## Abstract

We theoretically investigate the sequential two-step up-conversion of correlated photon pairs with positive and negative energy correlations, in terms of how the up-conversion efficiency depends on the incident pulse delay. A three-level atomic system having a metastable state is used to evaluate the up-conversion efficiency. It is shown that a photon pair with a positive energy correlation can drastically enhance the up-conversion efficiency compared with uncorrelated photons and correlated photons with a negative energy correlation.

© 2010 Optical Society of America

## 1. Introduction

Up-conversion is a well-known nonlinear optical process and has a wide variety of applications in the chemical and biochemical fields, such as coherent visible and near-ultraviolet emission [1, 2], solar cell systems [3], and time-resolved microscopes [4]. In contrast to two-photon or multi-photon absorption, up-conversion requires an absorbing center having metastable intermediate states between the ground state and an emitting state so that incident photons can be absorbed sequentially rather than simultaneously. Thus, for efficient up-conversion, controlling the time delay between incident photons is important.

Recently, up-conversion of quantum-correlated photons has been reported [5]. Correlated photons are well known for the enhancement of two-photon absorption due to the coincidence originating from the quantum correlation [6, 7, 8]. However, by using a correlated photon pair with time delay, efficient up-conversion can be achieved owing to long correlation time of quantum correlation, though the coincidence is sacrificed. Thus, correlated photons can enhance the efficiency of the two-photon process not only in simultaneous absorption but also in sequential excitation. Therefore, by combining with conventional optical methods, such as the pump-probe method, correlated photons are applicable to various fields as well as up-conversion process.

The above property of correlated photons is limited to a photon pair with a negative energy correlation. Owing to the recent development of microfabrication of nonlinear optical crystals, we can prepare a correlated photon pair with a positive energy correlation by extending the conventional phase-matching condition [9]. In fact, generation of the correlated photons has been demonstrated [10, 11, 12], where the two-photon joint amplitude can be controlled by artificially designed material with quasi phase matching. The correlated photon pair with a positive energy correlation has an inherent time delay between its two constituent photons; two photons always appear symmetrically about the mean time of arrival of the two photons. Since efficient up-conversion requires sequential excitation rather than simultaneous excitation, the inherent time-delay property could work well for the up-conversion process and a further enhancement of the up-conversion efficiency could be expected.

In this study, we therefore analyze in detail the up-conversion of two correlated-photon pairs with positive and negative energy correlations, in terms of how the up-conversion efficiency depends on the incident pulse delay. In particular, we investigate the dynamics of up-conversion process by exactly two photons in real time and evaluate the enhancement factor by directly comparing the up-converted photons obtained from uncorrelated and correlated photon pairs. We show that correlated photons with a positive energy correlation can drastically enhance the up-conversion efficiency while correlated photons with a negative energy correlation can also enhance the efficiency compared with uncorrelated photons. The condition for each correlated photon pair to achieve higher efficiency is different; the photon pair with a negative energy correlation requires a large delay, and the photon pair with a positive energy correlation requires only small delay and reaches its peak at zero delay.

The rest of this paper is organized as follows. In Sec. 2, a theoretical model of an atom-cavity system is introduced and the formulation of correlated photon pairs is given. In Sec. 3, we analyze in detail the quantum dynamics of atomic states driven by correlated photon pairs and the dependence of up-conversion efficiency on pulse delay. In Sec. 4, we summarize our results.

## 2. Model

#### 2.1. One-dimensional atom model

As a model system, we consider a one-dimensional photon field interacting with an atom-cavity system, as depicted in Fig. 1(a). Two incident photons propagate parallel to the *r*-axis and penetrate into a one-sided microcavity (at dumping rate *κ*). The two photons interact with an atom inside the cavity through a cavity field (at coupling rate *g*) and an up-converted photon returns to the initial photon field. The atomic system inside the cavity is a modified three-level system including a metastable state: the ground state |*g*〉, the two intermediate states |*m*_{A}〉 and |*m*_{B}〉, and the excited state |*e*〉. |*m*_{A}〉 absorbs photons and its population rapidly decays to |*m*_{B}〉 through multiphonon non-radiative relaxation. |*m*_{B}〉 is metastable and becomes a step ladder in the transition to |*e*〉. The energies of the atomic system are denoted by *ω*_{mA(B)} and *ω _{e}* (in units of

*h̄*).

