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June 22-26, 2009 Olomouc, Czech Republic

11th International Conference on Squeezed States and Uncertainty relations. June 22-26, 2009 Olomouc, Czech Republic. Extraction of a squeezed state in a field mode via repeated measurements on an auxiliary quantum particle. B. Bellomo , G. Compagno

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June 22-26, 2009 Olomouc, Czech Republic

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  1. 11th International Conference on Squeezed States and Uncertainty relations June 22-26, 2009 Olomouc, Czech Republic Extraction of a squeezed state in a field mode via repeated measurements on an auxiliary quantum particle B. Bellomo, G. Compagno Dipartimento di Scienze Fisiche ed Astronomiche, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy, H. Nakazato Department of Physics, Waseda University - Tokyo 169-8555, Japan K. Yuasa Waseda Institute for Advanced Study, Waseda University, Tokyo 169-8050, Japan

  2. P= ..…. U(t) U(t) U(t) U(t) P P P P P P Purificationation through Zeno-Like Measurements (i) System A System B H. Nakazato, T. Takazawa and K. Yuasa, Phys. Rev. Lett. 93, 2003 U(t) total time evolution operator Repeated measurements on A: Projection on a given state B Total system state after N measurements B

  3. Purificationation through Zeno-Like Measurements (ii) Dynamics of B governed by projected evolution operator H. Nakazato, T. Takazawa and K. Yuasa, Phys. Rev. Lett. 93, 2003 Spectrum of Vt: |g0|>| g1|> …>| gn| Not measured system B is driven toward a pure state provided largest eigenvalue of Vt ,g0: unique, discrete, and nondegenerate Purified state independent from the initial state of system B, but dependent on Hamiltonian, state |F0 and time interval t Purification is achieved quickly if |g1/g0|  1 Purification occurs with a certain probability Possibility to control the state of a quantum system on which we have no direct access

  4. System P Free particle: continuum spectrum interacting with Field mode: discrete infinite number of levels Free particle + Field mode + Interaction System F Description of the model Purification mainly analyzed on finite dimensional systems (discrete spectrum) Purification with continua spectra (measured system)? Conditions for distillation of Vmay or not be satisfied Hamiltonian Exact evolution – Time evolution operator B. Bellomo, G. Compagno and F. Petruccione, Phys. Rev. A 74, 052112 (2006)

  5. ..…. U(t) U(t) U(t) U(t) P P P P P P Measurement protocol Unitary evolution interrupted by projective measurements at intervals  Measurement protocol Field mode P= Particle wave packet Projection on ground state of an harmonic potential Particle is initially prepared in the ground state of an harmonic potential (Gaussian form) and repeatedly projected on it at intervals  Gaussian state Total system state after N measurements

  6. Projected evolution operator Unified single exponential Adimensional free parameters Time interval Coupling constant Wave packet width

  7. Squeezed state purification For our system (Continuum spectrum for P ) Projected evolution operator Vt has a discrete spectrum largest eigenvalue unique, discrete, and nondegenerate  field purification Squeezing parameter r=|| squeezed state Bellomo et al., in preparation The purification is achieved quickly if |1/0|<<1 (|Re k| >>1) Quick purification

  8. Features of purification process Independence from the initial state of the field Purification rapidity:|1/0|<<1 (|Re k| >>1) Value of squeezing parameter r Dependence from the initial state of the field Survival probability after N measurements Protocol success probability Fidelity after N measurements Distance of field state from purified state Compromise:after N measurements rapid purification (|Re k| >>1) with respect to decay of survival probability Other requests: final states sufficiently squeezed Achievable by appropriate choice of the parameters

  9. Purification speed vs coupling strength g and time interval t Strong dependence on adimensional free parameters g, t, Dp Quick purification |Re k| >>1 around with

  10. Purification speed vs Squeezing parameter Efficient Purification toward a sufficiently squeezed (high values of Tanh r) Comparison between purification speed vs squeezing parameter p/2<wt <p Tanh r> 0.5 (suffciently high squeezing) Values of parametrs

  11. Survival probability vs Fidelity Distillation with a good probability of success of the measurement protocol? Comparison between survival probability and fidelity for a given wt Field initially in a coherent state (a= 1) High values of Ft(N) for Pt(N) far from 0 Pt(N)vs Ft(N) Values of parametrs wt=9p/10

  12. Conclusions • Two interacting systems; repeated measurements on the first one • Purification of non-measured system achievable also when the measured system has a continuum spectrum First system: free particle (continuous spectrum); Second system: field mode (infinite discrete spectrum) • Field mode is purified into a squeezed state • Efficient purification may occur with good squeezing • Optimization: Frequency measurement 1/ t linked to frequency mode w Coupling strength must be larger than some threshold

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