Enhancing sensitivity to rotations with quantum solitonic currents

Quantum mechanics is characterized by quantum coherence and entanglement. After having discovered how these fundamental concepts govern physical reality, scientists have been devoting intense efforts to harness them to shape future science and technology. This is a highly nontrivial task because most often quantum coherence and entanglement are difficult to access. Here, we demonstrate the enhancement of the sensitivity of a quantum many-body system with specific coherence and entanglement properties. Our physical system is made of strongly correlated attracting neutral bosons flowing in a ring-shaped potential of mesoscopic size. Because of attractive interactions, quantum analogs of bright solitons are formed. As a genuine quantum-many-body feature, we demonstrate that angular momentum fractionalization occurs. As a consequence, the matter-wave current in our system can react to very small changes of rotation or other artificial gauge fields. We work out a protocol to entangle such quantum solitonic currents, allowing them to operate rotation sensors and gyroscopes to Heisenberg-limited sensitivity.

Quantum mechanics is characterized by quantum coherence and entanglement.After having discovered how these fundamental concepts govern physical reality, scientists have been devoting intense efforts to harness them to shape future science and technology [1].This is a highly nontrivial task because most often quantum coherence and entanglement are difficult to access [2].Here, we demonstrate the enhancement of sensitivity of a quantum many body system with specific coherence and entanglement properties.Our physical system is made of strongly correlated attracting neutral bosons flowing in a ring-shaped potential of mesoscopic size.Because of attractive interactions, quantum analogs of bright solitons are formed [3][4][5].As a genuine quantum-many-body feature, we demonstrate that angular momentum fractionalization occurs.As a consequence, the matter-wave current in our system can react to very small changes of rotation or other artificial gauge fields.We work out a protocol to entangle such quantum solitonic currents, allowing to operate rotation sensors and gyroscopes to Heinsenberg-limited sensitivity.
Phase coherence is one of the most pervasive concepts in science and technology.In classical physics, coherence leads to interference.With classical interference, we fabricated devices for every-day life, for example, to manipulate sound waves or for audio-video transmissions.With quantum mechanics, we discovered that also massive particles can be coherent.The technological progress that followed has had a disrupting impact in shaping the world as we know it now, with electronics, computer science, photonics, etc.The nature of quantum coherence, though, poses challenging questions when it is referred to many-particle systems because of entanglement-induced non-local correlations.Such features have been of central importance in quantum optics [6], mesoscopic physics [7], and quantum material science [8], and they are now at the heart of quantum technology.Indeed, the defining goal of quantum technology is to realize new concepts of quantum devices and simulators harnessing quantum coherence and entanglement [1].
A natural way to access the resources needed for quantum technologies is to refer to quantum many-body systems.Several options have been studied so far, with the different choices implying a quantum technology with different features.For example, superconducting circuits and circuit QED rely on the quantum coherence resulting from the specific electronic (pairing) correlations occurring in superconductors [9]; with similar logic, quantum devices robust to imperfections and noise have been conceived as based on the braiding properties of the quasiparticles in the topological matter as provided e.g. by quantum Hall systems or other topological matter [10].
increases in quantized steps as a function of Ω [26,27]; correspondingly, the amplitude of persistent currents displays periodic oscillations with Ω [28,29], with a periodicity that Leggett proved to be fixed by the effective flux quantum of the system, irrespective of particle-particle interactions [24].
Below, we demonstrate that for strongly correlated one-dimensional bosons with attractive interactions, the very nature of flux quantum is nontrivial, due to the formation of many-body bound states.This feature has dramatic effects on the persistent current that oscillates with a periodicity N times smaller than in the standard case corresponding to repulsive interactions.Remarkably, the periodicity depends on interaction, which leads to an extension of the Leggett theorem.We show how our system can be harnessed to construct specific entangled states of persistent currents characterised by sensitivity to the effective magnetic field reaching the Heiseberg limit (quantum advantage).
Before treating the general case of the lattice ring, we will first assume that the density N/L of bosons, where N is the particle number and L = 2πR is the perimeter of the ring of radius R, is small enough to describe the system through the continuous Bose-gas integrable theory or equivalently the Lieb-Liniger model [30].For such systems, we can apply exact results [31].
