’Three-Orders Time-Limit’ for Phase On the ’Three-Orders Time-Limit’ for Phase Decoherence in Quantum Dots Decoherence in Quantum Dots

The   notion   of   a   quantum   dot   [1–3]   comprises   various   nanometre-size   semiconductor structures,   manufactured   by   means   of   different   technologies   and   resulting   in   special limitations   on   the   carrier   dynamics   (electrons   and   holes),   as   well   as   excitations   of electron-hole  pairs  (excitons).   The  Coulomb  energy  scales  withe  QD  size  d  as  1/d  (and  is of the order of meV for QDs), while kinetic energy scales as 1/d2, which leads to the shell properties of dots, distinct in comparison to atoms (more complicated Hund-type rules for QDs), since both energies remain within mutual proportions of d, which favours Coulomb energy  for  dots  in  contrast  to  atoms  [1]  at  the  scale  of  meV  orders.   The  nanometre-scale limitations on quantum dynamics result in kinetic energy quantization,

In the case of QDs, quantization energy locates thus within a range accessible for control by means of external fields (electric and magnetic), in contrast to atoms (for the latter, quantum state control by means of external fields requires such values that are beyond the reach of present technologies).  This advantage of QDs—which are relatively easy to create due to a variety of existing technologies in addition to their parameters’ flexibility and the possibility of immersing them in various media or even creating or modifying them by means of external fields—makes them very promising objects of new nanotechnologies and spintronic practical projects. Various  semiconductor  materials  may  be  used  to  create  QDs.     Note  that  insulator  or metallic nanoparticles are also manufactured (however, collective electron liquids in metallic nanoparticles manifest distinct physical properties in comparison with semiconductor QDs, which  explains  why  metallic  nanostructures  are  not  called  QDs).   For  opto-electronic use, semiconductor  dots  seem  best-suited  due  to  their  localization  within  other  nanostructures (e.g.,  in  quantum  wells),  with  well-established  technologies  for  control  over  such  systems. Semiconductor   QDs   may   be   manufactured  by   means   of   etching   technologies   after   a high-resolution  photolithographic  process  (with  the  use  of  an  ion  or  electron  beam)  has been  applied  (ordinary  optic  lithography  with  a  resolution  of  up  to  200-300  nm  is  not sufficient).    Other  technologies  used  here  include  among  others  the  Stransky-Krastanov dot  self-assembling  method  consisting  in  the  application  of  epitaxy  layers  by  MBE  or MOCVD [MBE, Molecular Beam Epitaxy; MOCVD, Metal Organic Chemical Vapour Deposition]. Various  lattice  constants  in  successive  epitaxy  layers  result  in  the  spontaneous  creation  of nanocrystals on the ultra-thin so-called ’wetting layer’. Electrical focusing in a quantum well [1, 4, 5] consists of yet another promising technique which, despite being at an early stage (due to a lack of sufficiently precise electrodes), offers the highest dot parameter flexibility and allows for dots to switch on/off within the working time periods of devices based on them [1, 4–6].

The  possibility  of  control  over  the  quantum  states  of  carriers  in  QDs  and  their  coherent (deterministic,  controllable)  time  evolution  are  vital  for  nanotechnological  and  spintronic applications  (especially  where  this  concerns  so-called  ’single-electron’  or  ’single-photon’ devices) as well as for the quantum processing of information.  The absence of decoherence or its significant reduction up to the lowest possible level,  at least within the time periods of  control  realization,   is  essential  for  all  these  applications.     However,   decoherence  is unavoidable  due  to  irreducible  dot-environment  interactions  (there  is  no  means  of  a  dot’s total isolation).  In the case of nanostructures, QDs offer a new class of physical phenomena within the decoherence and relaxation range, entirely distinct from analogous processes in bulk materials or in atomic physics.  This is due to the characteristic nanoscale-confinement energy, reaching values close to the typical energy parameters of crystal collective excitations (the  energy  characteristics  of  band  acoustics  and  optical  phonons).   This  convergence  of energy scales results in resonance effects, which is different from what is observed in atomic physics where the scale of the atom-confinement energy is three orders of magnitude higher than the energy of crystal collective excitation,  resulting in the weak influence of phonons on atomic states (included as a very small perturbation only). Specific decoherence effects in QDs result from a strong (resonance) coupling effect between the carriers trapped in them and the sea of various types of phonons (as well as with other collective excitations, or with local degrees of freedom, e.g., related to admixtures). This is why the frequently-used notion of an ’artificial atom’ in reference to QDs is, to some extent, misleading. The  same  reasons  are  responsible  for  the  fact  that  QD  modeling  which  does  not  account for  environment-induced  collective degrees  of  freedom  may  give  rise  to  false  conclusions since  significant  hybridization-induced  (decoherence)  changes  of  energy  levels  can  reach up  to  10%.   This  reduces  the  modeling  fidelity  if  the  environmental  effects  are  neglected. Therefore,  the  current  physics  of  nanostructures  should  embrace  the  recognition  of  the complex decoherence and relaxation effects observed in QDs for trapped carriers’ spin and charge, which are essentially different from what is observed in bulk materials and atoms.

