Efficient quantum computing using coherent photon conversion
N. K. Langford, S. Ramelow, R. Prevedel, W. J. Munro, G. J. Milburn, A. Zeilinger
aa r X i v : . [ qu a n t - ph ] J un Efficient quantum computing using coherent photon conversion
N. K. Langford , , , ∗ , S. Ramelow , , R. Prevedel , , W. J. Munro , G. J. Milburn , & A. Zeilinger , , ∗ Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics,University of Vienna, Boltzmanngasse 5, A-1090, Vienna, Austria Institute for Quantum Optics and Quantum Information (IQOQI),Austrian Academy of Sciences, Boltzmanngasse 3, A-1090, Vienna, Austria Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK Institute for Quantum Computing, University of Waterloo, Waterloo, N2L 3G1, ON, Canada NTT Basic Research Laboratories, NTT Corporation,3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan Centre for Engineered Quantum Systems, University of Queensland, St Lucia, 4072, Queensland, Australia
Single photons provide excellent quantum information carriers, but current schemes for prepar-ing, processing and measuring them are inefficient. For example, down-conversion provides her-alded, but randomly timed single photons, while linear-optics gates are inherently probabilistic.Here, we introduce a deterministic scheme for photonic quantum information. Our single, versatileprocess— coherent photon conversion —provides a full suite of photonic quantum processing tools,from creating high-quality heralded single- and multiphoton states free of higher-order imperfec-tions to implementing deterministic multiqubit entanglement gates and high-efficiency detection.It fulfils all requirements for a scalable photonic quantum computing architecture. Using photoniccrystal fibres, we experimentally demonstrate a four-colour nonlinear process usable for coherentphoton conversion and show that current technology provides a feasible path towards deterministicoperation. Our scheme, based on interacting bosonic fields, is not restricted to optical systems,but could also be implemented in optomechanical, electromechanical and superconducting systemswhich exhibit extremely strong intrinsic nonlinearities.
Photonic qubits were used in the earliest demonstrations of entanglement [1] and also to produce the highest-qualityentanglement reported to date [2, 3]. One of the key challenges for photonic quantum information processing (QIP)is to induce strong interactions between individual photons, which cannot be realised with standard linear opticalcomponents. The scheme proposed by Knill, Laflamme and Milburn for linear optics quantum computing [4, 5](LOQC) managed to sidestep this problem by using the inherent nonlinearity of photodetection and nonclassicalinterference to induce effective nonlinear photon interactions nondeterministically. Alternatively, in the one-waypicture of quantum computing, the required nonlinearities are replaced by off-line probabilistic preparation of specialentangled states followed by detection and feed-forward [6–8].Nonlinear optics quantum computing (NLOQC) takes a different approach by directly using intrinsic nonlinearitiesto implement multiphoton interactions. NLOQC schemes have been proposed using different types of optical nonlin-earities, including cross-Kerr coupling [9, 10] and two-photon absorption [11]. Since then, more complete multimodeanalyses of the cross-Kerr NLOQC schemes suggest that they cannot in fact produce phase shifts large enough forNLOQC because of spectral correlations created between the interacting fields [12–14]. Other work, however, showsthat these difficulties can be circumvented in the related case of strong χ (2) interactions by carefully engineering thephase-matching conditions [15].Here, we introduce an alternative nonlinear process— coherent photon conversion (CPC)—based on creating an en-hanced, tunable nonlinearity by pumping higher-order nonlinear interactions with bright classical fields, allowing thepotential for multiparty control of coherent nonlinear dynamics. A simple example of CPC gives an effective quadratic(three-wave mixing) nonlinearity by pumping one mode of a four-wave mixing interaction. We show that this exampleis an extremely versatile process which provides a range of useful photonic QIP tools, including deterministic two-qubit entangling gates based on a novel form of effective photon-photon interaction, high-quality heralded sources ofmultiphoton states and robust and efficient single-photon detection. Such tools are valuable building blocks in manyoptical quantum-enabled technologies. In particular, we outline a new approach for optical quantum informationprocessing based on CPC that fulfills all of the DiVincenzo criteria for a viable implementation of quantum compu-tation [16]. Moreover, it can also be used to produce heralded multiphoton entanglement, create optically switchablequantum circuits, convert arbitrary input states into high-purity, probabilistic single- and multiphoton Fock