In this chapter we have investigated the transient behaviour of the atom when the EIT probe field (Ωa) is suddenly turned on. One of the most striking features is the
large and relatively long duration transient absorption of the EIT probe. This occurs due to the finite time required to transfer atomic population from the previously uncoupled state |1i into the new dark state of the Λ subsystem. The rate of this process is limited by the slow relaxation of the bright state superposition of the ground states. Thus, the transient absorption is a precursor of the establishment of electromagnetically induced transparency in the Λ subsystem.
Expressions for the transient cross-phase modulation on the Ωa and Ωc probe
transitions are also derived. In both cases it is seen that XPM arises on a time scale dictated by the relaxation of the atom into the EIT state. However, the interpretation of these results in terms of the bright/dark state partly dressed basis leads to some surprising results. In the case of the Ωa transition, XPM is found to
be generated by the coupling between the dark state and the excited state |4i. This is identical to the explanation of XPM given in the steady state regime (subsection 3.2.1). For the Ωc transition however, it is seen that the coupling between the bright
state and |4i gives a significant transient contribution. Although it is sufficient to view XPM as arising to due the perturbation of the Λ-atom dark state in the steady-state regime, this is not a complete explanation when time-dependent fields are considered.
From the form of the susceptibilities it can also be seen that the rise time is pro- portional to the magnitude of the nonlinear susceptibility. Thus, larger nonlineari- ties will take longer to become established. This could have significant implications when trying to achieve strong nonlinear interactions between short optical pulses. Furthermore, since the atom is slow to respond to changes in the probe field, this indicates that there could be limitations and interesting non-adiabatic behaviour of the atom when using short pulses on the EIT probe transition. The non-adiabatic behaviour of the N-configuration atom will be investigated further chapter 5.
Chapter 5
Slowly Pulsed Cross-Phase
Modulation
So far we have considered the interaction of an ensemble of identical atoms with spa- tially and at least piece-wise temporally constant electromagnetic fields. In many respects this is a very reasonable approximation. Consider first the temporal varia- tion of the field.
From the Weisskopf-Wigner theory (2.33), we have seen that the radiative decay rate of a transition at optical frequencies will be on the order of tens of MHz. Typ- ically the pulse duration used in experiments will be of the order of many microsec- onds, although much shorter is possible. Nonetheless, it would seem reasonable to assume that the atom has sufficient time to relax into a quasi-steady state. As noted in the previous chapter the actual atomic dynamics in the Λ- and N-configuration systems is dictated by the much slower relaxation of the bright-state superposition of the ground states. Therefore, even when using relatively long duration pulses then non-equilibrium effects should be taken into consideration.
A second and related issue is the assumption that the electromagnetic field is constant across the dimensions of the atomic sample. Again, we consider typical parameters: sample length, ls = 10−3m and pulse duration τ = 1µs. In free space
we would therefore expect the pulse length to be of the order
lp =c×τ ≈300m. (5.1)
72 CHAPTER 5. SLOWLY PULSED CROSS-PHASE MODULATION
Clearly, the pulse length is much greater than the atomic sample length. However, due to the remarkable properties of the Λ atom we will see that pulses propagating through the sample will undergo compression by several orders of magnitude: this is a natural consequence of slow-light propagation in the atomic ensemble. In turn, this turns out to be a non-equilibrium effect of the slow bright-state relaxation rate.
5.1
Slowly Varying Envelope Approximation
During the calculations in this thesis, and indeed for most experiments undertaken, we are able to work within the slowly-varying envelope approximation. That is, we assume that variations in the classical field amplitude occur on a length scale much longer than the wavelength of the light. We begin with the Maxwell wave equation:
∂2 ∂t2 −c 2 ∂2 ∂z2 E(z, t) =µ0c2 ∂2 ∂t2P(z, t). (5.2) The electric and polarisation fields have the form
E(z, t) = E0(z, t)
2 exp[kz−ωt+φ(z, t)] +c.c. , (5.3)
P(z, t) = P0(z, t)
2 exp[kz−ωt+φ(z, t)] +c.c. . (5.4) where the coefficients E0(z, t) are real, slowly-varying functions of space and time. The corresponding polarisation terms P0(z, t) may be complex, since the induced polarisation will generally not be in-phase with the applied electromagnetic field. Making these assumptions we find coupled first-order wave-equations for the ampli- tude and phase of the electromagnetic field [42]:
∂ ∂t +c ∂ ∂z E0(z, t) = − ω 20 Im [P0(z, t)], (5.5) E0(z, t) ∂ ∂t +c ∂ ∂z φ(z, t) = ω 20 Re [P0(z, t)]. (5.6) When employing a semi-classical approximation we want to relate the macroscopic polarisation of the material to the off-diagonal elements of the density matrix. In this case we have
Im [P0(z, t)] = 2N