In the frame rotating at frequency Ω, the Lieb-Liniger Hamiltonian reads ĤLL = , where m and the p i 's are respectively the mass and the momentum of each particle, L z = N j=1 L z,j is the total angular momentum of the N particles and g is the interaction strength.The Lieb-Liniger Hamiltonian can be recast to with a constant E Ω = −N mΩ 2 R 2 /2.Here, we assume periodic boundary conditions.Using a transformation to Jacobi coordinates (see Methods) ξ l and their canonically conjugate momenta Q l , where ξ N = X CM and Q N = P CM where X CM = (1/N ) j x j and P CM = j p j are, respectively, the coordinate and momentum of the center of mass, we find that in the Hamiltonian (1) only the center-of-mass momentum is coupled to the artificial gauge field Ω. Correspondingly, the manybody wavefunction can be written as Ψ(x 1 , ..., x N ) = e i(PCM−N mΩR)XCM/ χ relative (ξ 1 , ..., ξ N −1 ).In this case, P CM = /R can take any value allowed by quantization of momentum in the ring ( being an integer).
The ground-state energy reads , where E int is the interaction energy of the fluid, which does not depend on Ω.
For repulsive interactions, independently of the interaction, E GS results periodic in Ω with period Ω 0 = /mR 2 (see Methods).Therefore, the persistent current in the rotating frame I p = −(Ω 0 / )∂E GS /∂Ω reflects the center-of-mass quantization, and displays the characteristic sawtooth behaviour versus Ω [24], corresponding to a staircase behaviour of angular momentum L z .
For attractive interactions the ground state is a manybody bound state, i.e. a 'molecule' made of N bosons, corresponding to the quantum analog of a bright soliton [3][4][5].This picture arises from the exact Bethe ansatz solution; within the regime of validity of the string hypothesis [4] (see Methods) the ground state energy for arbitrary Ω reads where the second term accounts for the interaction energy E int .The above equation shows that attracting bosons behave as a single massive object of mass M = N m under the effect of the artificial gauge field.The energy displays a 1/N -periodicity as a function of the artificial gauge field, Ω, in units of Ω 0 corresponding to fractionalisation of angular momentum per particle.In analogy to the fractional quantum Hall effect, in our system, the elementary particles carrying a fraction of quantum of angular momentum are parts of composite objects.We shall see, however, that our composite object displays a very specific dependence on the interplay between interaction and system size.
any finite interaction).As an effect, the internal structure of the many-body bound state is affected by the interplay between interaction and artificial gauge field Ω (since P CM depends on Ω, and the internal structure depends on P CM ).Here, we find that the periodicity of the persistent current for lattice rings does depend on interaction.We remark that such a 'non-perfect' fractionalization (see Fig. 2(b)) is observed for solitons that are properly formed in the system (i.e. when the system size is larger than the correlation length of the densitydensity correlations).Fig. 2 shows our numerical results (confirmed by exactly solving the BHM in the 2-particle sector-see Methods) for the ground-state energy, persistent currents and angular momentum: also in the lattice nonintegrable case the 1/N periodicity in Ω/Ω 0 of the persistent currents emerges, as well as fractionalization of angular momentum.Indeed, these features, though, are affected by the interplay between system size and interaction strength.The 1/N periodicity is found when interactions are sufficiently large: In these conditions, the 'size of the many-body bound state', defined as the typical decay length of the density-density correlations [5], is smaller than the size of the system.Upon decreasing the interactions, the many-body bound state spreads more and more over the sites making the solitonic nature of the state less and less pronounced (see Methods).We remark that all the observed features are purely quantum many-body effects tracing back to specific quantum correlations: Indeed, mean-field Gross-Pitaevskii equation (corresponding to a non-entangled ground state) provides persistent currents displaying no fractionalization, independently on the strength of the interaction (see Fig. 2,  c).Remarkably, the afore discussed angular momentum fractionalization and persistent current periodicity emerge in the time-of-flight (TOF) distributions of the atoms after releasing the trap confinement and switching off interactions.We obtain it from n(k) = |w(k)| 2 j,l e ik•(xj −x l ) a † j a l , where x j indicate the position of the lattice sites in the plane of the ring and w(k) is the Fourier transform of the Wannier function of the lattice [32].Instead of the characteristic widedependent minimum ('hole') arising for zero or repulsive interactions [26,27], we find no clear hole at k = 0 for the attractive case -Fig.3.Such a feature is due to the reduction of coherence implied by the solitonic manybody bound state.Despite the seemingly featureless momentum distribution, we find that fractional steps of the mean-square radius of the distribution for Ω/Ω 0 = /N [33].This effect provides the univocal signature of 1/N fractionalization of angular momentum in the presence of a many-body bound state.

