Harnessing temporal modes for multi-photon quantum information processing based on integrated optics

Abstract

In the last few decades, there has been much progress on low loss waveguides, very efficient photon-number detectors and nonlinear processes. Engineered sum-frequency conversion is now at a stage where it allows operation on arbitrary temporal broadband modes, thus making the spectral degree of freedom accessible for information coding. Hereby the information is often encoded into the temporal modes of a single photon. Here, we analyse the prospect of using multi-photon states or squeezed states in different temporal modes based on integrated optics devices. We describe an analogy between mode-selective sum-frequency conversion and a network of spatial beam splitters. Furthermore, we analyse the limits on the achievable squeezing in waveguides with current technology and the loss limits in the conversion process.

This article is part of the themed issue ‘Quantum technology for the 21st century’.

1. Introduction

Quantum information science promises great improvements over classical systems in communication, metrology and computing. However, the requirements to actually achieve a point at which quantum systems start to be truly superior to classical systems are high. Very good control over the used quantum states is necessary, as well as a high dimensionality of the quantum system.

Squeezed states in continuous-variable systems are an intriguing resource for quantum information science as such systems are scalable in principle. Entanglement between 10 000 temporal modes has been achieved [1]. In the spectral domain up to 60 modes have been entangled in a continuous-wave frequency comb structure [2] and around 10 modes in a pulsed, ultrafast system [3]. Using homodyne detection, the full multipartite entanglement could be verified in such a system [4]. If the squeezing can reach certain thresholds, error correction for quantum computing becomes possible [5], which greatly motivates technological improvement to increase the squeezing values.

Additionally to generating entanglement in many spectral modes, it would be useful to manipulate the modes after the generation process. One candidate to achieve this goal is the quantum pulse gate (QPG) [6,7], an engineered sum-frequency conversion (FC). It is designed to operate on a single arbitrary temporal mode, which can be set by the spectral shape of the pump pulse. Operating with single photons, it can build up a complete framework for quantum information science [8]. Such a QPG was implemented in lithium niobate waveguides and characterized with classical light [9,10]. Furthermore, it has been applied for temporal state tomography of single photons [11]. The advantage of waveguides over an implementation in bulk is that the waveguide can be designed to support in the spatial domain only one mode at the longest wavelength. This greatly simplifies the spatial properties, allowing to reduce the process to one single spatial mode. The advantage of a single-pass process with broadband modes as opposed to a cavity system is that it is compatible with single photon detectors and applicable for the development of practical integrated devices. This promises quantum manipulations that go beyond the Gaussian regime. For example, photon subtraction from multimode quantum fields using a QPG becomes possible [12]. The disadvantage of single-pass configurations on the other hand is that they do not have the intrinsic spectral filtering of a cavity. Therefore, much care has to be taken into the spectral engineering of the nonlinear process.

Using squeezed states in combination with the QPG requires that the spectral properties of the squeezed states are adapted to the QPG. Single-pass parametric down-conversion (PDC) could be used to this end. Spectral engineering of pure quantum states generated by PDC in single-pass configurations has been demonstrated in potassium dihydrogen phosphate (KDP) [13] and in potassium titanyl phosphate (KTP) at telecom wavelengths [14–16]. The process has further been implemented in KTP waveguides [17], which restrict the process to a single spatial mode and increase the process efficiency due to the strong confinement along the whole propagation length. This allowed one to generate mean photon numbers of 20 while maintaining the single-mode character [18]. The fact that the single-mode states can be driven to such high mean photon numbers suggests that strong squeezing in controlled temporal modes is feasible in such waveguides.

In this publication, we review the engineering of PDC and FC with the focus on integrated devices. Restricting ourselves to waveguides, we do not consider spatial correlations and assume that all processes happen in one spatial mode only. We then present a spatial analogy of the pulse gate, which allows us to develop simple interpretations of its performance on multi-photon states. For the prospect of combining it with squeezed input states, we review the limitations on integrated PDC sources and estimate the achievable squeezing with realistic sources. Whereas high squeezing can be generated in principle with integrated devices, the current limitation for combining it with the pulse gate for quantum information science lies in the limited conversion efficiencies of experimentally demonstrated pulse gates.

