Quantum theory of light-matter interaction: Fundamentals

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Quantum theory of light-matter interaction:

Fundamentals

Lecture 12

Coherent transient phenomena

Mihály Benedict

University of Szeged, Dept. of Theoretical Physics, 2014

Table of contents

1 Introduction

2 Specific solutions of the Bloch equations Fully coherent case, with exact resonance:

Fully coherent case, out of resonance, the Rabi solution Solution without external field with phase decay

3 Coherent transients with propagation The area theorem

Self-induced transparency

4 Superfluorescence and coherent amplification

5 Questions, references

Introduction

Introduction

In this Lecture we repeat first the interaction of the two-level system with an external (nearly) resonant field in the Bloch-vector formalism, due to F. Bloch, Nobel prize winner for inventing NMR, which is actually the Rabi method applied to nuclear spins in solids.

We consider then two coherent effects: free induction decay and photon echo.

Then we combine coherent interaction with field propagation in an extended pencil shaped medium. Pioneering work in this field was done by S. McCall and E. Hahn (the inventor of spin echo) who developed the theory and performed the first experiments in 1967 on coherent field propagation effects.

Introduction

Maxwell- Bloch equations

The optical Bloch equations for two-level atoms with relaxation terms have been obtained previously for two-level atoms.:

U˙ =−∆V−U/T₂, (U)

V˙ = ∆U+ Ω_rW−V/T₂, (V)

W˙ =−Ω_rV−(W+1)/T₁. (W) These are to be completed with the equation for the slowly-varying field envelopeE₀(z,t)determiningΩ_r=dE₀(t)/~,as well as with one for the slowly-varying phase of the field,φ(z,t) :

∂

∂z+ 1 c

∂

∂t

E₀(z,t) = k 2₀Nd

Z

g(∆)V(∆)d∆, (Amp)

E₀ ∂

∂z+ 1 c

∂

∂t

φ(z,t) =− k 20

Nd Z

g(∆)U(∆)d∆. (Phase)

Introduction

Comparison with semiclassical laser theory

The second set of equations are the consequences of Maxwell

equations for the field strength in the SVEA. Therefore the above set of equations is frequently called the system of Maxwell-Bloch equations, which are to be solved self-consistently in the sense explained in Lecture 9.

These are essentially the same equations as those used in semiclassical laser theory. We use however a different notation here, because in dealing with coherent interactions this is the customary one.

In laser theory emphasis was on mode decomposition and on the stationary regime,here we will mainly (but not exclusively) consider processes, during which atomic coherence is maintained, which is the time scale shorter than the relaxation times:T₁andT₂.

Note also that in the source term of the equations (Amp) and (Phase) for the field ("Maxwell part") the average over the inhomogeneous broadeningg(∆)is explicitly included.

Specific solutions of the Bloch equations Fully coherent case, with exact resonance:

Bloch equations without propagation

In the next few slides we study the Bloch equations without propagation effects, so the field strengthΩ_r(t)is taken at a given position of a group of atoms extending much less than the wavelength of the field.

Specific solutions of the Bloch equations Fully coherent case, with exact resonance:

Bloch vector and Bloch sphere

The optical Bloch equations for two-level atoms, in a certain spatial point in the sample, without the relaxation terms are given by

U˙ =−∆V, (U0)

V˙ = ∆U+ Ω_rW, (V0)

W˙ =−Ω_rV. (W0)

This is the fully coherent case withT₁ =∞,T₂=∞, conserving U²+V²+W²=const=1

(from the initial condition). The vectorU= (U,V,W)moves along the surface of a sphere called theBloch sphere.

Specific points:W =−1:ground state,W=1:fully excited state.

Problem: Invert the definitions U=b₁b^∗₂+b^∗₁b₂,V=i(b^∗₁b₂−b₁b^∗₂), W =|b2|²− |b1|²as given in the end of Lecture 8,where b1and b2 are the modified interaction-picture amplitudes of the levels introduced there. Show that this is possible only up to a common phase factor of the amplitudes.

