3 Enhanced and Selective Two-Photon Excitation of Molecular …
47
using the following Schrödinger equation,
|ψ(t) = exp
−i H
t
|ψ(0),
where |ψ(0) is the initial state at t = 0 and H
is the total Hamiltonian of the system.
Using the photon dispersion relation of ω = ck = k, the initial state |ψ(0) is given
by
|ψ(0) =
1
√
2
∫ dk ∫ dk
ψ 2 p
k, k
ˆ
a
†
(k) ˆ
a
†
(k)|0 ⊗ |g,
where ψ 2 p
k, k
is the two-photon wavefunction and ˆ
a
†
(k) is the creation operator
of a photon with wavenumber k. The molecular system is assumed to be in the ground
state |g at t = 0.
To show the usefulness of entangled photons, we consider two-photon pairs as
incident two-photon pulses, namely uncorrelated photons and entangled photons, for
comparison. The uncorrelated photons are corresponding to classical light, given by
ψ 2 p
k, k
= ψ 1 p (k)ψ 1 p
k
,
where ψ 1 p (k) is the one-photon wavefunction. The uncorrelated photons literally
have no correlation between photons and are separated into the direct product of the
one-photon wavefunction, which is the characteristic of classical light. For entangled
photons, we adopt two photons with energy anticorrelation, given by
ψ 2 p
k, k
= ψ 1 p (k)δ
k + k
− 2k 0
,
where δ(·) is the Dirac delta function and k 0 is the central wavenumber of incident
photon pulse. In contrast to uncorrelated photons, the entangled photons cannot
be separated into the direct product of ψ 1 p (k) and have energy anticorrelation so
that the energy sum of two photons can be always 2k 0 , which is ensured by the
δ function. In actual photons generated experimentally, entangled photons are not
described by δ function but have a certain amount of width owing to spontaneous
emission. In this study, we approximate the δ function by a Gaussian function as
∝ exp
−
k + k
− 2k 0
2 /4σ
2
s
, for simplicity. In the limiting case of σ s → 0, the
Gaussian becomes δ function.
Figure 3.4a, b shows the two-photon joint spectra for uncorrelated and entangled
photons, respectively. For ψ 1 p , a Gaussian form with pulse width of σ = k 0 /20π
is used and σ s is set to σ/7. In contrast to Fig. 3.4a, ellipsoidal photon distribution can be found in Fig. 3.4b, which indicates the energy anticorrelation of two
photons, conserving the total energy of 2k 0 . Figure 3.4c shows corresponding spectra
of Fig. 3.4a, b. Intriguingly, one can find that two spectra are identical in spite of
the presence or absence of quantum correlation. Thus, the only difference between
47
using the following Schrödinger equation,
|ψ(t) = exp
−i H
t
|ψ(0),
where |ψ(0) is the initial state at t = 0 and H
is the total Hamiltonian of the system.
Using the photon dispersion relation of ω = ck = k, the initial state |ψ(0) is given
by
|ψ(0) =
1
√
2
∫ dk ∫ dk
ψ 2 p
k, k
ˆ
a
†
(k) ˆ
a
†
(k)|0 ⊗ |g,
where ψ 2 p
k, k
is the two-photon wavefunction and ˆ
a
†
(k) is the creation operator
of a photon with wavenumber k. The molecular system is assumed to be in the ground
state |g at t = 0.
To show the usefulness of entangled photons, we consider two-photon pairs as
incident two-photon pulses, namely uncorrelated photons and entangled photons, for
comparison. The uncorrelated photons are corresponding to classical light, given by
ψ 2 p
k, k
= ψ 1 p (k)ψ 1 p
k
,
where ψ 1 p (k) is the one-photon wavefunction. The uncorrelated photons literally
have no correlation between photons and are separated into the direct product of the
one-photon wavefunction, which is the characteristic of classical light. For entangled
photons, we adopt two photons with energy anticorrelation, given by
ψ 2 p
k, k
= ψ 1 p (k)δ
k + k
− 2k 0
,
where δ(·) is the Dirac delta function and k 0 is the central wavenumber of incident
photon pulse. In contrast to uncorrelated photons, the entangled photons cannot
be separated into the direct product of ψ 1 p (k) and have energy anticorrelation so
that the energy sum of two photons can be always 2k 0 , which is ensured by the
δ function. In actual photons generated experimentally, entangled photons are not
described by δ function but have a certain amount of width owing to spontaneous
emission. In this study, we approximate the δ function by a Gaussian function as
∝ exp
−
k + k
− 2k 0
2 /4σ
2
s
, for simplicity. In the limiting case of σ s → 0, the
Gaussian becomes δ function.
Figure 3.4a, b shows the two-photon joint spectra for uncorrelated and entangled
photons, respectively. For ψ 1 p , a Gaussian form with pulse width of σ = k 0 /20π
is used and σ s is set to σ/7. In contrast to Fig. 3.4a, ellipsoidal photon distribution can be found in Fig. 3.4b, which indicates the energy anticorrelation of two
photons, conserving the total energy of 2k 0 . Figure 3.4c shows corresponding spectra
of Fig. 3.4a, b. Intriguingly, one can find that two spectra are identical in spite of
the presence or absence of quantum correlation. Thus, the only difference between