2. Engineered χ⁽²⁾ processes

The generation of squeezing as well as frequency conversion are second-order (χ⁽²⁾) nonlinear processes. Quite generally for any χ⁽²⁾ process in the undepleted-pump approximation, the Hamiltonian is bilinear in the creation and annihilation operators. For parametric down-conversion and frequency conversion, it can be written as

2.1

and

2.2

where a(ω),b(ω),c(ω),d(ω) are the usual annihilation operators at frequency ω and f is the joint spectral amplitude function (JSA). For PDC the JSA describes the distribution of the generated signal and idler photons and for FC it can be interpreted as the transfer function from mode 2 to 1. In both cases, all spectral properties are fully described by the JSA. To further understand the spectral properties of a given process, it is useful to perform a Schmidt decomposition

2.3

where g_k(ω) and h_k(ω) are orthonormal functions and . These functions define broadband operators and . The orthonormality allows one to rewrite the processes into a sum of independent processes:

2.4

and

2.5

The factor is often referred to as the squeezing parameter and the frequency-conversion efficiency is given by for each mode k. To use both processes for quantum information tasks, we need to be able to control the shape of the broadband modes as well as their coefficients. In particular, the QPG is a frequency conversion with only one coefficient c₀=1 and c_k=0 for k>0.

(a). spectral properties in the low-gain regime

In the low-gain or low-conversion-efficiency regime, the JSA can be described in terms of a pump-amplitude function α and a phasematching function ϕ by [6]

2.6

where

2.7

Here v₁,v₂ and v_p are the group velocities v_g=∂ω/∂k of the three fields. The two equations (2.7) can be interpreted as the momentum and energy conservation conditions, where the + sign is for PDC and the − sign for FC. Equation (2.6) gives a good idea of the knobs we can use to adjust the processes and thus the JSA, since it depends on the

• — choice of material and wavelength combinations setting the group velocities,

• — pump spectral profiles setting the function α, and

• — the poling pattern and length of the crystal setting the function ϕ.

We want to be able to control the number of modes and their shapes. For PDC, this is possible if ϕ(ω₁,ω₂) is a positively correlated function, which translates to the condition v₁<v_p<v₂. Fortunately, this condition is possible for type II processes in KTP and KDP and was exploited in [13–18]. So far, other than Gaussian pump shapes have not been applied but it is possible to generate a well-defined number of modes by changing the pump shape alone [8]. We illustrate this in figure 1, where we show the JSAs for the first three Hermite–Gaussian pump shapes and the corresponding squeezing values in each mode.

Figure 1.

Modelled JSA for waveguided type II PDC in KTP at telecom wavelengths and the corresponding squeezing strength per mode. Left to right: Gaussian pump, first-order Hermite–Gaussian pump, second-order Hermite–Gaussian pump. Using this scheme, we can generate a low number of strongly squeezed states.

The considerations in the engineering of frequency conversion are very similar. For the QPG, mode selective operation is the goal and can be achieved by group velocity matching of the input and the pump v_p=v₂. In that case, the phasematching takes a horizontal shape and the pump shape defines the mode that is converted; see [6] for a detailed description. The result is that it is possible to convert any mode to one particular output mode, given by the phasematching function.

(b). spectral properties in the high-gain regime

Unfortunately, the simple picture presented above for engineering the JSA does not hold in the high-gain regime due to the time-ordering effect. Time ordering here means that the Hamiltonian, which contains the electric fields of the process, does not commute with itself at different times or positions in the crystal. Full understanding of this effect is currently investigated in several groups [19–23].

