26.1 Introduction

Quantum information exploits quantum systems and their features to explore a wide range of applications, ranging from communication to computation, from metrology to simulation (1). Starting from the analogy with classical information theory, the basic tools for any information protocol are a physical system that carries the information and a network where the information can be manipulated, transported, and shared between different parties.

In the quantum realm, information is carried by quantum systems and a network corresponds to many physical modes, or levels, that the quantum system can occupy; links between these modes are represented by unitary operations. It is clear that implementing a quantum network requires control over a large number of modes, that is, control over a high‐dimensional degree of freedom of the quantum system and arbitrary connections between these modes. Among the various physical systems that can be used for quantum information – spins, atoms, ions, superconducting qubits – we focus on single photons. We present in the next section one of the most powerful approaches to implement quantum networks by accessing a high‐dimensional degree of freedom for single photons: multiplexing. The first scheme for multiplexing has been implemented in the spatial domain: here, spatial modes are adopted to define a quantum network. This approach has been widely used both in bulk and integrated photonic circuits (2); however, with the development of fast electrooptical devices, another approach has emerged as prominent in this field. This is time multiplexing, the exploitation of discretized time as network. This implementation has shown great advantages over the previous technique, in particular, in terms of stability and resource efficiency. We present in the following the basic idea of multiplexing in the spatial and time domains and show some applications of this approach as a tool to implement quantum networks.

26.2 Multiplexing

Networks comprising a large number of modes are equivalent to high‐dimensional quantum systems and can be implemented in various ways. In optics, there are essentially two ways of creating such a quantum system: the first one is to exploit a discrete high‐dimensional degree of freedom of the system itself, for example, orbital angular momentum (3) or time–frequency modes (4) of single photons. The second approach is to discretize a continuous degree of freedom external to the system, such as space or time. This second approach has some advantages over the first one, essentially in that we are able to experimentally address space and time with standard optical elements. Here, we are interested in this second approach and will start first from the spatial degree of freedom and then translate the concept to the time domain.

What does discretizing a continuous degree of freedom mean? It means that we allow the physical system to occupy not the complete space (or time) but just a well‐defined number of binned positions (or time bins).

When using photons as our quantum system, a specific position in space is represented by a spatial mode, that is, the direction of propagation of the light. When a photon is generated, it propagates along a single spatial direction; however, we can allow the photon to travel along more than one path, that is, we allow for spatial multiplexing. New spatial modes can be provided by letting the photon impinge on a balanced beam splitter (BS). The BS is a standard optical device that deflects the photon on a different spatial mode or leaves it in the original one, both actions occurring with 50% probability. If we represent the spatial position of a photon with a vector , where is a position in space, the action of the BS is described by the matrix

26.1

The action of this device is to take the photon in – for example, position – and transform it in a superposition of positions and , that is, . The basic scheme of spatial multiplexing with one BS is presented in Figure 26.1a. With this optical device, we introduce a new spatial mode for our quantum system, which will now have dimension equal to 2, which is usually called dual‐rail encoding. By using more than one BS in a cascaded configuration, we can provide as many spatial modes as we need and therefore expand the dimensionality of the system.

Spatial multiplexing is very powerful when only few modes are adopted, corresponding to systems with low dimensionality. The main limitations are size and stability of the bulk architecture and the requirement of many optical components. Integrated photonic circuits have overcome the stability problem; however, the need of a large number of optical components and detectors still remains an intrinsic limitation for spatial multiplexing.

We can overcome these issues by discretizing a different degree of freedom: the time. This is known as time multiplexing, and it is achieved by allowing the photon to travel only at specific times, which are called time‐bins. To understand how this works, let us translate the concepts we have described so far in the spatial degree of freedom to the time domain: what was a position in space is now associated to a specific time‐bin. New positions in time are created by introducing time delays with respect to the original time‐bin. An optical scheme to achieve this task is shown in Figure 26.1b: here a photon (or a pulse) is split into two different spatial modes of unequal length and then recombined such that the pulse that travels the long path acquires a delay with respect to the pulse traveling the short one. This is a time splitter (TS) and its operation can still be described by Eq. 26.1, but it now acts on the time degree of freedom, that is, . We can now measure the arrival time of the photon at the detector ( or ) to infer its time‐bin. As in the spatial case, by introducing different delays, we can increase the number of time‐bins and, consequently, increment the dimensionality of our system.

Once we are able to create different time‐bins, we can also connect them and create a network (Figure 26.1c), such that the light can propagate from one mode to another: each time‐bin represents a node of the network and the links are provided by TSs (photons interfere at the TS). In principle, it is possible to create as many links as we need and arbitrarily connect the nodes.

