Alex Zazunov Alex Zazunov LPMMC, Grenoble Denis Feinberg (LEPES - - PowerPoint PPT Presentation

Phonon Squeezing in a Superconducting Molecular Transistor

Alex Zazunov Alex Zazunov

LPMMC, Grenoble Denis Feinberg (LEPES CNRS, Grenoble) Thierry Martin (CPT, Marseille)

AZ, DF, & TM, Phys. Rev. Lett. 97, 196801 (2006)

Contents

• Introduction:
• squeezed states in quantum mechanics
• generation of squeezed states of nanomechanical resonators
• Andreev bound states in superconducting molecular junctions
• Effective Hamiltonian for Andreev levels
• Phonon squeezing in the ‘toy-polaron’ model
• Variational ansatz approach
• Dissipation and detection of squeezed phonon states
• Summary

Introduction: Squeezed states in QM

Harmonic oscillator: Minimum-uncertainty states: displacement operator squeezing operator The variance of one quadrature component is reduced below the zero-point noise level

Phase-coherent superposition

• f number (Fock) states

p q p q p q

Squeezing effect results from the quantum nature of conjugated dynamical variables (vacuum state) |α |

Generation of squeezed states

• f nanomechanical resonators
• ‘Quantum phonon optics’ (in bulk materials)
• Coherent quantum behavior of micron-scale mechanical systems

with high quality factors (approaching Q~100000 below 50 mK):

• entanglement and quantum superposition of spatially separated states [1]
• minimum-uncertainty squeezing of the vibrational motion [2]

[1] Armour, Blencowe, & Schwab, PRL (2002) [2] Blencowe & Wybourne, Physica B (2000); Rabl, Shnirman, & Zoller, PRB (2004); Zhou & Mizel, PRL (2006)

• Applications: weak force detection,…, quantum computing (with continuous variables)

Two phase qubits coupled through a piezoelectric resonator

Cleland & Geller, PRL (2004) Hu & Nori, PRB (1996)

Canonical squeezing Hamiltonian:

• parametrically driven nonlinear potential of cantilever

(starting from the GS of harmonic oscillator) Blencowe & Wybourne, Physica B (2000)

• resonator weakly coupled to a charge qubit (2-level system)

Zhou & Mizel, PRL (2006) Rabl, Shnirman, & Zoller, PRB (2004)

Superconducting Molecular Transistor Φ (magnetic flux)

• Josephson transport through a vibrating molecule
• ‘Molecule’ = resonant electron level coupled to a local vibration
• rf SQUID with a suspended CNT (vibration=bending mode)
• Strong electron-phonon interaction (g ~ Ω)

Vg

Sapmaz et al., PRL (2006)

Andreev bound states

• no phonons:
• degeneracy point:
• ‘slow’ single-electron tunneling,

‘fast’ electron correlations in the Cooper pair Two-level Andreev bound-state system = phase-dependent hybridization of 0- and 2-electron (degenerate) states

• f the resonant level due to Cooper-pair tunneling:

Δ −Δ

After tracing out the bulk degrees of freedom:

Electron self-energy due to tunneling: Large Δ limit (no quasiparticle tunneling): Effective Hamiltonian: Effective current operator:

Spin – single-boson representation

Pauli spin notation for the electron states: (Andreev level spectrum in the absence of phonons) The parity symmetry: Current operator:

Antiadiabatic limit:

Rotation to the polaron basis: The ground state (to lowest order in Γ/Ω):

Josephson current is strongly suppressed for α>1: Entanglement in the charge-state basis (no squeezing)

Spatially separated (α>1) phonon ‘cat’ states in the current-state basis: Squeezing for Γ∼Ω (α<1):

Adiabatic approximation (λ<1, α>1): λ = electron localization energy / delocalization energy

Anharmonic potential for phonon coordinate q:

• single well for λ<1/2
• double well for λ>1/2

Toy-polaron model

• D. Feinberg, S. Ciuchi, & F. de Pasquale,
• Int. J. Mod. Phys. (1990)

Squeezing appears near the bifurcation point: Bohr-Sommerfeld rule: the zero-point energy

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 1.2 1.4 1.6 1.8 2

Squeezing is a crossover phenomenon: It is maximal for intermediate coupling parameters

• f the polaron problem.

λ Δp Δx Δp λ ΔxΔp remains close to one up to the bifurcation point (λ~0.5), while Δp<1 (squeezing)

Squeezing is tunable with the superconducting phase difference:

its intensity depends on φ, in accordance with the effective polaronic interaction

Γ/Ω = 0.5, 2, 4 g=0.9 Ω

Josephson current vs e-ph interaction (φ=π/2) λ Γ/Ω = 0.5, 2, 4

The inflexion region corresponds to the polaron crossover regime where optimum squeezing is achieved.

γ = 0.5 and 0.75 Γ=4 Ω, g=1.5 Ω

• junction asymmetry leads to strong and nearly harmonic squeezing
• weak φ−dependence

Wigner distribution function of the squeezed phonon state Δp = 0.69, Δx Δp = 1.15 g = 2Ω, Γ = 5Ω Δp Δx Δp Δx

Variational ansatz approach

• Strong electron-phonon interacting system (small-polaron problem)
• The ground state of the system (junction + electrodes):

The free energy: δ, η - variational parameters

The variational parameters are obtained by minimizing the free energy: Renormalized tunneling width:

• Good agreement with the exact large Δ calculations
• Δ<< Γ: squeezing is still present provided that g>Ω

Dissipation and detection of squeezed phonon states

Sources of dissipation:

• decoherence of Andreev levels (external flux fluctuations)
• coupling of the local vibration to other phonon modes

Carbon nanotubes: large enough quality factors, Q ~ 1000 Resonant Raman Scattering: spectroscopy of the squeezed states (photon is reemited with excitation of the phonon mode) RR emmision lines (Stokes) at energies Projection of the ground state on the single-electron subspace:

with the intensity

NB Squeezing is the ground state property protected by the gap in the excitation spectrum

Summary

• Josephson transport through a vibrating ‘molecule’ (suspended nanotube)
• Josephson effect triggers coherent phonon fluctuations and generates

non-classical phonon states (‘cat’ states, squeezed states)

• squeezing occurs for a wide range of parameters and is maximal

in the polaron crossover regime (Γ ∼ Ω ∼ g)

• squeezing is tunable with the phase difference (flux)
• minimum-uncertainty state with ~40% squeezing can be obtained