Filtration and Extraction of Quantum States from Classical Inputs
aa r X i v : . [ qu a n t - ph ] A ug Filtration and Extraction of Quantum States from Classical Inputs
Chang-Ling Zou , , , Liang Jiang , ∗ Xu-Bo Zou , , † and Guang-Can Guo , Key Lab of Quantum Information, University of Science and Technology of China, Hefei 230026 Department of Applied Physics, Yale University, New Haven, CT 06511, USA and Synergetic Innovation Center of Quantum Information & Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China (Dated: October 10, 2018)We propose using nonlinear Mach-Zehnder interferometer (NMZI) to efficiently prepare photonicquantum states from a classical input. We first analytically investigate the simple NMZI thatcan filtrate single photon state from weak coherent state by preferrentially blocking two-photoncomponent. As a generalization, we show that the cascaded NMZI can deterministically extractarbitrary quantum state from a strong coherent state. Finally, we numerically demonstrate that thecascaded NMZI can be very efficient in both the input power and the level of cascade. The protocolof quantum state preparation with NMZI can be extended to various systems of bosonic modes.
Introduction.-
Integrated photonics can achieve un-precedented interferometric stability [1, 2] and build largescale interferometers [3, 4]. However, reliable quantumstate preparation for integrated photonics remains an im-portant challenge, because interferometers and coherentinput states are insufficient for quantum states prepa-ration. We may use either post-selection or nonlinearinteraction to overcome this challenge. The approach ofpost-selection only requires linear optical elements andphoton detectors, but the preparation of quantum stateis probabilistic and conditioned on the outcome of theprojective measurement [5–7]. The approach of nonlin-ear interaction assisted by an ancillary two-level system(TLS) can deterministically prepare arbitrary quantumstate of the photonic mode [8–10], but it requires strongcoupling between the optical mode with the single TLS,which is experimentally challenging for integrated pho-tonics. Alternatively, we may consider using the nonlin-ear optical waveguide combined with ultra-stable inter-ferometers to achieve reliable quantum state preparation,without requiring TLS [8–10], post-selection [11–13], norfeedback/feedforward control [14].In this Letter, we propose to use interferometry com-bined with Kerr nonlinearity to filtrate single photons orextract any desired quantum states from coherent stateinput, as illustrated in Fig. 1(a). We first present theidea of quantum state filtration (QSF) of single pho-tons, which keeps the desired single photon componentby blocking the undesired component to a different port.We then generalize the idea to quantum state extraction(QSE), which not only keeps the desired component, butalso extracts the desired component from the undesiredcomponent before blocking/redirecting the residual pho-tons.
Single photon filtration.-
We first consider the simpletask of QSE of single photons. As shown in Fig. 1(b), weuse a nonlinear Mach–Zehnder interferometer (NMZI),with a Kerr nonlinear medium in one of the arms. SinceKerr nonlinearity can induce photon number dependentphase shift, we can design the NMZI to induce destruc- (a)(d)(e) (b) AB BS1 BS2 K e rr |a Laser || Q SF / Q S E (c) Figure 1: (color online) (a) Schematic illustration the arbi-trary quantum state filtration and extraction from coherentstate input. (b) The configuration for QSF of single photonfrom coherent state input | α i , using the simple NMZI (con-sisting of Mach–Zehnder interferometer, and Kerr mediumand phase shifter). (c) Three processes for two photon out-put of Path A for weak coherent input. (d) The probabilitiesof n photons output of Path A against the phase differencebetween two arms φ , with ϕ = 0 . α = 0 .
