1. Introduction

Since its birth approximately 40 years ago (Dubetskii et al., Reference Dubetskii, Kazantsev, Chebotayev and Yakovlev1984), the field of atom interferometry has matured significantly. The current state and prospects in this area are presented, for example, in reviews (Tino and Kasevich, Reference Tino and Kasevich(ed)2014) and proposals (Canuel et al., Reference Canuel, Abend, Amaro-Seoane, Badaracco, Beaufils, Bertoldi, Bongs, Bouyer, Braxmaier, Chaibi, Christensen, Fitzek, Flouris, Gaaloul, Gaffet, Garrido Alzar, Geiger, Guellati-Khelifa, Hammerer, Harms, Hinderer, Holynski, Junca, Katsanevas, Klempt, Kozanitis, Krutzik, Landragin, Roche, Leykauf, Lien, Loriani, Merlet, Merzougui, Nofrarias, Papadakos, Pereira Dos Santos, Peters, Plexousakis, Prevedelli, Rasel, Rogister, Rosat, Roura, Sabulsky, Schkolnik, Schlippert, Schubert, Sidorenkov, Siemß, Sopuerta, Sorrentino, Struckmann and Tino2020; Zhan et al., Reference Zhan, Wang, Ni, Gao, Wang, He, Li, Zhou, Chen, Zhong, Tang, Yao, Zhu, Xiong, Lu, Yu, Cheng, Liu, Liang, Xu, He, Ke, Tan and Luo2020; Badurina et al., Reference Badurina, Bentine, Blas, Bongs, Bortoletto, Bowcock, Bridges, Bowden, Buchmueller, Burrage, Coleman, Elertas, Ellis, Foot, Gibson, Haehnelt, Harte, Hedges, Hobson, Holynski, Jones, Langlois, Lellouch, Lewicki, Maiolino, Majewski, Malik, March-Russell, McCabe, Newbold, Sauer, Schneider, Shipsey, Singh, Uchida, Valenzuela, van der Grinten, Vaskonen, Vossebeld, Weatherill and Wilmut2020; Battelier et al., Reference Battelier, Bergé, Bertoldi, Blanchet, Bongs, Bouyer, Braxmaier, Calonico, Fayet, Gaaloul, Guerlin, Hees, Jetzer, Lämmerzahl, Lecomte, Le Poncin-Lafitte, Loriani, Métris, Nofrarias, Rasel, Reynaud, Rodrigues, Rothacher, Roura, Salomon, Schiller, Schleich, Schubert, Sopuerta, Sorrentino, Sumner, Tino, Tuckey, von Klitzing, Wörner, Wolf and Zelan2019; El-Neaj et al., Reference El-Neaj, Alpigiani, Amairi-Pyka, Araújo, Balaž, Bassi, Bathe-Peters, Battelier, Belić, Bentine, Bernabeu, Bertoldi, Bingham, Blas, Bolpasi, Bongs, Bose, Bouyer, Bowcock, Bowden, Buchmueller, Burrage, Calmet, Canuel, Caramete, Carroll, Cella, Charmandaris, Chattopadhyay, Chen, Chiofalo, Coleman, Cotter, Cui, Derevianko, De Roeck, Djordjevic, Dornan, Doser, Drougkakis, Dunningham, Dutan, Easo, Elertas, Ellis, El Sawy, Fassi, Felea, Feng, Flack, Foot, Fuentes, Gaaloul, Gauguet, Geiger, Gibson, Giudice, Goldwin, Grachov, Graham, Grasso, van der Grinten, Gündogan, Haehnelt, Harte, Hees, Hobson, Hogan, Holst, Holynski, Kasevich, Kavanagh, von Klitzing, Kovachy, Krikler, Krutzik, Lewicki, Lien, Liu, Luciano, Magnon, Mahmoud, Malik, McCabe, Mitchell, Pahl, Pal, Pandey, Papazoglou, Paternostro, Penning, Peters, Prevedelli, Puthiya-Veettil, Quenby, Rasel, Ravenhall, Ringwood, Roura, Sabulsky, Sameed, Sauer, Schäffer, Schiller, Schkolnik, Schlippert, Schubert, Sfar, Shayeghi, Shipsey, Signorini, Singh, Soares-Santos, Sorrentino, Sumner, Tassis, Tentindo, Tino G, Tinsley, Unwin, Valenzuela, Vasilakis, Vaskonen, Vogt, Webber-Date, Wenzlawski, Windpassinger, Woltmann, Yazgan, Zhan, Zou and Zupan2020; Tino et al., Reference Tino, Bassi, Bianco, Bongs, Bouyer, Cacciapuoti, Capozziello, Chen, Chiofalo, Derevianko, Ertmer, Gaaloul, Gill, Graham, Hogan, Iess, Kasevich, Katori, Klempt, Lu, Ma, Müller, Newbury, Oates, Peters, Poli, Rasel, Rosi, Roura, Salomon, Schiller, Schleich, Schlippert, Schreck, Schubert, Sorrentino, Sterr, Thomsen, Vallone, Vetrano, Villoresi, von Klitzing, Wilkowski, Wolf, Ye, Yu and Zhan2019; Abe et al., Reference Abe, Adamson, Borcean, Bortoletto, Bridges, Carman, Chattopadhyay, Coleman, Curfman, DeRose, Deshpande, Dimopoulos, Foot, Frisch, Garber, Geer, Gibson, Glick, Graham, Hahn, Harnik, Hawkins, Hindley, Hogan, Jiang, Kasevich, Kellett, Kiburg, Kovachy, Lykken, March-Russell, Mitchell, Murphy, Nantel, Nobrega, Plunkett, Rajendran, Rudolph, Sachdeva, Safdari, Santucci, Schwartzman, Shipsey, Swan, Valerio, Vasonis, Wang and Wilkason2021).

