Major advancements in quantum metrology over the past few decades have placed optical lattice clocks [1,2] and atomic interferometers [3,4] as enabling technologies for fundamental physics research. One successful maturation of quantum sensing has been the ever increasing precision of matter wave sensors capable of measuring inertial forces [5], rotations [6,7], and gravity gradients [8,9]. Atomic interferometers have also established constraints on fundamental constants [10,11], set certain dark energy bounds [12], and provided the most rigorous studies of the equivalence principle [13,14]. While atomic interferometers have proven to be excellent precision measurement devices in the laboratory, many forefront objectives such as simultaneous sensing in multiple dimensions [15,16] and demonstrations of practical field-deployable architectures [17,18] have remained an ongoing challenge.
Recent advances in atomic clocks suggest that a powerful path forward is one which makes use of optical lattices for control, and indeed recent experimental results have shown that an emergent class of atom interferometers using threedimensional optical lattices may offer an alternative solution [19]. These have been termed Bloch-band interferometers (BBIs) since they accumulate the inertial phase in the Bloch band eigenstates of an optical lattice, rather than in free space as in traditional atom optic devices. The use of stationary eigenstates of the system allows for phase accumulation over long interrogation times, a necessary condition for precision metrology applications. In addition, an optical lattice can impart significant impulse force, offering the potential for a robust platform for sensing in the face of dynamically harsh environments with inherent thermal and vibrational noise. Furthermore, this system offers the opportunity for dynamic reconfigurability of the sensing functionality without the need for any modifications to the physical device itself [20].
In this paper, we propose a basis for atom interferometry by exploring a universal gate set of operations that can be sequenced to build any inertial sensing device. We believe it is useful to point out the natural connection between this new design philosophy and the universal gate sets of quantum computing [21,22], in which a handful of elementary operations acting upon qubits can be applied sequentially to execute a quantum algorithm of arbitrary complexity. In the case of quantum logic gates, a finite number of qubits connect to the input and a unitary operator maps to the output by modifying the quantum amplitudes of each state according to the gate function. Here, we do the same thing, but instead of quantum logic gates, our gates are elementary input-output operations that perform unitary transformations on quantum matter waves between lattice eigenstates [23][24][25]. The concept of universality is established in a metrological context, meaning the ability to sense potential fields in three dimensions of arbitrary functional form, including derivatives. This framework unifies a wide variety of sensors including accelerometers, magnetometers, gravimeters, gyroscopes, and gradiometers.
While some of the gates we present are familiar from other contexts, such as beam splitters and mirrors, here we also present atom optic components such as "Split and Hold" and "Echo" that broaden the class of potential sensors. Each gate is associated with a control function that specifies a
(a) (b) (c) ConducƟon Band Valence Band
FIG. 1. Bloch-band interferometry (BBI). (a) Band energies as a function of quasimomentum for a lattice depth 10E r (dashed line), the exploded graph magnifying the small avoided crossing of the two conduction bands. (b) Example sequence of a BBI sensor circuit producing an accelerometer with a hold stage. The tiles are specific matter wave gates that will all be described in this paper. (c) The timing protocol of the control function that results, φ(t ), showing the lattice shaking during the gate operations, with the 2π radian range corresponding to moving the lattice by a single optical wavelength. The lattice is fixed in space during most of the sequence, with the atoms in either the conduction or valence band eigenstates.
timing protocol for the lattice position [26,27] and is designed using machine-learning and quantum optimal control algorithms. For each gate, we provide high-fidelity examples of these protocols and show their performance through numerical simulation. To demonstrate that they work in practice, we implement all the gates and confirm their anticipated function in our rubidium Bose-Einstein condensation experiment (BEC) using a one-dimensional optical lattice [28]. We do this by imaging directly the evolution of the BEC in situ, while the atoms are held in the optical lattice, and by measuring the momentum distribution after time-of-flight (TOF) expansion. Following presentation of the gate set, we explicitly show how the components can be sequenced in atom optic circuits to construct a variety of basic sensing devices and evaluate the resulting sensitivity for parameter estimation.
