Time-bin entanglement in the deterministic generation of linear photonic cluster states Open Access

Quantum information theory and processing are poised to fundamentally transform computational paradigms, surpassing the limitations of classical systems.¹ Central to this quantum leap is the development of reliable and deterministic photon emitters,^2–17 which are essential for the encoding, transmission, and manipulation of quantum information in measurement-based quantum computing^18,19 and quantum communication.^20–27 Multi-photon states with significant quantum correlations or entanglement are a critical ingredient for numerous quantum protocols,^22,28,29 and the quest for their robust generation forms a challenging field of research.^30,31 In addition to polarization^28,32 and other degrees of freedom,^33–36 quantum correlations can be encoded in the form of time-bin entanglement^37–41 of photonic qubits,^38,42 which holds great potential for applications in quantum teleportation and quantum key distribution.^39,43,44

In the present work, we delve into the generation of time-bin entangled photonic states with a Λ-type electronic system. The system enables the individual excitation of the system’s ground states to corresponding excited states to produce photons, while the coherent superposition of the ground states can be controlled through optical driving of the lambda transitions. This approach is quite broadly applicable to systems exhibiting lambda configurations, including, but not limited to, color centers,^43,45–47 stacked quantum dots,^48,49 and quantum dot molecules,^50,51 commonly using trion-based spin-hole qubits.^43,52 Entanglement between individual time bins is achieved by alternating photon emission and ground state rotations, which, in theory, allows for the generation of complex cluster and graph states.^53,54 In the present work we go beyond the development of a schematic protocol and actually implement, benchmark, and optimize a specific excitation scheme. We explicitly assess quantum correlations of the emitted photons via time-shifted second- and third-order correlation functions in advanced numerical simulations. This rigorous analysis allows us to estimate the lower bounds of time-bin entanglement by calculating expectation values of stabilizer generators.³⁹ We also explore photon indistinguishability, uncovering the capability to generate sequences of photonic qubits with robust time-bin entanglement and high indistinguishability, even from lossy quantum emitters.^55,56 In comparison with our analysis of photon correlation functions, we demonstrate that gate fidelities are generally not a sufficient measure of system performance. With losses, even for almost perfect gate fidelities and excitation–emission protocol implementation, quantum correlations in the generated photon emission can in fact be very low. Our microscopic theory includes dynamical light-field-induced shifts and different types of losses, which leads to predictive capabilities for realistic systems and structures. The present work not only advances our understanding of photonic entanglement generation but also reinforces the foundational techniques necessary for the next generation of quantum information processing.

The design of the emitter structure is pivotal, as it must exhibit a Λ-shaped energy configuration of the underlying electronic states. Such a configuration enables us to achieve dual objectives: first, to excite individual electronic states, leading to the emission of photons via radiative decay; and second, to infuse the emitted photons with the desired quantum phase information, a direct consequence of the coherences between the states within the Λ system, which is schematically shown in Fig. 1. The capability to instill coherence of the quantum state is essential for the generation of photon entanglement. The temporal entanglement of photonic qubits, as produced by this structure, plays a foundational role in a wide array of quantum protocols, and thus constitutes a key mechanism for advancing the field of quantum information science. For example, specific types of quantum dot molecules are known to exhibit such energy structure, making them a prime solid-state candidate for realizing the Λ-type configurations required.

Accounting for the ground state splitting, the resonant pulse frequencies are ω and ω − ω_GS, respectively. Since the coupling ratio is not an explicit part of our investigation, we standardize the pulse coupling for each transition to a value of one, acknowledging that selecting other couplings high-fidelity rotations can also be achieved, albeit at different detuning values.

We simulate the temporal dynamics of the system using the von Neuman equation, $∂ρ∂t=−iℏ[H,ρ]+∑ÔγO22ÔρÔ†−Ô†Ôρ−ρÔ†Ô$⁠, with Lindblad-type losses, where $Ô$ denotes any of the available system state operators and γ_O is the corresponding loss rate. The density matrix ρ in matrix representation is propagated in time using explicit numerical integration methods and using the Hamiltonian in Eq. (A3). We distinguish between the following decays or loss channels: the radiative decay of the cycling transition (γ_Cyc), the decay of both the target and unwanted state population $(γT,UR)$⁠, and the spin dephasing rate of the ground states $(γT,UD)$⁠. Unless specified otherwise, we assume that both $γT,UD$ and $γT,UR$ are set to zero.

Figure 4 shows that the choice of detuning is arbitrary for the resulting phase dependency, as we can freely chose other pulse phases to compensate for different choices of pulse detunings. We find that high fidelity π/2 rotations are always possible, while high fidelity π rotations are limited by system losses.