states,and implement an improved form of down-conversion with higher emission probabilities and lower higher-order terms.In standard photonic systems, cubic (four-wave mixing) nonlinearities in optical fibres have produced some of thehighest-brightness photon pair sources [17, 18], and the available precise dispersion engineering and fibre structuringtechnologies have allowed optimisation of these sources to produce ultrabright high-purity heralded single photons [19,20]. Here, we carry out proof-of-principle experiments using χ (3) photonic crystal fibres which demonstrate the four-mode interaction underlying CPC via photon doubling of a weak input state. Critically, we show that we can tune andenhance the effective χ (2) interaction strength by varying the classical pump power. We also use these experimentsto characterise the strength of the corresponding CPC interaction and outline how to reach the deterministic regimewith current technology. I. COHERENT PHOTON CONVERSION
The fundamental process underlying coherent photon conversion is a nonlinear interaction between m bosonic modeswhich coherently converts single excitations in some of the modes (depending on the precise form of the interaction),into single excitations in the remaining modes. A key principle of CPC is that this basic nonlinearity can in turn begenerated by pumping some modes of a higher-order nonlinearity with strong classical fields. This induces an effectivecoupling between the quantum modes which can be tuned (and enhanced) by the classical pumps. Indeed, it may bepossible to produce an effective interaction which is stronger than naturally occurring couplings of the same form. Asimilar effect is achieved in photon pair sources based on four-wave mixing in photonic crystal fibres, where very highpair rates are achieved with very low pump powers [17, 18]. To illustrate the potential of CPC, here we focus on anovel case which has very interesting properties for applications in quantum optics.Consider a standard four-wave mixing interaction involving four distinct frequency modes: H = γab † c † d + γ ∗ a † bcd † , (1)where the strength, γ , arises from the third-order ( χ (3) ) nonlinearity. Pumping one of the modes ( d ) with a brightclassical beam, E , yields the effective second-order interaction:˜ H = ˜ γab † c † + ˜ γ ∗ a † bc, (2)where ˜ γ ∝ γE . This now looks like a standard three-wave mixing Hamiltonian. To explain the basic CPC operation,here we use a simple single-mode “time-evolution” model where the modes satisfy energy conservation, ∆ ω = ω a − ω b − ω c + ω d = 0, and we for the moment ignore any higher level of sophistication introduced by imperfect phase-matchingand photon loss.The key to understanding how this process works, and its potential, is to look at its effect on input Fock states ofthe form | n a n b n c i . In particular, the Hilbert space defined by the set of states ˜ H j | n a n b n c i (for all integers j ) is agreatly restricted subspace of dimension ( n a + min( n b , n c )+1). Consequently, if the quantum state starts in a productnumber state, it will evolve entirely within this finite subspace and will therefore exhibit the collapses and revivals inindividual population elements which are characteristic of coherent quantum processes. The most important exampleof this for our scheme is the case which evolves within a novel two-dimensional subspace, {| i , | i} : specifically,˜ H | i ∝ | i and ˜ H | i ∝ | i . Thus, the coupling induced by the Hamiltonian then drives Rabi-like oscillationsbetween the two basis states, i.e., given | i as an input, the output state evolves as: | ψ ( t ) i = cos (Γ t ) | i + i ˜ γ | ˜ γ | sin (Γ t ) | i , (3)where Γ = | ˜ γ | / ~ .Interestingly, standard single photon up-conversion, a special case of CPC (indeed the simplest case), is one of thesmall subset of CPC processes with purely classical analogues: if a classical input field is used, then, provided this“pump” field remains undepleted throughout, the input field will undergo complete coherent oscillations between thetwo frequency modes. By contrast, if a classical input is used in the above example, then no coherent oscillationswill be observed—the output is the well-known two-mode squeezed state of a parametric down-conversion source. Inother words, in most cases a key element of CPC operation is the use of quantised inputs. II. TOOLS FOR QUANTUM COMPUTING
