Atom mesoscopic field entanglement

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Atom / mesoscopic field entanglement

Entangling a qubit with a large system

• When is a coherent field classical ?

How to prepare large Schrödinger cats and study
their decoherence
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The strangeness of the quantum

• Superposition principle and quantum interferences
• Feynman Youngs slits experiment contains all
  the mysteries of the quantum

Shimizu et al 1992
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The strangeness of the quantum
a thought experiment about
complementarity
(Bohr-Einstein debate, Solvay 1927)

• Microscopic slit set in motion when deflecting
  particle. Which path information and no fringes
• Macroscopic slit insensitive to interfering
  particle. No which path information fringes are
  observed.
• Wave and particle are complementary aspects of
  the quantum object.

Particle/slit entanglement
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The strangeness of the quantum

• Entanglement
• Two systems after an interaction described by a
  single global state
• No system has a state of its own (only a density
  operator)
• Measurements performed on the two systems are
  correlated in a non classical way (no hidden
  variables).
• Quantum correlations irrespective of the distance
  between entangled systems
• At the heart of quantum non-locality
• Einstein did not like thatand he was wrong (Bell
  inequalities violation)

?YABgt ? ?YAgt ?YBgt
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The strangeness of the quantum

• Quantum/classical limit
• No quantum superpositions at macroscopic scale
• The "Schrödinger cat"
• No macroscopic entanglement.
• We only observe a very small fraction of all
  possible quantum states
• WHY ??

• Decoherence
• A macroscopic system is strongly coupled to a
  complex environment entanglement with
  environment
• In all models, this coupling
• leaves only a few states intact (preferred basis)
• destroys very rapidly the quantum superpositions
  of these states
• Decoherence

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Outline of this talk

• Two experiments on complementarity, entanglement
  and mesoscopic state superpositions using
  circular Rydberg atom and millimeter-wave
  cavities
• A complementarity experiment at the
  quantum/classical boundary
• A realization of Bohrs thought experiment based
  on Rabi oscillation and Ramsey interferometry
• Atom/mesoscopic field entanglement induced by
  photon graininess
• A surprising insight into the fundamental Rabi
  oscillation phenomenon
• A new tool to prepare Schrödinger cat states
  and to investigate their decoherence

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A complementarity experiment at the
quantum/classical boundary
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A modern version of Bohrs proposal

• Mach Zehnder interferometer

• Interference between two well-separated paths.
• Getting a which-path information?

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A modern version of Bohrs proposal

• Mach-Zehnder interferometer with a moving beam
  splitter

• Massive beam splitter negligible motion, no
  which- path information, fringes
• Microscopic beam splitter which path
  information and no fringes

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Complementarity and entanglement

• A more general analyzis of Bohrs experiment
• Initial beam-splitter state
• Final state for path b
• Particle/beam-splitter state
• Particle/beam-splitter entanglement
• (an EPR pair if states orthogonal)
• Final fringes signal
• Small mass, large kick
• NO FRINGES
• Large mass, small kick
• FRINGES

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Entanglement and complementarity

• Entanglement with another system destroys
  interference
• explicit detector (beam-splitter/ external)
• uncontrolled measurement by the environment
  (decoherence)

Complementarity, decoherence and entanglement
intimately linked
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A more realistic system Ramsey interferometry

• Two resonant p/2 classical pulses on an atomic
  transition e/g

R1
R2
Which path information? Atom emits one photon in
R1 or R2 Ordinary macroscopic fields (heavy
beam-splitter) Field state not appreciably
affected. No "which path" information
FRINGES Mesoscopic Ramsey field (light
beam-splitter) Addition of one photon changes
the field. "which path" info NO FRINGES
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Experimental requirements

• Ramsey interferomtery
• Long atomic lifetimes
• Millimeter-wave transitions
• Circular Rydberg atoms
• p/2 pulses in mesoscopic fields
• Very strong atom-field coupling
• Circular Rydberg atoms
• Field coherent over atom/field interaction
• Superconducting millimeter-wave cavities

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Circular Rydberg atoms

• High principal quantum number
• Maximal orbital and magnetic quantum numbers
• Long lifetime
• Microwave two-level transition
• Huge dipole matrix element
• Stark tuning
• Field ionization detection
• selective and sensitive
• Velocity selection
• Controlled interaction time
• Well-known sample position
• Atoms individually addressed
• (centimeter separation between atoms)
• Full control of individual transformations

Complex preparation (53 photons ! ) Stable in a
weak directing electric field Single atom
preparation
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Superconducting cavity

• Design
• Open Fabry Perot cavity with a "photon
  recirculating ring"
• Compatible with a static electric field (circular
  state stability and Stark tuning)
• Very sensitive to geometric quality of mirrors

• Highly polished niobium Mirrors
• Cavity Damping time 1 ms

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General scheme of the experiments
Rev. Mod. Phys. 73, 565 (2001)
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The resonant interaction between an atom and a
field results in the Rabi oscillation

