# Deterministic generation of high-dimensional entanglement between distant
atomic memories via multi-photon exchange

A. Gogyan<EMAIL_ADDRESS>Institute for Physical Research, Armenian
National Academy of Sciences, Ashtarak-2, 0203, Armenia  S. Guérin
<EMAIL_ADDRESS>Laboratoire Interdisciplinaire Carnot de Bourgogne,
CNRS UMR 6303, Université Bourgogne Franche-Comté, BP 47870, F-21078 Dijon,
France  Yu. Malakyan<EMAIL_ADDRESS>Institute for Physical Research,
Armenian National Academy of Sciences, Ashtarak-2, 0203, Armenia

###### Abstract

Promising access to high-speed quantum networks relies on the creation of
high-dimensional entangled memories that provide quantum communication with
higher capacity of noisy quantum channels, thereby reducing the transmission
time of information. Yet, the distribution of multidimensional entanglement
between remote memory nodes is still faintly investigated. We propose an
experimentally feasible protocol of deterministic generation of high-
dimensional entanglement between distant multi-level atoms confined in high-
finesse optical cavities and driven by laser pulses. Three-dimensional
entanglement is generated deterministically between remote atoms by triggering
a two-photon wavepacket, which mediates a superposition of states chosen by
the laser pulse parameters. The efficient transfer of atomic states between
remote nodes allows the construction of a three-dimensional quantum repeater,
where the successful creation of entanglement can be verified by a simple
method for reliable measurement of atomic ground Zeeman states.

###### pacs:

42.50.Ex, 03.67.Lx, 42.50.Pq, 32.80.Qk

##  I INTRODUCTION

Quantum entanglement is a fundamental resource to realize quantum networks 1 ;
2  and long-distance secure key distribution 3 ; 4 . In the quantum internet 5
; 6 , quantum repeaters (QRs) are a promising approach, enabling long-distance
quantum communication over lossy channels, where the entanglement distribution
between distant matter nodes is usually obstructed by the exponential loss of
photons and ubiquitous decoherence over communication line. The QRs comprise
local quantum memories, which are designed to store, process and release the
quantum information in a reversible manner, while quantum channels link the
nodes by transmitting photonic states with high transfer fidelity 7 ; 8 ; 9 ;
10 . A number of QR protocols have been proposed during two last decades 9 ;
10 . However, the predicted times in these schemes for the overall
entanglement distribution at large distances are very long compared to the
realistic quantum memory lifetimes. The communication rate is limited not only
by the time required to correct the operational errors and photon losses, but
mainly due to the probabilistic nature of both the entanglement generation
caused by ineffective atom-photon coupling in matter nodes and entanglement
swapping due to the incomplete Bell-state detection 9 ; 10 ; 11 ; 12 ; 13 . As
a result, a desired outcome is obtained only after many unsuccessful attempts
necessitating the need for long-lived quantum memories. To date, the efficient
coupling between a single photon and quantum emitters has been implemented in
optical cavities 14  and in one-way all-photonic QRs where the emitter spin
and photons are deterministically coupled 15 ; 16 ; 17 ; 18 ; 19 . Note,
however, that the complexity of all-photonic structures yet poses a major
obstacle to the realization of large-scale QRs. Another way to significantly
improve QR efficiency, which is widely studied in recent years 20 ; 21 ; 22 ;
23 ; 24 ; 25  (for an overview see 24 ), is the creation of high-dimensional
(HD) entangled memories and photonic states offering high data capacity and
noise immunity in quantum communication. In this regard, the orbital angular
momentum (OAM) states of photons provide ample opportunities 23 ; 24 ; 25 ; 26
, since the OAM mode-space is practically unbounded, which allows encoding
information with very high degrees of freedom. Nevertheless, the
reconstruction of the density matrix for high-dimensional entanglement of OAM
states is difficult due to the growing amount of measured quantities as the
system increases in size.

Our proposal to overcome these challenges is to construct a QR protocol that
combines (i) the deterministic generation of high-dimensional entanglement
over the elementary links via multi-photon exchange and (ii) near-
deterministic swapping of entanglement between the entangled pairs. In this
paper, we focus on the scheme of entanglement generation between two distant
nodes.

The proposed scheme consists of single $F$-Zeeman-structured multilevel atoms,
with $-F\leq m_{F}\leq+F$, trapped in high-finesse optical cavities. This
system has an important advantage compared to the ensemble-based protocols, in
which the deterministic storage and retrieval of a single photon is
deteriorated by multi-photon errors. The latter are amplified, when the
probability of conversion of the atomic spin excitation into photons is
increased thanks to collective interference. This imposes significant
limitations on the performance of the protocols based on a single-photon
detection 9 .

The cavity enhanced atom-photon interaction opens a way for efficient
interconversion of photonic and atomic states enabling faithful transfer of
quantum states between remote atoms in a small-scale network. Here we focus on
two-photon transfer, because in this case, as shown in a recent work on the
deterministic transfer of photonic qutrit states between two distant atoms 27
, the information is not less protected from transmission losses even over
large distances than the photonic polarization qubit in conventional
protocols.

For alkali atoms, the corresponding configuration of atom-photon interaction
is shown in Fig. 1 for the $S_{1/2}(F=1)\rightarrow P_{1/2}(F^{\prime}=2)$
transition, where the upper states $F^{\prime}=2$ and $F^{\prime}=1$ are well
separated. In the sending node A, the atom is initially prepared, e.g., in the
ground state $m_{F}=-1$ (Fig. 1, left) and interacts with a linearly polarized
control laser pulse $\Omega_{1}(t)$ and a $\sigma^{-}$-polarized cavity mode
in Raman configuration with intermediate state $F^{\prime}=2$. Hereafter, all
physical quantities in the first and second cavities are labeled with indices
1 and 2, respectively. This interaction coherently distributes the atomic
population over the Zeeman sublevels $m_{F}=-1,0,1$ with different amplitudes,
meanwhile generating two photons in the cavity. Atomic population amplitudes
and photon temporal profiles are deterministically controlled by the laser
pulse under realistic high signal-to-noise ratio. An external magnetic field
applied parallel to the cavity axis removes the Zeeman degeneracy to finely
tune the $\sigma^{-}$-polarized cavity photons to the two-photon Raman
resonance on the $m_{F}\rightarrow m_{F}+1$ transition, as well as to quench
the decoherence of the atomic superposition state by ambient magnetic fields.
The cavity photons are efficiently coupled to a low-loss single-mode optical
fiber as a multi-photon wave packet. The fiber transmits it to the receiving
system B (Fig. 1, right), where a single atom, initialized in the state
$m_{F}=+1$, interacts with a linearly polarized $\Omega_{2}(t)$ laser pulse.
It induces a controlled reversed absorption of the incoming photons with unit
probability, thus creating a three-dimensional entanglement between three-
level ground states of the sending and receiving atoms. The shape and
intensity of the $\Omega_{2}(t)$ laser pulse is derived from the key
requirement to entirely exclude the photon leakage from the second cavity
indicating that the quantum output field is zero at all times. Importantly,
the absorption amplitudes of the photons in the receiving node are not
sensitive to their spectral properties due to the integral dependence on the
photon temporal profiles (see Sec. IIIA). Under these conditions, the
entanglement generation between nodes is much faster than the decoherence time
of entanglement. To the best of our knowledge, this proposal is the first
protocol for entanglement creation between atomic qutrits, which can be
directly extended to higher dimensional $(d>3)$ atomic states using the
results of Ref.31 .

Different from the probabilistic protocols 9 , the entanglement between two
distant memories is implemented in our scheme by direct transfer of photons,
and does not arise in a ”heralded” way. At the same time, the heralding of the
successful entanglement makes the process resistant to operational errors such
as fluctuating photon arrival times and detector inefficiency 9 ; 12 ; 13 ; 28
; 29 . However, the implementation of this approach in our case of multiphoton
absorption is not yet available. Instead, to verify the successful storage of
incoming photons into the second node we propose a simple method for reliably
measuring the ground state of the receiving atom.

Figure 1: (Color online) (a) Entanglement generation between remote atoms via
two-photon exchange. The insets show the atomic level structure and
interaction of the atoms with laser fields $\Omega_{1,2}$ and external
magnetic field B applied along the cavity axis. The sequence of
$\sigma^{-}$-polarized cavity photons (thin red lines) are generated in the
left and absorbed in the right cavities. The initial populations of atomic
states are shown by filled circles

This paper is organized as follows. In the next section, we present the
interaction setup and solve the evolution equations for the atomic state
amplitudes and the cavity field. The necessary conditions for deterministic
generation of $\sigma^{-}$-polarized cavity photons in the sending node A are
derived. We calculate the populations of ground Zeeman states of the atom for
different durations of the Gaussian pumping laser and obtain the temporal
profiles of emitted photons. These data are important to control the
entanglement between the atoms. Here we find also the final state of quantum
field produced from the node A. In Sec. III, the complete absorption of
incoming photons at the node B is calculated under strongly prohibiting the
photon loss from the second cavity. We demonstrate the creation of three-
dimensional entanglement between the atoms with maximally entangled two-qutrit
state. Here we also present a new scheme for measuring the atomic ground
states, which is designed for testing the robustness of our scheme against the
noisy local operations and photon losses in the quantum channels. We summarize
the results in Sec. IV.