The basic framework of BBI is sketched out in Fig. 1. A one-dimensional optical lattice contains a periodic potential with at least one discrete translational symmetry V (x + jb) = V (x), where V (x) is the potential at position x, b is the periodicity, and j is an integer. A typical optical lattice is composed of pairs of monochromatic counterpropagating lasers of lattice wave vector k L = 2π/λ, defined by the optical wavelength λ. The resulting periodicity of the sinusoidal optical intensity pattern is b = λ/2. The eigenstates of such a periodic system are the Bloch states, which can be written as
where ϕ (n) (x + jλ/2) = ϕ (n) (x) has the same periodicity as the potential. The locus of eigenenergies of these states form bands labeled by the band index n, a non-negative integer, as well as the continuous quasimomentum q that takes possible values spanning the Brillouin zone, q ∈ [-hk L , hk L ]. In Fig. 1(a) we illustrate the energy dispersion of the lowest five Bloch bands for a lattice of depth 10E r , where E r = h2 k 2 L /2m and m is the atomic mass. The three lowest bands, i.e., |0 , |1 , and |2 , which we refer to as valence band states, have low curvature, a consequence of the fact that the atoms do not possess enough energy to propagate between lattice sites without tunneling. The conduction band states, |3 and |4 , display essentially the quadratic energy dispersion of free particles implying that in these states atoms are able to transit without tunneling and can cover large distances. Accessing near-free-particle states is important for matter wave sensing since the ability to split the atomic wave function and create a large spatial separation between wave packets is critical for achieving high sensitivity in many precision metrology applications.
To preview our gate-based sensor framework we show an example sensor protocol-namely an accelerometer with a hold stage. In the sequence shown in Figs. 1(b) and 1(c), an atom wave packet is split along two paths, held at its furthest extents, released, propagated back, and recombined to give quantum interference and parameter readout. The sequence is symbolically represented as an abstract sensor circuit, with each tile representing an elementary gate. Later we will analyze a universal set of gates, including the ones shown here, and for each demonstrate its functionality. As is apparent in this example, the gate transformations are generated by rapidly varying control functions of the lattice position that are produced by optimization algorithms [29]. The gate operations occupy only a small portion of the total sequence time. In between gates, the lattice position is held stationary and the atoms are in Bloch eigenstates: either in the conduction band states |3 and |4 where they propagate or in the valence band states |0 and |1 where they are held stationary.
A qubit is the quantum version of a classical bit that can be in one of two states and allows for the possibility of a quantum superposition. A qudit is simply the d-dimensional generalization. It may not be apparent a priori that lattice-based interferometry, which deals with matter wave interference, can be well described by a qudit representation. This comes about because the potential operator acts on a Bloch eigenstate without changing the quasimomentum. In other words, if one begins with a specific q, e.g., q = 0 for a Bose-Einstein condensate, only the discrete set of states with that q and all possible values of the non-negative band index n are dynamically coupled. This is true even if the Hamiltonian has a time-dependent lattice amplitude or phase. The way to break the quasimomentum invariance is to break the spatial periodicity of the potential function, which is assumed to be never the case for the gates presented in this work.
This means that the subspace in which the dynamical matter wave state resides is a family of states in a countably infinite manifold with fixed q and n = 0, 1, 2, . . .. The dimension is simplified further due to the fact that the manifold can often be truncated at some upper value of the band index, since higher band index corresponds to higher energy. For our purposes, with a lattice depth of 10E r , only the states presented in Fig. 1(a) are relevant and higher energy bands are not occupied. The system is therefore well described as a qudit with d = 5 and all our matter wave gates are well described as unitary rotations that operate in this low-dimensional subspace.
This concept of a universal gate set for quantum metrology is inspired by the architecture of quantum computation. In quantum computing, if a finite set of gates can carry out any unitary operation to an arbitrarily high fidelity, then this set is said to be a "universal gate set." This is significant since any quantum algorithm can be performed by a universal gate set. In quantum metrology, the concept of universality is to employ gates to construct input-output protocols that embed sensitivity to arbitrary signals. In both quantum computation and quantum metrology, one sequences together elementary unitary gates to build circuits that can be used for universal functionality.
We define metrological universality here as the ability to detect any signal through measurement of its associated potential function. Since any analytic potential can always be Taylor expanded at each point in space in terms of a series of derivative orders, this means that universality requires demonstrating sensitivity to any vector differential. Curls, divergences, and gradients are all vector differentials and so this notion unifies an entire class of sensing where an electromagnetic or gravitational field, or inertial effect, appears in a quantum Hamiltonian through a potential function. This includes vector accelerometry and gravimetry, multiaxis rotation sensing with gyroscopes, and tensor gradiometry, magnetometry, and electrometry, among others.
In the following sections, we demonstrate the universality of our gate set in our qudit subspace through example, that is, by the explicit construction of the important sensing circuits to measure accelerations, gravity gradients, and rotations. We calculate each protocol's response to an applied signal through the path integral formalism of matter-wave interferometry [30] and numerically show that these phases are detectable by our momentum observables.
