## Abstract

Pair creation by spontaneous parametric down-conversion (SPDC) has become a reliable source for single-photon states, used in many kinds of quantum information experiments and applications. In order to be spectrally pure, the two photons within a generated pair should be as frequency-uncorrelated as possible. For this purpose most experiments use narrow bandpass filters, having to put up with a drastic decrease in count rates. This article elaborates (theoretically and by numerical evaluation) the alternative method to engineer a setup such that the SPDC-generated quantum states are intrinsically pure. Using pulsed pump lasers and periodically poled crystals this approach makes bandpass filtering obsolete and allows for significantly higher output intensities and therefore count rates in the detectors. After numerically scanning all common wavelength regimes, polarisation configurations and three different non-linear crystals, we present a broad variety of setups which allow for an implementation of this method.

© 2016 Optical Society of America

## 1. Introduction

Spontaneous parametric down-conversion is a quantum-mechanical process where a photon of high energy (usually referred to as *pump*) spontaneously decays into two daughter photons (*signal* and *idler*) by interaction with a non-linear optical medium. In order for the three fields not to interfere destructively within the crystal, they can be phase-matched via the quasi-phase-matching technique, where the crystal is composed of thin layers with alternating non-linearity coefficient. This approach allows for phase-matching at any collinear three-wave mixing process. In general the frequency spectra of signal and idler are mutually correlated which negatively affects quantum purity. As pointed out in this article, this results in poor Hong-Ou-Mandel visibility [1] when two independent SPDC sources (or photons from different pairs of the same source) are used. By insertion of narrow bandpass filters into signal and idler channel the spectral correlations can be removed under the cost of discarding a better part of the down-converted photons and undermining heralding efficiency, as illustrated in Fig. 1(a). The approach discussed in this article aims to design the SPDC setup such that the output radiation is *a priori* uncorrelated, as depicted in Fig. 1(b), so no photons have to be discarded by filtering. This can be achieved by mutual matching of pulsed pump lasers (centre wavelength and spectral width) and periodically poled non-linear crystals (material, crystal type, poling periodicity, temperature). While the postulates of quantum mechanics, combined with numerical mathematics, show how the purity of down-converted photons can be evaluated [2–5], first experimental tests of this approach yielded promising results [5–12]. In our research we investigated phase-matching conditions and quantum state purity of all types of down-conversion in all common wavelength regimes and in three kinds of periodically poled crystals: *potassium titanyl phosphate* (KTiOPO_{4}, ppKTP), *lithium niobate* (LiNbO_{3}, ppLN) and *lithium tantalate* (LiTaO_{3}, ppLT).

## 2. The SPDC amplitude

The quantum state that describes SPDC reads as

*𝒩̃*is a normalisation constant,

*d*

_{eff}is the effective non-linearity coefficient of the crystal,

*L*is the interaction length (i.e. the length of the non-linear crystal), ${a}_{s/i}^{\u2020}$ are the creation operators of a signal and idler photon respectively,

*μ*is the pump envelope amplitude and

*ψ*is the phase-matching envelope amplitude:

**k**representing the phase-mismatch:

*λ*and

_{p/s/i}*n*are the wavelengths and refractive indices respectively. In the case of collinear quasi-phase-matching the mismatch reads

_{p/s/i}*m*is an integer and Λ is the poling periodicity of the crystal. In order to achieve quasi-phase-matching, i.e. to reset the phase mismatch within the crystal periodically, the sign of the effective non-linearity

*d*

_{eff}is alternated in multiples of the length Λ. The refractive indices are subject to the wavelength- and temperature-dependent Sellmeier equations for respective crystal and polarisation direction. Therefore the phase-matching amplitude

*ψ*is unique for each setup of down-conversion type, wavelengths, crystal and temperature.

## 3. Factorability

The Hong-Ou-Mandel visibility of two interfering substates *ρ _{A}* and

*ρ*with respective purities

_{B}*P*and

_{A}*P*reads as [2]

_{B}*purity P*and

*indistinguishability*‖

*ρ*−

_{A}*ρ*‖ required for high visibilities. In our case

_{B}*A*and

*B*represent photons from two

*independent*SPDC sources. (Note that Hong-Ou-Mandel interference of signal and idler from the

*same*generated pair does not require high purity and is therefore not embraced in this paper.) Consider the special case of an interference experiment with two independent

*identical*sources (or subsequent photons from the same source which is equivalent). If we let the two signal photons interfere and use the idler photons for heralding, the purities and the state operators of the interfering photons coincide (

*P*=

_{A}*P*=:

_{B}*P*,

*ρ*=

_{A}*ρ*) and the visibility depends on the purity only: In the case where one pair’s signal and the other pair’s idler of are brought to interfere, the distinctness term ‖

_{B}*ρ*−

_{A}*ρ*‖ does not necessarily cancel and the expression reads where we used the fact that for identical sources signal and idler carry the same purity,

_{B}*P*=

_{s}*P*=

_{i}*P*, as derived below.

