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Measurement-Induced Long-
Distance Entanglement with
Optomechanical Transducers
Ondřej Černotík and Klemens Hammerer
Leibn...
Quantum information
2
Processing
Superconducting qubits,
trapped ions, …
Schoelkopf
Blatt
Transfer
Light
Zeilinger
Storage...
SC qubits can be interfaced with light
using spin ensembles.
3
C. O’Brien et al., PRL 113, 063603 (2014)
K. Xia & J. Twaml...
Mechanical oscillators can also mediate
the coupling.
4
T. Bagci et al., Nature 507, 81 (2014)R. Andrews et al., Nature Ph...
Mechanical oscillators can also mediate
the coupling.
5
K. Xia et al., Sci. Rep. 4, 5571 (2014)
K. Stannigel et al., PRL 1...
Vg
Mechanical coupling
Single-qubit readout
Two-qubit readout Implementations
This talk will be about:
Coupling light and SC qubits to
mechanical oscillators
Vg
8
a x
!,  ⌦, ¯n
!(x) ⇡ !(0) +
d!
dx
x
Cavity frequency:
g0 =
d!
dx
xzpf =
!
L
xzpfCoupling strength:
xzpf =
r
~
2m⌦
x = x...
⌦
Strong coupling can be achieved using
laser driving.
9
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: s...
Strong coupling can be achieved using
laser driving.
10
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: st...
⌦
Strong coupling can be achieved using
laser driving.
11
Optomechanical coupling is weak
g0 = !
xzpf
L
⇡ 25 Hz
Solution: ...
' = '1 '2
Josephson junction is a basic building
block of SC circuits.
12
Superconductor
Insulator (∼ 1 nm)
Superconductor...
' = '1 '2
Josephson junction is a basic building
block of SC circuits.
13
Superconductor
Insulator (∼ 1 nm)
Superconductor...
Charge qubit is a voltage-biased JJ.
14
Electrostatic energy:
ECoulomb = 4EC(N Ng)2
Ng =
CgVg
2e
, EC =
e2
2C
Cg
Vg
Ng
Ene...
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...
Mechanical coupling is achieved using a
mechanically compliant capacitor.
16
Charge qubit with a movable gate
H = 4EC[N Ng...
Optical readout of a
superconducting qubit
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 ⇢ =...
We adiabatically eliminate the transducer
degrees of freedom.
19
d⇢q = ( meas + mech)D[ z]⇢qdt +
p
measH[ z]⇢qdW
meas = 16...
Intermezzo: Adiabatic elimination of
Gaussian quantum systems
System Transducer
System
Gaussian systems are described by
quadratic Hamiltonians, linear jumps, and
homodyne measurements.
21
d⇢ = i[H, ⇢]dt +
X
n...
Dynamics can be described using
statistical moments of canonical operators.
22
Mean values:
Covariance matrix:
d⇢ = i[H, ⇢...
Dynamics can be described using
statistical moments of canonical operators.
23
Mean values:
Covariance matrix:
d⇢ = i[H, ⇢...
Description using statistical moments
enables adiabatic elimination.
24
OC et al., PRA 92, 012124 (2015)ˇ
System Transduce...
Two-qubit readout using light
Joint measurement on two qubits can
generate entanglement between them.
26
1
z + 2
z
| 0i = (|0i + |1i)(|0i + |1i)
!
8
<
:...
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 +
✓...
We can postselect entangled two-qubit
states.
28
We can postselect entangled two-qubit
states.
29
We can also include optical losses.
30
We can also include optical losses.
31
Experimental implementations of
half-parity measurements
The mechanical system can be formed by
a nanobeam.
33
G. Anetsberger et al., Nature Phys. 5, 909 (2009)
J. Pirkkalainen et...
The mechanical oscillator can be a
membrane.
34
R. Andrews et al., Nature Phys. 10, 312 (2014)
T. Bagci et al., Nature 507...
Mechanical oscillators can also couple to
flux qubits.
35
F. Xue et al., NJP 9, 35 (2007)
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 (2...
Summary
37
Vg
Mechanical oscillators can mediate
interaction between light and SC qubits.
38
• Strong optomechanical cooperativity,
• Su...
More complex schemes can be designed.
39
• Measurement feedback
• Two-mode optomechanical driving
• More experimental impl...
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Measurement-induced long-distance entanglement with optomechanical transducers Slide 1 Measurement-induced long-distance entanglement with optomechanical transducers Slide 2 Measurement-induced long-distance entanglement with optomechanical transducers Slide 3 Measurement-induced long-distance entanglement with optomechanical transducers Slide 4 Measurement-induced long-distance entanglement with optomechanical transducers Slide 5 Measurement-induced long-distance entanglement with optomechanical transducers Slide 6 Measurement-induced long-distance entanglement with optomechanical transducers Slide 7 Measurement-induced long-distance entanglement with optomechanical transducers Slide 8 Measurement-induced long-distance entanglement with optomechanical transducers Slide 9 Measurement-induced long-distance entanglement with optomechanical transducers Slide 10 Measurement-induced long-distance entanglement with optomechanical transducers Slide 11 Measurement-induced long-distance entanglement with optomechanical transducers Slide 12 Measurement-induced long-distance entanglement with optomechanical transducers Slide 13 Measurement-induced long-distance entanglement with optomechanical transducers Slide 14 Measurement-induced long-distance entanglement with optomechanical transducers Slide 15 Measurement-induced long-distance entanglement with optomechanical transducers Slide 16 Measurement-induced long-distance entanglement with optomechanical transducers Slide 17 Measurement-induced long-distance entanglement with optomechanical transducers Slide 18 Measurement-induced long-distance entanglement with optomechanical transducers Slide 19 Measurement-induced long-distance entanglement with optomechanical transducers Slide 20 Measurement-induced long-distance entanglement with optomechanical transducers Slide 21 Measurement-induced long-distance entanglement with optomechanical transducers Slide 22 Measurement-induced long-distance entanglement with optomechanical transducers Slide 23 Measurement-induced long-distance entanglement with optomechanical transducers Slide 24 Measurement-induced long-distance entanglement with optomechanical transducers Slide 25 Measurement-induced long-distance entanglement with optomechanical transducers Slide 26 Measurement-induced long-distance entanglement with optomechanical transducers Slide 27 Measurement-induced long-distance entanglement with optomechanical transducers Slide 28 Measurement-induced long-distance entanglement with optomechanical transducers Slide 29 Measurement-induced long-distance entanglement with optomechanical transducers Slide 30 Measurement-induced long-distance entanglement with optomechanical transducers Slide 31 Measurement-induced long-distance entanglement with optomechanical transducers Slide 32 Measurement-induced long-distance entanglement with optomechanical transducers Slide 33 Measurement-induced long-distance entanglement with optomechanical transducers Slide 34 Measurement-induced long-distance entanglement with optomechanical transducers Slide 35 Measurement-induced long-distance entanglement with optomechanical transducers Slide 36 Measurement-induced long-distance entanglement with optomechanical transducers Slide 37 Measurement-induced long-distance entanglement with optomechanical transducers Slide 38 Measurement-induced long-distance entanglement with optomechanical transducers Slide 39
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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.