The dynamics of the Jaynes-Cummings hamiltonian
Y(t) gt exp (- iHJC t/h) Y(0)gt
Harmonic oscillator coupled to spin 1/2 like
system
Cavity QED, ion trap physics, laser spectro.,
quantum info.
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Resonant atom-cavity interaction Rabi
oscillation in vacuum

• Initial state e,0gt
• Oscillatory Spontaneous emission and strong
  coupling regime.

p/2 pulse
Vacuum Rabi frequency W 50 kHz
In a large coherent field, Rabi frequency becomes
W ? n
m
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An object at the quantum/classical boundary

• Coherent field in a cavity
• State produced by a classical source coupled for
  finite time to the cavity mode field defined by
  
• complex amplitude a
• A picture in phase space (Fresnel plane)

• From quantum to classical
• Vacuum or
• small field
• Large quantum fluctuations. A field at the
  single-photon level is a quantum object
• Large field
• Small quantum fluctuations. A field with more
  than 10 photons is almost a classical object.

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Bohrs experiment with a Ramsey interferometer

• Illustrating complementarity Store one Ramsey
  field in a cavity
• Initial cavity state
• Intermediate atom-cavity state
• Ramsey fringes contrast
• Large field
• FRINGES
• Small field
• NO FRINGE

From quantum to classical
classical
Atom-cavity interaction time Tuned for p/2
pulse Possible even if C empty
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Quantum/classical limit for an interferometer

• Fringes contrast versus photon number N in first
  Ramsey field

• Fringes vanish for quantum field
• photon number plays the role of the
  beam-splitter's "mass"
• Also an illustration of the DNDF uncertainty
  relation
• Ramsey fringes reveal field pulses phase
  correlations.
• Small quantum field large phase uncertainty
  and low fringe contrast

Nature, 411, 166 (2001)
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Entanglement between a mesoscopic coherent field
and a single atom
The Ramsey interference experiment shows that,
during a p/2 pulse, the atom and the field do not
get entangled when n gtgt1 NO ENTANGLEMENT during
time t p/2 p / 2 W ?n
Atom and field get however ENTANGLED if they are
coupled for a longer time, of the order of 2p/W
Atom dipole states
t gt 2p/W
egt a gt ? Y atom gt a gt Y
-atom gt a - gt
a
Coherent field split into two components
Mesoscopic superposition of coherent states with
opposite phases
Rabi oscillation collapse and revivals revisited
a -
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Atom/mesoscopic field entanglement induced by
photon graininess
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To be classical a field in a cavity must be
coherent and contain many photons on average.
Correspondance principle a
coherent field with many photons has small
relative fluctuations and behaves asymptotically
classically
The interaction with an atom, which can emit or
absorb at most one photon, is expected to leave a
large field practically  unperturbed  and
the  atom field system  unentangled
a (0)gt Yatom(0)gt ? a (t)gt Y(a) atom(t)gt

How large must the photon number be for this
classical limit to be valid?
It depends on how long the interaction lastsA
large field exhibits quantum features if the
interaction with the atom has enough time to
create entanglement.and if there is no
decoherence
Mesoscopic physics in Quantum Optics
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Rabi oscillation is a quantum interference effect
Coherent field amplitude a ?n

Classical limit n ? 8 W
(vacuum Rabi frequency) ? 0 WR W ?n
finite
e gt a gt ?cos (W R t / 2) e gt - i sin (WR t
/ 2) g gt a gt
Yat(t)gt exp (-i WR t / 2) (egt ggt) exp
( i W R t / 2) (egt - ggt)
At classical limit, field unperturbed.
Atomic Rabi oscillation is a quantum interference
between two pathes corresponding to the atomic
dipole states (egt ggt)
time t
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Rabi oscillation in a mesoscopic field

• An interesting situation
• A complex Rabi oscillation signal
• Collapse
• Dispersion of Rabi frequencies
• Revivals
• Finite number of frequencies
• Direct consequence of field
• quantization (photon graininess)

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Collapse and revival in cavity QED

• Rabi oscillation in a 0.85 photons coherent field
  (Brune et al PRL 76, 1800)
• Also observed for in
• closed cavities (Rempe, PRL 58, 353)
• ion traps (Meekhof, PRL 76, 1796)
• What about larger ns?
• What about the field evolution in this complex
  Rabi oscillation process ?

