Modeling and Control of the LKB Photon-Box:
1Spin-Spring Systems
Pierre Rouchon
pierre.rouchon@mines-paristech.fr
Würzburg, July 2011
1LKB: Laboratoire Kastler Brossel, ENS, Paris.
Several slides have been used during the IHP course (fall 2010) given with Mazyar Mirrahimi (INRIA) see:
http://cas.ensmp.fr/~rouchon/QuantumSyst/index.html
Outline
1 Spring systems: the quantum harmonic oscillator
2 Spin-spring systems: the Jaynes-Cummings model
Harmonic oscillator2(1): quantization and correspondence principle
Classical Hamiltonian formulation of dtd22x =−ω2x
d
dtx =ωp= ∂H
∂p, dtdp=−ωx =−∂H
∂x, H= ω
2(p2+x2).
Quantization: probability wave function|ψit ∼(ψ(x,t))x∈Rwith
|ψit ∼ψ(,t)∈L2(R,C)obeys to the Schrödinger equation (~=1 in all the lectures)
idtd |ψi=H|ψi, H =ω(P2+X2) =−ω 2
∂2
∂x2 +ω 2x2 whereHresults fromHby replacingx by position operator
√2X andpby impulsion operator√
2P=−i∂x∂ .
PDE model:i∂ψ∂t(x,t) =−ω2∂∂x2ψ2(x,t) +ω2x2ψ(x,t), x ∈R.
2Two references: C. Cohen-Tannoudji, B. Diu, and F. Laloë.Mécanique Quantique, volume I& II. Hermann, Paris, 1977.
M. Barnett and P. M. Radmore.Methods in Theoretical Quantum Optics.
Oxford University Press, 2003.
Harmonic oscillator (2): annihilation and creation operators
Averaged positionhXit =hψ|X|ψiand impulsionhPit =hψ|P|ψi3: hXit = √1
2
Z +∞
−∞
x|ψ|2dx, , hPit =−√i
2
Z +∞
−∞
ψ∗∂ψ
∂xdx. Annihilationaandcreationoperatorsa†:
a=X +iP= √1
2
x+ ∂
∂x
, a†=X −iP = √1
2
x − ∂
∂x
Commutation relationships:
[X,P] = 2i, [a,a†] =1, H =ω(P2+X2) =ω
a†a+1
2
. SetXλ= 12 e−iλa+eiλa†
for any angleλ:
h
Xλ,Xλ+π
2
i
= 2i.
3We assume everywhere that for eacht,x7→ψ(x,t)is of the Schwartz class (fast decay at infinity + smooth).
Harmonic oscillator (3): spectral decomposition and Fock states [a,a†] =1 and Ker(a)of dimension one imply that thespectrum ofN=a†aisnon-degenerateand isN. More we have the useful commutations for any entire functionf:
a f(N) =f(N+I)a, f(N)a†=a†f(N+I).
Fock statewithnphoton(s): the eigen-state ofNassociated to the eigen-valuen:
N|ni=n|ni, a|ni=√
n|n−1i, a†|ni=√
n+1|n+1i.
Theground state|0i(0 photon state or vacuum state) satisfies a|0i=0 and corresponds to theGaussian function:
|0i ∼ψ0(x) = 1
π1/4exp(−x2/2).
The operatora(resp. a†) is the annihilation (resp. creation) operator since it transfers|nito|n−1i(resp. |n+1i) and thus decreases (resp. increases) the quantum numbernby one unit.
Harmonic oscillator (4): displacement operator Quantization of dtd22x =−ω2x−ω√
2u H=ω
a†a+1
2
+u(a+a†).
The associated controlled PDE i∂ψ
∂t (x,t) =−ω 2
∂2ψ
∂x2(x,t) +
ω
2x2+√ 2ux
ψ(x,t).
Glauberdisplacement operatorDα (unitary) withα ∈C: Dα=eαa†−α∗a=e2i=αX−2ı<αP
FromBaker-Campbell Hausdorf formulavalid for any operators AandB,
eABe−A=B+ [A,B] +2!1[A,[A,B]] +3!1[A,[A,[A,B]]] +. . . we get theGlauber formulawhen[A,[A,B]] = [B,[A,B]] =0:
eA+B =eAeB e−12[A,B].