Among other applications, atom interferometers (AIs) are now proposed for use as inertial measurement units (IMUs) in navigation.

The precision accuracy of AIs allows us to hope for significant progress, improving the accuracy and duration of the navigation. These applications are based on the possibility of using an AI as an accelerometer (Kasevich and Chu, Reference Kasevich and Chu1991). If the atoms move with acceleration $\boldsymbol{a}$, then the AI phase is given by

(1)\begin{equation} \phi =\mathbf{k}\cdot \mathbf{a}T^{2}, \end{equation}

where $\mathbf {k}$ is the effective wave vector of the Raman pulses and $T$ is the delay time between these pulses. The error of the acceleration measurement is

(2)\begin{equation} \delta a=\phi_{{\rm err}}/kT^{2}, \end{equation}

where $\phi _{{\rm err}}$ is the accuracy of the AI phase measurement. To date, for example, the accuracy of differential acceleration measurements has reached

(3)\begin{equation} \delta a=1.4\,{\rm ppt} \end{equation}

at the delay time

(4)\begin{equation} T=955\,{\rm ms}. \end{equation}

For navigation applications, a system of six AIs was created and tested (Canuel et al., Reference Canuel, Leduc, Holleville, Gauguet, Fils, Virdis, Clairon, Dimarcq, Borde and Landragin2006). However, the direct use of a precise AI as an IMU is difficult. As a matter of fact, the IMUs data should be substituted into the navigation equations, and for an accurate numerical solution of these equations, the IMUs measurements should be repeated with a time step of at least $\tau \sim 1$ ms (Ahmed and Cuk, Reference Ahmed and Cuk2005). To do this, the interrogation time of the AI must be no more than $\tau$, and therefore $T\lesssim \tau /2$. Thus, with the same accuracy of the phase measurement $\phi _{{\rm err}}$ and the effective wave vector $k$, the acceleration measurement error increases by six orders of magnitude to the level of $5$ ppm. With such an inaccuracy, atomic interferometers will be even an order of magnitude worse than conventional accelerometers (Jekeli, Reference Jekeli2001) and will not result in any progress in navigation.

Two methods have been proposed to avoid this difficulty. In one method, AIs are used in a hybrid mode together with conventional IMUs (Canciani, Reference Canciani2012; Lautier et al., Reference Lautier, Volodimer, Hardin, Merlet, Lours, Pereira Dos Santos and Landragin2014; Cheiney et al., Reference Cheiney, Fouché, Templier, Napolitano, Battelier, Bouyer and Barrett2018; Wu et al., Reference Wu, Guo, Feng, Chen, Yuan and Zhang2020; Wang et al., Reference Wang, Kealy, Gilliam, Haine, Close, Moran, Talbot, Williams, Hardman, Freier, Wigley, White, Szigeti and Legge2021). Another option Kasevich and Dubetsky (Reference Kasevich and Dubetsky2005) is related to the fact that one can restore the position of the platform directly from the AI phase without measuring the atomic acceleration and, in principle, without any use of conventional IMUs. To understand why this is possible, one notes that Equation (1) is valid only for the uniform accelerated motion of the atom. If the atomic cloud moves in the platform frame, then, ignoring the quantum effects, for the phase instead of that from Equation (1), one gets

