Density matrix operator analysis

To conclude this chapter, a third approach to the calculation of echo signals in a two-level system is considered, as a prelude to choosing a methodology in Chapter 5 that will be applied throughout the remainder of the book. This one is based on formal integration of the equation of motion for the density matrix [4.19] and highlights

ê₂

t I(t )

0

Stimulated echo

τ τ

2 2 2

ê₃

ê₁

ê₃

ê₁

ê₂

ê₃

ê₁

ê₂

ê₃

ê₁

ê₂ ê₃

ê₁

ê₂ p p p

Figure 4.11 Illustration of the pulse sequence and Bloch vector dynamics during three-pulse stimulated photon echoes.

exponential forms of the evolution operators based on the angular momentum oper-ator. In quantum mechanics, angular momentum operators are the generators of rotations (see Appendix H). So their appearance here cements the idea that coherent interactions between light and matter in nonlinear and quantum optics are unitary transformations in Hilbert space. This accounts for their prominence in coherent laser spectroscopies of many types [4.18], as well as in the fields of coherent control and quantum information science [4.20], where precise manipulation of populations and superposition states is critical to accuracy.

A few introductory remarks are needed, since multiple representations are used in this section. The interaction picture is used in addition to the Schr¨odinger representa-tion for reasons that will shortly be made clear. In the Schr¨odinger picture, the density matrix satisfies the equation

i
dρ^S

dt = [H, ρ^S]. (4.3.35)

If H is time-independent (H = H0), this equation has the particularly simple solution ρ^S(t− t0) = U0(t− t0)ρ^S(t0)U₀⁺(t− t0), (4.3.36) where

U₀(t− t0) = exp[−i(t − t0)H₀/
]. (4.3.37) Exercise: Verify that the transformed density matrix in Eq. (4.3.36) is a solution of the master Eq. (4.3.35) using direct differentiation of Eq. (4.3.37).

Photon echoes 93 During periods when the Hamiltonian is not time-independent, for example during an optical pulse when H(t) = H0+ V (t), the density matrix Eq. (4.3.35) becomes difficult to solve. The two parts of the Hamiltonian cause essentially different dynamics.

This is the motivation behind the interaction picture covered in Chapter 2, which removes the evolution of the state vector due solely to the static Hamiltonian (see Eq. (2.4.13)). The equation of motion for the density matrix in the interaction picture is

i
dρ^I

dt = [V^I, ρ^I], (4.3.38)

where V^I(t− t0) = U₀⁺(t− t0)VU (t− t0). This is often easier to solve than Eq. (4.3.35). The formal solution to Eq. (4.3.38) is

ρ^I(t− t0) = U_I(t− t0)ρ^I(t₀)U_I⁺(t− t0), (4.3.39) where

U_I(t− t0) = exp[−i

t

t₀

V^I(t^)dt^]. (4.3.40)

The integral in the evolution operator above is readily evaluated in the interaction picture. This is due to the fact that the integrand consists of the slowly varying amplitude of the optical interaction V^I = µE0σ1/2 in the rotating wave approximation (RWA), not the rapidly varying interaction Hamiltonian of the lab frame. The operator UI(t− t0) describes the temporal evolution from t to t0, but at the same time can be interpreted as a simple rotation through an angle θ about axis ˆn in Hilbert space:

UI(θ) = exp[−iσ1θ/2] = exp[−iˆn · ˆS1θ/
], (4.3.41) A square pulse of duration ε and area θ(ε) = µE0ε/
is assumed, and the spin angular momentum ˆn· ˆS1=
σ1/2 has been introduced formally as the generator of rotations in two-level spin space. That the operator in Eq. (4.3.41) does indeed cause rotations like the matrices of the last section is justified more thoroughly in Ref. [4.21] and Appendix H.