The DiVincenzo criteria describe the basic conditions for a viable implementation of quantum computing [16]. Themajor unresolved challenges for photonic quantum information are good multiphoton sources, reliable multiqubitinteractions, and robust, high-efficiency single-photon detection. We show here that CPC provides tools to solve allthree of these issues (Figs 1 and 2), all derived from a single process just by choosing different interaction strengths.Figure 1a illustrates how CPC directly implements a two-qubit controlled-Z (CZ) gate between the photons in thetwo modes b and c . The key insight is that CPC, like any coherent process which cycles between two orthogonal states,exhibits geometric (Berry’s) phase effects [21–23] (cf. the all-optical switch demonstrated in Ref. [24]). Therefore, for t = π/ Γ, an input state | i will undergo a full oscillation and pick up a π phase shift, giving the final state −| i . (c) | 〉 | 〉| 〉| 〉| 〉 …… π p p α | 〉+α | 〉+α | 〉+α | 〉 (a) CPCE d CPC π /2 | 〉 a | 〉 a | 〉 a E d E b E c | 〉 b | 〉 c CPCCPC α | 〉+α | 〉+α | 〉−α | 〉 (b) FIG. 1: Fulfilling the DiVincenzo criteria with CPC. (a) Deterministic controlled-phase gate. A “ π ” CPC interaction ( t = π/ Γ)is a novel type of effective photon-photon interaction, which implements an entangling CZ gate between two logical statesof frequency non-degenerate photons (e.g., polarisation or spatial encoding). (b) Scalable element for deterministic photondoubling. A “ π/
2” CPC interaction ( t = π/ Because this phase shift only occurs when two single photons are present, this controlled phase shift can be used toimplement a maximally entangling CZ gate with 100% efficiency. Note that this is a truly non-classical geometricphase which has no equivalent with classical input states. This CZ gate can also be switched very fast optically(by switching the bright classical pump beam in and out, cf. Ref. [24]), allowing the fast, real-time “rewiring” ofoptical quantum circuits. This may have application in various adaptive quantum schemes such as quantum phaseestimation or adaptive quantum algorithms and might be particularly useful in wave-guide and integrated-opticsarchitectures [25].If the input state is only allowed to undergo half an oscillation ( t = π/ coherently and deterministically (i.e. with 100% efficiency) into two single photons—a deterministic photon doubler [26](or alternatively a deterministic two-photon absorber in the reverse direction). Figure 1b illustrates one method forimplementing a scaleable photon doubler, allowing them to be chained together to create an arbitrary number ofphotons (see Supplementary Information). This efficient photon doubling cascade (Fig. 1c) can be used to createa high-quality, scaleable source of multiphoton states from any source of genuine single photons (on-demand orheralded). Note that the photon doubler can also be used in conjunction with existing methods to create arbitrary,heralded (perhaps also nonlocally prepared) entangled Bell-type two-photon [27] and GHZ-type three-photon [28]states (see Supplementary Information). Indeed, these tools can directly implement the encoding step for a simple9-qubit error-correction scheme [29].The same photon doubling cascade (Fig. 1b) can also be used to perform high-efficiency, low-noise detection withreal-world noisy, inefficient detectors (the photon doubler implements the quantum copier from [30]). The cascadecreates n copies of any single photon arriving at the detector and a detection event can then be defined to be a k -fold coincidence between any k detectors at the output. By choosing n and k appropriately, we can simultaneouslyboost the detector efficiency and reduce the noise from dark counts (see Supplementary Information). Interestingly,this technique can also produce marked improvements in detector characteristics, even when the photon doublingoperates at lower efficiencies ( η dbl < η dbl > / (2 − η ), where η is the single-detector efficiency. For example, for η = 50%, the doubling efficiency wouldneed to be greater than 2/3. Moreover, if the remaining mode can be detected as well, the final efficiency can beimproved for any value of η dbl .Finally, CPC can also be used to create a high-fidelity source of heralded single photons which could be used toseed the efficient photon doubling cascade described above. As noted previously, higher-order input states of theform | n a i will evolve within a restricted, ( n a +1)-dimensional Hilbert space. As with the qubit case, this leads tocoherent oscillations of population, as illustrated in Fig. 2a for n a = 1, 2 and 3, but their complexity increases rapidlywith larger n a , because the evolution is governed by an increasingly complicated distribution of eigenfrequencies popu l a t i on π π π π Γ τ p p p (a) (b) FIG. 2: Heralded single-photon source. (a) Evolution of | n a i populations under the CPC interaction for n a =1, 2 & 3.(b) Number-state populations after each filtering step for t = π/ Γ, giving | i a . Combined with a single photon-doubling stepand given a weak coherent state with | α | = 1 . ∼
