Field of the Invention

The present invention generally relates, in a first aspect, to a quantum sensing device, comprising a Bose-Einstein condensate (BEC) trapped in a two-dimensional ring trap, and more particularly to a quantum sensing device made to sense a phenomenon based on the existence of non-linear interactions and or their eventual modification.

A second aspect of the present invention relates to a system comprising the quantum sensing device of the first aspect of the invention.

A third aspect of the present invention relates to a method for measuring, comprising using the quantum sensing device of the first aspect of the invention.

Background of the Invention

Pushing the limits of sensing technologies is one of the main challenges in modern physics, opening the door to high-precision measurements of fundamental constants as well as applications in many different areas of science. Specifically, the development of highly-sensitive compact magnetic field sensors enables from detecting extremely weak biologically relevant signals to localize geological structures or archaeological sites [1]. In this context, superconducting quantum interference devices (SQUIDs) [2, 3] and atomic [4-1 1] and nitrogen-vacancy diamonds [12, 13] magnetometers are the three main approaches that allow achieving, in a non-invasive way, unprecedented sensitivity to extremely small magnetic fields.

In particular, the extraordinary degree of control of ultracold atomic systems [14, 15] makes them ideal platforms for precision measurements [16]. There are basically two types of ultracold atomic magnetometers depending on whether the magnetic field drives the internal or the external degrees of freedom of the atoms. The former are typically based on the detection of the Larmor spin precession of optically pumped atoms while the latter encode the magnetic field information in the spatial density profile of the matter wave. Atomic magnetometers with Bose-Einstein condensates (BECs) have been investigated, for instance, by using stimulated Raman transitions [17], probing separately the different internal states of a spinor BEC after free fall [18], or measuring the Larmor precession in a spinor BEC [19-23]. In the latter case, sensitivity can be increased by probing spin-squeezed states [24] In [25], the possibility of taking profit of Feshbach resonances to use a two-component BEC as a magnetometer was also outlined. Ultracold atomic magnetometers based on detecting density fluctuations in a BEC due to the deformation of the trapping potential have also been demonstrated [26-28].

Ring-shaped potentials for ultracold atoms are a particularly interesting trapping geometry for quantum sensing and atomtronics [29, 30]. Ring potentials are currently implemented by means of a variety of techniques, such as optically plugged magnetic traps [31 ], static Laguerre-Gauss Beams [32], painting [33, 34] and time-averaged potentials [35, 36] or conical refraction [37] In fact, persistent currents have been observed in BECs confined in annular traps [38, 39] and it has also been shown that their physical behaviour is in close analogy to that of SQUIDs [40-44, 49-52] It has also been suggested [35, 44] that BECs in this trapping geometry could be used as rotation sensors, which have already been realized with superfluid ³ He [45] and have been proposed for matter waves based on the Sagnac effect [46-48].

Patent JP5561717B2 discloses a quantum sensing device that comprises the features included in the preamble clause of claim 1 , i.e. a Bose-Einstein condensate (BEC) trapped in a two-dimensional ring trap and distributed according to a spatial atomic density profile, wherein said spatial atomic density profile has a structure that moves induced by a phenomenon to be sensed. That structure is a dark soliton which moves due to a phase difference in the BEC, at both“sides” of the soliton, with a speed that also depends on the speed of sound. Dark solitons in BECs can decay due to thermal fluctuations, quantum fluctuations, or transversal excitations, if the system implementing the device is not purely unidimensional. Also, depending on the system dimensionality and the system dynamics, such a quantum sensing device is very sensitive to interactions with sound. Such a device is designed to measure field gradients, which induce a phase difference in the BEC, at both“sides” of the soliton.

It is necessary to provide an alternative to the state of the art which covers the gaps found therein, by providing a quantum sensing device, a system comprising the device, and a method for measuring using the device which do not possess the above mentioned drawbacks of the existing proposals.

Description of the Invention

To that end, the present invention relates, in a first aspect, to a quantum sensing device, comprising a Bose-Einstein condensate (BEC) trapped in a two-dimensional ring trap and distributed according to a spatial atomic density profile, wherein said spatial atomic density profile has a structure that moves induced by a phenomenon to be sensed.