Limitations on the quantum processing of information

Unavoidable decoherence—uncontrolled quantum information leakage into the surrounding environment  due  to  the  system’s  interaction  with  the  environment—perturbs  the  ideal quantum  procedures  which  ensure  the  running  of  quantum  schemes  [7–12].   If,  however, decoherence  is  kept  below  a  certain  threshold,  quantum  error  corrections  can  be  made  by applying  so-called  ’quantum  error  correction schemes’  [13],  which  enables  the  realization of  any  quantum  procedures  of  a  quantum  computer  or  any  other  deterministic  quantum project.In classical information processing, quantum error correction consists in multiplying classical information and verifying by comparison the multiplied (redundant) classical registers with arbitrary  frequency,  errors  which  appear  from  time  to  time  are  identified  and  corrected immediately.  In the quantum case, the multiplication of quantum information is impossible (No-Cloning theorem) and quantum error correction is based on a different scheme:

•   Seeking more decoherence-resistant areas of the Hilbert space (multi-qubit states which, in a pair of qubits, record symmetrically both “true” and “false” are decoherence-resistant, e.g., singlet-type qubit states;  information (or quantum states) symmetrization requires, however,   the   multiplication   of   quantum   registers,   which   makes   decoherence   rise exponentially).

•  Attempting  the  replacement  of  an  information  carrier  for  a  more  decoherence-resistant one (e.g., temporarily, a state can be teleported onto a more resistant carrier).

In  order  to  satisfy  quantum  error  correction  requirements,  DiVincenzo  formulated  a  set of  conditions  [7,  14–16]  which  allow  for  the  possibility  of  the  implementation  of  quantum error  correction  (the  typical  decoherence  time  must  be  at  least  of  six  orders  longer  than the  typical  times  of  quantum  procedures).    None  of  the  currently  suggested  solutions for  quantum  computers  have  satisfied  these  time  restrictions.   This  situation  may  follow from  the  fact  that  the  same  interactions  which  allow  for  qubit  control  (logical  operations) are  also  responsible  for  decoherence.    The  stronger  (energetically)  the  interaction  is,  the faster  the  logical  operations  can  be  carried  out.    However,  the  same  interaction  couples the  system  with  the  surrounding  environment  and  produces  strong  decoherence  effects. In  nanotechnological  and  optical  projects  involving  quantum  computers  (multi-qubit),  the difference  in  the  time-rate  of  quantum  operations  in  relation  to  decoherence  still  do  not exceed three orders of magnitude.

However, it is expected that further intensive research in this area should result in:

   Finding another method of quantum error correction (despite great efforts, there is still no relevant solution).

•   Finding  a  combined  solution  with  qubit  conversion  (between  a  fast,  controlled  carrier and a decoherence-resistant one—unfortunately, qubit conversion is

     also   inconveniently long-lasting).

•   Finding   global,    topological   and   thus   decoherence-resistant   carriers   of   quantum information in them.

•   Braid  groups  (and  non-Abelian  anyons)—herein,  the  durations  of  logical  operations are  expected  to  be  of  30  orders  of  magnitude  greater  than  those 

   of  decoherence processes [17] (however,  this is still unclear and it is doubtful if it is  experimentally viable).

•   It is hoped that superconductive states may satisfy the DiVincenzo conditions as they have non-local properties to a significant extent.

In the case of quantum cryptography, equipment requirements [18] are more easily met in respect  to  decoherence  and  this  is  why  this  quantum  technology  (public  key  distribution via a quantum channel) has been used in practice in optical systems [fibre-optical ones over distances of 100 km (up to 1000 km), and outdoors over distances of 2 km].

