Experimental High-Dimensional Entanglement by Path Identity

TL;DR

This paper presents a scalable experimental method to generate high-dimensional entangled photon pairs using path identity, enabling advances in quantum communication and fundamental tests with higher-dimensional quantum states.

Contribution

The authors introduce a new scalable approach to produce high-dimensional entangled photons via path identity, adaptable to various degrees of freedom and integrated devices.

Findings

Successfully created three-dimensional entangled states.

Demonstrated increase in entanglement dimensionality while maintaining quality.

Method is adaptable to different degrees of freedom and scalable for integrated devices.

Abstract

Versatile and high-brightness sources of high-dimensional entangled photon pairs are important for emerging quantum technologies such as secure quantum communication. Here, we experimentally demonstrate a new scalable method to create photon pairs carrying orbital angular momentum that are entangled in arbitrarily high dimensions. Our method relies on indistinguishable photon pairs created coherently in different sources. We demonstrate the creation of three-dimensionally entangled states and show how to incrementally increase the dimensionality of entanglement. The generated states retain their quality even in higher dimensions. In addition, the modular structure of our approach allows for generalization to various degrees of freedom and even implementation in integrated compact devices. We therefore expect that future quantum technologies and fundamental tests of nature in higher dimensions will benefit from this novel approach.

Tables

Table 1. Table 1: Fidelities F ​ ( | ψ ⟩ , ρ ) = Tr ​ ( | ψ ⟩ ​ ⟨ ψ | ​ ρ ) 𝐹 ket 𝜓 𝜌 Tr ket 𝜓 bra 𝜓 𝜌 F(|\psi\rangle,\rho)=\text{Tr}(|\psi\rangle\langle\psi|\rho) between several two- and three-dimensionally entangled states | ψ ⟩ ket 𝜓 |\psi\rangle and their experimental realizations ρ 𝜌 \rho . States | ψ 1 ⟩ ket subscript 𝜓 1 |\psi_{1}\rangle , | ψ 2 ⟩ ket subscript 𝜓 2 |\psi_{2}\rangle and | ψ 3 ⟩ ket subscript 𝜓 3 |\psi_{3}\rangle form an orthonormal set of maximally entangled states in three dimensions ( ω = e 2 ​ π ​ i / 3 𝜔 superscript 𝑒 2 𝜋 i 3 \omega=e^{2\pi\mathrm{i}/3} ). State | ψ 5 ⟩ ket subscript 𝜓 5 |\psi_{5}\rangle is a manifestation of our ability to control not only relative phases in the quantum state, but also relative magnitudes.

State	Fidelity $F$
$|{\Phi }^{+}⟩=1/\sqrt{2}\left(|0,0⟩+|2,2⟩\right)$	$0.904\pm 0.005$
$|{\Phi }^{-}⟩=1/\sqrt{2}\left(|0,0⟩-|2,2⟩\right)$	$0.891\pm 0.005$
$|{\psi }_1⟩=\frac{1}{\sqrt{3}}\left(|0,0⟩+|2,2⟩+|-2,-2⟩\right)$	$0.870\pm 0.005$
$|{\psi }_2⟩=\frac{1}{\sqrt{3}}\left(|0,0⟩+\omega |2,2⟩+{\omega }^{-1}|-2,-2⟩\right)$	$0.852\pm 0.007$
$|{\psi }_3⟩=\frac{1}{\sqrt{3}}\left(|0,0⟩+{\omega }^{-1}|2,2⟩+\omega |-2,-2⟩\right)$	$0.903\pm 0.006$
$|{\psi }_4⟩=\frac{1}{\sqrt{3}}\left(|0,0⟩-|2,2⟩-|-2,-2⟩\right)$	$0.890\pm 0.004$
$|{\psi }_5⟩=\frac{1}{\sqrt{22}}\left(2|0,0⟩+3|2,2⟩+3|-2,-2⟩\right)$	$0.848\pm 0.008$

Equations

$$|\psi ⟩=\frac{1}{\sqrt{2}}(|0,0⟩+e^{i\phi }|1,1⟩).$$

$$|\psi ⟩=\sum _{\ell =0}^{d-1}{c}_{\ell }|\ell ,\ell ⟩,$$

$$|\psi ⟩=\alpha \underset{\text{crystal A}}{\underbrace{|0,0⟩}}+\beta e^{i{\phi }_1}\underset{\text{crystal B}}{\underbrace{|2,2⟩}}+\gamma e^{i{\phi }_2}\underset{\text{crystal C}}{\underbrace{|-2,-2⟩}}.$$

