Knight/paper about CQED

Entangling atoms and photons in CQED

Peter KnightImperial College London

With Ed Hinds, Martin PlenioSusana Huelga, Almut Beige, Stefan Scheel, plus many others

Outline

• Spontaneous emission in confined geometries

Spontaneous emission in a cavityspontaneous emission in front of mirrors

• Entanglement via dissipation

EntanglementEntangling a pair of atoms in a cavity Remote entanglementchip realization (Hinds Group)

• entanglement and Jaynes Cummings High Q cavities nonclassicality

Cavity QED - highlightsInitial ideas:Purcell 1947Quantum Mode Confinement: KleppnerShifts: Power; Barton; Babiker…enhancement and suppression (Hulet, Kleppner,

Hinds, Haroche.....)Rydberg Atoms:Haroche; Walther, Meschede,

Rempe..Optical Transitions: Kimble, Rempe… atom chips (Schmiedmayer, Reichel, Hinds,

Westbrook, Vuletic....)

How does the atom know the mirror is there? Interference: as in Gerhard Rempe’s talk

Emission,propagation, reflectionand re-excitation in acavity: Buzek…plk, Phys Rev A60, 582, 1999

Mirror from first principles• a set of many two-level atoms can play the role of a "mirror".

We 1584 two-level atoms positioned at the center of the 2-D cavity to form a "mirror". All atoms have the same decay constants & resonance frequencies. They are exactly on resonance with the central frequency of the photon wave packet, which propagates towards the atoms from the left. We take into account 256 x 256 modes of the em field from which the photon wave packet is created. These 65 536 modes interact with the 1584 atoms in RWA. We show the propagation (t=0.0) & reflection (t=20.0) of the photon wave packet on the mirror. At t=6.0 and t=16.0 the incoming & reflected parts of the photon interfere & result in intensity oscillations. At t=20.0 the photon has reflected from the mirror. The atoms up- and right from the mirror are analyzer atoms used to detect the spectra of the radiation.

Beam splitter from first principles• we just showed that it is possible to build an almost perfect mirror using

two level atoms. There are several ways how to modify the "mirror" configuration to obtain a beam splitter: the most satisfactory results were obtained by detuning the atoms.

• The total number of two-level atoms is 881 . The atoms are detuned below the central frequency of the photon wave packet. This wave packet is again composed of 256 x 256 modes of the 2-d cavity. Then solve the Schroedinger equation numerically.

• at t=20.0 the original wave packet is split into two parts propagating up and to the right. The energy is divided equal: the atoms form a 50-50 beam splitter.

• this beam splitter has its own internal degrees of freedom and transiently becomes excited.

• Nevertheless, after a while the atoms have completely emitted the excitation & the beam splitter is in its ground state - at this point it is completely disentangled from the one-photon radiation field which is now in a pure superposition state with two macroscopically distinguishable components (reflected &transmitted). We see that when the photon wave-packet is close to the atoms (the beamsplitter) the incident and reflected components of the wavepacket interfere which results in an oscillatory pattern. The wavelength of the photon in the present case has been chosen shorter than in the mirror simulation - this results in less

pronounced interference.

Jaynes-Cummings modeland single-mode-atominteractions.

so what is quantum and what is classical?

• is there a classical world distinct from the quantum?

• are there intrinsically classical states (coherent, thermal...)?

• I believe not - one needs to look harder to uncover the universality of “quantum-ness”

• and here we can look at how it all started: discreteness!

• in principle we can observe this in any realization of a radiation field

• and in practice with a micromaser, as developed by Herbert Walther and Serge Haroche, this has been done, revealing discreteness even in that most “classical-like” of fields, a coherent state

17

W Zurek: Physics Today

being discrete?

• Jaynes-Cummings model

18J H Eberly et al theory ENS: quantum Rabi oscillations

resolve

• its easy to show a coherent state is mathematically equivalent to the vacuum field plus a classical field of suitable amplitude using Roy Glauber’s displacement operator

• classical fields just drive sinusoidal Rabi oscillations

• but whenever the atom is in the excited state the vacuum interaction kicks in and generates a spontaneous term, and them then on a cascade of quantum-generated eigenvalues which beat together and cause collapses (and because discrete, revivals)

• Einstein was right: quantum randomness and spontaneous processes are key! 19

its spontaneous emission!

collapse and field states

Buzek, Knight review Progress in Optics

observed by Haroche group

echo

Coherent state superpositions in dissipative environments

AVidiella Barranco, V Buzek & PLK, Phys Rev A45,6579,(1992)

Dying cat?