Appendix A: Separation of center-of-mass and relative coordinates
We detail here the coordinate transformation to centerof-mass and relative coordinates.We introduce the Jacobi coordinates with the Jacobi matrix given by is the center-of-mass coordinate we want to separate out.The Jacobi matrix (A1) is however not orthogonal (ie it is not a rotation).Nonetheless, the matrix M Jac can be easily converted to a pure rotation R Jac via the rescaling: where diag(. . . ) is a diagonal matrix, with the numbers in the parenthesis specifying the diagonal matrix elements.Indeed one can straightforwardly verify that R Jac •R Jac = 1, where R Jac is a transpose of R Jac .We define then the coordinates Note that z N = √ N X CM .Let us now introduce one final transformation, which brings us back to X CM one of the variables, while keeping the Jacobian determinant of the transformation equal to one: This defines the relative and center-of-mass coordinates used in the main text.
By a similar procedure one can identify the transformation to the Jacobi momenta Q l , canonically conjugate to ξ l , where Q N = N j=1 p j = P CM is the center-of-mass momentum.In particular, by introducing a set of momenta P z = R Jac p, with the same Jacobi matrix R Jac as the one used for spatial coordinates, one can show that Q l = αP z l for l = 1, ...N − 1 with α = N −1/[2(N −1)] , and where m is the mass of the relative problem, M = N m is the total mass.

Appendix B: Exact Bethe Ansatz results for the continuous ring
We start from the Lieb-Liniger model Eq.( 1) of the main text, where we drop the constant E Ω : For the Lieb-Liniger model, the total momentum and energy are respectively, where the k j are obtained by solving the Bethe equations where c = 2mg/ 2 , L = 2πR is the ring circumference and I j is a set of integer (semi-integer) numbers defining the state of the system.For repulsive interactions, all the k j 's are real.For 2lπ/L ≤ Ω ≤ 2(l + 1)π/L, the ground states can be obtained by I j = −(N − 1)/2 + j + , with integer , yielding a center of mass momentum given by P CM = j k j = N /R, as readily follows by noticing that arctan[(k j − k )/c] is an odd function.
For repulsive interactions, the allowed values for the center of mass are integer multiples of 2p F , where p F = N/2R, yielding E GS = N 2 2mR 2 ( − Ω/Ω 0 ) 2 + E int with Ω 0 = /mR 2 .The ground state energy hence results periodic in Ω with period Ω 0 and the persistent current, obtained as I p = −(Ω 0 / )∂E GS /∂Ω, clearly reflects the center-of-mass quantization for any value of interaction strengths.
For attractive interactions the Bethe equations of ( B1) admits complex solutions and the ground state corresponds to a many-body bound state: k j = κ − i(n + 1 − 2j)g/2, j = 1 . . .n.Such n string solutions holds also for Ω = 0, since the scattering matrix is not affected by Ω.The ground state of ( B1) is made of a single n = Nstring, yielding Eq.( 2) of the main text.Here we point out that string hypothesis holds for cL → ∞.The finite size corrections to the string solutions (for recent references, see [37,38]) can affect the interaction energy E int .

