Limitations of two-level emitters as non-linearities in two-photon controlled phase gates
Anders Nysteen, Dara P. S. McCutcheon, Mikkel Heuck, Jesper Mørk, Dirk R. Englund
LLimitations of two-level emitters as non-linearities in two-photoncontrolled phase gates
Anders Nysteen, Dara P. S. McCutcheon, Mikkel Heuck, Jesper Mørk, and Dirk R. Englund DTU Fotonik, Department of Photonics Engineering,Technical University of Denmark, Building 343, 2800 Kgs. Lyngby, Denmark Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory andDepartment of Electrical and Electronic Engineering, University of Bristol,Merchant Venturers Building, Woodland Road, Bristol BS8 1FD, UK Department of Electrical Engineering and Computer Science,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: December 5, 2018)We investigate the origin of imperfections in the fidelity of a two-photon controlled-phase gatebased on two-level-emitter non-linearities. We focus on a passive system that operates withoutexternal modulations to enhance its performance. We demonstrate that the fidelity of the gateis limited by opposing requirements on the input pulse width for one- and two-photon scatteringevents. For one-photon scattering, the spectral pulse width must be narrow compared to the emitterlinewidth, while two-photon scattering processes require the pulse width and emitter linewidth to becomparable. We find that these opposing requirements limit the maximum fidelity of the two-photoncontrolled-phase gate to 84% for photons with Gaussian spectral profiles.
I. INTRODUCTION
Key requirements for the successful implementationof photonic quantum computing architectures are i)efficient sources of single indistinguishable photons,and ii) a method to coherently interact two such pho-tons [1–7]. Since these requirements were first stated,single photon sources have steadily improved [7–10],with the most promising platforms based on few-level-emitters, most notably semiconductor quantumdots [11–13] which now boast near-unity indistin-guishability with (source to first objective) efficien-cies above 70%. Generating photon–photon inter-actions can be achieved by ‘off-line non-linearities’consisting of measurements and feed-forward [1–4,7], or deterministically using ‘in-line’ non-linearitiesbased on a non-linear material through which twoor more photons interact [14–17]. These in-line non-linearities can in principle also be generated by few-level-emitters [18–20], suggesting a quantum photonicarchitecture in which few-level-systems act as bothphoton sources and photon couplers.Experimentally, probabilistic photonic gates havebeen demonstrated using off-line non-linearities inboth free-space [7, 21, 22] and in integrated plat-forms [5, 23]. Strong in-line non-linearities and photonswitching have been achieved using Rubidium atomsstrongly coupled to optical cavities [17, 24–26], quan-tum dots in photonic crystal cavities [27–30], and ni-trogen vacancy centers in diamond [31]. The poten-tially deterministic nature of few photon in-line non-linearities makes this approach particularly attrac-tive for the realisation of photonic gates, and a num-ber of proposals have been put forward to constructcontrolled-phase gates on various platforms and withvarious degrees of complexity [14, 15, 32–36]. Someproposals even have the potential to operate at nearunity fidelities by using distributed interactions [36]or pulse reshaping techniques [35], though these ap- proaches have the challenges of high complexity andpotentially high losses. Ultimately the usefulness ofa photonic gate in future quantum computing archi-tectures will depend on