We are now ready to write down the solution for the density matrix of a two-level system subjected to an arbitrary pulse sequence in terms of evolution operators. In principle, the temporal evolution is given by the appealingly simple expression

ρ(t) = U (t− t⁰)ρ(t0)U⁺(t− t⁰), (4.3.42) provided there is no dissipation in the system and we can determine the evolution operator U (t− t⁰). The initial matrix is assumed to correspond to an equilibrium state, with populations of ρ11and ρ22 in the ground and excited states, respectively. Hence

ρ^I(t0) = ρ^S(t0) =

ρ₂₂ 0 0 ρ11



=1

2(σ0− σ3). (4.3.43) We consider the same sequence of two square pulses separated by the time interval τ that was analyzed in previous sections. The pulses are applied from t = t₀to t₁= ε₁

and from t1= ε1 to t2= τ . Each is followed by a free precession period, first from t1= ε1 to t2= τ and then from t2= τ to t3= τ + ε2 as in Fig. 4.7. The system is strongly driven for brief moments and evolves freely at other times. The problem of determining the overall operator U (t− t0) is therefore rather complex, but can be subdivided conveniently on the basis of time periods when the Hamiltonian is H0

versus when it is H₀+ V (t). That is, the problem may be broken down into discrete periods during which the system is either perturbed or not.

During the free precession periods, evolution may be readily described in the Schr¨odinger picture using Eq. (4.3.36). During the pulses however, when the Hamil-tonian is time-dependent, this expression is no longer valid. It is simpler to calculate the driven dynamics in the interaction picture and subsequently transform the result back to the Schr¨odinger lab frame to finish the calculation in a single frame. In this approach, time periods when radiation is present are handled in the interaction picture.

Driven dynamics may be analyzed separately and then transformed to the Schr¨odinger picture using the result of the exercise below.

Exercise: Show directly from Eq. (2.4.13) that the density matrices in the Schr¨odinger and interaction pictures are related by

ρ^S(t) = exp[−iH0t/
]ρ^I(t) exp[iH0t/
]. (4.3.44) Following this strategy, a decomposition of the evolution operator is made into four time periods, and also into suitable representations. This gives the following result:

U (t− t0) = U (t− t3)U (t3− t2)U (t2− t1)U (t1− t0) Substitution of Eqs. (4.3.46)–(4.3.49) into Eq. (4.3.45), followed by the combination of Eq. (4.3.45) with Eq. (4.3.42), yields an expression for the density matrix after the pulses have been applied. Here h.c. stands for the Hermitian conjugate of the entire product of operators acting on ρ(0) from the left.

Photon echoes 95 To simplify Eq. (4.3.50) and compare it with earlier results, it is necessary to commute the second and third terms in ρ(t). For this purpose, commutation relations of the Pauli matrices must be taken into account, and the identity in Problem 4.9 is helpful. Using the result

Only terms with the time argument (t− 2τ) can contribute to echo formation.

Consequently, the density matrix reduces to ρ(t) = sin² Now, only the σ₃ term in ρ(0) leads to nonzero contributions to ρ(t) in Eq. (4.3.51), since the unit matrix portion σ₀ allows conjugates to combine and cancel. Fur-thermore, terms containing σ₁²ρ(0)σ²₁ or σ₁ρ(0)σ₁ are proportional to σ₃. These terms are diagonal and cannot give rise to radiation. Diagonal terms of ρ(t) do not appear in the calculation of polarization in Chapter 3. They do not correspond to oscillating dipole moments and cannot contribute to radiant emission by the sample.

Dropping the non-radiant terms, we therefore find ρ(t) = sin²(θ2/2) In this result for ρ(t), which determines the signal field (see Eq. (3.9.16)), all the basic features of two pulse echoes established in earlier sections of this chapter are again evident. The echo appears at t = 2τ and the echo amplitude is maximum for pulse areas of θ₁= π/2 and θ₂= π.

