From linear optical quantum computing to Heisenberg-limited interferometry

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1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS J. Opt. B: Quantum Semiclass. Opt. 6 (004) S796 S800 PII: S (04) From linear optical quantum computing to Heisenerg-limited interferometry Hwang Lee,PieterKok,,Colin P Williams and Jonathan P Dowling,3 Quantum Computing Technologies Group, Section 367, Jet Propulsion Laoratory, California Institute of Technology, MS 6-347, 4800 Oak Grove Drive, CA 909, USA Hewlett Packard Laoratories, Bristol BS34 8QZ, UK jonathan.p.dowling@jpl.nasa.gov Received 6 Novemer 003, accepted for pulication 5 Feruary 004 Pulished 7 July 004 Online at stacks.iop.org/joptb/6/s796 doi:0.088/ /6/8/06 Astract The working principles of linear optical quantum computing are ased on photodetection, namely, projective measurements. The use of photodetection can provide efficient nonlinear interactions etween photons at the single-photon level, which is technically prolematic otherwise. We report an application of such a technique to prepare quantum correlations as an important resource for Heisenerg-limited optical interferometry, where the sensitivity of phase measurements can e improved eyond the usual shot-noise limit. Furthermore, using such nonlinearities, optical quantum non-demolition measurements can now e carried out easily at the single-photon level. Keywords: quantum imaging, quantum metrology, shot-noise limit, Heisenerg limit, quantum computing. Effective nonlinearities from projective measurements Looking ack, scalale quantum computation with linear optics was considered to e impossile due to the lack of efficient two-quit logic gates, despite the ease of implementation of one-quit gates. Two-quit gates necessarily need a nonlinear interaction etween the two photons, and the efficiency of this nonlinear interaction is typically very tiny in ulk materials []. However, Knill et al recently showed that this arrier can e circumvented with effective nonlinearities produced y projective measurements [], and with this work scalale linear optical quantum computation (LOQC) ecomes areality. Let us consider the Kerr nonlinearity, which can e descried y a Hamiltonian [3] H Kerr = hκâ â ˆ ˆ, () where κ is a coupling constant depending on the third-order nonlinear susceptiility, and â, ˆ and â, ˆ are the creation and 3 Author to whom any correspondence should e addressed. annihilation operators for two optical modes. One convenient choice of the logical quit can then e represented y the two modes containing a single photon, denoted as 0 L = 0 l k L = l 0 k, where l, k represent the relevant modes, and we have used the notation L for a logical quit, in order to distinguish it from the photon-numer states k. For atwo-quit gate, let us assign modes, for the control quit, and 3, 4 for the target quit. Suppose now only the modes, 4 are coupled under the interaction given y equation (). For a given interaction time τ,thetransformation can e written as 0 L 0 L 0 L 0 L 0 L L 0 L L L 0 L L 0 L L L e iϕ L L, () (3) /04/ $ IOP Pulishing Ltd Printed in the UK S796

2 From linear optical quantum computing to Heisenerg-limitedinterferometry Ψin BS BS BS3 Ψ out D 0 D Figure. Adiagram for the nonlinear sign gate. Conditioned upon aspecific detector outcome, the desired output state can e otained y choosing appropriate transmission coefficients of the eam splitters. The success proaility of the gate operation is /4, ut we always know when it succeeds. Q BS NS BS Q NS Q Q Figure. Nondeterministic conditional sign-flip gate. The relevant optical modes are assigned as {,, 3, 4} from the top. When the modes, and 3 contain one photon each,,,3 ( L L ), it ecomes, 0,3 0,,3 after the first eam splitter (BS ). Passing through the nonlinear sign gates, it yields, 0,3 + 0,,3.The second eam splitter (BS conjugate to eam splitter ) then puts this into,,3.oviously, all other input states, 0 L L, 0 L L, L 0 L,arenot changed. where ϕ κn a n τ and n a = â â, n = ˆ ˆ. This operation yields a conditional phase shift [4]. When ϕ = π, we have the two two-quit gate called the conditional sign-flip gate. A typical two-quit gate, controlled-not (CNOT), is then simply constructed y using the conditional sign flip and two one-quit gates (e.g., Hadamard on the target, followed y the conditional sign flip and another Hadamard on the target). In order to have ϕ π at the single-photon level, however, a huge third-order nonlinear coupling is required [5]. Instead, Knill, Laflamme, and Milurn devised a nondeterministic conditional sign flip gate using nonlinear sign gate defined y α 0 + β + γ α 0 + β γ. (4) The