Appendix C: Numerical Methods
Here we present the numerical techniques that have been used to obtain the results presented in this paper.We solve the eigenvalue problem by writing the Hamiltonian, Ĥ, as a matrix H ij in the Fock basis.This basis is then hashed in a more efficient form [39] in order to write the Hamiltonian in a sparse way.In particular, our numerical code is written in Python and the sparse Hamiltonian is diagonalized using ARPACK within the SciPy library.We have performed simulations with N s = 11 to N s = 24 sites and N = 2 to N = 6 particles, with a Hilbert space dimension up to 10 6 , for different values of the flux Ω/Ω 0 .Simulations have also been benchmarked with DMRG [40] data.After solving the eigenvalue problem, the correlation function C lk = a † l a k is calculated using the ground state of the system and is used to obtain the time-of-flight results of Fig. 3.

Appendix D: Two-particle exact solution
In the N = 2 sector, the Bose-Hubbard model the many-body wavefunction can be obtained using the coordinated Bethe Ansatz approach.Therefore, the groundstate energy and correlation functions can be accessed exactly.We generalize Ref. [41] to include the presence of an artificial gauge field in the Hamiltonian.Here, we gauge away the Peirerls factors in the Hamiltonian and we impose twisted boundary conditions: âNs+1 = e 2πiΩ/Ω0 â1 .A general two particle state can be written as: where φ jk is the two-partcile wavefunction, symmetric under the exchange of j and k, and normalized to unity.
The energy of the system is found by solving the timeindependent Schrödinger equation Ĥ|φ = E|φ using the Bethe Ansatz technique.In the center-of-mass and relative discrete dimensionless coordinates X = (j + k)/2, x = j −k and P = p 1 +p 2 , p = (p 1 −p 2 )/2 the wavefunction φ jk reads: The energy eigenvalues of the two-particle system are given by E = −4J cos( P +Ω 2 ) cos(p).The center of mass momentum is obtained by imposing twisted boundary conditions and quantization of the ring: For the BHM the relative momentum p is obtained by the condition: with It is interesting to compare the BH and the Lieb-Liniger pictures.In the latter case, the equations to solve are with Note that, in contrast with the BH case, Eqs.(D6), (D7) are decoupled, ie the center of mass momentum P decouples to the relative momentum.As a result, the imaginary part of the momentum p is independent on Ω; this feature implies that the periodicity of the ground state energy does not change with the interaction strength.For the BHM, instead, P and p are coupled; this feature has a clear effect in the periodicty of the ground state energy.In conclusion, in the BHM the dependence of the periodicity on interactions is an effect of the coupling between center of mass and relative momentum.
Note that by solving Eqs.(D4), (D5) becomes fully determined.Thus, the time of flight images can be then readily evaluated by: e ik•(xj −x l )+iΩ(j−l)/Ω0 n φ * jn φ nl (D8) Appendix E: Finite-size effects In order to relate the size of the many-body bound state and the periodicity of the currents we analize the dependence of the ground-state energy on the artificial gauge flux Ω/Ω 0 for various values of interaction strength U and different system sizes N s .
We estimate the spatial size associated to the manybody bound state by studying the exponential decay of the density-density correlations [5] We quantify the quality of the 1/N periodicity of the ground-state energy E(Ω) by calculating such that E(U, N s , N ) = 0 corresponds to a perfect 1/N periodicity.Figure (5)(a) shows the density plot E(U, N s , N ) for a fixed number of particles N = 2.In this figure, we show that for large U and a sufficiently large system size, the periodicity of the ground-state energy is increased by a factor N with respect to the noninteracting case.In Fig. 5)(b) we calculated the threshold for which the minimum of the N -time periodicity is obtained within an error of 0.1%, i.e.E < 10 −3 , for different number of particles (corresponding to the solid line in Fig. 5)(a)).Finally we compare the densitydensity correlations n j n k for two different points in the density plot shown in (a), one within the region where the current presents N -time periodicity and one above the threshold.Indeed, comparing Fig. 6 and Fig. 7, we can demonstrate that the size of the soliton, which depends on U for a fixed number of particles, must be smaller than N s in order to observe the enhanced sensitivity presented in this paper.where Ĥtot = ĤBH +∆ 0 nj with ĤBH taken at Ω = Ω 0 /2 and |ψ(0) is the ground state of the pre-quench Hamiltonian ĤBH + ∆ 0 nj taken at Ω = 0.The fidelity is then obtained as F = | ψ(t)|ψ N OON | 2 and the current is given by I(t) = −iJ j ψ(t)|a † j+1 a j − H.c.|ψ(t) The oscillation period of the current can be modified by tuning the barrier strenght.The physical parameters we used to obtain the data shown in Fig. 4, panel (b) are obtained as follows: for each value of N we choose U in order to have the same spatial size of the many-body bound state as obtained by the study of the density-density spatial correlation function.The final choices are summarized in the Table I