56% and virtually no higher-order photon-number terms ( < . (e.g., n a = 3 in Fig. 2a; see Supplementary Information for details). As more competing frequencies come into play,for higher orders these oscillations are characterised by collapses and revivals in the input state population at oftenirregular times. Remarkably, these frequencies are incommensurate with the frequencies from other orders, so therevivals occur at different times for different input states (Fig. 2a).Consider therefore an input state in mode a which is a superposition (or mixture) of different n a , e.g., | ψ (0) i a = P j α j | j i a or | ψ (0) i a = | α i a (a “classical” coherent state). When the | i term has undergone one com-plete oscillation (i.e., same as the CZ gate, t = π/ Γ), all other terms will, with non-zero probability, have convertedinto states with photons in modes b and c , which can be rejected via spectral filtering. Applying this process repeat-edly will suppress all contributions from other orders, leaving only the | i a state with a finite probability (Fig. 2b).(Note: by detecting the dump port of the filtering step with high efficiency and rejecting events which lead to clicksin these arms, this acts like a pure Fock-state filter [31].) By combining this with a single coherent photon-doublingstep, it then becomes a heralded single-photon source.Currently, spontaneous parametric down-conversion (SPDC) and spontaneous four-wave mixing (SFWM) providethe best available sources of heralded single photons, but the performance and achievable rates of these sources areintrinsically limited by the effects of higher-order photon-number terms [32]. By contrast, given a simple weak coherentstate with | α | = 1 . ∼
56% and virtually no higher-order photon-number terms ( < . | i ) and to implement an improved form of down-conversion which can provide substantially higherpair-emission probabilities with much higher “heralded” state fidelity than a standard down-conversion source withcomparable emission rates (see Supplementary Information). III. EXPERIMENTAL COHERENT PHOTON CONVERSION
There are many different media that can be used to provide the type of χ (3) nonlinear interaction required forCPC. Some examples which are very promising for strong χ (3) interactions are standard optical fibres [33], photoniccrystal fibres [34], silicon waveguides [35] and EIT media [36, 37]. Although the χ (3) nonlinearity for a given materialis normally much weaker than the χ (2) nonlinearity for the same material (often by many orders of magnitude), theenhancement by the classical field can, for a sufficiently strong pump, result in an effective χ (2) nonlinear interactionwhich is stronger than the available natural χ (2) interaction. There are several key advantages to using such a pumped χ (3) interaction to produce an effective lower-order nonlinearity. Firstly, materials with inversion symmetry have no χ (2) nonlinearity ( χ (2) = 0), but all materials possess some χ (3) nonlinearity. Using the classical pump creates aquadratic nonlinearity which is tunable and no longer limited by fixed material properties. Finally, and perhaps mostimportantly, conservation of energy allows the four-mode interaction to take place between nearly degenerate frequencymodes, which makes CPC compatible with standard telecom-band fibre-based implementations, unlike standard χ (2) interactions, where the pump must by definition be roughly double the frequency of the other two photons.There are several possible types of four-wave mixing interactions, all of which occur simultaneously according tovarious selection rules given by symmetries in the χ (3) nonlinear susceptibility tensor and governed by phase-matching SMF SMFSMF S M F BeamDump B ea m D u m p s NF NFLP LP BP IF
MCA
SMF couplerpoln analysis Si - APDfilters
Beam Analysis
PCF
Beam Preparation
FIG. 3: Schematic of the experiment. The nonlinear medium is a standard commercial, polarisation-maintaining photoniccrystal fibre (PCF: 1.8 µ m core, nonlinearity ∼
95 (W km) − ). The χ (3) nonlinearity is pumped by a frequency-doubled pulsedneodymium vanadate laser (Nd:YVO : 532 nm, 7.5 ps, 76 MHz) to create the desired tunable effective χ (2) nonlinearity. Acontinuous-wave (cw), external-cavity diode laser (710 nm, ∼ × photons per Nd:YVO pulse) provides the input state inmode a which we use to characterise the strength of the CPC interaction. From estimated dispersion for the PCF, birefringentphase matching is satisfied for the following four-mode interaction: 532 nm (H) + 710 nm (H) →
504 nm (V) + 766 nm (V),where H and V denote horizontal and vertical polarisation, respectively. The 532 nm and 710 nm input beams are spatiallyfiltered with single-mode fibres (SMF) before being combined on a notch filter (NF) and coupled into the PCF. The beamsemerging from the output are then spectrally separated using a range of filters (NF, LP: long-pass, BP: band-pass) and a simplemonochromator (a rotating interference filter, IF), passed through polarisation analysers, and finally analysed in coincidenceusing time-to-amplitude conversion and a multichannel analyser (MCA). conditions. Some of these processes, like cross-Kerr and self-Kerr phase modulation, are automatically phase-matched,but they do not inhibit coherent photon conversion, because they only modify the phase-matching conditions of thetarget process, rather