement backaction can be evaded when the measurement rate is suitably modulated.

1. Quantum force sensing with
optomechanical transducers
Ondřej Černotík
Institut für Theoretische Physik, Leibniz Universität Hannover
Friedrich-Alexander-Universität, Erlangen, 19 July 2016

2. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
D. Rugar et al., Nature 430, 329 (2004) B. Abbott et al., PRL 116, 061102 (2016)
M. Wu et al., 1605.03138
L. de Lépinay et al., 1604.01873
We use mechanical oscillators for sensing
with remarkable precision.
2

3. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Standard quantum limit bounds the
sensitivity of displacement measurements.
3
A. Clerk et al., RMP 82, 1155 (2010)
M. Aspelmeyer et al., RMP 86, 1391 (2014)
H = !mb†
b + g(a + a†
)(b + b†
)

4. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Standard quantum limit bounds the
sensitivity of displacement measurements.
4
A. Clerk et al., RMP 82, 1155 (2010)
M. Aspelmeyer et al., RMP 86, 1391 (2014)
S2
imp =

16g2
✓
1 + 4
!2
2
◆
m(!) = [m(!m !)2
im !] 1
S2
ba =
4~2
g2

✓
1 + 4
!2
2
◆ 1
| m(!)|2

5. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
I will talk about:
5
-
OC and K. Hammerer, PRA (accepted)ˇ
Entanglement of
superconducting qubits
Optomechanical single-photon detection

6. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Entanglement of superconducting qubits
6
-
OC, K. Hammerer, PRA (accepted)ˇ

7. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Superconducting systems are among the
best candidates for quantum computers.
7
Schoelkopf
• 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)

8. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
We can generate entanglement by
measurement and postselection.
8
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

9. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
We want to extend the distance over
which the qubits become entangled.
9
-
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)

10. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Optomechanical transducer acts as a
force sensor.
10
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†
)

11. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
The thermal mechanical bath affects the
qubit.
11
mech = S2
f (!) =
2 2
!2
m
¯nDephasing rate:
⌧meas <
1
mech
! C =
4g2
 ¯n
>
1
2

12. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
The system can be modelled using a
conditional master equation.
12
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)

13. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
The transducer is Gaussian and can be
adiabatically eliminated.
13
OC et al., PRA 92, 012124 (2015)ˇ
2 qubits
Mechanics,
light

14. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
We obtain an effective equation for the
qubits.
14
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)

15. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Optical losses introduce additional
dephasing.
15
(1 ⌧) measD[ 1
z]⇢q
p
⌘ measH[ 1
z + 2
z]⇢q

16. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
A transmon qubit can capacitively couple
to a nanobeam oscillator.
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= 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 (accepted), 1512.00768ˇ

17. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Optomechanical single-photon detection
17

18. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
We measure the force with which a
photon kicks the oscillator.
18
Hint = g0a†
2a2(b + b†
)
a1 a2
d⇢ =

2
[a†
1a2 a†
2a1, ⇢]dt
+ D[a1 + a2]⇢dt
+ D[a†
2a2]⇢dt
+
p
⌘ H[a†
2a2]⇢dW

19. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
The measurement results in quantum
Zeno effect.
19
Coupling
Measurement
MeasurementS0 =
1
2
(a†
1a1 + a†
2a2)
S1 =
1
2
(a†
1a2 + a†
2a1) ⇠ x
S2 =
1
2i
(a†
1a2 a†
2a1) ⇠ y
S3 =
1
2
(a†
1a1 a†
2a2) ⇠ z
J. Gambetta et al., PRA 77, 012112 (2008)
[Si, Sj] = i"ijkSk
[S0, Si] = 0

20. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
We estimate the error by calculating the
(Gaussian) signal-to-noise ratio.
20
˙⇢ =

2
[a†
1a2 a†
2a1, ⇢] + D[a1 + a2]⇢ + D[a†
2a2]⇢
J(T) =
Z T
0
dt S(t)I(t) I(t), S(t) / n1(t)
SNR =
s
Z T
0
dt n2
1(t)
+
p
H[a†
2a2]⇢dW/dt
Idt = 2
p
ha†
2a2idt + dW
Dark counts
Missed photons
Average error

21. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
Modulated
measurement
We can partially avoid the measurement
backaction.
21
Off–on measurement
Modulated
measurement
(t) = sin(!t)
(t) = ⇥(t t0)

22. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
We also need better control over the
photon to improve incoupling.
22

23. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
• Strong optomechanical cooperativity,
• Sufficient qubit lifetime
I talked about:
23
-
OC and K. Hammerer, PRA (accepted)ˇ
Entanglement of
superconducting qubits
C =
4g2
 ¯n
>
1
2

24. Cernotík (LUH): Quantum force sensing with optomechanical transducersˇ
I talked about:
24
Optomechanical single-photon detection
• Quantum Zeno effect limits the efficiency
• We need backaction evasion and better incoupling