*ω*

_{mA}and

*ω*–

_{e}*ω*

_{mB}are set to the central energy

*k*

_{0}of incident photons so that |

*m*

_{A}〉 and |

*e*〉 can be resonantly excited.

In the bad cavity regime of *κ* ≫ *g*, photons inside the cavity are emitted so rapidly from the cavity that the cavity field can be adiabatically eliminated. In this case, the atom-photon interaction can be characterized by a single effective emission rate, Γ ≡ *g*^{2}/*κ*. If Γ is much larger than the spontaneous emission rate of the atomic system into free space (non-cavity modes), the system can be reduced to a one-dimensional input-output system with negligible radiative losses [Fig. 1(b)], called the one-dimensional atom model.

#### 2.2. Hamiltonian and quantum dynamics

Setting natural units of *h̄* = *c* = 1, the Hamiltonian of the whole system is given by

*k*,

*p̂*is the annihilation operator of a phonon with energy

_{q}*q*, and

*γ*is the non-radiative decay rate. Eq. (2) is a fully quantum-mechanical description of the Bixon-Jortner theory [14], based on quantum noise [15]. In this study, we ignore the transitions between |

*g*〉 and |

*m*

_{B}〉 and between |

*m*

_{A}〉 and |

*e*〉, for simplicity.

The dynamics of the whole system can be calculated from the Schrödinger equation,

where |Ψ(0)〉 is the initial state of the whole system, given by*ψ*

_{2p}is the two-photon joint amplitude of the incident pulse. The whole wave function is normalized to be 〈Ψ|Ψ〉 = 1. The populations of the excited and intermediate atomic states are described by 〈

*e*〉 = Tr[|

*e*〉〈

*e*||Ψ(

*t*)〉〈Ψ(

*t*)|] and 〈

*m*〉 = Tr[|

_{ℓ}*m*〉〈

_{ℓ}*m*||Ψ(

_{ℓ}*t*)〉〈Ψ(

*t*)|], respectively.

#### 2.3. Quantum-correlated photons

For comparison, we consider three photon pairs which are classically indistinguishable from each other: An uncorrelated photon pair, corresponding to classical light, given by

*ψ*(

*k*

^{(′)}) is the one-photon pulse and ${r}_{0}^{(\prime )}$ is the spatial center position of the wave packet at

*t*= 0. We define the delay between the two pulses as Δ ≡ |

*r*

_{0}–

*r*′

_{0}|. The correlated photons described by Eq. (6) are referred to as a twin-beam (TB) state and can be obtained from spontaneous parametric down-conversion [16].

*δ*(

*k*+

*k*′ − 2

*k*

_{0}) indicates the negative energy correlation of two photons: One photon with energy

*k*

_{0}−

*k*is accompanied by another photon with energy

*k*

_{0}+

*k*, conserving the total energy of 2

*k*

_{0}. By Fourier transforming to the time domain, this property implies that the photon pair has time coincidence, as has been measured in the famous Hong-Ou-Mandel experiment [17]. The correlated photons described by Eq. (7) are referred to as a difference-beam (DB) state, and the photon pair can be obtained by extending the conventional phase-matching condition [9].

*δ*(

*k*–

*k*′) indicates the positive energy correlation of the two photons, in other words, the two photons have the same frequency. In the time domain, this property means that one photon at

*t*

_{0}–

*t*is accompanied by another photon at

*t*

_{0}+

*t*, where

*t*

_{0}is the mean time of arrival of the two photons. Thus, the DB photons have an inherent time delay between the two photons.