(5a)\begin{align} \phi & =\mathbf{k}\cdot \mathbf{p}, \end{align}

(5b)\begin{align} \mathbf{p} & =\mathbf{x}(t_{3}) -2\mathbf{x}(t_{2}) + \mathbf{x}(t_{1}), \end{align}

where $\mathbf {x}(t)$ is the classical trajectory of the cloud in the platform frame. Knowing the trajectory at the previous moments $t_{1}$ and $t_{2}=t_{1}+T$ and measuring the phase of the AI, one restores the trajectory at the moment $t_{3}=t_{1}+2T$. At the same time, there is no need to solve navigation equations, and as a result, the restriction on the feasible delay time $T$ is removed. One sees that when using an AI as an IMU, one actually does double the work: first, by measuring the trajectory, $\mathbf {x}(t_{3})$, to determine the acceleration, and then from solving the navigation equations to determine the trajectory.

The error of measuring the position $\mathbf {x}(t_{3})$ is given by

(6)\begin{equation} \delta_{x}\sim \phi_{{\rm err}}/k\sim 1.3\text{pm}, \end{equation}

where to estimate the parameter $\phi _{{\rm err}}/k$, one can use Equations (2), (3), (4). Despite the fact that at each step the error determined by Equation (6) does not depend on $T$, the total error $\Delta X$ during navigation time $t_{n}$ accumulates after repeated measurement of phases at time multiples of $T$. For the given $t_{n}$, the number of steps is

(7)\begin{equation} n\sim t_{n}/T. \end{equation}

The use of AIs with a longer interrogation time T leads to a reduction in the number of steps and, consequently, to a reduction in the accumulated error. Thus, one sees that, as proposed by Kasevich and Dubetsky (Reference Kasevich and Dubetsky2005), the rejection of the use of AIs as accelerometers allows the use of an unprecedentedly high accuracy of trajectory measurement by Equation (6) for an unprecedentedly accurate navigation.

Atom interference is an exclusively quantum phenomenon. Nevertheless, Equation (5) for the phase is purely classical. This means that instead of quantum objects, atoms, one can use classical material points, which in the future we will call particles. We do not specify the kind of particles. However, we assume that the mass of the particle $M$ is sufficiently large to neglect the recoil velocity $\hbar k/M$, where, in our case, $\hbar k$ is the change in momentum that accompanies the measurement of the particle position.

Like in the patent of Kasevich and Dubetsky (Reference Kasevich and Dubetsky2005), in this paper, we propose to abandon the use of accelerometers, gyroscopes, and navigation equations, and instead measure the trajectories of particles that are launched in the platform system from certain points, at certain speeds, and with a time interval $T$. We will develop a theory of such navigation here and perform numerical simulations. The obvious advantages of classical objects are a dramatic mitigation in the requirements of high vacuum, avoiding the necessity to solve the problem of the phase multiple ambiguity, and the absence of a recoil effect. The obvious disadvantage of our method is that for classical objects, the accuracy by Equation (6) of their position measurement is hardly achievable at the present time. We have, however, developed methods here that will significantly relax the requirements for $\delta _{x}$. The ultimate goal of numerical simulation is to answer the question of what the accuracy of measuring the coordinates and velocities of particles should be to achieve a given accuracy of navigation $\Delta X$ for a given period of time $t_{n}$. Specifically, we will find the answer to this question for

(8a)\begin{align} t_{n} & =1\,\text{h}, \end{align}

(8b)\begin{align} \Delta X & =10\,\text{m}. \end{align}

The accelerometer in conventional navigation measures the acceleration of particle $\mathbf {a}$ in the platform system. For $\mathbf {a}$ in the absence of rotation, one has

(9)\begin{equation} \mathbf{a}=\mathbf{g}-\mathbf{A}, \end{equation}

where $\mathbf {g}$ and $\mathbf {A}$ are the gravitational field of the Earth and the acceleration of the platform. Knowing $\mathbf {g}$ and measuring $\mathbf {a}$, one obtains the acceleration of platform $\mathbf {A}$, and by substituting it into the navigation equation

(10)\begin{equation} {\mathbf {\ddot X}}(t) = {\mathbf A}, \end{equation}

one can determine the trajectory of the platform $\mathbf {X}(t)$.

In Kasevich and Dubetsky (Reference Kasevich and Dubetsky2005), it is actually proposed to replace the accelerations $\mathbf {a}$ and $\mathbf {A}$ with the trajectories $\mathbf {x}(t)$ and $\mathbf {X}(t)$. The procedure for determining the position of the platform by measuring the position of the particle in the absence of rotation of the platform is illustrated in Figure 1.