It is also noteworthy that small dephasing rates of the transition states have a marginal negative impact on the rotation fidelities. Conversely, even relatively low decay rates of the target and unwanted states considerably diminish the rotation fidelities. Previous research has corroborated these findings, although it did not assess the translation of rotation fidelities into quantum correlations in the emitted photonic states, an analysis we undertake in the following with the present study.

Time bins serve as a fundamental concept in our study, essentially acting as discrete temporal slots into which photons can be emitted. These bins are critical for encoding quantum information, where the presence or absence of a photon in a specific time slot represents a logical 1 or 0. We define early and late time bins to create a pair that forms a single photonic qubit. The early time bin, denoted as |early⟩ = |1_τ=i0_τ=i+1⟩ = |0⟩, corresponds to the emission of a photon in the initial time slot, whereas the late time bin, |late⟩ = |0_τ=i1_τ=i+1⟩ = |1⟩, represents photon emission in the subsequent time slot. This binary representation allows us to encode and manipulate quantum information across a sequence of time bins, effectively creating a framework for generating complex quantum states. It is crucial for the photons of these time bins to demonstrate particular phase relationships to ensure the quantum correlations are high. We use the notation R_ϑ(Θ) to denote rotations in the Bloch sphere of the two ground states, with ϑ being the angle in the x–y plane and Θ being the angle in the z direction. As an explicit example, in the present study, we limit our investigation to the following excitation–emission scheme to generate a linear cluster state:^50,59,60

• Start or prepare the system in $|Ψ⟩=12|G1⟩+|G2⟩$

• Step 1: drive Ω_Cyc using a π pulse, followed by emission of first photon.

• Step 2: perform R_ϑ(π) through the Λ system.

• Step 3: drive Ω_Cyc using a π pulse, followed by emission of the second photon.

• Step 4: perform R_ϑ(π/2) through the Λ system.

• Repeat steps 1–4 N times.

• Perform a final Z projection to decouple the emitter.

Steps 1–4 are repeated N times to create an N qubit linear cluster state. The final step, involving the removal of the emitter from the cluster state, is not always necessary and depends on the specific desired state.⁵³ The photon emission occurs through spontaneous radiative emission from the corresponding excited state. The x–y angle ϑ is tunable by selecting appropriate pulse phases, while Θ can be selected varying the pulse amplitudes or detunings. We consistently select a bin length that accurately encompasses the emitted photons, in accordance with γ_Cyc. For a radiative decay rate of γ_Cyc = 1.2 µeV (≈2 GHz), we find that a time bin length of T = 4 ns suits the emission scheme well as it incorporates both the emission process as well as a time buffer to apply the rotation pulses. In addition, we choose rotation pulses of the highest fidelity, ensuring that the pulse bandwidth is appropriately matched to the selected splittings. In Fig. 2(b), the emission of up to three photons is shown; it is important to note although the system or protocol is not constrained to this particular number of photons.

We observe that the electronic coherences disappear immediately upon the application of the π-extraction pulse through the cycling transition, provided the system is in an equal superposition of both ground states. Further Θ = π or Θ = π/2 rotations through the Λ system are then not visible within the electronic coherences and, in an ideal case, leave the ground state population unchanged. For this reason, these pulses are referred to as transitionless.⁴³ This phenomenon is by design, as it facilitates the transfer of coherences from the electronic states onto the emitted photons.

Although our research focuses on the previously described scheme, it is feasible to adopt alternate excitation–emission protocols by adjusting the rotation pulses and with expanding the present Λ system accordingly. Such modifications can then lead to the generation of higher-order cluster,^8,61 Greenberger–Horne–Zeilinger (GHZ),^62–66 or graph states,⁶⁷ expanding the scope of potential quantum states achievable. Moreover, while our focus is on time-bin entanglement, the adoption of polarization entanglement is conceivable,⁵⁴ albeit with the trade-off of potentially constraining the diversity of quantum states that can be generated. Furthermore, the approach to measuring and characterizing cluster states significantly influences their classification. Specifically, the “horseshoe” configuration presents an interesting case within the broader category of 1D linear cluster states. This distinction highlights the complexity in cluster state classifications and emphasizes the nuanced understanding required to fully leverage their potential in quantum information science .⁶⁰ 