In contrast to the quantum sensing devices known in the prior art, particularly in contrast to that disclosed by JPJP5561717B2, in the quantum sensing device of the present invention, in a characteristic manner, the above mentioned structure is a minimum density region that rotates around the centre axis of the ring trap in the presence of non- linear interaction from said phenomenon, and that is created from an imbalanced superposition of two counter-propagating orbital angular momentum (OAM) modes of said trapped BEC.

As stated above, the device disclosed by JPJP5561717B2 is designed to measure field gradients, which induce a phase difference in the BEC, at both“sides” of the soliton. Instead, the present invention provides a device which is able to measure a uniform field that uniformly modifies the atom-atom interaction along the ring.

For a preferred embodiment, said minimum density region is a minimum density line, or nodal line.

The quantum sensing device of the first aspect of the invention further comprises, for an embodiment, measuring means configured and arranged to monitor the minimum density region while rotating, and to perform measurements of said phenomenon based on the monitored minimum density region rotation.

Different kinds of phenomena can be measured with the sensing device of the present invention, including but not limited to at least one of the following phenomena: two body interactions of the BEC atoms, scalar magnetic fields and rotations, a combination of both, and any other phenomena modifying the non-linear two body interactions of the BEC atoms.

Both, internal magnetic fields (such as those used to trap the BEC, when is the case) and external magnetic fields (such as the earth magnetic field) can be sensed and measured with the quantum sensing device of the first aspect of the present invention.

The above mentioned measuring means generally comprises processing means for processing data representative of said measurements by means of at least one computing algorithm running therein.

Other embodiments are described with reference to claims 6-12, including the provision of specific analytical expressions obtained as described in the following, to be processed by the computing algorithm of the sensing device of the first aspect of the present invention.

For an embodiment, said computing algorithm process said data by means of at least the following expression: where _2d is the non-linear interaction parameter, W is the rotation frequency of the minimum density region, D is the chemical potential difference of the BEC states with one unit of OAM and the BEC states with three units of OAM, I is the integral of the square of the BEC radial density probability, and ni _± is the population imbalance for a pi ₊ population of the state |1 ,+) and a pi- population of the state 11 ,->. The equational development resulting in expression (10), and also those resulting in the expressions which will be shown below, will be described in detail in a subsequent section referring to the detailed description of several embodiments.

The quantum sensing device of the first aspect of the present invention, for an embodiment, is configured and arranged to sense phenomena modifying the non-linear two-body interactions of the BEC atoms by calculating g _{2 d} with expression (10) by means of said computing algorithm.

Any phenomena whose presence modifies the non-linear two-body interactions of the BEC atoms can be sensed, such as, but not limited to, electric, magnetic, microwave or optical fields. For example, in the presence of a magnetic field, g ₂ d is modified but it is not sensitive to its direction so the absolute strength of the external magnetic field can be measured with the sensing device of the first aspect of the invention. In addition, there is no need of an external magnetic field gradient to induce a variation in the rotation of the minimum density region, a uniform external magnetic field that varies in time would give rise to this variation.

However, in addition to the absolute value of a magnetic field, a gradient thereof can be measured by an appropriate arrangement of several quantum sensing devices defined according to the first aspect of the invention, particularly by a series connection of said devices, no matter the relative orientation between them (which is a big difference with the prior art magnetic sensor arrangements).

For an embodiment, the above mentioned measuring means comprises fluorescence imaging means (such as a CCD camera) configured and arranged to acquire fluorescence imaging information of the spatial atomic density profile of the BEC, and one or more processors operatively connected to said fluorescence imaging means and being configured and arranged to process data representative of said fluorescence imaging information to obtain parameters W, /, and ni _± .

Alternatively to the fluorescence imaging means, other measuring means not using fluorescence techniques but being capable of measuring the density profile of the BEC can be used according to the present invention, and therefore the one or more processors operatively connected to said other measuring means are configured and arranged to process data representative of the density profile measurement information to obtain parameters W, I, and ni _± .

The quantum sensing device according to the first aspect of the present invention is configured and arranged to sense scalar magnetic fields, for an embodiment, by means of the at least one computing algorithm running in the measuring means relating changes of the rotation frequency of the minimum density region to variations of the modulus of a magnetic field.

