Thermally Activated Propagation of Quantum

“Thermally Activated Propagation of Quantum Advection Modes” Kamil Walczak Department of Chemistry and Physical Sciences, Pace University, 1 Pace Plaza, New York, NY 10038 Momentum Fluxes in Quantum Wells

INTRODUCTION

Pumping Heat by Brownian Boosts

Nanoscale Heat Conduction Single-phase lagging model with thermal memory kernel:

We derive new expressions for quantum fluxes associated with transfer of momentum and transfer of energy (heat). Those formulas may be used to build relevant models and simulate nanoscale transport phenomena from first principles. All formal derivations are based on continuity equations (which represent local conservation laws), while the generally applicable expressions are valid for quantum carriers confined within arbitrary potential and affected by time-varying perturbations. Specifically, transport properties of a closed quantum system to which energy is stochastically pumped by Brownian particle immersed into a thermal bath are discussed. Further, we use formula for quantum energy flux to formulate nonlinear quantum transport theory within single-phase lagging model with thermal memory kernel. Moreover, the detailed analyses of qubit dynamics and the associated quantum fluxes are performed within the newly developed formalism by using the concept of quantum advection modes.

t   jNE ( t  )     ( t  t ' )T( t ' )dt ' 

Taylor expansion of nonlinear energy heat flux (RTA):

   2  3  jNE ( t  )  jE   jE  jE  jE  ... 2 6 The Cattaneo-Vernotte correction to the linear Fourier' term:



2                    jCV   jE  e          2m



Energy conservation law (source-involved case):

1  1  T    jE  q   The generalized heat equation for temperature distribution:

    1  q T  T  T  ...   ( t  t ' )T( t ' )dt '   ... 2     2

Quantum Momentum Flux

Figure 1: Momentum fluxes for qubits prepared in superposition of two states.

Definition of momentum operator in quantum mechanics:

Quantum Energy (Heat) Flux

 pˆ   i







      ˆ i   V   H    2m  2

 ˆp     pˆ d  i   d

The expectation value of total energy (Hamiltonian):

The momentum density (vector) in quantum systems:







    i      ˆ M  e  p  m        2





    M  jM   V





Under the non-steady-state conditions (during the process of pumping heat), a quantum system in the excited state relaxes via higher energy level rather than the lower one.

The continuity equation for energy (with a source term):

Ballistic Fourier’s law is still applicable to heat conduction processes in closed nanosystems with thermal conductance which strongly depends on environmental damping.

The formula for quantum energy flux:

Statistical analysis on histograms in the energy pumping process by Brownian boosts show shifted Lorentzian distribution for heat flux and Gaussian for probability flux.













     E    jE   V

The formula for quantum momentum flux (scalar):

 

All momentum and energy fluxes oscillate in time with characteristic frequencies related to initial conditions in which qubits were prepared (specific superpositions).

The energy density in quantum systems:



  d pˆ   V  F dt



 ˆ   ˆ H    Hd  i   d 



Figure 3: (a) Snapshot of the potential, (b) demonstration of ballistic Fourier’s law via linear trend on the plot of the averaged heat flux versus temperature difference, (c) thermal equilibration, (d) unusual thermal relaxation process via higher excited state, (e) histogram for energy flux (shifted Lorentzian), (f) histogram for probability flux (symmetrical Gaussian).

CLOSING REMARKS



i     ˆ   E  e  H  m        2

Integrating the above equation, we obtain Ehrenfest relation:

     jM  e       2m





The continuity equation for momentum (with a source term):

2

Energy Fluxes in Quantum Wells

Time-Dependent Schrödinger Wave Equation:

The expectation value of momentum: 

t



2         jE  e     2m



Figure 2: Energy fluxes for qubits prepared in superposition of two states.