In this tutorial, I will give an overview of hybrid quantum systems and their applications in quantum technologies. I will start by reviewing their individual components, focusing primarily on the theory of superconducting circuits, cavity optomechanics, and electromechanics. Afterwards, I will discuss a few applications of hybrid systems composed of these components. In particular, I will explain how opto-electro-mechanical systems can be used to achieve frequency conversion between microwaves and light and how electromechanical systems can be used to couple mechanical motion to superconducting quantum bits.

1. Hybrid Quantum Systems
Interfacing optical, electrical, and mechanical
degrees of freedom
Ondřej Černotík
Leibniz Universität Hannover
Nová Lhota, 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

3. 3
Investigating the quantum boundary
Zurek, quant-ph/0306072

4. Toolbox Applications
Superconducting qubits
Optomechanics
Electromechanics
Photon conversion
Coupling SC qubits to
mechanics

5. Toolbox Applications
Superconducting qubits
Optomechanics
Electromechanics
Photon conversion
Coupling SC qubits to
mechanics

6. Superconducting qubits

7. Why solid-state cavity QED?
7
Cavity QED
Strong coupling:
• large atoms,
• tight field confinement.
H = g( +a + a†
),
g / d/
p
V
Rempe

8. Why solid-state cavity QED?
8
Solid-state solution:
• artificial atoms,
• strip-line cavities.
Boissonneault et al., PRA 79, 013819 (2009)

9. Josephson junction
9
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:
V =
~
2e
˙', I = I0 sin '
Josephson relations
˙I = I0 cos(') ˙'
V =
~
2e
1
I0 cos '
˙I = L(') ˙I
Bennemann & Ketterson, Superconductivity (Springer)

10. Phase qubit: current-biased JJ
10
~2
2EC
¨' +
~2
(2e)2R
˙' +
@
@'
EJ
✓
cos '
I
I0
'
◆
= 0
I
'
Energy
EJ ECRequires
Two-level approximation:
H =
~!
2
z
! =
r
EJ EC
2
"
1
✓
I
I0
◆2
#1/4

11. Phase qubit: current-biased JJ
11
~2
2EC
¨' +
~2
(2e)2R
˙' +
@
@'
EJ
✓
cos '
I
I0
'
◆
= 0
EJ ECRequires
Two-level approximation:
H =
r
EJ EC
2
"
1
✓
I
I0
◆2
#1/4
z =
~!
2
z

12. Charge qubit: voltage-biased JJ
12
Electrostatic energy:
ECoulomb = 4EC(N Ng)2
Ng =
CgVg
2e
, EC =
e2
2C
Cg
Vg
Ng
Energy
Bennemann & Ketterson, Superconductivity (Springer)

13. Charge qubit: voltage-biased JJ
13
Total Hamiltonian: H = 4EC(N Ng)2
+ EJ cos '
Two-level approximation:
H = 2EC(2Ng 1) z
EJ
2
x
EC EJ

14. Flux qubit
14
Total magnetic flux: = 0
⇣
n
'
2⇡
⌘
0 =
~
4⇡e
flux quantumL
H =
Bz
2
z
Bx
2
x
Bennemann & Ketterson, Superconductivity (Springer)
Energy
|0i |1i
'

15. Flux qubit
15
H = ECN2
EJ cos
✓
2⇡
0
◆
+
( x)2
2L
U(') =
2
0
4⇡2L
(' 'x)2
2
EJ cos '
EJ
2
0
4⇡2L
, x ⇡
1
2
0, EJ EC
H =
Bz
2
z
Bx
2
x

16. Some experiments in circuit QED
16
• 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
Kelly et al., Nature 519, 66 (2015)

17. Cavity optomechanics

18. Radiation pressure
18
• J. Kepler (1619): Light from the Sun pushes comet
tails away
• J.C. Maxwell (1865): Momentum of EM waves
connected to the Poynting vector

19. Radiation pressure
19
• Enables laser cooling and trapping
• Controlling mechanical oscillations by light (or vice
versa) → optomechanics