According to an embodiment, the atomic species forming the BEC are stable across Feshbach resonances, and the at least one computing algorithm is adapted to calculate the modulus B of said magnetic field by processing said data by means of at least the following expression:

wherein So is the value of the magnetic field at a Feshbach resonance, d is the width of the Feshbach resonance, % is the s-wave scattering length, and d _s is the background scattering length.

The above expression is always valid, regardless of whether the magnetic field strength is constant (and thus the minimum density region rotates at constant speed) or it changes with time (in which case the rotational speed of the minimum density region would also be time dependent).

The at least one computing algorithm is adapted, for an embodiment, to calculate magnetic field variations for a variable speed of rotation of the minimum density region, by processing said data by means of at least the following expression:

where / is the integral of the square of the BEC radial density probability, rii _± is the population imbalance for a pi ₊ population of the state |1 ,+) and a pi- population of the state 11 ,-), B is the modulus of the magnetic field to be sensed, D is the chemical potential difference between the chemical potentials of the the BEC states with one unit of OAM and the BEC states with three units of OAM, a _s is the s-wave scattering length, N is the number of atoms of the BEC, m is the mass of the atoms of the BEC, U is a parameter of the four-state model (FSM) Hamiltonian, and W is the experimentally measured rotation frequency of the minimum density region. The device of the first aspect of the present invention can measure both constant and variable rotation speeds. For constant speeds one must use expression (10) while by means of expression (1 1 exp) what is analyzed is the variation of the rotation speed for variations of the magnetic field

As indicated above, the quantum sensing device of the first aspect of the present invention is also capable of sensing rotations. To that end, for an embodiment, the quantum sensing device is configured and arranged to be attached to an external rotating object, and to sense the rotation speed Q _ext of said external object when attached thereto, by means of the measuring means being adapted to:

- when the non-linear interaction parameter g2 _d = 0, directly determine that said rotation speed Q _ext is the same as the rotation speed of the minimum density region, or - when the non-linear interaction parameter g2 _d ¹ 0, determine that said rotation speed Q _ext calculating the same with said at least one computing algorithm running in the measuring means by processing said data by means of the following expression: Q _ext = W . QFSM, where W is the measured rotation frequency of the minimum density region and

CF _S M is the rotation frequency of the minimum density region calculated in the context of the FSM and given by:

Preferably, all or some of the components of the quantum sensing device of the first aspect of the present invention are integrated into a chip

The present invention also relates, in a second aspect, to a system comprising the quantum sensing device of the first aspect of the invention, and further comprising excitation means configured and arranged to create the above mentioned imbalanced superposition of two counter-propagating orbital angular momentum (OAM) modes of the trapped BEC.

For an embodiment, the system of the second aspect of the present invention also comprises means for creating the two-dimensional ring trap, and the trap itself, generally a magneto-optical trap for capturing and cooling the atoms of the BEC, and also a vacuum chamber for the BEC.

In a third aspect, the present invention also relates to a method for measuring, comprising using the quantum sensing device of the first aspect of the invention, to sense a phenomenon inducing a non-linear interaction in the quantum sensing device. The method of the third aspect of the invention comprises performing the measuring functions carried out by the measuring means of the quantum sensing device according to any of the embodiments described above.

BRIEF DESCRIPTION OF THE FIGURES

In the following some embodiments of the invention will be described with reference to the enclosed figures. They are provided only for illustration purposes without however limiting the scope of the invention.

Figure 1 shows a sketch of the physical system under consideration on which the present invention is based. A BEC formed by N atoms is loaded in an annular trap, with a pi ₊ population of the state |1 ,+> and pi- population of |1 ,->. The interference between these two counter-rotating modes yields a minimum line in the probability density. R is the radius of the annulus and s is the width of the radial harmonic potential.

Figure 2. (a) Time evolution of the population of the states involved in the dynamics of the system of Figure 1. (b) Snapshots of the density profile for different instants of the dynamical evolution (c) Time evolution of the real part of the coherence between the |1 ,+) and |1 ,-) states. The points correspond to the numerical simulation of the GPE, while the continuous lines are obtained by solving the FSM equations. The considered parameter values are R = 5, g _2d = 1 , for which U = 0.0128, pi = 0.529 and m ₃ = 0.699,

Figure 3. (a) Rotation frequency of the nodal line as a function of g _{2 d} for different values of n- _\± obtained with the FSM (continuous lines) and full integration of the GPE (points) (b) Relative error committed in the determination of W using Eq. (9) as a function of the ab initio values of g _2d and n- _\± in the simulation.