Full text

Introduction.—The transition from two- to multi-dimensional entangled quantum systems brings about radical improvements in the distribution and processing of quantum information. Such systems play an important role in secure communication over high capacity quantum channels Hu et al. (2018). They offer improved noise resistance and increased security against eavesdropping Huber and Pawłowski (2013); Cerf et al. (2002). Furthermore, multi-dimensionally entangled quantum systems are beneficial for experiments that are interesting from a more fundamental perspective, such as tests of local realism Collins et al. (2002); Ryu et al. (2013); Lawrence (2014); Vértesi et al. (2010). If we think of teleporting the entire information of a single or even several photons in the future Wang et al. (2015); Malik et al. (2016); Erhard et al. (2018a), such multi-dimensionally entangled sources will be indispensable. In view of these perspectives and applications, high quality sources for multi-dimensional entanglement are highly desirable.

Various degrees of freedom, such as frequency Kues et al. (2017), time-bin Thew et al. (2004); de Riedmatten et al. (2004); Tiranov et al. (2017) and path Schaeff et al. (2015); Wang et al. (2018), have been employed so far for the generation of high-dimensionally entangled states. In the following, we focus on the orbital angular momentum (OAM) of photons, but our technique is valid for other degrees of freedom as well. The OAM of photons is an in principle unbounded discrete quantity and as such has been used extensively Allen et al. (1992); Molina-Terriza et al. (2007); Rubinsztein-Dunlop et al. (2017); Padgett (2017); Erhard et al. (2018b) to prepare high-dimensionally entangled photonic states. In the traditional way, the OAM-entangled photon pairs are produced in a single spontaneous parametric down-conversion (SPDC) process Mair et al. (2001). Albeit convenient, this process exhibits several drawbacks. For example, photon pairs generated in this way have a non-uniform distribution of OAM Miatto et al. (2011); Dada et al. (2011); Romero et al. (2012); Krenn et al. (2014). The maximally entangled states can then be generated either by post-processing techniques, such as Procrustean filtering Bennett et al. (1995); Vaziri et al. (2003), or by pre-processing of the pump beam. In a recently demonstrated approach Kovlakov et al. (2018); Liu et al. (2018), a superposition of OAM modes is imprinted by holograms into the pump beam, which translates via down-conversion into maximally entangled states of two photons.

In this paper we present the first experimental demonstration of a new method of generating high-dimensionally entangled states. Multiple SPDC processes are employed, but none of them directly produces entangled states. The method relies on the concept known as entanglement by path identity Krenn et al. (2017a, b), which was discovered recently with the help of a computer program Krenn et al. (2016). Our technique displays several important advantages over the traditional approach. Our source of entangled photon pairs is versatile as both magnitudes and phases in a high-dimensional quantum state can be adjusted completely arbitrarily. Furthermore, the experimental implementation of our source has a modular structure. The modularity, together with the fact that adding one dimension to the entangled state increases the count rates accordingly, makes our approach scalable. High brightness of our source is ensured as all photons are produced already in the desired modes and no photons have to be discarded by post-selection.

This work is organized as follows. After a brief introduction to the concept of entanglement by path identity, we describe the main features of our experimental implementation. Then we demonstrate the scalability and versatility of our method by generating several different states in two and three dimensions. We verify the quality of the produced entangled quantum states using quantum state tomography.

Entanglement by path identity.—Consider a simple experimental setup consisting of two nonlinear crystals that are aligned in series and coherently emit photons via SPDC, as shown in Fig. 1(a) and (b). The pump power for both crystals is set sufficiently low such that events when either crystal emits multiple photon-pairs as well as events when both crystals simultaneously generate a photon-pair can be neglected. The propagation paths of the down-converted photons coming from the two crystals are carefully overlapped. As a result, once the photon pair leaves the setup, no information can be obtained, not even in principle, in which crystal the pair was created Wang et al. (1991); Herzog et al. (1994); Weinfurter et al. (1995). The down-conversion processes in both crystals are adjusted such that photon pairs may be emitted only into the fundamental mode $|0,0\rangle$ with zero quanta of OAM 111Parameters of the two SPDC processes are chosen such that photon properties such as frequency, polarization, and OAM are identical for both crystals and also higher-order OAM modes are highly suppressed.. Importantly, no entanglement is generated by either of the two crystals. (In practice, a small contribution of higher-order OAM modes is also present. For the detailed discussion see Supplementary sup .)