So no matter whether the experiment relies on interference of two signal photons or mutual interference of signal and idler, a high spectral purity *P* plays a key role in achieving high HOM-visibilities.

#### 3.1. Pure and mixed states

The density operator of any quantum state reads

with ∑

_{j}*p*= 1 and |Ψ

_{j}*〉 are pure quantum states. The operator*

_{j}*ρ*represents a pure state when there is only one non-vanishing

*p*such that

_{j}*ρ*= |Ψ〉〈Ψ|, otherwise we speak of a mixed state. The purity of a state

*ρ*is evaluated by tracing over its square: In case of a bipartite quantum state

*ρ*we can trace over one subsystem in order to obtain the quantum state of the respective other subsystem:

_{AB}*can be expressed in terms of a shared complete set of orthonormal basis states:*

_{AB}*Schmidt decomposition*of the state |Ψ〉

*. The vectors |*

_{AB}*α*〉

_{j}*and |*

_{A}*β*〉

_{j}*are the so-called*

_{B}*Schmidt modes*of the respective subsystems and

*λ*are the

_{j}*Schmidt coefficients*which add up to one: ∑

_{j}*λ*= 1. The pairs |

_{j}*α*〉

_{j}*⊗ |*

_{A}*β*〉

_{j}*form a complete set of basis states of the total system. The Schmidt decomposition provides an intuitive measurement of entanglement within a pure state: If the state is entangled, then there is more than just one term present in Eq. (11).*

_{B}When we build the density operator *ρ* = |Ψ〉〈Ψ| of the state (11) and perform a partial trace over the subsystem *B* we arrive after straightforward calculation at

*λ*= 1 yielding maximum purity in both subsystems:

*P*=

_{A}*P*= 1.

_{B}#### 3.2. Factorability of the SPDC amplitude

When we summarise pump and phase-matching envelope amplitude into a single function, the *joint spectral amplitude* (JSA)

*ω*and

_{s}*ω*in the joint spectral amplitude

_{i}*f*(

*ω*,

_{s}*ω*)

_{i}*do*indicate a frequency entanglement of signal and idler photons. In order to investigate the entanglement of the SPDC-amplitude |Ψ〉—a pure bipartite state—we would like to express it in terms of a Schmidt decomposition: with Schmitdt modes |

*s*〉 and |

_{j}*i*〉, representing the respective subsystems, signal and idler. We perform the Schmidt decomposition numerically by making a

_{j}*singular value decomposition*(SVD) of the discretised JSA. For this purpose we first split the relevant range of signal and idler frequencies into discrete values

*ω*and

_{s,m}*ω*. Then we express the state as a sum over all possible combinations of signal and idler eigenfunctions, |

_{i,n}*s̃*〉 and |

_{m}*ĩ*〉, weighted by their respective joint amplitude

_{n}*f*(

*ω*,

_{s,m}*ω*) each of which is calculated numerically:

_{i,n}*f*(

*ω*,

_{s,m}*ω*) can be understood as components of an

_{i,n}*M*×

*N*-matrix

*ℱ*, each row (column) of which represents a particular discretised signal (idler) frequency:

*ℱℱ*

^{†}is the partial trace of the total state over the idler subsystem [4] which is in turn just the definition of the reduced density operator of the signal subsystem: Analogously

*ℱ*

^{†}

*ℱ*represents just the opposite: We now express the respective amplitudes

*f*(

*ω*,

_{s,m}*ω*) in Eq. (19) as components of

_{i,n}*ℱ*and obtain

*U*and

*V*

^{†}and a diagonal matrix

*D*, such that where the columns of

*U*represent the eigenvectors of

*ℱℱ*

^{†}=

*ρ*and the columns of

_{s}*V*(or rows of

*V*

^{†}) are the eigenvectors of

*ℱ*

^{†}

*ℱ*=

*ρ*. The entries in

_{i}*D*are real, positive, appear in descending order on the diagonal and represent the eigenvalues of the eigenvectors described by the columns of

*U*and

*V*:

*d*=

_{j}*D*are just the coefficients of the diagonal matrix. It is important to note that the

^{jj}*j*

^{th}column of

*U*and the

*j*

^{th}row of

*V*

^{†}are associated with the same eigenvalue

*d*, i.e. the columns of

_{j}*U*and

*V*have a shared spectrum. We now replace

*ℱ*by its SVD representation, so Eq. (22) becomes

*d*. As long as the squares of the coefficients sum up to one, this is just the unique expression for the Schmidt decomposition of |Ψ〉. So after

_{j}*D*has been normalised such that $\sqrt{\text{Tr}({D}^{2})}=1$ we can identify its entries with the Schmidt coefficients: ${d}_{j}=\sqrt{{\lambda}_{j}}$. Furthermore we define yielding the desired expression (18). The purity of

*ρ*and

_{s}*ρ*can then be easily obtained by Eq. (9), only using the entries

_{i}*d*of the diagonal matrix

_{j}*D*.

Experimentally, a separable JSA can be achieved by appropriate matching of the pulse duration and the crystal length. Fig. 2(a) depicts the *joint spectral intensity* |*f*(*ω _{s}*,

*ω*)|

_{i}^{2}(JSI) in the case of a narrow-band CW laser impinging a short crystal, resulting in anti-correlated signal and idler spectra. Conversely, a broadband femtosecond laser impinging a long crystal would result in correlated daughter photons as shown in Fig. 2(b). When pulse duration and crystal length are mutually matched, we can achieve frequency-uncorrelated signal and idler spectra as illustrated in Fig. 2(c) and Fig. 2(d).

## 4. Numerical evaluation

Unfortunately, the generation of intrinsically pure quantum states is only possible for a limited amount of setups; as pointed out above, the JSA depends on the mutual relation of pump envelope intensity and phase-matching envelope intensity. The latter depends ultimately on the material’s Sellmeier equations which provide the refractive index for a given polarisation, wavelength and temperature. As it turns out, each crystal allows for generation of intrinsically pure down-converted photons only in the case of very specific polarisation- and wavelength configurations.

Using numerical calculations we investigated the suitability of three periodically poled nonlinear crystals, potassium titanyl phosphate (KTP), lithium niobate (LN) and lithium tantalate (LT), for all kinds of down-conversion processes which carry a non-vanishing effective non-linearity (Table 1). (All calculations were performed using our own Matlab-based program
`QPMoptics` which is available from the authors.) This systematic and extensive search for setups which allow for generation of intrinsically pure quantum states is—to our knowledge—unprecedented so far. This way we were able not only to confirm the choice of specific crystals in recent experiments but also to find other promising (type 0, I and II) setups that were previously unknown and will be presented in the succeeding subsections. (For the sake of a concise presentation we here omit the discussion of configurations with very low effective non-linearity, namely *o* → *o* + *o*, *e* → *o* + *o* and *o* → *o* + *e* in LT. Moreover, our calculations revealed that there are no opportunities for designing a setup with intrinsically pure daughter states exploiting type I SPDC in ppKTP. Therefore this setup will neither be discussed any further in this paper.)

#### 4.1. Parallel polarisation

In order to generate photon pairs with same polarisation, down-conversions of type 0 and type I can be used. Type 0 SPDC with polarisations *e* → *e* + *e* gives access to the effective nonlinearity coefficient *d*_{eff} = *d*^{33} which carries by far the highest value in all three crystals. Setups which allow for intrinsically pure type 0 SPDC are depicted in Fig. 3. As the plots show, these setups generate photons with signal or/and idler in the mid-infrared regime and are therefore of little relevance due to poor detection efficiencies. However, type I down-conversion (*e* → *o* + *o*) in ppLN, depicted in Fig. 4, does offer configurations of low spectral correlation with wavelengths which are more accessible for today’s detectors.

The plots in Figs. 3 and 4 depict wavelength configurations which are promising regarding spectrally uncorrelated daughter states. Note however that, as an additional requirement for high purity, the laser’s bandwidth and the crystal length have to be matched to each other. The dependence of purity on these parameters varies with the involved wavelengths and has to be evaluated for each individual setup. As an example, Fig. 5 illustrates the function *P*(*τ*, *L*) for the type I down-conversion 750nm → 1200nm + 2000nm in ppLN.