20. Basics of cavity optomechanics
20
H = ~!a†
a + ~⌦b†
b + ~g0a†
a(b + b†
)
Aspelmeyer, Kippenberg,
Marquardt, RMP 86, 1391 (2014)
a x
!,  ⌦, ¯n
H = ~!(x)a†
a + ~⌦b†
b
Hamiltonian:
!(x) ⇡ !(0) +
d!
dx
x
Cavity frequency:
g0 =
d!
dx
xzpf =
!
L
xzpfCoupling strength:
xzpf =
r
~
2m⌦
x = xzpf (b + b†
),

21. Mechanically mediated nonlinearity
21
! = !(x), x = x(a†
a)
! = !(a†
a)→ Kerr nonlinearity
Optomechanical interaction:
Formally: diagonalise the Hamiltonian
U = exp
⇣g0
⌦
a†
a(b b†
)
⌘
H ! U†
HU = ~⌦b†
b + ~
✓
!
g2
0
⌦
◆
a†
a ~
g2
0
⌦
a†
a†
aa
Fabre et al., PRA 49, 1337 (1994)

22. Linearised dynamics
22
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: strong optical drive a ! ↵ + a
Aspelmeyer, Kippenberg,
Marquardt, RMP 86, 1391
(2014)
Interaction Hamiltonian
Hint = ~g0(↵ + a†
)(↵ + a)(b + b†
)
= ~g0{↵2
(b + b†
) + ↵(a + a†
)(b + b†
) + a†
a(b + b†
)}
⇡ ~g0
p
n(a + a†
)(b + b†
)

23. Linearised dynamics
23
H = ~ a†
a + ~⌦b†
b + ~g(a + a†
)(b + b†
)
= ! !L, g = g0
p
n
Hint ⇡ ~g(a†
b + b†
a)
Red-detuned drive:
Optomechanical cooling
Full Hamiltonian:
!!L
⌦
Aspelmeyer, Kippenberg,
Marquardt, RMP 86, 1391
(2014)
= ⌦

24. Linearised dynamics
24
H = ~ a†
a + ~⌦b†
b + ~g(a + a†
)(b + b†
)
= ! !L, g = g0
p
n
Hint ⇡ ~g(ab + a†
b†
)
Blue-detuned drive:
Two-mode squeezing
Full Hamiltonian:
! !L
⌦
Aspelmeyer, Kippenberg,
Marquardt, RMP 86, 1391
(2014)
= ⌦

25. Linearised dynamics
25
H = ~ a†
a + ~⌦b†
b + ~g(a + a†
)(b + b†
)
= ! !L, g = g0
p
n
Hint ⇡ ~g(a + a†
)(b + b†
)
Resonant drive:
Position readout
Full Hamiltonian:
! = !L
Aspelmeyer, Kippenberg,
Marquardt, RMP 86, 1391
(2014)
= 0

26. 26
Aspelmeyer, Kippenberg,
Marquardt, RMP 86, 1391
(2014)

27. Some optomechanical experiments
27
• Ground state cooling of mechanics
Chan et al., Nature 478, 89 (2011)
• Quantum coherent coupling
Verhagen et al., Nature 482, 63 (2012)
• Ponderomotive squeezing of light
Brooks et al., Nature 488, 476 (2012); Safavi-Naeini et al., Nature 500, 185
(2013)
• Observation of back-action noise
Purdy et al., Science 339, 801 (2013)
• Quantum feedback control
Wilson et al., Nature 524, 325 (2015)

28. Electromechanics

29. Capacitive electromechanical coupling
29
H = ~!a†
a + ~⌦b†
b + ~g0a†
a(b + b†
)
x
H = ~!(x)a†
a + ~⌦b†
b
Hamiltonian:
!(x) =
1
p
LC(x)
= !(0) +
d!
dx
x
Circuit resonance:
g0 =
!
2C
dC
dx
xzpfCoupling strength:
xzpf =
r
~
2m⌦
x = xzpf (b + b†
),

