Exploring the Quantum-Classical Transition Using Optomechanical Systems Paul Nation Korea University National Taiwan University December 17, 2013 Phys. Rev. A 88 , 053828 (2013)
How does the classical world emerge from the underlying rules of quantum mechanics?
Quantum-Classical Crossover: ?
Quantum-Classical Crossover: Can we push the boundary higher? ?
Quantum-Classical Crossover: Can we push the boundary higher? ? - In principle yes! One of the goals in nanomechanics: a b J 50 µ m J R S 60 μ m Mavalvala, MIT Etaki, Nature Phys. (2008) O’Connell, Nature (2010)
Schrödinger’s Cat (1935): - Death of cat entangled with the quantum mechanical decay of radioactive atoms. - If atom has 50% chance of decay then state of cat is: 1 i + 1 p p | ψ i cat = 2 | 2 | i
1 i + 1 p p | ψ i cat = 2 | 2 | i
1 i + 1 p p | ψ i cat = 2 | 2 | i | ψ i cat = | i
1 i + 1 p p | ψ i cat = 2 | 2 | i
| ψ i cat = | i
| ψ i cat = | i - When Schrödinger looks he is making a measurement .
| ψ i cat = | i - When Schrödinger looks he is making a measurement . - Is the cat simultaneous dead and alive before I measure?
- Absolutely not! The Environment is always making measurements. Gas molecules Photons - Many di ff erent environments, all too complicated to keep track of the dynamics. - Interaction with the environment leads to classicality, (loss of entanglement, superpositions, coherence,...) - Larger objects -> more environ. interactions. - Can make quantum objects behave classical. IBM, 2013.
- Absolutely not! The Environment is always making measurements. Gas molecules Photons - Many di ff erent environments, all too complicated to keep track of the dynamics. - Interaction with the environment leads to classicality, (loss of entanglement, superpositions, coherence,...) - Larger objects -> more environ. interactions. - Can make quantum objects behave classical. IBM, 2013.
- Can not get rid of all environment e ff ects. Gravity may be ultimate environment! - Must find balance between quantum dynamics and environmental e ff ects. Quantum E ff ects in Massive Objects: - Must minimize the coupling to the environment. - Low temperatures. - In vacuum. - Want quantum dynamics that are clearly distinguishable from classical motion. - Want massive object, but simple to model theoretically. Mechanical Oscillator + Nonlinear Interaction
Optomechanics: x ˆ b, ω m , Γ m E, ω p , κ ˆ a, ω c Comet “tail” due to radiation pressure - Interaction between mechanical oscillator and optical of light. cavity via radiation pressure generated by a laser. - Retardation e ff ects give rise to nonlinear Give to ~ ω m Take from ~ ω m oscillator interaction. oscillator - Changing the laser frequency with respect to the optical cavity resonance Red detuned Blue detuned frequency leads to cooling or heating of the resonator. 0 ω m − ω m Laser detuning ∆ - Same dynamics in many quantum optics related fields.
Macroscopic Mirrors Microscopic Mirrors Suspended Pillars Trampoline Resonators Membranes Microtoroids Double-disk Resonators Near-field Resonators Freestanding Waveguide Optical Resonators Superconducting Circuits Photonic Crystals a 5 μ m Photonic Nanobeam “Zipper” Cavity Cavity Nanorods Cold Atom Cavities
Grams Macroscopic Mirrors Microscopic Mirrors Suspended Pillars Trampoline Resonators Membranes Microtoroids Double-disk Resonators Near-field Resonators Freestanding Waveguide Optical Resonators Superconducting Circuits Photonic Crystals a Zeptograms 5 μ m Photonic Nanobeam “Zipper” Cavity Cavity Nanorods Cold Atom Cavities
Grams Macroscopic Mirrors Microscopic Mirrors Suspended Pillars Trampoline Resonators Same physics over 20 orders of Membranes Microtoroids Double-disk Resonators Near-field Resonators magnitude! Freestanding Waveguide Optical Resonators Superconducting Circuits Photonic Crystals a Zeptograms 5 μ m Photonic Nanobeam “Zipper” Cavity Cavity Nanorods Cold Atom Cavities
Applications of Optomechanics: - Ground state cooling of mechanical oscillators. - Quantum limits on continuous measurements. - Sensitive force, mass, and position detection. - Nonclassical states of light and matter. - Entangled states of light and matter. - Quantum information processing and storage. In general, Optomechanical Interaction Nonclassical mechanical states - Want to find simple analogue quantum system that leads to nonclassical oscillator states?