nonlinear sign gate can e implemented nondeterministically y three eam splitters, two photodetectors, and one ancilla photon [6] (see figure ). The implementation of a conditional sign flip gate is then made y the comination of the nonlinear sign gate and the physics of the Hong Ou Mandel (HOM) interferometer [7]. For two aritrary quits Q =α 0 0 L + α L = α α 0, Q =α 0 0 L + α L = α α 3 0 4, the transformation of applying a condition sign flip gate can e written as Q Q α 0 α 0 0 L 0 L + α 0 α 0 L L + α α 0 L 0 L α α L L = α 0 α 0 0,, 0, + α 0α 0,,, 0 + α α 0, 0, 0, α α, 0,, 0, (6) where the modes and are designated for the control quit, and 3and4areforthetargetquit. A sign change happens (5) a a Figure 3. Mach Zehnder interferometer with two additional eam splitters, which direct the reflected eams to photodetectors. Conditioned on a specific outcome of photodetection, a desired output state can e prepared in the mode a and. only when there is one photon in mode and one photon in mode 3. The implementation of the desired operation is achieved y two 50:50 eam splitters and two nonlinear sign gates (see figure ), with proaility of success /6. Effectively then, a Kerr nonlinearity can e generated y linear optics and projective measurements. The proaility of success then can e oosted y using a gate-teleportation technique and a sufficient numer of ancilla photons. It has also een demonstrated that such a nondeterministic two-quit gate can e made for quits defined y the polarization degree of freedom [8, 9]. A general formalism for the effective photon nonlinearities generated y conditional measurement schemes in linear optics has een developed in some of our recent work [0]. Naturally, we emphasized that the aility to discriminate the numer of incoming photons plays an essential role in the realization of such nonlinear quantum gates in LOQC [ 3].. Optical lithography eyond the diffraction limit Since the projective measurement can produce an effective photon photon interaction, it can e a useful tool to manipulate quantum correlations etween photons. A particularly interesting type of quantum state of light is the maximally entangled photon-numer state. In our recent work, it has een shown that the Rayleigh diffraction limit in optical lithography can e overcome [4] y using a quantum state of light of the following form: ( N, 0 a + 0, N a ), (7) where a, denote two different paths. It is well known that the N = path-entangled state of equation (7)can e generated using an HOM interferometer and two single-photon input states. A 50:50 eam splitter, however, is not sufficient for producing path-entangled states with a photon numer larger than two [5]. Onthe other hand, the generation of these states with N > seemstoinvolvealargekerr nonlinearity, which makes their physical implementation very difficult [6]. Using the technique of projective measurement, we have shown that y conditioning on single-photon detection, the generation of path-entangled photon-numer states is possile for more than two photons [7, 8]. Figure 3 depicts a simple Mach Zehnder type interferometric scheme for producing such a state with N = 4, using dual Fock state inputs, N a N. S797

3 HLeeet al Suppose that we have the 3, 3 state asthe input entering into the modes a and. Then, the first eam splitter transforms 3, 3 into a linear superposition of 6, 0, 4,,, 4, and 0, 6. After passing through the two intermediate eam splitters, and if one and only one photon is counted at each detector, the state is then projected onto an equal superposition of 3, and, 3. Simply, the states 6, 0 or 0, 6 are discarded y this feedack from the photodetectors, since they cannot yield a click at oth detectors. The 4, and, 4 states, on the other hand, lose one photon in each arm of the interferometer and are projected to 3, and, 3, respectively. Thus, just efore the last eam splitter, we have a superposition of 3, and, 3 with a known phase. We use an appropriate phase shifter in one of the two arms of interferometer so that the state after the projective measurement is reduced to 3,, 3. Consequently after the last eam splitter, we get the desired state 4, 0 0, 4. We have further shown that it is possile to produce any twomode, entangled, photon-numer state with only linear optical devices conditioned on photodetection [8]. Although the proaility of success generally decreases exponentially as N increases [8 ], it was shown that the scaling can e suexponential in N y using quantum memory []. For some applications, however, it can already e useful to have fourphoton entanglement. Quantum interferometric lithography is such an example. Our approach has een used in a recent experiment to produce maximally entangled three- and fourphoton polarization states [3]. 