The expectation values $⟨X̂(1)Ẑ(2)⟩$ and $⟨Ŷ(1)Ẑ(2)⟩$⁠, as well as $⟨Ẑ(1)X̂(2)Ẑ(3)⟩$ and $⟨Ẑ(1)Ŷ(2)Ẑ(3)⟩$ can be obtained by taking the real or imaginary part in Eqs. (7) and (8), respectively. This corresponds to setting φ = 0 (φ = π/2) in the $Φ̂$ operator. Later generator expectation values can be calculated by shifting t₁ appropriately, indicated by the superscript indices. It should be noted that to classify other graph or cluster states, one may need to evaluate different stabilizer expressions.⁶² Using optimal detunings for π and π/2 rotations, we carry out explicit simulations of the scheme outlined in Sec. III, as shown in Fig. 2(b). We adjust the relative phases Θ_i of the pulses that interact with each ground state, effectively tuning the rotation angle ϑ in the x–y plane. The phase dependency observed from these adjustments is shown in Fig. 4. Importantly, the magnitude of $⟨Φ̂(1)Ẑ(2)⟩$ and $⟨Ẑ(1)Φ̂(2)Ẑ(3)⟩$ consistently remains close to unity, with neither value falling below 0.95 for either property. This suggests that the chosen phase of the rotation pulses predominantly influences the phase of the stabilizer generator expectation values rather than their magnitudes. Furthermore, it is important to recognize that the magnitudes approach unity exclusively under conditions of ideal pulses and for a lossless system. In scenarios deviating from these ideals, the magnitudes can drop significantly below one. Based on the data presented in Fig. 4, we deduce that it is feasible to adjust the rotation angle to optimize either the real or the imaginary component of the associated stabilizer generator expectation values, without compromising the overall magnitude of these values, as long as all rotations to generate the cluster state occur in the same plane. In essence, it is essential for all rotation pulses to be phase locked, although the specific phase value itself is inconsequential.

We now explore how rotation fidelities directly influence the quantum characteristics of the generated N-photon states, specified by the stabilizer generator expectation values. Given that it is always possible to identify a pulse phase optimizing both characteristics simultaneously, we select an arbitrary phase for our analysis, focusing solely on the magnitude of these properties. Initially, we introduce additional radiative loss exclusively to the transition states, previously set to zero, and adjust it from zero to 2.5 µeV. The radiative loss rate for the cycling transition is kept constant. In Fig. 5, we observe that increased radiative losses in the transition state correspond to a direct decrease in the magnitudes of the stabilizers, exhibiting an almost linear relationship for the losses shown. Subsequently, we apply spin dephasing to the ground state coherences, again with rates ranging from zero to 2.5 µeV. We note that these values lead to dephasing times significantly exceeding the typical achievable values for spin qubits⁷² and provide a solid estimation for the lower bound for the achievable entanglement. In this scenario, we notice a rapid exponential decline in the stabilizer values even for relatively small dephasing rates. It is important to note that this detrimental effect of the ground state dephasing is not apparent in the temporal dynamics of the system, which for small loss values closely resembles the zero-dephasing behavior shown in Fig. 2(b). It is only for losses exceeding 3 µeV that a noticeable difference also becomes visible for the system dynamics.

It becomes evident that although the rotation fidelity somewhat diminishes with increasing losses, the nonlinear decay of quantum correlations with increasing spin dephasing does only emerge in a detailed analysis of the quantum state properties through stabilizer generators. Note that we find that the selection of an optimal bin length is also influenced by the system’s loss parameters (not shown), with systems experiencing higher levels of loss potentially benefiting from shorter time bins, which helps mitigate the adverse effects of dephasing over extended periods of time, optimizing the overall performance of the system under such conditions.

In Fig. 6, we calculate stabilizer generators for up to the 15th expression. Notably, these stabilizers maintain a consistent behavior across a spectrum of loss conditions, meaning that all stabilizers remain close to their mean value. In other words, the standard deviation of all the stabilizers of a given linear cluster state is small. This consistency is crucial, as it demonstrates the potential for creating large cluster states; diminishing effects from losses do not appear to accumulate through the protocol cycle. The stabilizers remain constant because the loss between the first and second Z projections within each stabilizer expression is consistent, making the starting time of the stabilizer evaluation insignificant. This stability plays a critical role in the practicality of developing complex cluster and graph states, even within lossy environments. In addition, it proves advantageous for the numerical analysis that only the first three stabilizer generator expectation values need to be assessed, since due to their local nature, the subsequent values resemble the initial ones almost exactly.

Although not visualized in this work, our simulations reveal the indistinguishability of the single photons generated in the system to be inherently high, a direct consequence of the photon emission originating from a modified two-level system.^73,74 Consequently, the indistinguishability is lowered for non-zero spin dephasing of the ground states, again closely mirroring an isolated two level system. This resemblance of a two-level system ensures that each emitted photon possesses uniform phase and spectral characteristics, critical factors that enhance their indistinguishability.⁷⁵ In quantum information processing, the ability to produce highly indistinguishable photons is paramount, as it facilitates efficient quantum interference, a cornerstone for quantum computing algorithms and photonic quantum information protocols. When dealing with indistinguishable photons, it is also possible to expand the foundational cluster state into multidimensional states through specific measurements.¹⁸ For the case with strong radiative losses and dephasing shown in Fig. 6, the indistinguishability and single-photon purity of the qubit states each stay above 95%, approaching unity values when losses approach zero.

