A robust approach for time-bin encoded photonic quantum information protocols
Simon J. U. White Emanuele Polino Farzad Ghafari Dominick J. Joch Luis Villegas-Aguilar Lynden K. Shalm Varun B. Verma Marcus Huber Nora Tischler




These authors contributed equally to this work




e.polino@griffith.edu.au



f.ghafari@griffith.edu.au





Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology  Griffith University   Yuggera Country    Brisbane  Queensland 4111 Australia






National Institute of Standards and Technology  325 Broadway  Boulder Colorado 80305 USA


Atominstitut Technische Universität Wien Stadionallee 2 1020 Vienna Austria
 Institute for Quantum Optics and Quantum Information (IQOQI) 
Austrian Academy of Sciences Boltzmanngasse 3  1090 Vienna  Austria



n.tischler@griffith.edu.au

Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology  Griffith University   Yuggera Country    Brisbane  Queensland 4111 Australia







Quantum states encoded in the time-bin degree of freedom of photons represent a fundamental resource for quantum information protocols. Traditional methods for generating and measuring time-bin encoded quantum states face severe challenges due to optical instabilities, complex setups, and timing resolution requirements. Here, we leverage a robust approach based on Hong-Ou-Mandel interference that allows us to circumvent these issues. First, we perform high-fidelity quantum state tomographies of time-bin qubits with a short temporal separation. Then, we certify intrasystem polarization-time entanglement of single photons through a nonclassicality test. Finally, we propose a robust and scalable protocol to generate and measure high-dimensional time-bin quantum states in a single spatial mode. The protocol promises to enable access to high-dimensional states and tasks that are practically inaccessible with standard schemes, thereby advancing fundamental quantum information science and opening applications in quantum communication. 


A robust approach for time-bin encoded photonic quantum information protocols
    Nora Tischler
    April 29, 2024
=============================================================================





Photonic quantum technologies rely on the capability to manipulate and measure quantum states of photons <cit.>.
Among the variety of photonic degrees of freedom that can be used to encode quantum states, one of the most noise-resilient and reliable is time. This degree of freedom was first proposed by Franson to test quantum nonlocality <cit.>. Here, quantum states can be encoded using the arrival time of photons, i.e. time bins, which offer an infinite Hilbert space to encode information in high-dimensional states <cit.>. 
Time-bin states have been shown to be highly robust against turbulence and are suitable for long-distance transmissions in fibers <cit.>, free-space interconnects <cit.>, and ground-satellite links <cit.>.
 These properties make time-bin encoding a promising choice for quantum communication
<cit.> and may be the foundation for the future quantum internet <cit.>.

An increasingly important challenge is the robust and efficient generation and measurement of time-bin encoded states. To date, the most common techniques are based on unbalanced interferometers followed by time-resolved detection <cit.>, using post-selection for the recombination after an interferometer, or active switching protocols <cit.>. Unfortunately, interferometric techniques can suffer from severe limitations, where large differences in interferometer arm lengths are required for detectors to resolve neighboring time bins. 
Although state-of-the-art superconducting single-photon detectors and electronics can reach a jitter and time resolution on the order of a few picoseconds <cit.>, standard technology and electronics available to most laboratories limit the overall time resolution to the order of nanoseconds. Consequently, the use of large unbalanced interferometers is often practically unavoidable. This leads to poor scalability in optical setups due to phase instabilities during long integration measurements, and impractical setups for high-dimensional states (requiring larger interferometers exacerbates these difficulties).
Other time-bin measurement techniques do exist, such as those based on nonlinear optical interactions, including pulse-shaping in conjunction with sum-frequency generation <cit.>. Alternatively, using both optical nonlinearity and interferometers allows one to resolve short time delays <cit.>. 
These approaches, however, come at the cost of increased complexity and expense, limiting their suitability for most applications.
To avoid the use of unbalanced interferometers, frequency-based techniques can be employed, but they only allow for particular measurements in the time-bin space, such as mutually unbiased bases <cit.>.
As a result, the practical drawbacks of standard measurement techniques pose severe limitations on the robustness, feasibility, and accessible Hilbert space dimensionality of time-bin states used in real experiments.

In this work, we employ a measurement scheme based on Hong-Ou-Mandel (HOM) interference <cit.> that overcomes the intrinsic limitations present in standard approaches and enables different tasks.
The HOM effect is based on the quantum interference between indistinguishable photons and is a building block for realizing interaction between photons in multi-photon protocols <cit.>. 
Here, we apply the HOM measurement scheme to harness time-bin quantum encoding in a robust way: That is, the scheme enables short temporal delays, manipulated by reliable, versatile, and low-noise setups. This allows us to advance time-bin based photonic information protocols in three core directions. 