Multiplexing can be used for different purposes: sources (5–7), circuits (8–10) and detection (11–15). In the next sections, we address two of the main applications where time multiplexing is adopted: photon‐number‐resolving (PNR) detection and quantum walks.

26.3 Photon‐Number‐Resolving Detection with Time Multiplexing

First, we look into time multiplexing as an efficient resource to perform photon‐number‐resolved detection.

When having a quantum state, it is desirable to determine its photon number distribution or in general we can think about discriminating between single‐ and multiphoton events. To achieve this, we need PNR detection. Generally speaking, there exist two main tasks for which photon‐number‐resolving detectors are needed. The first case is when the initial photon number distribution is either unknown or –as in quantum cryptography (1) – is to be confirmed by ensemble measurements. In the second case, the incident state is well known from the beginning, and one wants to detect the photon number on a single‐shot basis to address states individually. These tasks can be achieved with PNR detectors; however, such devices usually work at cryogenic temperatures and are then expensive. More standard equipment for single‐photon detection are avalanche photodiodes (APDs), which are able to detect a single‐photon event but not to distinguish between one or more photons impinging on it. Here, we present a scheme for achieving photon‐number‐resolved detection with just APDs and passive optical components (16).

Suppose we have a quantum state with an unknown photon number distribution and we want to retrieve such a distribution with ensemble measurements. To achieve this aim, we must be able to measure (i.e., discriminate) different photon numbers; however, we only have APDs. The simplest way to achieve this is to use multiplexing. Let us start again from spatial multiplexing and split the quantum state into different spatial modes, each of them connected to an APD. An optical setup to achieve this is shown in Figure 26.2a: here the state is divided into different spatial modes by BS. We know that a photon impinging on a BS has 50% probability of being reflected or transmitted, and this applies to every photon present in the initial state. Imagining an ideal setup (perfect BS and no losses), after splitting times our initial state (which corresponds to layers of cascaded beam splitters), the probability of having one photon in a certain spatial mode is and the probability of having two photons will be . In general, the probability for photons to end up in the same mode is . By adapting the number of modes to the expected maximum number of photons present in the initial state, it is possible to minimize the probability of having more than one photon in each mode and thus arbitrarily reduce the error on the photon number estimation. This architecture has been implemented in integrated setups (17); however, while it is, in principle, good to perform photon number resolved detection, the architecture still requires as many APDs as the number of spatial modes making the complete setup not easily scalable to a large number of modes and arbitrarily expensive.

A better solution for photon‐number‐resolved detection, first introduced in (14), is provided by time multiplexing. A scheme of the optical setup is shown in Figure 26.2b: here we use only two different spatial modes and split the input pulse into many time‐delayed pulses. In the first step, the input signal is divided by means of a BS into two signals that are launched into fibers of unequal lengths. These pulses become delayed with respect to each other by a specific time delay , and their subsequent combination at another BS yields two pulses in each of the two output channels. By iterating this setup with delays twice as long as in the preceding step, further doubling of the number of temporal output modes can be achieved. This scheme performs the same task as the one reported in Figure 26.2a but it is resource efficient as we only need two APDs and for modes one needs only BSs instead of as for spatial multiplexing.

Now we look into a practical implementation of this scheme. Indeed, while the ideal scheme presented here is very easy, an experimental implementation requires a careful characterization of the BS reflectivities and of the losses of each channel, such that the data can be analyzed correctly and the actual number of photons present in the initial state can be estimated. A wrong estimation can occur because high photon numbers result in a nonnegligible probability that two photons remain together in one time‐bin and are counted as one. Moreover, losses play a crucial role, as they change the photon number distribution of the input state. Therefore, the number of stages of the fiber configuration and the transmission efficiency essentially limit the incident photon numbers that can be reliably distinguished by the setup. To find the optimal number of stages, we need to know the probability of obtaining counts from incident photons . This probability is indeed related to the input photon number distribution through

26.2

It is possible to calculate from the real values of the BS's reflectivities and the losses in each fiber and from these values estimate the maximum photon number that can be reliably measured. Achilles et al. (14) have implemented an eight‐time‐bin detector using the scheme of Figure 26.2b and were able to measure states with average photon number with 99% accuracy. The performance of the setup then limits the photon number resolution; however, this resolution is also strictly related to the task we want to perform.

