Quantum Particles Slow Down with Distance in New Simulations

Giuseppe Del Vecchio of the Paris Cité University in collaboration with Tata Institute of Fundamental Research and Raman Research Institute, Paris-Saclay University and colleagues investigate the behaviour of noninteracting fermions within a one-dimensional quantum system exhibiting a spatially varying effective mass, a phenomenon termed “sluggish quantum mechanics”. Particle motion is progressively suppressed with increasing distance in this model, derived from an inhomogeneous tight-binding system. Through exact calculations of eigenfunctions and quantum propagators, both with and without an external potential, the team determined the ground-state wavefunction and joint probability density for N fermions. Their analysis reveals a new correlation kernel governing the many-body quantum probability density, differing from standard Bessel or Airy kernels typically observed in trapped fermion systems, and offering insights relevant to engineered optical lattices with position-dependent tunneling.

Discovery of a novel correlation kernel for non-monotonic density profiles in position-dependent mass systems

For the first time, a correlation kernel differing from standard Bessel or Airy kernels has been determined. It achieves a sum of two Bessel kernels with distinct indices, previously known only as standard kernels. This breakthrough unlocks the analysis of systems with a non-monotonic density profile and a vanishing minimum at the origin, scenarios inaccessible to standard kernels. The work details a position-dependent effective mass, scaling as |x|α with α greater than zero, representing ‘sluggish quantum mechanics’ where particle motion diminishes with distance, contrasting with systems assuming constant mass. The underlying theoretical framework stems from the BenDaniel-Duke form of the Schrödinger equation, which arises when considering the continuum limit of an inhomogeneous tight-binding model. This model allows for a spatially modulated hopping amplitude between lattice sites, directly translating into the position-dependent effective mass. The significance of this lies in providing a tractable mathematical description of systems where the ease with which a particle moves is not uniform throughout space.

The new correlation kernel arises from analysing non-interacting spinless fermions trapped within a potential, specifically a harmonic potential of the form Vext(x) = ½meffω²|x|α+². This setup, termed ‘sluggish quantum mechanics’, models systems where particle motion decreases with distance from the origin due to a position-dependent effective mass scaling as |x|α, where α exceeds zero. Detailed calculations revealed the many-body quantum probability density follows a determinantal point process, allowing determination of average density and higher-order correlations for any number of fermions. The determinantal point process is a powerful tool in quantum statistical mechanics, enabling the exact calculation of many-body correlation functions given the single-particle wavefunctions. Consequently, the scaled average density profile exhibits a non-monotonic shape with a vanishing minimum at the origin for all values of α greater than zero. This non-monotonicity is a direct consequence of the suppressed particle motion at larger distances, leading to a depletion of particle density away from the origin. Currently, however, these findings describe only one-dimensional systems and do not yet demonstrate a clear pathway towards realising similar effects in more complex, three-dimensional materials. Extending this model to higher dimensions presents significant mathematical challenges due to the increased complexity of the Schrödinger equation and the resulting wavefunctions.

The value of α in the effective mass scaling, |x|α, plays a crucial role in determining the degree of ‘sluggishness’. A larger α value indicates a more rapid suppression of particle motion with increasing distance. The researchers found that for any α greater than zero, the resulting density profile exhibits the characteristic vanishing minimum at the origin. This is in stark contrast to systems with constant effective mass, where the density profile typically exhibits a monotonic decay or a Gaussian shape. The exact form of the correlation kernel was derived through a careful analysis of the single-particle wavefunctions obtained by solving the Schrödinger equation with the position-dependent mass. This involved expressing the wavefunctions in terms of special functions and then calculating the corresponding correlation functions. The resulting kernel provides a complete description of the many-body quantum probability density, allowing for the prediction of various observable quantities, such as the average density and the probability of finding particles at specific locations.

Characterising fermion correlations in position-dependent mass optical lattices

Precise theoretical models are essential for predicting and interpreting experimental results in engineered quantum systems, particularly optical lattices. A new correlation kernel, a mathematical tool describing particle relationships, has now been characterised for fermions experiencing a position-dependent effective mass, extending beyond established Bessel or Airy kernel descriptions. This advance, however, rests on a simplified scenario, examining only non-interacting, spinless fermions, raising questions about its applicability to realistic materials where particle interactions and spin are unavoidable. Optical lattices, created by interfering laser beams, provide a highly controllable environment for studying quantum phenomena. By carefully tuning the laser parameters, researchers can engineer the potential experienced by atoms or other particles, effectively creating a lattice structure. The ability to control the potential allows for the investigation of various quantum effects, including those related to position-dependent mass.

Despite the fact that these calculations involve simplified fermions without interactions or spin, the development of this new correlation kernel remains a valuable step forward. Understanding particle relationships within these engineered systems, even in idealised conditions, provides a key benchmark for more complex modelling. The detailed correlation kernel provides a better understanding of how fermions behave in engineered quantum systems. This kernel describes particle relationships where motion is restricted, differing from previously known models used for these scenarios. Future work will need to incorporate the effects of interparticle interactions and spin to provide a more realistic description of these systems. These interactions can significantly alter the correlation functions and lead to new and unexpected phenomena. The inclusion of spin would also introduce additional degrees of freedom and further complicate the calculations.

A new correlation kernel for describing non-interacting fermions in one-dimensional systems with spatially varying effective mass has been established, differing from those previously used for constant mass scenarios. Determining this kernel, a function defining particle proximity, allows for a more accurate understanding of many-body quantum probability densities, particularly within engineered optical lattices where precise control over quantum environments is possible. The findings reveal a non-monotonic density profile with a vanishing minimum at the origin, a characteristic absent in systems modelled with standard kernels. The implications of this work extend beyond fundamental quantum mechanics, potentially influencing the design of novel quantum devices and materials. By manipulating the effective mass of particles, it may be possible to create new types of quantum sensors or to enhance the performance of existing quantum technologies. The ability to control particle motion at the nanoscale could also lead to the development of new materials with tailored properties.

The researchers successfully developed a new correlation kernel to describe the behaviour of up to N non-interacting fermions in a one-dimensional system where effective mass varies with position. This is important because it provides a more accurate model of particle relationships than previously available, particularly for engineered quantum systems. Analysis revealed a non-monotonic density profile, meaning the density of particles does not change uniformly across space. The authors intend to incorporate interparticle interactions and spin into future investigations to create a more complete description of these systems.