In the experiments reported here, the sequence begins with the production of a rubidium-87 Bose-Einstein condensate (BEC) which is produced all optically via forced evaporation in a crossed-dipole trap (CDT). A detailed description of the experimental system can be found in Refs. [19,31]. For the results of this paper we have chosen to produce condensates of order 4 × 10 4 atoms at 10 nK. Although we can create condensates that are many times larger, we intentionally do not do so to simplify interaction effects. After forming the condensate in the CDT, the BEC is adiabatically loaded into a one-dimensional optical lattice aligned in the vertical direction. The loading is performed by linearly ramping up the lattice laser intensity in a total time ranging from 400 µs to 1 ms. The lattice beams are generated using a 30 W 1064 nm fiber laser. In the same apparatus, we can perform atom interferometry experiments with three-dimensional lattices, but for the gate protocols demonstrated here, which are all one dimensional, only two counterpropagating (i.e., not retroreflected) laser beams on a single axis are used.
Each lattice beam passes through its own independent acousto-optic modulator (AOM) that is utilized for active control of the intensity and phase of the light. The gate protocols are applied to the atoms by updating the radio frequency input sent to the lattice AOMs. We sample the theoretically calculated protocols at 50 ns resolution and experimentally create the signals by imprinting the sampled waveform onto a 80 MHz carrier, as needed for the AOM drive, using an arbitrary waveform generator (AWG). Lattice depths were calibrated using Kapitza-Dirac diffraction [32] or by observation of the highest fidelity of applied protocols.
We determine the momentum transformation that results from the applied gate protocol sequence through absorption imaging. This means that we expose the atoms after evolution to a resonant 780 nm laser beam directed orthogonal to the lattice direction and image the shadow of the cloud on a CMOS camera. The pixel values on the camera correspond to the integrated column density (optical density) through the quantum gas at each location. In this paper, we perform absorption imaging in two ways; both in situ imaging of atoms in the optical lattice which yields the position distribution probabilities, as well as imaging after time of flight (TOF) expansion of 10-15 ms, which reveals the momentum distribution probabilities following applied operations. Successful adiabatic loading into the lattice is confirmed by observing the diffracted momentum components following TOF and comparing with what is expected for the ground state |0 at the ascribed lattice depth. Since the lattice photons can only transfer quantized momentum, observed diffraction orders are easily discriminated after TOF and are separated by 2 hk along the vertical axis.
The one-dimensional optical lattice potential, V (x, t ), at each point in space x and time t is given by
where V 0 is the lattice depth (i.e., the potential difference between intensity minima and maxima) and φ(t ) is the control function. In the theoretical optimization process all gates were designed for a lattice depth of V 0 = 10E rec , that is, the band structure in Fig. 1(a). Interestingly, for any given gate, we have typically been able to find numerous control functions that look qualitatively very different but produce similar functionality when implemented in both numerical simulation and in experiment. We have used a variety of optimization methods to produce the control functions, including quantum optimal control (QPRONTO) [33,34] and machine learning (reinforcement learning) [31,35]. Illustrative examples of control functions are given in Fig. 2. These examples are not the only solutions and a different initialization of the neural network in reinforcement learning, or a different seed function in quantum optimal control, will usually lead to a different result. However, all the protocols shown produce gates that have in excess of 90% fidelity with respect to their anticipated function (often in excess of 99%). For the purposes of this paper, the exact method employed to produce high fidelity, short duration (on the order of 100s of microseconds) control functions is not essential. The effective design of control functions is a topic in its own right and will not be extensively explored here [36]. However, we point out that the optimization process to produce the gates can incorporate a variety of additional constraints that may respond to a specific intended application. A few examples of constraints are insensitivity to noise, insensitivity to nuisance parameters such as lattice depth error [37], broad dynamic range with respect to an inertial signal, and insensitivity to mean fields and other interaction effects. We now describe the elementary unitaries (gates) and observe their operation directly from the spatial evolution of the BEC in the lattice. Since absorption imaging is destructive, we run multiple experiments and take snapshots of the in situ density for an array of hold times following the application of the gate protocol. The momentum transformation is revealed via TOF by imaging both before and after applying the shaking waveform.
time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs) time (µs)
FIG. 2. Universal gate set for atom interferometry. For each protocol the symbol (gray) as well as its reverse (blue) is shown. The waveforms are some examples of control functions for lattice shaking protocols as produced by both reinforcement learning and by quantum optimal control methods, with the horizontal axis giving times in µs and 2π on the vertical axis giving a spatial shift of one lattice wavelength. Reverse waveforms are generated by playing the control backwards
The desired functionality of the beam splitter is a unitary operation that causes a wave packet to split and separate in a superposition of taking two alternate paths. For our situation, the input is the ground state of the lattice |0 and is macroscopically occupied by the BEC. The conduction band states |3 and |4 are attractive output beam splitter candidates since they are eigenstates comprised primarily of an equal superposition of ±4 hk L momentum components, each with 47.4% occupation, and have sufficient kinetic energy to exceed the maximum of the optical potential allowing atoms to move through the lattice. Our designated Bloch-band transformation for the beam splitter is therefore one of two options: with the first illustrated in Fig. 3. The two possibilities arise from simply permuting the two output channels, a common theme for most of the components we will show.
