Linear quantum measurement theory of matter-

Yiqiu Ma,1, 2, ∗ Xiang Li,2 Shengjun Yang,3 and Yanbei Chen2, † 1Center for Gravitational Experiment, Ministry of Education key Laboratory for fundamental physical quantity measurement, School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, China 2Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA 3College of Science, Southern University of Science and Technology, Shenzhen,518055, China (Dated: November 25, 2019) The theory of linear quantum measurement has been developed for analysing the sensitivities of experimental devices that measure extremely weak signals, such as gravitational . It has successfully contributed to the theoretical understanding of interferometer gravitational-wave detectors (used by LIGO, VIRGO and KaGRA) and helped many important experimental upgrades. In this work, we establish a linear quantum mea- surement theory for another kind of measurement device— interferometers, which has been widely discussed as an important platform for many high-precision experiments. This theory allows us to account for both and fluctuations, and leads to a detailed analysis of back-action in matter-wave interferometry (action of light back onto the ) and its effect on dynamics and measurement noise. From this analysis, we obtain a Standard Quantum Limit (SQL) for matter-wave interferometry. A comparison between the LIGO detector and matter wave interferometer is also given from the perspective of quantum measurement.

I. INTRODUCTION though the multi-stage vibration isolation technique has been applied [12]. For space-borne optical GW detector such as The detection of gravitational waves (GWs) from merging Laser Interferometer Space Antenna (LISA), the test masses binary black holes [1] and merging binaries [2] are also connected to the satellite platforms thereby the ran- by an international network of gravitational-wave detectors dom motion of the satellites will be transfered onto the test (LIGO, VIRGO and KAGRA) opened the era of gravitational masses and contaminate the GW signal. However, for the wave astronomy (in this paper we shall use "LIGO detector" atom interferometer, since the atoms are free-falling during to refer to a detector that is used by LIGO [3], VIRGO [4] the interferometry process, they are less sensitive to the seis- and KaGRA [5]). This detection is also a milestone in the mic perturbation (or the satellite motion in the space case). development of high-precision measurement physics, making The laser noise can be removed by designing the detector con- LIGO detector the most sensitive instrument that human be- figuration with common mode rejection. More sophisticated ings ever built. Parallel to LIGO detector where the under- designs such as implementing the large momentum transfer line principle is the interference of the electromagnetic waves, technique or optical cavities have been also discussed [33–37]. other concepts of GW detectors have also been proposed, even Typically, experimental devices such as GW detectors that before the first detection event. One particular attractive con- targeted on measuring extremely weak signals can be even cept is the atom-interferometer GW detector, first raised by affected by the . The theory of quantum Dimopoulos et. al [6, 7] and later enriched by many further measurement developed from 1960s is a framework to anal- discussions [8–16]. Different from the LIGO, the physical yse how quantum mechanics affects the sensitivity of an ex- principles under the atom interferometer GW detector is the perimental device [38]. The early resonant bar GW detectors interference of the matter waves, rather than the light waves. and the current laser interferometer GW detectors have been The concept of atom interferometer can be traced back to extensively studied and understood using this quantum mea- the 1930s, when Rabi demonstrated that the atoms’ internal surement theory framework [38–40]. For atom interferome- quantum states can be altered using rf resonance [17]. In try, although the effect of quantum noise has been discussed 1949, Ramsey firstly created and detected long-lived coher- by various authors [41–44], a complete analysis under quan- ent superposition of internal quantum states [18]. These pio- tum measurement theory has not been discussed in the current literatures. Establishing such a theory will provide important arXiv:1911.08697v2 [quant-ph] 22 Nov 2019 neering works pave the way for the further development of a field named atom optics, namely, one can manipulate coher- insights in understanding the atom interferometer. Here, it ent beams of atoms as manipulating that of light fields [19]. is useful to briefly overview such a framework, based on the Atom interferometry is an art of atom optics and an important block-diagram shown in Fig. 1. In this framework, a quantum experimental platform for high-precision measurement, which is now being used for measuring earth’s acceleration G Probe ↵xˆFˆ Detector yˆ and testing fundamental physics [20–32]. The advantage of the proposed application of atom inter- xˆ (Z,ˆ Fˆ) ferometer in GW detection is mostly at low frequency (below 10 Hz), which can be understood as follows. Because the test masses are connected to the ground through suspension sys- FIG. 1. Block diagram of a linear quantum measurement device. tem, the sensitivity of a laser interferometer GW detector is seriously contaminated at low frequencies partly through the measurement device is divided into probe and detector, where coupling of the test masses with the sesmic oscillations, al- the probe dynamical quantityx ˆ is linearly coupled to G—the 2 information to be measured. The probe and detector are cou- The Hamiltonian describing the Raman interaction happens pled through linear Hamiltonian Hˆint = αxˆFˆ. The informa- in an atom interferometer has the following structure: tion of G will flow into the detector through− probe-detector ih Z ∞ ˆ ¯ † † interaction and then be read out asy ˆ(t): Hopt = [∂xaˆcxaˆcx ∂xaˆpxaˆpx] + h.c, 2 ∞ − −  Z t  † † (3) Hˆa = h¯ωAAˆ Aˆ + h¯ωBBˆ Bˆ, yˆ(t) = Zˆ(t)+α xˆzero(t) + xsig(t) + α dt0χm(t t0)Fˆ(t0) , ∞ − H h Aˆ Aˆ† Bˆ Bˆ† a a† a a† − (1) int = ¯ χ( + )( + )( ˆc + ˆc)( ˆp + ˆp), x x Fˆ x where ˆzero, sig, are the zero-point fluctuation of ˆ, the sig- where the Hopt describes the free control lighta ˆc and passive nal, and the back-action force, α is the coupling strength and lighta ˆp in x space; Hˆa describes the whole atom clouds at the χm(t t0) is the dynamical response function of the probe. two different− levels and thereby does not depend on Braginsky− et.al shows that for measuring the GW tidal force, x; Hˆint describes the atom-light Raman interaction at one spe- the zero-point fluctuation of test masses does not contribute to cific spacetime location, the derivation of such a four-field in- the final sensitivity therebyx ˆzero can be simply ignored [45]. teraction Hamiltonian is shown in the Appendix. The Aˆ and Bˆ ˆ ˆ If Z and F have no correlation, the sensitivity will be limited are the effective annihilation operators for the energy level 1 by the so-called standard quantum limit (SQL), given by (in and 2 , respectively. Their corresponding number operators| i the frequency domain): | i † † are NˆA = Aˆ Aˆ and NˆB = Bˆ Bˆ and we have the commutation re- † SQL lation [Aˆ,Aˆ ] = 1 (the same for Bˆ). For the continuous optical Sxx (Ω) = 2h¯ χm(Ω) , (2) † | | fields, we have [aˆcx,aˆcx ] = δ(x x0) (the same fora ˆpx) and 0 − ˆ R † where Ω is the angular frequency of GWs. For advanced it is related to particle numeber by N = dxaˆcxaˆcx. The pre- SQL 2 LIGO, the SQL is given by Sxx (Ω) = 2h¯/(mΩ ), with the cise definition of these effective operators will be presented in mass of the test mirrors denoted by m. Section IV. In this work, we set up a quantum measurement theory framework for analysing the physics of atom interferometer, which is based on the interaction between atom cloud and two optical fields (passive and control laser). It is straightforward to extend our result to other atom interferometer configura- tions.

II. EFFECTIVE HAMILTONIAN OF AN ATOM INTERFEROMETER

In an atom interferometer GW detector, the GW informa- FIG. 2. A four-boson Raman interaction kernel and its corresponding WKB trajectory. For a detailed discussion of this process and the tion is carried by the light field in the TT gauge, thereby the description using field theory, see the Appendix. light field corresponds to the probe and the detector corre- sponds to the atom cloud in the above quantum measurement model. Concretely speaking, the atom cloud, as a phase me- Note that under the rotating wave approximations, only the ter, records the optical phase (more precisely, the phase dif- terms satisfying ωA + ωc = ωB + ωp will be kept while those ˆ ˆ † † ference between the control and passive fields as we shall non-rotating wave terms such as ABaˆcaˆp etc. can be safely ig- see) imposed by the signal. In this section, we will estab- nored, which leads us to a simpler form of interaction Hamil- lish an one-dimensional effective Hamiltonian for analysing tonian: this system. Real systems are three-dimensional therefore Hˆ = h¯ χAˆ†Bˆaˆ†aˆ + h.c, (4) this one-dimensional model is obtained by reducing a three- int c p dimensional system by paraxial approximation. This Hamil- and the corresponding equations of motion for atomic clouds tonian can be derived from first principle and the details are are given by: given in Appendix. In this section, we are going to show how ˙ † ˙ † the back-action effect manifests itself in atom interferometers. Aˆ = iχBˆaˆ aˆp, Bˆ = iχAˆaˆcaˆ . (5) − c − p Solving these equations perturbatively by writing the oper- ˆ ¯ ˆ ˆ ¯ ˆ A. Effective Hamiltonian and dynamics of an interaction ators as: A = AA +AA and B = AB +AB (where the magnitude ˆ ¯ kernel of AA/B is small compare to the AA/B) leads to the zeroth-order equations:

The basic physical process happen in a typical atom inter- ˙ iϕ ˙ iϕ A¯A = iΩA¯Be , A¯B = iΩA¯Ae− . (6) ferometer is a four-boson interaction, where the atomic tran- − − sition between energy level 1 and 2 happens through cou- where Ω = χ a¯ca¯p is the Rabi-frequency and ϕ = ϕc ϕp pling to an intermediate energy| i level| i3 by control and pas- which is the phase| | difference between control field and− pas- sive fields, as shown in Fig 2. | i sive field. 3

Then the rest terms satisfy the following equations: which describes the pondermotive force exerted by the mean optical fields on the atoms.    † iϕ iϕ  Aˆ A¯ (aˆ a¯ e p + aˆ a¯ e c ) A/B ˆ A i B c p − p c (3) The F have the form: L = χ ¯ | | iϕp † | | iϕc dy AˆB − AA(aˆc a¯p e + aˆp a¯c e− ) | | | | (7)  iϕ  2 A¯ e A χ 2 2 † 2 B F = ( a¯ a¯ )(A¯ A¯ Aˆ + A Aˆ ), + Ωϕs ¯ iϕ . dy p c A B B B A AAe− 2 | | − | | | | (14) − 2 B χ 2 2 † 2 ˆ F = ( a¯p a¯c )(A¯BA¯AAˆ + AA AˆB). in which the differential operator L takes the form of: dy 2 | | − | | A | |  iϕ  (†) (†) ˆ ∂t iΩe These forces, which depends on the operators Aˆ ,Aˆ will L = iϕ , (8) A B iΩe− ∂t modify the Rabi-dynamics of the atom fields. In case of balanced passive and control fields (i.e. a¯p = and the right hand side of these equations describes the in- | | a¯c = a¯L), these dynamical back-actions will vanish. In the fluence of optical fields to the evolution of atom fields. The |following| sections, we will focus on the configuration with ϕs ϕ is the signal phase carried by the optical field, and we  balanced passive and control fields. expand it to the linear order to obtain the signal terms. The solution of these dynamical equations can be expressed The solutions for the optical fields, to the leading order, are in a more convenient way by using the following basis: (1) given by: iϕ/2 iϕ/2 Aˆ = (AˆAe− AˆBe )/√2 for atom fields; (2)a ˆc1 = ± iϕc † iϕc ±√ iϕp † iϕp √ a¯cout = a¯cin, a¯pout = a¯pin. (9) (aˆce + aˆce− )/ 2,a ˆp1 = (aˆpe− + aˆpe )/ 2,a ˆc2 = iϕc † iϕc iϕp † iϕp (aˆce aˆce− )/(√2i),a ˆp2 = (aˆpe− aˆpe )/(√2i) for and the rest terms satisfy: optical− fields; (3) the common and differential− modes of in- † coming optical fields:a ˆ in1/2 = (aˆcin1/2 aˆpin1/2)/√2. Un- aˆcout = aˆcin iχ(A¯∗ A¯Baˆp + A¯∗ a¯pAˆB + A¯Ba¯pAˆ + A¯∗ A¯Ba¯p), ± ± − A A A A der these basis, the equations of motion Eq. (7) can be trans- † aˆpout = aˆpin iχ(A¯AA¯∗ aˆc + A¯Aa¯cAˆ + A¯∗ a¯cAˆA + A¯AA¯∗ a¯c). formed to: − B B B B (10) (∂t iΩ)Aˆ (t) = ΩϕsA¯ (t) ± ± ∓ ∓ (15) The first terms in the brackets of the r.h.s of the Eq. (10) iχa[A¯ (t)aˆ+in1 + iA¯ (t)aˆ in2], p ± ∓ − are much smaller than the rest terms (the ratio N /N ∓ a L where a¯ . Then the solution can be written as signal where N and N are the atom number and photon∼ number χa = χ L a L and noise parts, respectively. The signal part is: in the pulse), which can be ignored. Also note that the op- tical operators on the r.h.s of the above equations are de- Z t iΩ(t t0) ¯ fined at the interaction point, in principle the Eq. 7 should be As (t) = Ωϕs dt0e∓ − A (t0) ± ∓ t0 ∓ (16) solved in the way that we substitutea ˆp = (aˆpin + aˆpout)/2 and = ϕsA¯ (0)sinΩ(t t0). aˆc = (aˆcin + aˆcout)/2. However, since the atom-light interac- ∓ ∓ − tion is weak, if we ignore the term involving high orders of The noise part is: χ, the r.h.s. of Eq. (10) can be simply written in the way that ˆ iΩt ˆ ˆ aˆ aˆ ,a ˆ aˆ . A (t) = e∓ A (t0) + A opt(t), (17) c = cin p = pin ± ± ± Substituting Eq. (10) into the equations of motion for the where atom field Eq. (7), we obtain: Aˆ opt(t) = Aˆ am(t) + Aˆ ph(t), (18)  ˆ  Fˆ A FA FA   ¯ iϕ  ± ± ± ˆ AA flu + cl + dy ABe L ˆ = ˆ B B B + Ωϕs ¯ iϕ . (11) with AB Fflu + Fcl + Fdy AAe− − Z t ˆ iΩ(t t0) ¯ Here the terms on the r.h.s can be understood as "optical force" A am(t) = iχa dt0e∓ − A (t0)aˆ+in1(t0), ± ∓ t0 ± acting on the atomic fields, explained in detail as follows: Z t (19) ˆ A/B Aˆ (t) = χ dt e iΩ(t t0)A¯ (t )aˆ (t ). (1) The Fflu have the form: ph a 0 ∓ − 0 in2 0 ± ± t0 ∓ − A i † i ˆ ¯ ϕp ϕc iΩ(t t ) Fflu = iχAB( a¯p e− aˆcin + a¯c aˆpine ), Here, the e 0 is the free propagator of the atom opera- − | | | | (12) ∓ − B iϕp iϕc † Aˆ Aˆ Aˆ Fˆ = iχA¯A( a¯p e aˆcin + a¯c e− aˆ ), tors . The ph and am are the quantum optical noise flu − | | | | pin ± ± ± contribution to the atom clouds evolution, while the Aˆ (t0) is which is the optical Langevin force acting on the atom fields, the initial quantum fluctuation of atom field. ± due to the randomness of the incoming states of optical fields. A/B (2) The Fcl have the form: B. Back-action noise χ2 FA = A¯ 2A¯ ( a¯ 2 a¯ 2), cl B A p c For the atom interferometer systems (both for a single atom 2 | | | | − | | (13) 2 interferometer and for the GW detector configuration involv- B χ 2 2 2 F = A¯A A¯B( a¯p a¯c ), cl 2 | | | | − | | ing a pair of atom interferometers), there exists such situations 4 that the same control fields connects several different intera- III. INTERFEROMETRY SOLUTION tion kernels. For example, in Fig. 3, the two π/2 processes are connected by the same control field. In the GW detec- This section will give the solution of the atom interferom- tor configuration proposed by Dimopoulos et.al (see Fig. 4), eter. As an example, we only show the solution of which the all the interaction kernels of the two atom interferometers are signal is only contributed from the optical phase imprinted on connected by the control fields. In these cases, the quantum the atom cloud during the atom-light interaction. This is ac- fluctuation of the first interaction kernel (e.g. denoted by a) tually the situation for the proposed atom interferometer GW can be carried by light field (probe) and then affects the sec- detectors. In many other important applications, the signals ond interaction kernel (e.g. denoted by b), and finally affects are carried by the atom fields themselves. For example, the the output atom fields (detector). This would lead to a “back- atom interferometry gravity meter is based on the principle action" noise, somewhat similar to the optomechanical system that the gravitational acceleration will affect the propagation that the quantum fluctuation of light (detector) will be carried phase of the atom fields. In an atom interferometer GW detec- by the test masses (probe), and then affect the output light field tor, this effect is the physical origin of the gravity noise, which (detector). Formally, to analyse such a system, we have to du- is an important issue that needs to be taken care for the design plicate the Hamiltonian, and the interaction should happen at since the local gravitational field can not be screened. In this two different spacetime points: paper, we will not discuss these issues (and all the classical noise sources) since they are not the subject of the quantum ˆ(a)† ˆ(a) † (a) ˆ(b)† ˆ(b) † (b) Hint = χA B (aˆcaˆd) + χA B (aˆcaˆd) . (20) measurement theory. For simplicity, we also do not consider | | the effect such as distortion of the atom cloud for simplicity Following the same approach discussed in the last section, and we assume that the free propagation of atom fields is co- one can write down the equations of motion for atom clouds of herent. the second interaction kernel in an almost identical form. The only difference is that the optical fields operators on the r.h.s of the atom equations of motion (Eq. (7)) can be connected to A. Input-output relation the optical fields flying out of the atom-light interaction region of the first interaction kernel, that is: At the detection stage of an atom interferometer, firstly the particle numbers of A and B atom species are detected re- (b) (a) spectively, and then the signal is extracted from their differ- aˆcin = aˆcout. (21) ence. Since the detected quantity ∆Nˆ = NˆA NˆB is in the T − (AˆA,AˆB) basis while the formulae of optical noise and back- Substituting this relation into the Langevin force T action terms are more concise in the (Aˆ+,Aˆ ) (see Eq. (19)), Eq. (12)for the second interaction kernel, and keeping − only those terms which due to the atom fluctuations brought we will use the transformation matrix between these two ba- from the first interaction kernel, we obtain the “back-action sis, defined as: force” acting on the atom fields of the second interaction 1  eiϕ/2 eiϕ/2  kernel as: T(ϕ) = iϕ/2 iϕ/2 , (25) √2 e− e− − (b) (b) (a) (a) (a) (a)† 2 ¯ ¯ ˆ ¯ ˆ T [FA ]BA = χa AB (AB ∗AA + AA AB ), and the transfer matrix of atom field in the Aˆ Aˆ basis is (22) ( A, B) (b) 2 (b) (a) (a)† (a) (a) given by: [F ]BA = χ A¯ (A¯ Aˆ + A¯ ∗Aˆ ). B − a A B A A B  iϕ  cosθ j isinθ je j In writing down these expressions, we have used the condi- M(θ j,ϕ j) = iϕ j − , (26) isinθ je− cosθ j tions of balanced control/ passive a¯p = a¯c = aL and − | | | | the optical field strength for these two interaction kernels are where θ j = Ωt j. (b) (a) identical aL = aL = aL. Adding these back-action force terms into the atom dynam- For the beam-splitting process (named as step-1), we have: ical equations Eq. (11) for the second interaction kernel, inter-  (1)    A (t) A(t0) grating the equations and expressing these back-action equa- (1) = M(θ1,ϕs1). , (27) ˆ B (t) B(t0) tions in terms of A ,aˆ1,2c/p, we have the back-action force ˆBA ± contributions A as : in which t0 is the initial time of the interrogation process, and ± ¯ ˆ A/B(t) can be decomposed into A/B(t) = AA/B(t) + AA/B(t), 2 t χ Z (b) (b) where A¯ (t) is the mean value of the atom field while AˆBA(t) = a dt e iΩ(t0 t)[A¯ (t ) ˆ(t )+A¯ (t ) ˆ(t )]. A/B 0 ± − 0 A 0 0 B 0 ˆ ± ∓ 2 t0 ± ∓ AA/B(t) is the perturbation around the mean value. At step- (23) ˆ 1, AA/B(t) contains the quantum fluctuation of atom field and where also the quantum fluctuation of light field, given as: a a a a ˆ ¯( ) ˆ( )† ¯( ) ˆ( ) " (1) #   " (1) # A (t) = A+ (t)A (t) + A ∗(t)A+ (t) h.c, Aˆ (t) Aˆ (t ) Aˆ (t) − − − (24) A A 0 +opt (a) (a)† (a) (a)† (1) = M(θ1,ϕs1). + T(0). (1) . (28) ˆ ¯ ˆ ¯ ˆ ˆ AˆB(t0) ˆ B(t) = A+ (t)A+ (t) A (t)A (t) + h.c. AB (t) A opt(t) − − − − 5