Figure 4. (a) Example of A1 and A2 integration areas to experimentally determine the population imbalance (b) Relative error committed in the determination of g _2d using the full experimental protocol described in the following section as a function of the ab initio values of g _2d and pi ₊ in the simulation.

Figure 5 schematically shows the quantum sensing device of the present invention, for an embodiment.

DETAILED DESCRIPTION OF SOME EMBODIMENTS

In the present section, in order to provide a full support for the present invention, a detailed description of the quantum sensing device of the first aspect of the invention is provided, for some embodiments, including the description of the physical system on which the device is based, a quantum sensing protocol including the equational development of the analytical expressions identified above, mathematical simulations of the system thereof, and an experimental fluorescence imaging protocol to obtain some of the parameters of the given analytical expressions, being both protocols to be implemented for performing the sensing and measuring actions according to the present invention.

QUANTUM SENSING DEVICE

A. Physical system

It is considered herein a BEC formed by N atoms of mass m confined in a 2D annular trap of radial frequency w. Within the mean-field approach, the dynamics of the system is governed by the 2D Gross-Pitaevskii equation (GPE):

where the energies are expressed in units of hoc, the distances in

and time in 1/w. V (r) = 1/2(r- R) ² , where R is the radius of the ring, is the trapping potential. The non-linear parameter is given by

8ppiw _z

92 d = Na _s Ί h where _<¾ is the s-wave scattering length and w _z is the trapping frequency in the z direction. Note that the wave function fulfils the normalization condition

This system supports stationary states with a well-defined total OAM / and positive or negative winding number, which is herein denoted as \l, ±). The wave functions of such states can be written as

where f {r) is the radial wave function.

B. Dynamics in the weakly interacting regime

As initial state an imbalanced superposition of the |1 ,+) and 11 ,-) states, with m _{± º} pi ₊ - pi being the population imbalance is herein considered. Such state could be realized for instance by preparing the BEC in the ground state of the ring, imprinting a 2p round phase and momentarily breaking the cylindrical symmetry of the potential to induce a coupling between the degenerate states of positive and negative circulation [53, 54]

Due to parity reasons, the non-linear term in the GPE can only excite OAM states with odd total OAM /. Thus, we can write the total wave function at any time t as

F (G, ί) = y b \ί)Fΐb \P - (4) l odd /3 å==t

Since being focused on the weakly interacting regime, it is considered herein that the only higher energetic states with a relevant role in the dynamics are |3,+) and |3,-). In order to simplify the forthcoming analytical expressions, it is assumed that the radial wave functions to be the ground state one, i .e.fi(r) =f ₀ (r) is taken in Eq. (3). This is an excellent approximation as long as the width of the BEC ground state fulfils OBEC « R, which is always the case in the weakly interacting regime.

The time evolution of the probability amplitudes a _\± {t) (l = 1 ,3) is obtained by substituting (4) into the GPE (1 ):

where the four-state model (FSM) Hamiltonian reads

where p _t±j± º c _± a‘ _± with i,j = 1 , 3 are the density matrix elements, /i _/ the chemical potential of the l = 1 , 3 OAM states, these parameter definitions, the validity condition of the weakly interacting regime reads (jis - mi) º D » U. Within this regime, Fig. 2(a) shows a typical temporal evolution of the populations of all the OAM states involved in the dynamics considering as initial state an imbalanced superposition of the |1 ,+) and |1 ,-) states. The continuous lines have been obtained by solving numerically the FSM, Eq. (5), and the insets show the comparison with the results obtained by full numerical integration of the 2D GPE (points). For all the populations, an excellent agreement between the results obtained with the two different methods is found. Despite the fact that the populations of the different OAM states present only very small fluctuations, the initial state is not in general a stationary state of the system because the minimum appearing in the density profile rotates at a constant speed. This fact can be appreciated in Fig. 2(b), where the density profile is shown for different times. At t = 0, the density profile has a minimum density line at = 0, and as time marches on this line rotates in the x -y plane. The fact that the minimum density line rotates means that there is a time- dependent relative phase a(t) between the ai ₊ (t) and ai (t) coefficients, so that the state of the system evolves in time a