than competing with it. Of the other processes, careful engineering of the scheme and thenonlinear medium should be able to ensure that only one is phase-matched and the others can be neglected.In these first proof-of-principle experiments, we aim to demonstrate the viability of the nondegenerate four-modenonlinear interaction underlying the above CPC process. In particular, we wish to demonstrate the tunability ofthe effective χ (2) interaction and to estimate its efficiency to determine whether current technology offers a feasiblepath towards the deterministic regime. To provide the required χ (3) nonlinearity, we use a standard commercial,polarisation-maintaining nonlinear photonic crystal fibre (PCF) and pump it with a 532 nm pulsed laser. We useweak coherent states from a 710 nm diode laser in mode a and characterise the CPC interaction strength via theresulting double-pumped down-conversion pair source. With horizontally polarised input photons at 532 nm and 710nm, birefringent phase matching leads to vertically polarised output photons at 504 nm and 766 nm.The basic experimental design (Fig. 3) consists of three main stages: 1) state preparation, where the input beamsare prepared (polarisation and spatial mode) and combined; 2) nonlinear interaction, where the beams are coupledinto the nonlinear PCF for the CPC interaction; and 3) beam analysis, where the beams are separated and analysedafter exiting the PCF. Figure 4a illustrates the signal observed for around 90 mW of 532 nm and 8.5 mW of 710 nmaverage power in the fibre (both horizontally polarised), with both detectors set to analyse V. The diode laser deliversaround 10 photons per pulse during the 532 nm pulses, which are more than three orders of magnitude stronger. Thebackground signals measured when only one beam is present show clearly that the observed peak is a combined effectof both input beams. The full time trace allows us to correct very precisely for the periodic background and isolate thesignal that arises just from the CPC interaction (Fig. 4b). Polarisation and spectral measurements (see SupplementaryInformation) show that the interaction is strongly polarisation sensitive, making it suitable for implementing a CZgate directly in the polarisation degree of freedom, and that birefringent phase-matching ensures that the outputphotons emerge in two distinct spectral bands. Together, these measurements confirm that the observed photon pairsresult from the target four-colour nonlinear interaction which underlies our four-mode CPC process.Figure 4c shows the dependence of the pair-production rate on the pump power, which exhibits a linear trendwith exponential saturation at higher pump powers. The saturation arises predominantly from two technical effects,detector and counting saturation and reduced performance at high powers of generic single-mode fibres for spatialfiltering, both of which can be addressed in future experiments. We can now use the results from Fig. 4c to experi-mentally estimate the nonlinear interaction strength, Γ t , that appears in Eq. 3. The linear fit (for the points up to
10 20 30 40 50 60 700500100015002000 time delay (ns) r a w c o i n c i den c e s ( i n s ) (a) 10 20 30 40 50 60 702004006008001000 time delay (ns) c o rr e c t ed c o i n c i den c e s ( i n s ) (b)0 0 20 40 60 80 10000.511.522.53 x 10 pump power at 532nm (mW) c o rr e c t ed c o i n c i den c e s ( i n s ) (c) FIG. 4: Experimental results. (a) Photon doubling signal resulting from the four-mode nonlinear interaction which underliesour four-mode CPC. The signal is only observed when both input beams are present. The strong periodic background is causedby accidental coincidences from single photons created by Raman scattering from both beams, although very few accidentalsarise from just the 710 nm input as it creates very few Raman photons at 504 nm. (b) The full MCA trace allows us tocorrect very precisely for this periodic background and isolate a background-subtracted signal that arises just from the CPCinteraction. (c) The pair-production rate depends linearly on the pump power with some saturation at higher pump powers.The saturation arises predominantly from detector and counting saturation and the reduced performance at high powers of thegeneric single-mode fibres used to spatially filter the 532 nm beam before it is coupled into the PCF. In particular, the detectorsaturation results mainly from unwanted Raman scattering and it should be possible to dramatically suppress this effect bycooling the PCF. The linear fit (for the points up to 20 mW of pump) corresponds to a pair detection rate (for 8.5 mW 710nm) of 1 . ± .
02 pairs per second per mW of 532 nm pump power.
20 mW of pump) corresponds to a pair detection rate (for 8.5 mW 710 nm) of 1 . ± .
02 pairs per second per mWof 532 nm pump power. Taking into account the measured losses due to coupling and optical elements in the beamanalysis circuit ( ∼ .
6% in arm 1; ∼ .
