1

Nature of laser light

David PeggCentre for Quantum Computer Technology

Griffith University, Brisbane

John JeffersUniversity of Strathclyde, Glasgow

2

Is laser light in a coherent state?Traditional answer: Laser light is in a coherent

state , with , of definite, butunknown, phase. Thus represent by a mixture:

This can be derived by standard treatments.However we can also write as a mixture:

- a state of definite, but unknown, photon number.

!

"##$

!

2ˆ

2

0

dF %=

!

nnn

e

n

n

F !"

=!

ˆ

22 #

$#

!"" ie=

3

Does it matter what state it is in?

Yes, if we wish to use the traditional assumption toinfer from the results of an experiment that wehave e.g. prepared a squeezed state (Mølmer,PRA, 55, 3195, who conjectured that the coherentstate description is a “convenient fiction”). Sameapplies to continuous variable teleportation(Rudolph and Sanders PRL, 87, 077903).

Can we tell whether it is a coherent state of unknownphase or a photon number state of unknownnumber? Not without more information.

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Can use measurement information orpreparation information.

Measurement informationDo experiments on the light.Photon counting? No, clearly does not

distinguish.Split beam and let beams interfere

(interferometer)? No, can be described interms of interfering amplitudes for photonpaths.

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Excite atoms, (e.g. π/2 pulse), making useof Rabi precession? No, get Rabiprecession with a number state.

Interfere with another laser? See later.

Disrupt phase of laser? (Photon numberstates have random phase anyway, soshould not be affected. Coherent stateswould have π/2 pulse disrupted.) No(surprisingly).

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Preferred ensemble fallacyIn general we measure probabilities and

expectation values. These depend only on and not on its partitioning.

This may indicate that no measurementinformation can tell us whether the laser is in acoherent state or not. So we must turn topreparation information.

!

)ˆˆ(Tr AA !=

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Preparation informationAlice prepares a spin-1/2 system in +z or –z state

with equal probability. The density operator forthis is

In (a), the coefficients are preparation probabilities;in (b), they are not.

)(

1

)(ˆ

21

21

21

21

21

byyyy

azzzz

!!+++=

=

!!+++="

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Without sufficient knowledge of thepreparation we might describe the state aseither a state of definite but unknown z or astate of definite but unknown y.

If we find out that the magnetic field in Alice’sStern-Gerlach apparatus is in the z-direction, then we would know that thelatter description is a fiction, even though itmay be convenient to use this descriptionfor predicting results of future experimentson the system.

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Can we prepare light in a coherent state?

From a classical source? Yes.From a quantum oscillator? Yes, if oscillator is in a

coherent state itself.From a group of excited atoms in a cavity? Yes, if

the atoms are in an eigenstate of which is alinear combination of terms of the type

(lasing transition states of i th atom)

c

egii

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Does a normal laser prepare such anatomic state?

No (extremely unlikely)Perhaps a small accidental approximately

coherent state is amplified by the laser?No – amplification adds phase noise,

cannot achieve a state with well-definedphase appropriate for an intense coherentstate.

A coherent state from a laser does appearto be a fiction which cannot be exposedby future measurements.

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What state is prepared by a laser?

An entangled state of the field and the source(atoms + excitation mechanism) with randomoptical phase.

Can we reduce this state by tracing over the sourcestates?

Yes, then we obtain the density operatorHowever is itself a fiction, indeed one that can

be exposed by measurement information(measuring the atom states).

F!

F!

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How can we explain interference effects?

Leaking mirror, interaction Hamiltonian terms(energy conserving).

Initial state evolves after short time toinclude a term

Detection of one photon thus reduces intra-cavitystate from to (unnormalised)

++! aaaa

ooˆˆ andˆˆ

00ˆoo

!

11ˆˆˆoo

aa+!"

! +aa ˆˆˆ!

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One photocount registered. Measured state is

when evolved back to just outside cavities.Interaction Hamiltonian term:

( ) ( )oo

i

ooboa

i

oboavacbeae

+!+! +=+ ˆˆ10012

1

2

1 ""

a

b

)]ˆˆ)(ˆˆ()ˆˆ)(ˆˆ[(ˆˆˆˆ21

o

i

o

i

o

i

o

i

ooobeabeabeabeabbaa

!!!! ""+++=++"++"+++

=o

f

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Evolves from to include term

Thus effect of detecting first photocount is to change to (unnormalised), giving

changes from 1/2π (random) to

vacvacoo

!

)ˆˆ()ˆˆ(ˆ)ˆˆ()ˆˆ(ˆˆˆˆo

i

oo

i

o

i

oo

i

ooobeavacbeabeavacbeabbaa

!!!! " ++++=++#++#+++

fbeabeafo

ii

o)ˆˆ(ˆ)ˆˆ( ++

++! "" #

! )ˆˆ(ˆ)ˆˆ( ++++ beabea

ii !! "

ba

i

ba

i

nn

beabea

+

++=

+!+ )ˆˆ(ˆˆ)ˆˆ(ˆ1

"" ###

)cos(1

2

1)(

2/12/1

!""

#$+

+=$ba

ba

nn

nnP

)(!P

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First photocount sharpens phase differencedistribution.

For sharp number state distribution and:

bann =

!

!21

!"

)(!P

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First photocount also increases probability thatsecond photocount will be at same detector.

Narrow number state distributions:

(c.f. Hong-Ou-Mandel “dip”)

bababa

baba

nnnnnn

nnnn

P

P

422

22

11

12

+!!+

!!+=

3

1

1112/ =PP

: 1==bann 0/

1112=PP

:1>>=bann

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The measurement itself creates the quantitybeing measured, that is, the well-definedphase difference between the two laserbeams.

For coherent states, the measurement doesnot alter the states and they retain theirwell-defined phase difference. Onlyretrodiction is involved.

This gives rise to the illusion that the laserbeams are in coherent states.

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Disrupt phase during a π-pulse

Strong coherent state similar to classical lightwith definite phase. Thus shift of phase willdisrupt π-pulse.

Photon number state has no well-definedphase to disrupt – therefore phase shift willnot affect π-pulse?

No – same result in both cases. π-pulse isdisrupted.

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Before pulse the atom-field state isPhase shift does not physically alter

this state.

After pulse the state isPhase shift does not physically alter this state.

During pulse atom-field state is in asuperposition of and .

Phase shift does physically alter this state.

Obtain same result as for coherent state.

gn

en 1!

)ˆexp( !ni

gn en 1!

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ConclusionMølmer’s conjecture that describing laser

light as a coherent state is a “convenientfiction” appears to be correct.

However describing by a number state isalso a fiction.

We do not need either to predictinterference between two lasers. Causeof interference is the measurementprocess itself.

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?Interpretation of experiments:

Squeezed statesContinuous variable teleportationOptical homodyne tomography

State measurementPhase measurementCoherent state qubits

Quantum state engineeringMeasurement by retrodictive state engineering

etc.