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Robust Entanglement in Cavity Opto-mechanics
Paolo Tombesi
Università di Camerino David Vitali Claudiu
Genes Stefano Mancini Stefano Pirandola Vittorio
Giovanetti Luciano Ribichini Andrea Mari Giulia
Gualdi
Daniel Walls University of Auckland Gerard J.
Milburn University of Queensland Antoine
Heidmann, ENS Lab Kastler Brossel Markus
Aspelmeyer University of Vienna Seth Lloyd, MIT

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OUTLINE

• Some history
• The system
• Reductionism
• Linearized quantum Langevin equations
• Cooling to the ground state (self-cooling)
• Optomechanical Entanglement
• Tripartite entanglement (Stokes-mirror-antiStoke
  s)
• Many mechanical modes (two-mode case)
• Output modes
• Conclusions

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Aspelmeyer
Heidmann
Harris
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Standard way to operate a quantum memory state
transfer from light to matter Alternative
solution use entanglement between light and the
mechanical mode (more powerful and flexible for
quantum information processing)
System of interest driven, high-finesse
Fabry-Perot cavity, with a vibrating micro-mirror
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doubly clamped free-standing Bragg mirror (Wien)
cantilever mirror (LKB, Paris)
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The Fabry-Perot description also applies to other
different systems
silica toroidal optical microcavities (Caltech,
Munich, Nat. Phys. April 08), still coupled by
radiation pressure
nanomechanical resonator capacitively coupled to
a superconducting, transmission-line microwave
cavity of C. Regal et al, Nat. Phys. 08)
D. Vitali, P. T., M. J. Woolley, A. C. Doherty,
and G. J. Milburn, Phys. Rev. A 76, 042336 (2007))
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MIM
membrane in the middle scheme by Thompson et
al., Nature, March 08
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The System
In general, many cavity modes interact with many
vibrational modes of the micro-mirror
normal vibrational modes un(r) of the micromirror
Optical and vibrational modes interact through
radiation pressure
surface deformation field of the micro-mirror
Radiation pressure field on the mirror, caused by
the optical modes of the cavity (with length L),
with annihilation operator ak, frequency wk, and
field transverse spatial structure nk(r)
overlap between optical and vibrational modes
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Simplification of the system

• We now assume that
• The external laser (with frequency wL ? wc)
  drives only a single cavity mode a
• Mechanical frequencies Wn ltlt c/2L cavity
  free-spectral range ? scattering into the other
  cavity modes is negligible ? Single optical
  cavity mode description

Q mirror transverse deformation x (r) averaged
over the spatial structure of the driven optical
mode, v02(r) . It is a superposition of the
excited normal modes
Using a bandpass filter in the detection scheme
one can isolate a single mechanical resonance
? Q ? q coordinate of the observed normal mode,
with given frequency and damping The
micro-mirror is equivalent to a single damped
quantum harmonic oscillator
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With dimensionless operators (a,a)1, (q,p)i
Finally the Hamiltonian
Noises acting on the system
Gaussian, generally non-Markovian, It is
Markovian in the case Qm gtgt 1
Gaussian, Markovian
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In a frame rotating at ?L
opto-mechanical coupling
Amplitude quadrature Phase quadrature
Amplitude noise Phase noise
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GROUND STATE COOLING OF THE MICRO-MIRROR

• Gaussian steady state, due to
• Linearized dynamics
• all quantum noises are Gaussian

