the excited state and thus insensitive to its decay. This so-called dark state,
|D = Ω|2Ω|1|2−+Ω|1Ω|2|2 1 2
(5.50)
is an eigenstate of the effective Hamiltonian with zero energy: ˆ eff D = 0. H
|
(5.51)
A system in state D has vanishing absorption and polarization on the 1 to 3 transition, but finite polarization on the forbidden transition. Quantitatively, the density matrix for the dark state,
|
|
|
ρ = D D ,
| |
(5.52)
has off-diagonal elements ρ21 =
∗
− |Ω1|Ω2 2+Ω|1Ω2|2 .
(5.53)
In the limit that Ω1 Ω2 , we reproduce our previous result. Consequently, we see that the vanishing absorption on resonance is associated with the production of dark states which are decoupled from both light beams. This can also be viewed as a quantum interference phenomenon, where simultaneous excitations of 3 by Ω1 and Ω2 destructively interfere, leaving the excited state unpopulated by the applied fields.
|
Dispersive properties of the dark resonance: ”slow light” Thus far we have considered only the imaginary part of the probe susceptibility in the vicinity of the dark resonance. The real part leads to dispersive effects, since for ∆2 = 0 and δ small Re[χ]
∝ Re[ ρΩ311 ] ≈ |Ω2|2 +δ γ 12γ 13 .
(5.54)
In the situation relevant to the dark resonance, the slope of the refractive index n Re[χ] can be controlled by changing the intensity of Ω2 . In particular, this slope determines the group velocity since
∝
vg =
dω c = dn dk n(ω) + ω dω
88
(5.55)
(where n(ω) is the index of refraction) and n(ω) = 1 + Re[χ]/2. The slope dn 0 and Ω2 0: dω diverges as γ 12 c vg (5.56) γ 13 ν¯ 3π(N/V )(λ/2π)3 1+ |Ω2|2 +γ 13γ 12
→
→
≈
so that the group velocity is reduced to zero as Ω2 0 (see problem set 3 for a derivation of this formula). Simultaneously, the width ∆w of the the resonance also vanishes, since
→
Ω2 |2 | ∆w ≈ → 0. γ 13
(5.57)
Note that when the linewidth is dynamically reduced while a pulse is travelling through the medium, careful considerations show that the spectrum of the pulse itself is also dynamically reduced. In particular, if the pulse spectrum fits inside the transparency window initially, it fits in the window at all times and no dissipation is taking place (see problem set 3 and references therein for further details). Since the two-photon resonance can be much narrower than the singlephoton linewidth, the group velocity of the probe pulse Ω1 can be reduced substantially below the speed of light. As resonant probe light enters the medium it compresses by a factor vg /c, then slowly propagates through the atomic cloud, dragging along a spin coherence associated with the dark state. Since absorption vanishes, virtually all of the probe light is transmitted, with a significant delay due to the slow group velocity. This phenomenon is known as “electro-magnetically induced transparency” or “slow light”. Preparation of the dark state Our discussion has not explained how the system is prepared in the dark state. In fact, if the probe light turns on slowly enough, the system will adiabatically follow from the initial state 1 to D , in essence preparing itself. This phenomenon is an example of stimulated Raman adiabatic passage (STIRAP), whereby slowly varying applied fields can be used to efficiently and robustly manipulate the populations and atomic coherences of a system. Unlike π pulses based on Rabi oscillations, which are exquisitely sensitive to pulse amplitude and timing, STIRAP techniques do not depend on the details of pulse shape. To understand the basic mechanism for dark state formation, consider the asymptotic behavior of the dark state under different applied powers:
| |
Ω2
→ 0, Ω1 fixed ⇒ |D = −|2 89
(5.58)
Ω1
(5.59) → 0, Ω2 fixed ⇒ |D = |1. The physics here is fairly obvious, since |1 does not couple to Ω2 and vice versa. Consequently, if the system starts out in the state | 1 illuminated
only by Ω2 , it is already in the dark state. By slowly decreasing Ω2 while increasing Ω1 , the atomic state will adiabatically follow the dark state, finally ending up in 2 when Ω2 = 0. Since the dark state contains no component of the excited state, and does not couple to the light, the atomic state will be free from dissipation or decoherence from the excited state. Such manipulation is known as a “counter-intuitive pulse sequence,” since the light is initially applied on the transition which is unpopulated.
|
5.3
Parametric Processes
Parametric processes represent an important category of coherent interactions between atoms and light. In such systems, we are interested primarily in the propagation and interaction of applied and generated light fields; the nonlinear atomic response is important only because it mediates interactions between optical fields. No energy is exchanged between photons and atoms, but atomic nonlinearities can lead to mixing between applied fields or even generation of fields at new frequencies. In practice, the nonlinear medium need not b e atomic; in the most general treatment, optical fields impinge on a “nonlinear black box” which can give birth to higher frequency harmonics or sum and difference frequencies.
