A Decade of Advancement of Quantum Sensing and Metrology in India Using Cold Atoms and Ions

Abstract

Ultracold atoms and ions provide novel test-beds for quantum sensing and metrology. Technological advancement towards the construction of ultrastable and narrow linewidth lasers, optical frequency combs, and microwave electronics made it possible to realize atom interferometers with state of the art sensitivity and the most accurate clocks in the world based on long-lived quantum states of atoms and ions. Such systems have been utilized in applications starting from geodesy and navigation to the measurement and redefinition of fundamental constants. The field awaits a more enriched future in addition to the proposed applications of such systems in improving the Global Positioning System, secured communications, and exploring mineral and underground water resources. Here, we briefly report on the initiatives and current status of India in this research field in view of the global progress.

1 Introduction

In the last few decades, there has been a rapid progress in our understanding of fundamental quantum mechanics which has led to the development of novel ultra-sensitive measurement techniques based on advanced quantum sensors. The fragility of quantum states and their tendency to undergo decoherence when exposed to the environment can limit the performance in applications like quantum simulations and computation¹. On the other hand, such extreme sensitivity of the quantum devices to their environment is exploited to construct quantum sensors. The promising testbeds for quantum sensors range from atomic vapor cells², atomic clocks³, atom interferometers (AI)⁴ to nitrogen vacancy (NV) centers⁵, trapped ions⁶ and superconducting qubits⁷. Several of these sensors have moved beyond the proof of concept laboratory phase and are being employed in field applications. Advances in laser spectroscopy with atomic vapor cells have led to them being used for ultra-sensitive magnetic field sensing². Vapor cell magnetometers also find applications in non-invasive probing of brain activity⁸, testing battery cells⁹, etc. Using Rydberg states, these vapor cells are also being used to carry out low intensity RF field sensing with an ultra-wide bandwidth and dynamic range¹⁰. Efforts are now underway to make these atomic vapor cell-based sensors more compact for ease of commercial adaptation¹¹. Apart from neutral atoms, a trapped ion based platform has also been demonstrated as a quantum-enhanced displacement and AC electric field sensor¹². In a similar direction, spin-based quantum sensor technology based on NV centers is already mature enough to be commercially offered to perform high-resolution mapping of magnetic fields⁵. The spin qubit in an NV center is well protected from environmental decoherence¹³ and can be probed with broad and less coherent lasers compared to atomic sensors¹⁴. In addition, the low working distance for the detection and their capability of detecting magnetic signals at cellular level make them ideal candidate for cutting edge biomedical applications¹⁵ and single-molecule detection¹⁶.

Quantum sensors have been proposed to be useful on geological length scales. Compact atom interferometers are being used to map the sub-surface features such as old tunnels, mineral and oil reserves, monitoring groundwater aquifers¹⁷, performing navigation in GPS denied environments¹⁸, rotation sensing¹⁹, etc. An intense global effort is underway to miniaturize AIs for broader applicability²⁰. Another class of quantum sensors which operates in this regime are the state-of-the-art optical atomic clocks. These clocks are sensitive enough to detect gravitational redshifts at the centimeter scale. This makes them useful for applications like chronometric leveling²¹.

Quantum sensors are also being used to study fundamental physics like searching for dark matter²², measuring changes in fundamental constants²³, detection of gravitational waves²⁴ etc. These experiments have paved the way in the search for new physical theories beyond the standard model²⁵.

In this article, we will describe the progress of the development of such quantum sensors in India. In particular, we would like to highlight the work done at IISER Pune in the direction of developing AI-based gravimeters and rotations sensors, Strontium lattice-based optical atomic clock, and a calcium ion trap.

The article is organized as follows. A brief overview of the global status and efforts in the direction of atom interferometry and atomic clocks are described in Sects. 2 and 3 respectively. In Sect. 4, we describe the experiments carried out at the laboratory of IISER Pune towards realization of atom interferometry-based quantum sensing. In Sect. 5, we report the status of the construction of the Strontium (Sr) based atomic clock being built in our laboratory (Table 1).

Table 1: Performance of quantum sensors for the measurement of various physical quantities.

2 Atom Interferometers

Atom interferometers (AI) have shown promising performance over an optical interferometer (OI) in precision measurement. Theoretically, the AI’s sensitivity is about ten orders of magnitude larger than the OI²⁹ due to the mass of the interfering atoms compared to photons. However, given the much larger interferometric area of OI over an AI, practically about 3-4 orders of magnitude gain in sensitivity can be achieved. In the last three decades, the sensitivity of AIs has been improved by replacing hot atomic sources with laser-cooled and ultracold atomic samples^30,31,–32. At present, cold atom-based interferometers have been successfully realized in the demonstration of gravimetry^33,34,–35, magnetometry^28,36, rotation sensing^19,37, inertial navigation^38,39, measurement of fundamental constants^40,41 and tests of general relativity^42,43. AIso, based on cold atoms are limited by the coherence length and large thermal expansion of atomic clouds, which degrade the contrast of the interference signal leading to low visibility⁴⁴. On the contrary, a Bose-Einstien Condensate (BEC) has a longer spatial coherence and hence offers high brightness. The mean-field energy of BEC arising from high number density can create interaction-induced dephasing, which may degrade the interference contrast. Such problems can be overcome by giving sufficient time of flight before the AI sequence is applied⁴⁵. BEC-based AIs have been successfully employed both in an atom chip, and hybrid trap^46,47,–48 and have shown accuracies below sub$-\mu$Gal (1Gal = $10^{-2}\ \text {m/s}^2$)⁴⁶. BEC-based AI needs more complex electronics and optics compared to cold atoms, and continuous worldwide effort is going on to reduce the duty cycle and the Size, Weight and Power (SWAP) of the sensor head for transportability and deployment in space. In the last few years, the AI community is slowly shifting from cold atoms to BEC because it has several advantages as shown in various experiments^46,47,–48. In section. 4.2 we report on our demonstration of a BEC-based gravimeter with $^{87}\text {Rb}$ in conjugation with the Bragg diffraction technique.

The development of the Inertial Navigation System (INS) depends on the accuracy of gyroscopes and accelerometers which predicts the position of a moving object precisely. This system has huge benefits in the fields of defense and transportation, and thus it promotes the advancement of better inertial senors. The implementation of cold AI gyroscope is used to demonstrate the navigation system without GPS signals for an underwater vehicle⁴⁹. In Sect. 4.3 we report on our efforts to demonstrate a BEC-based rotation sensor.