##  II DETERMINISTIC PRODUCTION OF TWO-PHOTON PROPAGATING STATE FROM THE
SENDING CAVITY

###  II.1 Model

To describe the production of the cavity photons in the sending node A in Fig.
1, we use the method for deterministic generation of a stream of multiphoton
pulses in a single atom-cavity system proposed in 30 ; 31 . The off-resonant
Raman scattering of the linearly polarized $\Omega_{1}$ laser field generates
$\sigma^{-}-$polarized cavity photons on the transitions
$|F^{\prime}=2,m_{F^{\prime}}=m_{F}\rangle_{1}\rightarrow|F=1,m_{F}+1\rangle_{1}$
with the coupling strength $g$, thus transferring the atom into the next
ground-state Zeeman sublevel $m_{F}+1$. The quantization axis is taken
parallel to the cavity axis and to the external magnetic field, which is
chosen strong enough to prevent the generation of $\sigma^{+}-$polarized
cavity photons due to their off-resonant interaction with the atom. The one-
photon detunings $\Delta$, taken the same for both atoms in the nodes A and B,
are chosen (i) small enough compared to the upper level hyperfine splitting,
but (ii) much larger than the cavity decay rate $k$, the natural spontaneous
decay rate $\gamma_{\text{sp}}$ of the atom and the Rabi and Larmor
frequencies: $|\Delta|\gg
k,\gamma_{\text{sp}},\Omega_{1}(t),|\Delta_{B}^{(F,F^{\prime})}|$. Here
$\Delta_{B}^{(F,F^{\prime})}=g_{L}^{(F,F^{\prime})}\mu_{B}B/\hbar$ is the
Zeeman splitting of the ground and excited states in the magnetic field $B$,
with $g_{L}^{(F,F^{\prime})}$ the Landé factor and $\mu_{B}$ the Bohr
magneton. These conditions allow one to neglect the spontaneous losses from
upper levels and dephasing effects induced by other excited states. In this
far off-resonant case and for slowly varying laser fields:
$d\Omega_{1}(t)/dt\ll\Delta\Omega_{1}(t)$, one can adiabatically eliminate the
upper atomic states and define the effective Raman atom-photon coupling

$G_{1}=g\Omega_{1}/\Delta.$ (1)

We describe the laser pulse as

$E_{1}(t)={\mathcal{E}}_{1}f_{1}^{1/2}(t)\exp(-i\omega t)+h.c.,$ (2)

where $f_{1}(t)$ features its temporal profile of duration $T_{1}$ and
${\mathcal{E}}_{1}$ is the peak amplitude defining the one-photon detuning
$\Delta=\omega_{FF^{\prime}}-\omega$ and the peak Rabi frequency
$\Omega_{1}=\mu_{F,F^{\prime}}{\mathcal{E}}_{1}/\hbar$. For simplicity, we
assume here that the dipole matrix elements $\mu_{F,F^{\prime}}$ of the
$|F^{\prime},m_{F^{\prime}}\rangle\rightarrow|F,m_{F}\rangle$ transitions do
not depend on the magnetic quantum numbers. This actual dependence can be
easily taken into account in the numerical simulations as is shown in Ref. 31
. Hereafter the phase of ${\mathcal{E}}_{1}$, serving as the phase reference,
is taken zero without loss of generality.

The process of producing photons from the sending node is deterministic if it
can be controlled by the laser pulse $\Omega_{1}(t)$, for which two basic
requirements must be met. First, the generated photons must leave the cavity
before they can be reabsorbed by laser-stimulated Raman scattering on
$|F=1,m_{F}+1\rangle_{1}\rightarrow|F=1,m_{F}\rangle_{1}$ transitions, that
is, the two-photon Rabi oscillations with frequency $G_{1}$ must be slower
than the rate of photon leakage from the cavity

$|G_{1}|<k.$ (3)

Secondly, all relaxation processes including the optical pumping from
$|F,m_{F}\rangle_{1}$ into states $|F,m_{F}\pm 1\rangle_{1}$, and the loss of
atomic populations due to spontaneous decay of the upper state
$|F^{\prime}=2\rangle_{1}$ into the states outside the system should be
negligible compared to the cavity photon generation rate
$\alpha_{1}=4G_{1}^{2}/k$, that is $\alpha_{1}f_{1}(t)\gg\Gamma_{1}(t)$, where
$\Gamma_{1}(t)=\frac{\Omega_{1}^{2}}{\Delta^{2}}f_{1}(t)\gamma_{\text{sp}}$ is
the total spontaneous decay rate induced by the $\Omega_{1}(t)$ pump pulse 31
. This defines the signal-to-noise ratio or the single-atom-cavity
cooperativity $C_{c}$ 32

$R_{\text{sn}}=C_{c}=\frac{\alpha_{1}f_{1}(t)}{\Gamma_{1}(t)}=\frac{4g^{2}}{k\gamma_{\text{sp}}}\gg
1.$ (4)

This condition is clearly fulfilled in high-finesse optical cavities with
$g>k,\gamma_{\text{sp}}$.

After adiabatic elimination of the upper states, the effective interaction
Hamiltonian in RWA takes the form

$\displaystyle H$
$\displaystyle=\hbar\Biggl{[}\sum_{m_{F}=-F}^{F}\biggl{(}\frac{g^{2}}{\Delta}a_{1}^{{\dagger}}a_{1}+f_{1}(t)\frac{\Omega_{1}^{2}}{\Delta}\biggr{)}\sigma_{m_{F},m_{F}}$
$\displaystyle+G_{1}f_{1}^{1/2}(t)\sum_{m_{F}=-F}^{F-1}\biggl{(}a_{1}^{{\dagger}}(t)\sigma_{m_{F}+1,m_{F}}(t)+h.c.\biggr{)}\Biggr{]},$
(5)

where $\sigma_{i,j}(t)=|i\rangle_{1}\langle j|_{1}$ and
$a_{1}(t),a_{1}^{\dagger}(t)$ are the atomic and photonic mode operators in
the first cavity, respectively. The two first terms in Eq. (5) describe the
Stark shifts of atomic ground states induced by the cavity field and laser
pulse $\Omega_{1}(t)$, respectively. With the adopted approximation of equal
dipole moments, the Stark shifts induced by the laser field have no influence
on the photon generation, while the Stark shift $g^{2}/\Delta$ is the same for
all transitions and can be included into the cavity mode frequency:
$\omega_{c}\rightarrow\omega_{c}-g^{2}/\Delta$. With the adopted approximation
of equal dipole moments, the Stark shifts induced by the laser field have no
influence on the photon generation, since their difference between two
neighboring atomic ground states is zero for all transitions $m_{F}\rightarrow
m_{F}+1$, while the Stark shift $g^{2}/\Delta$ can be included into the cavity
mode frequency. In real atoms, where the dependence of $g$ and $\Omega_{1}$ on
Clebsch-Gordan coefficients leads to non-vanishing Stark-shift difference, the
latter can be made negligibly small as compared to the cavity decay rate $k$
by appropriately choosing the system parameters 31 , [Sec. IIIC].

###  II.2 Evolution of output photonic state

The elementary link in our QR protocol is a cascade system consisting of two
consecutive cavities in a chain, where the subsequent cavity is driven by
photons coming from the previous cavity 33 ; 34  without photon reflection.
Our goal is to absorb the incoming photons in the second cavity with a unit
probability, which requires precise knowledge of the imperfect control of the
wave functions of emitted photons by the $\Omega_{1}(t)$ laser in order to
correct operational errors, primarily fluctuations in the time-of-arrival of
photons, as indicated in Introduction.

Here we obtain the output state of the emitted photons from their total flux
defined in units of photons per unit time by

$\frac{dn_{\text{out}}(t)}{dt}=\langle
a^{{\dagger}}_{1,\text{out}}(t)a_{1,\text{out}}(t)\rangle,$ (6)

where $n_{\text{out}}(t)$ is the mean number of photons emitted from the first
cavity in the time interval $(-\infty,t]$. The output field
$a_{1,\text{out}}(t)$ is connected to the input $a_{1,\text{in}}(t)$ and
cavity mode $a_{1}(t)$ operators by the input-output relation
$a_{1,\text{out}}(t)-a_{1,\text{in}}(t)=\sqrt{k}a_{1}(t)$ 35 , and $a_{1}(t)$
is obtained through the atomic operators from the Heisenberg-Langevin equation
along with the Hamiltonian (5) and in the adiabatic limit $kT_{1}\gg 1$ as 31

$a_{1}(t)=-\frac{2iG_{1}}{k}f_{1}^{1/2}(t)\sum_{m_{F}=-F}^{F-1}\sigma_{m_{F}+1,m_{F}}(t)-\frac{2}{\sqrt{k}}a_{1,\text{in}}(t).$
(7)

In the sending node, the input field $a_{1,\text{in}}(t)$ is in the vacuum
state $\langle a^{{\dagger}}_{1,\text{in}}(t)a_{1,\text{in}}(t)\rangle=0$ and
will be ignored in further calculations.

Then, combining Eqs. (6) and (7), we have

$\frac{dn_{\text{out}}(t)}{dt}=\alpha_{1}f_{1}(t)\sum_{m_{F}=-1}^{0}\langle\sigma_{m_{F}}(t)\rangle,$
(8)

where the equations for the Zeeman sublevel populations
$\langle\sigma_{m_{F}}(t)\rangle\equiv\langle\sigma_{m_{F},m_{F}}(t)\rangle$
with arbitrary $F$ were derived in 31  from the master equation for the whole
density matrix of the system. For $F=1$, they are solved from the initial
conditions $\langle\sigma_{-1}(-\infty)\rangle=1$,
$\langle\sigma_{0}(-\infty)\rangle=\langle\sigma_{1}(-\infty)\rangle=0$. Using
condition (4) to neglect the spontaneous losses, we get

$\displaystyle\langle\sigma_{-1}(t)\rangle=e^{-\vartheta(t)},$ (9a)
$\displaystyle\langle\sigma_{0}(t)\rangle=\vartheta(t)e^{-\vartheta(t)},$ (9b)
$\displaystyle\langle\sigma_{1}(t)\rangle=1-\sum_{m_{F}=-1}^{0}\langle\sigma_{m_{F}}(t)\rangle,$
(9c)

where the variable

$\vartheta(t)=\alpha_{1}\int\limits_{-\infty}^{t}f_{1}(t^{\prime})dt^{\prime}$
(10)

is proportional to the $\Omega_{1}(t)$ pulse energy confined in the
$(-\infty,t]$ interval. Meanwhile, in the absence of Rabi oscillations, the
atomic coherence is always zero
$\langle\sigma_{m_{F},m_{F}^{\prime}}(t)\rangle\equiv 0,m_{F}\neq
m_{F}^{\prime}$.