When the incident state is well known and we want to measure the photon number on a single‐shot basis, as for example, for conditional state preparation (18), the requirements are more strict. This is because in this case the performance of the setup has to be evaluated on the confidence that counts have actually been triggered by photons. This confidence can be associated with the conditional probability , which measures the probability that detection events are caused by photons. Obviously, this probability is affected by the performance of the setup as well as from the statistics of the initial state. Achilles et al. (14) have shown that a lossless setup with eight time‐bins allows good discrimination of low photon numbers for coherent states with mean photon numbers up to . The distribution of the input light into a limited number of time bins can therefore be adopted if the ratio of maximum photon number to time‐bins is sufficiently small. Not surprisingly losses will decrease the confidence for measurements on single quantum systems, which restricts the setup's use to photon number distributions with low‐enough mean values. Concerning the experimental tailoring of the timings, one has to keep in mind that APDs possess a so‐called dead time: when an APD is triggered, it remains “blind” for some time thereafter (in the order of several nanoseconds) and then any photon impinging on the detector during this time cannot be registered. This implies that the distance between the different time bins must be longer than the dead time such that when a possible photon in the subsequent time‐bin arrives at the detector, the APD is ready again to measure it. The dead time then directly gives a lower bound on the difference in length of the fibers composing the time‐multiplexed detector. For example, a typical dead time of 100 ns requires a difference in the fiber length larger than 15 m. With the present technology, it has been shown that time multiplexing for photon number detection can be pushed to the measurement of photons with 256 time‐bins (19), a limit due to dispersion and losses in fibers. However, with the development of single‐photon detectors with much lower dead times, dispersion and losses can be reduced and this architecture pushed even further.

26.4 Quantum Walks in Time

26.4.1 A Classical Motion – The Random Walk

Interestingly, the BS cascade from Figure 26.2a resembles the classical Galton board (see Figure 26.3a). Here, a classical particle – a ball – starts from the top and has a 50% chance to fall either to the right or to the left at each pin. This is repeated in several layers of pins – the steps – until the ball ends up in one of the bins. In order to observe the final probability distribution, the experiment must be performed many times. It turns out that the probability distribution of finding the ball in bin is given by the binomial distribution

26.3

with for the unbiased Galton board, which converges into the normal distribution for steps. The maximum of the distribution, that is, the expectation value, is , which was the walker's starting point, since the walker has, on average, taken the same number of jumps to the right as to the left. The variance of the binomial distribution is proportional to , which means that its width grows very slowly with the number of steps. In other words, the expected translation distance grows with , which characterizes a diffusive motion in analogy to physical diffusion processes, for example, Brownian motion.

Such a Galton board is a well‐known implementation of a one‐dimensional random walk. A random walk is an established mathematical model of a random movement with broad applications in various fields; for example, the description of the probability distribution of general measuring results, prediction theories of financial markets and stock price evolutions, the motion of suspended particles in a fluid – the Brownian motion –, growth processes and population evolution and many more. The common theme is an evolution consisting of discrete time steps, which are unpredictable and can be described by a random movement. This random motion in the simple picture can also be illustrated in another way: Imagine streets with crossroads in an unknown city and a pedestrian who does not know his way home. Thus, at each crossroad all he can do is to toss a coin and depending on the outcome he will either turn to the left or to the right, trying to find his destination. The idea of a coin toss will become important for the transfer of the random walk idea into the quantum regime.

26.4.2 The Random Motion in the Quantum Domain: The Quantum Walk

In the following, we will introduce the discrete‐time quantum walk (DTQW) (20–22), the quantum analogue of the classical random walk. Both can be realized in Galton‐board‐like implementations, which are presented side by side in Figure 26.3 (23,24).

In quantum mechanics, the state of a particle is modeled by a wave function with describing the probability of finding the particle in a particular state x. Its discrete time evolution is determined by a unitary operator , which maps the wave function of the time to the next point in time according to In contrast to the classical walker, whose path is defined by a coin toss, the quantum wave function is split up at each crossing, that is, the walker takes all paths simultaneously. In such a coherent evolution, the state is given by a superposition of components originating from different paths. Where two or more paths cross, again interference effects occur and determine the dynamics of the quantum walker. Note the difference in the beam splitter cascades in Figures 26.2a and 26.3b without and with crossing paths. This capability of walking several paths simultaneously must be taken into account when translating a random walker into the quantum domain. In order to introduce the coin, in analogy with the classical walk, the Hilbert space of a DTQW is given by a tensor product of position and coin space . Here represents the position space and the position of the walker. is the so‐called coin space of an internal degree of freedom. The vector is a quantum counterpart to the classical coin (which is usually two dimensional for a one‐dimensional quantum walk, since there are jumps to the right and to the left possible). In contrast to the classical walker, whose coin can either be heads or tails enforcing it to jump to the left or to the right, the quantum coin can be in a superposition of different coin states determining a superposition of paths. Depending on the experimental implementation, such an internal state can be realized, for example, by a spin, the orbital angular momentum, or polarization. Since we will present a photonic implementation, we will in the following use the convention and for horizontal and vertical polarizations representing the coin space in order to unify the notation. Concretely, the quantum walker's state for a one‐dimensional walk at a time looks like