Figure 1:

Principle of AI: a Diagram of laser cooling and trapping of atoms in 3 dimensions, b Diagram of BEC in a crossed dipole trap and AI of BEC after it is released from the dipole trap.

Figure 2:

Schematic of AI: a Schematic diagram of an atom interferometer, b Pulse sequence of atom interferometer and c Relaunch mechanism using Bloch oscillation and double Bragg mechanism to increase the sensitivity. Figure adapted from Ref⁴⁷.

2.1 Global Status of Compact Quantum Gravimeter

Quantum gravimeters are the first of their kind to be commercially available. The first quantum gravimeter was developed by Achim Peters et al.³⁴ where they had shown a sensitivity of around $3\times 10^{-9}$ $\ {\text{ m }s^{-2}}/\sqrt{{\text{ H }z}}$. Figure 1 shows a schematic diagram of ultra-cold atom-based gravimeter. Later in the 2000s and afterwards, many groups worldwide such as SYRTE⁵⁰, UK-HUB⁵¹, AOSense⁵², Muquans⁵³, and Msquared Lasers⁵⁴ developed a few compact and commercial designs for quantum gravimeters. The group of Prof. E. M. Rasel worked towards minimizing the state-of-art design by performing the gravity measurements on an atom chip based on ultra-cold atoms⁴⁷. Figure 2 is the schematic diagram for the atom chip experiment.

Few application-based quantum gravimeters are deployed in a ship for marine mapping⁵⁵ and in aircraft for airborne gravity mapping⁵⁶. The recent work carried out by the group of Prof. Kai Bongs showed the use of a transportable quantum gravimeter (Fig. 3) to detect aquifers and underground tunnels, map water table, and many more, which shows the real potential of such quantum sensors in field applications¹⁷.

Figure 3:

Gravity gradiometer: a Gravity gradiometer built by Kai Bong’s group, b Gravity cartography from Ref. 17.

2.2 Emerging Techniques to Enhance the Sensitivity in Atom Interferometer

In the last decade, there have been huge efforts to develop new techniques to enhance the sensitivity of cold atoms as well as ultracold atom-based interferometers. Cutting-edge techniques such as large momentum transfer, Bloch oscillations, double Bragg diffraction, and momentum entangled state preparation for the interferometers are a few of them. There have been efforts to improve precision, accuracy, stability, and in decreasing the size of AIs. Here we discuss briefly how entanglement and large momentum transfer play crucial roles in boosting the sensitivity of AIs.

Figure 4:

Spin squeezing in an atom interferometer: a Schematic diagram of an atom interferometer, b and c Coherent Spin State evolution of non-entangled states and entangled states, d and e The output signal for phase measurement above and below SQL. Figure adapted from Ref⁵⁷.

2.2.1 Quantum Entanglement in Atom Interferometry

The sensitivity of an atom interferometer is limited by the standard quantum limit (SQL). Quantum entanglement of atoms in an interferometer provides a promising tool to enhance the sensitivity beyond SQL through relative phase measurements. The early experiments which demonstrated the spin squeezing and momentum entanglement for metrological measurements are given in^58,59,–60.

In an atom interferometer the relative phase difference between two momentum states is determined through some observable $\hat{S}$ related to $\phi$ as $\Delta \phi =\frac{\sqrt{Var(\hat{S})}}{\mid \partial \hat{S}/\partial \phi \mid } = \frac{\zeta }{\sqrt{N}}$, where $\zeta$ quantifies the role of quantum entanglement through enhancement of sensitivity. Figure 4 shows the diagram of a quantum enhanced signal in atom interferometry. In the case of Quantum Gravimeter, where atoms fall under gravity in the standard Mach-Zehnder interferometer configuration, one can relate the acceleration due to gravity with the phase of an interferometer as $\phi ={{\varvec{k}}}\cdot {{\varvec{g}}} T^2$. For a quantum-enhanced signal through quantum entanglement, the sensitivity in acceleration due to gravity can be seen as $\Delta g = \frac{\zeta }{\sqrt{N}k_{\mid \mid }T^2}$ where N is the number of atoms, $k_{\mid \mid }$ is the component of k parallel to g. Similarly for a rotation sensor where the phase shift is given as $\phi =2m\varvec{\Omega } \cdot {{\varvec{A}}}/\hbar$, the sensitivity scales as $\Delta \Omega = \frac{\zeta \hbar }{\sqrt{N}2mA_{\mid \mid }}$, where $A_{\mid \mid }$ is the component of A parallel to $\varvec{\Omega }$. Recent developments such as the work reported in⁶¹ show through numerical simulation of spin squeezed ultracold atoms that the sensitivity gain can reach upto 14 dB beyond the SQL limit. The experimental work on the realization of momentum entanglement in atom Interferometry is shown in⁶². The article reports on the demonstration of spin squeezing of – 3.1 dB in ultracold atoms. Thus, the coming decades can see the quantum-enhanced atomic inertial sensors in a more matured form.

Figure 5:

Multi-photon momentum transfer of BEC done at Atomic Physics and Quantum Optics Lab (APQOL) in IISER Pune.

2.2.2 Large Momentum Transfer

Another way to enhance the sensitivity is to increase the interferometric area through multi-photon excitation. Methods like multiple two-photon Raman or Bragg transitions, double Raman or Bragg diffraction, Bloch oscillations or a combination of Bragg and Bloch oscillations^{63,64,65,66,–67} allow improvements in the sensitivity by increasing the effective separation between the two interfering atomic samples. The disadvantage of this technique, especially with cold atoms, is the addition of extra phase noise due to the multi-port nature induced by the effective multi-photon process. Figure 5 shows multi-photon Bragg diffraction of Bose-Einstein condensate realized in our experiment.

2.3 Global Status of Atomic Gyroscope

From the first atom interferometer in 1991⁶⁸ to date the atomic gyroscope has improved a lot. Early 2000s has seen the development of the first atom gyroscope by Gustavon et al.⁶⁹ who measured the Earth’s rotation rate and demonstrated the short-term sensitivity of $6\times 10^{-10} \ {\text{(rad/s) }}/\sqrt{{\text{ H }z}}$ for 1 s interrogation time (Fig. 6).

