Effects of coherence on quantum speed limits and shortcuts to adiabaticity in many-particle systems

We discuss the effects of many-body coherence on the speed of evolution of ultracold atomic gases and the relation to quantum speed limits. Our approach is focused on two related systems, spinless fermions and the bosonic Tonks-Girardeau gas, which possess equivalent density dynamics but very different coherence properties. To illustrate the effect of the coherence on the dynamics we consider squeezing an anharmonic potential which confines the particles and find that the speed of the evolution exhibits subtle, but fundamental differences between the two systems. Furthermore, we explore the difference in the driven dynamics by implementing a shortcut to adiabaticity designed to reduce spurious excitations. We show that collisions between the strongly interacting bosons can lead to changes in the coherence which results in different evolution speeds and therefore different fidelities of the final states.


I. INTRODUCTION
While the Heisenberg energy-time uncertainty relation is often viewed as a purely fundamental restriction on quantum mechanical measurements, it has also implications for dynamical processes. This was first formally recognized by Mandelstam and Tamm (MT) [1,2], who used the standard deviation of the energy to introduce the lower bound, τ QSL ≥ π/(2∆H), for a quantity that has become known as the quantum speed limit (QSL) [3,4], and which describes the minimum time required to transform a given quantum state into a final one with unit fidelity. In the last few years, QSLs have been extensively studied, in particular for applications in quantum computing [5], quantum metrology [6], quantum optimal control [7] and quantum thermodynamics [8]. Various improved bounds and alternative derivations have been proposed, including generalizations to interacting many-body systems [9], mixed states [10,11] and open systems [12,13].
Among all possible dynamical processes driven by timedependent Hamiltonians, adiabatic evolution and quench dynamics have in the past received a large amount of attention. The first one happens on infinitely slow time-scales and allows for high fidelities to be achieved, whereas the second one describes an instantaneous change that does not usually end in an eigenstate of the system. More recently the field of shortcutsto-adiabticity (STA) has shown how one can construct dynamical processes that lead to adiabatic fidelities on finite time scales [14,15]. Up-to-now, however, most of the shortcuts suggested and investigated are for single particle systems [16], where the fidelity between the achieved final wavefunction and the target wavefunction is a good indicator for the success of the shortcut as the density distribution is usually the only interesting quantity. But many-particle systems are more complex, as non-local correlations between the particles need to be taken into account and evolved on the given timescales. Therefore the speed at which correlations can spread in manyparticle quantum systems becomes important [17][18][19][20][21] and one can expect the speed limit to depend on the coherence inherent in the system.
Applying and testing this idea by designing shortcuts for many-particle systems is a formidable problem as it requires one to solve the exact many-particle model. This is not possible in general, but recently experimental progress has allowed to realise the textbook example of a strongly-correlated bosonic quantum gases in one dimension, the so-called Tonks-Girardeau (TG) gas. This model, even though it describes the physics of strongly interacting particles, is solvable due to the existence of a Bose-Fermi mapping theorem [22,23], which also implies that the fermionic counterpart is exactly solvable. Since the coherences in the TG case are √ N times larger than the coherence of the Fermi gas, where N is the number of particles, these models offer insight into two interesting regimes which are physically connected. We find that the MT bound holds for all local properties of these two systems, but the coherence properties in each system needs to be carefully analysed. We also show that the trace distance between the initial and the final reduced single particle density matrices is a good figure of merit for the success of the shortcut [11]. While creating a shortcut for the dynamics of a many-particle system is not an easy task, the Bose-Fermi mapping theorem allows us to essentially treat this as a single particle problem which can be described by a Lagrangian variational method [24,25]. We show that one can then create approximate many-body STAs that can prevent dynamical excitations in the entire system [26] which can lead to high-fidelity dynamics on short time scales.