After the step-1, we have θ1 = π/4 with t1 = π/(4Ω), and atom information of step-2a and impose a “back-action" on the A(t) and B(t) fields start to separate spatially. The π/2 the step-2b. Here the effect of this back-action is denoted by processes for A-channel and B-channel connected by the con- ˆ(2) A BA(t), whose concrete representation can be derived from trol light happen sequentially and they should be treated in- Eq.± (23) dividually. Let us denote the π/2 processes of the A and B As shown in Fig.3, only one component of the output channels to be the step-2a and step-2b, respectively. Clearly, fields from step-2a/2b participates the recombination stage, the initial conditions of the step-2a and step-2b processes are while the other component is left unmeasured. For those re- (1) ˆ(2) T ˆ(2) (1) T ˆ(2) [A (t1),AB ] and [AA ,B (t1)] respectively, where AA/B combined components, we form a new input field column (2b) (2a) T are the field fluctuations injected at the π/2 steps, shown in for the recombination stage as: [A (t2),B (t2)] ,where Fig.3. t2 = π/(2Ω). The fields evolution at the recombination stage now can be x written as: 3  (3)   (2b)  A (t) π A (t2) (3) =M(θ3, + ϕs3). (2a) aˆ (3) B (t) 2 B (t2) p 3 (30) 2b " ˆ(3) # π A+opt(t3) 2a + T( ). (3) , 2 Aˆ (t3) (3) aˆ (2 opt c p b − aˆ ) 2a 2b which completes its recombination process at t = t3 = (2) π/(4Ω). Note that this equation can be expanded perturba- c aˆ (2 1 aˆ p a ) tively, since the signal terms containing phase ϕs and the noise (1) 1 aˆ (1) t aˆ c p terms are small compared to the expectation values. The re-

(2) (2) sults are given as follows. Aˆ AˆBini Aˆ B A Mean field—Expanding the output atom fields Eq. (30) to Atom Interferometer Mach-Zender Interferometer the• zeroth order, we obtain the final mean field as: " # A¯(3)(t ) 1  eiπ/4  FIG. 3. Atom interferometer and optical Mach-Zender interferom- A 3 = A¯ (0). (31) ¯(3) e iπ/4 A eter: a comparison. Left pannel: space-time diagram of the π/2 AB (t3) −√2 − processes happen in an atom interferometer.Right pannel: an optical Mach-Zender interferometer. On the optical Mach-Zender interfer- Signal field— Expanding the output atom fields Eq. (30) to ometer, part of the quantum noise injected at 2a and 2b stages are re- the• first order, we obtain the signal field as: flected away and left unmeasured. Similar situation also happens in the atom interferometer, where part of the atom noise of A chan-     A/B AAs 1 2 iϕs1 (1 + i)ϕs2 + ϕs3 nel injected to 2b/a interaction kernels will be reflected away and left = A¯A(0) − . (32) ABs 2 iϕs1 + (1 + i)ϕs2 ϕs3 unmeasured. However, the difference is, in the atom interferome- − − ter, the “mirrors" that reflects the matter waves are not uncorrelated Atom noise— Similarly, the noise contributed by the atom as in the optical Mach-Zender interferometer. The control field that • connects the interaction kernel 2a/b is the same field. fluctuations can be written as: " # Aˆ(3)(t ) eiπ/4  1 1Aˆ  During the π/2 processes, the corresponding transfer ma- A 3 = Aini ˆ(3) i i Aˆ trices are given by: AB (t3) − √2 Bini | − {z } atom shot noise Step-2a: (33) 2 2 " (2) (2)† # " # " (2a) # iχ A¯ (0) Aˆ Aˆ  (2a)  A(1) t ˆ a A B + B A (t) ( 1) A+opt(t) (2) (2)† . = (θ2,ϕs2). + (0). , − √ ˆ ˆ B(2a) t M ˆ(2) T ˆ(2a) 4 2Ω AB AB ( ) AB A opt(t) − − − | {z } Step-2b: back action noise " # A(2b) t  ˆ(2) Optical noise—The formulae for optical noise are more ( ) AA • b = M(θ2,ϕs2). complicated since they contain contributions from four dif- B(2 )(t) B(1)(t ) 1 ferent steps and the results are: " b # Aˆ(2 ) (t) + Aˆ(2) (t) + (0). +opt +BA , 1 T ˆ(2b) ˆ(2) ˆ(3) ˆ(1) ˆ(2a) ˆ(2a) ˆ(2b) ˆ(2b) A opt(t) + A BA(t) AA/B(t3) = [ 2iA+opt + A+opt A opt A+opt + A opt 2 − − − ± − − − (29) ˆ(3) ˆ(3) + (1 i)(A+opt + A opt)], ± − where the upper indices a/b denotes the A/B channels, re- (34) spectively. Since step-2a and step-2b are connected by the same control light, the control light after step-2a will carry Substituting the Eqs. (18) and (19) leads to the representation 6 of the above formula in terms of incoming optical noise fields: in which the first, second and third term are the orders of mag- nitude of the errors contributed by back-action noise, atom ¯ Z π/2Ω ˆ(3) χaA (2a) (2b) shot noise and purely optical noise, respectively. Apparently AA/B(t3) = dt0 sin2Ωt0[aˆ 2 (t0) iaˆ 2 (t0)] 2 0 − ± − the first and second terms have a trade-off when Na = NL, χ A¯ Z π/4Ω therefore the error has a minimum value i3π/4 a (3) 2iΩt0 (3) + e± dt0[aˆ+1(t0) + e aˆ+1(t0)] √2 0 2 1 χ A¯ Z π/4Ω [σs ]mim , (42) i3π/4 a (1) 2iΩt0 (1) ∼ NL + e dt0[aˆ+1(t0) + ie aˆ+1(t0)], √2 0 (35) which is actually the photon shot noise (usually Ωla/c = π/2 or π/4, i.e. Ωla/c 1). This corresponds to the standard quantum limit given∼ in Eq. (2). B. Standard quantum limit for a single atom interferometer. It is important to note that this Standard Quantum Limit can only be understood in the sense of extrapolation. Actually when Na NL, the linear approximation we used in analysing Using Eqs. (31)-(34), we can compute the particle numbers ∼ † † the atom-light interaction will not be valid. The real atom NA = A A and NB = B B after the recombination completes and expand to the first order of perturbation: interferometer does not work in this fully-nonlinear region. Therefore for real device, even in the most ideal situation, this 1 2 1 2 Standard Quantum Limit is not accessible as that of LIGO. It N A¯ (0) A¯ (0)(ϕs1 2ϕs2 + ϕs3), (36) A/B ≈ 2 A ∓ 2 A − only gives a bound to the device sensitivity.