3. Phase-noise reduction eyond the shot-noise limit In a typical optical interferometer, in which ordinary coherent laser light enters via one input port, the phase sensitivity in the shot-noise limit scales as ϕ = / N, where N is the mean numer of photons. Over the last two decades, a lot of effort was devoted to overcoming this limit, due to the ovious practical applications. In the early 980s, Caves first demonstrated that squeezing the vacuum noise in the unused input port of an interferometer causes the phase sensitivity to eat the standard shot-noise limit y / N / N in the limit of infinite squeezing [4]. Bondurant and Shapiro [5] proposed a multifrequency squeezed state interferometer for this same purpose. Haus pioneered in the generation of squeezed light in optical fires [6] as well as in Mach Zehnder interferometers [7], towards achieving the goal of Heisenerg-limited interferometry. On theother hand, in 986 it had een suggested y Yurke and y Yuen thatthe phase-noise reduction can also e achieved using inputs with numer eigenstates incident upon oth input ports of a Mach Zehnder interferometer [8, 9]. In particular, Yurke and collaorators showed that if the photons entered into each input port of the interferometer in nearly equal numers with a certain type of correlation, then, it was possile to otain an asymptotic phase sensitivity of /N,theHeisenerg limit [30]. The so-called Yurke state is of the form [ N, N a + N, N a ], (8) where a, denote the two input modes. Then, in the early 990s, Holland and Burnett proposed Heisenerg-limited interferometry y the use of so-called a a Figure 4. Asimplepath-entanglement generator. A Yurke-type quantum correlation etween the two modes can e produced with a dual Fock state. Suppose we post-select the outcome, conditioned upon only one photon detection y either one of the two detectors. Duetothe 50:50 eam splitter in the midway, it is not possile to tell whether mode a or lost one photon. The fundamental lack of which-path information provides the entanglement etween the two output modes. For two-fold coincidence detection, the two detected photons are from either mode a or mode, whicheliminates the possiility of peeling off one photon from each mode. dual Fock states of the form N, N a [3]. Such a state can e approximately generated y degenerate parametric down conversion or y optical parametric oscillation. In a conventional Mach Zehnder interferometer, only the difference of the numer of photons at the output is measured. However, to otain increased sensitivity with dual Fock states, some special detection scheme is required, for which Hall and co-workers proposed a comination of a direct measurement of the variance of the difference current as well as a dataprocessing method ased on Bayesian analysis [3]. Other types of special input states have een proposed for achieving the Heisenerg-limited phase sensitivity [33 35]. In particular, the Yurke state approach has the same measurement scheme as the conventional Mach Zehnder interferometer; a direct detection of the difference current [36]. It is, however, not easy to generate the desired correlation in the input state. On the other hand, the dual Fock state approach finds a rather simple input state, ut requires a complicated data processing method. However, y a simple utilization of the projective measurements with linear optical devices, it is possile to generate a desired correlation in the Yurke state directly from the dual Fock state. Let us consider a linear optical setup depicted in figure 4. For agiven dual Fock state input N, N a,theoutput state conditioned on, for example, a two-fold coincident count is given y [ N, N + N, N ]. (9) It is not difficult to see that the condition of the coincident detection yields that either one of the two modes efore the eam splitter must contain two photons while the other mode contains no photon. This is an inverse-hom situation where the detection of only one photon at each input mode cannot contriute to the coincident detection. Consequently, the coincident detection results in a situation where the main modes a and can only lose two photons or none at all. Here the proaility success of this event can e optimized y choosing the reflection coefficient of the first eam splitters. c d φ S798