The high fidelity of the control function that was found for this transformation was confirmed by performing a numerical simulation. In the simulation, a quantum wave function was prepared in the Bloch ground state with a spatial extent of approximately 20 lattice sites. The time-dependent Schrödinger evolution was then solved in momentum space with the beam splitter protocol shaking the lattice position according to Eq. ( 2). The gate protocol lasted for slightly more than 150 µs and, after application of the gate, the lattice was held stationary and the evolution was tracked numerically for 7 ms more. The anticipated wave packet splitting of the condensate is clearly seen in the bottom portion of Fig. 3. We validated the application of the beam splitter protocol experimentally and the results of this study are shown in Fig. 4. In the false color absorption images, the red regions have high atomic density and blue regions have essentially no atoms. In this paper, all false color images of this type are experimental and are averages of 3-5 experimental shots. When imaged, the condensate is observed to split into two clouds that linearly separate with a measured velocity commensurate with ±4 hk L momenta. The in situ expansion within the lattice shows good agreement with theory and displays a separation of order 200 µm for a 7 ms hold. The momentum components that were observed by TOF before and after the gate protocol also agree well with the defining Bloch state transformation of a beam splitter. In this paper, the TOF images showing the results of the applied protocols in momentum space (originally vertical) have all been rotated 90 • (shown horizontal) to enable easier comparison to the predicted theory.
The symmetric 50/50 beam splitter can be generalized, since it is possible to design transformations that send unequal proportions of atoms along the two paths. The limiting case would be to use only one path, i.e., an asymmetric or 100/0 beam splitter. In terms of Bloch state transitions, the target state for the asymmetric beam splitter is
to send all the atoms along one path and
to send all the atoms along the other path. We derived a shaking function for this transformation and in Fig. 5 show the numerical simulation to confirm its functionality. In this simulation, the control function shakes the lattice for less than 200 µs and then the lattice is held stationary to illustrate the subsequent evolution. Even though there are two separate design problems corresponding to the left and right paths, only one protocol has to be learned, since changing the sign φ(t ) ⇒ -φ(t ) changes the direction.
As for the 50/50 beam splitter, we also validated the operation of the asymmetric beam splitter in experiment. The in situ evolution shown in Fig. 6 demonstrates that the atoms (b) (a) (c) FIG. 6. Asymmetric beam splitter experiment. (a), (b) As in Fig. 4 but applying the asymmetric beam splitter shaking function in experiment and validating its operation. In (c), the bar graph for the momentum probability for (|3 -|4 )/ √ 2 is in good agreement with that observed experimentally after the gate is applied.
are primarily transferred into a -4 hk L momentum component and propagate through the lattice in a single direction for 7 ms. Small amounts of occupation are observed in other momentum orders, which is in agreement with the numerical simulation of Fig. 5. The anticipated gate behavior is confirmed by the momentum probabilities after TOF.
Enclosing a space-time area is necessary to build an inertial sensor and this requires a mirror so that the two distinct wave packet components that have been separated can be brought back together and recombined to produce the interference signal. The defining transformation is actually embedded in the design rules for the asymmetric gate just described, since (|3 + |4 )/ √ 2 was the target state to travel in one direction and (|3 -|4 )/ √ 2 was the target state to travel in the other. Comparing these, and ignoring the global phase, it is evident that a mirror is well described as a device that satisfies one of two design options
related by permuting the output labels. We refer to this as target operator design since, for either option, two rules have to be simultaneously satisfied. This differentiates it from the previous two gates that were targeting a single state. Here we stay within a SU(2) subspace, i.e., the conduction band manifold {|3 , |4 }, so this transformation can also be recognized as a single qubit gate. In fact, in the language of quantum information science, these possible transformations are referred to as the Pauli Z-gate and the Pauli Y -gate, respectively. In Fig. 7, the numerical simulation is shown using the shaking function that was found for this design. To make its functionality transparent, we solve the Schrödinger evolution for a composite waveform that involves stitching together a sequence of gates. We begin with a BEC of finite extent in the ground state of the lattice, apply the beam splitter shaking function, evolve for 4 ms, apply the mirror protocol with the Bloch transformation described in this section, and then propagate for a further 4 ms. The reflection of the momentum components by the mirror is evident.
When we validated the operation in experiment (Fig. 8), the anticipated space-time diamond pattern is observed in the in situ image sequence, showing the wave packet components separating along two paths, being reflected, and coming back together, all while being confined to the optical lattice. We point out that this is almost a complete Mach-Zehnder interferometer that misses only the final recombination step in which the interference is formed. In the experimental TOF validation of the momentum transformation, we replaced the symmetric beam splitter with an asymmetric beam splitter so that it was clear the atoms were being reflected. Similar results were seen for atoms sent along the other path.