the ease with which it can beexperimentally realised and repeated, and the maxi-mum efficiency and fidelity that it can achieve.In this work we analyse the performance of per-haps the simplest deterministic passive controlled-phase gate, which acts on two uncorrelated indistin-guishable photons in a dual-rail encoding. The ide-alised gate we consider uses the in-line non-linearitiesof two two-level-emitters embedded in loss-less waveg-uides. The fundamental operating principle of thegate relies on the saturability of a two-level-emitter,which means that the phase imparted onto a photonor photons scattering on such an emitter depends onhow many photons are present [19, 20, 37, 38]. Al-though it has been shown that such a gate can neverperform with perfect fidelity [38, 39], the purpose ofthis work is to understand the limits and origins of itsimperfects with a view towards improved future im-plementations. Even in the loss-less case where thegate is fully deterministic, we show that the maxi-mum gate fidelity is limited to 84% for single photonswith Gaussian spectral profiles. This number is de-termined by opposing requirements on the spectralwidth of the input photons; one-photon scattering re-quires spectrally narrow photons so that the greatestfraction is strictly resonant with the emitters, whiletwo photon scattering requires photons with spectralwidths similar to the emitter linewidths, which max-imises saturation effects. Although the fidelities wecalculate are significantly less than unity, in contrastto other schemes, the present one does not use dy-namical photon capture methods [34], uses only two(identical) emitters per gate [36], and does not takeadvantage of possible pulse reshaping techniques [35],all of which are likely to introduce additional losses.This paper is organized as follows. In Section II a r X i v : . [ qu a n t - ph ] A p r the basic gate structure and components are intro-duced, and the gate operation in an idealised case isdiscussed. In Sections III and IV a more realistic sce-nario is analysed and the linear and non-linear gateoperations are described. A general fidelity measureis considered in Section V to quantify the gate perfor-mance, and we conclude our findings in Section VI. II. THE CONTROLLED-PHASE GATE c on t r o l (cid:1) /2 (cid:1) /2 '0''1' s i gn a l '0''1' (cid:1)(cid:1) c on t r o l '0''1' s i gn a l '0''1' '0''1''0''1'chiraltwo-way (b)(a) c on t r o l (cid:1) /2 (cid:1) /2 '0''1' s i gn a l '0''1' (cid:1)(cid:1) c on t r o l '0''1' s i gn a l '0''1' '0''1''0''1'chiraltwo-way (b)(a) c on t r o l (cid:1) /2 (cid:1) /2 '0''1' s i gn a l '0''1' (cid:1)(cid:1) c on t r o l '0''1' s i gn a l '0''1' '0''1''0''1'chiraltwo-way (b)(a) c on t r o l (cid:1) /2 (cid:1) /2 '0''1' s i gn a l '0''1' (cid:1)(cid:1) c on t r o l '0''1' s i gn a l '0''1' '0''1''0''1'chiraltwo-way (b)(a) Quantum Emitter c on t r o l (cid:1) /2 (cid:1) /2 '0''1' s i gn a l '0''1' (cid:1)(cid:1) c on t r o l '0''1' s i gn a l '0''1' '0''1''0''1'chiraltwo-way (b)(a) Directional Coupler c on t r o l (cid:1) /2 (cid:1) /2 '0''1' s i gn a l '0''1' (cid:1)(cid:1) c on t r o l '0''1' s i gn a l '0''1' '0''1''0''1'chiraltwo-way (b)(a) Phase Shifter c on t r o l (cid:1) /2 (cid:1) /2 '0''1' s i gn a l '0''1' (cid:1)(cid:1) c on t r o l '0''1' s i gn a l '0''1' '0''1''0''1'chiraltwo-way (b)(a) Mirror