(show thatCt =et(A+B)−etAetBe−
t2 2[A,B]
satisfiesdtdC= (A+B)C)
Harmonic oscillator (5): identities resulting from Glauber formula
WithA=αa†andB=−α∗a, Glauber formula gives:
Dα=e−
|α|2
2 eαa†e−α∗a=e+
|α|2
2 e−α∗aeαa† D−αaDα =a+α and D−αa†Dα=a†+α∗. WithA=2i=αX ∼i√
2=αx andB=−2ı<αP∼ −√
2<α∂x∂ , Glauber formula gives4:
Dα =e−i<α=αei
√
2=αxe−
√ 2<α∂x∂
(Dα|ψi)x,t =e−i<α=αei
√2=αxψ(x −√ 2<α,t) Exercice
For anyα, β, ∈C, prove that Dα+β=e
α∗β−αβ∗ 2 DαDβ
Dα+D−α=
1+α∗−α2 ∗
1+a†−∗a+O(||2)
d dtDα
D−α= αd
dtα∗−α∗d dtα 2
1+ dtdα
a†− dtdα∗ a.
4Remember that a time-delay ofrcorresponds to the operatore−rdtd.
Harmonic oscillator (6): lack of controllability
Take|ψisolution of thecontrolled Schrödinger equation idtd |ψi= ω a†a+12
+u(a+a†)
|ψi. Sethai=hψ|aψi. Then
d
dthai=−iωhai −iu.
Froma=X +iP, we havehai=hXi+ihPiwhere
hXi=hψ|X|ψi ∈RandhPi=hψ|P|ψi ∈R. Consequently:
d
dthXi=ωhPi, dtdhPi=−ωhXi −u.
Consider thechange of frame|ψi=e−iθtDhai
t |χiwith θt =
Z t
0
|hai|2+u<(hai)
, Dhai
t =ehaita†−hai∗ta, Then|χiobeys toautonomous Schrödinger equation
i dtd |χi=ωa†a|χi.
The dynamics of|ψican be decomposed into two parts:
acontrollable part of dimension twoforhai
an uncontrollable part of infinite dimension for|χi.
Harmonic oscillator (7): coherent states as reachable ones from|0i Coherent states
|αi=Dα|0i=e−
|α|2 2
+∞
X
n=0 αn
√
n!|ni, α∈C
are the states reachable from vacuum set. They are also the eigen-stateofa: a|αi=α|αi.
A widely known result in quantum optics5: classical currents and sources (generalizing the role played byu) only generate classical light (quasi-classical statesof the quantized field generalizing the coherent state introduced here)
We just propose here a control theoretic interpretation in terms of reachable set from vacuum6
5See complementBIII, page 217 of C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg.Photons and Atoms: Introduction to Quantum
Electrodynamics.Wiley, 1989.
6see also: MM-PR, IEEE Trans. Automatic Control, 2004 and MM-PR, CDC-ECC, 2005.
Atom-cavity coupling
Thecomposite systemlives on the Hilbert space
C2⊗L2(R;C)∼C2⊗l2(C)with theJaynes-Cummings Hamiltonian HJC= ωeg
2 σz+ωc(a†a+1
2) +iΩ(t)
2 σx(a†−a), withωc/2π≈ωeg/2πaround 50 GHz.
Gaussian radial profile of the cavity quantized mode (fort =0 atom at cavity center):
Ω(t) = Ω0exp − vt
w 2!
wherev is the atom velocity (250 m/s),wis the width (6 mm),Ω0/2π around 50 kHz. Thus we have alsoΩ(t)ωc, ωeg.
Jaynes-Cumming model: RWA
We consider the change of frame:
|ψi=e−iωct(a†a+12)e−iωctσz|φi.