To make gradiometers that measure differential operators, we need another type of beam splitter that symmetrically splits wave packet components that are already propagating in the conduction band. Here the desired transformation can be defined by one of two alternate transformations,
that operate entirely within the SU(2) manifold of conduction band states. These are also single qubit gates familiar in quantum information science where they are known as Hadamard H gates.
We generated a shaking function for this target operator and demonstrated its application by numerical simulation, as shown in Fig. 9. In this simulation we solved the Schrödinger evolution for a sequence involving a beam splitter, propagation for 3 ms, conduction-band beam splitter, and additional propagation for a further 3 ms. The splitting of wave packets already propagating in the conduction band is evident.
The corresponding experimental demonstration for this sequence is given in Fig. 10 showing that the sequence functions as anticipated. The initial beam splitter divides the BEC into two parts and the atom clouds are split a second time by the application of the conduction-band beam splitter at 3 ms. At the end of the sequence, three peaks are visible in the in situ image, with a large central peak that is due to atoms that have taken either the upper or lower middle paths meeting in the center. When observing the experimental TOF image, as for the mirror, we employed an asymmetric beam splitter to start with so that the splitting in the conduction band was evident in the resulting momentum probabilities.
A that is created by leaving the optical lattice potential on during the entire interferometry sequence is the ability to transfer atoms back into the nonpropagating valence band. A split and hold gate is a target operator transformation that performs this by the following two alternate design options:
representing a conduction to valence band mapping. This offers the intriguing possibility of transforming atoms during an interferometry sequence into states where they can be held and accumulate inertial phase for extended periods without requiring an increase in the footprint of the device. Confining atoms in an optical lattice at controllable distances is comparable to the cavity-mediated interferometers [38,39], which offer enhanced interrogation time and increased immunity to high frequency phase noise.
Once confined in the valence states of the lattice after the application of the split and hold gate, the wave function is no longer in an eigenstate. Instead the amplitudes for |0 and |1 will evolve with a relative phase that accumulates at a rate dependent on the energy spacing of the lowest two levels. This can equivalently be viewed by the manifestation that the total probability density will oscillate back and forward in the potential wells. Since the optical lattice potential is sinusoidal and the bottom of the potential wells is approximately harmonic, this is often referred to as a vibrational oscillation in analogy with molecules.
The numerical simulation for the shaking function that was found for this Bloch transformation is shown in Fig. 11. The sequence used to demonstrate the functionality of the split and hold protocol is the application of a beam splitter, propagation for 4 ms, split and hold, and then an additional 4 ms of hold time to view any subsequent evolution. Before the split and hold gate, the atoms transport through the lattice in the conduction band and after the split and hold they are localized in the potential wells in the valence band states where they are frozen in place and no longer travel through the lattice.
Experimental results of the application of the split and hold gate is shown in Fig. 12. After a beam splitter and subsequent 4 ms propagation, the split and hold gate is applied and the atoms are held in place for a further 4 ms hold. Atoms that are not faithfully returned to the trapped bands are faintly discernible by their increasing separation at long times.
The echo gate allows one to confine atoms in lattice potential wells with the split and hold protocol and then release them in a manner that does not require a specific hold time. If the exact vibrational spacing of the lowest two valence states were perfectly known, and the hold time could be adjusted to a half-integer of the vibrational period, then the echo gate would not be needed. However, this may not be possible even in principle since spatial variations of the vibrational spacing due to lattice depth variation or other effects may lead to inhomogeneous dephasing. In this case, the necessary hold time designed into the protocol may not be single valued for different lattice sites.
To provide an effective solution to this problem, we draw inspiration from spin-echo protocols [40,41] that introduce a π pulse at the middle of the interferometer sequence to compensate for inhomogeneous dephasing. This design protocol is defined by the two possibilities
which is simply a mirror in the valence band which reflects the oscillation of the wave packets in the potential wells. In the terminology of quantum information science, the echo gate can be recognized as a single qubit Pauli Z-gate or Pauli Y -gate in the |0 and |1 subspace. The intended application of the echo gate is to apply it at the half-time of a hold phase which we will see later through its application in various sensing circuits. (See Fig. 13.)
We have solved for a shaking function for the echo transformation, validated it in numerical simulation, and demonstrated its function in experiment as shown in Fig. 14. In this sequence, an initial BEC was transformed from the ground state into the first vibrational state by the application of an echo gate and then brought back to the ground state by a second application.