5.3.1
Example: the cascade system
We will first examine how parametric processes in the cascade level configuration can be used to generate higher-frequency fields. The situation we will consider has two fields 1 and 2 near resonance with the two-photon transition ν 1 + ν 2 ω12 . By driving this two-photon process, we hope to induce the atom to emit radiation at frequency ν 1 + ν 2 . From symmetry considerations, one might argue that the 2 1 transition is dipole forbidden if the two-photon process is dipole-allowed. Nevertheless, by breaking the associated symmetry (in this case, inversion symmetry), it is possible to have both single-photon and two-photon dipole allowed transitions between 1 and 2 . Such symmetry breaking can be accomplished in a variety of ways. The hydrogen atom furnishes a particularly simple example: if we apply a static electric field to the hydrogen
≈
E
E
|→|
|
|
90
ˆ when the system is in state The expectation value of the observable O ψ is
|
Oˆ
=
ψ|Oˆ |ψ.
(1.6)
ˆ is diagonal in the basis When the operator O
{|n}, we have
ˆn = nn O ˆ = O
| O| ⇒ On|cn|2 =
n
(1.7)
O
n pn
(1.8)
n
ˆ . No where n denotes the nth eigenvalue of the operator O Note te that that if the the ˆ , we always get the same result upon repeated system is in an eigenstate of O ˆ. measurements of the observable O
O
1.2.4 1.2.4
Uncertai Uncertaint nty y principle principle
In general operators do no commute, i.e. ˆ B ˆ ] = iC ˆ [A,
(1.9)
ˆ = 0, the operators Aˆ and B ˆ do not have the same eigenstates. so that for C Heisenberg’s uncertainty principle states that for any two operators Aˆ ˆ , we have and B
ˆ B ˆ ] ≥ 21 [A,
∆A ∆B
·
where ∆A ∆A =
1.2.5 1.2.5
ˆ (A
(1.10)
− Aˆ)2.
Time evoluti evolution: on: Schrodinge Schrodinger’s r’s picture picture
In non-relativistic quantum mechanics, the dynamics of the system is described by Schrodinger’s Schrodinger’s equation: equation: in Schrodinger’s Schrodinger’s picture the state vecvector describing the state of the system evolves in time, while the operators describing the observables corresponding to possible measurements on the system are stationary. The state vector evolves in time according to d i ψ dt
|
ˆ ψ = H
|
(1.11)
ˆ is the Ham where H H Hamilt iltoni onian an of the system system.. The station stationary ary solutio solutions ns of ˆ Schrodinger Schrodinger’s ’s equation equation are given by E n where H E n = E n E n . Any
|
3
|
|
Hydrogen atom in a DC electric field
Cascade system: frequency sum generation
|2p〉
|2〉
hybrid state
|2s〉
2 ν2
Ε
Ε 2
3 ν3
Ε 3
|i〉
Ε
Ε 1 ν1
Both singleand two-photon transitions are dipole allowed.
intermediate states
ν2
ν3
|1〉
Ε 1 ν1
|1s〉
Figure 5.6: Left: The cascade level system used for sum-frequency generation. Right: A physical implementation of the cascade level system in the Hydrogen atom. A DC field mixes the 2s and 2 p states so that both the single- and two-photon transitions are dipole allowed. This DC Stark shift breaks the inversion symmetry of the Hydrogen atom, which makes second-order nonlinear processes possible.
|
|
atom, the linear Stark shift mixes the 2s and 2 p levels so that the resulting hybrid state does not have a well-defined parity. Selection rules are thereby relaxed, so that a far-off-resonant two-photon transition can indeed induce single-photon transitions back to 1s . Sum frequency generation has an intuitive semiclassical explanation. Suppose you drive the Stark-shifted Hydrogen atom with two fields with frequencies ν 1 + ν 2 = ω1s→2s . The resulting off-diagonal atomic coherence ρ12 leads to a polarization which oscillates at the sum frequency ν 1 + ν 2 . From a semiclassical perspective, the oscillating polarization will emit radiation at the sum frequency. Note that the polarization is proportional to both applied fields P ν 1 +ν 2 µ12 1 2 , (5.60)
|
|
|
∝ E E
and we will introduce a nonlinear susceptibility as the proportionality constant P ν +ν χ(2) = 1 2 . (5.61) 0 1 2
E E
Whenever a medium has χ(2) = 0, application of two fields can result in generation of the sum frequency.
91