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Rabi oscillation in a mesoscopic coherent field
(n large but finite, lowest non
trivial order in n n)
Y atfield (t)gt exp (- i ?R t/2) (e- i? t /
4?n e gt ggt) a e - i? t / 4?n gt
exp ( i ?R t/2) (e i? t /
4?n e gt - ggt) a e i? t / 4?ngt
Field splits into two components with opposite
phases
Atomic dipole phases rotate in opposite
directions
Field and atom states are locked in phase two by
two field-atom entanglement
Two different time scales phase drift velocity
is n time slower than Rabi oscillation W/4?n
WR /4n

Effects due to photon number graininess, vanish
at classical limit

(J.Gea-Banacloche, Phys.Rev.A 44, 5913 (1991).
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Atom-field states evolution
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Link with Rabi oscillation

• Rabi oscillation quantum interference between
  and
• Contrast vanishes when
• A direct link between Rabi collapse and
  complementarity

• Fast preparation of large Schrödinger cat states
• Another illustration of complementarity
• A surprising insight in the simple Rabi
  oscillation phenomenon

Atom-field decorrelation Unconditional
preparation of the field In a  phase
Schrödinger cat state 
Field state merge again Quantum revival of Rabi
oscillation
Quantum Rabi oscillation and progressive collapse
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Field phase distribution measurement

• Homodyning a coherent field

• Inject a coherent field agt

• Add a coherent amplitude aeif
• Resulting field (within global phase) a(1-eif)gt
• Zero final amplitude for f0

• Probe final field amplitude with atom in g
• Pg1 for a zero amplitude
• Pg1/2 for a large amplitude

• More generally Pg(f) reveals field phase
  distribution
• In technical terms, Pg(f)Q distribution

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Phase splitting in quantum Rabi oscillation

• Timing

• Inject a coherent field

• Send a first atom Rabi oscillation and phase
  shift

• Inject a phase tunable coherent amplitude

• Send an atom in g final amplitude read out

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Mean photon number n 33
No atom (homodyning of initial field)
1 atom interacting during t
32 ms
1 atom interacting during t
52 ms
Phase splitting DF W t /4?n
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Auffeves et al, PRL 91, 230405 (2003)
Field Wigner function (n 40, t 32
ms)
Distance in phase space D2 20 photons
Position of Homodyning peaks (degrees)
DF W t/4?n (degrees)
Large Schrödinger cats, D2 varying from 20 (t
32 ms) to 40 (t 52 ms)
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Schrödinger cat decoherence

• The mesoscopic coherence is destroyed as soon as
  an information about the fields phase in the
  cavity leaks into the environment
  (This is complementarity again)

When field components are well separated, this
happens basically when a single photon is
escaping from the cavity
(?a eij gt ?a e-ij gt ) ?0gtenv ? ?a eijgt ?b
eij gt ?a e-ijgt ? b e-ijgt
These states are quasi orthogonal as soon as ? b
?? 1, i.e. as soon as about 1 photon is lost in
environment
TDecoherence Tcav / 2n sin2j 2 Tcav/D2
Tcav 1 ms, n 20 and j 45 degrees ?
Tdecoherence 50 ms
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Test of coherence induced quantum revivals
Initial Rabi rotation, Collapse And slow phase
rotation
Stark pulse (duration short compared to phase
rotation). Equivalent to a Z rotation by p
Reverse phase rotation
Recombine field components and resume Rabi
oscillation
A spin echo experiment
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Separation and recombination of field components
by Stark switching
Rabi oscillation revivals
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Perspectives

• Rydberg atoms and superconducting cavities
• Towards a two-cavity experiment
• Creation of non-local mesoscopic Schrödinger cat
  states
• Non-locality and decoherence (real time
  monitoring of W function)
• Complex quantum information manipulations
• Quantum feedback
• Simple algorithms
• Three-qubit quantum error correction code

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The Quantum Entanglement Project at ENS in Paris

• 
  S.Haroche
• ENS and Collège de France,
  Paris
• J.M. Raimond M. Brune G. Nogues
• A. Rauschenbeutel P. Bertet
  S. Osnaghi
• A. Auffeves P. Maioli T. Meunier
• P. Hyafil J. Mozley S. Gleyzes
• S. Kuhr P. Milman L. Davidovich

SupportJST (ICORP, Japan), EC, CNRS, UMPC, IUF,
CdF
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Why explorations of the quantum world ?

• A fundamental interest
• Better understanding of quantum postulates
• Superposition
• Measurement
• Entanglement and non-locality
• Exploration of the quantum-classical boundary
• And by realizing these experiments, physicists
  relish in the pride of proving Schrödinger
  wrong.....

• Promising applications
• Use quantum weirdness to realize new functions
  for information transmission or processing
• From bits (0 or 1) to qubits (0gt and 1gt)
• Quantum cryptography
• Quantum teleportation
• Quantum information processing
• Quantum computing
• All rely on sophisticated quantum entanglement
  manipulations

we never experiment with just one electron or
atom or (small) molecule. In thought-experiments
we sometimes assume that we do this invariably
entails ridiculous consequences.  (Schrödinger
British Journal of the Philosophy of Sciences,
Vol 3, 1952)
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An elementary quantum eraser

• Another thought experiment

Two interactions with the same beamsplitter
assembly erase the which path information and
restore the interference fringes
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Ramsey quantum eraser

• A second interaction with the mode erases the
  atom-cavity entanglement
• Ramsey fringes without fields !
• Quantum interference fringes without external
  field
• A good tool for quantum manipulations

?g,1gt
Atom found in g one photon in C whatever the
pathno info and fringes
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From Dream to Reality