The wave functions $\Phi_{\text{I,II}}(t)$ of the two emitted photons can be
derived from the total flux (8) as a sum of the first and second photon
fluxes, proportional to the atomic populations $\langle\sigma_{-1}(t)\rangle$
and $\langle\sigma_{0}(t)\rangle$, respectively:

$\displaystyle\frac{dn_{\text{I}}(t)}{dt}$ $\displaystyle=$
$\displaystyle|\Phi_{\text{I}}(t)|^{2}=\alpha_{1}f_{1}(t)e^{-\vartheta(t)},$
(11a) $\displaystyle\frac{dn_{\text{II}}(t)}{dt}$ $\displaystyle=$
$\displaystyle|\Phi_{\text{II}}(t)|^{2}=\alpha_{1}f_{1}(t)\vartheta(t)e^{-\vartheta(t)},$
(11b)

where $\Phi_{\text{I,II}}(t)$ having the phase of control field
$\Omega_{1}(t)$ are real.

Having thus obtained the photon wave functions, it remains to construct the
final state of the sending cavity and outgoing photons. In the model we
consider, the basis states of the sending system at asymptotic times
$t>k^{-1}$ are the tensor product state of (i) atomic ground Zeeman states
$|m_{F}\rangle$ with (ii) photonic states $|j\rangle$ corresponding to free
propagating $\sigma^{-}$-polarized $j$ photons of frequency $\omega_{c}$,
which are coupled to a single-mode optical fiber, and (iii) the vacuum cavity
mode state $|0\rangle_{1c}$. The final state $|\Psi_{1,\text{fin}}(t)\rangle$
can be then expanded in this basis as an entangled state between the atom and
the outgoing light, of the form

$|\Psi_{1,\text{fin}}\rangle=\biggl{(}\sum_{m_{F}=-1}^{1}\beta_{m_{F},j}|m_{F}\rangle_{1}|j=m_{F}+1\rangle_{ph}\biggr{)}|0\rangle_{c},$
(12)

where the real coefficients $\beta_{m_{F},j}$ are probability amplitudes of
the atom asymptotically in the state $|m_{F}\rangle_{1}$ corresponding to the
photon number $j=m_{F}+1$. Since the second index $j$ of $\beta_{m_{F},j}$ is
uniquely determined by $m_{F}$, we will safely omit it below.

The photonic states
$|j=1\rangle_{ph}=|1_{\Phi_{\text{I}}}\rangle|0_{\Phi_{\text{II}}}\rangle$ and
$|j=2\rangle_{ph}=|1_{\Phi_{\text{I}}}\rangle|1_{\Phi_{\text{II}}}\rangle$ can
be defined as the product of single-photon states with wave functions
$\Phi_{\text{I}},\Phi_{\text{II}}$ introduced in Eqs. (11). These states are
defined as $b_{i}^{\dagger}|0_{\Phi_{i}}\rangle=|1_{\Phi_{i}}\rangle$, where
the independent operators $b_{i},i=I,II$ have the standard boson commutation
relations $[b_{i},b^{\dagger}_{k}]=\delta_{ik}$. In this case, the operator
$a_{1,\text{out}}(t)$ is expressed through $b_{i}$ as 36

$a_{1,\text{out}}(t)=\sum_{i=\text{I}}^{\text{II}}\Phi_{i}(t)b_{i},$ (13)

which gives

$a_{1,\text{out}}(t)|n_{\Phi_{\text{I}}}\rangle|n_{\Phi_{\text{II}}}\rangle=\sum_{i\neq
k}n_{\Phi_{i}}^{1/2}\Phi_{i}(t)|n_{\Phi_{i}}-1\rangle|n_{\Phi_{k}}\rangle.$
(14)

Using the relations $\beta_{m_{F}}^{2}=\langle\sigma_{m_{F}}(\infty)\rangle$,
followed from Eq. (12), and Eqs. (9), $\beta_{m_{F}}$ are found to be

$\displaystyle\beta_{m_{F}=-1,j=0}=e^{-\vartheta(\infty)/2},$ (15a)
$\displaystyle\beta_{m_{F}=0,j=1}=[\vartheta(\infty)e^{-\vartheta(\infty)}]^{1/2},$
(15b)
$\displaystyle\beta_{m_{F}=1,j=2}=[1-(1+\vartheta(\infty))e^{-\vartheta(\infty)}]^{1/2}.$
(15c)

We show in the next section that these coefficients determine the amount of
entanglement between the sending and receiving atoms. The dynamics of the
atomic populations $\langle\sigma_{m_{F}}(t)\rangle$ is shown in Fig. 2 for
three durations $T_{1}$ of a Gaussian $\Omega_{1}(t)$ laser pulse
$f_{1}(t)=e^{-(t/T_{1})^{2}}$, where the realistic parameters
$(g,k,\gamma_{\text{sp}},\Omega_{1},\Delta^{F}_{B},\Delta^{F^{\prime}}_{B},\Delta)=2\pi\times(12,3,5.87,7,-12,4,100)$MHz
37 ; 38  were used to ensure a high cooperativity in Eq.(4): $C_{c}\sim 30$.
We list the numerical results for $\beta_{m_{F}}^{2}$ in Table 1. For the case
of $T_{1}=0.12\mu s$, the photon wave functions $\Phi_{\text{I,II}}(t)$, which
are real and positive, are shown in Fig. 3. We observe that the photons have
different pulse areas, which can be an issue for creating the maximal
entanglement between the atoms. This is discussed in Sec. IIIA.

Figure 2: (Color online) Evolution of atomic populations of the ground Zeeman
states $|m_{F}=-1\rangle_{1}$ (red), $|m_{F}=0\rangle_{1}$ (blue) and
$|m_{F}=+1\rangle_{1}$ (black) for three durations $T_{1}=0.75\mu$s (a);
$0.22\mu$s (b) and $0.12\mu$s (c) of Gaussian laser pulse
$f_{1}(t)=e^{-t^{2}/T_{1}^{2}}$ (dashed line). The atom is initially prepared
in the states $|F=1,m_{F}=-1\rangle_{1}$. For the parameters see the text.

Figure 3: (Color online) Wave-functions of the photons (indicated on the
curves) corresponding to the dynamics shown in Fig. 2 c.

##  III ENTANGLEMENT GENERATION VIA ELEMENTARY LINKS

For the system of cascade memories in Fig. 1, assuming no losses in the
communication channel, the final state Eq. (12) serves as an input state for
the receiving node, where the atom with the same level configuration is
initially prepared in the state $|F=1,\bar{m}_{F}=1\rangle_{2}$. Thus, the
initial state of the receiving system is given by

$|\Psi_{2,\text{in}}\rangle=|\bar{m}_{F}=1\rangle_{2}\Bigl{[}\sum_{m_{F}=-1}^{1}\beta_{m_{F}}|m_{F}\rangle_{1}|j=m_{F}+1\rangle_{ph}\Bigr{]}.$
(16)

The atom interacts on the transition
$|F=1,\bar{m}_{F}=1\rangle_{2}\rightarrow|F^{\prime}=2,\bar{m}_{F}^{\prime}=1\rangle_{2}$
with the linearly polarized laser pulse of frequency $\omega$, duration
$T_{2}$ and the peak Rabi frequency $\Omega_{2}$, which can be different from
$\Omega_{1}$. Similarly, the same magnetic field as in the sending node is
applied to provide two-photon resonant Raman transition with $\sigma^{-}$
polarized cavity photons.

The key requirement for complete storage of incoming photons in the receiving
node is to prevent reflection or leakage of photons from the second cavity,
which implies that the output quantum field is always zero. This amounts to
formally setting

$a_{2,\text{out}}(t)=0$ (17)

in the input-output relation
$a_{2,\text{out}}(t)-a_{2,\text{in}}(t)=\sqrt{k}a_{2}(t)$ for the second
cavity. Then, using $a_{2,\text{in}}(t)=a_{1,\text{out}}(t-\tau)$, we have

$a_{2}(t)=-\frac{1}{\sqrt{k}}a_{1,\text{out}}(t-\tau),$ (18)

where $\tau=L/c$ with L the distance between the nodes is the time needed for
the photons to travel from one cavity to the next. In the theory of cascade
systems, which is also our scheme, it is assumed that the dynamics of the
receiving node does not depend on L 33 ; 34 . However, this is true only if
the path length fluctuations change $\tau$ insignificantly. Due to these
fluctuations the incoming photons undergo the phase noise during the travel
time. In a long fiber, the phase noise has been measured over the scale of
photon travel time in 39  using Mach-Zehnder interferometry. The results show
that the phase over 36 km long interferometer remains stable at a level of
0.1rad for the duration of around 100$\mu$s, which corresponds to a path
length fluctuation of several tens of nm for a wavelength 1550nm and even less
for visible light. Obviously, this induces a negligible change in the delay
time, which justifies the above assumption. This allows safely omitting $\tau$
in all equations.