Fock state decay

Jumping cats

-Tregenna, Kendon, Knight: quantum interferences-Bartlett, Sanders, Knight: realization in CQED: a quantum Galton quincunx

classical quantum

Barry SandersViv Kendon

Ben Tregenna

Steve Bartlett

Why are these walks interesting? Motivation:quantum interferencesrelate to algorithmic speed-up, can engineer coin decoherence and study transition to classical, ….

Quantum Quincunx

• Head for L, Tail for R:

Superposition of heads and tails goes to superposition of left and right: interferences!

Quantum superpositions with coin flip:

Quincunx in a Cavity: quant-ph/0207028• A two-level atom is the coin: a conditional Stark term shifts the phase

of the cavity field clockwise or counterclockwise depending on the state of the atom as shown by Serge and the ENS group

• Sequence of atoms for repeated random phase shifts: classical walk.• The cavity field initially has a “sharp” phase distribution (NB domain

is a circle, not a line).• variance grows linearly with number of atoms: phase diffusion

Recycling the Coin: quantum walk• The quantum ‘coin’ corresponds to the two levels of the atom, and

this coin can be recycled to give a quantum walk.• The cavity evolution is interrupted by periodically spaced

Hadamard transformations, which ‘flip’ the coin: F(ϕ) operates on the atom-cavity system in between these Hadamard coin flips.

• It is conceivable to apply (FH)15 within the timescale of an experiment

• . For small ϕ, the results are similar to 1D random walk, but large ϕ is also possible: random walk on a circle.

Quantum Quincunx• Use a single atom (recycle the

coin)• Use π/2 pulse to implement

quantum coin flip• Quantum phase diffusion =

quantum quincunx• Phase spreads quadratically

faster

Conditional phase shifts?• Conditional phase-shift operator• Cavity prepared in a coherent state and the atom in

either the + or – state:• The phase of the cavity coherent state undergoes a

random walk in discrete steps of ϕ.• Ideally consider the (un-normalized) phase state

Wigner functions, α=3, phase step 2π/6?

T=4 T=3

T=2T=1

Note fringes as well as displacements-see via quadraturemeasurements

Quadrature variances

Entanglement: History

"When two systems, …… enter into temporary physical interaction due to known forces between them, and …… separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled."

Schrödinger (Cambridge Philosophical Society)

Schrödinger coined the term “entanglement” in 1935

Entanglement

Superpositions:

Superposed correlations:

Entanglement(pure state)

Separability

Separable states (with respect to the subsystems A, B, C, D, …)

Separability

Separable states (with respect to the subsystems A, B, C, D, …)

Everything else is entangled e.g.

Measures of entanglement: Schmidt decomposition- Ekert & PLK Amer J Phys 63, 415 (1995)

Bipartite pure states:

Schmidt decomposition

Positive, real coefficients

Measures of entanglement




Same coefficients

Measure of mixedness

Reduced density operators





Same coefficients

Measure of mixedness

Reduced density operators

Unique measure of entanglement (Entropy) Phoenix & Knight, Annals of Physics (N.Y.) 186, 381 (1988)

Example

Consider the Bell state:

Example


This can be written as:

Example


This can be written as:

Maximally entangled (S is maximised for two qubits)

“Monogamy of entanglement”


Bipartite mixed states:

Entanglement of formation

von Neumann entropy

Minimum over all realisations of:

• Average over pure state entanglement that makes up the mixture

• Problem: infinitely many decompositions and each leads to a different entanglement

• Solution: Must take minimum over all decompositions (e.g. if a decomposition gives zero, it can be created locally and so is not entangled)

Entanglement swapping C.H. Bennett et al., PRL 70,1895 (1993)

• With use a Bell-state measurement, it is possible to swap entanglement.