Specific solutions of the Bloch equations Fully coherent case, with exact resonance:

Resonant coherent solution

Fully coherent case:T₁=∞,T₂ =∞; and on exact resonance∆ =0 U(t)≡0,V²(t) +W²(t) =1, implying the following parametrization:

V=−sinΘ, W=−cosΘ, for W(0) =−1, V=sinΘ, W=cosΘ, for W(0) =1,

the first used for atoms initially in the ground state, while the second for atoms initially in their excited state

Considering theW(0) =−1 case:

V˙ = Ω_rW=−Θ˙ cosΘ, W˙ =−Ω_rV=−Θ˙ sinΘ.

Which means thatΘ = Ω˙ _r(t), and

Θ = Zt

0

Ω_r(t⁰)dt⁰,

Specific solutions of the Bloch equations Fully coherent case, with exact resonance:

Pulse angle and Rabi oscillations

The expression

Θ = Zt

0

Ω_r(t⁰)dt⁰, is called theBloch angleand

A:= Θ(∞)

is thepulse area. For a givenΩ_r(t), the solution of the Bloch equations is V(t) =−sin

Rt 0

Ω_r(t⁰)dt⁰, W =−cos Rt 0

Ω_r(t⁰)dt⁰

for the ground state att=0,and with reversed sign for the excited case. The Bloch vector turns around theUaxis by an angle

Θ = Zt

0

Ω_r(t⁰)dt⁰.These are the Rabi oscillations in their cleanest form.

Specific solutions of the Bloch equations Fully coherent case, out of resonance, the Rabi solution

Rabi solution

Fully coherent caseT₁=∞,T₂ =∞; out of exact resonance: ∆6=0.

Solution, which can be obtained directly by solving the system : U˙ =−∆V, V˙ = ∆U+ Ω_rW, W˙ =−Ω_rV,

or rewriting the Rabi solution forb₂andb₁of Lecture 8 in terms ofU, VandW. For the initial conditionsU(0) =V(0) =0,W(0) =−1,this gives withΩ =p

Ω²_r + ∆²:

U(t,∆) = ∆Ω_r

Ω² (1−cosΩt), V(t,∆) =−Ω_r

Ω sinΩt, W(t,∆) =−1+Ω²_r

Ω²(1−cosΩt).

Specific solutions of the Bloch equations Solution without external field with phase decay

Free induction decay

Assume that the atoms are first excited by a short pulse of areaΘ₀such that its bandwidth is sufficiently large, and all the atoms with different

∆’s can be treated as resonant ones. After this the system evolves freely (Ωr=0) from

U(0,∆) =0, V(0,∆) =−sinΘ₀, W(0,∆) =−cosΘ₀, and we assume thatT₁ =∞,T₂ <∞.

Then according to the Bloch equations, an individual atom with detuning∆evolves according to

U(t,∆)−iV(t,∆) =isinΘ₀e^{−(1/T2+i∆)t}, W=W(0).

Problem: Prove by substitution the validity of this solution.

Assume for simplicity1/T₂=0,and show that an atom with detuning∆ precesses in the UV plane with an angular velocity of∆.

Specific solutions of the Bloch equations Solution without external field with phase decay

Free induction decay II

As different atoms have different detunings∆, an oriented collection of Bloch vectors, all pointing in theU(0)direction initially, suffer a different phase shift∆tand the average value ofV, i.e. the

macroscopic polarization becomeszero in a short time.

This effect is called free induction decay (FID).

Assuming a Gaussian Dopplerg(∆)line this is seen mathematically from:

P(t) =Nd Z T^∗

√

2π exp(−T^∗2∆²/2) [U(t,∆)−iV(t,∆)]d∆ =

=NdsinΘ₀e^−t2/2T∗2.

For a typical room temperature gas a visible transition has a width 1/(2πT^∗) =1.5 GHz, so that the time constant of thisdephasing mechanism is T^∗=10⁻¹⁰s.