phase difference is due to the non-linear interaction, and can be understood as a consequence of the presence of off-diagonal terms in the FSM Hamiltonian (6). In order to determine the time dependence of a, in Fig. 2(c) the temporal evolution of the real part of the coherence pi _+i - = ai ₊ (t)a‘_(t) is plotted. It can be observed that it oscillates harmonically, which means that a evolves linearly with time. The oscillation frequency of the coherence corresponds to the rotation frequency of the minimum density line.

From the FSM, the oscillation frequency pi _+i. can be obtained by solving the von

Neumann equation After assuming pi _+i+ = pi ₊ and p _{\.\~ =} p _\ - to be constant and neglecting all terms ^(°3± W) , one arrives at a linear system of three coupled differential equations:

The characteristic frequencies k of the system of equations (7) are obtained by solving the eigenvalue equation

Since D « U in the weakly interacting regime, the term proportional to pi ₊ ri-U ² can be neglected in front of the others. The three eigenvalues that are obtained after solving Eq. (8) are imaginary. The eigenmode associated to the eigenvalue of lowest modulus h has a predominant component of ₊ - ( t ), allowing us to write p i _+i _ (t) ~ p i _+i _ (0)0+ Thus, the rotation frequency of the nodal line is QFSM = - ko, where the subscript indicates that the rotation frequency has been obtained in the context of the FSM. In the limit D » W, the rotation frequency of the nodal line is given by

Uni±

¾SM — (9)

2(1 + ¾)

Note that, although the l = 3 states are nearly not populated during the dynamical evolution, the parameter D, which contains the chemical potential mi, plays a significant role in the expression of the rotation frequency (9). Thus, these states must be taken into account for an accurate description of the dynamics of the system. QUANTUM SENSING PROTOCOL

A. Sensing of two-body interactions:

Recalling that the parameter U of the FSM Hamiltonian (6) is given by

U (9) allows to express the interaction parameter g _2d as

The relation (10) constitutes the basis to use the physical system under consideration as a quantum sensing device. By determining the parameters appearing on the right hand side, one can infer the value of g _2d and thus, from Eq. (2), either the s-wave scattering length or the number of atoms forming the BEC.

In Fig. 3(a), W is plotted as a function of g _2d for different values of ni _± , computed using (9) (continuous lines) and the full numerical integration of the 2D GPE (points), showing an excellent agreement between the two methods for low non-linearities and dί

population imbalances. For g _2d < 4, Fig. 3(b) shows the relative error - , where C _GPE is the rotation frequency of the nodal line obtained from the GPE and 8W = | CFSM - CGPE|, as a function of the ab initio values of ni _± and g _2d in the numerical simulation, finding a maximum relative error of 10 ² . Since all the treatment developed so far is valid for low values of g _2d , this sensing device could be used for dilute BECs.

The rotation frequency of the minimum density line W, can be measured by direct imaging in real time of the density distribution of the BEC. If the coherence time of the BEC is T, in order for this measurement to be possible the condition Ww 1/ t must be fulfilled, since otherwise the rotation would be so slow that it could not be appreciated during the time that the experiment lasts. The upper limit of observable relevant values of W is imposed by the regime of validity of the model. If the interaction is too large, the assumptions of the FSM model are no longer valid and it is thus not possible to relate the rotation frequency of the nodal line to the nonlinear interaction parameter using (10). The rest of parameters appearing on the right hand side of (10) can be determined experimentally from fluorescence images of the BEC. A specific protocol to measure the population imbalance rii _± , the integral of the radial wave function /, and the chemical potential difference D, has been designed by the present inventors and is disclosed below at the end of this section.

B. Sensing of magnetic fields

Assuming that the total number of atoms of the BEC N and the trapping frequency in the z direction c¾ are precisely known quantities, eqs. (10) and (2) together with the protocols to measure hi _± , I and D allow to determine the scattering length a _s at zero magnetic field. Alternatively, if the scattering length is a known quantity, the measurements of W, I and D can be used to determine ni _± through the aforementioned relations.