Quantum dots—the prospective technology for quantum gates

The  idea  of  employing  quantum  evolution  for  information  processing  corresponds  with the  feasibility  of  the deterministic  control  over  a  quantum  system  in  order  to  execute  a previously  designed  quantum  algorithm.    However,  such  a  deterministic  evolution  (also called  ’unitary’  or  ’coherent’)  requires  a  totally  isolated  quantum  system.   Unfortunately, no quantum system can be totally isolated from the environment.  Any quantum system is susceptible  to  the  environment’s  influence.   In  consequence,  unitary  or  coherent  evolution is  perturbed,  and  quantum  information  undergoes  uncontrollable  and  irreducible  leakage into  the  environment.   Therefore,  the  feasibility  of  the  construction  of  a  scalable  quantum computer  is  seriously  hindered  due  to  decoherence  phenomena.    The  better  recognition of  decoherence  processes  in  quantum  systems  may,  however,  enable  the  development  of new technologies transcending these limitations and facilitating the attainability of quantum gates. Quantum state decoherence progresses along two channels:  relaxation,  i.e.,  quantum state annihilation; and dephasing, i.e., phase relations change within a quantum state description. Relaxation  (or  ’amplitude  decoherence’)  is  related  to  the  decrease  in  time  of  the  diagonal elements of the quantum state density matrix, whereas dephasing (or ’phase decoherence’) corresponds to the reduction of the off-diagonal elements of the density matrix.  Both types of  decoherence  are  caused  by  interaction  with  the  environment  and  they  become  more significant the stronger the interaction is. Solid-state  technology  (which  is  promising  for  new  practical  realizations  of  the  quantum processing  of  information  using  nanometre-scale  semiconductor  QDs)  is  burdened  mostly with  phase  decoherence  processes.    Both  the  charge  (i.e.,  orbital)  and  the  spin  degrees of  freedom  of  quantum  states  in  QDs  undergo  dephasing  due  to  their  environment (however,  it  should  be  emphasized  that  the  spin  degrees  of  freedom  seem  to  be  more decoherence-resistant  than  orbital  degrees  of  freedom,  since  they  are  less  susceptible  to direct  crystal  phonon-induced  interaction;  however,  spin  requires  much  longer  periods  of time-control than orbital degrees of freedom due to weaker interaction with spin).

Below we present a decoherence analysis, in particular the phase decoherence of the charge (orbital) degrees of freedom and degrees of freedom of spin of excitations localized in QDs, dealing with the issues associated with limitations on the feasibility of QIP. In the case of semiconductor QDs, decoherence is unavoidable due to strong dot-environment interaction   (there   are   no   means   for   the   perfect   isolation   of   a   dot).      In   the   case   of nanostructures—QDs  included—there  appears  to  be  a  new  class  of  physical  phenomena related to decoherence and relaxation,  distinct from analogous processes in bulk materials and atomic physics. This is due to characteristic meV-scale energy resulting due to nanoscale confinement, reaching values close to the typical energy parameters of band phonons in the surrounding medium.  This coincidence of energy scales results in resonance effects, which is different from what is observed in atomic physics.  For atoms, the incommensurability of the atom-confinement energy and phonon energy is of three orders of magnitude, resulting in a weak phonon impact on atomic states.  Specific decoherence effects in QDs result from strong  and  resonant  coupling  between  the carriers  trapped  in  dots  and  the  sea  of  various types  of  collective  excitations  in  the  surrounding  medium,  which  highly  modifies  the  QD states.   Hybridization-induced  changes  of  energy  levels  reach  up  to  10%.   Therefore,  the decoherence and relaxation effects observed in QDs and for trapped carrier spin or charge (which are essentially different from what is observed in bulk materials and atoms), seem to be of central importance for any possible QD applications, including QIP.