Then the ∆N = NB NA, which is the atom number difference at states 2 and 1 −. , is linearly proportional to the signal: | i | i IV. BACK-ACTION IN ATOM INTEFEROMETER PAIRS ¯2 ∆Nsignal = AA(0)(ϕs1 2ϕs2 + ϕs3). (37) − − The back-action effect discussed in the last section, as we Similar methods can be used to treat the quantum optical have mentioned, also exists for the system of a pair of atom noise and the quantum atom noise, the latter of which is given interferometers. The detector configuration of atom interfer- by: ometer pair was proposed to measure low frequency GWs. Control fields carrying the atom information of the first inter- 3A¯3 ˆ ¯ ˆ ˆ† χa A(0) ˆ(2) ˆ(2)† ferometer imposes back-action on the second interferometer. ∆Natom = AA(0)(ABini + ABini)+ (AB + AB ). | {z } 2√2Ω The calculation follows the same logic as in the above sec- atom shot noise | {z } back-action noise tions, which is straightfoward but a bit tedious. We only give (38) the final results and discussions here. and the optical noise is given by: The ∆Nˆatom for the second interferometer is given by:

A¯A(0) (1) (1) (3) 2 ¯3 ˆ i3π/4 ˆ iπ/4 ˆ √ ˆ (II) (II) (II)† χa AA(0) (2II) (2II)† ∆Nopt = [e Aopt + e Aopt+ 2Aopt+ ∆Nˆ = A¯ (0)(Aˆ + Aˆ )+ (Aˆ + Aˆ ) √2 − − atom A Bini Bini B B (39) | {z } 2√2Ω 1 ˆ(2a) ˆ(2a) ˆ(2b) ˆ(2b) atom shot noise of AIII | {z } + (Aopt Aopt+ + iAopt + iAopt+] + h.c. back action noise inside AIII √2 − − − χ2A¯3 (0) a A [(Aˆ(I) + Aˆ(I)†) + i√2(Aˆ(2I) Aˆ(2I)†) Now, normalising the particle number difference ∆N by the 2Ω Aini Aini A − A signal coefficient, the estimator of the signal can be written (Aˆ(I) + Aˆ(I)†) + √2(Aˆ(2I) + Aˆ(2I)†)]. as: Bini Bini B B | − {z } 1 back action noise brought from AII ∆N = (ϕ 2ϕ + ϕ ) + (∆Nˆ + ∆Nˆ ), (40) est s3 s2 s1 ¯2 opt atom (43) − AA(0) Here the indices I and II here stand for the first (AI ) and sec- Here, we can approximate ϕ 2ϕ + ϕ ϕ¨ T 2, where T I s3 s2 s1 s ond (AI ) interferometers. The first term in the above ∆N is the interrogation time of the− atom interferometer.≈ II atom is the atom shot noise of the second interferometer while the To estimate the scaling of the error contributed by these second term is the back action noise contributed by atom fluc- noises, we need to map the parameters in the effective Hamil- tuations of step-2a of the second interferometer, similar to tonian model to the experimental parameters. It is easy to the second term in Eq. (38). The last term in Eq. (43) rep- prove that χ2 Ω χ2 a¯ 2 Ω Ωa c N using the rela- a / = L / = ( / )/ L resents the back-action imposed by the first interferometer via tion χ a¯ 2 Ω and the| fact| that the photon number in the L = control light. This result is obtained under the condition that rectangular| | pulse is N = a¯ 2l /c where l is the width of L L a a ¯(II) ¯(I) ¯ the optical pulse. Then the| scaling| of the error can be esti- AA (0) = AA (0) = AA(0), that is, the two atom interferome- mated as: ters have the same atom initial states [46]. The signal field is given by:  2 2 NA Ωla 2 1 σ + + , (41) (II) (II) (II) s ∼ N2 c N N ∆NII = A¯2 (0)(ϕ 2ϕ + ϕ ). (44) L A L signal A s1 − s2 s3 7

x AIII

Control

Aˆ(2II) B ˆ(2II) AA Aˆ(II) ˆ(II) Aini AII ABini

Passive lights

ˆ(2I) AB ˆ(2I) AA

ˆ(I) ˆ(I) t ABini AAini

FIG. 4. Atom interferometer detector configuration: this is the configuration proposed by Dimopoulos et.al. Two atom interferometers are connected by the control lights. The control fields flying out of the first atom interferometer will carry the corresponding atom informations and impose a back-action noise on the second atom interferometer. The scales in this figure (for illustrative purpose) does not reflects real situation. The distance between two interferometers in the real device is much larger than the length scale of atom interferometers themselves.

Clearly, the noise described by Eq. (43) and Eq. (38) are the same in terms of the orders of magnitudes. It is inter- ˆ(I) ˆ(I)† esting to note that, according to the general theory of linear correlated, since there are terms with (ABini + ABini) and quantum measurement, there exists a fundamental quantum (Aˆ(2I) +Aˆ(2I)†) in Eq. (43). This simply means that the two in- B B limit which is the so-called quantum Cramer-Rao bound [47]. terferometers are entangled via the coupling to the same con- Eq. (42) is also the quantum Cramer-Rao bound of the atom trol light fields, if the control light does not decohere strongly interferometer since the signals directly couple to the optical during its propagation between two interferometers. fields (probes) in the TT gauge and the probes’ fluctuations The optical noise in the case of atom interferometer pair here are determined by their own initial quantum states. consists of the contribution of three control lights and eight passive lights, which is very cubersome and not very interest- ing in the aspect of quantum measurement theory— simply contains the initial quantum fluctuation of the probes. We are V. DYNAMICS OF THE EFFECTIVE OPERATORS — A not going to show it here. MORE EXACT TREATMENT Finally, for extracting the GW signal, we need to substract the measurement results of the two interferometers. Sup- ˆ ˆ pose ϕGWi = kxGWi for a GW-induced optical phase modu- The exact definition of the operators AA,AB etc, and the 2 lation, then ϕGW1 2ϕGW2 + ϕGW3 kaGWT , where aGW = coupling strength χ in the effective Hamiltonian can be de- 2 − ∼ ω hGWL is the tidal acceleration (for a monochromatic grav- termined by using a field theory approach developed in the itational wave with frequency ω and detector baseline length Appendix. This field theory approach is based on the follow- L) and T is the interrogation time. A more detailed calcula- ing action: tion [6] showed that the full result (for a monochromatic GW Z wave with frequency ω and strain hGW) is: 2 Sint = g d xφA(x)φB(x)φc(x)φp(x), (46)   2 ωT sinωL ϕGW khGWLsin sinωt. (45) ∼ 2 ω where the coordinates x represents (t,z). The relationship be- in the limit of ωT 1 (which can be easily satisfied form low tween g and the physical quantities describing the atom-light frequency GW), it reduces to the result here. For the noise interaction such as the atom dipole moments, frequencies of part, it is easy to prove that the Eq. (41) and Eq.(42) are still different energy levels, etc. is given in the appendix. The cor- 8 responding equations of motion are given by: amplitude operators. Expanding the right hand side to the first order, we obtain: ˜+ ˜+ ˜+ ˜ (∂t + vA∂z)φ = gAφ φp φc−, A B Z z ˜+ ˜+ ˜ ˜+ ¯+ ¯+ (∂t + vB∂z)φB = gBφA φp−φc , gc dyφp φB (t + y,y)δφA−(t + y,y) (47) ε ˜+ ˜ ˜+ ˜+ − z (∂t + ∂z)φc = gcφA−φB φp , Z + g dyφ¯+φ¯ (t + y,y)δφ +(t + y.y) (50) ˜+ ˜+ ˜ ˜+ c p A− B (∂t ∂z)φp = gpφA φB−φc . ε − Z−z ¯+ ¯ + ˜+ + gc dyφB (t + y,y)φA−(t + y,y)δφp (t + y,y). Here the φ j ( j = A,B,c, p) are the slowly varying amplitude ε − field operators of the positive branch defined through φˆ+(x) = j The classical atomic fields can be written as (under the slow ˜+ iω j0(t t j0)+ik j0(z z j0) φ j (x)e− − − where ω j0 are the frequency of motion approximation): the free fields φˆi and related to the wave vector k j0 through +( ) ( ) 2 2 2 ¯ − ¯ ∗ ω j0 = k j0 + m j0 (for optical fields, the masses are zero). The φA/B (t,y) = fa(y)αA/B(t), (51) vA/B is the WKB velocity of atom wave packet A/B and the two optical fields are propagating along the opposite direc- where tions. The coupling constants are defined as: g = ig/(2ω ). j j0 1/2   ∆a 1 The φˆ− is the corresponding negative branch field operators. 2 2 j fa(y) = exp ∆ay . (52) As shown in details in Appendix, the initial states of the mean (2π)1/4 −4 optical fields can be treated as plane waves, the initial states of Since y [ ε,ε] and ε 1, we can expand the mean atom fields are zero and a Gaussian profile for the φB ∈ −  and φ , respectively. This Gaussian profile, in the spacetime- A α¯ (t + y) α¯ (t) + yα¯˙ (t). (53) domain is given by: A/B ≈ A/B A/B ˙ 1/2  2  Note that yα¯ A/B(t) Ωyα¯ B/A(t) and Ωy 1, we can sim- α¯ A∆A ∆A 2 | | ∼  φ¯A = exp [z z0 va(t t0)] , (48) plify the above terms to be: (2π)1/4 − 4 − − −  Z z Z z  ¯+ + where the α¯ A is the coherent amplitude and the 1/∆A is the gcφp α¯ B(t) dy fa(y)δφA−(t,y) + α¯ A∗(t) fa(y)δφB (t,y) width of the Gaussian profile. ε ε Z z − − 2 + + gc dyα¯ A∗(t)α¯ B(t) fa (y)δφp (t + y,y). ε − A. Perturbative solution to the optical fields: effective (54) operator for atoms Now let us define the effective atom operators to be: Typically, in an interferometric process, the light-atom in- Z z teraction time is very short compared to the free evolution δΦA±/B(t) = limz ε dy fa(y)δφA±/B(t,y). (55) → ε time of the atom cloud, and the centre of mass velocity of − the atom cloud is very low, typically 2 cm/s. Therefore to As we shall see later, these effective operators have a nice the leading order, we can treat the atom∼ center of mass motion property that the commutation relation of the associated cre- to be static during the interaction process, that is, vA vB 0. ation and annihilation operators normalises to one. The We also note that the spatial size of optical fields≈ are much≈ Physical interpretation of the effective operators is that it de- larger than the size of the atom cloud, therefore we can ap- scribes the whole wavepacket of the atom field. Using these proximate the mean value of the optical fields to be almost effective operators, the input-output relation for a light ray constants during the interaction process. For this calculation, passing through the atomic cloud can be written as (for the we only care about control field because it transfers noise to passive field, it can be calculated in the same way): the next atom-light interaction kernel, while different kernels + + ¯+ + interact with different passive fields, as shown in Fig. 3. δφcout(t) δφc (t) = gcφp [α¯ B(t)δΦA−(t) + α¯ A∗(t)ΦB (t)] − in Z ε These equations can be solved in a perturbative way. For 2 + + gcα¯ A∗(t)α¯ B(t) dy fa (y)δφp (t + y,y), the equation of motion of the control field, the first order per- ε turbation formal solution is given by: + + +− + δφ (t) δφ (t) = gpφ¯ [α¯ ∗(t)δΦ (t) + α¯ A(t)Φ−(t)] pout − pin c B A B + + Z ε δφc (t + z,z) δφc (t ε, ε) = 2 + Z z − − − + gpα¯ A(t)α¯ B∗(t) dy fa (y)δφp (t y,y). + + (49) ε − gc dyδ[φA−(t + y,y)φB (t + y,y)φp (t + y,y)], − ε (56) − where the atom cloud mostly distributed in [ ε,ε], as shown The negative frequency branches simply obey the Hermitian in Fig. 5 (we move the coordinate origin to the− atom center of conjugate of the above equations. The ratio between the sec- mass position). For brevity, we remove the tilde on the op- ond term and the first term on the r.h.s of the equation is p erators. In the following, all operators are the slowly varying Na/NL 1, therefore can be safely ignored. ∼  9