A. Quantum Speed Limits
Mandelstam and Tamm's QSL describes the minimal timescale for the unitary dynamics of quantum systems based on the variance of the Hamiltonian, ∆H = H 2 − H 2 , and is given by where the Bures angle, B(ρ 0 , ρ τ ) = arccos √ F(τ) , allows one to generalise the QSLs to arbitrary initial and final states, ρ 0 and ρ τ respectively, with F(τ) = tr √ ρ 0 ρ τ √ ρ 0 2 being their fidelity. For pure states this reduces to the overlap of the two many-body states, F(τ) = | Ψ 0 |Ψ τ | 2 , and even though the MT bound describes the timescales for pure states well [27], extensions to mixed states can yield tighter bounds based on coherences [28]. Therefore to derive a QSL that takes the coherence within the many-body state into account one must start from the density matrices of the initial and final state and quantify the connection between these two in terms of the trace distance.
This can be done by starting from the geometric formulation of the QSL using the Schatten-1-norm [4,11,28] with p = 1, which gives One can see that this is characterized by the trace distance T D (ρ τ , ρ 0 ) = Tr (ρ τ − ρ 0 ) 2 and the trace norm of the rate of change of the density matrixρ τ . For bosons this requires the use of the commutator i ρ B τ = [H τ , ρ 0 ], whereas the anticommutator i ρ F τ = {H τ , ρ 0 } has to be used for fermions. Similar to the variance in the MT bound, the trace norm describes the energy excitations of the density matrix. Even though there are an infinite family of bounds to define the QSL, e.g., the Bures angle, the quantum Fisher information and the Wigner-Yanase information, they are all bounded by the norm of ρ τ [29]. Therefore, the QSL which is characterized by Schatten-1-norm is tighter than others [11]. In the next section we will use this measure of the QSL to quantify the dynamics of the reduced states of two related systems, spinless fermions and the strongly interacting TG gas.

B. Coherent Quantum Gases
Let us begin by describing the many-body state. We consider a gas of N interacting bosons of mass m trapped in a quartic trap and assume tight transverse trapping potentials such that the motion of the particles is confined to one dimension. The system can then be described by the Hamiltonian where λ(t) is a tunable strength of the potential and which can be experimentally realized by propagating a blue-detuned Gaussian laser along the axial direction [30]. Our choice of the quartic potential is motivated by exploring dynamics away from the well known and extensively studied harmonic oscillator [16,[31][32][33][34], allowing us to study a more general case where the exact dynamics may not be known [35]. We assume that the interaction between the bosons is pointlike and controlled by the 3D scattering length via g = where d ⊥ is a length scale characterising the strong transversal confinement and the constant C is given by [36]. In general this model is not exactly solvable for arbitrary values of g, however, the solution becomes tractable in the TG limit of g → ∞. In this regime the interaction terms in the Hamiltonian can be replaced by a constraint on the many-body bosonic wavefunction given by which is formally similar to the Pauli principle for identical fermions. This allows to map the strongly interacting bosons onto a gas of non-interacting and spin-polarised fermions which are described by the many-body wavefunction where the ψ n (x j ) are the single particle eigenstates of the trapping potential. To obtain the TG many-body wavefunction one needs to symmetrize the fermionic state, is the unit anti-symmetrisation operator [37]. Therefore, in this hardcore limit, calculating the dynamical evolution of the entire strongly interacting gas only requires evolving the singleparticle states ψ n (x, t), which are governed by the dimensionless single particle Hamiltonian, We choose to rescale our system with respect to a harmonic oscillator of frequency ω 0 as this provides a convenient basis for discussing the dynamics of the individual single particle states. Indeed, while the width of the quartic trap single particle states are narrower than the harmonic oscillator eigenstates (see Fig. 1) they can be approximately mapped to one another by applying a scaling factor which will be introduced in the next section. We therefore express lengths in units of d = √ /(mω 0 ), time in units 1/ω 0 , energy in units of ω 0 andλ(t) is the time-dependent trap strength in units mω 3 0 / . For simplicity of notation in the following sections we will drop the tilde for the scaled variables.