Phase decoherence of orbital degrees of freedom in nanostructures

Orbital  degrees  of  freedom  pertain  to  charge-type  excitations,  such  as  electrons  and  holes and charge-balanced electron-hole pairs—excitons. As charge carriers, these excitons interact with the electric field of the electromagnetic wave and so they can be controlled by means of  quantum  optics  methods.   Charge-type  excitations  can  be  localized  in  nanometre-scale artificial  structures  manufactured  within  various  semiconductor  heterostructures,  namely in   QDs.      Excitons   attract   special   interest   as   they   can   be   precisely   controlled   by   an electromagnetic wave within the visible (or near infrared) light range corresponding to the typical energy gap separating electron states from hole states in semiconductors (a typical material  is  GaAs  and  QDs  will  be,  e.g.,  self-assembled  structures  of  GaAs/InAs  type). By  accommodating  the  energy  of  (incident  light)  photons  with  the  energy  of  the  exciton, an  exciton  state  in  the  Rabi  oscillation  regime  can  be  created  in  which  the  superposition state  of  the  charge  qubit  spanned  on  the  states  |1  >  (no  exciton  in  a  QD)  and  |2  >  (one exciton  in  a  QD)  can  be  selected.   The  techniques  of  ultra-high-frequency  laser  impulses (measured in femtoseconds) and the resulting application of a high intensity beam allowing for  high-frequency  Rabi  oscillations  [19]  has  attracted  a  lot  of  interest  in  QIP  research. This  interest  has  been  centred  on  the  fact  that  the  lifetime  of  the  excitons  in  the  dots  is measured in nanoseconds (this may suggest a difference of six orders of magnitude between the  control  time  and  the  amplitude decoherence  time,  which  is  required  by  DiVincenzo’s criteria [7, 13–16, 20]). Nevertheless, in QDs the interaction between the excitons (electrons and holes) and phonons of the surrounding crystal is unavoidable and must be accounted for in all considerations, thus  diametrically  changing  this  ostensibly  convenient  situation.   Phonons  are  quanta  of the  crystal  oscillations;  acoustical  phonons  refer  to  the  oscillations  of  the  density  type  (all the  atoms  in  the  unit  cell  oscillate  in  the  same  direction)  and  optical  phonons  are  related with polarization oscillations (the opposite-sign ions in the unit cell oscillate in the opposite directions; polarization oscillations in such ionic crystals can be excited by means of light, and thus they are called ’optical’ phonons). Both types of phonons can interact with charge-type degrees of freedom in QDs.  Phonons can be of transversal or longitudinal polarization, but these are the longitudinal modes (LO and LA for optical and acoustic phonons, respectively) that contribute most substantially to the interaction with the electrons/excitons [21].

In  polar  materials  (e.g.,  GaAs,  a  weakly  polar  semiconductor),  LO  phonon  interaction prevails.    The  interaction  of  charges  with  LO  phonons  is  characterized  by  means  of  the dimensionless Fröhlich constant [21, 22]. The higher the constant value, the stronger that the interaction  is  between  the  charges  and  the  LO  phonons,  and  for  the  semiconductor  GaAs three-dimensional  (bulk)  the  constant  averages  out  at  around  0.06.    For  QD  GaAs/InAs, experiments  (infrared  absorption  in  a  magnetic  field  and  the  broadening  of  the  satellite luminescence  peak  connected  to  LO  phonons,  expressed  quantitatively  via  the  so-called ’Huang-Rhys factor’ [23]) show a double value of the constant, which suggests a substantial increase of the interaction with LO phonons. This phenomenon has been explained [24] with regard  to  a  certain  ambiguity  [21]  in  the  definition  of  LO  phonon-electron  interactions  in crystals.  The  interaction  between  an  LO  phonon  and  an  electron  leads  to  the  polarization of the crystal lattice by the moving electron.  This polarization (i.e., an appropriate packet of optical phonons) is dynamic and leads to a reverse interaction with the polarization-inducing electron.    It  can  be  thought  of  as  being  composed  of  two  components:   an  inertial  one, which  lags  behind  the  moving  electron;  and  a  non-inertial  one,  which  accompanies  the moving electron.  The latter component should be contained in the total crystal field which defines the electron itself (the electron in the crystal is not a free particle and includes, by its definition, the periodic crystal field—thus it can be characterized by the effective mass and quasi-momentum instead of momentum).  The necessity of extracting only the inertial part of the polarization from the total interaction of the electron and the LO phonons leads to the above-mentioned ambiguity in the definition of the electron-LO phonon interaction.  When the electron is trapped in a QD, it moves with a quasi-classical velocity [25] which exceeds the velocity of a free band electron.  Thus, it better escapes from the dynamic polarization, which results in an increase in the inertial part of polarization and the interaction between the electron and LO phonons in QDs.  The more localized the electron in a smaller QD, the bigger the quasi-classical velocity of the electron and the bigger the increase in the interaction with LO phonons. The quantitative analysis of the problem agrees well with the experimental data. It should be emphasized, however, that the marked increase in the value of the Fröhlich constant in QDs parallels the increase in the decoherence of electron/exciton states in dots due to the increase in the interaction between the small system of the QD and the sea of LO phonons in the crystal.