If the scattering length depends somehow on the modulus of the external magnetic field B, turning on the field will be translated into a variation of W. Thus, the system could be used as a scalar magnetometer by relating changes on the frequency of rotation of the minimal line to variations of the modulus of the magnetic field. Taking into account that I and D are almost independent of _2d and thus of B in the regime of interaction strengths for which the model is valid, combining eqs. (2) and (9) one can evaluate the sensitivity that this magnetometer would have as

Note that expression (1 1 ) differs from expression (1 1exp) shown in a previous section, in that the latter refers to the experimentally measured rotation frequency W of the minimum density region, while the former refers to the calculated rotation frequency WrbM-

Since U « D must be met in order for the model to be valid, one can define a threshold limit for the sensitivity by taking II/D = 1 in (1 1 ). Defining the aspect ratio L º w _z/ w and changing the differentials in (1 1 ) by finite increments, the following upper threshold for the sensitivity in magneticfield variations ABm as a function of the change in the rotation frequency of the nodal line is found:

From eq. (12), it can be observed that the sensitivity is improved by having a large number of condensed particles and a strong dependence of the scattering length on the magneticfield modulus. However, since the parameterg2 _d « Na _s needs to be small in order for the model to be valid, it is also required that the scattering length takes small values.

In the presence of a Feshbach resonance, the scattering length depends of the magnetic field modulus as

From expression (13), the following expression for measuring B can be derived:

Where a _s is the background scattering length, Bo is the value of B at resonance and d is the width of the resonance. Thus, by placing the magnetic field close to the resonant value Bo, one could in principle meet both the requirement that the scattering length is small and that it depends strongly on the magnetic field modulus. However, in most cases this procedure would have the inconvenient that close to a Feshbach resonance the three-body losses are greatly enhanced, limiting the lifetime of the BEC and hindering the measurement procedure. Nevertheless, some atomic species such as ⁸⁵ Rb [55], ¹³³ Cs [56], ³⁹ K [57] or ⁷ Li [58] have been reported to form BECs that are stable across Feshbach resonances, so they could be potential candidates for using the system as a magnetometer. Additionally, the BECs formed by these species have lifetimes on the order of a few seconds. Taking into account the trapping frequency ®, in units of which is expressed DW in (12), is typically of the order of a few hundreds of Hz for ring-shaped traps, this means that long integration times could be used in the imaging procedure and thus good sensitivities in DW could be achieved. These atomic species have, however, the drawback that they typically form BECs with a low number of particles, which limits the sensitivity of the magnetic field sensing.

As a last remark, it must be pointed out that after measuring the scattering length, far from the resonant field Bo if the line of minimal density rotates at a constant speed the relation (13) can be inverted to infer the absolute value of the magnetic field. C. Sensing of rotations

It is considered herein the case when the BEC is placed in a reference frame rotating at an angular frequency Q _ext , which is positive (negative) if the rotation is clockwise (counter-clockwise). Now the dynamics is governed by the modified GPE

L = -ih - where is the z component of the angular momentum operator. The ideal instance for using the system under study as a sensor of rotations is the non-interacting limit _2d = 0. In that case, it can be easily shown that the effect of the external rotation is to make the line of minimal density rotate at an angular speed Q _ext , which can be directly measured in experiments.

In the weakly interacting regime, the system under study can still be used as a sensor of external rotations. In that case, the only difference in the dynamics with respect to the case when there is no external rotation is that the rotation frequency of the nodal line is shifted precisely by a quantity Q _ext . Thus, if g _{2 d} is known and /, ni _± and D are measured using, for example, the protocol provided below, the system under consideration can be used as a sensing device for external rotations by computing the external rotation as Q _ext = W - QFSM, where W is the rotation frequency of the nodal line observed in the experiment and QFSM is given by (9).