Bosons in the TG limit share many properties with spinless fermions, such as equivalent densities and thermodynamic observables [23], and they also possess identical fidelities as the symmetrisation operator vanishes when taking the many-body overlap Ψ 0 |Ψ τ . In fact, the fidelity between two states can be conveniently written as where σ 1(2) denotes a permutation in N indices, P i j = ψ i |φ j , and ψ i and φ j are the single particle states of the fermionic states Ψ 0 and Ψ τ respectively. The equivalent fidelities of the TG and Fermi gas therefore result in identical QSLs in terms of the MT bound which is shown in Fig. 2. The QSL decreases as the system size grows which is due to linear growth of the energy variance with N, and is also a manifestation of the orthogonality catastrophe whereby the overlap of two manybody states decays with increasing particle number [38,39]. While the dynamics of the TG and Fermi pure states are identical, the mixed reduced states of both systems differ drastically. This is due to interactions being present only in the bosonic state, which results in different momentum distributions and coherences that can be gleaned from the reduced single particle density matrix (RSPDM) [40][41][42] which describes the two-point correlations in the gas after tracing out all particles but one, ρ 1 (x, x ) = +∞ −∞ Ψ(x, x 2 , . . . , x N )Ψ * (x , x 2 , . . . , x N )dx 2 . . . dx N . While the RSPDM of the TG gas is sensitive to the phase of the single particle wavefunctions due to interparticle collisions, the Fermi gas is not as the particles do not interact, allowing one to write the RSPDM for the fermions as whereby the phase information between the fermions is lost. Although the RSPDM of both the TG gas and the spinless fermions do not possess off-diagonal long range order, the TG gas possesses a larger off-diagonal contribution than the fermions, showing that it possesses significantly more coherence (see Fig. 3). Indeed, one may quantify the coherence from the largest eigenvalue θ 0 of the RSPDM via . The coherence of the TG gas scales as θ 0 ∼ √ N (compared to the coherent Bose-Einstein condensate with θ 0 = N), while the Fermi gas is comparatively incoherent with θ 0 = 1. This inherent difference in the coherence of the two species results in a subtle distinction in FIG. 2. QSL time for a quench from λ i = 1 to λ f = 8 as a function of particle number N for the MT bound (stars) and the Schattenp-distance based bound for both fermions (red circles) and TG gas (blue triangles). the QSL (see Fig. 2) where the consistently lower values for the fermions suggests that it reaches the target state faster than the TG gas. We will explore the relation between the QSL and the dynamics in the next section.

III. DRIVEN DYNAMICS AND SHORTCUTS TO ADIABATICITY
While QSLs are certainly useful indicators of the dynamical properties of quantum states, we will now qualify these observations by exploring the many-body dynamics of several example cases. Specifically we consider dynamically driving the system through a ramp of the trapping potential λ(t) to find the timescales needed to reach adiabatic behaviour, which is related to the QSL, and can be quantified through the different fidelity measures. For the pure states we rely on the many-body fidelity defined in Eq. (8) and calculate the overlap between the driven state Ψ(t f ) after a ramp duration t f and the target eigenstate Ψ τ as F(t f ) = Ψ(t f ) Ψ τ 2 . For mixed states we will use the trace distance as the fidelity measure, T D (t f ) = Tr (ρ(t f ) − ρ τ ) 2 , with ρ τ being the RSPDM of the respective target state. We choose to focus on squeezing the trap λ(t f ) > λ i , and optimize the ramp by employing an STA technique, which will minimize unwanted excess excitations in the system allowing one in theory to achieve perfectly the target groundstate for any finite timescale. However, for many-body systems which are not scale invariant or are in anharmonic trapping potentials, only approximate STAs can be designed, which will not suppress all excitations of the many-body state. Nevertheless, they still allow one to find a close to optimal driving ramp allowing us to quantify the QSL. We stress that the designed STA is based on the single particle states of the system and will be the same for the Fermi and TG gases, ensuring that any discrepancy is due to the different coherences inherent in the respective systems.