FIG. 2 .
FIG. 2.Fractionalisation of angular momentum in a Bose gas with attractive interactions.Average angular momentum per particle (main) and ground-state energy (inset) for bosons on a lattice ring as a function of artificial gauge field, from numerical exact diagonalization calculations: a) at varying particle number, for chosen values of interaction strength as indicated on the figure, b) for various values of interaction strength at fixed N = 4. Panel c) shows the corresponding predictions from the mean-field Gross-Pitaevskii equation for zero and finite attractive interactions indicated by the dimensionless parameter g0 = mgL/ 2 .The angular momentum per particle is obtained as = M ef f M

FIG. 3 .
FIG. 3. Time of flight expansion of the Bose gas after releasing the atoms from the ring trap.a-b-c Density plot of the TOF expansion for different values of the artificial gauge field in a system of N = 4 particles and L = 11 lattice sites.d Renormalized width σTOF/σTOF(Ω = 0) of time of flight density distribution, n(k), for different number of particles, and interactions U/J = −0.6,U/J = −1 and U/J = −3 respectively.For the sake of graphical clarity, each curve is offset by 0.05.Note how the TOF density distribution width abruptly changes with the increase of the strength of the artificial gauge field, and how the sensitivity proportionally increases with the number of particles.In all the calculations we have approximated the Wannier functions with Gaussians functions with width δ = a/ √ 2π with a the lattice spacing.

FIG. 4 . 1 √ 2 (
FIG.4.Creation of entangled states of angular momentum with quantum solitons.a Exact many-body dynamics of the current (in units of the hopping constant J) following a quench from Ω/Ω0 = 0 to Ω/Ω0 = 1/2.Here we set L = 28, N = 3, U/J = −0.51 and ∆0/J = 0.015.At one quarter of oscillation period, the superposition |ψ = 1 √ 2 (|Lz = 0 + |Lz = N ) is formed, with a fidelity very close to 1.No fine tuning of parameter is required.b Scaling of the quantum Fisher information with particle numbers, showing that it reaches the Heinsenberg limit FQ ∝ N 2 .The system parameters are described in the supplementary material.