Figure 6:

Rotation sensing based on atom interferometry: a Schematic design of two MOTs launched in vertical direction adapted from SYRTE⁷⁰, b Interferometric pulse scheme for three components of acceleration and rotation, c Acceleration and rotation signals extracted from half sum and half difference of the phase shifts of two interferometers⁷¹.

Figure 7:

Interleaved atom interferometry: a Zero dead-time cold-atom gyroscope operating in a four pulse method. The last $\pi /2$ pulse is shared among the previous and the current atomic cloud making it in a continuous mode¹⁹, b Rotation rate measurements data for about 32.5 h using the mentioned setup⁷².

Figure 8:

Differential atom interferometry: a The top panel displays how a BEC is split into two wave-packets using double Bragg diffraction (green arrows)⁷³. Two independent Mach-Zehnder interferometers are realized in a vertical direction parallel to gravity using three successive Bragg pulses (red arrow) separated by an interferometric time T. The interferometer is sensitive to both acceleration and rotation. The bottom figure shows the absorption images of the BEC for different phases in the experiment. Twin-lattice atom interferometer: b This is formed using co-propagating Bragg beams of two different frequencies with linear orthogonal polarization. Quarter wave plate in front of the retro-reflected mirror changes the polarization to generate two different co-moving optical lattices in opposite directions. Figures adapted from Refs. a⁷³ and b⁷⁴.

They were able to segregate the phase shift observed due to the rotation and the acceleration by making two counter-propagating atomic beam geometry. Further improvements in long-term sensitivity were introduced by Durfee et al.⁷⁵ eliminating the phase shifts due to non-inertial effects such as magnetic field and AC stark shift. They reversed the propagation vector $k_{eff}$ periodically, which introduced the opposite sign in the inertial phase while maintaining the sign of the non-inertial phase. This technique helped in removing the systematic effects and long-term drift allowing the system to increase the sensitivity to $3\times 10^{-10}\ {\text { (rad/s)}}/\sqrt{\text {Hz}}$ for an interrogation time of 5 h.

These initial atomic gyroscopes were in space domain where Raman lasers were kept running continuously with a velocity selection of atomic beam source. Space-domain atomic interferometers were later replaced by Time-domain atomic interferometers where laser-cooled atoms replaced atomic beam source. In an experiment carried out at SYRTE⁷¹, two cold atomic clouds were used, which were launched in curved parabolic trajectories in a counter-propagating directions, and three pulsed Raman beams were used instead of continuously running beams in three orthogonal directions. This experiment done in SYRTE reported short-term acceleration and rotation sensitivity of $4.7\times 10^{-6} \ \text {m/s}^2/\sqrt{\text {Hz}}$ and $2.2\times 10^{-6}\ {\text {(rad/s)}}/\sqrt{\text {Hz}}$. In a similar experiment reported by A. Gauguet et al.⁷⁰, the long term stability after an integration time of 1000 s was shown to be limited by the quantum projection noise in the system, $1\times 10^{-8}\ {\text {(rad/s)}}/\sqrt{\text {Hz}}$.

The latest generation of atomic gyroscope uses a four pulse method, and a cold atomic cloud is launched vertically upwards. Figure 7 from Dutta et al.¹⁹ displays the schematic of a zero dead-time gyroscope. Such continuous operation has helped in the improvement of short-term sensitivity to $100\times 10^{-9}\ \text { (rad/s)}/\sqrt{\text {Hz}}$. Implementation of interleaved atom interferometry where three atomic clouds are interrogated simultaneously has enhanced the sensitivity to $3\times 10^{-10}\ \text { (rad/s)}/\sqrt{\text {Hz}}$ with a sampling rate of 3.75 Hz and an interrogation time of 801 ms in Savoie et al.⁷² which is at par with the fiber-optics gyroscope.

Along with these, a few compact designs based on atom chips by Gersemann et al.⁷³ and Gebbe et al.⁷⁴ have demonstrated initial rotation sensing with double Bragg diffraction (DBD). Figure 8 summarizes the results from these two different experiments.

3 Atomic Clock

At the dawn of the civilization, humans used the periodic movement of celestial objects in the sky for timekeeping. This was primarily done for keeping track of the seasons for farming and carrying out important ceremonies. As human civilization progressed, these celestial clocks proved to be insufficient for applications that required higher operating frequency than days. This requirement led to the pursuit of development of better clocks such as flow-based clocks, pendulum clocks, quartz clocks, and atomic clocks. Today’s state-of-the-art atomic clocks operate at a frequency many orders of magnitude higher than even a quartz clock and are stable enough to lose only a few seconds over millions of years³. The development of such highly stable and accurate clocks has been behind many key technologies which require synchronization such as GPS-based navigation⁷⁶, high-speed data networks, and high-frequency trading⁷⁷.

Figure 9:

The clockwork for optical atomic clocks: The figure shows the major components required for the realization of an ultra-stable optical frequency standard using neutral atoms or ions. The laser (oscillator) to be stabilized is referenced to an ultra-stable high finesse cavity with the help of Pound–Drever–Hall (PDH) locking. This provides a short-term stable reference to the laser on the order of $\sim$100 s of seconds. The long-term frequency reference is provided by interrogating a sub-Hertz atomic transition. The frequency comb is then used to measure the frequency offset of the laser with one of the spectral lines of the comb. These clocks have phenomenal stability which can be specified by the fractional frequency deviations $\Delta f/ f \sim 10 ^ {-18}$ (after a few thousand seconds of averaging)⁷⁸.

Clocks based on the principles of atomic physics are the current state-of-the-art as their oscillator frequencies are based on narrow linewidth transitions between the two states of an atom or a molecule. This makes them ideal frequency references as the clocks do not undergo aging which is typical in mechanical clocks. The frequency of such an oscillator is also universal in nature which is highly favorable for them to be used as a frequency standard. It is due to these attractive features that the current primary frequency standard is based on the hyperfine transition in Cs-133 and is used to define the SI second⁷⁹.