1. (e) Thesecond-order correlation function ( g (2) ) of light output of PathA against φ for various ϕ with α = 0 . tive interference at the output port when there are twophotons. More specifically, with a vacuum input at PathA (upper path) and a coherent state input at Path B(lower path), the input state to the filtration is | ψ i in = | vac i A ⊗ | α i B , (1)where | α i = e −| α | / P ∞ n =0 α n n ! ( b † ) n | vac i , and a ( a † ) and b ( b † ) are annihilation (creation) operators for Paths Aand B, respectively. Each beam splitter (BS) induces aunitary evolution, U BS ( θ , ) = e iθ , ( a † b + ab † ) , (2)with θ and θ for BS1 and BS2, respectively. The evo-lution in the nonlinear Kerr medium in Path A is U K ( φ, ϕ ) = e iφa † a + iϕa † a † aa , (3)where ϕ is the Kerr coefficient and φ is linear phase shift(relative to Path B). The final output state of the singlephoton filtration is | ψ i out = U BS ( θ ) U K ( φ, ϕ ) U BS ( θ ) | ψ i in = ∞ X p =0 ∞ X q =0 µ p,q ( a † ) p ( b † ) q | vac i , (4)with µ p,q = P pl =0 P l + qn = l λ n,p + q − n (cid:0) nl (cid:1)(cid:0) p + q − np − l (cid:1) × ( − n − l (sin θ ) p + n − l (cos θ ) q − n +2 l . The probabilityof p photons at the output of Path A is P p = h p | Tr B {| ψ i h ψ |} | p i = ∞ X q =0 p ! q ! | µ p,q | . The second-order correlation function [15] is g (2) = (cid:10) a † a † aa (cid:11) h a † a i = P ∞ p =2 p ( p − × P p ( P ∞ p =1 p × P p ) , (5)which characterizes the generated single photon state.For a weak coherent input | α | ≪
1, we have P ≪ P and can safely neglect the probability of multiple photons( P n ≥ ). By considering the leading contribution, we have g (2) ≈ P P ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ , µ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − − e i ϕ (1 − ηe − iφ ) (cid:12)(cid:12)(cid:12)(cid:12) , (6)where µ , = α cos θ cos θ ( e iφ − η ) and µ , = ( α cos θ cos θ ) [ − ηe iφ + e i φ + ϕ ) + η ] for the simpleNMZI with η = tan θ tan θ . The three terms in µ , cor-respond to three different processes with two photons atthe output of Path A, as shown in Fig. 1(c). The inter-ference of these three processes can be controlled by thelinear phase shift ( φ ) and nonlinear coefficient ( ϕ ). Theoptimal condition for µ , = 0 is ηe − iφ = 1 ± p − e i ϕ , (7)which can always be fulfilled as long as ϕ = 0, so thatthe leading contribution to g (2) can be eliminated.Fig. 1(d) shows the probability of n photons at the out-put of Path A ( P n ) depending on the linear phase shift φ ,with parameters ϕ = 0 . η = | −√ − e i ϕ | and α = 0 . P is greatly suppressed for φ ≈ . π , while the dominant single photon emission P ≫ P , , is not significantly affected. In Fig. 1(e), the relation be-tween g (2) and φ are plotted for different values of non-linear coefficient ϕ , with α = 0 . η given by optimalcondition from Eq. (7). We find good agreement betweenthe approximated analytical solution from Eqs. (6)&(7)(solid lines) and the exact numerical solution from Eq.(5) (dashed lines). With increasing nonlinear coefficient ϕ , the deviation from g (2) = 1 becomes more signifi-cant, due to the Fano interference of the three processes(Fig. 1(c)) contributing to µ , . These Fano-like curvesshow sub-Poisson statistic with g (2) ≈ φ close tothe optimal condition (Eq. (7)), where the two photonoutput can be totally forbidden due to destructive in-terference. Meanwhile, we can also find the constructiveinterference of the two-photon output, which gives rise tosuper-Poisson statistic ( g (2) (0) ≫
1) output. Comparingthe curves with different nonlinear effect coefficients, thesingle photon filtration is more sensitive to phase φ forsmaller ϕ , indicating the crucial role of nonlinearity.For QSF of single photon, the fidelity is F = P =(cos θ cos θ ) | α | (cid:12)(cid:12) − e i ϕ (cid:12)(cid:12) . The optimal condition re-quires η = tan θ tan θ ≈
1, we have | cos θ cos θ | < and P < ϕ | α | /
2, which implies that the fidelity depends onboth the the Kerr nonlinearity coefficient and the inten-sity of the coherent state input. QSF with simple NMZIcannot suppress the components with n > (cid:12)(cid:12) α (cid:12)(cid:12) ≪