First, we provide an optical experiment, using time-bin qubits generated by intrinsically stable unbalanced interferometers. 
We demonstrate that the measurement approach is reliable and can be used to perform high-quality quantum-state tomography with an average fidelity between reconstructed and target states of ⟨ F ⟩=99.55±0.07%. Next, we use the versatility of the protocol to perform a noncontextuality test for the intrasystem entanglement between the polarization and time degrees of freedom of single photons. Violating a Bell-like inequality, we unambiguously demonstrate the presence of entanglement between these degrees of freedom. 
Finally, we propose a scheme, based on quantum walk dynamics, to generate high-dimensional time-bin quantum states. The proposed scheme allows for arbitrary measurements, which are increasingly difficult in high dimensions, enabling high-dimensional quantum communication protocols and nonlocality tests.


By improving the robustness, versatility, and accessible dimensionality of time-bin encoded states, this work represents an advance toward scalable quantum technology protocols.





Measurement scheme
The goal here is to use HOM interference to perform arbitrary projective measurements on an unknown target time-bin encoded quantum state |Ψ_target⟩. 
Suppose the temporal wavefunction of a reference state is prepared in the state |Φ_ref⟩. Then, one performs a HOM experiment (see Fig. <ref>), interfering the two photons on a beam splitter (BS). The probability P_ab of measuring a coincidence event between the two outputs of the BS (anti-bunching), given that the photons are indistinguishable in all other degrees of freedom besides time and path, will depend on their overlap <cit.>, which in the case of pure reference state reads:

    P_ab= 1- | ⟨Φ_ref|Ψ_target⟩|^2/2  .


One can realize arbitrary projections on a target (pure or mixed) photon by preparing a suitable (pure) reference photon and performing a HOM interference experiment. This kind of measurement is general and can be applied, in principle, to any photonic degree of freedom (for polarization and hybrid polarization-time states see Refs. <cit.>). 
Furthermore, different kinds of light sources can be used as a reference <cit.>.

The minimum delay required in principle to resolve two time bins is limited only by the coherence length of the employed single photons. This minimal delay maximizes the robustness against noise during preparation, transmission, and measurement.

Although we focus on time-bin qubits in the first part of this work, the scheme can be directly generalized to arbitrary high-dimensional states. This is because relation (<ref>)  provides information on the overlap between photonic states, independently of their dimensionality. Hence, given the ability to prepare arbitrary reference states, the measurement scheme applies to any time-bin encoded quantum state (qudit).


Measurements of time-bin qubits 
 To verify the experimental performance of the scheme depicted in Fig. <ref>, we carry out an experiment using single photons (see Fig. <ref>). 
A pair of photons is generated using spontaneous parametric down-conversion (SPCDC) by a pulse at time t_0, in a separable state of time and polarization |t_0⟩_s|H⟩_s ⊗|t_0⟩_i|V⟩_i, where the subscripts indicate signal and idler photon, respectively (see Supplemental Material). 






The polarization of two photons is independently manipulated using waveplates along their respective paths (yellow and blue panels in Fig. <ref>), preparing the arbitrary polarization state |Ψ⟩ =cos(α) |H⟩+e^i ϕsin(α)|V⟩, which is then directly mapped to time-bin encoding by means of an intrinsically stable asymmetric interferometer, based on beam displacers (BDs).
Photons in one polarization state experience a temporal delay ≃8 ps (state |t_1⟩), relative to the undisplaced polarization (state |t_0⟩). 
After the asymmetric interferometer the state of each photon will be entangled in the polarization and time degrees of freedom of the same particle. To generate pure time-bin qubits, the polarization information is erased by a fixed half-waveplate (HWP) at 22.5^∘ followed by a polarizing beam splitter (PBS). The complete evolution probabilistically maps (with probability p=1/2) the polarization to pure time-bin qubits for each photon: |Ψ⟩ = |t_0⟩⊗(cos(α) |H⟩+e^iϕsin(α)|V⟩) 
⟹(cos(α) |t_0⟩+e^iϕsin(α) |t_1⟩)⊗|H⟩. 



 To perform HOM measurements, the photons are recombined in a single path and, after a final HWP at 22.5^∘, interfere in a final PBS.
 





To demonstrate tomography of the time-bin encoded qubits using the HOM scheme, we reconstruct a number of (approximately) pure and mixed states. First, we prepare 15 states, including the six mutually unbiased bases (MUBs) states  and nine randomly sampled pure states, and perform tomographic reconstruction. Furthermore, we prepared 33 states with varying amounts of mixture, as depicted in Fig. <ref> a). To generate the necessary mixture, we prepared pairs of pure states and switched between them periodically throughout the measurement. 
For each tomography, we prepare the reference photon in the six states of the three MUBs. This measurement set is over-complete, and one can also use a smaller set using different strategies (see Supplemental Information I). We then reconstruct the target state using standard maximum likelihood estimation <cit.>. For both pure and mixed states, we obtained a mean experimental fidelity <cit.> of ⟨ F^ exp⟩ =0.9955±0.0007 with respect to theoretical targets (see the Supplemental Material for individual fidelities). 