To design the STA we use a variational method whereby an ansatz for the n th single particle state is chosen to minimize the effective Langrangian of the system [24,25]. For our case a good ansatz is given by choosing the harmonic oscillator eigenstates, as they approximately describe the n th -eigenstate of the quartic potential [43,44] where A n = 1/2 n √ πn!a n (t) is the normalization condition, H n the Hermite polynomials and b(t) the chirp. The scaling factor a n (t) plays an important role as it describes the rescaling of each single particle state in the quartic trap with respect to the corresponding harmonic oscillator eigenstate which has a fixed lengthscale d . This therefore allows us to map the quartic potential to this paradigmatic optimisation problem of a single particle in a harmonic trap, which has been studied extensively previously [32,33].
Calculating the Langrangian using the ansatz in Eq. (10), we explicitly get where B(n) = 3(2n 2 + 2n + 1)/8. See the Appendix (A) for its derivation for arbitrary power law potentials. After calculating the Euler-Lagrange equations with respect to a n (t) and b(t) we get the Ermakov-like equation [26,45] a n (t) + 3(2n 2 + 2n + 1)a 3 n (t)λ(t) 2n The change λ → λ + δλ, is closely coupled to a change in the scaling factor a n such that it induces an energy shift in the single particle states, which is associated with the adiabatic invariant [46]. To speed up the frequency changes from λ i to λ f , we reverse engineer the scaling factors a n (t) using an interpolating function a n = 5 ξ=0 d ξ t ξ , with the initial scaling factor a n (0) = [(2n + 1)/8B(n)λ i ] 1/6 and the final scaling factor a n (t f ) = (2n + 1)/8B(n)λ f 1/6 with the usual boundary conditions at the beginning and end of the ramp,ȧ n (0) =ä n (0) = 0 andȧ n (t f ) =ä n (t f ) = 0.
In the following we will assume the total number of particles in the system is fixed at N = 50 and we design STAs using FIG. 4. (a) Fidelity F versus STA duration t f after using the STAs λ n=49 (t) (black solid line) and λ n=0 (t) (dotted green line). The total particle number is fixed at N = 50 and the inset shows the respective shortcut pulses for t f = 2. (b) Infidelity versus particle number N for t f = 2. (c) Scaling factor a n (0) as a function of n. The inset shows the difference in a n (0) between neighbouring single particle states, ∆a n = a n+1 − a n . the two boundaries of the Fermi sea for the ansatz in Eq. (12): i) the STA λ n=0 (t) which uses the lowest energy single particle state (n = 0 state) at the bottom of the Fermi sea as the ansatz and ii) the STA λ n=49 (t) which uses the largest energy single particle state which sits at the Fermi surface (n = 49 state at zero temperature). In the inset of Fig. 4(a) we show the designed STAs for a ramp from λ i = 1 to λ f = 8 over a time of t f = 2. While possessing a similar form, the ramps clearly differ at the beginning and end points, with the λ n=49 (t) having a smaller slope compared to λ n=0 (t). Fig. 4(a) shows the overlap fidelity as a function of t f , and one can see that for slow ramps (t f ≥ 4) the fidelity of the two STAs are equivalent and the ramps can be considered adiabatic. However, for shorter ramp times the STA λ n=49 (t) shows a clear advantage by possessing unit fidelity already for t f ∼ 1, while the STA λ n=0 (t) results in distinct oscillations of the fidelity for short times. Indeed, in Fig. 4(b) we show the resulting infidelity as a function of N, and while the STA λ n=0 (t) is effective for small particle numbers (N < 6) it gets increasing worse as the system size grows. In comparison the STA λ n=49 (t) improves as N → 50 and is efficient for most N. It should be not surprising that the STA designed for higher energy states performs better for larger systems, as near their Fermi surface the scaling factors of the single particle states become comparable with values a n ∼ (3λn) −1/6 , see Fig. 4 (c). Actually, for single particle states with n 20 the differences in a n between consecutive states is less than 1% (see inset), suggesting that the dynamical timescales of these higher lying states are closely related. Therefore, with similar scaling factors and thus equivalent dynamics described by Eq. (12), a large majority of particles in the Fermi sea are optimally driven by the STA λ n=49 (t). In comparison the scaling factor of the groundstate is much larger, a 0 ∼ (3λ) −1/6 , and a n varies greatly between particles in the low energy states of the trap. As the scaling factors of the higher energy particles are therefore quite different from a 0 (t), the STA based on λ n=0 (t) is ineffective.