It was known that the atomic clocks would perform better if the reference transition was in the optical regime instead of the microwave regime as they would possess 10$^4$ times higher operating frequency. Even though research was being pursued towards building such clocks, moving into the optical regime had a practical difficulty. It was very hard to measure the optical frequency and also to compare two such clocks as their operating frequency would be several 100’s of Tera-Hertz, which is far beyond the bandwidth of conventional electronics ($\sim 100$ GHz). There were a few devices e.g. “frequency chains” which achieved this but were immensely complex in design⁸⁰. The breakthrough in optical clocks came with the discovery of octave-spanning optical frequency combs⁸¹. The frequency spectrum of these lasers was a comb of spectral lines in frequency space with a separation (several 100 MHz) fixed by the repetition rate of the femtosecond laser. Thus two lasers operating at different wavelengths could beat against each other “virtually” using the frequency comb as an intermediate step⁸². The optical frequency comb now allows a very convenient solution for frequency measurement and comparison of optical clocks and thus completes the optical clock package as described in Fig. 9. Optical atomic clocks have been demonstrated to be several times more stable than microwave ones and a few of them form the secondary representations of the SI second⁸³. Research on long-distance comparison of such clocks is being carried out using optical fiber networks⁸⁴, satellite transmission⁸⁵ and free space propagation⁸⁶. There are also versions of these clocks that are transportable, which is essential to use them for measuring the geodetic height of arbitrary places⁸⁷.

There are primarily two types of optical atomic clocks being pursued globally viz. (a) the neutral atom optical lattice clock and (b) trapped-ion clocks.

3.1 Optical Lattice Clock

The first atomic clocks, based on a beam of neutral atoms utilized the technique of Ramsey-Borde interferometry (RBI) where the transition could be resolved more precisely by encoding the internal state onto the momentum state⁸⁸. Atomic beam clocks primarily suffered from limitations related to the high Doppler shift associated with the technique and large size of the apparatus. On the other hand, technology related to cooling and confinement of neutral atoms and ions was developing rapidly. Thus the next generation optical atomic clocks have started operating by loading laser-cooled atoms into a magic optical lattice to avoid Stark shift⁷⁸. These atoms are then prepared in the lowest motional state of the optical lattice by side-band cooling. The low temperatures thus achieved, leads to a drastic reduction in the Doppler shift and no longer dominates the uncertainty budget of the clocks. Such optical lattice clocks are now one of the best available clocks routinely performing at the $10^{-18}$ level of instability⁸⁹ (Fig. 10).

Figure 10:

Towards trapped Strontium optical lattice clock: Image of a vacuum chamber containing a cloud of laser-cooled Strontium atoms at IISER-Pune.

Such an optical lattice clock is being constructed at IISER-Pune using Strontium atoms. Further information about this is given in Sect. 5.

3.2 Trapped Ion Clock

In a trapped ion clock, typically a single ion or two ions are confined in a Paul-type ion trap. Ion confinement in a Paul trap is achieved with the help of a combination of static and oscillating electric fields⁹⁰. Several key advances in ion trap quantum computation technology have worked in the favor of the development of this system as an atomic clock⁹¹. Typically a single ion is trapped for interrogation. Relative to an optical lattice clock, the trapped ion clock benefits from low dead time as the same ion can be trapped and interrogated for days due to the high confinement potential. However, ion clocks have a relatively low signal-to-noise ratio (SNR), as in this case the clock spectroscopy signal is obtained from a single ion as opposed to several thousand atoms in lattice clocks. This leads to the optical atomic clocks being better in terms of accuracy⁹⁰ as it can reduce the quantum projection noise limit by loading $\sim 10^{4}$ atoms for clock spectroscopy. Despite this, the trapped ion clocks are still one of the most stable clocks currently available. The physics package of the trapped ion clock systems is also relatively easier to miniaturize and efforts are underway for delivery and collection of light from the ions via photonics fabricated in the trap structure itself⁹². Several promising trapped ion-based clocks such as highly charged ion clocks⁹³ and nuclear clocks⁹⁴ are being pursued and they promise further improvement in the performance of such clocks.

Figure 11:

Towards trapped ion based optical clock: a EMCCD image of the fluorescence at 397 nm from $\sim$1000 $^{40}$Ca$^+$ ions trapped in a PCB-based surface ion trap at IISc Bangalore shown in (b). The energy level diagram of the calcium ion is shown in (c). Transitions at 397 nm, 866 nm and 729 nm constitute the cooling, repumping and clock transition, respectively. The clock transition has a theoretical linewidth of $\sim$ 136 mHz.

Efforts are underway to construct such a clock at IISER-Pune. The preliminary steps of constructing and testing an ion trap were done in collaboration with IISc Bangalore. A printed circuit board (PCB) trap was made for trapping calcium ions at IISER-Pune. This design was inspired by the ‘Bastille’ design of Innsbruck and used vacuum-compatible RF laminate (Rogers 4350B). The trap was then tested at IISc Bangalore and fluorescence of the trapped ions at 397 nm transition was observed through an EMCCD camera as shown in Fig. 11. The trap worked at an RF frequency of $\sim$11 MHz, a peak-to-peak RF voltage of $\sim$1000 V, and an end-cap DC voltage of $\sim$100 V. The trap lifetime was suspected to be limited by surface dielectric charging and RF noise pick-up on DC trapping electrodes. An optical atomic clock with Ytterbium ions is also bieng constructed at Inter-university Center for Astronomy and Astrophysics (IUCAA), Pune - India.

4 Towards Realization of a Gravity and Rotation Sensor at IISER Pune

In this section, we discuss briefly the important experiments that are carried out at the laboratory of IISER Pune, India, towards the construction of atom interferometers and describe the use of such quantum sensors for the precision measurement of absolute gravity and for rotation sensing based on atom interferometry.

4.1 Towards Multiport Atom Interferometer: Diffraction of an Atom Laser in Raman–Nath Regime

Figure 12:

Quasi-continuous atom laser: Left panel: Atomic flux for three different outcoupling rates: a $1.39\times 10^6$ atoms/s, b $2.91\times 10^5$ atoms/s, c $1.71\times 10^5$ atoms/s. Right panel: Absorption images of the atom laser corresponding to the flux rates displayed in the left panel. The images are taken at a 40 ms, b 200 ms and c 300 ms after the beginning of the outcoupling process. Color scale represents optical density. Figure is taken from Ref.⁹⁵.

The initial efforts at the laboratory of IISER Pune were focused on the demonstration of the diffraction of an atom laser in the Raman Nath regime. The experimental setup has been described in⁹⁵. We produce a $^{87}$Rb BEC with 7$\times$10$^4$ atoms in a hybrid trap (a crossed optical dipole trap in conjugation with a quadrupolar magnetic trap). A well-collimated flow of coherent ultracold atoms is outcoupled from the trap by the ‘spilling’ method. Flux of such an atom laser can be controlled by manipulating the rate of change of optical trap depth. Figure 12 displays different flux rates of outcoupling along with the absorption images of the atom laser at different times after outcoupling.