26.4

with the time‐dependent amplitudes and . In the DTQW, the evolution unitary for one discrete time step consists of a coin operation , the analogue of the classical coin toss, and a subsequent step operation realizing the conditioned spatial movement

26.5

The step operator is given by

26.6

Consequently, the part of the walker's wave function in horizontal polarization undergoes an increase of position by one (jump to the right), while the part in vertical polarization has the position decreased by one (jump to the left). In general, the step operator carries the information about the connections (edges) between possible positions given by the underlying graph. Here, we present the walk on a one‐dimensional line, but one can also think of walks on higher‐dimensional grids ( 10,25), nonregular graphs (26,27), or percolated grids with missing edges (28,29).

The quantum coin toss takes place only in the coin space leaving the position space unaffected. In the chosen polarization basis and the coin operation is in general a two‐dimensional polarization rotation by an angle . According to the experimental implementation used in (30), we define its action as

26.7

Note that depending on the device realizing it in the experiment the specific shape of the matrix can differ.

Due to the interferences, the spatial distribution of a quantum walk is rather counterintuitive compared to the random walk, see Figure 26.4. Due to destructive interference, the probability of finding the walker in the centre of the distribution is small, instead there are two side lobes at the edges of the distribution. This is reflected in the width of the distribution, which grows faster with step number than in the classical case, that is, .

This faster spread is one of the main features of a quantum walk, which makes it highly attractive in many different contexts. It has applications ranging from energy transport in quantum biology (31,32), solid‐state phenomena based on topological phases (33–35), quantum search algorithms (36–38), graph isomorphism problems (39) to quantum simulation (10) and quantum computation (40,41).

26.4.3 Experimental Implementation of Quantum Walks

Due to their compatibility with various theoretical models, several groups are working on the experimental realization of quantum walks using very different platforms. Current experimental implementations include nuclear magnetic resonance (42,43), trapped ions (44,45), atoms (46,47), photonic systems ( 23, 24 48–50) and waveguides (51–59). Sticking to the original motivation of the Galton board, a spatial arrangement of beam splitters for a photonic walker would be the straightforward approach ( 23, 24). However, this idea has a big drawback: with each new step, a new layer of beam splitters has to be added and aligned, and for each position a separate detector is necessary. In general, the number of resources grows quadratically with step number. Additionally, the phase stability, which ensures the coherence properties of the walk, is very critical. In such an arrangement, the coherence can only be guaranteed for the first few steps due to inevitable inaccuracies in alignment or small errors of the components. Therefore, a new approach must be applied and we have one already at hand: as explained above, there is a direct correspondence between spatial and temporal modes. Thus, we translate the quantum Galton board (Figure 26.3b) into the time domain (8). Concretely, we apply the time‐multiplexing scheme as shown in Figure 26.5.

Here, a photonic walker – a single photon or a weak coherent laser pulse – with its polarization as the coin state performs a quantum walk in the time domain. A standard wave plate (here a HWP) implements the coin operation by rotating the initial polarization (here: horizontal) into a superposition of and according to Eq. 26.7. At a polarizing beam splitter (PBS1), the pulse is split according to its polarization and routed afterward through two spatial paths of different lengths in order to introduce a well‐defined time delay between the two polarization components; see Figure 26.5. This splitting operation in combination with the time delay and the subsequent merging of the paths at the second PBS constitutes one shift of the time‐multiplexed quantum walk implementing Eq. 26.6. We interpret the part arriving earlier as having undergone an increase in the position by one and the component arriving later as having been subjected to a reduction of the position by one. Now one time step in the quantum walk evolution according to Eq. 26.5 is completed. For the second step, the two wave packets travel through the same configuration of elements. They acquire the same temporal delay , which induces an overlap at the output port of PBS2 of the packet that took the long path at step 1 and the short path at step 2 and vice versa. The coinciding packets correspond to the two parts of the wave function arriving at position from different directions after two steps in a spatial quantum walk. From this step on, interference effects occur whenever two wave packets have been delayed by the same multiples of . Hence, every spatial position is uniquely represented by its arrival time including the requisite interference effects. Such a mapping from spatial into the temporal domain does not automatically reduce the number of employed resources, since the repeated action for each further step again requires more optical elements, which scales still linearly with . The main improvement regarding resource efficiency and coherence properties can be achieved by applying a loop architecture. This means that the pulses leaving at PBS2 are fed back into PBS1 for performing the following step in the quantum walk evolution. Thus, all the different contributions pass a constant number of the same optical elements over and over again, which ensures a perfect matching of the delay times as well as maximal phase stability between the interfering pulses and extensive homogeneity of the setup. The physical implementation of the time‐multiplexing feedback loop is presented in Figure 26.6.