Figure 1. (Colour online) Exploiting the measurement of the particle position in the platform frame for navigation. The trajectories of the platform and the particle are shown with solid and dotted curves

We are interested in the trajectory of the platform $\mathbf {X}(t)$ in the coordinate system $\{ X,Y,Z\}$ rotating with the Earth. On the platform, one installs a local frame $\{x,y,z\}$ with the origin at the point $\mathbf {X}(t)$ and sensors that measure the position of the particle $\mathbf {x}(t)$ in that local frame. At the moment of time $t_{1}$, one launches the particle. After the launch, the particle is completely decoupled from the platform and moves along the trajectory $\mathbf {X}_{E}(t)$ in a vacuum under the influence of only the rotating Earth. Knowing the potential of the Earth's gravitational field and the velocity of the Earth's rotation $\vec {\Omega }_{E}(t)$, one can obtain the trajectory $\mathbf {X}_{E}(t)$. At time $t_{2}$, one measures the position of the particle $\mathbf {x}$. Then from Figure 1, one gets for the platform trajectory

(11)\begin{equation} \mathbf{X}(t_{2}) =\mathbf{X}_{E}(t_{2}) -\mathbf{x}. \end{equation}

The position of the particle at the time of launch in the system $\{ X,Y,Z\}$, obviously coincides with the position of the platform, $\mathbf {X}_{E}(t_{1}) =\mathbf {X}( t_{1})$. To determine the trajectory $\mathbf {X}_{E}(t)$, it is also necessary to determine the initial velocity of the particle, $\mathbf {v}_{E}(t_{1})$, which depends on the velocity of the platform $\vec {V} (t_{1})$, the velocity of the particle $\mathbf {v}$ in the platform frame, the initial orientation of the platform, the rotation matrix $R(t_{1})$, and the initial rate of the platform rotation $\mathbf {\Omega }(t_{1})$. It is also necessary to take into account that at the moment $t_{2}$, the orientation of the system changes, the rotation matrix $R( t_{2}) \not =R( t_{1})$, and one must rotate the vector $\mathbf {x}$ from the frame $\{x,y,z\}$ in which it is measured to the frame $\{ X,Y,Z\}$. Below we will derive an obvious generalisation of Equation (11) taking into account all these factors. However, here we we would like to note that for successful navigation restoring the position of the platform, it is also necessary to restore the parameters

(12)\begin{equation} Z=\{ \mathbf{V}(t), \mathbf{\Omega }(t), R(t) \}. \end{equation}

However, to demonstrate the power of the navigation method we are considering here, let us assume that these parameters are known exactly. We also assume that the particles are launched with a step $T=300$ ms, and the position of the particle is measured with accuracy [standard deviation (SD)]

(13)\begin{equation} \delta_{x}=1\,\mu \text{m}. \end{equation}

Then the time dependence of the navigation error is shown in Figure 2.

Figure 2. (Colour online) Time dependence of the navigation error (green curve) $\Delta X(t)$ for an ensemble of 30 randomly generated platform trajectories $\mathbf {X}(t)$. The red curve is the average error in the ensemble $\langle \Delta X(t) \rangle$ and the black curve is the SD $\sqrt{\left\langle \left[ \Delta X\left( t\right) \right] ^{2}\right\rangle -\left\langle \Delta X\left( t\right) \right\rangle ^{2}}$

We assume that the accuracy of measuring the trajectory of a classical particle using Equation (13) is six orders of magnitude worse than the accuracy of measuring the trajectory of a quantum atomic cloud using Equation (6), which is achievable by atomic interferometry methods (Dubetskii et al., Reference Dubetskii, Kazantsev, Chebotayev and Yakovlev1984). Nevertheless, ideally, the accuracy of navigation in ${1}$ h does not exceed $150\,{\mu }$m and practically remains the same for different trajectories.

Following Jekeli (Reference Jekeli2001), in the future we will assume that the north-east-down (NED) local frame is installed on the platform. Likewise for GPS, we propose to install in the NED frame a sensor system to measure the distance to the particle, which we will call the local positioning system (LPS), see Figure 3. As in GPS, we will call these sensors satellites.

Figure 3. (Colour online) LPS – measurement of the particle's coordinates. From the distances to three red satellites, one determines two positions of the particle, true white and false red. Similarly, for green satellites, one gets true white and false green. The true solution is obviously the same for the red and green satellites

Our code can be used for any satellite location. However, in the simulation, we use only satellites located on the frame axes. When three satellites are located on the positive semi-axes of the system, the vectors connecting the particle and the satellites do not lie in the same plane, and the coordinates of the vector $\mathbf {x}$ can be obtained from the distances to the satellites. The equations for $\mathbf {x}$ have two solutions, one true, the other false. To overcome this ambiguity, one can use an additional system of satellites located, for example, symmetrically on negative semi-axes and choose as true the solution that is the same for both satellite systems. During the simulation, pseudo-random corrections were made to the exact distances using a normal random number generator (Box and Muller, Reference Box and Muller1958) with SD $\delta _{x}$.