The energy scale corresponding to the nanometre localization of electrons (excitons) in QDs ranges  from  a  few  to  several  tens  of  meV.  The  same  energy  scale  also  characterizes  the phonons in crystals,  in which the energy of LA phonons ranges from 10 to 20 meV at the edge of the Brillouin zone and the energy of LO phonons at the centre of the Brillouin zone (a gap in the LO phonon spectrum at point Γ [21, 22]) reaches 30 meV. Thus, in the case of QDs we deal with a strong coupling regime for an interaction of QD charge degrees of freedom with  phonons  (of  all  types).   The  same  energy  scale  of  both  types  of  excitations—local  in QDs and collective in the surrounding crystal—results in the strong mutual hybridization of these excitations or in the dressing of electrons (holes) or excitons with phonons and in the creation of composite particles (quasi-particles)—polarons [21, 22, 26–29]. The creation of polarons in QDs is a strongly decoherent process (much more than it is in bulk materials).  The dynamics of this process can be investigated by employing the Green function technique [28].  By means of this technique, the correlation function of the exciton (electron) in the QD can be expressed, which defines the overlap (the scalar product) of the state of the carrier gradually dressed by phonons with the initial state of the bare exciton (or electron) in the dot. Thus, it is possible to quantitatively characterize the leakage of quantum information (fidelity loss) due to the entanglement (in a quantum sense) of the QD’s charge with the deformation and polarization degrees of freedom of the whole crystal,  which are entirely beyond our control.

The inertia of the crystal lattice is so disadvantageous that it makes it impossible to maintain the coherence of orbital degrees of freedom dynamics (the unitary quantum evolution of the excitations) within the time periods required by the DiVincenzo conditions. The typical times of dressing charge-type excitations with phonons are located within the time-range of single picoseconds,  which  is  the  middle  of  the  six-orders  time  window  between  the  amplitude decoherence  time  for  excitons  in  QDs  and  the  time-scale  of  the  quickest  techniques  for their  excitation.   On  both  sides  of  this  window,  there  appear  windows  of  three-orders  of magnitude,  which  precludes  the  implementation  of  the  quantum  error-correction  scheme due to the non-fulfilment of the DiVincenzo conditions. These  strongly  unfavourable  estimations  indicate  that  it  may  be  impossible  to  scale  a quantum  computer  in  a  QD  technology  with  only  by  light  control  unless  more  effective quantum error-correction schemes would be proposed [7, 13, 14].

It should be emphasized that LA phonons are of greater importance to the process of dressing the excitons with phonons (polaron decoherence effects), despite the fact that their interaction with  excitons  is  energetically  much  weaker  (at  least  by  one  or  two  orders  of  magnitude) than in the case of LO phonons.  Strong dephasing due to LA phonons corresponds with a wide linear dispersion of acoustic phonons,  which in turn leads to a more immediate and significant induced change in the wave functions of charge-type excitations in QDs than in other phonon modes.LA  phonons-induced decoherence  (phase decoherence  or  dephasing,  corresponding  to the reduction of the off-diagonal elements of the density matrix [8–12]) is—and as can be shown by means of a microscopic analysis—a relatively fast process and its time-scale is of the order of  the  ratio  of  the  dot  diameter  and  the  sound  velocity  (it  is  of  the  order  of  picoseconds). Acoustic phonons are especially inconvenient as they are present in any crystal (as well as in any amorphous material),  and this is why the above-presented mechanism of decoherence is  unavoidable  by  its  nature  [strong  dephasing  also  exists  at  a  temperature  of  0  K  due  to phonon emission; at higher temperatures, the dephasing effects are enhanced due to phonon absorption effects, which become more important with the increase in temperature].Strong decoherence restrictions on the quantum evolution of the charge degrees of freedom in QDs encouraged the researchers to concentrate their attention on the spin degrees of freedom in nanostructures (spin does not interact directly with phonons) instead of pursuing the idea of constructing an quantum computer based on QDs that is only controlled by light [7, 14, 28].