Multiple pulse photon echoes like the stimulated echo illustrated in Fig. 4.11 pro-vide a useful method of storing and retrieving information on demand and also of

processing information. For example, information in the form of amplitude modulation of the input light can be stored as an index or population grating in the medium and recalled at a later time (on demand) by the third pulse in a three-pulse echo sequence. High-fidelity applications of this kind have in fact been demonstrated by the retrieval of long pulse sequences constituting time-encoded bit strings [4.22].

High-speed, large bandwidth spectrum analysis can also be performed using coherent transients [4.23].

This subsection has presented yet another way to calculate the signal amplitude of photon echoes, adding to the methods covered in Sections 4.3.1 and 4.3.2. This third method was based on the formally concise expression (Eq. (4.3.42)), which yields the density matrix ρ(t) directly. While appealingly simple at the outset, this formula contained an exponential form of the evolution operator and matrix representations of the SU(2) Pauli spin operators which introduced inconvenient limitations. Explicit use had to be made, for example of the SU(2) commutation relations, as well as expansions of the exponential functions in terms of the Pauli spin matrix operators.

A moment’s pause makes one realize how awkward it would be to have to use complex functions of higher-order spinors, for example SU(n) spin matrices for an n-level system, to describe systems with more than two levels. Also, dissipation has been completely ignored in the treatment of this subsection. To include relaxation processes would greatly complicate the arguments of the temporal evolution operators even within a single time segment. This combination of drawbacks argues against adopting this approach as a general tool for analysis. Similar reservations can be leveled at the other approaches explored in Sections 4.3.1 and 4.3.2. Consequently, for the remainder of this book, we shall turn to solutions of the complete, differential master equation in component form. This will avoid restricting ourselves to a small number of energy levels or fields, and will include population and coherence decay processes which govern crucial aspects of the dynamics of real systems in a consistent manner.

Problems

4.1. Apply the evolution matrix U (t) defined by R(t) = U (t)R(0), where R is the Bloch vector, to find the radiant polarization P of a two-level system at an arbitrary time after applying a pulse as described below. Notice that the evolution period R(t) = U (t)R(0) of interest consists of two parts, namely nutation and precession.

(a) Assume the system is initially in the ground state and calculate the off-diagonal matrix element < ˜ρ₁₂> and the Doppler-averaged macro-scopic polarization at times after the application of a single rectangu-lar pulse that are short compared to T₂ in the limit of small detuning ((∆/ΩR) < 1 and (∆²/Ω²_R) << 1). (Ignore population and polarization relax-ation of the elements of R themselves.)

(b) If the system is initially prepared in a state characterized by R₁(0) = 1, then what will the polarization be at short times after the pulse?

Problems 97 4.2. Using the evolution matrix approach, show that the application of a single resonant π-pulse to a ground state atom leaves the system in an inverted state, by removing all population from the initial state.

4.3. An atomic ensemble is subjected to a resonant π/2 pulse. Consider spatial phase in the derivation of the evolution matrix to decide if it is possible to irradiate the ensemble with a second π/2 pulse that may be co- or counter-propagating with respect to the first and to controllably direct the final population to either the ground state or the excited state.

4.4. The Maxwellian distribution of velocities of atoms in a gas gives rise to a Gaussian distribution of resonance frequencies D(ω) that Doppler-broadens the atomic transition linewidth.

(a) Given the probability of finding atoms of resonant frequency ω in the interval dω is D(ω) = C exp5

−Mc²(ω₀− ω)²/2ω²₀k_BT6

dω, find the full width at half maximum intensity (FWHM) of the Doppler-broadened line.

(b) Collisions provide a second source of line-broadening on optical transitions that is inversely proportional to the time τ0between collisions. The full width at half maximum of the collisional contribution is just twice the collision rate given by

γcoll= 1

τ₀ = 4d²N V

πkBT M

1/2

.

Show that Doppler and collisional contributions to the transition linewidth are equal at a gas density for which the volume per atom is close to λd², where λ is the optical wavelength of the transition and d is the (average) distance between the atomic centers during collision.