Suppose now that two mode-shifters are inserted into the setup. These add an extra quantum of OAM to each photon originating in the first crystal and thus act as the only source of which-crystal information. As the down-conversion processes in the two crystals are (apart from the OAM) indistinguishable, the resulting state of a detected photon pair is a coherent superposition

[TABLE]

In the formula above, $\varphi$ is the phase between the two SPDC processes imparted by a phase-shifter and numbers in ket vectors refer to the OAM quanta of respective photons.

The generation of entangled states as described above is a specific example of the concept termed entanglement by path identity. This concept can be readily generalized for production of high-dimensionally entangled states Krenn et al. (2017a). When the number of crystals in the series is increased to $d$, and the number of phase- and mode-shifters is accordingly increased to $d-1$, high-dimensionally entangled states of the following form are produced

[TABLE]

where $d$ is the state dimension and $c_{\ell}$ are complex amplitudes, see Fig. 1(c) and (d). The magnitudes of $c_{\ell}$ can be set by pumping each crystal independently with properly adjusted power. By using different mode-shifters for either of the two photons in a down-converted pair, completely arbitrary states can be created. Interestingly, the widely used cross-crystal scheme is the simplest example of the above approach, where two-particle states are entangled in polarization Hardy (1992); Kwiat et al. (1999); Zhong et al. (2018).

Setup.—The experimental implementation presented here is based on the scheme in Fig. 1(c) with two main modifications. First, the pump and down-converted beams for each crystal are separated by two Mach-Zehnder interferometers. This way, phases as well as magnitudes of individual modes in the quantum state can be adjusted independently. In addition, there is an advantage that both wavelengths can be manipulated separately. Second, the mode-shifter is placed into the pump beam instead of the down-conversion beam. This measure was taken to preclude the mode-shifter from introducing undesirable OAM terms into the quantum state. For technical reasons, the down-converted photon pairs were not emitted in a perfectly collinear manner, but had a slight angular spread of roughly $1^{\circ}$. This leads to a non-perfect operation of the mode-shifter, which functions properly only when both photons propagate through its center (for details see [SeeSupplementalMaterialforthedetaileddescriptionoftheexperimentalsetup; spiralspectraofnonlinearcrystalsandcompletestatetomographydata]supp).

The setup, presented in Fig. 2, was designed to produce three-dimensionally entangled states. Each dimension in the generated quantum state corresponds to one of three nonlinear crystals A, B or C in the setup. In Fig. 2 this correspondence is emphasized by enclosing the crystals with associated elements into boxes 1st dim, 2nd dim and 3rd dim. The laser beam is split into three paths to pump each crystal separately. The pump beam for crystal A possesses zero quanta of OAM and so do the down-converted photons sup , which exit the crystal in state $|0,0\rangle$. (Apart from the predominant $|0,0\rangle$ component, also small but negligible contributions of higher-order OAM correlations are present in the photons’ state. For details see Supplementary sup .) The pump beam for crystal B acquires four quanta of OAM due to a spiral phase plate (SPP), which is inserted into the beam and plays the role of the mode-shifter. Consequently, each down-converted photon generated in crystal B carries two quanta of OAM and the pair is produced in state $|2,2\rangle$. Similarly, the pump beam for crystal C also acquires four quanta of OAM, but an additional mirror is used to invert the sign of the OAM value from $4$ to $-4$, effectively subtracting eight quanta of OAM. Down-converted photons coming from crystal C are then produced in state $|-2,-2\rangle$. The resulting quantum state reads

[TABLE]

Magnitudes $\alpha$, $\beta$ and $\gamma$ of the entangled state can be changed by adjusting the relative pump power for each crystal. The relative phases $\varphi_{1}$ and $\varphi_{2}$ are set by positioning two trombone systems that act as phase-shifters. By employing only the first two stages of the setup, namely parts in boxes 1st dim and 2nd dim, two-dimensionally entangled states are created.