We explicitly detail two classes of accelerometers, two classes of gravity gradiometers, and a matter wave gyroscope, each comprised of a sequence of elementary matter wave gates. For each sensing sequence, we present the time-ordered circuit and numerically simulate the space-time evolution of the atomic wave packet accounting for the full atom-lattice dynamics. The response of these protocols as a function of the parameter of interest is shown by the momentum-space interference pattern that results. In this interference pattern, the probability amplitudes for the discrete momentum orders can be considered to be the output channels of a multiport sensor from which the inertial phase is extracted.
For each sensor, we will provide an anticipated device sensitivity and its dependence on parameters, which can be calculated analytically from the path integral. For example, for a two-path atom interferometer, the relative inertial phase φ that is accumulated between the two arms is given by
written in terms of the action
where ± labels the classical paths. These paths are integrated over time t of the sequence, up to the terminal time τ . At each point in time, the action depends on the free Lagrangian of the matter-wave-lattice system L 0 and interaction potential that we wish to sense V I . To find analytical expressions for the phases, we integrate the classical paths as given by the stationary phase approximation [30]. We do this in the limit of long propagation times where the gates are effectively instantaneous.
From the numerical simulations for the complete sensor sequence, we calculate the resultant momentum probability fringes for each possible momentum diffraction order m, included as insets to the sensitivity scaling plots. The sensitivity scaling of each protocol is found by first calculating the classical Fisher information (CFI) [42,43], which characterizes the information gained about the value of a parameter β from a single measurement m. The CFI is defined as
where E β is the expectation value evaluated at β and P(m|β ) is the conditional probability of getting order m given β. This expression is for a single atom wave function; the sensitivity for an ensemble is given by
where N is the number of independent trials that can be extracted from a single absorption image. In this paper we will always assume N = 1000, a typical value measured in the current experiment [19]. Since the CFI is not constant across the entire scan, we plot the obtainable sensitivity as a shaded region, bounded by the maximum and minimum over the calculated scan range. The sensitivity improves as the square root of the shot number (number of experiments) due to statistical averaging.
The accelerometer sequence is a beam splitter, mirror, and recombiner with variable transport delays between operations, as shown in the matterwave circuit seen in Fig. 15(a). A simplified plan view of the 2D space-time area enclosed by the atoms during the applied sequence can be seen in Fig. 15(b). The accelerometer control function was created by stitching together the shaking sequences that were learned for the various components, where for the recombiner we used the reverse beam splitter and included delays of T = 3 ms between gates during which φ(t ) = 0. Starting from a BEC in the ground state of the lattice and numerically simulating this control function results in the wave function evolution shown in Fig. 15(c), where the familiar diamond pattern of the matter wave interferometer can clearly be seen.
For an acceleration a, the potential that we wish to sense is V I (x) = max, and the relative phase accumulated between the upper and lower arms of the accelerometer according to Eq. ( 11) is then given by
where p = 4 hk L . This formula has an intuitive interpretation as the product of the acceleration a with the space-time area of the diamond in units of h/m. It arises from the fact that the components of the wave packet in the two arms of the accelerometer travel through two different parts of the potential field which manifests as a phase difference in the matter wave amplitudes at recombination. We point out that this formula is fundamentally no different from a conventional light-pulse accelerometer. This design formula is a direct consequence of the matter wave physics [44] and de Broglie [45] relations that underpin all of the sensing devices. We scan the output as a function of a by adding V I (x) to the atom-lattice Hamiltonian and numerically simulating the evolution for the accelerometer control function. This allows us to calculate the expected sensitivity from the resulting classical Fisher information. We then plot the sensitivity as a function of repeated experiments (shot number), where we show the result in Fig. 15(d). The raw fringes for each momentum component used to calculate the CFI are also provided for reference and are given in the inset Fig. 15(e). The observed fringe period is in good agreement with Eq. ( 14).
An alternative approach to enhance the sensitivity to applied forces is a hybrid accelerometer that initially separates the atomic wave packet, but then holds the atoms at a fixed distance apart for extended interrogation times. This decouples the sensitivity from the physical dimensions of the sensor because the space-time area can be increased by simply holding the atoms for longer times instead of having them separate further apart. Since the gates are sequenced, this hold time can be programed in software. The matter wave circuit and its associated plan view are given in Figs. 16(a) and 16(b). The sequence is a beam splitter, split and hold, echo, reverse split and hold, and recombiner, with transport times occurring while atoms are in the conduction band and hold times while they are in the valence band.
The function of the echo pulse should now be transparent. The |1 state is of higher energy than the |0 state and so, during a hold, the relative phase of wave packet amplitudes in these states change in time. The echo pulse placed at the midpoint means that during the second part of the hold this phase accumulation is unwound. The result is that the combination of split and hold, echo, reverse split and hold replaces the mirror in the conventional geometry. 15(c), but for the accelerometer with hold circuit and using a hold time of 16 ms total. (d) Plot of the short-term sensitivity that results, the shaded region indicating the maximum and minimum CFI over the scan range. (e) Scan of the seven momentum state fringes obtained for a in the range -0.0003g to 0.0003g.