Table 1: Populations of the sending atom in Zeeman states $|F=1,m_{F}=-1,0,1\rangle_{1}$ for three durations $T_{1}$. The last column shows the amount of entanglement between the atoms. $T_{1}(\mu$s) | $\beta_{-1}^{2}$ | $\beta_{0}^{2}$ | $\beta_{1}^{2}$ | $E$
---|---|---|---|---
0.75 | 0.00065 | 0.0048 | 0.995 | 0.051
0.22 | 0.11 | 0.25 | 0.64 | 1.27
0.12 | 0.31 | 0.36 | 0.33 | 1.58

###  III.1 Photon storage in the receiving node. Maximally entangled state of
two atoms

The dynamics of photon absorption in the receiving system is governed by an
effective interaction Hamiltonian in the form

$H=\hbar
f_{2}^{1/2}(t)\bigl{[}G_{2}^{*}(\varrho_{0,1}+\varrho_{-1,0})a_{2}(t)+G_{2}a_{2}^{{\dagger}}(t)(\varrho_{1,0}+\varrho_{0,-1})\bigr{]}$
(19)

where $\varrho_{i,j}=|\bar{m}_{F}=i\rangle_{2}\langle\bar{m}_{F}=j|_{2}$ are
the operators of the receiving atom, $G_{2}=g\Omega_{2}/\Delta$ and $f_{2}(t)$
represents the time profile of the pumping laser in the second cavity.

In general, the state of the receiving system can be expanded as

$\displaystyle|\Psi_{2}(t)\rangle=\sum_{m_{F}=-1}^{1}\beta_{m_{F}}|m_{F}\rangle_{1}$
$\displaystyle\times\sum_{\bar{m}_{F}=-m_{F}}^{1}\gamma_{\bar{m}_{F},j}(t)|\bar{m}_{F}\rangle_{2}|j=\bar{m}_{F}+m_{F}\rangle_{ph},$
(20)

where $\gamma_{\bar{m}_{F},j}(t)$ are the amplitudes of the probabilities that
at time $t$ the receiving atom is in the state $|\bar{m}_{F}\rangle_{2}$ and
the number of incoming photons is $j=\bar{m}_{F}+m_{F}$ provided that the
first atom is in the state $|m_{F}\rangle_{1}$. They satisfy the normalization
equation

$\sum_{m_{F}=-1}^{1}\sum_{\bar{m}_{F}=-m_{F}}^{1}\beta_{m_{F}}^{2}|\gamma_{\bar{m}_{F},\bar{m}_{F}+m_{F}}(t)|^{2}=1.$
(21)

The structure of Eq. (20) is the result of the double action of the first term
of Hamiltonian (19) on the initial state (16) and shows that the photons are
partially entangled with both atoms, which is evidenced by the dependence of
the photonic state on magnetic numbers of two atoms. Initially, the photons
are entangled with the first atom shown in Eq. (16), but not with the second
one. However, under the conditions derived below, the photons are completely
absorbed by the second atom leading to the conversion of the atom-photon
entanglement into the entanglement between the two atoms with maximum value at
$\beta_{j}=1/\sqrt{3}$. The entire dynamics of this effect is captured by the
evolution of the coefficients $\gamma_{\bar{m}_{F},j}(t)$, as we show further.

We find $\gamma_{\bar{m}_{F},j}(t)$ from the Schrödinger equation for
$|\Psi_{2}(t)\rangle$ with the use of Hamiltonian (19), where the operator
$a_{2}(t)$ is replaced by $a_{1,\text{out}}(t)$ from Eq. (18) with $\tau=0$.
We recognize that the action of the latter on the two-photon state
$|1_{\Phi_{\text{I}}},1_{\Phi_{\text{II}}}\rangle$ generates, according to Eq.
(13), the superposition of two one-photon states

$a_{1,\text{out}}(t)|1_{\Phi_{\text{I}}},1_{\Phi_{\text{II}}}\rangle=\frac{1}{\sqrt{2}}\sum_{i=+,-}\Phi_{i}(t)|i\rangle,$
(22)

which arises in the equation for $\gamma_{\bar{m}_{F}=0,j=1}(t)$. Here
$\Phi_{\pm}(t)=\Phi_{I}(t)\pm\Phi_{II}(t)$ and
$|\pm\rangle=\frac{1}{\sqrt{2}}(|1_{\Phi_{I}},0_{\Phi_{II}}\rangle\pm|0_{\Phi_{I}},1_{\Phi_{II}}\rangle)$.
A key approximation to achieve the complete absorption of the photons consists
in considering

$\eta(t)\sim\zeta(t),$ (23)

with the variables

$\displaystyle\eta(t)=2\frac{|G_{2}|}{\sqrt{k}}\int\limits_{-\infty}^{t}f_{2}^{1/2}(t^{\prime})\Phi_{I}(t^{\prime})dt^{\prime},$
(24a)
$\displaystyle\zeta(t)=\frac{|G_{2}|}{\sqrt{k}}\int\limits_{-\infty}^{t}f_{2}^{1/2}(t^{\prime})\Phi_{+}(t^{\prime})dt^{\prime},$
(24b)

featuring the time integral of the photon fluxes multiplied by the profile of
the second laser driving the receiving node.

How to satisfy this approximation is discussed below. This implies taking only
the leading term $~{}\Phi_{+}(t)$ into account in Eq. (22) while integrating
the Schrödinger equation. This gives a simple form for the equations of
$\gamma_{\bar{m}_{F},j}(t)$:

$\displaystyle\frac{d\gamma_{0,0}(t)}{d\eta}=\frac{i}{2}\gamma_{1,1}(t)e^{-i\varphi_{2}},$
(25a)
$\displaystyle\frac{d\gamma_{1,1}(t)}{d\eta}=\frac{i}{2}\gamma_{0,0}(t)e^{i\varphi_{2}},$
(25b) $\displaystyle\frac{d\gamma_{1,0}(t)}{d\eta}=0,$ (25c)

and

$\displaystyle\frac{d\gamma_{-1,0}(t)}{d\zeta}=\frac{i}{\sqrt{2}}\gamma_{0,1}(t)e^{-i\varphi_{2}},$
(26a)
$\displaystyle\frac{d\gamma_{0,1}(t)}{d\zeta}=\frac{i}{\sqrt{2}}\biggl{(}\gamma_{1,2}(t)e^{-i\varphi_{2}}+\gamma_{-1,0}(t)e^{i\varphi_{2}}\biggr{)},$
(26b)
$\displaystyle\frac{d\gamma_{1,2}(t)}{d\zeta}=\frac{i}{\sqrt{2}}\gamma_{0,1}(t)e^{i\varphi_{2}},$
(26c)

where $\varphi_{2}$ is the phase of $\Omega_{2}$ field.

These equations are solved with the initial conditions
$\gamma_{1,i}(-\infty)=1,i=0,1,2,$ and zero for the remaining coefficients,
which are obtained by comparing Eq. (20) with the initial state (16). For
$\varphi_{2}=\pi/2$, we get

$\gamma_{1,0}(t)=1,\quad\gamma_{0,0}(t)=\sin[\eta(t)/2],\quad\gamma_{1,1}(t)=\cos[\eta(t)/2],$
(27)

and

$\displaystyle\gamma_{1,2}(t)=\frac{1}{2}\biggl{[}1+\cos\zeta(t)\biggr{]},$
(28a) $\displaystyle\gamma_{0,1}(t)=\frac{1}{\sqrt{2}}\sin\zeta(t),$ (28b)
$\displaystyle\gamma_{-1,0}(t)=\frac{1}{2}\biggl{[}1-\cos\zeta(t)\biggr{]},$
(28c)

which satisfy Eq. (21).

It is evident that the complete absorption of photons in the second cavity is
achieved, when the time profile of the second laser is such that both
integrals (24) are ultimately $\pi$:

$\eta(\infty)=\zeta(\infty)=\pi,$ (29)

because in this case only the amplitudes $\gamma_{i,0}(\infty)=1,i=1,0,-1,$
remain different from zero, indicating that, at large times $t>T_{2}$, the
quantum field is in the state $|j=0\rangle_{ph}$ with zero photon number,
while the second atom occupies the Zeeman states with $\bar{m}_{F}=-m_{F}$.

The described process is two consecutive $\pi$-pulse excitations of two Raman
transitions
$|F=1,\bar{m}_{F}=1\rangle_{2}\rightarrow|F=1,\bar{m}_{F}=0\rangle_{2}$ and
$|F=1,\bar{m}_{F}=0\rangle_{2}\rightarrow|F=1,\bar{m}_{F}=-1\rangle_{2}$ with
effective (two-photon) pulse areas $\eta(\infty)$ and $\zeta(\infty)$,
respectively, which ensure the photon storage into the ground states of the
second atom with unit probability.

Under condition (29), two atoms ultimately turn out to be in a pure entangled
state in the three dimensional space of atomic states

$\displaystyle|\Psi_{fin}\rangle_{12}=|\Psi_{2}(\infty)\rangle=\sum_{m_{F}=-1}^{1}\beta_{m_{F}}|m_{F}\rangle_{1}|-m_{F}\rangle_{2}.$
(30)

The entanglement between the atoms, characterized by the quantity
$E=-\sum_{m_{F}=-1}^{1}\beta_{m_{F}}^{2}\log_{2}\beta_{m_{F}}^{2}$ 40 , is
maximal for equal $\beta_{m_{F}}=\frac{1}{\sqrt{3}}$. This case leading to
$E_{\max}\simeq 1.58$ is realized with a high accuracy for $T_{1}=0.12\mu s$
as shown in Fig. 2c. The corresponding values of $\beta_{m_{F}}$ are given in
Table 1.

Figure 4: (Color online) Variables $\eta(t)$ (red) and $\zeta(t)$ (blue) as
functions of time for $\Omega_{2}=4\Omega_{1}$ and $T_{1}=T_{2}=0.12\mu$s. The
profiles of $\Omega_{1,2}(t)$ laser fields are shown by red and blue dashed
line, respectively.

The robustness of our results is defined only by the stability of laser pulse
energy, because as follows from Eqs.(10), (15), (24) and (27,28), the
coefficients $\beta_{i}$ and $\gamma_{ij}$, and hence, the entangled state of
the first atom and the outgoing photons, Eq.(12), as well as the final
entanglement between two remote atoms, Eq.(30), depend only on the total
energy of the pump pulses. The energy stability factor [(max - min)/mean] of
modern nanosecond lasers usually does not exceed 5$\%$ (see, for example, 41
). As calculations show, this is an acceptable value in order to neglect laser
energy fluctuations in our scheme.