Figure 13:

Diffraction of the atom laser: The upper panel displays a time of flight absorption image of the diffraction of the atom laser. The red curve drawn beside is a guide to the eye for the representation of the position of the diffracting lattice. The lower panel shows the relative strength of the diffraction of atoms in different momenta (up to $\pm 18 \ \hbar$k). Dots represent experimental data whereas the solid line is derived using numerical simulation. Figure taken from Ref.⁹⁵.

A 1D optical lattice detuned by 6.8 GHz from the trapping state is placed 3 mm below the trap center. The optical lattice has a beam waist of $\sim$ 700 $\mu$m and it is pulsed using the help of an acousto-optic modulator (AOM). When the atom laser interacts with the lattice, it shows the diffraction of atoms in different momentum states. The part of an atom laser that interacts with the lattice, encounters a spatial modulation of Rabi frequency due to the large Gaussian profile of the lattice laser in the vertical direction. One can demonstrate excitation to different momentum states of such an atom laser by controlling the pulse duration of the lattice laser. Such a diffraction pattern is shown in Fig. 13.

4.2 An Atomic Gravimeter

We discuss in the following sections the efforts going on at IISER Pune, towards building a gravimeter based on Bragg interferometry.

4.2.1 Basic Concept of Atom Interferometer-Based Gravimetry

The basic operation of BEC-based AI using Bragg diffraction to different momentum orders is described here very briefly. A complete theory about Bragg transitions can be found in references^96,97. In a Bragg transition, two momentum states $p_0$ and $p_0 +2n\hbar k$ (where $p_0$ is the initial momentum of BEC, k is the wavenumber of light and n is the order of Bragg transition) are coupled through a two-photon process⁹⁸. Our AI works in $(\pi /2-\pi -\pi /2)$ configuration similar to an Optical Mach-Zehnder Interferometer (OMZI). For atoms, a far-off resonant optical lattice works as a beam splitter and a mirror. One can consider a $\pi /2$ pulse as a beam splitter where the population of atoms is equally separated in two momentum states and a $\pi$ pulse is used as a mirror that transfers the atoms completely from one momentum state to another⁹⁹. To start the interferometric sequence, we turn off the trap to release the BEC under the gravitational field. After a time of flight of a few milliseconds, a $\pi /2$ pulse splits the atomic wave packet with a certain momentum in the vertical direction, which is then reflected by a $\pi$ pulse after a time interval (T) called interferometric time. In the end, after the same time interval, the last $\pi /2$ pulse is applied to recombine the population of both momentum states¹⁰⁰. A brief experimental procedure is shown in Fig. 14 based on references^47,48. To obtain an AI signal, the following methods are used: 

Case 1: If the relative acceleration between the optical lattice and atoms is uniform, then the path followed by atoms during interferometry will be identical, and the phase contribution will be zero. Thus the overall phase shift will be proportional to the individual phases obtained during light-atom interaction⁴⁸ along the gravitational direction which can be written as:

$$\begin{aligned} \Phi =n(\phi _1-2\phi _2+\phi _3)=2n\vec {k}.\vec{g}T^2 \end{aligned}$$

(1)

$\phi _1,\phi _2$ and $\phi _3$ are optical phases when atoms interact with Bragg pulses and n is the Bragg order. If we scan the phase of the last $\pi /2$ pulse , we will get an oscillation in the population of both momentum orders, and the signal will be proportional to:

$$\begin{aligned} P=N(1+C\cos (\Phi ))/2 \end{aligned}$$

(2)

where, N is the population of atoms and C is the contrast of signal.

Case 2: This method is used for measuring the acceleration due to gravity. The basic concept of measuring acceleration due to gravity is to balance the phase shift introduced during the AI sequence owing to different sources of accelerations. To nullify the Doppler shift of light in the frame of freely falling atoms, we accelerate the optical lattice at the frequency difference between the lattice beams, keeping the condition of resonance fulfilled for Bragg transition. By scanning the lattice acceleration around the value of local acceleration due to gravity, one can get an overall phase shift $\Phi =n(2{\vec{k}}.{\vec{g}}T^2-2\pi \alpha T^2)$ where $\alpha$ is the sweep rate (frequency chirp) which decides the acceleration of lattice. By scanning the parameter $\alpha$ around the value of gravitational acceleration, we obtain interference fringes. For an exact calculation of g, there will be an $\alpha$ (say, $\alpha _{0}$) which will balance the effect of gravity, and an overall phase shift $\Phi$ will be set to zero implying $\alpha =({\vec{k}}.{\vec{g}}/\pi )$. To get the value of $\alpha$, one has to take the interferometric signal for at least three different interferogram times (T), and all of those interferometric signals have a common minimum at $\alpha _{0}$.

Figure 14:

Schematic of the gravimeter: The figure shows the diagramatic representation of the BEC-based atom interferometer and its space-time trajectories. The blue dots represent BEC with ballistic expansion after its release from the trap. The experiment runs in three steps. (i) The conversion of mean field energy into kinetic energy of BEC during the time of flight $T_{tof}$, (ii) The MZI with different interferometric time (T) ranging from 1 ms to 100s of ms, (iii) The final pulse for the detection of BEC in different momentum states after allowing the wave packets to get resolved for $\tau _{sep}$ ms.