The system becomesidtd |φi=Hint|φiwith
Hint = ∆
2σz+iΩ(t)
2 (e−iωct|gi he|+eiωct|ei hg|)(eiωcta†−e−iωcta), where∆ =ωeg −ωc much smaller thanωc (∆/2πaround 250 kHz, same order asΩ(t))
The secular terms ofHint are given by (RWA, first order approximation):
Hrwa = ∆
2(|ei he| − |gi hg|) +iΩ(t)
2 (|gi he|a†− |ei hg|a).
SinceΩ(t) = Ω0exp
− vtw2
we have, with∆>0 of the same order ofΩ0, for allt,
dtdΩ(t)
∆Ω(t): adiabatic coupling between atom/cavity, called also dispersive interaction.
PDE model behindHrwa
The quantum state|φisatisfying
d
dt |φi=−i
∆
2(|ei he| − |gi hg|) +iΩ(t2)(|gi he|a†− |ei hg|a)
|φi
is described by two elements ofL2(R,C),φg andφe,
|φi= (φg(x,t), φe(x,t)) with kφgk2L2+kφek2L2 =1 and the time evolution is given by the coupled Partial Differential Equations (PDE’s)
∂φg
∂t =i∆2φg+Ω(t)
2√ 2
x − ∂
∂x
φe
∂φe
∂t =−i∆2φe−Ω(t)
2√ 2
x+ ∂
∂x
φg
sincea= √1
2 x +∂x∂
anda†= √1
2 x −∂x∂ .
Jaynes-Cumming model: adiabatic propagator
The unitary operatorUsolution of
d
dtUt =−i ∆
2(|ei he| − |gi hg|) +iΩ(t)
2 (|gi he|a†− |ei hg|a)
Ut
starting fromU−T =IwithvT/w 1 (Ω(−T)≈0), satisfies in a good approximation
UT =|gi hg|eiφ(N)+|ei he|e−iφ(N+1)
withφ(n)being the analytic function (light-induced phase):
φ(n) = 12 Z +T
−T
q
∆2+nΩ2(t)dt
Proof:for eachn, use invariance of the space(|g,n+1i,|e,ni) and the adiabatic propagator for the spin system with an
adiabatic Hamiltonian of the form ∆2rσz +v(t)2 σy.
Exercise on Jaynes-Cumming propagator (1)
Let us consider the Jaynes-Cumming propagatorUC
UC =e
−iτ
∆ |eihe|−|gihg|
2 +i
Ω |gihe|a†−|eihg|a
2
whereτis an interaction time,∆andΩare constant.
Show by recurrence on integerkthat
∆ |ei he| − |gi hg|
+iΩ |gi he|a†− |ei hg|a2k
=
|ei he|
∆2+ (N+1)Ω2k
+|gi hg|
∆2+NΩ2k
and that
∆ |ei he| − |gi hg|
+iΩ |gi he|a†− |ei hg|a2k+1
=
|ei he|∆
∆2+ (N+1)Ω2k
− |gi hg|∆
∆2+NΩ2k
+iΩ
|gi he|
∆2+NΩ2k
a†− |ei hg|a
∆2+NΩ2k .
Exercise on Jaynes-Cumming propagator (2)
Deduce that
UC =|gi hg|
cos
τ
√
∆2+NΩ2 2
+i
∆sin
τ
√
∆2+NΩ2 2
√∆2+NΩ2
+|ei he|
cos
τ
√
∆2+(N+1)Ω2 2
−i
∆sin
τ
√
∆2+(N+1)Ω2 2
p∆2+ (N+1)Ω2
+|gi he|
Ωsin
τ
√
∆2+NΩ2 2
√∆2+NΩ2
a†− |ei hg|a
Ωsin
τ
√
∆2+NΩ2 2
√∆2+NΩ2
whereN=a†athe photon-number operator (ais the photon annihilator operator).
Exercise on Jaynes-Cumming propagator (3)
In the resonant case,∆ =0, prove that:
UC =|gi hg|cos
Θ 2
√ N
+|ei he|cos
Θ 2
√N+1
+|gi he|
sin Θ
2
√ N
√ N
!
a†− |ei hg|a
sin Θ
2
√ N
√ N
!
and check thatUC†UC =I.