The different dependence of $\eta(\infty)$ and $\zeta(\infty)$ on wave
functions $\Phi_{I}(t)$ and $\Phi_{II}(t)$ gives rise to the question of
simultaneous fulfillment of equations (29), as was pointed out in Sec. IIB.
There are many possibilities to overcome this difficulty by correctly choosing
the parameters of the system, primarily for the $\Omega_{2}(t)$ laser field.
We exemplary show in Fig. 4 that one can reduce the contribution of
$\Phi_{I}(t)$ by delaying the laser pulse $\Omega_{2}(t)$ turning on with
respect to the incoming photon pulses (see Fig. 3). In this case, the
condition (23) is well satisfied for any time moment being insensitive to the
spectral properties of the photon pulses. In the two other cases of
$T_{1}=0.75\mu s$ and $T_{1}=0.22\mu s$ shown in Fig. 2, the conditions (29)
are implemented in a similar way, leading, however, to smaller entanglement
between the atoms due to inhomogeneous distribution of the sending atom
population over the Zeeman states (see Table 1).

###  III.2 Atomic state detection

In our protocol, where the photon losses are considered negligible and the
detectors have high efficiency, the entanglement succeeds with unit
probability. In practice, the robustness of the proposed repeater has to be
tested against inevitable noise in local operations and losses in quantum
channels, although the latter is tiny within the fiber-channel attenuation
length. The issues with photon losses occur at larger distances, this have
been thoroughly discussed in our previous work 27 . The effect of both photon
loss and imperfect factors can be addressed by photon-heralding measurements,
which, however, are challenging with our multiphoton protocol. Instead, we
propose here to verify the successful storage of incoming photons in the
second cavity by using a scheme for a reliable measurement of the ground
Zeeman state populations of the second atom.

Figure 5: (Color online) Setup for measuring the Zeeman states of the second
atom at the node B. The states $|\bar{m}_{F}=-1\rangle_{2}$ and
$|\bar{m}_{F}=+1\rangle_{2}$ of the receiving atom are converted into
$\sigma^{-}$\- and $\sigma^{+}$-polarized single photons, which are separately
detected at the output ports D1 and D2, respectively, passing through a
$\lambda/4$ plate and PBS.

The schematic setup for atomic state detection is shown in Fig. 5, where the
receiving atom is driven on the $S_{1/2}(F=1)\rightarrow
P_{1/2}(F^{\prime}=1)$ transition off-resonantly by a linearly polarized laser
pulse $\Omega_{3}(t)$. The latter generates a $\sigma^{+}$\- or
$\sigma^{-}$-polarized cavity photon with probabilities $\beta_{\mp 1}^{2}$,
respectively, dependent on the state of the atom. To detect these photons, we
propose to use a two-mode cavity similar to that employed in 42 ; 43 . The
first cavity mode is coupled to the atomic $F=1\rightarrow F^{\prime}=2$
transition (Fig. 1) and supports the efficient storage of incoming photons,
while the second mode with frequency of $\Omega_{3}(t)$ enhances the atomic
state readout signal. A weak magnetic field $B$ is applied to reduce the local
magnetic field fluctuations. The emitted photons pass through a quarter-wave
plate ($\lambda/4$), which converts left- and right-polarized photons to be
horizontally (H) and vertically (V) polarized, respectively. Upon reaching the
interface of a polarizing beam splitter (PBS), the H polarized photon is
reflected to the $D_{1}$ detector, while the V polarized photon goes straight
to the $D_{2}$ detector. Thus, the states $|\bar{m}_{F}=+1\rangle_{2}$ and
$|\bar{m}_{F}=-1\rangle_{2}$ of the receiving atom are selectively measured by
single-photon detection. In the ideal case, the number $N_{\pm}$ of the events
with detection of $\sigma^{+}$\- and $\sigma^{-}$-polarized photons should be
related as $N_{+}/N_{-}=\beta_{-1}^{2}/\beta_{1}^{2}$ according to Eq. (29).
Deviations from this value will make it possible to determine the sources of
imperfect transmission of photons. In this sense, the simplest situation to
analyze is given in Fig. 2a (see also Table 1, $T_{1}=0.75\mu$s), where only
the state $|\bar{m}_{F}=-1\rangle_{2}$ of the receiving atom is populated as
the result of two-photon absorption. Correspondingly, the fidelity of the
measured density matrix $\rho_{2}$ of the second atom with the calculated
state (29) contains only one term
$F=|_{2}\langle\bar{m}_{F}=-1|\rho_{2}|\bar{m}_{F}=-1\rangle_{2}|^{2}$ to be
analyzed statistically. In the state $|\bar{m}_{F}=0\rangle_{2}$, the atom
does not interact with linearly polarized $\Omega_{3}(t)$ field, so no signal
appears at PBS output. This state is then measured by applying a laser pulse
tuned to the $S_{1/2}(F=1,\bar{m}_{F}=0)\rightarrow
P_{1/2}(F^{\prime}=2,\bar{m}_{F}=0)$ transition with a proper detuning, which
triggers the emission of a $\sigma^{+}$\- or $\sigma^{-}$-polarized photon at
the frequency of the first cavity-mode on the transition
$(F=1,\bar{m}_{F}=-1)\rightarrow(F^{\prime}=2,\bar{m}_{F}=0)$ or
$(F=1,\bar{m}_{F}=1)\rightarrow(F^{\prime}=2,\bar{m}_{F}=0)$, respectively.
The detection of this photon without discrimination of its polarization
indicates that the atom is in the state $|\bar{m}_{F}=0\rangle_{2}$.
Furthermore, the entanglement created between the atoms according to Eq.(29)
can also be verified using the scheme in Fig.5 at both nodes A and B. Upon
converting the states of two atoms into the single photons and detecting the
latter as the $\sigma^{\pm}$ events, one can make sure that a
$\sigma^{+}(\sigma^{-})$ event at the node A is always accompanied by the
$\sigma^{-}(\sigma^{+})$ event at the node B. If the atoms are in the states
of $m_{F}=\bar{m}_{F}=0$, the absence of an output signal must be registered
simultaneously at two nodes. The tomography data for the last case can be
obtained by detecting again one photon from each node emitted from upper
states $(F^{\prime}=2,m_{F}=0)$ and $(F^{\prime}=2,\bar{m}_{F}=0)$ of the
atoms, as described above. Then, using only the successful attempts of two
photon detection, the density matrix of the system can be reconstructed, from
which the amount of entanglement can be obtained with high accuracy, if all
conditions we use in our calculations are experimentally kept. Finally, we
note an especially important possibility of functional integration of this
scheme into QR structure at the stage of entanglement swapping between the
entangled pairs.

##  IV CONCLUSION

We have proposed the first protocol for generating multidimensional
entanglement between remote atoms in the space of atomic states of higher
dimension than two. We have developed an experimentally feasible model for
deterministic generation of three-dimensional entanglement between two distant
atoms trapped in optical cavities and exchanging two photons. We have carried
out the detailed analysis of a lossless evolution of the entanglement up to
the creation of maximally entangled two-qutrit state of the atoms. In
realistic conditions, the maximal entanglement has been calculated numerically
to be 1.58, far above the two-dimensional limit of 1. The conditions for
deterministic production of two photons from the sending cavity and their
complete absorption by the receiving atom have been determined. The necessary
condition in the latter case has been formulated explicitly by Eq. (17), which
is a universal requirement to prohibit the reflection or leakage of incoming
photons from a cavity. To test the robustness of our scheme against the
imperfect local operations and photon losses in the fiber-quantum channel, we
have proposed a technique, which is based on a reliable measuring of ground
Zeeman states of the atoms and is simple to implement.

These results are the basis for performing an entanglement swapping operation
between entangled pairs, which is under investigation.

### Acknowledgments

This work was supported by the RA Science Committee, in the frames of the
research project 20TTAT-QTc004. We acknowledge support from the EUR-EIPHI
Graduate School (17-EURE-0002) and from the European Union’s Horizon 2020
research and innovation program under the Marie Sklodowska-Curie grant
agreement No. 765075 (LIMQUET).

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Wave Function of Particle and Coordinates Distribution in Relativistic Quantum
Theory

V. F. Krotov

Institute of Control Problems, Russian Academy of Sciences, Russia, 117997,
Moscow, Profso’uzna’a 65

The conditions for observation of the individual particle coordinates,
required by logic of the Special Relativity and filtering the quantum field
effects, are described. A general relation between the corresponding density
of probability and the wave function is found. It is a relativistic invariant
describing probability of the particle emergences in spacetime. This density
is concretized for bosons, both scalar and vector (including photon), charged
and neutral, also electron. The Heisenbergs uncertainty relations have been
approved in regards to relativistic particle. As applied to the quantum field,
this new construction is transformed to new characteristic of the particles
distribution in space-time, which complete distribution throughout impulses.
The operators of these distributions and the invariant relativistic
description for free quantum fields have been obtained. These new properties
of the particles and fields are proposed for experimental investigations.