4.2.2 Experimental Details

The experimental setup is almost similar to the system described in Ref.¹⁰¹. We create a BEC of $^{87}$Rb consisting of $5\times 10^4$ atoms in every 15 seconds in a crossed optical dipole trap. The temperature of the residual thermal component of the BEC is measured to be less than 100 nK. After turning off the dipole trap, we give a 2 ms time of flight to reduce the mean-field effect. The laser used for realizing the optical lattice is locked to the $5^{2}S_{1/2}, |F = 2\rangle \longrightarrow 5^{2}P_{3/2},|F'=2\rangle$ D2 transition of $^{87}$Rb. Since the BEC is prepared in the $5^{2}S_{1/2}, |F =1, m_{F} = -1 \rangle$ state, this laser is 6.8 GHz red-detuned from the closest accessible transition of an atom. Two counter-propagating laser beams constitute the optical lattice, and these beams are obtained from the first-order diffraction of two separate acousto-optic modulators (AOM), which are driven by phase-locked arbitrary function generators (AFG)¹⁰². AFGs provide good control over absolute frequency, frequency sweep rate, and the lattices’ phase. To drive the Bragg transition in $^{87}$Rb, the frequency difference in lattice beam should be $f=2n\omega _{R}$, where n is the order of Bragg transition and $\omega _{R}$ is two-photon recoil frequency. But when atoms are freely falling under gravity, Bragg resonance condition gets modified due to the time-dependent Doppler shift $\delta _{d}(t)= 2\pi \alpha _{0}t$, where $\alpha _{0} =(\frac{1}{\pi })({\vec{k}}.{\vec{g}})$ is named as frequency chirp. Modified Bragg transition in the laboratory frame is $f = 2n\omega _{R} + 2({\vec{k}}.{\vec{g}})t$⁴⁸. To maintain the resonant Bragg condition, we apply a 25.078 MHz/s sweep rate (determined by the approximate theoretical value of g in Pune, India) in one of the lattice beams while switching off the optical dipole trap. Square pulses with an on-time of 80 $\mu$s and 160 $\mu s$ have been used as $\pi /2$ and $\pi$ pulses respectively to drive the first-order Bragg transition. The time sequence is shown in Fig. 14. We consider case-1 for the first AI signal, and here we scan the relative phase of the last $\pi /2$ Bragg pulse from $(-2\pi$ to $2\pi )$. Figure 15 shows the population oscillation of the $p = 2\hbar k$ as a function of phase when the interferogram time (T) is 1 ms. A clear interferometric oscillation can be seen in Fig. 15 with a contrast of 18$\%$.

Figure 15:

Population oscillation of first order $|p=2\hbar k\rangle$ momentum state with the relative phase of last $\pi /2$ Bragg pulse for interferometer time T = 1 ms.

After obtaining the AI signal, we scan the acceleration of the optical lattice by changing the sweep rate during the interferometric pulses $(\pi /2-\pi -\pi /2)$. The data is taken for four different interferogram times T = 1, 2, 3, 4 ms such that a common minimum exist for all T’s when the lattice acceleration matches with the local acceleration due to gravity. Figure 16 shows interferometer signals for different interferometric times as a function of lattice acceleration, where a common minimum is distinguishable.

4.2.3 Results and Discussion

Figure 16:

Central fringe determination for four different pulse separation times $T=1,2,3,4$ ms. The points are experimental data where different colors depicts different interferometric times. The lines here depict the fitted curves.

Figure 17:

Interferometric fringes of diffracted order for T =10 ms. The inset picture shows the data for a precise scanning over the sweep rate around the local gravity.

Figure16 shows the population oscillation for different interferometric times. The intrinsic sensitivity limit for each interferometry can be given as:

$$\begin{aligned} (\Delta g/g)_{\text {limit}} = \sigma _{qpn}.\sigma _{g} = \frac{1}{C\sqrt{N}gk_{eff}T^2} \end{aligned}$$

(3)

where $(\Delta g/g)_{\text {limit}}$ is the sensitivity of system, $\sigma _{qpn}$ is the quantum projection noise, $\sigma _{g}$ is the response of the interferometer to changes in g and $k_{eff}=2k$. The calculated sensitivity limit for $T=4$ ms has been found to be $8.8\times 10^{-6}$ with 20$\%$ contrast for $5\times 10^4$ atoms. In Fig. 16 one can find the common minimum for T = 1, 2 ,3 and 4 ms. This is called the “central fringe”or local gravity point. The central fringe is shifted for each interferometric time considering the systematic bias shift for each of them. The sensitivity of the system is derived from the Eq. (2) which has been modified for normalised population as $P=(A+C\cos (\Phi ))/2$ where A is the fringe offset. Thus small changes in phase can be observed as $\Delta P= C\mathbf {k}.\Delta \mathbf {g}T^2$⁴⁸.

With an arbitrary value of T, one can also obtain the central fringe in principle. But a smaller value of T helps to have a coarse estimation of the value of g whereas a larger T can help us to achieve the same with higher accuracy. Figure 17 shows the interferometric signal of BEC-based atomic gravimetery performed in APQOL, IISER Pune. The interferometer is performed for T = 10 ms which gives a sensitivity of about 200 $\mu$gal.

4.3 Rotation Sensor Using Atom Interferometry

4.3.1 The Sagnac Effect

The interconnection between differential phase shift, inertial force and gravitational force depends on the interferometric pulse sequence. The timing and the particular combination of pulses determine the atomic trajectories and control the inscription of laser phase onto the interfering paths. We will briefly discuss two methods in the following subsection: the three pulse method, and the four pulse method, and their effects on linear accelerations and rotations from the differential phase shift.

Figure 18:

Schematic pulse diagram of BEC-based Atom Interferometer and its space time trajectories: a three pulse scheme, b four pulse scheme.

4.3.2 Three Pulse Method

Figure 18a shows the MZI where the phase shift is caused by the doppler effect considering the rotation with respect to lab frame. The phase shift for a MZI is

$$\begin{aligned} \Phi _{rot}=\phi _{1}-2\phi _{2}+\phi _{3} \end{aligned}$$

(4)

where $\phi _{i}$ is the phase experienced by atoms during ith pulse. Thus $\phi _{i}=\phi _{i}^0+\mathbf {k_{eff}\cdot x_{i}}$, where $\phi _{i}^0$ is the arbitrary laser phase, $\mathbf {k_{eff}}$ is the wave vector and $\mathbf {x_{i}}$ is the distance from the retro-reflective mirrors. Considering the rotation $\varvec{\Omega }$ about the central point of MZI induces a Doppler shift which modifies Eq. 4 into

$$\begin{aligned} \Phi _{rot}=-2\mathbf {k_{eff}}\cdot (\varvec{\Omega }\times \mathbf{v_{0}})T^2 \end{aligned}$$

(5)

Here, $\mathbf {A} = \frac{\hbar }{m} \mathbf {k_{eff}\cdot v_{0}}T^2$ is the interferometric area which is similar to the Sagnac effect in light interferometer. The generalised equation dealing with the principle of rotation involves

$$\begin{aligned} \Phi _{\text {Total}}=\mathbf {k_{eff}\cdot g}T^2 -\mathbf {2k_{eff}}\cdot (\varvec{\Omega }\times \mathbf {v_{0}})T^2-2\mathbf {k_{eff}}\cdot (\varvec{\Omega }\times \mathbf {g})T^3 + \phi _{i}^0 \end{aligned}$$

(6)

where $\mathbf {k_{eff}}\cdot \mathbf {g}T^2$ is induced due to gravity, $-2\mathbf {k_{eff}}\cdot (\varvec{\Omega }\times \mathbf{v_{0}})T^2$ is induced due to the rotation about any axis and $-2\mathbf {k_{eff}}\cdot (\varvec{\Omega }\times \mathbf{g})T^3$ arises only if the gravity g is not parallel to the rotation axis.