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Measurement-induced long-distance entanglement with optomechanical transducers

  1. 1. Measurement-Induced Long- Distance Entanglement with Optomechanical Transducers Ondřej Černotík and Klemens Hammerer Leibniz Universität Hannover Palacký University Olomouc, 23 September 2015
  2. 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. 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. 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. 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. 6. Vg Mechanical coupling Single-qubit readout Two-qubit readout Implementations This talk will be about:
  7. 7. Coupling light and SC qubits to mechanical oscillators Vg
  8. 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. 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. 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. 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. 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. 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. 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. 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. 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. 17. Optical readout of a superconducting qubit
  18. 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. 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. 20. Intermezzo: Adiabatic elimination of Gaussian quantum systems System Transducer System
  21. 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. 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. 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. 24. Description using statistical moments enables adiabatic elimination. 24 OC et al., PRA 92, 012124 (2015)ˇ System Transducer ⇢ x,
  25. 25. Two-qubit readout using light
  26. 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. 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. 28. We can postselect entangled two-qubit states. 28
  29. 29. We can postselect entangled two-qubit states. 29
  30. 30. We can also include optical losses. 30
  31. 31. We can also include optical losses. 31
  32. 32. Experimental implementations of half-parity measurements
  33. 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. 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. 35. Mechanical oscillators can also couple to flux qubits. 35 F. Xue et al., NJP 9, 35 (2007)
  36. 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. 37. Summary 37 Vg
  38. 38. Mechanical oscillators can mediate interaction between light and SC qubits. 38 • Strong optomechanical cooperativity, • Sufficient qubit lifetime C = 4g2  ¯n 1
  39. 39. More complex schemes can be designed. 39 • Measurement feedback • Two-mode optomechanical driving • More experimental implementations

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.

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