4 From linear optical quantum computing to Heisenerg-limitedinterferometry For the reflection coefficient of r = /N,the proaility of success when N is found as /(e ) [37]. Furthermore, using a stack of such devices with appropriate phase shifters, we have developed a method for the generation of maximally path-entangled states of the form equation (7) with an aritrary numer of photons [8]. 4. Single-photon QND measurement devices In quantum optics the quantum non-demolition (QND) devices are usually considered in the context of photon-numer measurements [38]. In 985 Imoto et al developed the asic idea of QND in quantum optics, which consists of coupling the signal eam to the meter eam in a nonlinear medium and the detection of the phase shift of the meter eam measures the numer of photons in the signal eam [39]. The readouts of the numer of photons in the signal eam are performed y phase-sensitive homodyne detection of the meter eam in interferometer arrangements. By the same token, as discussed in section, QND measurements at the single-photon level ecomes extremely difficult due to the tiny strength of the nonlinear interaction etween photons. In a recent experiment, a single-photon QND has een demonstrated y using a resonant coupling etween a cavity field and the meter atoms [40]. Such a QND device at the single photon level can provide a key tool for optical quantum information processing, perhaps most importantly in quantum error correction. In contrast to the cavity approach, we have proposed a proailistic device that signals the presence of a single photon without destroying it using the technique of projective measurement [4]. A simple way to perform a single-photon QND measurement is to use quantum teleportation. For example, amaximally polarization-entangled photon pair produced y aparametric down-converter can serve as a quantum channel. If the input state is in a aritrary superposition of zero and one photon with a fixed polarization, the detector coincidence in a Bell statemeasurement, signals the present of a single photon in the input and also the output states. Simply, a vacuum input can never yield a two-fold detector coincidence. This teleportation-ased QND scheme works only if the input states are restricted to one or zero photons. However, it reaks down if there are more than two photons in the input. Forexample, if the input state is of the form ψ in = c c + c, (0) then the two-photon term will contriute to the two-fold coincidence even when the output of the down-converter is avacuum, yielding a false identification of a single photon in the output state, conditioned on a detector coincidence. If we restrict the numer of photons in the input up to two, we can eliminate such a false identification y using an interferometric setup depicted in figure 5. In figure 5, we assume that the input state of the form equation (0) enters into in mode a, and we further prepare single photons for modes c and d. Assuming that the eam splitters are 50:50, the transformation of the proe photons in the modes c and d can then e written as ĉ ˆd 4 ( ˆ â + ˆd ĉ â ĉ +ˆ ˆd ). () a c a d Figure 5. QND measurement device for single-photon detection. Theinput state, of an aritrary superposition of 0,,and, enters into mode a, and an auxiliary single photon is prepared for oth modes c and d. Conditioned upon a detector coincidence in modes c and d,and no count in mode a,theoutgoing mode is a single-photon state. Then we post-select the photodetection outcome for one and only one photon counted at each detector. This condition requires that either two photons are in mode c or two photons are inmode d,whicheliminates the contriution from c 0 0 of the input state. For one-photon and two-photon input states, we have â (â ĉ )/, â (â â ĉ + ĉ )/. () Now the only two-fold coincidence in the modes c and d y a two-photon input is possile when the ˆ ˆd from equation () and â ĉ from equation () comine, yielding â ˆ ĉ ˆd.However,further post-selectingonthevacuum in the mode a eliminates this two-photon contriution to the twofold coincidence in c and d. Asingle photon in mode a yields a contriution ˆ ĉ ˆd, indicating that there is a two-fold coincidence in modes c and d,and a single photon in the output mode. As can e seen in equations (), (), the proaility of success for this interferometric device is given y /8. By adjusting the transmission coefficients of the eam splitters in modes c and d, the proaility of success can e increased further. Oviously, this scheme does not work when the incoming state has an unknown polarization. However, it turns out that a more sophisticated interferometric setup with polarization eam splitters can do the jo while preserving the unknown polarization [4]. Of course, such a scheme is not a full QND measurement of the photon-numer oservale, since it works for only zero, one, and two photons. It is, however, the optimal way to distinguish different photon numer states [4] and it can still play an important role in linear optical quantum computation, where up to only two photons are used in each logic gate [43, 44]. Furthermore, such a single-photon QND device can e used in various quantum communication protocols such as quantum repeaters [45, 46]. 