Lecture 22:Coherent States

Phy851 Fall 2009

Summary

• Properties of the QM SHO:

1−= nnnA 11† ++= nnnA

€

n =A†( )

n

n!0

€

ψ0(x) = πλ[ ]−1/ 2

e−x 2

2λ2

€

ψ1(x) = 2 πλ[ ]−1/ 22 xλe−x 2

2λ2€

ψn (x) = π 2n n!λ[ ]−1/ 2

Hn x /λ( ) e−x 2

2λ2

€

ψn (x) =2nxλψn−1(x) −

n −1nψn−2(x)

nnnH )2/1( += ωh

222

2

1

2Xm

m

PH ω+=

+= PiX

Ah

λλ2

1

( )†2

AAX +=λ ( )†

2AAiP −−=

λ

h

€

A† =12

Xλ− i λ

hP

+=2

1†AAH ωh

2/1+=Δ nX λ 2/1+=Δ nPλh

ωλ

m

h=

memorize

What are the `most classical’ statesof the SHO?

• In HW6.4, we saw that for a minimumuncertainty wavepacket with:

The uncertainties in position andmomentum would remain constant.

• The interesting thing was that this wastrue independent of x0 and p0, the initialexpectation values of X and P.

• We know that other than the case x0=0and p0=0, the mean position andmomentum oscillate like a classicalparticle

• This means that for just the right initialwidth, the wave-packet moves aroundlike a classical particle, but DOESN’TSPREAD at all.

2oscxλ

=Δosc

osc Mωλ

h=

‘Coherent States’• Coherent states, or as they are sometimes

called ‘Glauber Coherent States’ are theeigenstates of the annihilation operator

– Here α can be any complex number– i.e. there is a different coherent state for every

possible choice of α– (Roy Glauber, Nobel Prize for Quantum Optics

Theory 2005)• These states are not really any more

‘coherent’ then other pure states,– they do maintain their coherence in the

presence of dissipation somewhat moreefficiently

• In QM the term ‘coherence’ is over-used andoften abused, so do not think that it alwayshas a precise meaning

• Glauber Coherent States are very important:– They are the ‘most classical’ states of the

harmonic oscillator– They describe the quantum state of a laser

• Replace the number of ‘quanta’ with the numberof ‘photons’ in the laser mode

– They describe superfluids and super-conductors

1=ααααα =A

Series Solution

• Let us expand the coherent state onto energyeigenstates (i.e. number states)

• Plug into eigenvalue equation:

• Hit from left with 〈m|:

∑∞

=

=0n

n ncα

∑∑∞

=

∞

=

=00 n

nn

n ncncA α

ααα =A

∑∑∞

=

∞

=

=−00

1n

nn

n ncnnc α

∑∑∞

=

∞

=

=−→00

1n

nn

n ncnncm α

∑∑∞

=

∞

=

=−00

1n

nn

n nmcnmnc α

mm cmc α=++ 11

Continued

• Start from:

– The constant N(α) will be used at the end fornormalization

• Try a few iterations:

• So clearly by induction we have:

mm cm

c1

1+

=+

α

€

c0 = ! (α)

€

c1 =α1c0 =

α1! (α)

1−= mm cm

cα

€

c2 =α2c1 =

α 2

2 ⋅1! (α)

€

c3 =α3c2 =

α 3

3 ⋅ 2 ⋅1! (α)

€

c4 =α4c2 =

α 4

4 ⋅ 3 ⋅ 2 ⋅1! (α)

€

cn =α n

n!! (α)

Normalization Constant

• So we have:

• For normalization we require:

• Which gives us:

€

cn =α n

n!! (α)

€

α = ! (α) α n

n!n= 0

∞

∑ n

αα=1

€

= ! (α) 2 α∗mα n

m!n!n= 0m= 0

∞

∑ m n

€

= ! (α) 2 α2n

n!n= 0

∞

∑

€

= ! (α) 2e α2

€

! (α) = e−α 2

2n

ne

n

n

∑∞

=

−=

0

2

!

2

αα

α

Orthogonality

• Let us compute the inner-product of twocoherent states:

• Note that:

• So coherent states are NOT orthogonal– Does this contradict our earlier results

regarding the orthogonality of eigenstates?

nmnm

e

mn

nm

∑∞

==

∗+−

=

00

2

!!