Micromaser (single-atom laser): S - Interaction between a stream of excited two-level atoms and an optical cavity. D B - Only a single atom in the cavity at a given time. R 1 R 2 C - Amount of time atom spends in cavity called interaction time . Gleyzes, Nature (2007) - When cavity has a large quality factor, many interactions Real quantum laser. - Steady states of cavity are sub-Poissonian , i.e. nonclassical oscillator states. - Crucial parameter is the “maser pump parameter”: p N ex gt int / 2 θ = # of atoms passing atom-cavity coupling in cavity lifetime. - Varying pump parameter gives oscillations in cavity photon number that can be interpreted as phase transitions: “Thumbprint of the micromaser.”
Sub-Poissonian States: Oscillator Fano Factor : F = h ( ∆ ˆ N b ) 2 i / h ˆ N b i p Coherent (classical) state | α = Fock (quantum) state | 3 i 3 i Poisson Probability Probability distribution Number state Number state - Poisson: Variance equal to average. - Variance vanishes F= 1 F= 0 - Sub-Poissonian states are quantum oscillator states with F<1. - Strongly sub-Poissonian states characterized by negative Wigner functions .
Wigner Functions: - A quantum phase space (pseudo)probability density distribution. - Not a true probability distribution due to . - Can possess (nonclassical) regions where distribution is negative. p Coherent state | α = Fock (quantum) state | 3 i 3 i Positive Wigner func. Negative Wigner func. Positive Wigner func. - Negativity of Wigner function can be used as measure of nonclassicality.
Optomechanical Setup: x ˆ b, ω m , Γ m E, ω p , κ - Consider a single-mode, driven optomechanical system ˆ a, ω c ⇣ b + ⌘ ˆ a + ˆ b + ˆ ˆ b + ˆ a + ˆ a + ˆ � a + � H = − ∆ ˆ b + g 0 a + E a + ˆ ˆ ˆ Cavity HO Mech. HO Radiation pressure coupling Pumping of cavity - All constants measured in units of the resonator frequency. - Laser-cavity detuning given by . ∆ = ( ω p − ω c ) / ω m - Coupling constant measures oscillator displacement due to a single cavity g 0 photon in units of the zero-point amplitude: p x zp = ~ / 2 m ω m Key Idea : Consider high-Q resonator, , and low-Q cavity, with damping Γ m = Q − 1 m rate , driven by weak laser. κ Single-photon interaction! h ˆ N a i ⇡ h ( ∆ ˆ N a ) 2 i ⌧ 1
Semiclassical Dynamics: - Input-output theory gives Langevin equations of motion for Hamiltonian operators ( ). τ = ω m t d ˆ a ⇣ b + ⌘ a − κ ˆ b + ˆ d τ = i ∆ ˆ a − ig 0 a − iE ˆ 2 ˆ d ˆ b a − Γ m √ d τ = − i ˆ ˆ Γ ˆ a + ˆ b − ig 0 ˆ b − b in 2 - Classical nonlinear e ff ects can be studied in the steady state regime. - Steady state cavity energy given by: ¯ N a 2 g 2 � ¯ ; E 2 = ∆ 2 + κ 2 / 4 a + K 2 ¯ ⌘ (Kerr constant) N a − 2 ∆ K ¯ 0 N 2 N 3 � K = − ⇣ a 1 + Γ 2 m 4 - The renormalized cavity frequency can be defined by the detuning value at which is ¯ N a maximized. ω ω c “Spring-so fu ening”
- The semiclassical limit-cycle dynamics of both the cavity and oscillator found by assuming oscillator undergoes sinusoidal motion (Marquardt et al. PRL 2006): x ( τ ) = ¯ x + A cos( τ ) Static displacement Oscillation amplitude - Plug into Langevin equation for cavity amplitude and use Fourier series a ( τ ) ¯ solution: ∞ J n ( g 0 A ) a ( τ ) = e i ϕ ( τ ) X α n e in τ α n = − iE ¯ i ( n − ∆ + g 0 ¯ x ) + κ / 2 n = −∞ - Time-averaged response peaked at discrete values: X | α n | 2 h | ¯ a | 2 i = n x n labels oscillator sidebands, i.e. ∆ = n + g 0 ¯ n ω m Shi fu due to Kerr nonlinearity (as we will see) - Lineshape is Lorentzian, but peak is shi fu ed depending on . g 0
- Displacement and amplitude are found by self-consistently solving time A ¯ x averaged force balance: X | α n | 2 x ∝ K x = − 2 g 0 ¯ g 0 ¯ n and power balance equations: X α ∗ Γ m A = − 4 g 0 Im n +1 α n n - In general, there are multiple solutions to these equations ; multiple oscillator limit-cycles exist for a given set of parameters.
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