Lecture 22: Coherent States

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1 Leture : Coheret States Phy851 Fall 9

2 Summary memorize Properties of the QM SHO: A 1 A ψ (x) ψ (x) H P + m 1 X λ A + i P λ h H hω( +1/ ) [ π!λ] 1/ H x /λ 1 mω λ h ( A A ) P i ( A A ) X + H x λ ψ 1(x) hω A A + X A 1 X λ i λ h P 1 λ ( ) e λ ( ) A x λ 1 ψ (x) h mω! ψ (x) [ π λ] 1/ e x λ ψ 1 (x) π λ [ ] 1/ x λ e h ΔX λ +1/ ΔP +1/ λ x λ

3 What are the `most lassial states of the SHO? I HW6.4, we saw that for a miimum uertaity wavepaket with: Δx λ os The uertaities i positio ad mometum would remai ostat. The iterestig thig was that this was true idepedet of x ad p, the iitial expetatio values of X ad P. We kow that other tha the ase x ad p, the mea positio ad mometum osillate like a lassial partile This meas that for just the right iitial width, the wave-paket moves aroud like a lassial partile, but DOESN T SPREAD at all. λ os h Mω os

4 Coheret States Coheret states, or as they are sometimes alled Glauber Coheret States are the eigestates of the aihilatio operator A Here a be ay omplex umber i.e. there is a differet oheret state for every possible hoie of (Roy Glauber, Nobel Prize for Quatum Optis Theory 5) These states are ot really ay more oheret the other pure states, they do maitai their oheree i the presee of dissipatio somewhat more effiietly 1 I QM the term oheree is over-used ad ofte abused, so do ot thik that it always has a preise meaig Glauber Coheret States are very importat: They are the most lassial states of the harmoi osillator They desribe the quatum state of a laser Replae the umber of quata with the umber of photos i the laser mode They desribe superfluids ad super-odutors

5 Series Solutio Let us expad the oheret state oto eergy eigestates (i.e. umber states) Plug ito eigevalue equatio: Hit from left with m : A A 1 1 m 1 m m m m m

6 m+ 1 m m + 1 Cotiued m m1 m Start from:! () The ostat N() will be used at the ed for ormalizatio Try a few iteratios: 1 1 1! () So learly by idutio we have: 1! () 3 3 1! () ! ()!! ()

7 Normalizatio Costat!! () So we have:! ()! For ormalizatio we require: 1 m! () m! () m!!! m Whih gives us:! () e! () e e!

8 Orthogoality Let us ompute the ier-produt of two oheret states: β e e + β m + β m β m!! ( β)! m e + β + β Note that: e β e ( β ) β ( ) e ( + β +a β +β ) β So oheret states are NOT orthogoal Does this otradit our earlier results regardig the orthogoality of eigestates?

9 Expetatio Values of Positio Operator Lets look at the shape of the oheret state wavepaket Let ψ X ( x ) dx x ψ ( x) xψ ( x) Better to avoid these itegrals, istead lets try usig A ad A : ( ) X λ A + A Reall the defiitio of : A A X λ ( A + A ) λ ( + ) λ ( + ) X λre{ }

10 Expetatio Value of Mometum Operator We a follow the same proedure for the mometum: P i P h iλ h λ A A ( ) ( A A ) h ( iλ ) h λ Im { } X λre{ } Not surprisigly, this gives: 1 1 λ X + i P λ h

11 Variae i Positio Now let us ompute the spread i x: X ( ) λ A + A λ ( A + AA + A A + A A ) Put all of the A s o the right ad the A s o the left: This is alled Normal Orderig λ ( A + A A +1+ A A ) λ ( ) X λ ( + ) λ (( ) +1) + X λ X + ΔX X λ X Exatly the same variae as the groud state

12 Mometum Variae Similarly, we have: P h λ A A Normal orderig gives: ( ) h λ ( AA AA A A + A A ) P h λ ( AA A A 1+ A A ) h λ ( AA A A 1+ A A ) h λ h λ P ( + 1) (( ) 1) h iλ ( ) P P + h λ ΔP P h P λ

13 Miimum Uertaity States Let us hek what Heiseberg Uertaity Relatio says about oheret states: ΔX X λ X ΔP P h P λ ΔXΔP λ h λ ΔXΔP h So we see that all oheret states (meaig o matter what omplex value takes o) are Miimum Uertaity States This is oe of the reasos we say they are most lassial

14 Time Evolutio We a easily determie the time evolutio of the oheret states, sie we have already expaded oto the Eergy Eigestates: Let ψ ( t ) Thus we have: ψ () e! ψ ( t) e e! iω( + 1/ ) t Let e iωt / iω t e e iωt / e ( t) e e! ( e iω t ) iω t! ψ ( t) ( t) By this we mea it remais i a oheret state, but the value of the parameter hages i time

15 Why most lassial? What we have leared: Coheret states remai oheret states as time evolves, but the parameter hages i time as ( t) e iω t This meas they remai a miimum uertaity state at all time The mometum ad positio variaes are the same as the Eergy eigestate Reall that: X P λre{ } h λ Im { } So we a see that: 1 x λ + i p λ h x p ( t) ( t) X ( t) P ( t) We already kow that <X> ad <P> behave as lassial partile i the Harmoi Osillator, for ay iitial state. p x( t) x os( ωt) + si( ωt) p( t) p os( ωt) ωx ω si( ω ) t

16 Colusios The Coheret State wavefutio looks exatly like groud state, but shifted i mometum ad positio. It the moves as a lassial partile, while keepig its shape fixed. Note: the oheret state is also alled a Displaed Groud State

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