Noncontextuality test for hybrid entangled polarization-time single photon state
Intrasystem entanglement describes the entanglement between different degrees of freedom of a single particle <cit.>.
These hybrid entangled states <cit.> hold promise for efficient high-dimensional quantum information schemes <cit.>, as they can also be used to enlarge the overall Hilbert space of a single particle.
For photons, this kind of entanglement has been experimentally demonstrated through nonclassicality tests in  orbital angular momentum (OAM)-polarization states <cit.> and path-polarization <cit.>.


Our approach allows us to extend these kind of demonstrations, by performing a nonclassicality test that unambiguously shows the presence of intrasystem entanglement in the time-polarization degrees of freedom through the violation of a Bell-like inequality. 



 As described in the previous section, our experimental setup naturally produces time-polarization entangled single-photon states. For instance, if one prepares the photon in the |-⟩ polarization, the single photon entangled state (before the polarization erasure) is:


    |ψ⟩=1/√(2)(|H⟩|t_0⟩ - |V⟩|t_1⟩)  .



In the realm of quantum mechanics, entanglement and nonlocality are distinct yet interconnected phenomena <cit.>. Although the intrasystem entangled state in Eq. (<ref>) cannot generate nonlocality due to the lack of spatial separation between measurements <cit.>, it can give rise to a violation of classical noncontextual models <cit.>.





To demonstrate nonclassicality, and therefore certify time-polarization entanglement, we performed a Clauser Horne Shimony Holt (CHSH) <cit.> type test, measuring the quantity |S|=|⟨  A_0 B_0⟩-⟨  A_0 B_1⟩+⟨  A_1 B_0⟩+⟨  A_1 B_1⟩|. Here, the correlation terms ⟨ A_x B_y⟩ are calculated from the two-outcome measurements A_x and B_y (with binary measurement choices x and y) performed on the time and polarization degrees of freedom, respectively. The relation |S|≤ 2 holds for any noncontextual classical model that describes the correlations between measurement outcomes.
We carried out the time-bin measurements as described previously. To measure the polarization state of the single photons independently of the time, we adjusted the polarization erasure step (green region in Fig. <ref>) to instead perform projective measurements with the HWP and PBS.




 After generating the state described in Eq. (<ref>) and performing the optimal measurements on this state ({σ_z, σ_x } for A, and {σ_x + σ_z/√(2), σ_x - σ_z/√(2)} for B), we obtain a value of |S|^ exp=2.744 ± 0.006 (the error is estimated by considering the Poissonian fluctuations of the counts), without rescaling the counts for accidental coincidences or limited HOM visibility.
 This result indicates a violation of the noncontextual bound by over 100 standard deviations, unequivocally certifying the presence of intrasystem entanglement between the time and polarization degrees of freedom of the single photons.





Proposal for generating and measuring high-dimensional time-bin photonic states
Leveraging the measurement scheme demonstrated above, we also devise a procedure for generating and measuring high-dimensional quantum states, overcoming the standard practical limitations. 
The crucial aspect used for this task is that the HOM measurement  maps any projective measurement
along arbitrary high-dimensional qudits into the preparation of high-dimensional reference states. 



Our proposal exploits an efficient probabilistic scheme to generate arbitrary high-dimensional quantum states via quantum walk (QW) dynamics <cit.>, which has previously been realised in high-dimensional OAM states <cit.> and with 2D walks in the transverse momentum <cit.>.


The basic tool for this task is a one-dimensional, discrete-time QW <cit.> involving two quantum systems: a walker and a coin. 
The QW dynamics (see Supplemental Material IV)  can be exploited to prepare arbitrary qudit states in the walker space by choosing the optimal coin operations <cit.>, thereby changing the polarization at each step. Such dynamics produce a high-dimensional entangled target state in the coin-walker Hilbert space. The final pure walker state is probabilistically produced after projecting the coin state onto a suitable basis. Indeed, it is always possible to optimize the set of polarization operations that maximizes fidelity with an arbitrary state and the success probability at the same time <cit.>. Furthermore, there is evidence that the number of required steps grows only linearly with the dimension of the target state <cit.>.




Here, we consider the time degree of freedom of a photon as the walker, living in an infinite dimensional space H_ w spanned by discrete “position states” i.e., time-bin states {|t_i⟩_w}, with i ∈ℕ.
The coin then corresponds to the polarization state of the photon, in a two-dimensional space H_ c whose basis states, |↑⟩|H⟩ and |↓⟩|V⟩, denote the “movement directions” of the walker (see Refs. <cit.> for experimental realizations of time-bin-based quantum walks).