Here, we present an intuitive physical explanation of this effect. If we regard the fields in the time domain while they propagate through the crystal, we can understand why the shape of the fields changes with different conversion efficiencies. Let us consider the QPG, where the input field and the pump field are temporally short and travel at the same group velocities while the output field is significantly slower. In the low conversion limit, input and pump build up a temporally long, rectangular output pulse by converting the same amount of power over the whole propagation length. In the high conversion regime, more power is converted at the beginning of the crystal. However, the overlapping part of the converted field carries only little power and hence the majority of the converted light cannot contribute in seeding the process. That means that the converted field does not stimulate the enhancement from a^†|n〉 = when applying the Hamiltonian to the intermediate state in the crystal. Therefore, the input mode is converted at a constant rate over the crystal length, thus producing an exponential-like shape of the output pulse in time. We sketch this picture in figure 2. Now it becomes clear that the JSA is power dependent in the high-power regime, making equation (2.6) invalid. However, the bilinear form of the Hamiltonian remains despite this effect [19,22], such that the conversion process is still defined by a two-dimensional JSA. The only difference is that the JSA now becomes power-dependent and needs to be calculated numerically using one of the methods presented in [19–21].

Figure 2.

Schematic picture of the temporal modes at the end of the crystal for group velocity matched input fields, i.e. a QPG. Pump and input overlap perfectly along the entire crystal. The output mode, however, has only a very small overlap with the two input fields, thus only negligibly contributing to ‘seeding’ the process. Therefore, the converted power reduces in the high-efficiency regime since the input gets depleted, whereas it stays constant in the low-efficiency regime. This leads to entirely different spectral shapes in the two power limits. (Online version in colour.)

An intriguing question is how the JSA can be engineered in the high-power regime. Different strategies can be followed to avoid an unwanted change of the JSA. One solution is to use two or more crystals, of which each operates at low pump powers, as suggested in [24,25]. In between the crystals, all fields are shifted to their initial position in time. Therefore, the mode shapes do not change much and the effect is reduced. Alternatively, it might be possible that the time-ordering effect can be compensated by adapting the phasematching and pump functions. In the example above, an increasing conversion efficiency over the length of the crystal might compensate the depletion effect. However, a full numerical study of the phasematching properties is necessary to answer the question whether it is possible to fully compensate the time-ordering effect by adapting the phasematching and pump shapes of a single crystal.

3. Quantum information in the multi-photon regime with nonlinear processes

(a). Concept

Engineered frequency conversion makes the spectral degree of freedom accessible for quantum information science. Since it can selectively convert specific temporal modes, it can generate superposition states with respect to some basis or probe spectral correlations in a bipartite system. Weakly pumped PDC in the photon-pair regime already generates spectral entanglement. For example, the configuration depicted in the central plot of figure 1 is essentially a Bell state and can be probed by two pulse gates operating on superposition modes of A₀,A₁ and B₀,B₁, respectively [8].

However, the QPG is not limited to single photons only. In principle, more information can be encoded per temporal mode if we use multiple photons per mode. What happens if we send in higher photon numbers or squeezed states in different spectral modes into the QPG? In this section, we answer this question by providing a simple analogy between the QPG and a spatial network of beam splitters. In fact, even though we operate on the frequency degree of freedom, any FC is just a beam splitter transformation. Particularly for the QPG, only one term is non-zero in the Schmidt decomposition, i.e. the conversion can be modelled with only a single beam splitter. If we send in photons in different temporal modes, we have to transform these modes to the eigenmodes of the pulse gate by an appropriate unitary U^basis and apply a beam splitter with transmittance on one of them. We illustrate this in figure 3. The unitary U^basis is not entirely set by the QPG as only one of the modes is fixed, whereas the number of input modes can be unlimited. This means that only the first row of the unitary is defined by the single eigenmode of the QPG:

3.1

The other rows are free at this point but can be set by subsequent pulse gates or the measurement basis of some measurement that we do not consider here.

Figure 3.

(a) Spatial analogy of the QPG. The actual frequency conversion operates on a single mode with a conversion efficiency . Any input state thus undergoes a unitary U^basis of which the first output corresponds to the pulse gate mode. Effectively, this can be used to perform beam-splitter operations on the input modes. (b) Two examples for the beam-splitter operation: a beam splitter on two input modes and a superposition of many input modes. (Online version in colour.)