FIG. 1. (a) Schematic of the controlled-phase gate, whichuses chiral waveguides, directional couplers, phase shifters,and two identical quantum emitters. The central idea ofthe gate is that the directional couplers act as 50/50 beamsplitters, and as such the input state | c (cid:105)| s (cid:105) gives rise toa Hong–Ou–Mandel bunching effect which can access theinherent non-linearities of the emitters. Only the | c (cid:105)| s (cid:105) input state bunches in this way, while all others transformlinearly, thus providing a fundamental non-linear interac-tion which can realize a two-photon gate. We focus on thechiral setup illustrated in (a), though an equivalent schemecan be realized with convention bi-directional couplers asshown in (b). Note that the ’1’ arm for the control andsignal is interchanged at the output ports in both cases. The structure implementing the gate, shown inFig. 1, consists of two phase shifters, two directionalcouplers, and two two-level emitters, similar to thesystems in Refs. [1, 35, 40]. We focus here on two-level-emitters, though we note also that very high Q-cavities could also be used [41, 42], in which split-ting of the energy spectrum arises though the non- linear Kerr effect [43]. In Fig. 1(a) we envisage chiralwaveguides, for which propagation is permitted onlyin one direction. We note, however, that an equiv-alent scheme can be realized by using standard bi-directional waveguides with emitters or perfectly re-flecting mirrors placed at their ends, as illustrated inFig. 1(b). For concreteness we focus on the chiralsetup of Fig. 1(a), though all of our subsequent anal-ysis equally applies to the two-way setup in Fig. 1(b).The central idea behind the scheme is that the com-ponents and waveguides are arranged in such a waythat only the combined control and signal input state | c (cid:105)| s (cid:105) accesses the non-linearity of the two-level sys-tems.To gain some intuition, we first consider quasi-monochromatic input photons, having a bandwidthmuch narrower than that of the emitters. Sincethe state of one photon can affect the state of theother, we must in general consider how pairs of pho-tons are transformed by the gate components. Con-sider first the evolution of two photons in the state | c (cid:105)| s (cid:105) . From Fig. 1 we see that these photons eachpick up a phase of ϕ , producing the transformation | c (cid:105)| s (cid:105) → e ϕ | c (cid:105)| s (cid:105) . For input states | c (cid:105)| s (cid:105) or | c (cid:105)| s (cid:105) , the photon in the | (cid:105) state again picks up aphase of ϕ , while the other passes through the direc-tional couplers and a two-level emitter. The direc-tional couplers act as 50/50 beam splitters, affectingthe mode transformation (cid:34) a † a † (cid:35) −→ √ (cid:20) − i − i 1 (cid:21) (cid:34) a † a † (cid:35) , (1)where a † | φ (cid:105) = | c (cid:105) and a † | φ (cid:105) = | s (cid:105) with | φ (cid:105) de-noting the vacuum. In this simplistic monochro-matic scenario, let us assume a single photon inci-dent on the emitter acquires a phase of θ . Thenthe combined effects of the two directional couplersand the emitter cause the transformation | s (cid:105) →− ie i θ | c (cid:105) and | c (cid:105) → − ie i θ | s (cid:105) . Therefore the pho-tonic states transform as | c (cid:105)| s (cid:105) → − ie i ϕ e i θ | s (cid:105)| s (cid:105) and | c (cid:105)| s (cid:105) → − ie i ϕ e i θ | c (cid:105)| c (cid:105) . Considering now theinput state | c (cid:105)| s (cid:105) , we find that the action of the firstdirectional coupler is to give rise to the Hong–Ou–Mandel interference effect; immediately after the firstdirectional coupler we have a state proportional to(( a † ) + ( a † ) ) | φ (cid:105) , in which two photons are inci-dent on each emitter in superposition. We denote thephase acquired by a two-photon state passing throughan emitter as χ , and therefore find that