This polarization-time QW can be realized in a robust and straightforward way using only waveplates and birefringent elements.
The waveplates are used to generate arbitrary coin operations, and the birefringent elements introduce a relative temporal delay (larger than the coherence length of the photon), acting on the temporal degree of freedom according to the polarization value. Importantly, this implementation uses only a single spatial mode and requires no need for spatial interferometers at all. The coin projection can be implemented by a PBS, after which the arbitrary state in the time bin is generated (see Fig. <ref> a). The limit on the achievable dimension is given by the length of the QW. Importantly, the time-bin state can be arbitrarily adjusted by only tuning the waveplates, making it considerably more practical than changing large spatial interferometric setups. 




This scheme can be used for different tasks. One possible scenario is the preparation and measurement of a single photon in a high-dimensional state for general quantum communication.
Here, the sender uses one QW to prepare the message state to be sent, while the receiver uses a second QW to prepare the reference state of the photon, which is used to measure the message. The reference photon could be the twin photon generated by an SPDC source, or a photon generated by a source electronically synchronized with the source generating the first photon. The rapid development of on-demand single-photon sources, such as quantum dots, has already enabled several time-bin-encoded demonstrations <cit.>, and such sources represent promising candidates for implementing high-dimensional single-photon time-bin states within our approach.




A second scenario would be quantum key distribution (QKD), depicted in Fig. <ref> b): A pair of time-bin entangled photons are shared among two measurement stations. The entangled state of the photons generated using an SPDC source can be engineered by suitably manipulating the pump in the time-bin degree of freedom. Here, a QW could be used to generate the time-bin state |Ψ_p⟩=∑_j=1^n   a_j |t_j⟩, where a_j are arbitrary coefficients and ∑_j=1^n |a_j|^2=1. Considering the cases where the probability of generating a pair of photons in a pulse is very low (so neglecting higher-order terms in the SPDC process), the following entangled state will be generated:

    |Ψ⟩=∑_j=1^n   a_j |t_j⟩_s|t_j⟩_i .


 Then, the time-bin entangled photons are distributed to their respective measurement stations, where they can be measured using reference photons engineered by respective QWs (see Fig. <ref> b) and Supplemental Material Fig. S3). 
 


An attractive feature of this scheme is its scalability, as the number of optical elements scales linearly with the number of steps <cit.>. Therefore, the effectively certifiable dimension of the state is fundamentally limited only by the ratio between the pump pulse separation and the coherence length of the generated photons, and not by the time resolution of the detector.
 
 The generation and measurement of high-dimensional entangled states promises not only higher key rates, but indeed surpassing some of the fundamental bottlenecks of quantum communication, such as robustness of entanglement with respect to noise <cit.> 
 or new ways of testing the foundations of quantum mechanics <cit.>. 
 If the time bins can be temporally resolved by the detection apparatus, a significant benefit for QKD is the ability to perform multi-outcome measurements (those encoding the exchanged key), yielding log(d) bits for each coincident event. The test measurements can be implemented with the help of QW and would only require a small fraction of total rounds, without the usual restriction of unbalanced interferometers.




Discussions
Time-encoded quantum states represent an extremely powerful tool for photonic quantum information tasks. Although several approaches have been developed to manipulate and measure these states, the standard techniques available  are limited by instabilities and slow electronic devices. These shortcomings restrict the practicality and, therefore, the potential of time-bin encoding.

In this work, we used a scheme based on HOM interference to overcome some of these limitations and perform proof-of-principle experiments that highlight the versatility and high fidelity of this measurement approach.
The scheme enables the use of time-bin qudits with minimal temporal delay between bins, limited only by the coherence time of the photons. With the use of femtosecond sources of single photons, one could extend this technique to create states with minimal delays between neighboring time bins, potentially surpassing the temporal resolution of existing detection technology.


This robust and reliable approach allowed us to detect the nonclassicality of intrasystem polarization-time entangled states of single photons. Furthermore, leveraging the demonstrated measurement method, we propose an intrinsically stable single-spatial-mode scheme for generating and measuring high-dimensional time-bin quantum states.

By enhancing the robustness and enlarging the accessible dimensionality, this work provides a step towards the scalable implementation of high-dimensional time-bin qudits as quantum information carriers.  




Acknowledgments–
This work was supported by ARC Grant No. CE170100012.
L.V.-A. and D.J.J. acknowledge support by the Australian Government Research Training Program (RTP).
We thank Sae-Woo Nam for supporting this work through his development of superconducting-nanowire single-photon detectors and Alex Pepper for helpful discussions and support with the experimental setup. M.H. acknowledges support from the Horizon-Europe research and innovation programme under grant agreement No. 101070168 (HyperSpace). 






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