Let us discuss the example of sending an arbitrary temporal mode into the pulse gate. We have to decompose this mode into the mode that the pulse gate operates on and one orthogonal mode. If the pulse gate operates at full conversion efficiency, then it corresponds to a standard two-mode beam splitter with a splitting ratio equal to the mode overlap between the input mode and the QPG. Applying this to squeezed states, we can send in two separable squeezed states in orthogonal modes and set the pulse gate to operate on a superposition of both modes. This would implement a beam splitter, thus entangling the two modes to a two-mode squeezed state. In a higher-dimensional system, we can set the QPG to operate equally on all input states, thus entangling many modes with a single operation. Furthermore, by using subsequent pulse gates operating on different modes, we can build up an arbitrary unitary transformation and thus construct arbitrary cluster states. This can also be achieved through the measurement-based mode selectivity of homodyne detection [26]. However, the pulse gate operation is not limited to homodyne detection and can be combined with any measurement, including non-Gaussian operations like photon-number detection.

(b). Loss limits

Here we discuss the loss limits in integrated systems. How feasible is such an implementation with squeezed states experimentally? As soon as we increase the photon number above 1, linear optical losses start to play an important role in the system. Several sources for losses exist in the pulse gate. First of all, the linear optical losses from couplings and propagation losses need to be considered. Luckily, LiNbO₃ is one of the most commonly used materials for nonlinear optics and has a very advanced fabrication technology. The losses through a good sample can be as low as 0.016 dB cm⁻¹ [27] becoming almost negligible.

Free space coupling on the other hand is not negligible. If we use KTP for state generation due to the possibility to tune the number of generated modes there, and LiNbO₃ for the pulse gate, a few percent of losses will be unavoidable due to differences in the waveguide shapes and sizes. Couplings up to 92% have been achieved between Ti-indifused LiNbO₃ waveguides and single mode fibres [28]. Similar values should be reasonable for couplings between Rb:KTP and Ti:LiNbO₃ waveguides. Alternatively, we could generate PDC states in LiNbO₃ directly. Here, so far no single-mode states have been reported in the literature. In principle, however, this is not required with a QPG as it can be used as a mode selective filter. The challenge will be to generate large squeezing. The highest reported value in the literature for directly measured single-pass pulsed squeezing in LiNbO₃ is 5 dB [29]. This is also the highest reported squeezing for a single-pass source.

The third loss contribution is the QPG process efficiency. In scenarios where full conversion efficiency is necessary, the time-ordering effect complicates the design of the QPG and currently limits the conversion efficiency to below 90% [19,20,22] while at the same time making the process multimode. However, the time ordering effect is likely to be controllable with clever QPG designs as mentioned above. The experimental work in [10] goes into this direction by shaping the pump pulse into experimentally optimized modes. With this optimization the authors manage to increase the conversion efficiency up to 92%.

Yet another limitation is self-phase modulation of the pump [30]. This happens at high peak intensities. In principle, the nature of this effect is similar to the time ordering effect as spectral profiles, in this case the shape of the pump, change during the propagation through the crystal. It needs to be further investigated how this affects the conversion efficiency. If this effect limits the performance of the QPG, spectrally narrower pulses need to be used since self-phase modulation scales stronger with peak powers than the second-order nonlinear processes.

(c). Practical parametric down-conversion sources

As mentioned above, KTP allows one to engineer the mode distribution by shaping the pump of the process. This has so far been applied to generate single-mode PDC state. Our source [18,31] showed such single-mode characteristics and near perfect photon-number statistics. In figure 4, we show the measured joint spectral intensity (JSI) for two pump configurations: an adapted Gaussian pump and a narrow Gaussian pump. The measurement is performed with a time-of-flight spectrometer consisting of a dispersive fibre and an avalanche photodiode [32]. In the first case, there is essentially only one spectral mode, while in the second case, there is a distribution of spectral modes with decreasing Schmidt coefficients. The Schmidt decomposition of the anti-correlated JSI shows modes which resemble Hermite–Gaussian modes. Those modes could now be addressed individually by the QPG. This way we could operate on individual squeezers or perform operation on several to produce a large entangled state.