According to the spatiotemporal pulse dynamics theory [13], we define *ψ*(*k*) by the Fourier transformation of the space domain and choose *ψ*(*k*) having a Gaussian shape,

*σ*is the coherent length of the wave packet. The two-photon joint spectra |

*ψ*

_{2p}|

^{2}corresponding to Eqs. (5), (6), and (7) are shown in Figs. 2(a), 2(b), and 2(c), respectively. Intriguingly, the spectra of these photon pairs are identical in terms of classical electromagnetic field, as shown in Fig. 2(d). Thus, the only difference is the quantum correlation, which can be controlled only by quantizing light fields.

## 3. Results

The dynamics of atomic states induced by quantum-correlated photons can now be calculated by solving Eq. (3). In the actual calculation, we omit the degrees of freedom of polarization by assuming linearly-polarized light and numerically solve Eq. (3) by discretizing the photon fields. Further, we approximate the *δ* functions in Eqs. (6) and (7) by a Gaussian function having a width of 8*σ* ≈ 240*λ* in space domain, where *λ* = 2*π*/*k*_{0}, corresponding to a typical value for correlated photons. As a model of the atomic system, we adopt *ω*_{mA} = *k*_{0}, *ω*_{mB} = 0.8277*k*_{0}, and *ω _{e}* = 1.8277

*k*

_{0}[18]. Further, we choose relatively large damping rates, Γ

*= Γ*

_{mg}*≈ 0.0003*

_{em}*k*

_{0}, Γ

*= 5Γ*

_{eg}*, and*

_{mg}*γ*= 10Γ

*, in order to shorten the computational time. We should add that the value of Γ hardly affects calculation results unless Γ is comparable to*

_{mg}*σ*

^{−1}.

Figure 3 shows the population dynamics induced by uncorrelated photons, TB photons, and DB photons for different values of Δ. The parameters are the same as those in Fig. 2 except for Δ. The horizontal axis is the spatial center position *r* between incident photon pulses, normalized by *σ*, and the vertical axis is the population of the atomic system, where red and blue curves represent 〈*e*〉 and 〈*m*_{A}〉, respectively. As shown in Fig. 3(a), efficient excitation of |*e*〉 by uncorrelated photons requires a large value of Δ. For Δ = 0, while 〈*m*_{A}〉 reaches its peak, 〈*e*〉 is hardly excited. As Δ increases, however, the two photons become well separated so that |*m*_{A}〉 can absorb photons one by one, as can be seen in the splitting of 〈*m*_{A}〉 into two peaks. This separation between the two photons allows the first photon to induce enough energy transfer from |*m*_{A}〉 to |*m*_{B}〉 before the second photon arrives. As a result, |*e*〉 is efficiently excited by the second photon. This process is well known as the sequential two-step up-conversion. For TB photons [Fig. 3(b)], we can find a similar process to that shown in Fig. 3(a). The difference between Figs. 3(a) and 3(b) is that the response of the atomic states to TB photons is slower (broader) than that to uncorrelated photons. For DB photons [Fig. 3(c)], however, the response of 〈*m*_{A}〉 and 〈*e*〉 are different in quality from both uncorrelated photons and TB photons. The critical difference is that 〈*e*〉 reaches its peak for Δ = 0 and decreases as Δ increases, in contrast to the cases of uncorrelated photons and TB photons. In addition, the splitting of 〈*m*_{A}〉 into two peaks cannot be found for DB photons.

The slow response to TB photons is because that TB photons have effectively two pulse widths: A narrow width characterizing the coincidence of two photons constituting a photon pair, and a wide width characterizing the distribution of the photon pair [19]. In Fig. 3(b), the latter appears because the coincidence leads to atomic saturation and works against the sequential two-step up-conversion process. In contrast, DB photons have an inherent time delay between the two constituent photons, and this property induces efficient excitation of 〈*e*〉 for smaller Δ, especially for Δ = 0.