To find the phase shift due to rotation, usually dual interferometric loops are formed in two opposite directions. Thus one can simultaneously separate out the phase shift due to acceleration and rotation by having common and differential modes. Thus the phase shift for rotation is $\Phi _{\text {Total}}=\Phi _{rot}=-4\mathbf {k_{eff}}\cdot (\varvec{\Omega }\times \mathbf{v_{0}})T^2$ in differential mode.

Figure 19:

Schematic of the pulse diagram of simultaneous BEC-based Atom Interferometer and its space-time trajectories for the four pulse scheme. The orange and grey colored arrows indicate the Bragg pulses for splitting, redirecting, and merging. The orange beam is coupled with two different frequencies of $\omega _{1}$ and $\omega _{2}$ whereas the grey beam consists of one frequency $\omega _{0}$. The atoms start from a position (a) where a $\pi /2$ pulse is applied. The atoms split into two momentum states simultaneously and symmetrically with a recoil momentum of $\pm 2 \hbar k$. Subsequently, we apply $\pi -\pi -\pi /2$ pulses with a time interval of $T-2T-T$ after the first $\pi /2$ pulse. The red arrow and circles denote the trajectories of the negative momentum interferometer, whereas the blue arrows and circles denote the trajectories of the positive momentum interferometer.

4.3.3 Four Pulse Method

Figure 18b shows the RBI where the phase shift caused during the four interferometric pulses is

$$\begin{aligned} \Phi _{rot}=\phi _{1}-2\phi _{2}+2\phi _{3}-\phi _{4} \end{aligned}$$

(7)

Here we apply four pulse $\frac{\pi }{2}-\pi -\pi -\frac{\pi }{2}$ interferometer sequence where time between the consecutive pulses is given as T-2T-T. The differential phase shift in RBI is :

$$\begin{aligned} \Phi _{total}=4\mathbf {(k_{eff}} \times \mathbf{g)}\cdot \varvec{\Omega } T^3 + \phi _{i}^0 \end{aligned}$$

(8)

In case of four pulse interferometry the constant acceleration term cancels out since $\mathbf {k_{eff}}\cdot (\varvec{\Omega }\times \mathbf{v_{0})}$ reverses sign. In Eq. 8, the rotation term is related to the acceleration due to gravity rather than the initial velocity of atoms. Also, the above four pulse configuration is insensitive to any dc acceleration along the direction of Bragg pulses.

4.3.4 Simultaneous Atom Interferometer

Scheme for a simultaneous measurement of three components of acceleration and rotation is discussed in the article¹⁰³. Here we implemented simultaneous diffraction of atoms in two momentum states along one axis using double Bragg diffraction (DBD). Figure 19 shows the schematic of the four pulse method interferometer with Bose-Einstein condensate.

4.3.5 Experimental Method

Figure 20:

a BEC after $\pi$ pulse, b BEC after $\pi /2$ pulse, c Rabi oscillation for two different momentum states.

Figure 21:

Normalised population oscillation of $\pm 2\hbar k$ for four pulse method T = 1 ms.

The experimental schematic for the dual atom interferometer is described in Fig. 19. The initial experimental setup is explained in Sect. 4.2.2. Here, three counter-propagating laser beams constitute the optical lattice, and these beams are obtained from the first-order diffraction of two separate acousto-optic modulators (AOM), which are driven by phase-locked AFGs. This is implemented by passing one laser beam in an AOM fed with two frequencies produced by two different AFGs and the other beam through another AOM fed with a single frequency. The peak intensity of each beam is about 1 $mW/cm^2$ with a gaussian shape, $1/e^2$ radius of 700 $\mu$m. For generating the AI pulses $(\pi/2 -\pi -\pi -\pi/2 )$, we have used square pulses with an on-time of 50 $\mu$s for $\pi /2$ pulses and 100 $\mu$s for $\pi$ pulses to drive the first-order Bragg transition. Figure 20 shows the simultaneous transfer of atoms to two different momentum states after $\pi$, $\pi /2$ pulses and their corresponding Rabi oscillations.

4.3.6 Results

To realize the RBI, we apply four pulses of $(\pi /2-\pi -\pi -\pi /2)$ to the BEC as discussed in Fig. 19. We consider the case where we scan the relative phase of the last $\pi /2$ Bragg pulses from 0 to $\pi$. Figure 21 shows the population oscillation of the $p = \pm 2 \hbar k$ as a function of phase when the interferogram time (T) is equal to 1 ms. A clear interferometric oscillation can be seen in Figure 21 with a contrast of $\sim$10$\%$.

Future work will be focusing on improving the interferometric signal for higher interferometric time and reducing the phase noise in the interferometric signal. Different ways to load the BEC in waveguides, analogus to light in the fiber optic gyro will be explored.

5 Towards Realization of a Sr Lattice Clock at IISER Pune

The Atomic Physics and Quantum Optics Lab (APQOL), IISER Pune has recently designed and constructed a versatile system that can be used for an atomic clock as well as for a ‘node’ in a distributed quantum network. APQOL uses Strontium atoms due to the existence of long-lived atomic states which can be used as a reference for a clock laser. Currently, efforts are being made to cool Strontium atoms to a very low temperature which is necessary to overcome thermal decoherence and the Doppler shift of atoms that affect the performance of an optical atomic clock. The vacuum system constructed for the atomic clock maintains a very low pressure (< 1$\times$10$^{-10}$ mbar). Such a low pressure ensures near ideal conditions for the experiment and reduces the background collision shift of the atomic clock. An oven attached to the vacuum system is the source of atoms for the experiment [104]. Thermal atoms, after emerging from the oven, pass through the collimator. The process of collimation increases the atomic flux which is necessary for trapping a large number of atoms within a small timescale in the experiment. Collimated atomic beam passes through the ‘Zeeman slower’ and finally, reaches the science chamber where they are trapped in a magneto-optical trap (MOT). The atoms undergo a strong damping force in MOT and thus the temperature of the atoms enters the sub-Kelvin regime (Fig. 22).