6% in arm 2), this corresponds to a nonlinear interaction parameter insidethe PCF of Γ t ∼ × − per √ mW (estimated directly from Eq. 3). For a reasonable fibre-coupled pump powerof 1 W, this gives Γ t ∼ − . We might also expect to improve this by one or two orders of magnitude in futureexperiments with technical improvements, by specifically engineering the nonlinear and dispersion properties of longerPCFs and matching them with the optical wavelengths. Therefore, while reaching the deterministic regime (Γ t ∼ π/ χ (3) nonlinearity (which is proportional to γ ) is around 10 times larger than insilica [38]. IV. DISCUSSION
Coherent photon conversion is a classically pumped nonlinear process which produces coherent oscillations betweenorthogonal states involving multiple quantum excitations, providing a new way to generate and process complexmultiquanta states. Using higher-order nonlinearities with multiple pump fields also allows a mechanism for multipartymediation of the dynamics.One special case, based around a single pumped four-wave mixing process, on its own already provides a versatilearray of tools which could have significant impact as building blocks for many quantum technologies, includingquantum computing. In particular, we show that two of these building blocks are already sufficient to define a newCPC-based approach for photonic quantum computing that fulfills all of the DiVincenzo criteria, including providingdeterministic two-qubit entangling gates based on a novel type of effective photon-photon interaction induced byBerry’s phase effects, heralded multiphoton sources with almost no higher-order terms and efficient, low-noise single-photon detection using real-world detectors. Importantly, CPC could also provide strong benefits for optical QIPexperiments even for interaction strengths substantially less than is required for deterministic operation. For example,the photon doubler provides improved performance for single-photon detection, even when less than 100% efficient.Also, the photon doubler does not introduce any of the extra higher-order terms that limit the performance of down-conversion based photon sources [32]. As a result, even at low efficiencies, a CPC-based multiphoton source offers thepotential for higher multiphoton rates with much lower noise terms.Our experiments provide a proof-of-principle demonstration of the process underlying four-mode CPC, demonstrat-ing that an effective χ (2) nonlinearity can be produced and tuned in a material where such a nonlinearity in otherwiseunavailable. The results also suggest that operation efficiencies near 100% could be achieved by using current, albeitsophisticated PCF technology, e.g., based on chalcogenide glasses [38]. The availability of such nonlinearities in theoptical regime would open the possibility of large-scale QIP with single and entangled photons. Furthermore, sincefour-wave CPC is derived from a χ (3) nonlinear interaction where near-degenerate operation is possible, this processis therefore compatible with telecom technology (unlike normal χ (2) processes) and is also ideally suited to integratedoptics and waveguide applications.Finally, since our scheme is based only on interacting bosonic fields, its applications are not restricted to opticalsystems. It could also be implemented in optomechanical, electromechanical and superconducting systems whereextremely strong intrinsic nonlinearities involving vibrational [39–41] or matter-based [42, 43] degrees of freedom aremore readily available. Acknowledgements
The authors would like to acknowledge helpful discussions with T. Jennewein, A. Fedrizzi, D. R. Austin, T. Paterek,B. J. Smith, W. J. Wadsworth, M. Halder, J. G. Rarity, F. Verstraete and A. G. White. This work was supportedby the ERC (Advanced Grant QIT4QAD), the Austrian Science Fund (Grant F4007), the EC (QU-ESSENCE andQAP), the Vienna Doctoral Program on Complex Quantum Systems (CoQuS), the John Templeton Foundation andalso in part by the Japanese FIRST program. ∗ Corresponding authors: NKL (email: [email protected]) or AZ (email: [email protected]).
Appendix A: Supplementary Information1. Scaling up.
In order to achieve scaleable operation, we need a small number of basic units which can be simply connectedtogether in an appropriately designed network. For example, the wavelengths of the photons at the outputs of eachunit should be compatible with the input modes of the next unit. Here, we suggest two different approaches to achievethis goal. Both methods are based entirely on processes that use the same type of four-wave CPC interaction andcan in principle be made 100% efficient. Finally, both methods can also operate with four near-degenerate frequencymodes and are therefore compatible with the extensive standard telecommunications toolbox.
Method 1 (see Fig. 1b of main text), as explained already in the main text, builds the photon-doubling units fromtwo stages, both of which utilise the same CPC interaction. The first stage uses the basic CPC interaction to takea single photon at ω a and create single photons at ω b and ω c . The next stage modifies the same interaction simplyby adding an extra pump beam at either ω b or ω c , thus creating a process which implements single-photon frequencyconversion between the remaining mode ( c or b ) and the original mode, a . Thus, using the same CPC interactionand the same high-power pump at ω d , with the addition of two relatively low-power pumps at ω b and ω c , the outputphotons from the photon doubler can be individually converted back into photons at the original frequency, ω a . Therequired resources for this process scale linearly with the number of output photons (one CPC photon doubler andtwo CPC frequency converters per extra photon). The controlled-phase gate illustrated in Fig. 1a of the main text,which takes input photons at frequencies ω b and ω c , can be modified in a similar way using frequency conversionto build a unit which implements a controlled-phase gate between input photons at ω a . Of course, in “compiling”any larger network of such units, many of these frequency conversion stages are redundant and can be removed tominimise the total number