At the launching time, we need to restore the values of the parameters listed in Equation (12). Errors in restoring these parameters degrade the navigation accuracy by several orders of magnitude. These parameters can be restored using conventional IMUs. In this case, we come to another kind of hybrid. Our code can be included in the code of operation of such a hybrid. In this paper, however, we will consider the possibilities of reconstructing $Z$ by launching particles with different velocities and simultaneously measuring their coordinates and velocities through time $T$.

The most significant factor affecting the accuracy of navigation is the platform rotation. In the case of atomic navigation, it was proposed to use a gimbal (Kasevich and Dubetsky, Reference Kasevich and Dubetsky2008) to reduce the effect of rotation. In this paper, we also assume the presence of a gimbal, which we define as reference systems that perform only noisy rotations at small angles, with zero mean value and SD $\delta _{g}$. In the future, we will determine the quality of the gimbals, SD $\delta _{g}$, to achieve the ultimate goal of navigation set up in Equation (8).

An elementary loop of the navigation is the restoration of the position, velocity, orientation, and rate of rotation of the platform at time $t_{2}=t_{1}+T$ from their values at the time of the particle launch $t_{1}$. Then the restored parameters at the time $t_{2}$ are used for navigation at time $t_{3}=t_{2}+T$ and so on. We will call this method first-order navigation. Due to the large Doppler broadening of the spectral lines, one could not use both the atomic interferometry (Dubetskii et al., Reference Dubetskii, Kazantsev, Chebotayev and Yakovlev1984) and the Ramsey fringes (Ramsey, Reference Ramsey1949) in this method. We propose installing velocimeters together with satellites in the NED frame. Measuring the velocity of a particle in the platform frame will restore the velocity of the platform $V(t_{2})$, while measuring the distance difference between particles launched at different velocities will restore the orientation of the platform $R( t_{2})$, and finally, measuring the velocity difference of these particles will restore the rotation rate of the platform $\mathbf {\Omega }(t_{2})$.

The severe requirements on the platform velocity recovery can be mitigated in the case of the second-order navigation, when one measures the particle's positions at three times $\{ t_{1},t_{2}=t_{1}+T,t_{3}=t_{1}+2T\}$ and employs the second-order difference by Equation (5b) to restore the platform position $\mathbf {X}(t_{3})$. The reason here is that even with an arbitrary trajectory of the platform, the difference by Equation (5b) does not depend on the initial velocity of the particle if the platform does not rotate and the Earth's gravitational field is uniform. The weak inhomogeneity of the field and the noisy, small rotation of the gimbal lead to such a weak dependence of $\mathbf {p}$ on the platform velocity that there is no need for velocimeters.

The requirements for the quality of the gimbal can also be relaxed if one launches simultaneously three particles with different velocities and restores the orientation of the gimbal from the differences in their trajectories. In the case of atom interferometry, an iterative method for this kind of restoration was developed by Kasevich and Dubetsky (Reference Kasevich and Dubetsky2008). Here, we generalise this method to the case of LPS navigation (LPS-N).

We offer here five varieties of LPS-N:

1. 1.1 first-order LPS-N, the quality of the gimbal is high enough to neglect its rotation;

2. 1.2 first-order LPS-N, the quality of the gimbal is sufficient to neglect its rotation, but one does not neglect the rate of rotation, and restores it by applying data from velocimeters;

3. 1.3 first-order LPS-N, one restores the orientation and rotation rate of the gimbal by applying data from satellites and velocimeters;

4. 2.1 second-order LPS-N, the quality of the gimbal is high enough to neglect its rotation;

5. 2.2 second-order LPS-N, one restores the orientation of the gimbal from the satellite data.

The paper is arranged as follows. The exact expression for the trajectory of a particle in the field of the rotating Earth, when the potential of this field is expanded to the first-order gravity-gradient terms, is obtained in Kasevich and Dubetsky (Reference Kasevich and Dubetsky2005). For the completeness of this study, we derive this expression in the next section. The technique of restoring the orientation, rotation rate, velocity, and then the position of the platform is described in Section 3 for LPS-N of the first-order and in Section 4 for LPS-N of the second-order. The results of the simulation and their discussion are carried out in Section 5. In Appendix C, we describe the model of the Earth's gravitational field accepted in our code.