Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector Christoph F. Wildfeuer,1, ∗ Aaron J. Pearlman,2 Jun Chen,2, 3 Jingyun Fan,2, 3 Alan Migdall,2, 3 and Jonathan P. Dowling1 1Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 2Optical Technology Division, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, MD 20899-8441 3Joint Quantum Institute, University of Maryland, College Park, MD 20742 With photon-number resolving detectors, we show compression of interference fringes with in- creasing photon numbers for a Fabry-P´erotinterferometer. This feature provides a higher precision in determining the position of the interference maxima compared to a classical detection strategy. We also theoretically show supersensitivity if N-photon states are sent into the interferometer and a photon-number resolving measurement is performed. PACS numbers: 42.50.St, 42.50.Ar, 42.50.Dv, 85.60.Gz I. INTRODUCTION an interferometer, which was first demonstrated by Caves [10]. Another promising strategy for quantum sensors is to maintain a coherent laser light source, but replace the Interferometers with coherent light are one of the classical intensity measurement with a photon-number building blocks for high-precision metrology. Recent resolving detector, or employ other more complicated en- progress in the field of photon-number resolving detec- tangling measurements to improve the performance of tors has made it possible to explicitly measure the pho- the quantum sensor further [11]. In addition, the per- ton statistics of different quantum-light sources in inter- formance of different non-classical input states, together ferometric schemes [1–3]. One such detector, the tran- with a photon-number resolving detection scheme, may sition edge sensor (TES), is a superconducting micro- be investigated. Under some conditions, the resolution bolometer that has demonstrated very high detection ef- and, in particular cases, the sensitivity of these quantum ficiency (95% at λ = 1550 nm) and high photon number sensors may exceed the performance of ‘classical’ light resolution [4–6]. sources and detection schemes. We emphasize here and Expanding the average intensity of an interference pat- highlight later that while resolution and sensitivity are tern into its photon-number resolved components pro- related, they are not identical. For an overview of quan- vides a better understanding of the interplay of sensitiv- tum metrology applications, see Ref. [12]. ity and resolution of an interferometer. Using a TES, we are now able to observe photon-number resolved inter- Many authors have proposed resolution and sensitiv- ference fringes and learn how they differ from a classical ity enhancements in different types of interferometric photon-averaged signal. Although it is also possible to schemes, where a large variety of Sagnac, Michelson, obtain the photon-number resolved interference fringes Mach-Zehnder, and Fabry-P´erotinterferometers (FPI) with multiplexed single photon counter modules [7], it is are considered [13–21]. A notable example is the laser in- advantageous to use a photon-number resolving detector terferometer gravitational wave observatory (LIGO) that like a TES which provides a high detection efficiency. The consists of a Michelson interferometer with Fabry-P´erot TES offers the advantage of high fidelity detection (high cavities in each of the two arms to boost the overall sen- probability of detecting the correct number of incident sitivity of the device [22]. photons). On the other hand, photon-number resolv- We first theoretically investigate the photon-number ing configurations that rely on multiplexed single-photon resolved interference fringes of a Fabry-P´erotinterferom- counters, which although have made exceptional progress eter as we scan its phase. We then experimentally im- in recent years, still suffer from limited photon-number plement this with a TES. Similarly, Khoury et al. have resolving fidelity [8, 9]. reported the use of a visible light photon counter to moni- For stand-off applications, such as a laser ranging de- tor the output of a Mach-Zehnder interferometer [19]. We vice, it is typical to use coherent states, since they are compare resolution and sensitivity of the photon-number more robust under loss than nonclassical states of light. resolved interference pattern with the classical case where A known strategy to improve the sensitivity of an inter- a coherent state is sent through a FPI and only the av- ferometer is to squeeze the vacuum of the unused port of erage intensity is measured. We also theoretically inves- tigate the performance of resolution and sensitivity for a single-mode photon-number state |ni in combination with a photon-number resolving detection of the inter- ∗Electronic address: [email protected] ference fringes. 