1, which significantly limits thefidelity. Moreover, the fidelity of QSF is fundamentallylimited by the overlap between the input state and thetarget state, P succ < |h ψ out | ψ in i| , because it blocks allundesired components. To go beyond this limit, we needto generalize QSF to QSE, which not only keeps the de-sired component, but also extracts the desired componentfrom the undesired ones. Cascaded filtration.-
To implement QSE, we considerthe cascaded NMZI, with a series of NMZIs connectedsequentially. As shown in Fig. 2(a), the basic elementconsists of a BS ( θ ) followed by a linear phase shifter( φ ) and a Kerr medium ( ϕ ) in the upper path. Thebasic element can be represented by a standard two-portunitary [Fig. 2(b)] U ( φ, ϕ, θ ) = U K ( φ, ϕ ) U BS . (8)The cascaded MNZI with N elements can be character-ized by U N = Π Nl =1 U ( φ l , ϕ l , θ l ) . (9)For example, the simple NMZI (Fig 1(b)) consists of N =2 basic elements, with φ = ϕ = 0.The cascaded NMZI can not only keep the desiredsingle-photon component, but also extract the (desired)single-photon state from (undesired) multi-photon states,as long as there are enough photons in the undesired com-ponent. We numerically optimize the fidelity by tuning (a) (b)(c)(d) U( ) j,f,qj,f AB q Figure 2: (color online) Cascaded NMZI. (a) The basic ele-ment consists of a BS ( θ ) followed by a linear phase shifter( φ ) and a Kerr medium ( ϕ ) in the upper path. (b) Schematicrepresentation of the element for cascaded NMZI. (c) Fidelityof the single photon extraction increases with the number ofcascade elements, N . The parameters are optimized numeri-cally under the constrain P n ≥ < .
01 with ϕ = 0 .
1. (d) Fi-delity of Fock state extraction ( n = 1 , ,
3) increase with | α | for cascaded NMZI with N = 40. The results are obtainedby optimize the parameters of each unit under the constrainthat 1 − P − P n ≤ . the parameters of the N elements. As illustrated in Fig.2(c), the optimized fidelity of single photon extraction F = P increases with N monotonically, with asymptoticvalue F → − |h | α i| (dashed lines), because our pas-sive device cannot extract single photon from the vacuumcomponent. Furthermore, the cascaded NMZI can ex-tract Fock state | n i with n = 1 , , , · · · . The asymptoticfidelity of n -photon extraction is F → − P n − m =0 |h m | α i| ,which can be achieved for | α | ≤ . /n with cascadedNMZI of N = 40 elements, as shown in Fig. 2(d). Arbitrary state Extraction.-
Remarkably, the cascadedNMZI can extract arbitrary superposition of Fockstates with a large coherent state input ( | α | ≫
1) withalmost perfect fidelity. For θ l ≪ l = 1 , · · · , N ,almost all input photons will be guided in Path B,which effectively remains as a coherent state (with smalldeviation of O ( θ )) for all intermediate stages. The effectof each beam splitter to the upper path can be regardedas an effective displacement operation to Path A, as D ( ǫ l ) = e ǫ l a † − ǫ ∗ l a with ǫ l = αθ l and a small deviationof O (cid:0) ǫ l /α (cid:1) [16]. In addition, the linear phase shiftand Kerr nonlinearity can achieve the unitary evolution U K ( φ l , ϕ ) = e iφ l a † a + iϕa † a † aa . Hence, the cascadedNMZI of N elements can induce the unitary evolution (b) P r obab ili t y P P (a) F , Q | | Fidelity Purity
Figure 3: QSE for | ψ target i = ( | i + | i ) / √
2. (a) The prob-ability of | i and | i , (b) the Fidelity ( F ) and Purity ( Q ) ofthe output. The results are obtained by optimize the param-eters of a chain of N = 20 units under the constrain that1 − P − P ≤ . U K ( φ N , ϕ ) U ( ǫ N ) · · · U K ( φ , ϕ ) U ( ǫ ) U K ( φ , ϕ ) U ( ǫ ),which in principle can accomplish any desired unitarytransformation for sufficiently large N and carefully cho-sen { φ l , ǫ l } l =1 , ··· ,N [17–19]. Despite the large overheadin N , this provides a generic approach using cascadedNMZI to extract arbitrary superposition of Fock statesfrom a large coherent state with almost perfect fidelity.In practice, it is favorable to design the cascadedNMZI with a small number of elements. To illustratethe feasibility, we consider the target state | ψ target i =( | i + | i ) / √ N = 20 cascaded elements optimizethe fidelity by tuning parameters of { ϕ l , φ l , θ l } l =1 , ··· ,N .As illustrated in Fig. 3, we can improve the fidelity F and purity Q = Tr( ρ A ) of the extracted state by increas-ing | α | . Both F and Q are greater than 97 .