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When
F. Marquardt et al.,PRL 99 (2007) I. Wilson-Rae
et al., PRL 99, (2007)
one can approach ground-state cooling when h ? 1
and by minimizing A/G, which implies taking D ?
wm to maximize A Example (minimum at neff ? 0.1
i.e Teff ? 0.2 mK) for a realistic set of
parameter values
wm/2p 10 MHz, gm/2p 100 Hz, PL 50 mW, L
0.5 mm, m250 ng, cavity finesse 1.2 x 105
(starting from ñ1250 i.e. T ? 0.6 K)
A simple lower bound for neff can be found,
C. Genes et al., Phys. Rev. A 77, 033804 (2008)
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Going back to the Hamiltonian
Linearizing with respect to the steady state we
have a parametric like interaction
This interaction states that the destruction of a
mechanical quantum is accompanied with the
creation of an up-shifted photon
(anti-Stokes scattering) and a destruction of a
pump photon, while the creation of a phonon is
accompanied with the creation of a down-shifted
photon (Stokes scattering) and
the destruction of a pump photon
Then one can immediately see that energy can be
transferred between the mirror and the field
sidebands. An imbalance between the two
sidebands scattering rates is responsible
for either cooling (when anti-Stokes scattering
is stronger than Stokes one) or heating
We have a tripartite system S,AS,Mirror
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We have a tripartite system Stokes-AntiStokes-Mirr
or which can be described by A 6x6 readout
correlation matrix
We are interested in S-M, AS-M and S-AS
entanglement then we can use a bipartite Gaussian
state description whose CM
For bipartite Gaussian states, rT1 negative
(NPT) is not only sufficient but also a necessary
entanglement condition, and it can be expressed
in terms of V only (R. Simon, Phys. Rev. Lett.
84, 2726 (2000))
V provides also a quantitative measure of
entanglement Logarithmic negativity, EN
G. Vidal and R. F. Werner, Phys. Rev.A 65, 032314
(2002) G. Adesso et al., Phys. Rev. A 70,
022318 (2004)
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S-M
AS-M
S-AS
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Relaxing the single mechanical mode
description What if a nearby mechanical mode is
present ?
C.Genes et al. NJP 10, 095009 (2008)
Everything depends upon the frequency mismatch
between the two modes dw21 w2 w1
Cooling is not disturbed if the two modes are not
too close the two modes are even simultaneously
cooled
w2 1.7w1
F 3 104 , k ? wm
F 1.5 105 , k ? 0.2wm
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Cooling is inhibited when the frequencies are
close!
It happens when the modes are separated by less
than the effective mechanical width, dw21 lt G2
(net laser cooling rate)
w2 0.95w1
D2 w1
one mode only
C. Genes et al., arXiv0803.2788v1 quant-ph
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This inhibition is due to a classical destructive
interference phenomenon, similar to a classical
analogue of electromagnetically induced
transparency (EIT)
Two modes
when dw21 ? 0
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EFFECT OF NEARBY MODE ON ENTANGLEMENT
Similar to cooling the two modes are
simultaneously entangled with the cavity mode if
the are not too close dw21 gt G2
w2 1.5w1
one mode only
Entanglement is more fragile and more affected
than cooling
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EFFECT OF NEARBY MODE ON ENTANGLEMENT
The situation is more involved when the modes are
close dw21 lt G2
T 0
one mode only
Entanglement at T 0 increases at resonance
because the center-of-mass is strongly
entangled with the cavity
T 0.4 K
But entanglement at resonance is soon destroyed
by temperature due to the uncooled relative
motion
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C. Genes, A. Mari, P.T.,D. Vitali PRA 78, 032316
(2008)
ENTANGLEMENT WITH OUTPUT FIELDS
For quantum communication applications one
manipulates traveling rather than intracavity
photons ? it is more important to analyze the
entanglement of the mechanical mode with the
optical cavity output
By considering the output, one can manipulate a
multipartite system in fact, by means of
spectral filters, one can select many different
traveling output modes originating from a single
intracavity mode
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PROPER DEFINITION OF OUTPUT MODES
input-output relation
From the continuous cavity output aout(t), one
can extract N independent output modes by
selecting appropriate time (or frequency)
intervals
gk(t) causal filter function
C. Genes et al., arXiv0806.2045v1 quantph
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SELECTING ONE OUTPUT MODE
By appropriately choosing the center and the
bandwidth of the detected output mode, the
entanglement with the mechanical resonator can be
even distilled and increased
W output center frequency e wmt , t inverse
bandwidth
Entanglement with the mechanical mode is
optimally carried away by the output mode
centered at the Stokes sideband of the laser,
with width ? G
Optomechanical entanglement with the output mode
centered at W
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SELECTING TWO OUTPUT MODES
The system is actually in a tripartite-entangled
steady state, involving the nano-resonator and
the two (Stokes and antiStokes) sidebands
antiStokes
Stokes
Entanglement between the Stokes output mode and
the second output mode with center frequency W
Stokes and antiStokes are entangled
Cavity output noise spectrum, with the Stokes and
antiStokes peaks
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ENTANGLEMENT BETWEEN OUTPUT SIDEBANDS
Entanglement increases for decreasing bandwidth
It is very robust against temperature
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CONCLUSIONS

• The radiation pressure of the optical light
  within a cavity is able to prepare genuine
  quantum states of macroscopic mechanical
  oscillators
• One can create stationary optomechanical
  entanglement between the cavity mode and the
  moving mirror, robust against temperature
• One can cool to the ground state the mechanical
  oscillator in particular, self-cooling is more
  suitable in the good cavity limit k lt wm,
• Cooling and entanglement can be suppressed by the
  presence of a nearby mechanical resonance.
• Optomechanical entanglement can be optimally
  transferred to the output fields by
  appropriately choosing the center and bandwidth
  of the detected field entanglement can be
  distilled and increased

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