30. Parallel-plate capacitor
30
dC
dx
=
✏A
(d + x)2
Lehnert
x
d
C(x) =
✏A
d + x
Capacitance:
g0 =
!
2d
xzpf
Coupling strength:

31. Piezoelectric capacitor
31
O’Connell et al., Nature 464, 697 (2010)

32. Some electromechanical experiments
32
• Ground state cooling of mechanics
Teufel et al., Nature 475, 359 (2011)
• Coherent state transfer
Palomaki et al., Nature 495, 210 (2013)
• Electromechanical entanglement
Palomaki et al., Science 342, 710 (2013)
• Squeezing of mechanical motion
Wollmann et al., Science 349, 952 (2015)

33. Toolbox Applications
Superconducting qubits
Optomechanics
Electromechanics
Photon conversion
Coupling SC qubits to
mechanics

34. Photon conversion

35. Double state swap
35
Optomechanics with red detuning: state swap
Hint = ~g(a†
b + b†
a)
Mechanical oscillator coupled
to microwaves and light
State swap between
microwave and optical fields.
Hint = ~ge(a†
b + b†
a)
+ ~go(c†
b + b†
c)
Andrews et al., Nature Phys. 10, 321 (2014)

36. Double state swap
36
Swapping rates
e =
4g2
e
e
o =
4g2
o
o
e = o
e,o
¯n
> 1
Efficient transduction:
• impedance matching
• strong cooperativity
Andrews et al., Nature Phys. 10, 321 (2014)

37. Optical detection of radio waves
37
Bagci et al., Nature 507, 81 (2014)

38. Piezoelectric optomechanical crystal
38
Bochmann et al., Nature Phys. 9, 712 (2013)

39. Alternatives: Adiabatic transfer
39
Tian, PRL 108, 153604 (2012); Wang & Clerk, PRL 108, 153603 (2012)
A =
1
g0
( goa + gec)
B =
1
g0
p
2
(gea g0b + goc)
C =
1
g0
p
2
(gea + g0b + goc)
g0 =
p
g2
e + g2
o
Hint = ge(a†
b + b†
a) + go(c†
b + b†
c)
Strong coupling: normal modes
H = ⌦AA†
A + ⌦BB†
B + ⌦CC†
C

40. Alternatives: Quantum teleportation
40
Optomechanics with blue detuning: entanglement
Hint = ~g(ab + a†
b†
)
With state swap:
Entanglement between
light and microwaves.
Quantum information
transfer using
teleportation.
Barzanjeh et al., PRL 109, 130503 (2012)

41. Coupling SC qubits to mechanics

42. Coupling to a common microwave field
42
Lecocq et al., Nature Phys. 11, 635 (2015)
Qubit coupling Hq = ~J( +a + a†
)
Hq = ~ a†
a z
Electromechanical interaction Hem = ~g(a + a†
)(b + b†
)

43. Direct qubit-mechanical interaction
43
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:
Heikkilä et al., PRL 112, 203603 (2014)

44. Measurement and control of mechanics
44
O’Connell et al., Nature 464, 697 (2010)

45. Hint = gb†
b z
Measurement and control of mechanics
45
Lecocq et al., Nature Phys. 11, 635 (2015)

46. Measuring qubits with mechanics
46
LaHaye et al., Nature 459, 960 (2009)

47. Quantum networks with SC qubits
47
Yin et al., PRA 91, 012333 (2015) Stannigel et al., PRL 105, 220501 (2010)
OC & K. Hammerer, in preparationˇ

48. Summary

49. This talk
49
ElectromechanicsOptomechanics
Superconducting qubits
Photon conversion
Electromechanics with qubits
Superconducting
quantum networks

50. Not in this talk
50
Nitrogen-vacancy centres
Strain coupling
Maletinsky
Bleszynski-Jayich
Atomic ensembles
Treutlein
Magnetic coupling
Rugar
Arcizet