PACS 03+p; 03.65-w; 03.65 Ta

## 1\. Introduction

Model of the Quantum Particle (QP) combines the deterministic dynamics of Wave
Function (WF) and the statistical relation of WF with observed values: the
dynamic part of QPs description and the statistical one. While the former one
is quite well formalized, the second one for relativistic QP (RQP) causes
problems: no formal constructs with coordinates distribution properties, and
the restrictions on observation possibilities caused by effects of the Quantum
Field (QF) are available. These problems have caused a general opinion that
“in consistent relativistic quantum mechanics the particles coordinates at all
cannot serve as dynamic variables, $\ldots$ and WF, as information carriers,
cannot be present in it” [1], page 16. We have to add: relations of
uncertainty of Heisenberg are devoid of the theoretical rationale as applied
to RQP, as there is no law of coordinates distribution necessary for that. We
demonstrate that both problems mentioned above are solved when the statistical
part of RQP have been included in logic of the Special Relativity (SR)
sufficiently fully. Despite the common opinion, there appears a theoretical
possibility to preserve information properties of the WF in the extent
comparable to non-relativistic quantum mechanics (QM), and there are
observation procedures, filtering effects of QF. This inclusion require
correction of the full set of the observables and of the observation
procedures. A correction of the requirements of covariance follow from this.
Formal constructs expressed via WF are obtained, that possess all necessary
properties of probabilistic distributions of the observed values, including
particle coordinates. In simplest cases this expression is formally similar to
probability density of the coordinates of the non-relativisic particle, but
has different content. Other observation conditions, other transformational
properties, other meaning: probability density of the events in space-time,
instead of the probability density of positions in space. Instead of
stochastic dance of the particle, described by the flow of probability, we
have probabilistic distribution of the particle appearance in space-time ,
which can not be, generally, represented by the motion in space. On this basis
relations of uncertainty obtain the justification as applied to RQP. As
applied to the quantum field, this new construct is transformed to new
characteristic of the particles distribution in space-time, which complete
distribution throughout impulses. The operators of these distributions and the
invariant relativistic description for free quantum fields have been obtained.

## 2\.

Since the statistical part is sufficiently formalized only for the non-
relativistic QM, let us start here with defining its necessary details limited
to spinless QP. Let $\mathbf{x}=\\{x^{i}\\}=(x^{1},x^{2},x^{3})$ be the vector
of particles spatial coordinates, the element of the related Euclidian space
$\mathbf{E}$. Complex WF $\psi(\mathbf{x})$ is defined on the $\mathbf{E}$,
and is considered as the element of the Hilbert space $\mathbf{H}$ with the
norm $L_{2}(\mathbf{E})$ and respective product
$(\psi_{1},\psi_{2})={\displaystyle\int\limits_{\mathbf{E}}}\psi_{1}(x)\psi^{*}_{2}(x)d\mathbf{E}$,
where $d\mathbf{E}$ is elementary volume of $\mathbf{E}$. As a rule, the WF is
finite in a cube $\mathbf{V}\subset\mathbf{E}$: $\psi(\mathbf{x})=0$,
$\mathbf{x}\notin\mathbf{V}$, and this integral is defined.The evolution of
the WF $\psi(t,\mathbf{x})$ on the interval $(0,T)$ describes the QP movement.
It satisfies Schrodinger equation, whereas $\|\psi(t,\mathbf{x})\|$ is its
dynamic invariant, normalized by the condition $\|\psi(t,\mathbf{x})\|=1$.

On observation procedure. Full set of characteristics with values
$y=\linebreak=\\{y_{i}\\}$ is being observed. They are either components of
the vector $\mathbf{x}$, or the energy, impulse, and momentum. Let the
operations of realization of QP movement during the time $T$ be defined in
identical physical conditions, corresponding to given WF evolution
$\psi(t,\mathbf{x})$. At a given moment of time $t$ a single session of
observation (session) is being performed over each realization. Values $y$ in
each session are defined, and are random would they repeat. Observation of
trajectories $y(t)$ is also admissible in cases when they are identical to the
time series of results of sessions of the independent realizations. For each
fixed function $\psi(t,\mathbf{x})$ and each $t$ the averages $\langle
y\rangle\left[\psi(t,\mathbf{x})\right]=\\{\langle y_{i}\rangle(\psi)\\}$ are
defined, corresponding to infinite set of the sessions.

On formal description. Given fixed $t$ the averages $\langle
y\rangle\left[\psi(t,\mathbf{x})\right]=\\{\langle y_{i}\rangle(\psi)\\}$ are
Hermits Quadratic Forms (HQF) in $\mathbf{H}$. If $y=\mathbf{x}$, then the
existence of the probability density $f(t,\mathbf{x})$ and its relation to WF
$f(t,\mathbf{x})=|\psi(t,\mathbf{x})|^{2}$ are postulated. Correspondingly,
the probability measure
$P(Q/t)=P[Q,\psi(t,\mathbf{x})]=\linebreak={\displaystyle\int\limits_{\mathbf{E}}}f(t,x)d\mathbf{E}$
is defined. If $y=\\{y_{i}\\}$ is an aggregate of energy, impulse, and
momentum, than the equality $\langle y\rangle=J(\psi)$ is postulated, where
the right side is the aggregate of related dynamic invariants of the Wave
Equation (WE), HQF $J_{i}(\psi)=(\psi,L_{i}\psi)$, and $L_{i}$ is a respective
operator (operator of the$i$-th observable). In this case the family of
distributions $P(Q,\psi)$ is expressed through components of spectral
functions of the operators $L_{i}$ in a known manner.

There exist exceptions from the rule $\psi\in L_{2}(\mathbf{E})$, when the
norm is not defined. In this case the condition $P(\mathbf{E},\psi)=1$ does
not hold. But relative probabilities $P(Q_{1},\psi)/P(Q_{2},\psi)$ are
defined. Let us note that $\psi(\mathbf{x})\in L_{2}(\mathbf{V})$ in any fixed
cube $\mathbf{V}\subset\mathbf{E}$, and WF is the element of Hilbert space
$\mathbf{H}(\mathbf{V})$ with this norm. By fixing here $\|\psi\|=1$, one can
give $P(Q,\psi)$, $Q\subset\mathbf{V}$, the meaning probability measure of the
positions $\mathbf{x}$, and $f(t,\mathbf{x})$ — its density, with the
following amendment to the observation procedure: only the sessions
registering $\mathbf{x}\in\mathbf{V}$ are considered.

## 3\. Relativistic Particle. General Model Description

Let $E$ be real pseudo-Euclidian space with coordinate vector
$x=\linebreak=\\{x^{\alpha}\\}=(x^{0},\mathbf{x})$, $\mathbf{x}\in\mathbf{E}$;
$x^{0}=ct$, $t$ is time, $c$ is speed of light; metric tensor
$e=\\{e_{\alpha\beta}\\}$ is diagonal: $e_{00}=-1$, $e_{ii}=1$, $i>0$. On a
finite bar $V=(0,cT)\times\mathbf{V}\subset E$ the WF $\psi(x)$ is defined,
that maps $V$ to an Euclidian space $U$ with elements $u$ and product
$u_{1}u_{2}$. It satisfies the respective wave equation (WE). The WF is
considered as an element of the Hil-bert space $H(V)$ with the norm
$\|\psi(x)\|^{2}={\displaystyle\int\limits_{V}}u^{2}dE$, $u=\psi(x)$,
$dE=dx^{0}d\mathbf{E}$ (let us denote norm of the space $\mathbf{H}$ as
$\|\psi(t,\mathbf{x})\|$), and respective product $(\psi_{1},\psi_{2})$. For
each WF the identical physical conditions are defined, in which the RQP is
observed.

Observation procedure reproduces it for non-relativistic QP with the following
differences. Session is not caused by moment of time $t$. Thus, time is
excluded from the observation process as a parameter, but is present as a part
of argument $x$, and, possible, of the observed value $y$. Session lasts
during the time $T$. Other conditions will hold good. The main condition among
them is preservation of uniqueness of $y$ value at session of measurement in
new conditions. We will discuss this condition below in application to the
concrete observable values. A measure $P(Q,\psi)$ is defined for each WF, and
respectively, the average value $\langle y\rangle[\psi(x)]$.

Formal description reproduces the same for non-relativistic QP with the
following differences. HQF’s $P(Q,\psi)$, $J_{i}(\psi)=(\psi,L_{i}\psi)$, and
respectively, the operators of the observables are defined in $H$, but not in
$\mathbf{H}$. Characteristics $Y$, and thus, their average $J(\psi)$, possess
relativistic transformational properties. Therefore, $P(Q,\psi)$ is
relativistic invariant. The latter statement follows from equality
$J(\psi)=\langle y\rangle(\psi)={\displaystyle\int}ydP(\psi)$, since $J(\psi)$
and $y$ have the same tensor dimension. The dimension of probability density
$g(y,\psi)$ is defined by the equality $dP=gdY=inv$, where $dY$ is elementary
volume of the space of observable values. Detailed elaboration of model with
reference to concrete observable values and types of particles follows below.

## 4\. Coordinates Observation

On observation procedure. The full set of the observed is vector $y=x$ of
spatial-time coordinates of a particle; the result of each session is the
fixation of the event that a particle has appeared in the point $x$ of the
space-time $E$. A content of the session: let the measuring device be an
electronic microscope; a bunch of electrons probes the space, and at some
point in time there happens a collision with a particle-object; such
collision, generally, distorts the observation conditions, corresponding to
the given WF (the least case is the object-photon, that disappears as a result
of the collision); because of this, only this first collision has to be taken
into consideration; the position of the particle is identified with the point
on the screen, — a trace of the single scattered electron. The measurement
should also record the moment of collision for complete fixation of the event.
Only the sessions, that demonstrate events $x\in V$ has to be taken into
consideration. The causes of the distortion of this condition can be the
following: the session demonstrate an event $x\notin V$, the collision is not
the only, effects of the QF there are. Admissibility of the latter is not
evident, because QP and QF are different systems. But it is valid (see below,
item 6.4).

On the formal description. The function

$g(x)=\psi^{2}(x),\quad\|\psi(x)\|^{2}=1$ (1)

posses all necessary properties of the probability density of the events $x\in
V$. This density is a relativistic invariant (elementary volume $dE$ is
invariant). In simplest cases this expression is formally similar to
probability density of the coordinates of the non-relativistic particle, but
has different content. Other observation conditions, other transformational
properties, other meaning of the function $\psi^{2}(x)$: probability density
of the events in space-time, instead of the probability density of positions
in space. The change of meaning corresponds to the logic of SR and predicts a
new property of the RQP. Instead of stochastic dance of the particle,
described by the flow of probability, we have probabilistic distribution of
the particle appearance in space-time , which can not be, generally,
represented by the motion in space.