following thesecond directional coupler we have the transformation | c (cid:105)| s (cid:105) → ( − i) e i χ | c (cid:105)| s (cid:105) .Collecting these results and relabelling − i | s (cid:105) →| c (cid:105) and − i | c (cid:105) → | s (cid:105) we find | c (cid:105)| s (cid:105) −→ e ϕ | c (cid:105)| s (cid:105)| c (cid:105)| s (cid:105) −→ e i ϕ e i θ | c (cid:105)| s (cid:105)| c (cid:105)| s (cid:105) −→ e i ϕ e i θ | c (cid:105)| s (cid:105)| c (cid:105)| s (cid:105) −→ e i χ | c (cid:105)| s (cid:105) . (2)If the emitters acted as linear optical elements, wewould have χ = 2 θ . Absorbing the phases ϕ and θ into the definitions of | (cid:105) and | (cid:105) respectively, thetransformation is locally equivalent to the identity andtherefore does not mediate any two-photon interac-tion. However, if the emitter–photon interaction canbe tailored such that θ = ϕ and χ = 2 ϕ + π , the trans-formation in Eq. (2) becomes proportional to the de-sired control phase gate unitary diag(1 , , , − θ = ϕ and χ = 2 ϕ + π canbe met a controlled-phase gate is realized. Thoughwe do not expect this to be possible with perfect ac-curacy [38], in what follows we shall explore the dif-fering requirements on the pulse shape relative to theemitter linewidth which these conditions impose.In addition to the two-level-emitters, the other es-sential components of the gate are the directionalcouplers needed to produce the transformation inEq. (1) and induce the Hong–Ou–Mandel effect forthe input state | c (cid:105)| s (cid:105) . These components may berealized in various waveguide technologies, such assilica-on-silicon ridge waveguides [44], GaAs photonicridge waveguide circuits [45], photonic crystals waveg-uides [46], or silicon on insulator platforms [47], wherein all cases the length of the coupling region must beengineered such the symmetrical beam splitter rela-tion in Eq. (1) is achieved. We also note, that due tothe choice of directional coupler, the output port ofthe ‘1’ control and signal states are swapped, as indi-cated in Fig. 1(a). This amounts to nothing more thannotation, and could easily be rectified by introducinga crossover between the two ‘1’ outputs.For proper functionality of the gate, the input states | c (cid:105)| s (cid:105) , | c (cid:105)| s (cid:105) , and | c (cid:105)| s (cid:105) , which only experiencelinear scattering effects, and the input state | c (cid:105)| s (cid:105) ,which undergoes a non-linear transformation, must allprovide the desired output states in Eq. (2) when θ = ϕ and χ = 2 ϕ + π . These scattering-induced changesare investigated below, treating the linear and non-linear case separately. III. LINEAR GATE INTERACTIONS
Let us now consider the gate components inmore detail and analyse the conditions under whichthe scheme can be realized for more realistic non-monochromatic single photon inputs. We describe asingle photon in the | c (cid:105) state as | c (cid:105) = (cid:90) ∞−∞ d k ξ ( k ) a † ( k ) | φ (cid:105) , (3)where | φ (cid:105) is again the vacuum, ξ ( k ) is the spectralprofile of the photon satisfying (cid:82) ∞−∞ d k | ξ ( k ) | = 1,while a † ( k ) is the creation operator of photons in thecontrol ‘0’ waveguide with momentum k , satisfying[ a ( k ) , a † ( k (cid:48) )] = δ ( k − k (cid:48) ). We note that these con-ditions ensure the input state | c (cid:105) contains exactlyone photon, and we consider a rotating frame suchthat k is measured relative to the carrier momentum, k = ω /c . The simple transformations in Eq. (2)are not generally valid for photonic wavepackets com-prised by many k -modes because the phases ϕ and θ depend on