Specific solutions of the Bloch equations Solution without external field with phase decay

FID and photon echo

Specific solutions of the Bloch equations Solution without external field with phase decay

Photon echo

In order to understand this effect – discovered and demonstrated in 1966 as an analogue of the spin echo in NMR – we consider the situation where it shows up most notably.

A sharp pulse of areaπ/2 brings all atoms into the stateU(0,∆) =0, V(0,∆) =−1,W(0,∆) =0,i.e., the Bloch vector becomes horizontal for all atoms, which is an equal-weight superposition of the ground and excited states. In terms of thebamplitudes:b₁=1/√

2,b₂=−i/√ 2 After the excitation is stopped, FID begins and duringT^∗it smears out the polarization, as atoms with different∆move around with different angular velocities.

Specific solutions of the Bloch equations Solution without external field with phase decay

Photon echo II

Now after a time intervalt⁰ T^∗, but still withinT₂, a sharpπpulse is applied to the system, that rotates all Bloch vectors around theUaxis byπ, resulting inV(t⁰+,∆) =−V(t⁰−,∆),leaving the other components unchanged. This puts the slowly-precessing vectors ahead of the more rapid ones. Now FID continues with the same angular velocities∆for the individual atoms as before. As a

consequence, the Bloch vectors begin to gather, and to rephase again, so that at 2t⁰they build up again the macroscopic polarization. This shows up in the emission of an echo pulse of areaπ/2. The process can be followed, of course, mathematically by the sequence of the

appropriate transformations of theb_iamplitudes, or the components ofU.

Coherent transients with propagation

Coherent transients with propagation: Introduction

The propagation equation for the slowly-varying amplitude becomes after multiplying byd/~

∂Ω_r

∂z +1 c

∂Ω_r

∂t = k 2

Nd²

~ Z

g(∆)V(∆)d∆.

We shall assume here that the phase is constant, so we exclude the equation forφ.

On a time scale less thanT2,T1, and for atoms in exact resonance:

∆ =0,the values of theBloch angle,Θ, andVandWare given by

Θ(z,t) = Zt

−∞

Ω_r(t⁰,z)dt⁰,

W(Θ,∆ =0) =±cosΘ, V(Θ,∆ =0) =±sinΘ.

Coherent transients with propagation

Coherent propagation – continued

From now on the initial time instant is set to−∞(for convenience), and according to the usual convention the zero value ofΘis chosen to depend on the initial state:

Θ(−∞) =0, W(−∞) =−1 attenuator, Θ(−∞) =0, W(−∞) =1 amplifier,

as the medium initially in the ground state will absorb the radiation, while an inverted medium will amplify it. Here we assume that initially the value ofWis homogeneous along the pencil-shaped sample.

Coherent transients with propagation The area theorem

Definition of the pulse area

As we have seen, a key concept in coherent resonant optics is the area of the pulse, which now depends on the spatial coordinate:

A(z) := Θ(z,t=∞) =

∞

Z

−∞

Ω_r(t⁰,z)dt⁰.

A(z)obeys a simple differential equation. We will show that:

dA(z)

dz =±α_B

2 sinA(z)

with an appropriate constantα_B, which is the small signal absorption (-) (or gain +) coefficient for the inhomogeneously broadened line.

This is the so calledarea theoremobtained first by S. McCall and E.

Hahn in 1967 for coherent pulse propagation.

Coherent transients with propagation The area theorem

Derivation of the area theorem I

In order to prove the area theorem, integrate the propagation equation with respect tot, and note that the time integral of the term^∂Ω_∂t^r

vanishes due toΩ_r(z,t=±∞) =0.

∞

Z

−∞

∂Ω_r(z,t)

∂z dt= dA(z) dz = k

20

Nd²

~

∞

Z

−∞

dtd∆g(∆)V(z,t,∆).

We shall substitute hereVfrom the negative of the imaginary part of the time integral of the Bloch equation forU˙ −iV˙ giving:

V(z,t,∆) =Im



ie^−i∆t Zt

−∞

dt⁰Ω_r(z,t⁰)W(z,t⁰,∆)e^i∆t0



.

