Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving

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 , 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 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 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 observatory (LIGO) that like a TES which provides a high detection efficiency. The consists of a 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 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. Note that the right hand side of Eq. (7) is a binomial Pn F distribution with the property k=0 pk = 1. The detec- F tion probabilities pk in Eq. (7) are displayed in Fig. 3(b) (dashed lines) as a function of L/λ for a four-photon pk k = 3 k = 4 state and a photon-number detection for k = 1,..., 4. The mean photon counts at the output of the FPI are obtained from the expectation value

L/λ L/λ Xn † F F 2 haˆuaˆui = k pk = n|T | . (8) (b) k=1

The result of Eq. (8) is the same as in the classical case 2 FIG. 3: (a) shows the classical transmission function |T | (Eq. (5)) above, when we identifyn ¯ = n. However, the 2 2 displayed for a reflectivity of |r| = 70% (LHS) and |r| = photon-number resolving measurements in Eq. (7) show 90% (RHS) of the FPI-mirrors. Note that the transmission a different behavior. In particular, we consider the case peaks become narrower and the maxima slightly shifted to the left as the reflectivity increases. Also the minima go to k = n, i.e., we measure the same photon number in our zero for 90% reflectivity whereas they do not reach zero for detector as that of the initial input state. In this case lower reflectivities. (b) shows the probabilities of detecting Eq. (7) reduces to coh k photons, pk for a single-mode coherent input state |αi 2 F 2n with mean photon numbern ¯ = |α| = 4 incident on the FPI pn = |T | . (9) and a photon-number resolving measurement displaying k = 1, .., 4 (solid lines). The reflectivity |r|2 of the mirrors is 70%. It turns out that the transmission peaks become nar- F The dashed lines shows the transmission probabilities pk for rower as the photon number n increases (Fig. 4). We a single-mode photon-number state |4i. We observe that the also observe a similar interference pattern for the photon- Fock states (dashed lines) show transmission peaks that are number resolved peaks for k < n for the same reason as sharper in general. This effect is equivalent to operating at described earlier for coherent states. As in the coherent a larger reflectivity of the mirrors, which is demonstrated in case, for k < n the transmission probability |T |2k for k (a) for the classical curves. A major difference appears for photons is multiplied by the probability (1 − |T |2)n−k k = 4 where the transmission maxima reach one for the Fock state, whereas the coherent state stays low. Here and in the which is the probability that the other n − k photons are following, we choose a reflectivity of 70% as the important reflected. The multiplication of these two probabilities, features are more pronounced than for larger reflectivities. which have opposite functional forms, produces the dip in the middle of the maximum.

III. RESOLUTION AND SENSITIVITY FOR COHERENT STATES

For a nonclassical input we consider a single-mode Fock To quantify our results for different states and detec- state |ni in modea ˆ incident on the FPI, and the vacuum tion operators, we first calculate the uncertainty in de- 0 state |0i in modea ˆ . The input state |niaˆ|0iaˆ0 = |n, 0ia,ˆ aˆ0 termining the free spectral range (FSR) given by ∆L/λ, 4