k . In a large-scale system, the outputfrom one gate must function as the input to anothergate and they should therefore only differ by a time-translation, which in momentum space corresponds tothe transformation ξ ( k ) → ξ ( k )e iϕ ( k ) with ϕ ( k ) = ϕ + kL, (4)where L is an additional optical path length of the ‘0’waveguides, either induced by a change in the refrac-tive index of the material or by a longer arm length.When ϕ ( k ) is of the form in Eq. (4), the input state | c (cid:105)| s (cid:105) is described by a product of two single-photonstates of the form in Eq. (3), and we write the corre-sponding output state as | c (cid:105)| s (cid:105) → | ˜0 c (cid:105)| ˜0 s (cid:105) with | ˜0 c (cid:105) = − (cid:90) ∞−∞ d k ξ ( k )e ikL a † ( k ) | φ (cid:105) , (5)and a similar definition for | ˜0 s (cid:105) . Single photons withstates of this form will be considered our ‘ideal’ outputstates, since they are identical to the input state upto a linear frequency-dependent phase correspondingto a fixed temporal delay. The choice of ϕ = π hasbeen chosen in anticipation of the transformation ofthe | c (cid:105)| s (cid:105) state discussed below.We now consider changes to the two input stateswith a single photon in one of the ‘1’ arms, | c (cid:105)| s (cid:105) and | c (cid:105)| s (cid:105) . The photon in the ‘0’ arm is treated analo-gously to Eq. (5), while that in the ‘1’ arm instead in-teracts with an emitter. Photons passing through the‘1’ arms must also give rise to states differing from in-put states only by a time-translation. To see the con-ditions under which this is the case, we consider a non-monochromatic single photon as described by Eq. (3)scattering on a two-level emitter in a chiral waveg-uide. The photon will acquire a complex coefficient t ( k ) for each momentum component k , resulting in aphoton with spectral profile t ( k ) ξ ( k ). The frequency-dependent transmission coefficient is [48, 49], t ( k ) = k − ∆ − i(Γ − γ ) /v g k − ∆ + i(Γ + γ ) /v g , (6)where v g is the group velocity in the waveguide, ∆the momentum detuning of the emitter from the pulsecarrier frequency, Γ the emitter decay rate into waveg-uide modes, and γ the loss rate into modes outsidethe waveguide [49]. Recalling the effect of the direc-tional couplers, we find that the states transform as | c (cid:105)| s (cid:105) → − i | ˜0 c (cid:105)| ¯1 c (cid:105) and | c (cid:105)| s (cid:105) → − i | ¯1 s (cid:105)| ˜0 s (cid:105) , where | ¯1 c (cid:105) = (cid:90) ∞−∞ d k ξ ( k ) t ( k ) a † ( k ) | φ (cid:105) , (7)with a similar definition for | ¯1 s (cid:105) . In the loss-less case γ = 0 and | t ( k ) | = 1, meaning that (cid:104) ¯1 c | ¯1 c (cid:105) = 1 andthe output state contains exactly one photon. As pre-viously discussed, we can simply relabel what we refer k − ∆ (units of Γ / v g ) π π π /2 π /20 -2-4 0 2 4 ph a s e θ ( k )Norm. pulseLinearized θ ( k ) FIG. 2. Phase θ ( k ) acquired by a single-photonwavepacket component with moment k propagating in achiral waveguide scattering on a lossless resonant emitter(black solid line), together with a linear approximation, seeEq. (8) (green dashed line). By comparison, the spectrumof a resonant Gaussian wavepacket with spectral FWHMof σ = Γ /v g is shown (orange dotted line) with a scaledintensity to match the plotting window. to as the control and signal photons in the outputs,and absorb factors of − i in these definitions. We thenhave | c (cid:105)| s (cid:105) → | ˜0 c (cid:105)| ¯1 s (cid:105) and | c (cid:105)| s (cid:105) → | ¯1 c (cid:105)| ˜0 s (cid:105) .What is required, however, is that each photon hasa spectral profile