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Hybrid quantum systems

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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.

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Hybrid quantum systems

  1. 1. Hybrid Quantum Systems Interfacing optical, electrical, and mechanical degrees of freedom Ondřej Černotík Leibniz Universität Hannover Nová Lhota, 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
  3. 3. 3 Investigating the quantum boundary Zurek, quant-ph/0306072
  4. 4. Toolbox Applications Superconducting qubits Optomechanics Electromechanics Photon conversion Coupling SC qubits to mechanics
  5. 5. Toolbox Applications Superconducting qubits Optomechanics Electromechanics Photon conversion Coupling SC qubits to mechanics
  6. 6. Superconducting qubits
  7. 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. 8. Why solid-state cavity QED? 8 Solid-state solution: • artificial atoms, • strip-line cavities. Boissonneault et al., PRA 79, 013819 (2009)
  9. 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. 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. 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. 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. 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. 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. 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. 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. 17. Cavity optomechanics
  18. 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. 19. Radiation pressure 19 • Enables laser cooling and trapping • Controlling mechanical oscillations by light (or vice versa) → optomechanics
  20. 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. 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. 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. 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. 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. 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. 26 Aspelmeyer, Kippenberg, Marquardt, RMP 86, 1391 (2014)
  27. 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. 28. Electromechanics
  29. 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. 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. 31. Piezoelectric capacitor 31 O’Connell et al., Nature 464, 697 (2010)
  32. 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. 33. Toolbox Applications Superconducting qubits Optomechanics Electromechanics Photon conversion Coupling SC qubits to mechanics
  34. 34. Photon conversion
  35. 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. 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. 37. Optical detection of radio waves 37 Bagci et al., Nature 507, 81 (2014)
  38. 38. Piezoelectric optomechanical crystal 38 Bochmann et al., Nature Phys. 9, 712 (2013)
  39. 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. 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. 41. Coupling SC qubits to mechanics
  42. 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. 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. 44. Measurement and control of mechanics 44 O’Connell et al., Nature 464, 697 (2010)
  45. 45. Hint = gb† b z Measurement and control of mechanics 45 Lecocq et al., Nature Phys. 11, 635 (2015)
  46. 46. Measuring qubits with mechanics 46 LaHaye et al., Nature 459, 960 (2009)
  47. 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. 48. Summary
  49. 49. This talk 49 ElectromechanicsOptomechanics Superconducting qubits Photon conversion Electromechanics with qubits Superconducting quantum networks
  50. 50. Not in this talk 50 Nitrogen-vacancy centres Strain coupling Maletinsky Bleszynski-Jayich Atomic ensembles Treutlein Magnetic coupling Rugar Arcizet

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