The up-conversion efficiency can be evaluated directly by comparing the total intensities of obtained up-converted photons. Figure 4(a) shows the dependence of up-conversion efficiency *ξ* on Δ/*σ. ξ* is defined by the total intensity normalized by that obtained from uncorrelated photons,
$\int \mathit{dk}{|{\psi}_{\text{UC}}^{\text{TB}(\text{DB})}(k)|}^{2}/\int \mathit{dk}{|{\psi}_{\text{UC}}^{\text{uncorr}.}(k)|}^{2}$, where *ψ*_{UC} is the probability amplitude of the up-converted photon. Thus, *ξ* yields the enhancement factor of up-conversion by TB and DB photons. The parameters are the same as those in Fig. 3, and *ψ*_{UC} is calculated after the incident photons completely passes through an atomic system. For TB photons, *ξ* is smaller than one in the range of Δ/*σ* < 1 and gradually increases as Δ/*σ* increases. For DB photons, however, *ξ* reaches its peak at Δ/*σ* = 0 (*ξ* ≈ 6.5 for the present parameters) and decreases as Δ/*σ* increases. For Δ/*σ* > 1, both TB and DB photons achieve *ξ* > 1. Figure 4(b) shows the dependence of *ξ* on *σ*/*λ* for Δ/*σ* = 0 and 10. For any *σ*/*λ*, TB photons with Δ/*σ* = 0 cannot achieve *ξ* > 1. For Δ/*σ* = 10, however, *ξ* is improved to *ξ* > 1 and gradually increases as *σ*/*λ* decreases (*ξ* ≈ 3 at *σ*/*λ* ≈ 10 for the present parameters). In contrast, for DB photons, *ξ* increases as *σ*/*λ* decreases for both Δ/*σ* = 0 and 10. In particular for Δ/*σ* = 10, *ξ* drastically increases and reaches up to ≈ 50 at *σ*/*λ* ≈ 10 with the present parameters. Although we calculate only up to *σ*/*λ* ≈ 10 because of the difficulty due to the discretization of photon fields, *ξ* will continue to increase for further decreases in *σ*/*λ*.

Thus, TB and DB photons can achieve higher up-conversion efficiencies compared with uncorrelated photons, in particular the efficiency by DB photons can be much higher than that by TB photons. However, there are advantages and disadvantages to using TB and DB photons to achieve higher efficiency; TB photons require larger Δ, and DB photons require smaller Δ and reach its peak at Δ = 0. Although TB photons with small *σ* realize *ξ* > 1 for Δ > *σ*, we cannot expect to achieve much larger *ξ* seen in the two-photon absorption process [20] because the increase of *ξ* is gradual. DB photons are suitable for the sequential two-step up-conversion process, and a further enhancement of the efficiency can be expected if DB photons with smaller *σ*, ultimately a monocycle pulse with *σ* ≈ *λ*, can be prepared.

Discussion.

## 4. Summary and discussion

We have theoretically investigated the dynamics of the sequential two-step up-conversion of correlated photon pairs with positive and negative energy correlations, in terms of the dependence of the up-conversion efficiency on the pulse delay time. We have shown that correlated photons with a positive energy correlation can drastically enhance the up-conversion efficiency 50 times compared with uncorrelated photons, while correlated photons with a negative energy correlation can also enhance the efficiency. The condition for each correlated photon pair to achieve higher efficiency is different; a zero delay for correlated photons with a positive energy correlation and a large delay for correlated photons with a negative energy correlation.

Finally, we discuss non-resonant up-conversion. In a non-resonant process, for example, a photon having an energy resonant with the transition energy between |*m _{B}*〉 and |

*e*〉 is absorbed, and the transition from |

*g*〉 to |

*m*〉 takes place off-resonantly. The results obtained in this study can be easily applied to such a non-resonant process by controlling two-photon joint spectra. For DB photons, efficient up-conversion can be realized by replacing the

_{A}*δ*(

*k*–

*k*′) in Eq. (7) to a function with a broad width, keeping a positive energy correlation, so that two photons with different energies can be correlated. For TB photons, spectrum-controlled photon pair is useful [21], where spectral distribution of two photons can be controlled so as to have two peaks, e.g., corresponding to

*ω*–

_{e}*ω*

_{mB}and

*ω*

_{mA}.

We hope that the results of this study facilitate applications of correlated photons to optical devices based on the two-photon process.

## Acknowledgments

This work was supported by the JST PRESTO program and a Grant-in-Aid for young scientist (B) (No. 21760043) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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