2 II. QUANTIZED DESCRIPTION OF A As a classical baseline, we consider a single-mode co- FABRY-PEROT´ INTERFEROMETER herent state given by [24] ∞ 2 X k We start our investigation by deriving a quantum me- − |α| α |αi = e 2 √ |ki , (3) chanical description for the Fabry-P´erotinterferometer. k! Loudon first considered a quantum theory of the FPI for k=0 high-resolution length measurements [23]. The two in- (where |ki is a k-photon Fock state and α is the dimen- coming and two outgoing modes of the FPI can be quan- sionless electric field amplitude of the coherent state with tized as displayed in Fig. 1. the mean photon numbern ¯ = |α|2), which describes very well a single-mode laser above threshold. This state is incident on the FPI in modea ˆ, and vacuum |0i goes in 0 modea ˆ . The two-mode input state |αiaˆ|0iaˆ0 = |α, 0iaˆaˆ0 is then transformed with the BS transformation in Fig. 2, † † † from which we obtaina ˆ = T aˆu + Raˆd, where the sub- scripts u and d stand for up and down, respectively. Note that T and R satisfy the conditions |T |2 + |R|2 = 1 as well as TR∗ + RT ∗ = 0. We transform the incident coherent state |α, 0ia,ˆ aˆ0 by FIG. 1: (Color online) Fabry-P´erotcavity with complex am- the effective BS transformation and obtain the output plitudes R and T for reflected and transmitted modes, respec- of the FPI. An ideal k-photon detection is described by tively, and incident intensity I. Each mode can be assigned a the projector Cˆ = |kihk|. An approach including detec- mode operator marked by the hat to quantize the respective tion efficiencies is presented in Ref. [25]. Applying this mode, where the subscripts u and d stand for up and down. on modea ˆ , we obtain the photon-number resolved in- We assume, for simplicity, that both mirrors have identical u complex reflection and transmission coefficients denoted with terference fringes, which leads to the expression for the r and t. probability of detecting k photons ³ ´ coh ˆ coh The modes described in Fig. 1 can be transformed by pk = Tr Cρˆ an effective beam-splitter (BS) transformation, as is dis- ∞ X n¯j played in Fig. 2. For a FPI with two identical highly = e−n¯ |T |2k(1 − |T |2)j−k, (4) k!(j − k)! j=k whereρ ˆcoh is the reduced density matrix for the coherent state in modea ˆu. This result is displayed in Fig. 3(b) for a mean photon number of four (¯n = 4) and k = 1, .., 4 coh detected photons, which shows pk as a function of L/λ (or φ/2π). The structure of the transmission functions may be understood from interpreting the terms in Eq. (4). Each term in the sum represents the probability that k photons FIG. 2: (Color online) Effective beam-splitter for the Fabry- are transmitted through the FPI, multiplied by the prob- P´erotcavity with complex amplitudes R and T for reflected ability that j − k photons are reflected off the FPI. The and transmitted modes, respectively, which satisfy the condi- 2k tions |T |2 + |R|2 = 1 as well as TR∗ + RT ∗ = 0. transmission probabilities |T | have a maximum where the reflection probability (1 − |T |2)j−k has its minimum. The multiplication of both probabilities results in the reflecting mirrors, the transmission and reflectance func- additional minimum in the transmission probabilities for tions T and R are given by [23] k < n¯. √ We can also calculate the response of a non-photon- (1 − |r|2)e−2i 1−|r|2 T (r, φ) = √ , (1) number resolving detector by calculating the expectation |r|2e−2i 1−|r|2 e2iφ − 1 value √ √ −i 1−|r|2 −iφ iφ −2i 1−|r|2 X∞ |r|e (e − e e ) coh † coh coh 2 R(r, φ) = √ , (2) p = haˆuaˆui = k pk =n ¯|T | , (5) 2 −2i 1−|r|2 2iφ |r| e e − 1 k=1 where r denotes the complex reflectivity of the mirrors, which is proportional to the mean number of photons and φ = kL = 2πL/λ denotes a phase determined by the incident on the detector. We refer to this result as the wave-number k of the incoming light, and the distance L ‘classical’ signal that is usually associated with the out- between the two mirrors [23]. put of a FPI. 3 is then transformed to † n † † n 2 (ˆa ) (T aˆu + Raˆd) |T | |r|2 = 70% |r|2 = 90% |n, 0ia,ˆ aˆ0 = √ |0, 0ia,ˆ aˆ0 → √ |0, 0iaˆ ,aˆ n! n! u d µ ¶ 1 Xn n p = √ T `Rn−` `!(n − `)!|`, n − `i . ` aˆu,aˆd n! `=0 L/λ L/λ (6) (a) Suppose we also perform a photon-number resolving mea- surement on the transmitted photons. The result of this measurement is given by ³ ´ F ˆ F n! 2k 2 n−k pk = Tr Cρˆ = |T | (1 − |T | ) , (7) pk k = 1 k = 2 k!(n − k)! whereρ ˆF is the reduced density matrix for the Fock state in modea ˆu and the superscript F denotes Fock state.