The initial pulse enters the setup at a partially reflecting beam sampler and passes the coin operator, given by HWP and/or EOM. After the described splitting and time delay implemented by PBS and two single‐mode fibers (SMFs) of different length, a small portion of the wave function is coupled out and sent to another PBS and a pair of avalanche photodiodes (APDs). By such a design of the detection unit, we gain access to the walker's full temporal evolution over all time steps including polarization resolution where required. By closing the loop, we ensure that the pulses are fed back and the action of coin and step operator is repeated for further steps.

Even with such a setup design, the maximum observable step number will be limited. Here, typically it is not the lack of coherence that restricts the measurement of high steps. The losses due to the outcoupling to the detection and the unavoidable losses of the optical components will decrease the measurement signal exponentially in time. Depending on the initial power and success in the loss minimization, up to 28 steps of a quantum walk on the line were observed with the presented setup; see Figure 26.4 (9).

26.4.5 Increasing the Dimension – Discrete‐Time Quantum Walks in 2D

The presented quantum walk scheme has so far been restricted to the evolution in one dimension, that is, on a line. Now we want to increase the dimension and study a walk taking place on a two‐dimensional grid. This will offer increased applications as many physical phenomena cannot be simulated with a single walker in a one‐dimensional setting, such as multiparticle entanglement and nonlinear interactions (61).

The state of a 2D walker is given by

26.8

where and denote the position and the coin state for dimensions 1 and 2, respectively. The coin operator is here a unitary and the step operation comprises a shift in both, ‐ and ‐directions. It is worth noting that the state given in Eq. 26.8 has even another interpretation: and can also be understood as position and coin of two walkers in one dimension. Their description is completely equivalent, which enables us to simulate and study two‐particle dynamics including the creation of entanglement in bipartite systems with conditioned interactions such as strong nonlinearities or two‐particle scattering (10).

In order to inherit all the benefits of the time‐multiplexing scheme, we will design the new setup by extending the feedback loop of the 1D walk. Figure 26.7 shows a sketch of the actual implementation.

Comparing the setups shown in Figures 26.6 and 26.7, one can see that for the 2D walk a second free‐space path (labeled with ) is installed, connecting the two so‐far free ports of the two PBS. In addition to the time delay introduced by the two fibers, this free‐space mode will establish another time delay , which can be interpreted as a spatial step in the second dimension.

In the second free‐space mode, another pair of APDs is included, measuring the step‐ and polarization‐resolved state of the walker. The two‐dimensional polarization in addition to the two spatial modes and realizes the four‐dimensional coin state determining the motion on a 2D grid according to Eq. 26.8.

Again using the EOM as a dynamic coin, various controlled particle–particle interactions can be simulated. One example is the evolution of two‐particle quantum walks with short‐range interactions, the so‐called two‐particle scattering, where interactions occur only when both particles meet in space. Both walkers occupying the same position corresponds – in the 2D quantum walk picture – to the vertices on the diagonal of the grid. Hence, we can simulate a short‐range particle–particle interaction by implementing a special coin operation only on the diagonal positions while keeping all other positions unaffected. The resulting quantum walk is strongly confined to the main diagonal, which is a simulation of the creation of bound molecule states, predicted as a consequence of the two‐particle scattering (62).

These results prove the potential of quantum walk systems as a way of simulating and understanding complex quantum systems including multiparticle dynamics.

26.5 Conclusion

Many effects and applications in the field of quantum information rely on large networks. They can be high dimensional and very complex, on the one hand, while, on the other hand, a precise control and a high flexibility in tuning the network parameters are essential. Thus, one main experimental challenge in quantum information is the implementation of quantum networks of macroscopic size without losing the control of the microscopic features of the single nodes and edges. Here, we presented the idea of multiplexing with special emphasis on time multiplexing as one way to reach the demanding properties in terms of size, stability, and homogeneity of a network regarding two important applications: photon‐number‐resolving detection and quantum walk dynamics. While the first one is fundamental in quantum state tomography for quantum information technologies such as cryptography and quantum computation, the second one is a valuable platform for the investigation of transport mechanisms on complex structures. However, the presented approach of time multiplexing is not limited to the two exemplary applications, but it is widely applicable and a promising platform whenever large networks with high control are in demand.

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