In the accelerometer with hold, V I (x) is the same as in the normal accelerometer and the relative phase accumulated between the upper and lower arms is
or equivalently θ AH = θ A + (2apT T H )/ h, where the additional term represents the increased space-time area that increases linearly with the hold time.
The numerical simulation of the accelerometer with hold protocol that stitches together the gate shaking sequences for a transport time of 3 ms and a hold time of 16 ms is shown in Fig. 16(c) and the resulting scan is shown in Figs. 16(d) and 16(e). This can be compared with the conventional accelerometer geometry in Figs. 15(d) and 15(e). From the variation of the momentum fringes, it is apparent that the device has increased sensitivity and the results agree well with Eq. (15).
Gradiometers are sensitive to the gradient of an inertial force. This measurement is typically accomplished by comparing accelerometers or gravimeters that are some distance apart. In traditional atom gradiometers, two spatially separated atomic samples are simultaneously split, mirrored, and recombined, using common light pulses, such that readout of the interference fringes provides information about the gravity gradient. In contrast, what we demonstrate here is a gradiometer that self-references, i.e., enabling the measurement of gradients from a single atomic source.
The necessary beam splitter, conduction-band beam splitter, mirror, reverse conduction-band beam splitter, and reverse beam splitter gates are sequenced in the sensing circuit of Fig. 17(a). Following a beam splitter and propagation, application of the conduction-band beam splitter splits the wave packet into four distinct paths that propagate further. Application of a mirror, reverse conduction-band beam splitter, and reverse beam splitter manipulates the wave packets to enclose two spatially separated space-time diamonds, and then recombines all the components. This can be seen in the plan view given in Fig. 17(b). The recombination creates an interference pattern that reflects the relative accumulated phase difference between the upper and lower interferometers, thus implementing a self-referenced gradient measurement.
In the gradiometer, the potential that we sense is V I (x) = mg x 2 , where g is found from the gradient of the acceleration field and x 2 corresponds to the local behavior of this first derivative. The relative phase accumulated between the upper and lower diamonds of the gradiometer and measured at the recombination stage is found by computing the action along four paths:
with u ≡ upper and l ≡ lower, the first index the result of the beam splitter, and the second index the result of the conduction-band beam splitter. Evaluating this gives
where T 1 is the propagation time between the beam splitter and conduction-band beam splitter and T 2 is the propagation time between the conduction-band beam splitter and the mirror. The greatest sensitivity for fixed total time is achieved for T 2 = 2T 1 indicating that it is better to have the diamonds overlap than to be separated in space as is typical in a conventional gravity gradiometer arrangement. Another important result is that the sensitivity increases rapidly in the gradiometer as the third power of the time.
The numerically simulated wave packet evolution using the gradiometer protocol that is created by stitching together the shaking waveforms represented by the matter wave circuit seen in Fig. 17(a) is given in Fig. 17(c) for transport times of 3 ms in all stages. The scaling sensitivity and momentum state fringes are given in Figs. 17(d) and 17(e). The variation of the fringes agrees well and can be predicted by Eq. ( 17).
Finally we point out that, although we are not providing simulations of the multidimensional versions of these sensors in this paper for the case of brevity, the second beam splitter (conduction-band beam splitter) can obviously operate in another dimension to the original beam splitter. This would enable measurements of the off-diagonal terms in the gradient tensor.
Performing gradiometry with a designed hold stage offers the same benefits as the accelerometer with hold by allowing us to extend the interrogation time without needing to increase the size of the device. As we did for the accelerometer, the implementation is straightforward and simply involves replacing the mirror with a split and hold component set. The total sequence for a gradiometer with hold is beam splitter, conduction-band beam splitter, split and hold, echo, reverse split and hold, reverse conduction-band beam splitter, and recombiner, all intermediated by transport and hold steps. This sequence along with the plan view is shown in Figs. 18(a) and 18(b).
In the gravity gradiometer with a hold, the potential that we sense is again the same as the normal gradiometer. The relative phase accumulated between the upper and lower diamonds of the gradiometer is
or θ GH = θ G + (8g T 1 T 2 T H p 2 )/ hm. Similar to the accelerometer with hold, the added term arises from increasing the space-time area by storing the atoms in place.
After combining the component shaking functions by stitching them together in the circuit sequence to form a gradiometer with hold control function, we solve numerically the Schrödinger equation and the result is given in Fig. 18(c). The simulation was performed with 3 ms transport time in all stages and 8 ms hold time. The associated sensitivity scan is in Figs. 18(d) and 18(e). Note the increased sensitivity with respect to the gradiometer without hold [Fig. 17(c)]. The observed momentum fringes agree well with Eq. ( 18).