replace tilde with hat. Then the Eq. (56) can be translated to: passive field atom cloud control field iϕp † aˆcout aˆci = iχLe− [α¯ B(t)Aˆ + α¯ ∗(t)AˆB], − A A (59) iϕc † aˆpout aˆpi = iχLe− [α¯ ∗(t)AˆA + α¯ A(t)Aˆ ]. − B B Out p where we have χL = gc φ¯L ωL/ωa under the approximation | | that ω = ω = ω and φ¯+ = φ¯+ = φ¯ = a¯ /√2ω . Com- (y, t + y) c p L p c L L L paring this equation with| Eq.| (10),| | we know that the χ in the t effective Hamiltonian has the form:

χ g/(2ωL) = gc = gp. (60) → −

In B. Deriving the evolution of atom fields using field theory z ✏ ✏ Using Eq. (55), we integrate the perturbation equations of atom fields to obtain the perturbation equations of effective FIG. 5. Atom-light interaction kernel. The atoms are undergoing the atomic operators: state swapping interaction, and during the very short interaction time Z ε + iϕ + ¯+ ¯ + (typically µs), the A and B atoms does not have enough time to fly ∂t δΦA + iΩe δΦB = gaφp dz fa(z)ΦB (t,z)δφc−(t,z) apart. Our∼ effective atom operators Φ are defined through integration ε Z ε − over the atomic cloud profile on the spatial direction. The discussion ¯ ¯ + + + gaφc− fa(z)ΦB (t,z)δφp (t,z), in the Section II will reduce the problem to be an effective model ε describing the interaction of the effective atom operators with the − Z ε + iϕ + ¯ ¯ + + incoming light fields. ∂t δΦB + iΩe− δΦA = gaφp− dz fa(z)ΦA (t,z)δφc (t,z) ε Z ε − ¯+ ¯ + + gaφc fa(z)ΦA (t,z)δφp−(t,z). Furthermore, we introduce the creation and annihilation op- ε − (61) erators that correspond to those effective atom operators and also the optical operators. Remind that the operators here are Now, we substitute the formal solution of δφ into the above related to the creation and annihilation operators as follows: c±/p equations, which will reveal the structure of the optical back- action on the atom fields. Aˆ t a z t + A/B( ) + ˆc/pin( , ) Let us take the first term on the r.h.s of δΦA equation δΦA/B = , δφc/pin = , (57) √2ωa √2ωL as an example, it has the following form after substituting δφc−(t,z): where we take assumptions ωc0 ωp0 = ωL and note that Z ε ¯ + ¯ ≈ ¯+ ¯ + ¯+ ΦA/B = αA/B/(2ωa). Now we want to rewrite the equations gaφp dz fa(z)ΦB (t,z)δφc−(t,z) = gaφp αB(t) ε × of motion of the atom fields in a more concise way, using − Z ε  Z z these creation and annihilation operators. First, let us check 2 2 dz fa (z) δφc−in(t) + gcαA(t)αB∗(t) dy fa (y)δφp−(y)+ the dimension of the above defined creation and annihilation ε ε − − operators. We have Z z  ¯ + gcφp− dy fa(y)[αA(t)δφB−(y,t) + αB∗(t)δφA (y,t)] . ε Z − ˆ (62) AA/B(t) = dz fa(z)aˆA/B(z,t), and (58) † Note that the fa(y) takes a Gaussian form symmetric around [aˆA/B(z,t),aˆ (z0,t)] = δ(z z0). A/B − y = 0, which means that we can write: Z ε Z z where the commutation relation here is the standard one on 2 + 1 + dz fa (z) dy fa(y)δφA (y,t) = δΦA (t), (63) a time slice t. Using this commutation relation and the nor- ε ε 2 − − malisation condition for fa(z), it is straightforward to show ˆ ˆ† ˆ where we have used the normalisation condition for fa(z), and that [AA/B(t),AA/B(t)] = 1, and A is a dimensionless opera- 1/2 we have: tor,a ˆcin(z,t) has the dimension [length]− . The gravitational Z ε wave community is more familiar with the operator satisfying ¯+ ¯ + ¯+ † gaφp dz fa(z)ΦB (t,z)δφc−(t,z) = gaφp αB(t)δφc−in(t) [a˜(t),a˜ (t0)] = δ(t t0), it is important to note that this is an ε − − equal time commutation relation for propagating fields, and 1 2 2 + 1/2 gagc φ¯p [αA(t)αB(t)δΦ−(t) + αB(t) δΦ (t)]. the operators here are related bya ˜cin/aˆcin = c where c is 2| || | B | | A the . In the following, we will use thea ˜ and (64) 10