←− ∆L −→ ˆ ˆ2 ˆ 2 1/2 1.0 λ and ∆C = (hC i − hCi ) is the standard deviation of n = 1 the observable Cˆ [26]. coh 0.8 n = 2 Next we compute the sensitivity δLn¯ for a coherent F state with average photon numbern ¯ as a classical base- pn 0.6 n = 3 line, and compare it with the photon-number resolved for k = n transmission probabilities (Fig. 5). For a coherent state 0.4 input and a mean intensity measurement, Eq. (11) re- duces to 0.2 1 |T | coh √ δLn¯ = 2 . (12) 0.0 0.2 0.4 0.6 0.8 n¯ |∂|T | /∂L| L/λ This expression√ defines the shot-noise limit evidenced by the 1/ n¯ dependence. Improving the sensitivity beyond FIG. 4: (Color online) Transmission probabilities for a single- this is referred to as supersensitivity. For the photon- mode photon-number state and a photon-number resolving number resolved sensitivity δLcoh, Eq. (11) reduces to measurement, for photon numbers n = 1, 2, 3. The free spec- k tral range ∆L/λ is the distance between two adjacent max- q ima. Transmission probabilities are calculated for 70% reflec- pcoh(1 − pcoh) coh k k tivity of the mirrors. δLk = coh . (13) |∂pk /∂L| We observe that the uncertainty of a length measure- which is the dimensionless distance between two adja- cent interference peaks (Fig. 4). From the experimental data we can determine the variance of the transmission coh 0.5 δL peaks. The uncertainty of the absolute positions, or in k k =4 λ other words, the standard deviation of the mean (SDM) 0.4 value of the individual peaks one and two are given by =1 √ √ 0.3 k σL1 = σ1/ n1, and σL2 = σ2/ n2, where n1 and n2 are the total number of counts in peak one and two. For an explicit definition of the SDM σ see section V, which 0.2 Li n¯ =4 contains our experimental results. The uncertainty in determining the FSR is then given by 0.1 s 2 2 √ σ1 σ2 σ -0.1 0.0 0.1 0.2 0.3 σ∆L = + ≈ 2√ , (10) n1 n2 n L/λ where the approximation holds, if the two peaks have ap- FIG. 5: (Color online) Dimensionless uncertainty δLcoh/λ proximately the same variance and number of counts. A k (where L is the length change of the FPI and λ is the wave- smaller variance σ for the transmission peaks provides an length of the coherent laser beam) for a mean intensity mea- improvement in resolution, but increasing the number of surement withn ¯ = 4 compared to photon number resolving counts can provide a similar improvement. If the variance measurements k = 1 and k = 4. The solid line also repre- 2 shrinks by a factor of m, i.e., σ → σ/m, the√ uncertainty sents the shot-noise limit. The reflectivity |r| of the mirrors of the peak center in Eq. (10) becomes 2σ/(m2n)1/2. is 70%. This means that we need a factor of m2 fewer counts to obtain the original variance σ. In section V we show ex- ment δLcoh/λ for the photon-number resolved measure- perimentally that we can determine the position of the ment is always larger than for the average photon-number peaks from the photon-number resolved data obtained measurement. In other words the photon-number re- with coherent states with up to three times higher preci- solved interference fringes do not increase the sensitiv- sion, for the same optical power compared to the classical ity of the interferometer. However, we show in the next signal. section that increased sensitivity can be achieved by re- On the other hand, we can compute the uncertainty placing the input coherent state with a photon-number δL of a , which we also refer to as state. sensitivity, from the expression

∆Cˆ ¯ ¯ IV. SENSITIVITY FOR N-PHOTON STATES δL = ¯ ¯ , (11) ¯∂hCˆi/∂L¯ We propose an experiment with an N-photon state |Ni where hCˆi is the mean value of the detection operator incident on the FPI as displayed in Fig. 6. From the 5

F δLk λ k =4

=1 FIG. 6: (Color online) Transmission experiment with an N- k photon state incident on a Fabry-P´erotinterferometer and photon-number resolving detection of the output. n¯ =4

Fig. 3(b) (dashed line for k = 4 ), and Fig. 4 we con- L/λ clude that in addition to obtaining narrower transmis- sion functions as the photon-number increases (increased resolution), the amplitude at the maximum remains one. That is an indication that we have an additional bene- fit from using N-photon states as opposed to a coherent F state input. We not only increase the resolution, we also δLk obtain a higher sensitivity. To quantify this statement λ we calculate the sensitivity as defined by Eq. (11). The sensitivity is given by q pF(1 − pF) F k k L/λ δLk = F , (14) |∂pk /∂L|

F FIG. 7: (Color online) Dimensionless uncertainty δLF/λ where pk is taken from the expression in Eq. (7), which k simplifies for a |ki Fock state input and a k-photon de- (where L is the length change of the FPI and λ is the wave- tection to length of the light) for a mean intensity measurement with n¯ = 4 compared to photon number resolving measurements p |T |k 1 − |T |2k k = 1 and k = 4. The solid line also represents the shot- F noise limit. The reflectivity |r|2 of the mirrors is 70%. The δLk = 2k . (15) |∂|T | /∂L| shot-noise limit (solid line) is beaten by the k = 4 curve. We display the sensitivity as a function of phase (here scaled length L/λ) in Fig. 7, analogous to the coherent state case in Fig. 5. We observe that the shot-noise limit given by the black solid line is beaten by the k = 4 curve. through a single beam-splitter, described by the com- This means that we can achieve supersensitivity (beating plex transmittivity t, and reflectivity r, is just |t|2. If we the shot-noise limit) with a photon-number state input ask for the probability that n photons in a Fock state |ni and a n photon detection. traverse through the BS, we obtain |t|2n (compare with We can also investigate how the minimum phase un- Eq. (7)). In our quantum mechanical model for the FPI, certainty behaves as a function of the photon number we use an effective BS transformation where the matrix n for the photon-number state input or mean photon elements of the unitary BS transformation are given by numbern ¯ for coherent states, respectively. We see that the complex functions T and R defined in Eqs. (1) and (Fig. 8), as opposed to coherent states, a length mea- (2). We observe then the same functional behavior as for surement with photon-number states provides us with a the regular BS and Fock states. The transmission func- much smaller uncertainty δL in the few photon limit. tion for the FPI, given the n photons in a Fock state that Hence, the sensitivity of the FPI is increased. An al- have traversed the FPI, is |T |2n, as given by Eq. (7). As ternative way to describe the result can be formulated |T | becomes smaller than one, the probability of trans- in terms of the finesse F = FSR/FWHM of the FPI, mitting n photons given by |T |2n decreases more rapidly where FWHM stands for full width at half maximum. than that for single photons (n = 1) or a coherent state, The finesse F of the FPI, which for R > 0.5 can be ap- which manifests as narrower transmission curves. This proximated by F = πR1/2/(1−R) for classical detection, feature may find applications in interferometry for high- is essentially improved by using photon-number resolving precision length measurements as in LIGO for instance. detection without changing the reflectivity of the mirrors, The quantum light source may provide a high sensitivity since the FSR remains the same but the FWHM becomes at a much reduced optical power. The FPI can also be narrower with increasing photon number. nested in a Michelson or Mach-Zehnder interferometer, Finally, we provide an intuitive interpretation of our as has been implemented at LIGO, to boost the sensitiv- results. The probability that a single photon traverses ity and achieve an even higher resolving power. 6

