Linear optics and projective measurements alone suffice to create large-photon-number path entanglement

Abstract

We propose a method for preparing maximal path entanglement with a definite photon number N , larger than two, using projective measurements. In contrast with the previously known schemes, our method uses only linear optics. Specifically, we exhibit a way of generating four-photon, pathentangled states of the form |4, 0 + |0, 4 , using only four beam splitters and two detectors. These states are of major interest as a resource for quantum interferometric sensors as well as for optical quantum lithography and quantum holography.PACS numbers: 03.65. Ud, 42.50.Dv, 42.25.Hz, 85.40.Hp Quantum entanglement plays a central role in quantum communication and computation. It also provides a significant improvement in frequency standards as well as in the performance of interferometric sensors [1,2]. In this context, it has been shown that the Heisenberg limit for phase sensitivity of a Mach-Zehnder interferometer can be reached by using maximally entangled states with a definite number of photons N , that is, |N, 0 A,B +|0, N A,B . Here, A and B denote the two arms of the interferometer. These states, also called pathentangled photon number states, allow a phase sensitivity of order 1/N , whereas coherent light yields the shot-noise limit of 1/ √n , with mean photon numbern [3]. The use of quantum entanglement can also be applied to optical lithography. It has been shown recently that the Rayleigh diffraction limit in optical lithography can be beaten by the use of path-entangled photon number states [4]. In order to obtain an N -fold resolution enhancement, with quantum interferometric optical lithography, one again needs to create the N -photon path-entangled state given above. Due to interference of the two paths, one obtains an intensity pattern at the lithographic surface which is proportional to 1 + cos N ϕ, where ϕ parametrizes the position on the surface. A superposition of these states with varying N and suitable phase shifts then yields a Fourier series of the desired pattern, up to a constant [5].In view of these potential applications, finding methods for generating path-entangled states has been a longstanding endeavor in quantum optics. Unfortunately, with the notable exception of N = 2, the optical generation of these states seemed to require single-photon quantum logic gates that involve a large nonlinear interaction, namely, a Kerr element with χ (3) on the order of unity. Typically, χ (3) is of the order 10 −16 cm 2 s −1 V −2 [6]. This makes a physical implementation with previously known techniques very difficult [7][8][9]. Recently, however, several methods for the realization of probabilistic singlephoton quantum logic gates have been proposed, which make use solely of linear optics and projective measurements (PMs) [10][11][12]. PMs are performed by measuring some part of the system while the rest of it is projected onto a desired state (state reduction). Since the state obtained is conditioned on a measurement outcome, this method only works probabilistically. Such a protocol has been employed ex...

Projection measurement of the maximally entangledN-photon state for a demonstration of theN-photon de Broglie wavelength

“…The N-fold coincidence measurement of all N detectors will give P N , the outcome of a NOON state projection measurement. Notice that state projection methods have been proposed before [3,4,5]. But those methods require 2N photons for the generation of N -photon NOON state.…”

Conditional generation of arbitrary multimode entangled states of light with linear optics

“…It was shown recently that these states may be conditionally generated with the help of linear optics, single photon sources and photodetectors [14,15,16,17]. We then combine the modes 1 and 2 on a balanced beam splitter and measure the number of photons in output modes 1 and 2 by means of two photodetectors P D 1 and P D 2 with single-photon resolution.…”

“…Schemes for probabilistic preparation of Fock states [9], arbitrary superpositions of Fock states of single-mode field [10,11], and superpositions of classically distinguishable states (Schrödinger-cat-like states) [12,13], have been found. Several setups for the generation of twomode N -photon path-entangled states have been suggested [14,15,16,17,18]. Recently, the experimental conditional preparation of a single-photon Fock state with negative Wigner function has been reported [19].…”

Construction of a quantum repeater with linear optics

“…In particular, when arbitrarily many auxiliary modes are available, one does not need the very weak nonlinearities assumed to be essential for these purposes. Subsequently, it was shown that the same resources (linear optics and projective measurements) can be used to create highly nonclassical number states [2,3]. These states are important for applications such as quantum lithography and quantum interferometry [4,5].…”