5. Summary Linear optics with projective measurements can e used to replace the use Kerr nonlinearities and provide a much higher efficiency. Using this technique, we have studied the generation ofuseful photonic quantum correlations. The maximally path-entangled photon-numer states provide an essential way for optical lithography to proceed eyond the Rayleigh diffraction limit. The Yurke-type pathentanglement is of particular importance in Heisenerg-limited interferometry. Projective measurements also enale us to d c S799

5 HLeeet al construct a device that signals the presence of a single photon without destroying it. Single-photon non-demolition measurement is of great importance in quantum information processing with photons, since most error-correction codes in the presence of quit loss requires QND measurements [47]. Acknowledgments This work was carried out at the Jet Propulsion Laoratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The authors wish to thank C Adami, N J Cerf, J D Franson, GJMilurn, W J Munro, T B Pittman, and T C Ralph for helpful discussions. We would like to acknowledge support from the NASA Intelligent Systems Program, the National Security Agency, the Advanced Research and Development Activity, the Defense Advanced Research Projects Agency, the National Reconnaissance Office, the Office of Naval Research, and the Army Research Office. PK thanks the National Research Council and NASA Code Y for support. References [] Boyd R W 99 Nonlinear Optics (San Diego, CA: Academic) [] Knill E, Laflamme R and Milurn G J 00 Nature [3] Scully M O and Zuairy M S 997 Quantum Optics (Camridge: Camridge University Press) [4] Turchette Q A, Hood C J, Lange W, Mauchi H and Kimle H J 995 Phys. Rev. Lett [5] Milurn G J 989 Phys. Rev.Lett. 6 4 [6] Ralph T C, White A G, Munro W J and Milurn G J 00 Phys. Rev. A [7] Hong C K, Ou Z Y and Mandel L 987 Phys. Rev.Lett [8] Pittman T B, Jacos B C and Franson J D 00 Phys. Rev. A [9] Pittman T B, Jacos B C and Franson J D 00 Phys. Rev.Lett [0] Lapaire G G, Kok P, Dowling J P and Sipe J E 003 Phys. Rev. A [] Bartlett S D, Diamanti E, Sanders B C and Yamamoto Y 00 Proceedings of Free-Space Laser Communication and Laser Imaging II vol 48 (Bellingham, WA: SPIE Optical Engineering Press) (Preprint quant-ph/004073) [] Lee H et al 003 Towards photostatistics from photon-numer discriminating detectors J. Mod. Opt. at press (Lee H et al 003 Preprint quant-ph/0306) [3] Achilles D et al 003 Photon numer resolving detection using time-multiplexing, unpulished (Achilles D et al 003 Preprint quant-ph/03083) [4] Boto A N et al 000 Phys. Rev. Lett [5] Campos R A, Saleh B E A and Teich M C 989 Phys. Rev. A [6] Gerry C C and Campos R A 00 Phys. Rev. A [7] Lee H, Kok P, Cerf N J and Dowling J P 00 Phys. Rev. A [8] Kok P, Lee H and Dowling J P 00 Phys. Rev. A [9] Fiurásek J 00 Phys. Rev. A [0] Zou X B, Pahlke K and Mathis W 00 Phys. Rev. A [] Pryde G J and White A G 003 Phys. Rev. A [] Fiurásek J, Massar S and Cerf N J 003 Phys. Rev. A [3] Bouwmeester D 004 Nature Walther P, Pan J-W, Aspelmeyer M, Ursin R, Gasparoni S and Zeilinger A 004 Nature Mitchell M W, Lundeen S J and Steinerg A M 004 Nature 49 6 [4] Caves C M 98 Phys. Rev. D [5] Bondurant R S and Shapiro J H 984 Phys. Rev. D [6] Bergman K and Haus H A 99 Opt. Lett [7] Shirasaki M, Lyuomirsky I and Haus H A 994 J. Opt. Soc. Am. B 857 [8] Yurke B 986 Phys. Rev.Lett [9] Yuen H P 986 Phys. Rev. Lett [30] Yurke B, McCall S L and Klauder J R 986 Phys. Rev. A [3] Holland M J and Burnett K 993 Phys. Rev. Lett [3] Kim K, Pfister O, Holland M J, Noh J and Hall J L 998 Phys. Rev. A [33] Hillery M and Mlodinow L 993 Phys. Rev. A [34] Brif C and Mann A 996 Phys. Rev. A [35] Berry D W and Wiseman H M 000 Phys. Rev.Lett [36] Dowling J P 998 Phys. Rev. A [37] Lee H, Kok P and Dowling J P 00 J. Mod. Opt [38] Grangier P, Levenson J A and Poizat P J 998 Nature [39] Imoto N, Haus H A and Yamamoto Y 985 Phys. Rev. A 3 87 [40] Nogues G et al 999 Nature [4] Kok P, Lee H and Dowling J P 00 Phys. Rev. A [4] Kok P 003 IEEE Sel. Top. Quantum Electron [43] Pittman T B, Fitch M J, Jacos B C and Franson J D 003 Phys. Rev. A [44] O Brien J L, Pryde G J, White A G, Ralph T C and Branning D 003 Nature [45] Jacos B C, Pittman T B and Franson J D 00 Phys. Rev. A [46] Kok P, Williams C P and Dowling J P 003 Phys. Rev. A [47] Gingrich R M, Kok P, Lee H, Vatan F and Dowling J P 003 Phys. Rev. Lett S800