Phonon-induced dephasing of excitons localized in quantum dots

An  exciton  created  in  a  QD  by  means  of  an  non-adiabatic  process  (in  the  sub-picosecond order)  [4,  5,  30]  is  a  bare  particle  (an  electron-hole  pair)  which  is  gradually  dressed  with phonons until it becomes a polaron.  The time within which the polaron is created depends upon the lattice inertia.  It is relatively long and its accurate evaluation is an important task. The  process  of  the  hybridization  of  a  QD-localized  exciton  with  the  collective  excitations of  the  crystal  lattice  surrounding  the  QD  is,  in  fact,  a  time-dependent  evolution  of  a non-stationary  state,  which  at  the  initial  time  (the  moment  of  the  excitation’s  creation)  is identical  with  the  state  of  the  bare  exciton.   The  bare  exciton  is  not  the  stationary  state of  the  whole  system,  the  QD  exciton  and  the  sea  of  phonons  in  the  surrounding  crystal interacting with it (a polaron represents a stationary state of such a complex system).  The non-stationary initial state (the bare exciton) [the electric field of the e-m wave interacts with the charge and, in consequence, excites a bare electron from the valence band into the QD; the resulting hole is also captured by the QD—a bare QD exciton is thus created] undergoes further non-stationary evolution.  In the non-stationary state, the energy is not determined; however,  the  mean  energy  is  shared  over  time  between  the  subsystems,  the  QD  and  the phonon sea. The mean energy of a bare QD exciton is higher in comparison with the polaron energy (whose energy is lower and therefore the polaron is created by means of interaction with phonons due to energy minimization). The excess energy of the lattice deformation (for acoustical  phonons)  together  with  the  polarization energy  (for  optical  phonons)  is  carried outside  the  QD  by  the  LA  and  LO  phonons,  respectively  (by  their  wave  packets).   A  QD polaron  is  created—a  hybridized  state  of  an  exciton  dressed  with  an  LA  and  LO  phonon cloud [actually, the name of the polaron refers to electrons dressed with LO phonons [22]—a process dominating in strongly polar materials; here, the name refers generally to an electron or exciton dressed with all types of phonons].  The time-scale of QD polaron creation is of the same order as the time that a phonon-wave packet needs to leave the QD area. It should be emphasized that this process is not to be interpreted in terms of Fermi’s golden rule [in such an approach, quantum phase transitions resulting from a time-dependent perturbation refer to transitions between stationary states,  which is not the case here] [25].  The process of polaron creation is a non-stationary state evolution, in which the elementary processes of phonon absorption or emission contribute in the virtual sense (without energy conservation). Note that the polaron energy is shifted with respect to the bare QD exciton energy by a few meV [28], while the LO phonons energy has a much greater gap, h¯ Ω ‘ 36, 4meV (in GaAs). The kinetics of polaron creation correspond with the coherent evolution of an entangled state of two interacting systems, namely a QD exciton and the sea of phonons (of various types), and this state in non-separable [28].

The exciton-phonon system is represented by the following Hamiltonian:

Decoherence of the degrees of freedom of spin in quantum dots

Spin  do  not  interact  directly  with  with  phonons—the  spin  of  the  QD  excitations  interacts weakly with the lattice oscillations due to their links to orbital (charge) degrees of freedom via:

•   spin-orbit coupling [43],

•   specific Hund-like rules for multi-electron QDs [1]—the filling of the subsequent shell in the  multi-electron  QD  depends  upon  the  total  electron  (hole)  spin  of  a  given  shell  [the generalization of singlet and triplet states] which, in effect, link spin and orbital degrees of freedom.

Weak  spin  coupling  with  phonons  suggests  that  the  spin  of  a  QD  electron  constitutes  a well-isolated quantum system (an insignificant spin-orbit interaction results in an extremely slow spin decoherence, the same is due to weak interaction with nuclear spin) that is suitable for a qubit’s definition.  One can expect that for spin qubits in QDs, DiVincenzo’s conditions would  be  satisfied  [7,  14,  15].   Due  to  the  minor  influence  of  the  surrounding  medium, the QD’s spin coherence is maintained until the time period of order of µs [16].  However, a  difficulty  arises  when  Rabi  oscillations  are  implemented  (for  single  qubit  operations). Because of the low value of the gyromagnetic factor in semiconductors, qubit spin control (a qubit spanned across two spin orientations in an external constant magnetic field) via Rabi oscillations is extremely slow and the DiVincenzo conditions are again not satisfied (the Pauli term, gµBszB, leads to very slight Zeeman splitting of only 0.03 meV/T, in GaAs)

For two-qubit operations of spin qubits,  no such disadvantage exists—there is an effective procedure  for  switching  spin  qubit  interactions  on  and  off  [15,  16]  resulting  in  qubit entanglement control at the time-scale of picoseconds.  The idea of spin interaction control follows from the phenomenon of exchange interaction between two spins, being induced by strong Coulomb interaction [44].  The exchange energy for it is the singlet-triplet energy gap for the spin pair [44],  and consequently it is of (several) meV in magnitude,  resulting in a picoseconds  time-scale  for  the  control  of  the  entanglement  of  qubits.   The  scheme  of  this control relies upon the singlet and triplet states of an electron pair (each electron captured in an individual QD but located closely enough to maintain their quantum indistinguishability [their localized wave functions must overlap]) and their relation with the orbital structure of the corresponding wave functions. Due to the fermionic nature of electrons:

Dephasing induced by the dressing of QD exciton spin with magnons in a diluted magnetic semiconductor’s surroundings

An  interesting  question  arises  with  regard  to  the  QD  spin  in  the  magnetic  surroundings when  the  Pauli  term  causing  the  Rabi  oscillations  for  spin  can  be  strengthened  due  to  an increase in the effective gyromagnetic factor.  Spin does not interact directly with phonons, and  thus  it  is  free  from  phonon-induced  dephasing.   Nevertheless,  the  dephasing  role  of phonons may be played by spin waves in magnetically-ordered media which, on the other hand,  are  convenient  for  accelerating  single-qubit  QD  spin  control  to  the  level  required by  the  DiVincenzo  conditions.    Spin  waves  (frequently  called  ’magnons’)  are  collective spin-type  excitations  in  the  ferromagnetic  or  anti-ferromagnetic  medium  (or  in  any  other magnetically-ordered  spin  system),  and  possess  similar  band  properties  to  phonons  in crystalline structures. The spin-exchange interaction between the magnons and the local QD spin (of an exciton trapped in a QD) is relatively strong and causes the dressing of the QD spin with the magnons in a similar fashion to the dressing of the QD charges with phonons. The  opportunity  for  the  experimental  study  of  such  a  spin  dressing  phenomenon  may  be linked to the so-called ’diluted magnetic semiconductors’ of the type III-V (e.g., Ga(Mn)As) or  II-VI  (e.g.   Zn(Mn)Se).   In  these  magnetically  and  weakly  doped  semiconductors,  some relatively small part of the cations (usually a few %) is randomly substituted by transition metal ions (typically of Mn).  The admixture spins interact with the spins of band holes and as a result the ferromagnetic ordering of the admixture spins is observed.  The related Weiss field enhances the effective gyromagnetic factor in the Pauli term, describing the spin action of the external magnetic field conveniently for the acceleration of the control over the local QD spin.

Spin waves in the diluted magnetic semiconductor

To  describe  the  dephasing  of  QD  spin  caused  by  magnons  in  DMSs  quantitatively,  the analytical expression of the spin wave spectrum in the DMS is needed.  This can be found in the paper [64].  For the relevant theoretical description of the spin subsystem of the DMS, the  model  of  dopant  spin  exchange  mediated  by  band  holes  is  utilized  [58,  59],  assuming


In  conclusion,   we  can  state that  in  the  case  of  QDs  we  deal  with  a  specific  type  of phonon-induced phenomenon, which is essentially distinct from the phonon-induced effects in bulk semiconductors.  This difference is caused by the compatibility of the energy scale for  carriers  trapped  in  QDs  with  the  band-phonon  energy  scale.    Owing  to  this  energy coincidence,  the coupling of carriers in QDs with phonons always meets its strong regime limit.  This coupling cannot be treated perturbatively,  in general,  and the resonance effects are of primary importance, resulting in strong polaron-type modifications of the QD electron and exciton spectra.  The typical energy shift due to the formation of electron-polaron is of the order of 10%, while for exciton-polaron it is of the order of 5% with respect to the bare energy levels in QDs treated as separated from the environment. The confinement of carriers, as  in  the  case  of  QDs,  also  causes  the  significant  enhancement  of  the  effective  Fröhlich constant due to non-adiabatic effects, which additionally strengthens the electron-LO phonon interaction.  The dressing of the electrons/holes/excitons in QDs with band phonons from the  surrounding  crystal  induces  the  dephasing  of  charge  (orbital  degrees  of freedom)  in QDs  (the  off-diagonal  decoherence).   The  typical  time-scale  of  this  dressing  process  in  the case  of  the  formation  of  an  exciton-polaron  in  QDs  turns  out  to  be  of  the  one  picosecond time-scale  (for  a  typical  QD of  10  nm  diameter).   This  dephasing  is  caused  by  the  exciton dressing  with  LA  phonons.    Worth  noting  is  the  observation  that  the  dephasing  due  to LO  phonons  is  considerably  smaller  and  slower—of  the  100  ps  scale.   This  phenomenon is  caused  by  the  relatively  weak  LO  phonon  dispersion  near  the  Γ  point  (in  the  Brillouin zone), in contrast to the dispersion of LA phonons.  Nevertheless, the outflow of the excess polarization  energy  to  the  space  region  outside  the  QD,  as  a  typical  process  during  LO polaron formation, is eventually accelerated by the anharmonic coupling (LO-TA is the most important anharmonic channel in GaAs), which results in a few ps time-scale. It is important to note that these effects of QD-charge dephasing by band phonons refer not only to QDs but also to all nanostructures in solids, because the carrier localization (the space-confinement of trapped carriers) plays the essential role here.