Figure 22:

The vacuum assembly for atomic clock at APQOL, IISER Pune: a 3D-CAD design, b Real assembly.

5.1 The Oven

The oven designed for the experiment generates a high flux ($10^{13}$ atoms/second) of atoms. The reservoir of the oven is loaded with 2.5 gm of pure Strontium. A micro-drilled flange is placed in front of the reservoir. Micro-drills coarsely collimate the atomic beam. Then the atoms pass through a differential pumping tube (DPT). DPT also maintains the pressure difference between the oven and the science chamber of the vacuum assembly. The measured flux of atoms after the DPT is $\sim 10^{10}$ atoms/second (Fig. 23).

Figure 23:

Simulation of the performance of the oven: The figure shows the simulated flux of atoms (left axis) emanating from the oven and the lifetime of the oven (right axis) as a function of temperature The lifetime of the oven is calculated assuming that the oven is loaded with 5 g of Strontium atoms.

5.2 Transverse Cooling

The transverse cooling is used to optically collimate the atomic beam and to increase the atomic flux. Transverse cooling is performed by sending two mutually orthogonal light beams along two orthogonal directions perpendicular to the propagation of the atoms. Transverse cooling increases the flux of the atoms and thus enables capture of a large number of atoms in the trap (Fig. 24).

Figure 24:

The effect of transverse cooling on atomic beam: The beam is compressed when the transverse cooling is applied. a Atomic beam without transverse cooling, b Atomic beam with transverse cooling, c Plot of atomic beam fluorescence showing the effect of transverse cooling.

5.3 The Zeeman Slower

Thermal atoms have velocities around  500 m/s initially. Generally, the trap depth of MOT is around 30 m/s in terms of velocity. Therefore, to capture atoms in MOT, the average velocity of atoms should not exceed the velocity corresponding to the trap depth. The Zeeman slower uses a combination of varying magnetic field along the direction of the velocity of atoms and a counter-propagating laser light. The varying magnetic field compensates for the changing doppler shift as the atoms slow down, and keep the atoms continuously in resonance with the counter-propagating laser light. This ensures a constant deceleration of the atoms, thereby slowing them down form several 100’s of meters down to several 10’s of meters (Fig. 25).

Figure 25:

The magnetic field of the Zeeman slower and the velocity profile of the atoms: a Spatially modulated magnetic field in the Zeeman slower. The simulated magnetic field nearly follows the ideal field profile required for the slowing of atoms. b Simulated velocity profile of the atoms under the influence of the Zeeman slower field. The Zeeman slower brings the atomic velocity below the capture velocity.

5.4 The Magneto-Optical Trap (MOT)

The MOT is constructed with three pairs of mutually orthogonal counter-propagating laser beams and an anti-Helmholtz magnetic field. The spatially dependant magnetic field and the red detuned laser beams exert a restoring and a damping force on any atom moving away from the trap center. On reaching the center of the trap, atoms with velocities below the capture limit of the MOT are cooled down. This method of cooling (Doppler cooling) is limited by the random kicks on atoms due to spontaneous emission. Thus, the limiting temperature depends on the linewidth of the cooling transition.

To cool Strontium atoms, the following optical transitions of Strontium are used consecutively in two different stages. (i) $5s^2$ $^{1}S_{0} \longrightarrow 5s5p$ $^{1}P_{1}$ (461 nm, 32 MHz) and (ii) $5s^2$ $^{1}S_{0} \longrightarrow 5s5p$ $^{3}P_{1}$ (689 nm, 7.5 kHz). They are called blue MOT and red MOT, respectively. The Doppler limited temperatures corresponding to these transitions are 760 $\mu$K and 180 nK. Recently, APQOL IISER Pune has successfully demonstrated the blue MOT of Strontium atoms. The temperature of the atomic cloud in blue MOT is $\sim 1.5$ mK and the number of atoms in the trap is $\sim 10^8$.

The number of atoms trapped in the MOT is crucial for the experiment and it depends on multiple parameters. Therefore manual optimization of MOT is time-consuming. Moreover while optimizing, one can get trapped in one of the local maxima of the parametric space. These problems can be bypassed using machine learning algorithms. APQOL attempted to maximize the number of atoms in the blue MOT by implementing the neural network algorithm using the MLOOP python module¹⁰⁵. The program has successfully maximized the number of atoms within 50 iterations. In some of the cases, the number increased to almost five times the number of atoms in MOT at the initial stage. Figure 26 is the experimental realization of machine learning in optimizing the blue MOT.

Figure 26:

Implementation of machine learning to maximize the number of atoms in blue MOT: Fluorescence from trapped atoms in the MOT a Before neural network optimization and b After neural network optimization.

The immediate next step of our experiment is to load Strontium atoms into the red MOT which will cool the atoms further. Once the cooling is done, the atoms will be optically transported to a new chamber with a better vacuum to allow for further mechanical manipulation of atoms using optical tweezers. Clock spectroscopy will be performed to lock the clock laser and a frequency comb will be used to finally measure the optical frequency precisely.

6 Conclusion

To summarize, we have briefly covered the global status and ongoing efforts on the progress of different quantum sensors based on atoms and ions. The work cited in this article is not exhaustive but can be used to understand the growth of the field and the scope of progress in the coming years. We also described in detail the progress towards quantum sensors based on cold atoms and ions in the APQOL at IISER Pune, India. The progress of setting up a gravimeter involved exploring the generation of an atom laser and studying the diffraction of such a coherent flow of atoms in the Raman Nath regime. Then we included the results demonstrating the Bragg diffraction of a BEC and finally extended the work to realize a gravimeter with appreciable sensitivity. While part of the ongoing efforts of this laboratory is to improve the sensitivity of the gravimeter, we also described in brief our initial efforts towards achieving preliminary results of rotation sensing. Finally, we have described our parallel efforts in building a system that has the potential to be realized as a state of the art optical atomic clock based on Strontium atoms.