A study of the dynamics of an imbalanced superposition of the two degenerate counter-rotating l = 1 OAM modes of a weakly interacting BEC trapped in a 2D ring potential have been presented in this document to support the present invention, by providing an enabling disclosure. It has been found that the non-linear interaction induces a time-dependent phase difference between these two modes which leads to a rotation of the line of minimal atomic density of the BEC. The derived few state model provides a simple analytical dependence between the rotation frequency and the non-linear parameter which, for low non-linearities, perfectly matches with the ab initio numerical simulations. The measurement of the rotation frequency allows to use the system as a quantum sensor of two-body interactions, scalar magnetic fields, and rotations. The theoretical treatment exposed in this document can also be extended to a regime of higher interactions, where higher OAM modes are excited and a myriad of new physical scenarios opens up. Fluorescence imaging protocol

In the following lines, a detailed description is provided regarding how the values of the parameters appearing in the right hand side of Eq. (10) could be inferred experimentally by means of direct fluorescence imaging of the BEC.

1. Population imbalance

The population imbalance between the two l = 1 states can be determined from the density profile per particle at any time t, which can be obtained by fluorescence imaging. Since the wave function is given by

Thus, the atom density has a minimum at f=p/2 - W,ί and a maximum at f=- W,ί. Let us now consider the two integration regions A _\ and A2 shown in Fig. 4(a), which are arcs of radius p and angle 2Q centred around the maximum and minimum of intensity, respectively. The integrals of |y| ² over A _\ and A2 can be performed numerically and, for sufficiently small Q, they yield approximately

Thus, combining (A2a) and (A2b) one can determine the product of populations as f?i -Pi

which, together with the constraint p-i ₊ + pi- = 1 , allows to determine the population imbalance from a fluorescence image.

2. Integral of the radial wave function I

From equation (A1 ), we can write

4

From a fluorescence image, one can numerically perform the integral f d ² r\ \ over the whole space, from which the desired quantity can be calculated as

3. Chemical potential difference

The chemical potential of the angular momentum states can be decomposed into its kinetic, potential and interaction contributions. Since one can assume that the wave ώΐL. (t \ =

functions take the form ^ f , the potential and interaction contributions will be the same regardless of /, while the kinetic contribution is given by

Thus, the chemical potential difference is only due to the difference in the centrifugal terms of the kinetic energy

From Eq. (A1 ), one can see that the integral (A7) can be numerically performed after

†2 f _r ) _ \ Y(E,Y=- S¾,t) [ ² determining / 0 ( ^r ) from a fluorescence image as ^ 1+2 ri+ri _

In order to check the accuracy of the proposed experimental protocol, the present inventors have computed _2d using Eq. (10) and determined all the parameters on the right hand side following the above described numerical procedures, and later on comparing with the ab initio used value of _2d in the simulation. In Fig. 4(b) it is plotted the g

relative error committed as a function of the ab initio values of g ₂ d and p-i ₊ . In the

region g _{2d «} 1 and ni _{± «} 0.6, the relative error is minimal and it reaches very low values, on the order of 10 ⁵ . The maximum value of the relative error is about 10%, and is found for low values of ni _± . In the performed simulations, a grid of dimensions 24 x 24 and 1000 points in each spatial direction has been used. With higher grid precision, the relative error committed with the proposed protocol could prove to be even lower.

Finally, Figure 5 schematically shows an implementation of the quantum sensing device of the present invention, for an embodiment, which includes the following components:

- An ultra-high vacuum chamber 1 that allows optical access through optical viewports or windows 2 (more could be present but not shown).

- Multiple optical laser beams 3 that are used to initially cool and trap the atoms in a ring potential.

- A BEC 4 in the ring that is created at the centre of the vacuum chamber 1.

- A laser beam 5 to perform phase imprinting is used to excite the BEC into the 1= 1 orbital angular momentum (OAM) state. Lasers are also used to break the cylindrical symmetry of the system to create the population unbalance between the two counter- propagating BEC modes, which generate the nodal lines in the BEC 4 density profile.

- A CCD camera 7.

The presence of external fields (not shown) modifying the atom-atom interactions induce or modify the rotation of the nodal lines of the BEC 4. To image the spatial and temporal distribution of the atomic density, a laser beam (not shown) illuminates resonantly the BEC and the fluorescence light 6 is detected through one of the windows 2 where the CCD camera 7 is placed.

A person skilled in the art could introduce changes and modifications in the embodiments described without departing from the scope of the invention as it is defined in the attached claims, such as, for example, using mechanisms for creating the ring trap that are different to the ones described above, or using other types of detectors of fluorescence light, or even detectors for detecting the BEC density profile which are not based in fluorescence techniques.

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