Knowing $g(x)$, one can determine density $g_{1}(\mathbf{x})$ of probability
of positions $\mathbf{x}$, $g_{0}(x^{0})$ of time $x^{0}$, density of spatial
coordinates at a fixed moment $g_{1}(\mathbf{x}/x^{0})$:

$g_{1}(\mathbf{x})=\int\limits_{(0,\infty)}g(x)dx^{0},\quad
g_{0}(x^{0})=\int\limits_{\mathbf{E}}g(x)d\mathbf{E},\quad
g_{1}(\mathbf{x}/x^{0})=g(x)/g_{0}(x^{0}).$

Accordingly to it, we can proof distribution (1) without observing time but
observing $g_{1}(x)$.

4.1. Scalar Boson. Space $U$ is one-dimensional, real or complex, density:
$g(x,\psi)=|\psi(x)|^{2}$, $\|\psi(x)\|=1$, in a limited beam $V$.

4.2. Vector Boson. WF of such particles is the vector $u(x)$, mapping $E$ into
$E$, or into its complex analog $E^{*}$. We have:
$u^{2}=\mathbf{u}^{2}-(u^{0})^{2}$. Euclidean space $U$ is separated from $E$
by condition: $u^{2}\geq 0$. For the latter, it is necessary and sufficiently
that $u^{\prime 0}=0$ in a fixed frame of reference $x^{\prime}$. Thus, space
$U$ is defined accurate to transformation $u^{\prime}\to u$, i.e. to vector
$\mathbf{v}$ of speed of the system $x$ relatively to $x^{\prime}$. For
massive boson $x^{\prime}$ is the system of coordinates related to the
particle. Therefore, vector $\mathbf{v}$ is fixed, and space $U$ is Euclidean
section of the pseudo-Euclidean space $E$. Density: $g(x,\psi)=u^{2}(x)$.

Massless vector boson, photon, has not the system of coordinates related to
the particle, but the lack of the system $x^{\prime}$, $u^{\prime 0}=0$ is not
follow from this. Moreover, if this system is available, then it is not the
only, contrary to massive particle. Really, let’s consider a flat wave package
such that its wave vectors are parallel to a vector $\mathbf{k}$. Including
Lorentz condition in the set of field equations provides: $u^{\prime
1}=u^{\prime 0}=0$. Assuming $\mathbf{v}\uparrow\uparrow\mathbf{k}$, similar
to massive boson, it is easy to make sure that density is defined:
$u^{0}=u^{1}=u^{\prime 1}=u^{\prime 0}=0$, and space $U$ is a plane with basis
$u^{2}$, $u^{3}$, independently from $|\mathbf{v}|$. I. e., calibration
$u^{0}=u^{1}=0$ is invariant in subgroup
$\mathbf{v}\uparrow\uparrow\mathbf{k}$ of the Lorentz group. It seems natural
the following Postulate of photon: the system $x^{\prime}$, $u^{\prime 0}=0$
is available. It determines the density $g(x)=u^{2}(x)$ and consistently
minimizes distinction of bosons properties: the system of coordinates related
to the particle in this case is absent, but its property $u^{\prime 0}=0$ is
kept in any system, moving in parallel to wave vectors. But it is obtained
with destruction of the principle of gradient invariance of electrodynamics.
The latter is confirmed by its experience. But all of it deals with values and
distributions of tensions, energy, impulses, the moments and does not concern
distributions of the photons coordinates. Only an experiment can determine
alternative choice: either this principle is unapplicable here and (1) is
valid, or (1) is not valid

4.3. Electron. $U$ is Euclidean space with elements, $u=\\{u^{\alpha}\\}$,
$\alpha=\linebreak=1,2,3,4$, have transformational properties of the spinor,
and $(u)^{2}$ is time component of the vector. WF $u(x)$ is usually considered
as a trajectory in Hilbert space $\mathbf{H}(\mathbf{V})$ with norm
$L_{2}(\mathbf{V})$:

$\|u(t,\mathbf{x})\|^{2}={\displaystyle\int}u^{2}(t,\mathbf{x})d\mathbf{E};\quad
u^{2}=\sum_{\alpha}|u^{\alpha}|^{2}.$

It satisfies the Dirac’s WE, and $\|u(t,\mathbf{x})\|$ is its dynamic
invariant. These properties give the bases for equation:
$u^{2}(x)=g(\mathbf{x}/x^{0})$, $\|u(t,\mathbf{x})\|^{2}=1$, [2]. Let us
introduce a new WF $\psi(x)$, such that
$g(x)=g_{0}(x^{0})u^{2}(x)=\psi^{2}(x)$, $\|\psi(x)\|^{2}=1$. In virtue of WE
and typical boundary conditions it coincides with $u(x)$ accurate to
normalization. We have: $g_{0}(x^{0})={\rm const}=1/cT$,
$\psi=\linebreak=(cT)^{-1/2}u$, $\|\psi(x)\|^{2}=1$. While $T\to\infty$, $V\to
E$ the limit $\|\psi(x)\|^{2}$ is defined here, if similar integral over
$\mathbf{E}$ is defined. Densities $g(x)$, $g_{1}(\mathbf{x}/x^{0})$ coincides
accurate to normalization. They can be constructed both observing
$\mathbf{x}$, with parameter $x^{0}$, and observing directly $x$.

## 5\. Observation of energy-impulse

Let us begin consideration with real scalar boson, WF of which is defined in a
finite beam $V\subset E$. Observable value $y$ is a vector of 4-impulse
$p=\linebreak=(p^{0},\mathbf{p})\in E$. Its average is a dynamic invariant,
quadratic form in $H$. Their eigen WF form a family:
$\psi_{k}=a_{k}\exp(ip_{k}x/\hbar)$ with parameters $a_{k}>0$, $p_{k}\in E$,
is known discrete row, $p^{2}_{k}=-(mc)^{2}$, values $a_{k}$ are defined by
normalization “particle in unit volume”, and distribution is described in
terms of average quantities of particles $n_{k}$ with given 4-impulse $p_{k}$
as a prototype of QF. More precisely, $n_{k}$ is an average number of the
measurement sessions with result $p_{k}$. In respective space $l_{2}$ of
coefficients $C=\\{C_{k}\\}$ of decomposition
$\psi={\displaystyle\sum\limits_{k}}C_{k}\psi_{k}$: $n_{k}=|C_{k}|^{2}$. And
under an additional condition $\|C\|=1$ it is the unconditional distribution
of probability of a individual particle occurrences in space of the impulses,
the relativistic invariant. The traditional distributions coincide with the
latter accurate to normalization, but they are attributed with the sense of
conditional distribution at the moment of time $t$. This sense contradicts
their relativistic invariance, and moreover, generates the known
contradiction, [1]: it requires instant fixing of an impulse at measurements,
whereas restrictions of accuracy of RQP observation require a long fixing. In
case of complex WF the charge is added to energy and impulse, and in
multicomponent case spin is added. Let us emphasize, that presence of negative
frequencies in decomposition of WF and, accordingly, observation of a
(individual) particle with different values of charge in fixed pair of
sessions, is admissible, but not pair occurrence (such results are not taken
into account). It does not contradict the laws of conservation which should be
carried out only on the average. But it can be limited by external for QM
laws, us law of the electric charge conservation.

## 6\. Quantum Field . The occupation of space-time

New properties of RQP are transformed as applied to the QF in the form of
characteristics of the particles distributions into the space-time. A suitable
foundation for this: the Diracs and Jordans conception of QF as a system of
identical particles, [2]. Let’s consider a system of the N identical particles
with common wave functions $\psi(x)\subset H(V)$. Let $y=\\{y_{k}\\}$ be set
of the QP observables and $\langle y\rangle=\\{Y_{k}\\}$ is this of the
system. Let $\\{\psi_{i}\\}$ be the eigen basis of some physical quantity;
$n=\\{n_{i}\\}$ be the set of its occupation numbers, $n_{i}=0,1,2,\ldots,N$;
$\Phi(n)$ be the symmetrized (respectively, antisymmetrized) WF of the system
expressed in terms of $n$.

On observation procedure. The operations of the system realization in
identical physical conditions corresponding to given WF have been defined. A
single session of observation is being performed over each realization during
the time $T$. In each session appears, in general, not simultaneously, $N$
particles. In doing so, for every particle is fixed value y. Aggregate
characteristics of the system are expressed directly through these values. For
each WF their averages $\langle Y\rangle(\Phi)$ corresponding to infinite set
of sessions are defined. The time is excluded from the process of observation
as a parameter.