Coherent transients with propagation The area theorem

Derivation of the area theorem II

As everything except forie^i∆(t0−t)is real here, we can write

dA(z) dz =Im



i k 20

Nd²

~

∞

Z

−∞

dt Z

d∆g(∆) Zt

−∞

dt⁰Ω_r(z,t⁰)W(z,t⁰,∆)e^i∆(t0−t)





Interchange the integration with respect totandt⁰:

∞

R

−∞

dt Rt

−∞

dt⁰ =

∞

R

−∞

dt⁰

∞

R

t⁰

dt=

substitutes=t−t⁰giving

∞

R

t⁰

dte^i∆(t0−t)=

∞

R

0

e^−i∆sds=πδ(∆)−iPr_∆¹.

Coherent transients with propagation The area theorem

Derivation of the area theorem III

Performing the integration over∆by the Dirac deltaδ(∆) dA(z)

dz =g(0) k 20

Nd²π

~

∞

Z

−∞

dt⁰Ω_r(z,t⁰)W(z,t⁰,∆ =0)

=±α_B 2

∞

Z

−∞

dt⁰Θ(z,˙ t)cosΘ(z,t) =∓α_B

2 sinΘ(z,∞) =±α_B

2 sinA(z), where

α_B =πg(0)k ₀

Nd²

~ , dA(z)

dz =∓α_B

2 sinA(z)

and the sign depends on the initial condition, - for the attenuator and + for the amplifier

Coherent transients with propagation The area theorem

The constant in the area theorem

α_Bis just thelinear steady stateabsorption coefficient in the Boguer-Lambert-Beer law for an inhomogeneously broadened attenuator, where for small, but non-negligible 1/T₂the conditions W =−1,U˙ = ˙V=0 gives the stationary solution from the Bloch equations:V(∆) =−Ω_rT₂(1+ ∆²T²₂)⁻¹.

Then in the right hand side of the propagation equation we get:

Rg(∆)V(∆)d∆ =−Ω_rR

g(∆)T₂(1+ ∆²T²₂)⁻¹d∆≈ −πg(0)Ω_r, because the inhomogeneous lineg(∆)is usually much broader than the homogeneous one determined by the factor(1+ ∆²T₂²)⁻¹. Then the normalized function _T¹

2π 1

1/T²₂+∆² can be approximated in this respect by a Diracδ(∆).

So the linear and steady state limit ^∂Ω_∂t^r =0 for an attenuator gives

∂Ω_r

∂z =−πg(0) k 20

Nd²

~ Ω_r=⇒Ω_r(z) = Ω_r(0)e^−α2Bz, with the sameα_Bas in the area theorem.

Coherent transients with propagation The area theorem

Solution of the area theorem

The solution of

dA(z)

dz =∓α_B

2 sinA(z) A(z) =2 arctan{e^∓

αB

2 ztan(A₀/2)}, whereA₀is the pulse area at the entry face of the medium.

In the case of the attenuator (−sign), we see that there is a critical area A₀ =π, under whichA(z)diminishes during the propagation, while for initial pulse areasπ <A₀<3π,the limit for largezis 2π. More generally, for initial areas(2k−1)π <A₀<(2k+1)π,k=1,2. . . ,the limit is 2kπ, an even multiple ofπ,depending on the value ofA₀. For the amplifier, (+sign), a small area pulse is amplified so that the limit isπ, and more generally stable areas are(2k+1)π,withk=0,1. . .

Coherent transients with propagation Self-induced transparency

2 π secanthyperbolic pulse and self-induced transparency

The coherent M-B equations have exact analytic solutions, even in the presence of inhomogeneous broadening. The simplest one for

W(z,t=−∞) =−1 is the "2πsecanthyperbolic pulse" (of area 2π, hence its name):

Ω_r= dE0(z,t)

~ = 2

τsecht−z/v

τ .

The pulse duration,τ is arbitrary, but must obeyτ T₁,T₂to remain within the coherent regime. Its propagation speed is given by

v=c(1+α_Bcτ)⁻¹.

This group velocity can be slower than the speed of light by orders of magnitude ifα_Bcτ 1. In fact,v∼10⁻³chas been demonstrated experimentally, the first example of "slow light".

McCall and Hahn called the effect corresponding to the transmission of the 2πsech pulse asself-induced transparency (SIT).

Coherent transients with propagation Self-induced transparency

Solitons

Any pulse of initial areaπ <A₀<2πcan be shown to become a 2π sech pulse, after a sufficiently long propagation.

That is why the 2πsech pulse is the so calledone-soliton solutionof the inhomogeneously broadened Maxwell-Bloch equations, which became an important prototype of the so-calledintegrable nonlinear systems, and the discovery of McCall and Hahn contributed a lot to themathematical theoryof integrable systems and solitons, as explicit but complicated expressions for stablen-soliton solutions withntravellingpeakscould be found theoretically.

McCall and Hahn performed detailed experiments on coherent pulse propagation in ruby at low temperatures (to supress homogeneous relaxation) confirming their theoretical findings, and their work was followed by several other studies.

Coherent transients with propagation Self-induced transparency

Problems related to self-induced transparency

Calculate the FWHM of the2πsech pulse

Show that the Bloch angle corresponding to the above solution is Θ(z,t) =4 arctan[exp(t−t₀)/τ].

Find the corresponding expressions for V and W.

Consider the sharp line limit of the problem where g(∆) =δ(∆), and introduce the new variablesξ =x andτ =t−x/c. Show that the M-B system with propagation can be transformed into the form:

∂²Θ

∂ξ∂τ =−γsinΘ known as the sine-Gordon equation

Superfluorescence and coherent amplification

M-B system for the amplifier

We consider now in some more detail the coherent propagation in an initially inverted medium=a coherent amplifier.

The Maxwell-Bloch system withT₁=T₂ =∞, without inhomogeneous broadening:

U˙ −iV˙ =−i∆(U−iV)−iΩrW, W˙ =−Ω_rV,

∂Ω_r

∂z +1 c

∂Ω_r

∂t = k 2₀

Nd²

~ V.

Initial conditions for an inverted mediumU=V=0,W =1,and no field present:Ω_r=0, correspond to an unstable equilibrium state.

Quantum fluctuations that are neglected in this semiclassical picture will start the spontaneous emission of the atoms. Quantum field theoretical considerations show that this brings about fluctuations of the dipole moment amplitude such that√

U²+V²∼N^−1/2,whereN is the total number of atoms in the sample.

Superfluorescence and coherent amplification

Initial condition for the atoms

For simplicity, we have already omitted the equation for phase

modulation, and in accordance with this, we setU≡0, without loss of generality. ThenV=sinΘ,W=cosΘgivesΘ = Ω˙ _r.

The effect of quantum fluctuations can be simulated by a small initial

"tipping" angleΘ₀∼N^−1/21,and the system starts from V=sinΘ₀≈Θ₀andW =cosΘ₀ .1

In contrast to the attenuator, no explicit analytic solutions are known for this initial condition. We can however learn the physics by assuming a spatially homogeneous solution, i.e., setting ^∂Ω_∂z^r =0.

This leads to the pendulum-like equation:

Θ =¨ βsinΘ, with β = ω₀ 2₀

Nd²

~ ,

whereΘis measured from the upper unstable position, andω₀is the transition frequency.

Superfluorescence and coherent amplification

Energy release from the sample

Explicit periodic solution of this pendulum equation withΘ(0) = Θ₀ in terms of elliptic integrals are well known. The periodic solution lasting forever is however due to the spatially homogeneous ansatz, meaning also that the energy of the field and that of the atoms are exchanged periodically.