δL whose integral is proportional to the energy absorbed. λ Thus, by simply integrating the output pulses from the TES, we can resolve straightforwardly the number of pho- 0.015 tons absorbed in a given time window. In our setup, we amplify the TES signal and record it using a digital os- 0.01 |αi cilloscope. We then integrate each pulse and create a 0.005 histogram of these pulse integrals to observe the photon- |ni number resolved detection as shown in Fig. 10. We repeat 0 1 3 5 7 9 11 13 15 n¯

FIG. 8: (Color online) Comparison of the sensitivity δL/λ (Eq. (12)) for a coherent state (solid line) and mean intensity detection (shot-noise limit) as a function of mean number of detected photons versus a n-photon state |ni with n-photon resolving measurement (Eq. (15)) as a function of the pho- ton number, where we take n =n ¯, slightly away from the transmission maximum. The reflectivity |r|2 of the mirrors is 70%. The parameter of the phase is chosen for each n,n ¯, respectively, so that the sensitivity is at its minimum.

V. EXPERIMENTAL RESULTS FOR COHERENT STATES

We performed an experiment with an attenuated co- FIG. 10: Sample histogram of the output pulse integrals of herent pulsed laser diode at a fixed λ = the photon-number resolving TES used in our experiment. 1550 nm, a repetition rate of 50 kHz, and a pulse dura- The histogram indicates the probabilities of detecting each photon number, as well as the probability no photons were tion of 50 ps (Fig. 9). Our photon-number resolving TES detected (labeled “0”). The vertical lines show the thresholds detected on average four photons per pulse [5]. We used a between the individual photon-number peaks.

this procedure to obtain a histogram at each value of the piezo voltage and generate the curves shown in Fig. 11 for 1 ≤ k ≤ 7. The preliminary data support the theoretical predictions, as can be seen in Fig. 11. Unfortunately, although the stability of the FPI out- put was improved with temperature control, substantial FIG. 9: (Color online) Transmission experiment with a weak drift occurred during the data acquisition time of more coherent laser beam incident on a length tunable FPI and a than 30 minutes, resulting in smearing of the data. Also, fiber coupled TES. The light is collected and coupled into a the apparent classical signal from the photon-number re- single-mode fiber and transmitted to the detector. solved data is systematically underestimated. For weak laser pulses with a mean photon number of four, we would scanning Fabry-P´erotinterferometer that had originally need to include photon-number resolved data of up to been designed as a tunable filter with a FSR of 70 nm and k = 10 to cover 99% of the signal. Our measurements a FWHM of 0.15 nm to be used in locked mode. However, only cover 1 ≤ k ≤ 7, before the signal disappears into its feedback (locking) circuit locks only to the maximum the noise, which results in the reconstructed classical sig- of the transmission curve. Since we wanted to measure nal being 15% lower than expected. the entire transmission function, we had to use the FPI Note that from the theoretical curves shown in in the unlocked mode. The stability of the unlocked FPI Fig. 3(b), one can see that there is always a dip in the was initially poor due to ambient temperature fluctua- middle of the maximum whenever k < n¯. The dip be- tions. To circumvent this, we employed a thermo-electric comes less pronounced as k approachesn ¯, and disappears cooler to stabilize the temperature within 0.1 ◦C. For the for k ≥ n¯. This feature can be utilized as a diagnostic measurement, we attenuated the laser diode output, sent tool to boundn ¯, without doing any detailed fitting. For it through the FPI and then to the fiber-coupled TES. We example, from the data in Fig. 11, we can easily identify adjusted the distance between the mirrors of the FPI by 3 < n¯ < 4, just from the “dip characteristics” described tuning the voltage of the piezo-electric transducer inside above. We can therefore confidently justify the rejection the FPI. of the reconstructed numbern ¯0 = 2.6 from Fig. 11 (bot- Photon absorption in the TES creates a voltage pulse tom right), which corroborates the arguments given in 7