of nonlinear interaction steps used. Method 2 takes a slightly different approach (Fig. 5). So far, we have considered only the CPC process involvingfour distinct frequency modes (described by the Hamiltonian H = γa † bcd † + γ ∗ ab † c † d ), but it is also possible toimplement a special case of this process when the two modes b and c are degenerate. Indeed, this is just the standardfour-wave mixing interaction used in most spontaneous four-wave mixing (SFWM) sources. In this case, the fullHamiltonian is: H = γab † d + γ ∗ a † b d † , (A1)and when mode d is pumped by a bright classical laser beam, E d , the effective Hamiltonian reduces to:˜ H = ˜ γab † + ˜ γ ∗ a † b . (A2)This modified, degenerate form of CPC also implements a photon doubling process, but takes a single-photon input at ω a and produces instead a two-photon Fock-state in ω b . Then, by embedding this in two arms of an interferometer, the π /2 E d | (cid:3822) b | (cid:3822) a E d | (cid:3822) a | (cid:3822) b | (cid:3822) a | (cid:3822) b | (cid:3822) b a bb π /2 E c | (cid:3822) a | (cid:3822) b E c | (cid:3822) b | (cid:3822) a | (cid:3822) b | (cid:3822) a | (cid:3822) a b aa a b aab aa a) b) c) BS BS
FIG. 5: Method 2: a) Single photon doubler for mode a , producing two photons in the degenerate mode b . Separating the twophotons into different modes is achieved via a “reverse Hong-Ou-Mandel”-type interaction at a beam splitter ( | , i + | , i →| , i ). With the pump field, E c , energy conservation is given by ω a + ω c = 2 ω b . b) Analogous photon doubler for mode b , butwith the roles of modes a and b swapped ( a now degenerate). With the different pump field, E d , energy conservation is nowgiven by ω b + ω d = 2 ω a . c) Cascaded concatenation of a) and b) to achieve scaling up to high number of photons. two degenerate and indistinguishable photons can be deterministically separated into different modes via a “reverseHong-Ou-Mandel”-type interaction at a beam splitter ( | , i + | , i → | , i ). Each of these photons (at ω b ) canthen in turn be converted in a similar way into two photons at ω a using the same CPC interaction with a differenthigh-power pump frequency. By alternating between these two processes, the photon doubling cascade can thereforebe scaled up to larger systems. As with Method 1, this approach is built using units with the same type of CPCinteraction, this time simply with two different pump frequencies. Once again, this scaled process requires only linearresources to create n -photon states (3 CPC photon doublers per extra 3 photons).
2. Entanglement sources.
The cascaded photon-doubling technique for efficiently generating multiphoton states is not limited to generatingsimple pure product states. Figures 6a and 6b illustrate CPC circuits for generating both bipartite and genuinetripartite entanglement. For example, if two polarisation states of an input single photon are split up in a polarisation-sensitive interferometer (e.g., a Mach-Zender interferometer using polarising beam displacers, or a Sagnac source [3])and if a CPC photon doubling interaction is applied to both polarisations, then the same process can be used as asource of polarisation-entangled photons (Fig. 6a). The same result could also be achieved using a “crystal-sandwich”arrangement. Existing experiments have already repeatedly demonstrated that these techniques produce extremelyhigh-quality entanglement, and by tuning the polarisation of the input single photon, we can generate different (bothmaximally and non-maximally) entangled states. This state can even be prepared nonlocally (Fig. 6c). Finally,Fig. 6d illustrates how these basic building blocks can be combined to directly implement the encoding step of anerror-correction protocol using a simple 9-qubit code.
3. Efficient detectors.
Figure 7a shows the effective detector efficiency for a 3-step cascade, which produces 8 (2 ) photons, and various k ,where k -fold coincidences are defined to signal successful detection events. When k = 1, it is clear that the detectorefficiency can be greatly increased, although there is naturally some trade-off when coincidence detection is usedto suppress the dark counts. Figure 7b illustrates how this scheme would improve both count rates and signal-to-noise ratio in an example where the single-detector dark counts are as large as the signal. Interestingly, withoutusing coincidence detection to suppress it, the effective dark count noise is actually higher for the cascade than for anindividual detector. This effect is quite pronounced in the example shown, because we deliberately chose an extremely ω d ω c ω b ω a ω a ω a ω a - ω b α| H 〉+β| V 〉 ω d ω a - ω c ω a - ω c ω a - ω b PBDHWP @ 45º α| HH 〉 +β| VV 〉 DC χ (2) “1” » trigger photonheraldedphoton α| H 〉+β| V 〉 α| HH 〉+β| VV 〉| φ + 〉| 〉 optionalfeed-forward α| H 〉+β| V 〉 @ 22.5º@ 22.5º α| HHH 〉+β|
VVV 〉α| H 〉+β| V 〉 (a)(b)(c) (d) FIG. 6: CPC circuits for creating (a) bipartite and (b) tripartite entanglement. (c) A CPC circuit for generating arbitrary,nonlocally prepared, nonmaximally entangled states. (d) A direct implementation of the error-correction encoding step for asimple 9-qubit code. t o t a l de t e c t ed c oun t s Single detector efficiency, η S i ng l e de t e c t o r e ff i c i en cy , η e ff (a) (b) k = 1 k = 2 k = 8 k = 7 k = 6 k = 5 k = 4 k = 3 Single detector efficiency, η total countsdark counts FIG. 7: Efficient detection via a photon-doubling cascade. (a) Effective efficiency, η eff , for detector cascade with 3 cascade steps( n =8) and a k -fold coincidence “clicks” between k =1 (red) and k =8 (violet). (b) Predicted counts for a simulated probabilisticexperiment with 10 rep. rate, 10 − incident photon probability and 10 − (individual detector) dark count probability (i.e.,SNR=1), using a detector cascade with n =8 and k between 1 (red) and 8 (violet). high dark count rate, but it is already overcome with k =2 only. In many practical situations, however, the raw darkcount probability is much less than the signal rate and this effect is not significant.0 target interaction targetstate Γ τ ( × π ) transmission | i | i / √ | i > . | i > . | i > . | i > . | i > . > . | i > .