identical to an ‘ideal’ state, | ˜1 c (cid:105) or | ˜1 s (cid:105) , defined as in Eq. (5) with a † replaced with a † or a † . Considering again the loss-less case where γ = 0we can write t ( k ) = exp[i θ ( k )]. The phase θ ( k ) isshown as a function of k in Fig. 2. If the incomingsingle-photon has a carrier frequency correspondingto the emitter transition frequency, ∆ = 0, the phasecan be Taylor expanded around k/ ˜Γ = 0, producing θ ( k ) = π + 2 k ˜Γ + O (cid:18) k ˜Γ (cid:19) , (8)where ˜Γ = Γ /v g . Keeping the condition ϕ = θ in mindand comparing Eqs. (4) and (8), we see that a goodgate performance requires | k | (cid:28) ˜Γ, which correspondsto pulses with a spectrum that is much narrower thanthe emitter linewidth. For spectrally broader pho-tons for which ξ ( k ) extends beyond k ∼ ˜Γ, terms ofhigher order in k will have an influence and introduce L [ v g / Γ ] σ [ Γ / v g ] | h ˜1 | ¯ i | FIG. 3. Overlap between the ideal and scattered statefor logical inputs | c (cid:105)| s (cid:105) or | c (cid:105)| s (cid:105) as a function of theadditional optical length of the ‘0’ arms, L , and the inputspectral width σ defined in Eq. (9). chirping effects [19, 20].To illustrate this in more detail for a specific case,let us consider a Gaussian single-photon wavepacket,defined by the spectral profile ξ ( k ) = ( πσ (cid:48) ) − / exp[ − k / (2 σ (cid:48) )] , (9)where the spectral bandwidth (FWHM of the intensityspectrum) is σ = 2 (cid:112) ln(2) σ (cid:48) . Fig. 3 plots the magni-tude of the overlap between the desired (ideal) stateand actual state for a Gaussian spectrum as describedabove. As expected, the overlap increases when thespectral width, σ , is decreased. The optimum ad-ditional path length, L , approaches L = 2 v g / Γ as σ is decreased, which is expected from the linear termin Eq. (8). For larger spectral widths, the optimum L decreases because a straight line with a slope smallerthan 2 v g / Γ approximates the phase θ ( k ) better in thiscase, as seen in Fig. 2. IV. NON-LINEAR GATE INTERACTIONS
The non-linear interaction occurs for the input state | c (cid:105)| s (cid:105) , where two photons may be present at thescatterers simultaneously, introducing non-linear in-teractions through a two-photon bound state [37].The non-linear scattering is treated by the scatteringmatrix formalism following Ref. [37], and we includethe directional coupler when calculating the scatteredstate of the entire gate. The gate input consists of twouncorrelated identical photons which we describe by | ψ in (cid:105) = ∞ (cid:90) −∞ ∞ (cid:90) −∞ d k d k (cid:48) ξ ( k ) ξ ( k (cid:48) ) a † c1 ( k ) a † s1 ( k (cid:48) ) | φ (cid:105) , (10)where as before (cid:82) ∞−∞ d k | ξ ( k ) | = 1 to ensure | ψ in (cid:105) con-tains two photons. Following the action of the first di-rectional coupler, scattering on the two-level emitters,and passing through the second directional coupler,we find | ψ in (cid:105) → | ψ scat (cid:105) with the total scattered stategiven by | ψ scat (cid:105) = ∞ (cid:90) −∞ ∞ (cid:90) −∞ d k d k (cid:48) β scat ( k, k (cid:48) ) a † c1 ( k ) a † s1 ( k (cid:48) ) | φ (cid:105) , (11)where we have removed a factor of ( − i) to be consis-tent with our definitions of the output states, and β scat ( k, k (cid:48) ) = β linearscat ( k, k (cid:48) ) + 12 b ( k, k (cid:48) ) , (12)with the linear contribution given by β linearscat ( k, k (cid:48) ) = t ( k ) t ( k (cid:48) ) ξ ( k ) ξ ( k (cid:48) ) and a non-linear scattering contri-bution by b ( k, k (cid:48) ) = (cid:90) ∞−∞ d p ξ ( p ) ξ ( k + k (cid:48) − p ) B kk (cid:48) p ( k + k (cid:48) − p ) . (13)The