22

βαβα

βα

€

= e−α 2 + β 2

2α∗β( )

n

n!n= 0

∞

∑

€

= e−α 2 + β 2

2+α∗β

€

e− α−β2

= e− α∗−β ∗( ) α−β( )

= e− α 2 + β 2 +a∗β +β ∗α( )

= α β2

Expectation Values of PositionOperator

• Lets look at the shape of the coherentstate wavepacket– Let

– Better to avoid these integrals, insteadlets try using A and A† :

– Recall the definition of |α〉:

αψα xx =)(

)()( xxxdxX αα ψψ ∗∫=

€

X = αλ2A + A†( )α

ααα =A ααα ∗=†A

€

X =λ2

α Aα + α A† α( )

€

=λ2α α α +α∗ α α( )

€

=λ2α +α∗( )

€

X = 2λRe α{ }

Expectation Value of Momentum Operator

• We can follow the same procedure for themomentum:

• Not surprisingly, this gives:

€

P = −i h

2λα A − A†( )α

€

=2h2iλ

α Aα − α A† α( )

€

=2h2iλ

α −α∗( )

€

P =2hλIm α{ }

€

X = 2λRe α{ }

+= PiXh

λλ

α1

2

1

Variance in Position

• Now let us compute the spread in x:

• Put all of the A ’s on the right and the A† ‘s onthe left:– This is called ‘Normal Ordering’

€

X 2 = αλ2

2A + A†( )

2α

€

= αλ2

2A2 + AA† + A†A + A†A†( )α

€

= αλ2

2A2 + 2A†A +1+ A†A†( )α

€

=λ2

2α 2 + 2α∗α +1+α∗2( )

€

=λ2

2α +α∗( )

2+1( )

2

222 λ+= XX( )∗+= αα

λ

2X

2

22 λ=−=Δ XXX

Exactly the same variance asthe ground state |n=0〉

Momentum Variance

• Similarly, we have:

– Normal ordering gives:

€

P 2 = −h2

2λ2α A − A†( )

2α

€

= −h2

2λ2α AA − AA† − A†A + A†A†( )α

€

P 2 = −h2

2λ2α AA − 2A†A −1+ A†A†( )α

€

= −h2

2λ2α AA − 2A†A −1+ A†A†( )α

€

= −h2

2λ2α 2 − 2α∗α +α∗2 −1( )

€

= −h2

2λ2α −α∗( )

2−1( )

€

P =2h2iλ

α −α∗( )

2

222

2λh

+= PPλ2

22 h=−=Δ PPP

Minimum Uncertainty States

• Let us check what Heisenberg UncertaintyRelation says about coherent states:

• So we see that all coherent states (meaningno matter what complex value α takes on)are Minimum Uncertainty States– This is one of the reasons we say they are

‘most classical’

2

h=ΔΔ PX

λ2

22 h=−=Δ PPP

2

22 λ=−=Δ XXX

λ

λ

22

h=ΔΔ PX

Time Evolution

• We can easily determine the time evolution ofthe coherent states, since we have alreadyexpanded onto the Energy Eigenstates:– Let

– Thus we have:

– Let

0)0( αψ ==t

nn

en

n

∑∞

=

−=

0

02

!)0(

2

αψ

α

nen

et tni

n

n)2/1(

0

02

!)(

2

+−∞

=

−

∑= ωα α

ψ

nen

ee tni

n

nti ω

αω α −

∞

=

−− ∑=0

022/

!

2

€

= e−iωt / 2e−α 2

2α0e

−iω t( )n

n!n= 0

∞

∑ n

tiet ωαα −= 0)(

)()( tt αψ =By this we mean it remains in acoherent state, but the value ofthe parameter α changes intime

Why ‘most classical’?

• What we have learned:– Coherent states remain coherent states as time

evolves, but the parameter α changes in timeas

– This means they remain a minimumuncertainty state at all time

– The momentum and position variances are thesame as the n=0 Energy eigenstate

– Recall that:

– So we can see that:

– We already know that <X> and <P> behave asclassical particle in the Harmonic Oscillator, forany initial state.

tiet ωαα −= 0)(

€

X = 2λRe α{ }

€

P =2hλIm α{ }

+= 00

02

1pi

x

h

λλ

α

)sin()cos()()sin()cos()( 000

0 txtptptp

txtx ωωωωω

ω −=+=

)()(

)()(

0

0

tPtp

tXtx

αα

αα

=

=

Conclusions

• The Coherent State wavefunction looksexactly like ground state, but shifted inmomentum and position. It then moves as aclassical particle, while keeping its shapefixed.– Note: the coherent state is also called a

‘Displaced Ground State’