Looking ahead, an intriguing direction for further enhancement of the photonic qubit generation scheme presented would involve the incorporation of optical cavities for the cycling transition. While this approach has not yet been implemented in our current study, it would significantly reduce the lifetime of the excited states.³⁶ This would increase the repetition frequency of photon emission leading to increased source brightness and reducing the required time-bin length, mitigating the undesired influence of spin dephasing of the ground states. This could be instrumental in achieving scalability of the investigated scheme.

We conducted an in-depth theoretical study of the generation of time-bin entangled photons using a Λ-type electronic system, relevant for various physical realizations of emitters. Specifically, we focused our research on a system present in a quantum dot molecule composed of two individual quantum dot emitters, resulting in a deterministic source. Our microscopic numerical investigation reveals that conventional metrics used to analyze excitation–emission protocols, such as fidelities of quantum-state rotations,⁶⁸ fall short of fully assessing the usefulness of the correlated quantum states generated, especially when faced with imperfections and losses in a realistic setting. Even when high rotation fidelities are achieved, quantum correlations generated can be surprisingly low, diminishing the usefulness of the generated quantum state. In the present study, we use expectation values of stabilizer generators as a more robust measure of correlations in a multi-photon quantum state. These include full sensitivity to dephasing and losses, providing a more nuanced understanding of the system’s quantum dynamics and usefulness as a quantum resource. For the generation of linear photonic cluster states, we find that even minor rates of spin dephasing (that hardly affect protocol fidelities) can result in a very significant reduction in stabilizer generator magnitudes and, with it, reduction of quantum correlations.

Combining these insights, we explicitly numerically demonstrate the generation of large linear photonic cluster states comprised of a two-digit number of photonic qubits. These results require optimized timing and phases of laser pulses used for excitation and gating. Notably, maximizing the rotation fidelities toward unity values, where precision is crucial, appears as a requirement for the generation of large numbers of entangled qubits. In addition to high degrees of quantum correlations achieved, photons emitted in our calculations also show inherently high degrees of indistinguishability due to the emission resulting from an effective two-level system. High photon indistinguishability is a property essential for subsequent quantum interference measurements. Further increase in quantum correlations for given loss parameters could potentially be achieved by reducing lengths of time bins, for example, by the use of optical cavities for the cycling transitions, improving scalability of the generation of correlated multi-photon states for quantum information processing purposes. Furthermore, by calculating even higher order correlation functions, the evaluation of different, more complicated cluster states can be done, albeit with significantly increased numerical effort.

This work was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) through the transregional collaborative research center TRR142/3-2022 (231447078, Project C09), the Photonic Quantum Computing initiative (PhoQC) of the state ministry (Ministerium für Kultur und Wissenschaft des Landes Nordrhein-Westfalen), and with the computing time provided by the Paderborn Center for Parallel Computing, PC². We acknowledge the fruitful discussions with Stefan Schulz and Gediminas Juska from Tyndall National Institute and with Zahra Raissi from Paderborn University.

The authors have no conflicts to disclose.

David Bauch: Conceptualization (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Nikolas Köcher: Conceptualization (equal); Investigation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Nils Heinisch: Conceptualization (supporting). Stefan Schumacher: Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Time-bin entanglement in the deterministic generation of linear photonic cluster, at https://zenodo.org/records/10809130, reference number 10.5281/zenodo.10809130.

We utilize the well-established second- and third-order correlation functions, denoted as $G(2)$ and $G(3)$⁠, respectively. These functions are essential for understanding the dynamics of time-bin entangled photon pairs within a quantum system. By incorporating the bin length T into our calculations, we adapt these correlation functions to effectively capture the temporal shifts in photon emissions. The evaluation of $G(2)$ and $G(3)$ necessitates integration over time variables t₁, t₂, and additionally t₃ for $G(3)$⁠. The integration intervals are set by the time bin length T. To ensure accuracy and convergence of our results, we compute these correlation functions over a finely resolved grid, set to 240 × 240(×240) grid points for all the presented outcomes. This high resolution allows for a detailed examination of the quantum correlations present within our system.

For illustrative purposes, we provide examples for computing specific correlation functions.

The datasets employed for the visualizations presented in this study can be accessed via Ref. 76, while the code utilized to generate the data are accessible through Ref. 77.