¯ Here, we ignore the φp− term since it is much smaller than the with the time scale equal to the interrogation time scale of other terms. In a similar way, we have: the matter wave interferometry. Therefore, the quantum limit discussed here is the limitations to the data point recorded by Z ε ¯ ¯ + + ¯ + each individual interrogation process. gaφc− dz fa(z)ΦB (t,z)δφp (t,z) = gaφc−αB(t)δφpin(t) ε Measurement quantity— Gravitational wave informa- − • 1 2 2 + tion is carried in the curvature perturbation Ψ h¨, which cor- gagc φ¯c [αA(t)αB(t)δΦ−(t) + αB(t) δΦ (t)]. − 2| || | B | | A responds to the acceleration of the test masses.∼ For LIGO, we (65) have the equation of motion for the test masses as mx¨ mΨ mLh¨, where m and L are effective mass of the test masses∼ mo-∼ It is clear that those terms containing δφc/pin are the op- tion and the baseline length of the interferometer, respectively. tical noise while the rest containing δΦA/B is the dynamical The light field will directly carry the information of test mass back-action of the control/passive optical fields to the atom displacement and the interferometer is a displacement sen- dynamics, which is a manifestation of fluctuation-dissipation sor. However, for an atom interferometer, each interrogation theorem. Note that when φ¯c = φ¯p = φ¯L , the dynamical | | | | | | process directly records the acceleration as shown in Eq.(44), back-action contributed by passive and control beam would therefore, an atom interferometer is an acceleration sensor. cancel with each other, leaving only the optical noise. Test mass quantisation—As we have shown in Eq. (1), Finally, we have the equations of motion for atom fields as: test• mass quantisation will generally have an impact on the measurement result. LIGO’s measurement result will be in ∂ δΦ+ + iΩeiϕ δΦ+ = t A B principle affected by the test mass quantisation if we directly ¯ iϕp iϕc + gaφLαB(t)[e− δφc−in(t) + e δφpin(t)], apply Eq. (1). However, if we do the post-data processing to extract the curvature perturbation (acceleration) informa- ∂ δΦ+ + iΩe iϕ δΦ+ = t B − A tion, we are targeted on x(2∆t) 2x(∆t) + x(0). Using the ¯ iϕp + iϕc − gaφLαA(t)[e δφcin(t) + e− δφp−in(t)]. free mass evolution equationx ˆ(t) = Lh(t) + pˆ0t/m + xˆ0, it is (66) easy to prove that all the information about the initial test mass quantum state will be eliminated when we extract the acceler- Then we can obtain the exact form of equations of motion ation information. This important result has been obtained by for atom cloud: Braginsky et al. [45] The key for the elimination of test mass quantisation effect is the fact that the differential motion of ˆ iϕ ˆ iϕp † iϕc ∂t AA + iΩe AB = iχB(t)[e− aˆcin(t) + e aˆpin(t)], th test masses of the two arms is a single degree of freedom (67) ˆ iϕ ˆ iϕp iϕc † during the entire detection process. However, for atom inter- ∂t AB + iΩe− AA = iχA(t)[e aˆcin(t) + e− aˆpin(t)]. ferometer (Dimopolous configuration), four different pairs of where ϕ = ϕc ϕp is the phase difference between control laser beams are needed to complete one interrogation period and passive lights.− and they belong to different degrees of freedom. Therefore, the probe quantisation effect can not be removed in the same r way as LIGO, that is why we need to consider the quantum ωa ¯ χA/B(t) = ga φLαA/B(t). (68) cωL | | fluctuation of light field (or the probe in a more general sense) in discussing the sensitivity of the atom interferometer. p The α (t) = α (0)exp[ Ωt] and χ0 := ωa/cωL ga φ¯L. Us- Back-action—In LIGO, the back-action is contributed ing the± relation± Eq. (60),∓ we can also map Eq. (67)| to| Eq. (7). by• the radiation pressure force acting on the test mass. In the atom interferometer, the back action comes from the atom noise carried by the control fields linking two atom interfer- VI. COMPARISON WITH THE LASER ometers. In LIGO, the back action can not only be a noise INTERFEROMETER GW DETECTOR source, but also change the dynamical behavior of the system in certain parameter regions. For example. If the resonant fre- Now we can make some comparison between Laser Inter- quency of the interferometer does not match the carrier laser ferometer GW detector and atom interferometer detector from frequency, which can be done through tuning the signal recy- several different aspects: cling cavity, the dynamics of the test mass and optical field Discrete or Continuous—LIGO is a detector where the will change and create an optical rigidity for the test mass. In optical• field continuously monitors the position of the test atom interferometer, the state transition dynamics (Rabi rate) masses, recording the continuous waveform of the gravita- of the atom cloud can also be changed by the interaction be- tional wave. The continuity of such measurement is the rea- tween optical and atomic fields, if there is an intensity unbal- son for the existence of SQL. However, the atom interferom- ance between the control and passive fields, as we have shown eter works differently, in a somewhat discretised way. Basi- in details in Section II.A and Section V.B. cally, each interrogation process records one data point of the The above discussion qualitatively compares the physics of waveform time-series, and the measurements of different data atom interferometer GW detector with the LIGO detector. It points are mutually independent. To record the waveform of also worthes to comment the similarity between atom inter- the GWs, the interrogation process needs to be repeated many ferometer GW detector with the LISA detector. Unlike the times. For each data point, the measurement is continuous, Michelson type LIGO detector, the LISA detector is actually 11 a transponder system where pairs of test masses are connected to the other will not contribute to the sensitivity in the cur- by an optical link and the optical phase change due to the grav- rent design of atom interferometer GW detector. The reason itational waves are recorded with local interferometry set up are the following. (1) the real three-dimensional atom-light around each test mass. Atom interferometer also matches with interactions happen in a point-scattering way that the cross this picture, where two atom clouds are connected by an opti- sectional area of the light field is much larger than the that of cal link and the phase change of this optical link is recorded. the atom field. Therefore such a scattering itself is very lossy In LISA, only the optical sensing noise is worth to be consid- thereby noisy. (2) for detecting GWs, these two interferome- ered and the quantum back action is extremely weak since the ters must be separated in a relatively large distance so that the optical power recieved on each satellite is of picowatts. Zero- atom cloud of the second interferometer will only interacts point fluctuation of the test masses on LISA can be eliminated with a small patch of the large wavefront propagated from the since it is also a displacement sensor. For a detailed study first interferometer. The diffraction loss will erase most of the for the comparison between atom interferometer GW detector information of the atom cloud of the first interferometer car- with LISA detector, see [10, 48] ried by the light field. Here, we want to emphasize that the lose of back-action noise here does not imply that the current atom-interferometer designs are perfect, quite contrary, it im- VII. DISCUSSION AND CONCLUSION plies that current designs are too lossy to have the back-action issue. It is probably insightful to study the method to miti- gate these issues in the future. General understanding tells us In this paper, the matter wave interferometry, in particular that the capability of any type of interferometric experimen- the example of a one-dimensional model of atom interferom- tal platform is determined by the of every physical eter, is carefully added onto the jigsaw puzzle of the linear steps in the device and the goal for reaching a better sensi- quantum measurement theory. Previous studies on the atom tivity is practically realised by mitigating all the issues that interferometry [7, 49], although quite complete and careful, decohere the waves in the interferometer. To mitigate the first mostly worked in the Schrödinger picture and did not speak issue mentioned above, we have to design the system so that in the langauge of quantum measurement theory. Establish- we have a mode-matched atom-light interaction and thereby ing such a theory for matter wave interferometry can help the a more coherent scattering of light by atom cloud, which is LIGO community clearly understand the physics of atom in- very difficult under the current technology; while to mitigate terferometer GW detectors. Our result demonstrates in de- the second issue, one way is to place the two atom interferom- tail about how the light-atom interaction affects the dynam- eters in some optical cavity assisted structures. This idea has ical and noise behavior of atom cloud and gives the input- been proposed for the MIGA (Matter-wave laser Interferomet- output relation for the linear response of the device. Con- ric Gravitation Antenna) project [37]. Analysing the quantum cretely, we clarify how the concepts of detector shot noise, noise for such a device will be a future extension of this work. probe’s zero-point fluctuation, back-action noise and dynami- cal back-action manifest themselves in the atom interferome- try. Similar to LIGO case, we also obtain the formula for the Standard Quantum Limit in the atom interferometry. ACKNOWLEDGEMENT The configuration raised by Dimipolous et.al is the most original one proposed for detecting GWs. Other configura- Y.M. thanks Zebin Zhou for his constant support and en- tions were also raised [9, 11, 37], in particular, the so called couragement, especially for sharing his interests in this work single-photon atom interferometer, where the interferometric during the trip to Hanford. Y.M. also thanks Albert Roura process happen via transition of Rabi oscillation of two-level for a fruitful discussion on different types of atom interfer- atoms rather than Raman transition [9]. The advantage of this ometer and Haixing Miao for an insightful discussion on the configuration is that the optical noise (zero-point fluctuations fundamental quantum limit. Y.M. thanks Yuta Michimura and of probe) can be removed in the ideal one dimensional case Kentaro Somiya for discussions on the concepts of using atom through common mode rejection. In order to increase the interferomter to detect gravitational waves. Y.C. thank Rana fringe visiblity of the atom interferometer, the distinguishbiliy Adikhari for a discussion on the experimental feasibility. He of the atom cloud trajectory should be increased which means also thanks Mark Kasevich and Hoger Müller for interesting we need a larger momentum transfer [33–35]. The formalism discussions on this work. Y.M, X.L, and Y.C. thank Belinda developed in this paper can be extended to these configura- Pang for discussions on the early stage of this paper. Y.C., tions also. Moreover, the concept of optical-cavity assisted X.L., and Y.M. are supported by the National Science Founda- atom interferometry was also discussed and experimentally tion through Grants PHY- 1612816, PHY-1708212 and PHY- tested [36, 37], using our formalism to study these configu- 1708213, the Brinson Foundation, and the Simons Foundation rations will be a future work. (Award Number 568762). The analysis in this work is targeted on revealing the phys- ical principle. Therefore we focus on the simplest case where we ignore the classical noise sources and optical losses that Appendix A: Field theoretical formalism of atom interferometer would more seriously affect the sensitivity of atom interfer- ometry. One typical example is, the back-action noise carried Usually, the light-atom interaction happens in a Λ-level sys- by the control fields that propagate from one interferometer tem with lower energy levels 1 , 2 which are the hyperfine | i | i 12 structures and a higher energy level 3 , as shown in Fig. 2. where mi = m + h¯ωi with the physical meaning that the inter- The atomic system is interacting with| i optical field which is nal electron-nuclei interaction energy also contributes to the off-resonant with respect to the energy gap of 1 3 and “rest mass" of the atom. It is clear that such a Hamiltonian can 2 3 . The dynamics of this system can be reduced| i − | i to an be derived from a canonically quantised Klein-Gordon field effective| i − | i 1 2 dynamics by adiabatically eliminating the with action: energy level| i −3 | .i Moreover, the atom states 1 and 2 are Z   also associated| i with different linear center of| massi momen-| i 4 1 µ 2 2 Si = d x ∂µ φi∂ φi + mi φi . (A6) tum. The non-relativistic Hamiltonian of such a system, in a 2 single-particle formalism, can be written as: Since we are now discussing a one-dimensional mode of atom ∇2 interferometers, we will apply the para-axial approximation Hˆ = i + h¯ω i i + E (x)d 1 3 + E (x)d 2 3 in order to reduce the above 3+1 action to a 1+1 action by 2m ∑ i c 13 p 23 − i | ih | | ih | | ih | integrating out the transversal components. + h.c, The relationship between 1+1 fields and the 3+1 fields is (A1) given as follows. The plane wave expansion of the fields is:

Z Z 2 where m is the inertia mass of the atom with center of mass dkz iωt+ikzz d k i~k ~x φ3+1(x) = e− ⊥ e ⊥ ⊥ aˆk + c.c. momentum pi = ih¯∇i, ωi corresponds to the energy of the (2π)(2ω) (2π)2 − internal electron state and di j describes the dipole moment (A7) 3/2 1/2 of the electron and it is convenient to define them as real We thus have the dimensional relation: [ak] = [m] [Hz] . numbers. The summation is over all three energy levels. While for 1+1 fields, we have: Since the 1 2 transition is forbidden, therefore d12 = 0. | i − | i Z dk For extension to the multi-particle system and furthermore z iωt+ikzz φ1+1(x) = e− ak + c.c, (A8) establishing a field-theoretique approach, we need to do (2π)(2ω) z second quantisation [50]. 1/2 1/2 where [akz ] = [m] [Hz] . Denoting the cross sectional area of field beams as A, we then integrate out the transversal com- ponents and define: 1. Free fields Z d2k i~k ~x √ akz ⊥2 ake ⊥ ⊥ A, (A9) For doing second quantisation, we need to first distinct the ≡ (2π) group of non-interacting identical particles, or equivalently, so that free theory. Tracing out the internal energy levels for the non- Z interaction single particle Hamiltonian, we have: dkz iωt+ikzz φ3+1(x) = e− ak + c.c (2π)(2ω)√A z 2 (A10) ∇i Hˆi = h¯ωi , (A2) 1 − 2m = φ1+1(x). √A Assuming that there are N identical particles in group i, we have a multi-particle Hamiltonian− as: The 3+1 form for electromagnetic field can be written as:

3 s N  2  Z d k h¯ωk c/p ∇ai Eˆ a e i(ωkt kx) Hˆi = h¯ωi . (A3) c/p = 3 ˆk − − + h.c, (A11) ∑ (2π) ε0 a − 2m and following the sam eapproach, one can show that it can where a is the index of particles, and ∇ai only act on the coor- dinate of a-th particle belong to group i. be effectively described by a 1+1 scalar field under paraxial Following the standard method of doing second quantisa- approximation: tion, and choosing the orthnormal eigenfunctions of particles s to be their momentum eigenstates, we have: q h¯ωc/p Ec/p(z,t) = 2ωc/p φc/p(z,t). (A12) ε0A Z 3  2  d p p † Hˆi = h¯ωi + m + aˆ aˆp, (A4) (2π)3 2m p 2. Interaction fields where the rest mass term is also included here. Apparently, this is a non-relativistic Hamiltonia. The full relativistic form In a similar way, we can do second quantisation to the in- is: teraction Hamiltonian as follows:

Z 3 q  j d p 2 † i j Z iE t i ˆ m p2 aˆ aˆ (A5) ˆ 3 p ˆ iEpt Hi = 3 i + p p, H = d xaψ∗ (xa)e 0 Ec/p(xa)di jψp(xa)e− , (A13) (2π) p0 p p0 13

i 2 where Ep = mi + p /(2mi), and ψp(xa) is the momen- 3. Effective interaction tum eigenfunctions of atoms in the coordinate representa- x tion. Using box normalisation condition, we have ψp( a) = The above full interaction Hamiltonian can be further re- ipx V exp( ~~a)/√ a. Substituting the second quantisation form duced to an effective Hamiltonian describing the transition be- of the− electromagnetic field: tween energy levels 1 2 by integrate out the φ3(x) field, | i − | i 3 s which can be done solving Heisenberg equation of motions Z d k h¯ωk c p ˆ / i(ωkt kx) ˆ Ec/p = 3 aˆk e− − + h.c, (A14) for the field φ3(x) (under canonical quantisation). (2π) ε0 The Heisenberg equation of motion for φˆ3(x)corresponds to leads to: the above action is: 3 3 i j j Z d xad k c p ˆ † i ikza i(Ep Ep ωk)t / H =di jaˆ aˆp ψp∗ (xa)ψp(xa)e− e 0 − − aˆk 2 ˆ ˆ ˆ ˆ ˆ p0 p p0 (2π)3 0 ( + m3)φ3 = g13φAφc + g23φBφp, (A24) † + (aˆk aˆ ,k k,ω ω) + h.c. → k → − → − Written in the rotating frame of ω and k (they are the (A15) 30 30 Compton frequency and Compton wave vector for the atoms Performing the integral on xa, we have: on level 3 ), we have: | i s j i i j h¯ω0 di j i(E Ep ω )t j† i c/p i t ik z ˆ p − − p0 p ˆ ˜ ω30 + 30 Hp p = 2 3 e 0 − aˆp aˆpaˆp p φ3(x) = φ3(x)e− + h.c, (A25) 0 ε0Va (2π) 0 − 0 (A16)

+ (non rotating wave terms) + h.c. and the equations of motion for φ˜3 is given approximately by: The energy difference in the exponential can be written as: ig13 2  2  ˜ ˜ ˜ i∆ω3At i∆k3Az p (p + kc) (∂t + v∂z)φ3(x) = φcφAe − m m 2ω30 i + + ωc/p j + , (A17) (A26) 2mi − 2m j ig 23 i∆ω Bt i∆k Bz + φ˜pφ˜Be 3 − 3 , which clearly describes the energy transfer of the photon- 2ω30 absorption process. In practice, the non rotating-wave terms can be safely ignored. The total interaction Hamiltonian will where we have already ignored the non-rotating-wave terms, be: which oscillates at frequencies ω30 ω ω or ω30 + ± c/p0 ∓ A/B0 H = H , (A18) ωc/p0 + ωA/B0. The ∆ω3A/B,∆k3A/B are defined as: ∑ p,p0 p,p0 ∆ω3A/B = ω30 ωc/p ωA/B0 where p, p0 are discretised momenta. In the continuous case, − − we have: ω3A/3B ωc/p0 + vA/B(k30 kA/B0), (A27) s ≈ − − Z 3 3 j i ∆k3A B = k30 kc p kA B0, d p d p0 h¯ω0 di j j† i c/p i(E Ep ω )t / / / p − − p0 p − − H = 3 3 3 aˆp aˆpaˆp p e 0 − , (2π) (2π) ε0 (2π) 0 − 0 where ω are the energy level difference between 3 and + h.c 3A/B 1 , 2 . In ideal case, we assume that the phase matching| i (A19) | i | i is satisfied ∆k3A/B = 0. We also define ∆ω0 := ω3A ωc = ˜ − where for the atomic operators:a ˆp √Vaaˆp (note that the ω3B ωp and if we futher choose φ3 to work iin the rotating → − dimension changes correspondingly). frame of ∆ω0, the equation becomes: This Hamiltonian can be derived from the following inter- action action: g13 iv k t ∆ω φ¯ (x) + (∂ + v∂ )φ¯ (x) = φ˜ φ˜ e A c Z 0 3 t z 3 c A full 4 2ω30 Sint = g d xφA(x)φ3(x)φc/p(x), (A20) g (A28) + 23 φ˜ φ˜ eivBkpt , 2 p B where: ω30 s h¯ω0 di j 1/2 1/2 1/2 Under the adiabatic approximation that the typical time scale g = 9 (2ωA) (2ωc) (2ωc/p) . (A21) ¯ ε0 (2π) of varying φ3(x) is much smaller than 1/∆ω0, the dynamics of φ¯3(x) is slaved by the right hand side of the above equation: Reducing it to 1+1 D case, the g will change as: g g s 13 ivAkct 23 ivBkpt h¯ω d φ¯3(x) φ˜cφ˜Ae + φ˜pφ˜Be , (A29) 0 i j 1/2 1/2 1/2 ≈ 2∆ω ω 2∆ω ω g = 9 (2ωA) (2ωc) (2ωc/p) , (A22) 0 30 0 30 ε0A (2π) and we have an 1+1 interaction action: while the positive frequency branch of φˆ3(x) can be recovered Z Z by changing to the original frame: φ¯3(x)exp[i∆ω0t iω30t + S = g dzdtφ φ φ + g dzdtφ φ φ . (A23) − 13 A 3 c 23 B 3 p ik30z]. Substituting φˆ3(x) back into the interaction Hamilto- 14 nian Eq. (A23), we have: field operator given by:

Stark AB Z dk h i ˆ = ˆ + ˆ , j iω j(t t j0)+ik j(z z j0) Hint Hint Hint φˆj(x) = Aˆ(k j,t)e + h.c , 2π2ω − − − g2 Z j ˆ Stark 13 ˆ ˆ ˆ ˆ (B4) Hint = dzφA(x)φc(x)φc(x)φA(x) 2∆ω0ω30 ˆ ˆ † ˆ ˆ † where A(k j,t) = UI (0,t)A(k j,0)UI (0,t). 2 Z g23 + dzφˆB(x)φˆp(x)φˆp(x)φˆB(x), 2∆ω0ω30 Since all these fields have a WKB trajectory in real experi- Z ment, they can take a scale-separated form as: Raman g13g23 Hˆ = 2 dzφˆA(x)φˆc(x)φˆp(x)φˆB(x). int × 2∆ω ω 0 30 ˆ ˜ iω j0(t t j0)+ik j0(z z j0) (A30) φ j(x) = φ j(x)e− − − + h.c (B5) where the exponents describes the fast-oscillating part of the ˆ Stark Here, the Hint corresponds to the AC Stark shift of the mass field and the tilde operators describe the slowly varying ampli- of the atom fields, which is typically very small compare to tudes. Using the above definition, one can obtain the relation the internal energy and rest mass thereby negligible [51].The between Aˆ operator and φ˜ as: ˆ Raman Hint term is the one we are interested in, i.e. describing − the Raman transition between 1 and 2 induced by control Z dk A i(ω j ω j )(t t j )+i(k j k j )(z z j ) | i | i φ˜j(t,z) = [Aˆ(t,k j)e− − 0 − 0 − 0 − 0 and passive fields. This term can also be correspondence to an 2π2ω j effective four-scalar field interaction action: + h.c.] Z 2 (B6) Sint = g d xφA(x)φB(x)φc(x)φp(x), (A31) Also note that the transformation Uˆ (0,t), as an unitary trans- ˆ Raman where g represents the coupling coefficient in Hint . formation, will not affect the commutation relation, therefore we have:

† [Aˆ j(t,k),Aˆ (t,k0)] = 2ω jδ(k k0). (B7) j − Appendix B: Field Quantisation It is worth to note that in the spatial domain, the Aˆ and the φ˜ 1. Definition of field operators is related by (take A-field as an example):

(+) t φ˜ (x) = Aˆ(t,z z )/(2ωA0), (B8) As discussed above, The one-dimensional light-atom inter- − A action model can be described by the following interaction t t where z zA (zA = zA(t0) + vA(t t0)) comes from the propa- action as: gation and− the first t argument− comes from the perturbation − Z due to the 4-scalar interaction. 2 Sint = g d xφA(x)φB(x)φc(x)φp(x) (B1) where x = (t,z). These free scalar fields, after canonical quan- 2. Quantum states of fields tisation can be expanded as: In the experimental setup, the control/passive field can be Z dk j h i ˆ(0) iω j(t t j0)+ik j(z z j0) well approximated to have a rectangular profile, and the states φ j (x) = aˆ(k j)e− − − + h.c , 2π2ω j of, e.g. the control field is given by: (B2) Z  where thea ˆ(k j) is an annihilation operator which is covariant dkc †  ψc = exp αc(kc)aˆc(kc) h.c. 0 c , (B9) under Lorentz transformation and j = A,B,c, p. The ω j and | i 2ωc − | i k are related by dispersion relation ω2 = k2 + m2, while j j j j where for optical fields m j = 0. The t j0,z j0 determine the phase reference point. sinπaδk αc(kc) = α¯ c , δk = kc kc0, (B10) √aπδk − When we introduce the interaction, the Heisenberg opera- tors will be modified, according to: where 2πa is the width of this rectangular wave and α¯ c is the coherent amplitude. † (0) Oˆ(t,z) = Uˆ ( ∞,t)Oˆ (t,z)UˆI( ∞,0), (B3) It is easy to show the φ waveform I − − − (0) Z dk  1  sinπaδk where Oˆ (t,z) and Oˆ(t,z) are the operators whose evolution (+) i(k0+δk)(x t x0+t0) φ¯ = α¯ c e − − 2π 2ω √aπδk are governed by the free and full Hamiltonian, respectively. (B11) The Uˆ (t ,t ) is the evolution operator in the interaction pic- α¯ I 1 2 c ik0(x t x0+t0) Recta[x t x0 +t0]e − − . ture. Therefore, in the interaction case, one can have a full ≈ 2ω0 − − 15

Since the typical atom width is much smaller than the light These conditions, under non-relativistic approximation and pulse width, it can be approximated as almost a plane wave in mB mA, has the clear physical meaning of relativistic ≈ the atom-light interaction region by lima ∞a sinc(πax) = Doppler effect: δ(x). → × The initial state of atomic cloud is a Gaussian profile, given ωc0 ωp0 v(kA0 kB0), (B17) as: − ≈ − Z   dkA † where v vA vB is the approximate speed of atom. The ψA = exp αA(kA)aˆA(kA) h.c. 0 A , (B12) ≈ ≈ | i 2ωA − | i Hermitian conjugation of these equations can also be easily obtained. In the following, we are going to solve these with equation in a perturbative way.  2  α¯ A (kA kA0) αA(kA) = exp − , (B13) Appendix C: Mean field solutions 1/4 1/2 − 4∆2 (2π) ∆A A while the B-field is a vacuum initially. Here the α¯ A is the atom Typically, in an interferometric process, the light-atom in- coherent amplitude, ∆A is its width in the k-domain. teraction time is very short compare to the free evolution time of the atom cloud, and the centre of mass velocity of the atom cloud is very low, typically 2 cm/s. Therefore to the leading ∼ 3. Equations of motion: structures order, we can treat the atom centre of mass motion to be static during the interaction process, that is, vA vB 0. We also ≈ ≈ Following the standard canonical quantisation scheme, we note that the spatial size of optical fields are much larger than have the Heisenberg equations of motion for the atomic and the size of the atom cloud, therefore we can approximate the optical field as: mean value of the optical fields to be almost constants during the interaction process. 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( + mA)φA = gφBφcφp, φc = gφAφBφp The zeroth-order of the equations of motion is simple: (B14) 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( + mB)φB = gφAφcφp, φp = gφAφBφc, ¯+ ¯+ ¯ ¯+ ¯+ ∂t φA = (gAφp φc−)φB , (∂t + ∂z)φc = 0, 2 2 2 (C1) where  = ∂t ∇ (in 1-dimensional case here,  = ∂t ¯+ ¯ ¯+ ¯+ ¯+ 2 − − ∂t φB = (gBφp−φc )φA , (∂t ∂z)φp = 0. ∂z ). These equations have the same form as their classical − counterparts, this is because different φˆj belongs to different Hilbert space and their operators are commute with each other. We ignore the r.h.s. of the equation for the optical field be- Substituting Eq. (B5), we have the approximated equations for cause the photon number is much larger than the atom num- ¯ ¯ the slowly varying operator’s positive frequency part (we take ber. Since φp and φc are almost constant, therefore we can c = 1 and their Hermitian conjugates are ignored for brevity): rewrite the zeroth-order atom equations to be: ˜+ ˜+ ˜+ ˜ (∂t + vA∂z)φA = gAφB φp φc−, ¯+ iϕpc ¯+ ¯+ iϕpc ¯+ ∂t φA = iΩe φB , ∂t φB = iΩe− φA , (C2) ˜+ ˜+ ˜ ˜+ − − (∂t + vB∂z)φB = gBφ φp−φc , A (B15) ˜+ ˜ ˜+ ˜ (∂t + ∂z)φc = gcφA−φB φp−, where ϕpc is the phase difference between the control field ˜+ ˜+ ˜ ˜+ and passive field, Ω is the Ramsey frequency. Here, we make (∂t ∂z)φp = gpφA φB−φc . ¯ ¯ − use of the approximation gA = gB := ga thereby Ω = gaφpφc . The full solution of this equation is: | | Here, vA/B the WKB velocity of atom wave packet A/B and the two optical fields are propagating along the opposite direc- + + + iϕcp tions. The coupling constants are defined as: g j = ig/(2ω j0), φ¯ (t) = φ¯ (0)cosΩt iφ¯ (0)e sinΩt, A A − B (C3) where j = A,B,c, p. In deriving the above equations of mo- + + + iϕcp φ¯ (t) = φ¯ (0)cosΩt iφ¯ (0)e− sinΩt. tion, we take the leading non-relativistic approximation so that B B − A vA/B kA/B/mA/B and we also use the rotating wave approxi- mation,≈ namely, we only keep those terms satisfying: The gravitational wave signal will be carried by the ϕcp. Clearly, this is where the matrix M(θ,ϕ) in the main text kp0 kc0 = kA0 kB0, ωc0 ωp0 = ωB0 ωA0. (B16) comes from. − − − −

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