5% when | α | ≥ .
5. It’s intriguing that a high fidelity QSE ofsuperposition of Fock states can be achieved using a rea-sonable size coherent state and a finite-stage cascadedNMZI.
Discussion.-
The photonic integrated circuits (PICs)provides a promising platform for realizing cascadedNMZI, where arrays of beam splitters and phase shifterscan be integrated on a chip [3, 4]. QSE can providearbitrary input photon states for quantum informationprocessing [1, 7]. For experiment realization, the mostchallenging part is the Kerr nonlinear at single photonlevel ( ϕ = 0 . Conclusion.-
We have demonstrated that the simpleNMZI can filtrate single photon state from a weak coher-ent state. Using cascaded NMZI, we can reliably extractarbitrary quantum state from a strong coherent state.Since our scheme only requires Kerr nonlinearity, linearphase shifter and beam splitter, it can be implementedin superconducting circuits, coupled optomechanical sys-tems, as well as photonic integrated circuits.C.L.Z. thanks Hailin Wang and Hong-Wei Li for fruit-ful discussion. This work is supported by the “StrategicPriority Research Program(B)” of the Chinese Academyof Sciences (Grant No. XDB01030200), National BasicResearch Program of China (Grant Nos. 2011CB921200and 2011CBA00200). LJ acknowledges support from theDARPA Quiness program, the ARO, the AFSOR MURI,the Alfred P Sloan Foundation, and the Packard Foun-dation. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. 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Contents
References I. Limitation of Two-Unit Quantum State Filtration S2 II. The Dependence on Kerr nonlinearity S3 III. Imperfections S3 IV. Proof of High Fidelity
S42
I. LIMITATION OF TWO-UNIT QUANTUM STATE FILTRATION -2 -1 -3 -2 -1 -2 -1 -2 -1 -4 -3 -2 -1 -4 -3 -2 -1 (c)(d) n=3n=4n=2n=0 n=1n=1 (b) P r ob a b ilit y (a) n=1 g ( ) Eq. (5) P n 2 <0.01 P n 2 <0.001 P =0.1 =0.5 =1.0 =0.1 =0.5 =1.0 g ( ) P n 2 Figure S1: The probability of n -photon (a) and the photon statistic g (2) (b) at output for increasing input coherent laserintensity | α | . The solid lines are results for optimal conditions from Eq. (7), the dashed lines are numerically optimized resultsfor constrains that P n ≥ < .
01 and P n ≥ < . P (c) and g (2) (d) against the optimization constrain P ≥ Since the fidelity of single photon filtration (the brightness of the single photon source) is limited as P < ϕ | α | / | α | numerically. Shown in Fig. S1(a) by solidlines, the probabilities of Fock state outputs P n ( n = 0 , , , ,
4) against | α | is calculated, for ϕ = 0 . P increases linearly with | α | as expected, and g (2) also increases [Fig. S1(b)]. This means that the practical performance of the SPF deviated from the expectedperfect single photon source that g (2) ≈ | α | ≥
1. The imperfection for larger input coherent laser intensityshould be attributed to the contributions of multiple photons with n ≥
3. This can be inferred from the probabilityof P and P in Fig. S1(a), where the P is approximately linearly depends on P . This indicating that the 2-photonoutput comes from the 3-photon component of input state, which can’t be eliminated efficiently by two-unit QSF.Since the derivation of the optimal conditions is based on the assumption that | α | ≪
1, the optimal conditionmay not be valid when increase the input coherent laser power. Therefore, to gain P as large as possible but keepthe multiple photon probability as small as possible, we may optimize the parameters of QSF numerically for larger | α | . For example, we optimize the P under the constrain P n ≥ < . . | α | ≥
1, the performance of the QSF in both P and g (2) are improved. Forexample, with the input mean photon number | α | = 1, P = 0 .