4.5. A one-dimensional atomic medium consists of only four atoms located at irregular (non-periodic) positions z = 0, 3λ/5, 8λ/9, and 4λ/3. A linearly polarized field E(z, t) = ¹₂E₀x exp[i(ωtˆ − kz)] + c.c. is incident on the medium and excites a microscopic dipolar response ¯p = ε0χ^(e)E in each atom.¯

(a) Calculate the net (macroscopic) polarization ¯P (z, t) =!4

i=1p¯_i(z_i, t_i) of the system, being careful to account for the delay time associated with propaga-tion of a fixed phase point of the driving field from one atom to the next, as well as their positions.

(b) Compare the squared polarization| ¯P|², which is proportional to the intensity of scattered light from the system, to the value expected from four indepen-dent atoms. Show that the reemitted energy from the system has an intensity that is not linear in the number of emitters.

(This problem illustrates the fact that the incident light can coherently phase randomly positioned atoms, and that phasing of constituent dipoles plays an important role in determining the strength of the resulting macroscopic polar-ization.)

4.6. A sequence of pulses is applied to a two-level, Doppler-broadened medium.

Consider the pulses to be delta functions applied with an interval τ , as shown in

the accompanying figure. The pulse areas of the first and second pulses are θ1

and θ2, respectively.

t

?

0 t 2t

(a) Write out a simplified evolution matrix for nutation of the Bloch vector R(t) in the limit that Ω_R>> ∆ (i.e. Ω_R∼= Ω).

(b) Solve for the complete Bloch vector at t≥ τ using the matrix evolution operator method. (Hint: Do not set ∆ = 0 during free precession periods or you lose track of when various potential signal contributions peak in time.) Ignore decay between pulses and do not assume any special values for θ₁ and θ₂.

(c) When the Doppler width is broad compared to the homogeneous linewidth, averaging over the distribution yields a simplified polarization ¯˜ρ₁₂∝ iR2. Then, only R2 components contribute to the polarization. By examining your result for the final Bloch vector, determine whether an echo forms or not when θ₁= π and θ₂= π/2. Justify.

(d) Is this pulse sequence equivalent to a stimulated echo sequence of three π/2 pulses? Draw Bloch vector model pictures to explain result (c) further and justify your conclusion.

(e) Finally, a third pulse of area θ3= π is applied at t = 2τ to complete a π− π/2− π sequence. Calculate the resulting behavior for t ≥ 2τ and explain the result in words or pictures.

0

t

?

t 2t 3t

4.7. Assuming there is an allowed transition with a transition dipole moment µ between the two levels of a two-level system,

(a) Find the Bloch vector of a two-level system, initially in the ground state, that is irradiated by a resonant π-pulse of short duration τp(τp T1, T2).

Take the time of observation to be t = τ (where τ_p τ < T1, T₂).

(b) Calculate and explain the value of polarization P observed at time t = τ ? (c) Draw the Bloch vector in a suitable coordinate system at time t = τ .

References 99 (d) What is the excited state population at t = τ ?

(e) What is the sign of the absorption at t = τ ? Would you expect a probe pulse to experience loss (positive absorption) or gain (negative absorption) at this time?

(f ) If a second pulse, a π/2 pulse, is applied at time t = τ (τ << T₁, τ >> T₂), what is the polarization P (t) at the end of the second pulse?

(g) Assuming the system is inhomogeneously broadened, would you expect an echo at time t = 2τ ? Why or why not?

4.8. Using Bloch vector diagrams similar to Fig. 4.5, give an argument explaining why the Carr–Purcell sequence of pulses, with areas and timing described by π/2(t = 0), π(τ ), π(3τ ), π(5τ ), . . . , produces echoes at times 2τ, 4τ, 6τ, . . . , whose envelope decays with a characteristic time T1 instead of T2.

4.9. By expanding the leftmost exponential function as an infinite sum, and using commutation properties of the Pauli matrices, show that

exp

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5

Coherent Interactions of Fields

In document Nonlinear and Quantum Optics (Page 108-118)