We  have  observed  also  that  in  magnetically-ordered  media  (like  in  DMSs),  magnons  (spin waves)play  a  similar  destructive  role  to  phonons.   Spin  waves  cause  the  dephasing  of  the exciton  spin  in  QDs,  in  a  process  of  the  formation  of  excitons-magnetos-polarons by  the step-by-step  dressing  of  the  local  spin  of  exciton  in  QD  with  the  magnon  cloud  from  the surrounding magnetically-ordered medium. By using the Green function technique, we have estimated the time for the dressing of the local exciton spin with band magnons in the case of DMSs which surround QDs by analogy to the dressing of QD charge (i.e., QD orbital degrees of freedom) with band phonons from the surrounding crystal.  Nevertheless, the significant difference  between  these  two  phenomena  is  observed  and  elucidated,  namely,  in  the  case of  spin  dressing  two  magnons  are  needed  (creation  and  annihilation)  owing  to  the  spin conservation in the interaction vertex, which results in the complete freezing (vanishing) of the spin pure dephasing at T = 0. This is in contrast to the phonon-induced pure dephasing, which maintains strong even at T = 0.The  dephasing  scheme  for  the  exciton  charge  and  spin  in  QD  structures  is  important  for the feasibility assessment of QIP implementations in QDs. The picoseconds time-rate for QD charge dephasing probably precludes the feasibility of the implementation of error-correction schemes for all optically-controlled gates in QD technologies. The dressing of a localized spin with magnons in a DMS (i.e., the time corresponding to the formation of EMP in a QD) takes place at a longer time-scale in comparison to dressing charges with phonons, and the related time-scale is of the order of 150-200 picoseconds due to the relatively weak quadratic magnon dispersion,  similar to as was the case for LO phonons.  Nevertheless,  the time-rate for QD spin dephasing induced by magnons in the surrounding DMS is also inconvenient for QIP applications, similarly to the case of the dephasing of QD charges by phonons.  The overall time-scale for QD spin kinetics (QD/DMS-embedded structures) is shifted by three-orders of magnitude to longer periods in comparison to QDs’ orbital degrees of freedom,  though again with the same inconvenient dephasing time-rate falling right in the middle between the  control-timing  and  the  relaxation-timing  (which  does  not  allow  the  satisfaction  of  the DiVincenzo conditions).  In this way,  the ’three-orders time-limit’ caused by the dephasing phenomena is repeated for spin in the QD/DMS.

The pure dephasing of spin in QD/DMS structures disappears,  however,  at  T  =  0,  and is strongly suppressed in amplitude at low temperatures (in contrast to the charge dephasing), which supports expectations of some advantages of spin degrees of freedom in QDs for QIP applications. that the  p − d exchange between the band holes and the impure magnetic atoms may lead to  the  ferromagnetic  alignment  of  the  magnetic  dopants  (Mn).   Note  that  the  holes  taking part in the spin exchange with the dopant Mn ions cause an indirect exchange beyond the weak,  direct,  short-range  and  anti-ferromagnetic  exchange  between  the  magnetic  dopants [48]. The hole-induced indirect coupling, even for low concentrations of holes xp  (lower than the magnetic dopant concentration x) occurs strong and leads to the ferromagnetic ordering of  Mn  spins  even  at  relatively  high  temperatures,  ∼  110  K  in  Ga0.947Mn0.053As  [58,  60]. Let  is  be  emphasized  that,  in  III(Mn)V  DMSs,  the  Mn  atoms  of  the  magnetic  dopants  are simultaneously shallow acceptor centres, whereas in II(Mn)VI-type DMSs, the Mn dopants are not acceptors and the holes must be supplied by additional p doping. To  briefly  sketch  the  derivation  of  the  magnon  spectrum  in  DMSs,  let  us  set  out  the Hamiltonian for the DMS system with magnetic dopants Mn2+  (with spin S  =  5/2) in the form:

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