scatterer-dependent coefficient B kk (cid:48) pp (cid:48) is evalu-ated in Ref. [49] for a two-level system, B kk (cid:48) pp (cid:48) = i (cid:112) /v g π s ( k ) s ( k (cid:48) )[ s ( p ) + s ( p (cid:48) )] , (14)where s ( k ) = (cid:112) /v g k − ∆ + i(Γ + γ ) / ( v g ) . (15)The ideal output state in the non-linear case is | ˜1 c (cid:105)| ˜1 s (cid:105) = − ∞ (cid:90) −∞ ∞ (cid:90) −∞ d k d k (cid:48) e i ( k + k (cid:48) ) L a † c1 ( k ) a † s1 ( k (cid:48) ) | φ (cid:105) , (16)where the minus sign accounts for the required phaseflip that defines the controlled-phase gate.To gain some insight into how well the actual state, | ψ scat (cid:105) , approximates the ideal state in Eq. (16), weplot the magnitude of their overlap as a function of L and σ in Fig. 4, again for Gaussian input pulses. Incontrast to the one-photon scattering case in Fig. 3, wenow see that the largest overlap is observed for pulsewidths σ ≈ . /v g . This is because it is for thesewidths that the non-linearities are strongest and therequired π -phase shift can be generated, consistentwith the results in Ref. [20][51]. Furthermore, the op-timal value of L in this non-linear scattering case issignificantly lower than in the linear case. A compar-ison of Figs. 3 and 4 demonstrates that limitations inthe gate performance are expected to occur because ofthese different requirements on σ and L to optimallyapproximate the ideal output states in the linear andnon-linear cases, which we now explore in more detail. V. FIDELITY OF THE GATE OPERATION
In order to find the optimal spectral width σ andpath length difference L , we now consider the opera-tion of the gate as a whole. When incorporated intoa larger optical circuit, the logical input state of thegate will necessarily be unknown, and the gate must L [ v g / Γ ] σ [ Γ / v g ] | h ˜1˜1 | ψ s c a t i | FIG. 4. Overlap between the ideal and scattered state forthe | c (cid:105)| s (cid:105) input as a function of the additional opticallength of the ‘0’ arms, L , and the input pulse width, σ . L [ v g / Γ ] σ [ Γ / v g ] G a t e F i d e li t y FIG. 5. Gate fidelity as a function of the additional opticallength of the ‘0’ arms, L , and the input pulse width, σ . therefore be able to operate for any linear combina-tion of the four possible logical input states. As such,the gate performance must be quantified by a fidelitybased on a worst case scenario, in which the outputstate of the gate is compared to the ideal target outputstate, minimised over all possible input states. A gatefidelity meeting these requirements is defined as [50] F ( ˆ U , ˆ E ) ≡ min | ψ (cid:105) F s (cid:0) ˆ U | Ψ (cid:105)(cid:104) Ψ | ˆ U † , ˆ E ( | Ψ (cid:105)(cid:104) Ψ | ) (cid:1) , (17)where ˆ U and ˆ E describe the transformations of theideal and actual gate, respectively, and F s is the statefidelity defined by [50] F s (ˆ ρ, ˆ σ ) ≡ Tr (cid:26)(cid:113) ˆ ρ ˆ σ ˆ ρ (cid:27) , (18)for two density operators, ˆ ρ and ˆ σ . The arbitraryinput state, | Ψ (cid:105) , is given by | Ψ (cid:105) = (cid:0) α | s (cid:105) + β | s (cid:105) (cid:1) ⊗ (cid:0) ζ | c (cid:105) + ϑ | c (cid:105) (cid:1) = αζ | (cid:105) + αϑ | (cid:105) + βζ | (cid:105) + βϑ | (cid:105) , (19)where | s (cid:105)| c (cid:105) ≡ | (cid:105) etc. Using the definitions fromprevious sections, the ideal gate transformation isˆ U | Ψ (cid:105) = αζ | ˜0˜0 (cid:105) + αϑ | ˜0˜1 (cid:105) + βζ | ˜1˜0 (cid:105) − βϑ | ˜1˜1 (cid:105) . (20)If we neglect loss, the output states are pure andthe actual (possibly imperfect) transformation is de-scribed by ˆ E ( | Ψ (cid:105)(cid:104) Ψ | ) = ˆ T | Ψ (cid:105)(cid:104) Ψ | ˆ T † , withˆ T | Ψ (cid:105) = αζ | ˜0˜0 (cid:105) + αϑ | ˜0¯1 (cid:105) + βζ | ¯1˜0 (cid:105) + βϑ | ψ scat (cid:105) , (21)where | ψ scat (cid:105) is given by Eq. (11). For purestates, Eq. (18) simplifies to F s ( | a (cid:105)(cid:104) a | , | b (cid:105)(cid:104) b | ) = |(cid:104) a | b (cid:105)| ,and the state fidelity is therefore F s ( ˆ U | Ψ (cid:105)(cid:104) Ψ | ˆ U † , ˆ T | Ψ (cid:105)(cid:104) Ψ | ˆ T † ) = |(cid:104) Ψ | ˆ U † ˆ T | Ψ (cid:105)| = (cid:12)(cid:12)(cid:12) | αζ | + (cid:104) ˜1 | ¯1 (cid:105) (cid:0) | αϑ | + | βζ | (cid:1) −| βϑ | (cid:104) ˜1˜1 | ψ scat (cid:105) (cid:12)(cid:12)(cid:12) . (22)To find the fidelity of the gate for a given pulse widthand path length difference, this state fidelity must beminimised over all possible logical input states | Ψ (cid:105) parameterised by the coefficients α, β, ζ, ϑ . Since the | α | | ζ | S t a t e F i d e li t y FIG. 6. State fidelity as a function of input states ex-pressed by | α | and | ζ | for L = 0 . v g / Γ and σ = 1 . /v g corresponding to the maximum gate fidelity in Fig. 5. state fidelity only depends on the magnitude of the co-efficients and the signal and control input states bothmust be normalized, the minimization in Eq. (17) canbe carried out by varying only, e.g. | α | and | ζ | . Byperforming this minimization for different values of σ and L , the trade-offs due to the effects of linear andnon-linear scattering can be quantified. The result isshown in Fig. 5, where the gate fidelity is plotted as afunction of L and σ , again for Gaussian pulses. Theoptimum set of parameters is seen to be close to thatin Fig. 4 but shifted towards smaller pulse widths andlarger L , where the optimum was observed in Fig. 3.This trend is expected, since Eq. (22) effectively per-forms a weighted average of the overlaps in Figs. 3and 4. In order to confirm that the gate fidelityindeed corresponds to a worse case scenario, Fig. 6shows the dependence of the state fidelity on the in-put states for the optimum parameter set in Fig. 5. Itshows that the state fidelity approaches unity for thestate | c (cid:105)| s (cid:105) , and is above 84% for the entire statespace, as expected.Finally, we note that our formalism easily allows forspectra other than Gaussians to be considered. Mostnotably, we find that Lorentzian spectral profiles re-sult in a worse gate fidelity of F ≈ pulses achieve a fi-delity marginally better than Gaussian pulses, raisingthe gate fidelity by only 0 . VI. CONCLUSION
We have investigated in detail the feasibility ofusing two-level-emitter non-linearities to construct apassive two-photon controlled phase gate, elucidat-ing the non-linearity-induced changes in the spectrum.We find that these effects ultimately limit the fidelityof a controlled phase gate based on two-level-emitternon-linearities, giving F ≈
84% for Gaussian inputpulses, decreasing to F ≈
62% for Lorentzian spectra.We emphasise, however, that the scheme we considerrequires no dynamical capture of photons [34], usesonly two identical two-level-emitters, and does notmake use of pulse reshaping techniques. Althoughschemes making use of multiple non-linearities pergate [36], or gradient echo memory [35] to reversepulse shapes, predict theoretical fidelities approachingunity, these processes increase the complexity of thegate, and are likely to introduce additional losses. Ul-timately it seems likely that efficiency–fidelity trade-offs will be present in any gate scheme, and thesetrade-offs must be carefully considered in a larger pho-tonic network with a given application.
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