Figure 4.

(a) Measured JSIs for two Gaussian pump shapes of different widths. (b) The absolute-squared Schmidt modes of the measured JSI, assuming a flat phase, since the phase information is not present in this measurement. (c) The Schmidt coefficients c_k describing the relative amplitudes of each spectral mode.

The important question is how much squeezing could we generate using such a source? From the photon number of the states versus pump power, shown in figure 5 as well as the full photon-number distribution presented in [18], we believe that the spectral properties do not change at least up to mean photon numbers of 20, corresponding to 19 dB of squeezing. The time-ordering effect should show a deviation from the curve in figure 5, which does not happen. This might be explained by the fact that we use a broad spectral filter with a width very close to the generated spectrum: since time-ordering predicts a spectral broadening of the down-conversion modes, a spectral filter might suppress this effect. Nevertheless, even below this regime of up to 20 photons we expect very strong squeezing producible from the waveguide. In the implementation of [18], the squeezing would be limited by the total losses of around 35%, as other limiting factors such as noise can already be excluded from the clean photon-number distribution. These losses so far are mainly due to optics and couplings outside of the waveguide. In an integrated solution or an extremely loss-optimized set-up, the squeezing would ultimately be limited by the loss inside the waveguide. For KTP, even low loss waveguides still have losses above 0.2 dB cm⁻¹. One might expect that this sets a bound on the achievable squeezing similar to cavity-based systems where the limit is given by the round-trip loss. For a single-pass source however, such a hard limit is not present [33]. In figure 5, we plot the expected squeezing for PDC for realistic loss parameters, using the equation given in [33]. From these estimations, we can conclude that squeezing above 20 dB is experimentally feasible, provided that the detection itself allows one to measure such values. For comparison, the highest squeezing directly measured is 15 dB [34] in a continuous-wave cavity system.

Figure 5.

(a) Measured mean photon number with a single-mode fit , where α is a fit parameter. (b) Theoretical squeezing taking into account uniform losses along the waveguide. We assume here losses between 3.6% (0.2 dB cm⁻¹) and 10% (0.66 dB cm⁻¹). At a mean photon number of 20, squeezing of 17 dB is realistic. With increasing pump power, the Heisenberg limit V (x)V (y)=1 becomes increasingly exceeded. However, the squeezing is not limited in principle. Even 25 dB of squeezing in a waveguide might be experimentally feasible. (Online version in colour.)

4. Conclusion

We have analysed the prospect of bringing squeezing and mode-selective frequency conversion together in a waveguide system. We have presented an analogy between the QPG and spatial networks, simplifying the interpretation of the pulse gate when operating on multi-photon states. This allows one to visualize the operation needed for entangling squeezed states or constructing complex multimode continuous-variable states. We have further estimated the ultimate squeezing achievable in a KTP single-pass, single-mode waveguide. While it seems feasible to generate very large squeezing of the order of 20 dB, operating on these states with currently implemented pulse gates would significantly degrade the squeezing. This is mainly due to the complicated behaviour of the process when approaching high conversion efficiencies. Not only is the conversion efficiency limited to below 90%, the phasematching and thus the mode structure change significantly when approaching these values. In applications where only low conversion efficiency is required, e.g. mode-selective photon subtraction, this does not pose a problem and the phasematching can be engineered using a simple model without pump-power dependence. In the high-conversion regime, however, these limitations significantly reduce the performance of the system. Nevertheless, an experimental conversion efficiency of 90% is very promising, and further progress in theory and experiment is ongoing.

Authors' contributions

G.H., B.B. and C.S. developed the concepts presented in this article. V.A. obtained and analysed the experimental PDC data. G.H. and T.J.B. evaluated the theoretical squeezing limit. G.H. drafted the manuscript. C.S. supervised the work. All authors read and approved the manuscript.

Competing interests

The authors declare that there are no competing interests.

Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 665148, and the DFG via the Gottfried Wilhelm Leibniz-Preis.