#### 4.2. Orthogonal polarisation

Figure 6 depicts configurations of type II SPDC in ppKTP and ppLN which allow for generation of spectrally uncorrelated daughter photons. It turns out however that for periodically poled lithium niobate the generated photons carry wavelengths which are—in the single photon regime—almost impossible to detect with satisfying efficiency. As seen in the right plot in Fig. 6(a) the required poling periodicity of KTP crystals approaches infinity for a specific set of wavelength configurations. This feature opens opportunities for phase matched *and* intrinsically uncorrelated spectra *without periodic poling* of the crystal. The advantages of this property range from cost savings to higher SPDC efficiencies since the effective non-linear coefficient *d*_{eff} = *d*^{24} in bulk KTP goes up as high as to 3.9 pmV^{−1} [14] compared to 2.3 pmV^{−1} in ppKTP. A depiction of all possible pump- and signal wavelength configurations which do not require periodic poling can be found in Fig. 7. Unfortunately, as the plot indicates, the generated wavelengths lie in areas which are difficult to detect with common avalanche photodiodes (*λ _{s}* = 1000–2200nm,

*λ*= 1800–3200nm). Future developments and upcoming detection technologies, i.e. superconducting nanowire detectors, might soon make them more accessible.

_{i}Photon pairs which carry the same centre wavelength but orthogonal polarisation find many applications in quantum-based experiments. We therefore subjected the crystals to an in-depth investigation to find out about their properties in frequency-degenerate type II down-conversion (*λ _{s}* =

*λ*= 2 ×

_{i}*λ*,

_{p}*o*→

*o*+

*e*). As illustrated by the grey lines in Fig. 6, both crystals, ppKTP and ppLN, offer many opportunities for the case of frequency-degenerate pure quantum states. However, in order to achieve maximum spectral indistinguishability of signal and idler, the daughter fields should not only carry the same centre wavelength but also the same bandwidth. Fig. 8 shows a plot of configurations with phase-matched and pure output states in ppKTP for which

*λ*=

_{s}*λ*= 2 ×

_{i}*λ*. The plot illustrates that signal and idler states are—although pure—in general spectrally distinguishable due to their different bandwidths (compare Fig. 2(c)) which undermines visibilities in interference experiments. However, in the telecom regime (located in the neighbourhood of

_{p}*λ*= 770–800nm) the output spectra turn out to coincide sufficiently for most applications which justifies the choices that were made in recent experiments [5–12]. Maximal spectral indistinguishability can be achieved at the down-conversion 791nm → 2 × 1582nm. In the case of frequency-degenerate SPDC in ppLN we found the point of coinciding wavelengths

_{p}*and*bandwidths to lie at 1755nm → 2 × 3510nm. These wavelengths can be regarded as material constants of the respective non-linear materials.

Given a configuration of wavelengths, polarisations and crystal that in principle allow for generation of intrinsically uncorrelated daughter fields, the pulse duration *τ* and crystal length *L* have to be mutually matched to achieve high purity in the laboratory. Fig. 9 depicts how the purity behaves with respect to *τ* and *L* in two different examples, 791nm → 2 × 1582nm in ppKTP and 650nm → 1011nm + 1820.36nm in bulk KTP.

## 5. Conclusion

We discussed, analytically and numerically, the generation of photon pairs with high spectral purity, circumventing the need for bandpass filtering. We presented a broad variety of SPDC setups that allow for intrinsically pure quantum states most of which were unknown so far. We could confirm the well known property of ppKTP with Λ ∼ 46μm to generate uncorrelated daughter photons with SPDC in the vicinity of 780nm → 2 × 1560nm with the maximal spectral indistinguishability being located at 791nm → 2 × 1582nm. For type II SPDC in ppLN we found the point of maximum spectral purity and indistinguishability to lie at 1755nm → 2 × 3510nm. Moreover, we presented a variety of novel configurations with high intrinsic purity. Among others we discussed how KTP allows for generation of phase-matched and pure daughter fields without periodic poling, which gives rise to significant advantages such as cost savings and higher count rates.

## Acknowledgments

This work was funded by the Austrian Research Promotion Agency (Österreichische Forschungsförderungsgesellschaft, FFG; contract: 4642983, ‘KVQ’). Furthermore, we thank the members of the Philip Walther group at the University of Vienna for technical support and fruitful discussions and advice. P.W. acknowledges support from the European Commission with the projects EQuaM (No. 323714), GRASP (No. 613024), PICQUE (No. 608062), QuILMI (No. 295293) and QUCHIP (No. 641039) as well as the Austrian Science Fund (FWF) with the projects PhoQuSi (Y585-N20) and the doctoral programme CoQuS Complex Quantum Systems (W1210-N16).

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