047 and g (2) = 0 .
48 for the optimal condition from Eq.(7). With numerical optimization, we obtained the improved performance as P = 0 .
18 and g (2) = 0 .
48 for constrain P n ≥ ≤ .
01 and P = 0 .
10 and g (2) = 0 .
22 for constrain P n ≥ ≤ . P and g (2) for different constrain P n ≥ , there is a trade-off relation between P and g (2) due to the lack of ability to suppress 2-photon output from multiphoton input state.3 II. THE DEPENDENCE ON KERR NONLINEARITY (b) P (a) P N Figure S2: (a) The probability of single photon filtration by QSF against Kerr nonlinearity ϕ for different input coherent laserintensity | α | . (b) The probability of single photon extraction by N -unit QSE against unit number N for different ϕ . All resultsare optimized under constrains that P n ≥ < . In the main body of the paper, most results of QSF are studied for fixed nonlinearity parameter ϕ = 0 .
1. From theresults of QSF, the fidelity of single photon filtration is also limited by the Kerr nonlinearity in the NMZI. Therefore,we provide more results to study the dependence of the performance of QSF/QSE on ϕ .Shown in Fig. S2(a) are the fidelity of single photon filtration bu QSF against ϕ for different | α | . For both | α | = 0 . | α | = 1 .
0, the P increase with ϕ , confirms that the asymptotic formula P < ϕ | α | /
2, indicatingthat the performance of QSF by the simple NMZI can be improved by larger nonlinearity. By further increase ϕ , P reaches the saturation value when ϕ ≈ P by N -unit QSE against N for various ϕ are studied. It’s not surprising that the performanceof QSE is also improved by increasing N , and reaches the saturation value 1 − |h | α i| . Comparing different curves,the higher the nonlinearity is, the less units required to reach the maximum fidelity. È Α È P Figure S3: The P for different input coherent laser power. III. IMPERFECTIONS
To test the performances of QSE against the imperfections of parameters, we studied the performance of the singlephoton extraction with the optimized parameters for the QSE consist of N = 20 units with | α | = 1 . ϕ = 0 . Sample Number P P C o un t s Figure S4: (Left) The P for different samples of varied parameters. (Right) The histogram of the P for the random variationof parameters. In Fig. S3, the P for varying input coherent laser intensity is shown. In a wide range of input coherent state meanphoton number, the P is not deviated much from the optimized value.In Fig. S4, the P is tested with random perturbations of φ and θ from the optimized value. The perturbationsare randomly and uniformly distributed in the range from − . π to 0 . π . From the results, the performance of theQSE show certain tolerance to the parameter imperfections. IV. PROOF OF HIGH FIDELITY
As evidences shown in Figs. (2)&(3), the purity and fidelity of the QSF/QSE can approach unitary when | α | and N approaches infinity. Here, we provide the analytical analysis to support this argument. In the example, the targetstate is | ψ target i = ( | i + | i ) / √ . (S.1)In general, the output state can be represent by U | i A | α i B = | i A ∞ X n =0 c n ζ ,n | n i B + | i A ∞ X n =1 c n ζ ,n − | n − i B , (S.2)where U is unitary transformation, c n = α n √ n ! e −| α | / for coherent input | α i , and the coefficients satisfy (cid:12)(cid:12) ζ ,n (cid:12)(cid:12) + (cid:12)(cid:12) ζ ,n (cid:12)(cid:12) = 1.For N ≫
1, it’s possible to achieve near-perfect extraction of Fock states that generate ζ ,n = √ for all n . Then,we obtain the fidelity of the QSE as F = 12 (1 + ∞ X n =0 c n +1 c ∗ n ) . (S.3)For | α | ≫
1, it can be approximated as F ≈ − | α | , (S.4)where ∞ X n =0 c n +1 c ∗ n = ∞ X n =0 α √ n + 1 | α | n n ! e −| α | ≈ ∞ X n =0 [1 − n + 1 − | α | | α | + 38 ( n + 1 − | α | ) | α | ] | α | n n ! e −| α | =1 − | α | + 38 1 | α | ..