TABLE I: Resolution improvements. Theoretical and exper- exp imental results for the photon-number resolved SDM σk exp compared to the SDM σcl = 0.103 nm of the classical sig- nal. The theoretical results are calculated for a mean photon numbern ¯ = 3.9 and 91% reflectivity (fit-parameters determined from the experimental data). The expected clas- theo sical SDM is σcl = 0.0995 nm. All standard deviations are given in nm. k 1 2 3 4 5 6 7 theo σk 0.176 0.116 0.074 0.052 0.039 0.032 0.028 theo theo σcl /σk 0.6 0.9 1.3 1.9 2.5 3.1 3.6 exp σk 0.161 0.094 0.062 0.064 0.075 0.045 0.036 exp exp σcl /σk 0.6 1.1 1.7 1.6 1.4 2.3 2.9

σ

σcl/σk FIG. 11: (Color online) Transmission probabilities for a single-mode coherent state and a photon-number resolving measurement, for photon numbers k = 1, .., 7. The blue solid lines show the individual theoretical fits. The peaks become narrower as k increases. The fit yields the mean k photon numbern ¯ ≈ 3.9 and the reflectivity of the mirrors ≈ 91%. The graph at the bottom right is the reconstructed FIG. 12: (Color online) Figure of the standard deviations as classical signal from the photon-number resolved curves with a function of k presented in Table I. coh P7 coh p = k=1 k pk . The black solid line is the theoretical fit. From the fit the mean photon-number is determined as 0 n¯ = 2.6, which is not in good agreement with the mean for larger photon numbers. This feature allows us to photon-number obtained from the photon-number resolved determine the center of the peak positions with higher curves ofn ¯ ≈ 3.9 and reflects insufficient stability of the used accuracy than classically possible, if we are using the FPI. We also only have access to photon-number counts in the range 1 ≤ k ≤ 7 which amounts to reconstructing the photon-number resolved peaks for k ≥ 3. classical signal with an amplitude 15% too low.

VI. SUMMARY the caption of Fig. 11 and the above text. The photon-number resolved output of the FPI We have highlighted the difference between resolution (Fig. 11) shows narrower peaks with increasing photon and sensitivity in interferometry measurements with clas- number. To quantify any improvement in resolution we sical and nonlcalssical sources and detectors. In partic- compare the standard deviations of the photon-number ular we have shown that the resolution of a Fabry-P´erot resolved peaks σk to the standard deviation σcl obtained interferometer with weak coherent states and photon- from the classical transmission peak (Table I). number resolving detection is improved up to three times compared to a classical detection strategy. Here resolu- tion relates to how well two peaks can be seen as distinct 2 P 2 The SDM is defined by σ P= i pi(φi −µ) , where the as opposed to the sensitivity in finding the center of one mean µ obtained from µ = i piφi, pi is the normalized lone peak. The improvement in resolution is not to be probability defined by pi = fi/N, fi is the number of confused with the sensitivity shown in Fig. 5, which ac- counts for a particular phase φi = 2πLi/λ, and N is tually cannot be improved for the coherent state input the total number of counts for the kth transmission peak and a photon-number resolving measurement. We also σk or the classical signal σcl. For the photon-number show that by replacing the classical input state with a resolved peaks we observe that the SDM gets smaller photon-number state |ni incident on the FPI and per- 8 forming a photon-number resolved measurement, we ob- gram. A.J.P., J.C., J.F., and A.M. acknowledge the tain supersensitivity (beat the shot-noise limit, Fig. 8). A MURI Center for Photonic Quantum Information Sys- demonstration experiment to show this effect with pho- tems ARO/IARPA, the IARPA Entangled Source, and ton pairs from a spontaneous parametric down conver- the Intelligence Community Postdoctoral Research Asso- sion source or an optical parametric oscillator incident ciateship Programs for their support. We thank H. Lee, on a FPI should be well within reach. An interesting N. Sauer, W. W. Johnson, S. Polyakov, and S. W. Nam line of research would be to investigate generalized quan- for very helpful comments. C.F.W. thanks all the mem- tum metrology schemes other than the schemes above bers of the optical technology division at NIST, Gaithers- with coherent states and detection strategies based on burg for their generous hospitality. photon-number resolving detectors. Approved for Public Release, Distribution Unlimited.

Acknowledgments

C.F.W. and J.P.D. acknowledge the Defense Ad- vanced Research Projects Agency Quantum Sensors Pro-

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