Entangler 2Entangler 1

Bell-statemeasurement

Zeilinger entanglement swap

Photons now entangled

Entangled photon source

See Bose, Vedral and PLK, Phys Rev A,

• Pan & Zeilinger, “experimental entanglement swapping”, PRL 80, 3891 (1998)

Imperial College London

Bell projection allows entanglement swapping

Bell projectiononto

Goal: Entangle Atoms with photons Make Bell projection on photons

Obtain entangled atoms

Cavity QED

Single atom light-matter interfaces

Most schemes require

Also have strong cooling/trapping requirements: Lamb-Dicke limit

Open systems

• Quantum systems are never completely decoupled from their environment

• This coupling can be a big problem for QIP:

Reduces fidelity in `small scale’ applications Resulting errors can prevent quantum algorithms from working

Open systems

• However, more subtle effects are possible when we monitor the environment

Spontaneous emission

• Dynamics of an open system can be described by a Lindblad equation

• If the environment is monitored, we have the conditional evolution:

`No click’ case

Click in jth detector

Spontaneous emission

• This equation is invariant under the transformation

• This implies that the following conditional evolution is also valid:

`No click’ case

Click in jth detector

• Physically: the conditional evolution that we obtain depends on how we monitor the environment

• This has important consequences!


Projective Measurements

Projective Measurement


Spontaneous emission: Observation of a decaying atom

Initial state: State after no-click at time Δt:

No detection ever Atom in ground state

Each failure to detect provides information quantum state changes!


Ingredients:• Spontaneous decay• Interaction between atoms

Idea: take two atoms in an optical cavity

Symmetrical positions

M.B. Plenio, S.F. Huelga, A. Beige, P.L. Knight, Phys. Rev. A 59, 2468 (1999)

Photons may leakout of the mirrors

Use no-detection events to create entanglement


Creating entanglement in a lossy cavity




Now we wait and see …



No photon detected

In 50% of the cases we will never see a photon singlet state

One photon detected

ge eg

gg

ee


Many photons will be emitted

No photons will be emitted

Detector will see some photons after sufficiently long time

When no photons are seen, success prob = p

M.B. Plenio, S.F. Huelga, A. Beige, P.L. Knight, Phys. Rev. A 59, 2468 (1999)


Beamsplitters make Bell projections


S. Bose, P.L. Knight, M.B. Plenio and V. Vedral, PRL 58, 5158 (1999)

D.E. Browne, M.B. Plenio, and S.F. Huelga, PRL 91, 067901 (2003)

Click

Entangled

B. B. Blinov, D. L. Moehring, L.-M. Duan, and C. Monroe, Nature 428, 153 (2004)

Remoteion-ionentanglement

BS

D

D

coincident photon

detection

table1 table 2

A“quantummodem”linkingquantumcomputers

Experiments

• Ion trap


Chips at Imperial: Ed Hinds

Chips at Imperial

Chips at Imperial

metal slab

height

dissipation

resistivity of metal

fluctuating field outside

spin flips (and heating)

Johnson noiseHugely increases the spin flip rate

Henkel, Pötting and Wilkens Appl. Phys B 69,379 (1999)

Spin flip rate near a surface

skin depth

fast decay due topresence of surface modes

Slabs

• Scheel, Rekdal, Knight & Hinds, quant-ph/0501149; Phys Rev A

• Lin et al (Vuletic group), PRL 92, 050404 (2004

90 µm

concave mirrors etched in silicon (Southampton)

after coating with gold,can see focal spots under a microscope

entangling an atom with a photon

this allows coherent atom-photon coupling

F = 2

F = 35S

5Pe.g. photon pistol a la Rempe, Kimble etc

F = 2

F = 3

Coherent exchange of quantum information between atom and photon

optical interface to atom quantum memory

atoms can be entangled through a shared cavity photon as we saw earlier

an alternative quantum logic gate

entangling two atoms in an optical microcavity

conclusions• My own work was tremendously influenced by Serge, on the

vacuum, superradiance, cavities, JCM….

• I have shown you a little on atoms in cavities and near surfaces

• Funding:

Happy 66th year... Serge!

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Knight/paper about CQED

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