Finally, let us explore the dynamics of the reduced states of the fermions and the TG gas in terms of the trace distance, see Fig. 5(a). Implementing the optimal STA λ n=49 (t) for N = 50 particles we see that the target state is reached for similar timescales as the fidelity overlap (compare with Fig. 4(a)), with T D ≈ 0 for t f 1. However, a strong discrepancy between the results for the different statistics is also clearly visible. Firstly, the T D for the non-interacting fermions is always less than that of the TG gas, showing that it always gets closer to the target state for the same t f , echoing the result from the QSL calculation in Fig. 2. Secondly, the T D of the TG gas possesses distinct oscillations, unlike the fermionic case which is monotonically decaying with t f . The source of these oscillations is the scattering of the hardcore bosons off one another during the trap squeezing which results in changes in the coherence (see Fig. 5(b)). In comparison the fermions are non-interacting and just pass through each other, therefore their coherence is constant throughout the driving. These dynamical collisions between the TG particles have before been referred to as a many-body bounce and can be observed as the doubling of the breathing oscillations in the bosonic momentum distribution compared to the fermionic one [47].
The scattering between the particles in the TG gas also affects the QSL of this driven system, which can be calculated via the time-average of the commutator (anti-commutator for the fermionic gas) between the STA Hamiltonian H t and the time dependent RSPDM ρ t for each system The time average of the trace norm captures the nonequilibrium dynamics during the STA process, diverging as

IV. CONCLUSION
In this work we have explored the differences in the dynamics of many-particle systems of spinless fermions and hardcore bosons. Beginning with the QSL we have demonstrated that coherences play an important role in the evolution of the reduced state of both systems, and through exploring the driven dynamics we have shown that particle collisions in the TG regime hamper implementing STAs efficiently. We have also shown that using approximate single-particle STA techniques in many-body states can yield good results as long as an appropriate ansatz is chosen, which can be promising for applications to more complex systems. Finally, with the goal to control larger quantum systems for applications in quantum information and computation there is a need to go beyond just characterising the system through the fidelity and instead probe deeper into the coherences and correlations which can exhibit different dynamics.

ACKNOWLEDGEMENT
This work was supported by NSFC (11474193), SMSTC (18010500400 and 18ZR1415500), and the Program for Eastern Scholar. XC acknowledges the Ramón y Cajal program of the Spanish MINECO (RYC-2017-22482). TF acknowledges support under JSPS KAKENHI-18K13507, and TB, TF and JL acknowledge support from the Okinawa Institute of Science and Technology Graduate University.
In the case of compression the parameters a and λ are time dependent, but ξ = x 0 = 0 and do not change. Therefore the Ermakov-like Eq. (A10) can be written asä with D(n) = 2q!q2 n−2q+1 n!(2n + 1) n j n j 2 j! 2 j (s + j)! .
By interpreting a as the position of a classical particle, it is straightforward to find its potential energy U through the Newton equationä = −∂U/∂a from Eq. (A12). In order to find the minimum of the potential, we set ∂U/∂a = 0 and get This expression can be used to find the specific boundary conditions aṡ As there are infinite number of functions that satisfy these boundary conditions, we choose a polynomial ansatz of the form of a = 5 i=0 c i t i for simplicity in our work.