We use type II SPDC in all three crystals. In order to measure the entangled state, we first deterministically separate the two down-converted photons by a polarizing beam splitter. Two spatial light modulators in combination with single mode fibers are used to perform any projective measurement for OAM modes Mair et al. (2001). The single photons are then detected by avalanche photon detectors and simultaneous two-photon events are identified by a coincidence logic.

Finally, the resulting quantum states are characterized by complete quantum state tomography. We use a maximum-likelihood reconstruction technique Hradil (1997) to estimate the physical density matrices of the detected photon pairs.

Experimental Results.—The high flexibility of our setup in producing various states is demonstrated in Tab. 1, where fidelities for different three-dimensionally (and also two-dimensionally) entangled states are presented. These data demonstrate our ability to control the relative phases and magnitudes of the generated quantum states. Most notably, we are able to create three mutually orthogonal and maximally entangled states $|\psi_{1}\rangle$, $|\psi_{2}\rangle$ and $|\psi_{3}\rangle$ with an average fidelity of $87.5\pm 0.6\%$. The orthogonality of these states does not follow directly from the orthogonality of OAM modes, but indeed from differently adjusted phases in the quantum states. With the state $|\psi_{5}\rangle$ we demonstrate the ability to adjust relative magnitudes of terms in the quantum superposition. We do so by slightly reducing the probability amplitude for the zero OAM mode. The real parts of density matrices for three of the states presented in Tab. 1 are displayed in Fig. 3. There the measurement results (solid bars) are compared to the theoretical expectations (translucent bars). The average fidelity of three-dimensionally entangled states does not decrease significantly when compared to the average fidelity of $89.8\pm 0.5\%$ for two-dimensional states sup . The quality of entangled states is thus mostly unaffected when going from two to three dimensions and indicates the scalability of our approach for even higher dimensions. The main limitations for achieving higher coherence are slight distinguishability of the SPDC sources, which we attribute to small misalignments and imperfect transformations of OAM modes. However, none of these imperfections are of fundamental nature and we discuss possible improvements of our design in the following section.

Scalability.—The scalability of our scheme is enabled by the modular structure of the experimental setup. Adding a crystal and a mode-shifter results in an increase of the entanglement dimension by one. In order to further improve the scalability, some modifications to our experimental implementation can be made. We adopted the Mach-Zehnder interferometric configuration in our experiment. This gives us freedom to access and manipulate the pump and down-conversion beams separately with no need of custom-made components. Currently, the size of the setup is governed by the two 4f lens systems between the crystals. The distance between two successive crystals in our setup is 600 mm. Due to these large interferometers active stabilization is inevitable. However, scaling down the distances and employing integrated fabrication techniques as used in micro-chip fabrication leads to intrinsically stable interferometers. An alternative approach is to circumvent interferometers completely. For example, by utilizing wavelength-dependent phase-shifters and q-plates Marrucci et al. (2006). This approach leaves the pump beam unaffected while manipulating only the down-conversion beam Yan et al. (2015). Thus the pump beam can co-propagate along the same path as the down-conversion photons leading to intrinsic phase stability.

Conclusion.—In conclusion, we experimentally demonstrated entanglement by path identity Krenn et al. (2017a) as a new method of generating high-dimensionally entangled states. The OAM degree of freedom of photons was utilized in our setup. We showed the flexibility of our approach by producing various entangled quantum states in two and in three dimensions. Both magnitudes and phases of complex amplitudes in the quantum states can be adjusted to arbitrary values independently of each other. The experimental setup has a modular structure and in principle allows to generate entangled states in arbitrarily high dimensions.

The modularity of the setup and the fact that only three kinds of elements—crystals, mode- and phase-shifters—are needed is the core strength of our approach. It promises high brightness, scalability in the dimension, and versatility of generated states. A very appealing feature of our method is that different families of spatial modes can be utilized. It is therefore possible to generate high-dimensionally entangled photon pairs in specialized modes, which are optimized, for example, for free-space communication or even for fiber-based systems.

This work was supported by the Austrian Academy of Sciences (OeAW), the European Research Council (SIQS Grant No. 600645 EU-FP7-ICT), and the Austrian Science Fund (FWF) with SFB F40 (FoQuS) and FWF project W 1210-N25 (CoQuS).

J. K. and M. E. contributed equally to this work.