A physically more relevant model is if we include the effect of the release of the field at the boundary of the sample, so the equation for the time derivative of the field strengthΩ_ris to be completed by a term

−κΩ_r=−κΘ,˙ describing these losses. It is reasonable to put the damping constant to be equal toκ=c/L, the rate at which a photon of velocitycleaves the sample of lengthL. The equation is now

Θ +¨ κΘ =˙ βsinΘ.

Superfluorescence and coherent amplification

Solution of the overdamped equation

In the strongly overdamped situation: Θ¨ κΘ,˙ the solutions of κΘ =˙ βsinΘfor the angle and the field strength are

Θ =2 arctan[exp(βt/κ)tan(Θ₀/2)], Ω_r= ˙Θ = d

~E₀= 1

T_Rsecht−t_d

T_R , (SRAD)

where the time constant of the process is T_R:= κ

β =2₀ ~c d²ω₀NL. while the so called delay time is

t_d=TRln(Θ₀/2).

This is a secanthyperbolic pulse again, but this time of areaπ.

Problem: Check the validity of the above solution and prove the relation for the delay time t_dwithΘ₀.

Superfluorescence and coherent amplification

Time constant of collective emission

The effect when a system of many atoms emits spontaneously and coherently is calledsuperradiation, in general, while the specific case of the pencil-shaped elongated system we considered above, is

sometimes calledsuperfluorescence.

This isNOTamplified spontaneous emission:

superfluorescence6=ASE≡superluminescencewhich is another incoherent process.

Compare the result forT_Rwithτ₀ =3πε_0d^~c_2ω³3 0

,the spontaneous decay time of a single atom. Noting thatω₀/c=2π/λ

T_R =τ₀8π 3

1 Nλ²L.

We see thatT_Ris smaller thanτ₀by the factor of∼ Nλ²L/8,where Nλ²Lis the number of atoms in the volumeλ²L.

Superfluorescence and coherent amplification

Superradiance

This means that

1 the collective spontaneous emission of atoms isfasterthan single- atom emission.

2 as by (SRAD) the amplitudeE₀ is proportional to 1/T_R ∼ Nλ²L, the intensity is proportional to thesquare of the number of atoms in the volumeλ²L.

3 the emitted pulse is delayed, and has a sharp peak, instead of the decaying exponential of a single-atom emission.

These effects have been first discussed by R. Dicke, who in 1954

developed amicroscopictheory of cooperative spontaneous emission of Natoms, and called the effectsuperradiance. The three characteristics enumerated above appeared in his theory quite differently from the way we obtained them. Since then a very large number of theoretical, as well as experimental work investigated superradiance.

Superfluorescence and coherent amplification

Coherent amplification and others

A related effect iscoherent amplification, when the system is initially in the inverted state, and in addition an incident pulse falls in that is to be amplified when passing through the sample. The important features of this and many other details on coherent transients can be learnt from the book [1].

In addition, there are a number of important effects discovered more recently in connection with coherent transients in three-level systems etc., and we can only enumerate a few:

Lasing without inversion

Electromagnetically induced transparency

Slowing down and stopping light in Bose-Einstein condensates For more details on these we can only refer to the original literature.

Questions, references

Questions

1 What is the physical meaning of the relaxation timesT₁andT₂?

2 What can we say about the length of the Bloch vector without re- laxation?

3 What do we mean by "free induction decay"?

4 What mechanisms appear in photon echo?

5 What kind of pulses appear in photon echo experiments?

6 What is the definition of pulse area?

Questions, references

Questions (continued)

7 What is the physical meaning of pulse area theorem?

8 What do we call "2πsecanthyperbolic pulse"?

9 What kind of solitons appear in inhomogeneously broadened Maxwell-Bloch equations?

10 What is the pendulum equation?

11 What are the characteristics of superradiance?

12 What is the difference between amplified spontaneous emission and superradiance?

Questions, references

References

1 M. G. Benedict et. al.,Superradiance, IOP (Bristol) (1996).

2 P. Meystre and M. Sargent, Elements of Quantum Optics, Spinger (Berlin, Heidelberg) (2007).