FIG. 5 .
FIG.5. a Density plot of the renormalized energy difference between the N -times periodicity and the nonrotating system for N = 2. Solid lines gives the threshold for which E(U, Ns, N ) < 10 −3 .In figure b we show the threshold given by condition E(U, Ns, N ) < 10 −3 for different number of particles and system sizes.
t e x i t s h a 1 _ b a s e 6 4 = " n G 3 R Q O t h h / A / U t N 9 F m B P u + q m c l U = " > A A A B / H i c b V A 9 T w J B E N 3 D L 8 Q v 1 N J m I z G x I n c G o y X R x h I T + Y h A y N w y 4 I a 9 v c v u n J F c 8 F f Y a m V n b P 0 v F v 4 X 7 5 B C w V e 9 v D e T e f P 8 S E lL r v v p 5 J a W V 1 b X 8 u u F j c 2 t 7 Z 3 i 7 l 7 D h r E R W B e h C k 3 L B 4 t K a q y T J I W t y C A E v s K m P 7 r M / O Y 9 G i t D f U P j C L s B D L U c S A G U S r c d w g f y B 4 k / 6 R V L b t m d g i 8 S b 0 Z K b I Z a r / j V 6 Y c i D l C T U G B t 2 3 M j 6 i Z g S A q F k 0 I n t h i B G M E Q 2 y n V E K D t J t P E E 3 4 U W 6 C Q R 2 i 4 V H w q 4 u + N B A J r x 4 G f T g Z A d 3 b e y 8 T / v H Z M g / N u I n U U E 2 q R H S K p c H r I C i P T K p D 3 p U E i y J I j l 5 o L M E C E R n I Q I h X j t J t C 2 o c 3 / / 0 i a Z y U v U r 5 9 L p S q l 7 M m s m z A 3 b I j p n H z l i V X b E a q z P B N H t i z + z F e X R e n T f n / W c 0 5 8 x 2 9 t k f O B / f 0 m O V r g = = < / l a t e x i t > a < l a t e x i t s h a 1 _ b a s e 6 4 = " Q 3 z D m v K k s t I r x n Y F z y 3 Z L c 4 e 6 0 s = " > A A A B / H i c b V A 9 T w J B E N 3 D L 8 Q v 1 N J m I z G x I n c G o y X R x h I T + Y h A y N w y 4 I a 9 v c v u n J F c 8 F f Y a m V n b P 0 v F v 4 X 7 5 B C w V e 9 v D e T e f P 8 S E l L r v v p 5 J a W V 1 b X 8 u u F j c 2 t 7 Z 3 i 7 l 7 D h r E R W B e h C k 3 L B 4 t K a q y T J I W t y C A E v s K m P 7 r M / O Y 9 G i t D f U P j C L s B D L U c S A G U S r c d w g f y B w l M e s W S W 3 a n 4 I v E m 5 E S m 6 H W K 3 5 1 + q G I A 9 Q k F F j b 9 t y I u g k Y k k L h p N C J L U Y g R j D E d k o 1 B G i 7 y T T x h B / F F i j k E R o u F Z + K + H s j g c D a c e C n k w H Q n Z 3 3 M v E / r x 3 T 4 L y b S B 3 F h F p k h 0 g q n B 6 y w s i 0 C u R 9 a Z A I s u T I p e Y C D B C h k R y E S M U 4 7 a a Q 9 u H N f 7 9 I G i d l r 1 I + v a 6 U q h e z Z v L s g B 2 y Y + a x M 1 Z l V 6 z G 6 k w w z Z 7 Y M 3 t x H p 1 X 5 8 1 5 / x n N O b O d f f Y H z s c 3 0 N O V r Q = = < / l a t e x i t > c < l a t e x i t s h a 1 _ b a s e 6 4 = " s n C 9 B C 5 c x n D 1 W 4 S h d f H M e 7 U g t R 8 = " > A A A B / H i c b V A 9 T w J B E N 3 D L 8 Q v 1 N J m I z G x I n c G o y X R x h I T + Y h A y N w y 4 I a 9 v c v u n J F c 8 F f Y a m V n b P 0 v F v 4 X7 5 B C w V e 9 v D e T e f P 8 S E l L r v v p 5 J a W V 1 b X 8 u u F j c 2 t 7 Z 3 i 7 l 7 D h r E R W B e h C k 3 L B 4 t K a q y T J I W t y C A E v s K m P 7 r M / O Y 9 G i t D f U P j C L s B D L U c S A G U S r c d w g f y B 4 m Y 9 I o l t + x O w R e J N y M e l 0 p V S 9 m z e T Z A T t k x 8 x j Z 6 z K r l i N 1 Z l g m j 2 x Z / b i P D q v z p v z / j O a c 2 Y 7 + + w P n I 9 v 0 / O V r w = = < / l a t e x i t > d < l a t e x i t s h a 1 _ b a s e 6 4 = " D q P 1 d r 9 H H 4 c b s

FIG. 6 .
FIG.6.Density-density correlations Cj,j+r for N = 4 within and outside the regime where the system presents an increase of the periodicity of the current.

FIG. 7 .
FIG. 7. a, b round state energy for different interaction as a of the flux.c, d current, calculated as I = −∂EGS(Ω)/∂Ω, for different interaction as a function of the flux.

TABLE I .
below.Choice of parameters for the study of the quech dynamics.At varying particle numbers, we have chosen the interaction stregth and the barrier strength in such a way that the many-body bound state has the same size.All the calculations are performed with Ns = 20.