Creating an atomic gyroscope sequence requires two components of a wave packet to traverse the perimeter of an area in opposite directions. This creates the same topology of atomic paths as seen in, for example, the Aharonov-Bohm effect, although the signal is generated here by rotation rather than the vector potential associated with a magnetic field. There are several simple configurations of gyroscopes that we can implement with our universal gate set. We will show one example here that is in the spirit of this paper by using only atom optic components. Our atomic gyroscope sequence is shown in Fig. 19(a). This is fundamentally a two-dimensional sensor that therefore requires two lines in the matter wave circuit, one for each dimension x and z. It involves sequenced gates with beam splitters, split and holds, echos, and asymmetric beam splitters, along with their inverses. The sequence within the brackets represents a single loop, which may be successively applied "N" times to increase the total enclosed area of the sensor [46]. The atoms are split in x, held in x while being transported in z, held in z while being released in x to exchange sides, again held in x while being transported back in z, and finally held in z while being released in x so the components can be recombined to produce interference. The numerical simulation of the action of all the control functions for the gates is given in Fig. 19(b) with a 3 ms half-side propagation time. The total enclosed area is 0.010 mm 2 per loop.
The 2D plan view is given in Fig. 19(c) showing the geometry, where in this case we show a two-dimensional space diagram and label the times it takes atoms to traverse each segment. A time-average "heat map" of the numerically simulated 2D wave packet evolution during the entire gyroscope sequence is given in Fig. 19(d).
In the gyroscope, the potential that we sense is due to a rotating, noninertial reference frame, and thus is two dimensional, depending on both x and z, V I (x, ẋ, z, ż; t ) = m (xżz ẋ), (19) where is the angular velocity of the rotation. Extending the path integral formulation to account for the second dimension gives a design equation for the relative phase accumulated between the clockwise and counterclockwise paths of the gyroscope. It is given by
where the times T 1 and T 2 are shown in Fig. 19(c). As was the case for the accelerometer, this equation has a simple interpretation of being the rotation parameter multiplied by twice the enclosed spatial area (twice due to atoms traveling in two opposite directions) in units of h/m. Physically this formula arises from the differential Doppler shifts of the matter wave as it traverses the loops, either in the same or opposite direction to the sense of rotation. The fringe frequency plotted as a function of the rotation angular velocity seen in Fig. 19(f) agrees well with the path integral formula in Eq. (20).
It is worth noting that there is nothing special about the square pattern shown here and other shapes or geometries are possible. For example, one can simply convey the atoms up and down in the z direction using a common translation technique in optical lattice experiments [47] and then apply the matter wave gates only in the x direction. This allows for more flexible patterns including circles that maximize the enclosed area for a given total time.
In this paper, we have presented the design and experimental realization of the matter wave universal gate set capable of constructing a myriad of inertial sensors. The faithful execution of every one of the gates was theoretically demonstrated via numerical simulation and then experimentally confirmed. Experimental validation was carried out by taking successive absorption images and tracking the evolution of the atomic density in the lattice and by observing their momentum state transformations from time of flight.
Using the universal gate set, we have demonstrated how to stitch together gates to produce accelerometers, gradiometers, and a gyroscope. Alongside each sensing circuit, we plotted the associated wave packet evolution from numerical simulation and calculated the expected sensitivity scaling in response to applied signals. Split and hold, conduction-band beam splitter, and echo gates offered design solutions that manipulate atoms in a way that is not directly accessible to conventional atom interferometer architectures.
As presented elsewhere, we have already experimentally demonstrated several of the sensors that we discussed here, including accelerometers in one dimension [31] and vector accelerometers in two dimensions [19]. Going forward, focus will be on realizing other types of sensor protocols as proposed in this paper. A principal goal will be to measure the sensitivity limits that can be achieved for precision sensing via this method of Bloch-band interferometry.
Even though this paper focused primarily on one dimension, we emphasize the straightforward extension of our gates to any or all axes of a 3D optical lattice. This natural extension to multiple dimensions would permit the device to be configured as a multiaxis gyroscope or even to measure the five independent components of the gradient tensor in a single instrument. Additionally, the flexible programing of Bloch-band gates could enable complex control over latticeconfined atoms for further studies of classical and quantum phase transitions, many-body physics, or the production of novel interferometer geometries.
We also emphasize that one can learn gates in which the lattice intensity is the control function or learn gates that incorporate an internal (spin) degree of freedom, opening up many possibilities to extend what we have presented. Finally, the use of entanglement generated by atom interactions would open up a host of new possibilities in leveraging quantum advantage for measurement gains.