4. Higher-order applications: Fock-state preparation.
Table I shows interaction lengths and target-state transmission probabilities for some early “revival” peaks fromdifferent photon-number input states, which have the potential to be used for Fock-state filtration. Because of thecomplexity of higher-order eigenvalues, longer interactions are generally required before a significant revival occurs,which is even then generally not 100% efficient. This lowers the success probability of the Fock-state preparation forhigher-order states, but for lower orders, quite pure Fock states can be prepared non-deterministically with relativelyfew interaction steps and quite high probability. For example, as mentioned in the main text, with ∼
56% probability,a single-photon state can be prepared with fidelity and purity greater than 99 .
6% using a weak coherent input state( | α | = 1 .
5) with only 5 steps.
5. Higher-order applications: improved down-conversion.
We now briefly outline how the CPC interaction can be used to implement an improved form of down-conversion.Specifically, consider an input state to mode a that contains higher-order Fock states, such as a weak coherent state orthe heralded single-photon state created by triggering from a down-conversion pair, and assume that the interactiontime corresponds to an integral number of complete oscillations of the | i input state (i.e., Γ τ =2 mπ/ √ | i term will have converted to | i with some finite probability, but the | i will haveremained unconverted with 100% probability, allowing the creation of correlated “down-conversion-like” photon pairs,with no | i term. Figure 8 shows the output of this process after each of the first three | i oscillation periods,given weak coherent input states with a range of average photon numbers. For both two and three periods (Fig. 8),the CPC process produces substantially higher pair-emission probabilities with much higher fidelity than a standarddown-conversion source with comparable emission rates. This technique can also provide improved higher-ordercharacteristics using a heralded single photon from standard down-conversion as the input.
6. Polarisation and spectral dependence of the CPC interaction.
Figure 9a shows the polarisation dependence of our four-wave mixing CPC interaction. The phase-matching in thePCF should ensure that the process can only take place for the correct combination of polarisations. The resultsshow that the target polarisation clearly gives the strongest signal, whereas combinations with equal polarizations(HHHH and VVVV) for all modes yield no coincidences within statistical error. For the other measured combinations,however, the signal did not completely vanish. We believe that this arises from a combination of two effects, namelyimperfect orientation of the polarisation-maintaining fibre FC-PC connectors and unwanted polarisation cross-talkfor the modes below the single-mode cut-off wavelength ( ∼ nm ). In other words, the signals still result from thetarget CPC interaction, but shows up in the incorrect polarisation because of polarisation cross-talk in the fibre eitherbefore or after the interaction.Figures 9b and c show the spectral and phase-matching dependence of our CPC interaction. Our single-photonspectrometers were constructed by placing rotating narrow bandpass interference filters (two per spectrometer) infront of multimode fibres. They were calibrated against a known spectrometer using bright light. The spectra1 |α| |α| |α| (a) (b) (c)single-pair probability fidelity total emission probabilitystandard down-conversion CPC “down-conversion” FIG. 8: Improved down-conversion using the CPC interaction. Overall single-pair probability per pulse (blue) and fidelity of a“heralded” state with the one-pair state (green) for a particular total photon-emission probability per pulse (red). Solid circlesshow the results for interaction times: (a) Γ τ =2 π/ √
6, (b) Γ τ =4 π/ √ τ =6 π/ √
6. For comparison, the lines show theresults for standard down-conversion with the same herald detection probability.
HH HH HH HH HV VH VV05k10k15kin c o rr e c t ed c o i n c i den c e s ( i n s ) HH HV VH VV VV VV VVout
503 504 5055k6k7k8k c oun t s wavelength (nm)504.1nm
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