Formal description. WF $\Phi(n)$ maps a set of values $n$ onto complex
Euclidean space $\Upsilon$ with the elements $\gamma$ and product
$\gamma_{1}\gamma_{2}$ and is considered as an element of a Hilbert space with
the product $(\Phi_{1},\Phi_{2})$, normalized: $\|\Phi(n)\|^{2}=1$. The
averages $\langle Y\rangle(\Phi)$ are the HQF: $\langle
Y\rangle(\Phi)=(\Phi,\Lambda\Phi)$, $\Lambda$ — corresponding operators. In
particular, $P(Q,\Phi)=\left(\Phi(n),f(n)\Phi(n)\right)$, where $f(n)=1$,
$n\in\linebreak\in Q$, $f(n)=0$, $n\notin Q$, is probability of the event
$n\in Q$. Second quantization reproduces nonrelativistic analog, including, in
addition to $\Phi(n)$, operators of the disappearance and the birth of
particle $a_{i}$, $a^{+}_{i}$ and they attributed to a point $x$, — wave
operators (WO):

$\Psi(x)=\sum_{i}\psi_{i}(x)a_{i},\quad\Psi^{+}(x)=\sum_{i}\psi^{*}_{i}(x)a^{+}_{i};$
(2)

with the follow differences: WO as functions of $x$ are defined in $H$, but
not in $\mathbf{H}$, and time is included in them symmetrically with spatial
coordinates; formalism must be relativistically invariant; if $\\{\psi_{i}\\}$
is a collection of the plane waves, then this basis is “doubled” in virtue of
the appearing states with negative frequencies and, accordingly, — additional
feature of QP, the charge, and the related components in (2) gain an unified
view: $\psi_{i}(x)a_{i}=\linebreak=\psi^{+}_{i}(x)b^{+}_{i}$,
$\psi^{+}_{i}(x)a^{+}_{i}=\psi_{i}(x)b_{i}$, where $b_{i}$, $b_{i}^{+}$, are
operators of the appearance and the birth of these particles. Also the
synthesis technique of the operators $\Lambda=\\{\Lambda_{k}\\}$ of the system
characteristics $Y=\\{Y_{k}\\}$ is reproduced, including a rule: we record the
average for an individual particle, and produce the replacement:

$\begin{array}[]{c}\langle
y\rangle(\psi^{*},\psi)=(\psi,L\psi)={\displaystyle\int}\psi^{*}(x)L\psi(x)dE;\quad\psi\to\Psi(x),\\\\[5.69054pt]
\psi^{*}\to\Psi^{+}(x),\quad\Lambda=\langle
y\rangle\left(\Psi^{+}(x),\Psi(x)\right).\end{array}$ (3)

In this product WO are considered as the elements of $H$. Respectively:
$\langle Y\rangle(\Phi)=(\Phi,\Lambda\Phi)$. Let $\\{\psi_{i}\\}$, be the
eigen basis of the observed, and $P_{i}$ be probabilities of the corresponding
values $y_{i}$ when an individual RQP is observed (or corresponding average
numbers of measurements). Then without using WO and (3) and making replacement
$P_{i}\to n_{i}$, we may write down

$\Lambda=\sum_{i}n_{i}y_{i};\quad\langle
Y\rangle(\Phi)=\biggl{(}\Phi,\sum_{i}y_{i}n_{i}\Phi\biggr{)}=\sum_{i}y_{i}\langle
n_{i}\rangle;$

$\langle n_{i}\rangle$ — the average occupation numbers.

Operators characterizing the coordinates distribution of particles are lacking
in RQM, as for the individual RQP, and (1), (3) fill this gap.

Operator $\Lambda(Q)$ of the particles amount in a domain $Q\subset E$ is the
follow:

$\Lambda(Q)=\left(\Psi^{+}(x),f(x)\Psi(x)\right)=\int\limits_{Q}\Psi^{+}(x)\Psi(x)dE,$
(4)

where $f(x)=1$, $x\in Q$, $f(x)=0$, $x\notin Q$. Operator
$\Lambda(\mathbf{Q})$ of the particles amount in a domain
$\mathbf{Q}\subset\mathbf{E}$ coincides with operator $\Lambda(Q)$,
$Q=\mathbf{Q}\times(0,T)\subset E$. The average ocupation number of the domain
$Q:\langle N\rangle(Q,\Phi)=\left(\Phi,\Lambda(Q)\Phi\right)$.

Let $S(\psi^{*},\psi)=(\psi,L_{S}\psi)$ be action functional of QP, and
$S(\Phi^{*},\Phi)=\linebreak=(\Phi,\Lambda_{s}\Phi)$ — of the system. Varying
the latter with respect to $\Phi^{*}(n)$, we get the following WE:

$\Lambda_{s}\Phi(n)=0,\quad\Lambda_{s}=\left(\Psi^{+}(x),L_{s}\Psi(x)\right).$
(5)

Let now the number $N$ be not fixed but varies from session to session. This
is consistent with the model of QF in the corpuscular concept with an accuracy
of the non-observed characteristics of the vacuum state. WF should now be
symmetrized also in $N$, [2]. And the rule (2), and respectively the concrete
representations of operators $\Lambda$, including $\Lambda(Q)$, remain valid
as well as WE.

6.1. Relativistic invariance of the field description. The number of particles
in a given state is a result of observation, which does not depend on the
choice of coordinates. Accordingly, a set of occupation numbers is
relativistic invariant, as well as operations on them $a_{i}$, $a^{+}_{i}$.
Therefore, WO have relativistic transformation properties of the particle WF
$\psi$, and operators $\Lambda$ have properties of their analogues $L$.
Furthermore,

$\begin{array}[]{c}\Lambda=\left(\Psi^{+}(x),L\Psi(x)\right)={\displaystyle\sum\limits_{ij}}l_{ij}a^{+}_{i}a_{j},l_{ij}=(\psi_{i},L\psi_{j});\\\\[5.69054pt]
\langle
y\rangle(\Phi)=(\Phi,\Lambda\Phi)={\displaystyle\sum\limits_{ij}}l_{ij}(\Phi,a^{+}_{i}a_{j}\Phi).\end{array}$

But $(\Phi,\Phi)$ is invariant, as a corresponding value of the probability
measure, as well as operators $a^{+}_{i}a_{j}$. Thus,
$(\Phi,a^{+}_{i}a_{j}\Phi)$ are invariants too, and the form
$(\Phi,\Lambda\Phi)$ possesses the relativistic transformation properties of
the form $(\psi,L\psi)$. This description is fully invariant, unlike
decomposition of field into system of the oscillators, which is invariant in
general, but contains noninvariant fragments. Also this decomposition is not
suitable to description of the coordinates distributions.

6.2. Representation of the Quantum Field Characteristics in Terms of
Occupation Numbers of the Impulse States. Let $\\{\psi_{i}(x)\\}$ be the eigen
basis of impulse, of spin and of charge. With regard to energy, impulse, spin,
charge (3) provides the textbook operators. Operator $\Lambda(Q)$ of the
particle amount in the domain $Q\subset E$ was given with (4), where basis
$\\{\psi_{i}(x)\\}$ in WO should be renormalized: $\|\psi_{i}(x)\|^{2}=1$
instead of “particle in the unit volume”. Let write (4) for the concrete
particles.

Scalar neutral boson. $a_{i}=b_{i}$, $\Psi^{+}(x)=\Psi(x)$;
$\Lambda(Q)=(1/2)\int\limits_{Q}\Psi^{2}(x)dE$.

Photon. In framework of model of item 4.2:
$\Psi^{+}(x)=\Psi(x)=\linebreak=\left(\Psi_{2}(x),\Psi_{3}(x)\right)$;
$\Lambda(Q)=(1/2){\displaystyle\int\limits_{Q}}\Psi^{2}(x)dE$. Here the WO
$\Psi_{2}(x)$, $\Psi_{3}(x)$ correspond to the components of basis $u^{2}$,
$u^{3}$.

The scalar charged boson.
$\Psi(x)={\displaystyle\sum\limits_{i}}a_{i}\psi_{i}(x)+b^{+}_{i}\psi^{*}_{i}(x)$,
$\Psi^{+}(x)=\linebreak={\displaystyle\sum\limits_{i}}a^{+}_{i}\psi^{*}_{i}(x)+b_{i}\psi_{i}(x)$;
$\Lambda(Q)={\displaystyle\int\limits_{Q}}\Psi^{+}(x)\Psi(x)dE$.

6.3. Representation of the Quantum Field Characteristics in Terms of
Occupation Numbers of the Space — time Cells. Let introduce into consideration
eigen basis of coordinates. For unification with discrete basis of the impulse
we make it at a prelimit level of Riemann integral sums. We split the beam $V$
into collection of beams $v(\xi)$ having volumes $w(\xi)$, every of them
marked with belonging to one value $x=\xi$. Define a set of functions
$\psi(x,\xi)$ with $\xi$ parameter: $\psi^{2}(x,\xi)=w^{-2}(\xi)$, $x\in
v(\xi)$; $\psi(x,\xi)=0$, $x\notin v(\xi)$. Approximate $\psi(x)$ with step-
function $\psi^{\prime}(x)=\psi(\xi)$, $x\in v(\xi)$. Functions $\psi(x,\xi)$,
$\psi^{\prime}(x)$ are the elements of finite-dimensional subspace
$H^{\prime}\subset H(V)$ with orthogonal basis $\psi(x,\xi)$ with accuracy of
approximation:

$\begin{array}[]{c}\left(\psi^{\prime}_{1}(x),\psi^{\prime}_{2}(x)\right)={\displaystyle\sum\limits_{\xi}}\psi^{\prime*}_{1}(\xi)\psi^{\prime}_{2}(\xi)w(\xi);\quad\|\psi^{\prime}(x)\|^{2}=1,\\\\[5.69054pt]
\|\psi(x,\xi)\|^{2}=w(\xi)^{-1};\\\\[5.69054pt]
\psi^{\prime}(x)={\displaystyle\sum\limits_{\xi}}\psi(\xi)\psi(x,\xi)w(\xi)\to\psi(x)={\displaystyle\int}\psi(\xi)\delta(x-\xi)d\xi,\\\\[5.69054pt]
w(\xi)\to 0.\\\\[5.69054pt]
P(Q,\psi^{\prime}(x))={\displaystyle\int\limits_{Q^{\prime}}}\psi^{\prime
2}(x)dE={\displaystyle\sum\limits_{\xi}}P(\xi),\
P(\xi)=\psi^{2}(\xi)w(\xi).\end{array}$ (6)

Here $Q$ is the minimal set of beams $v(\xi)$ covering $Q$. Accordingly,
complete the second quantization apparatus to the eigen basis of coordinates
$\psi(x,\xi)$: $n=\\{n(\xi)\\}$ — the set of occupation numbers of the beams
$v(\xi)$, $n(\xi)=\linebreak=0,1,2,\ldots,N$. Making the replacement
$P(\xi)\to n(\xi)$, we obtain the operator $\Lambda^{\prime}(Q)$ of the
particles amount in the domain $Q^{\prime}\subset E$:

$\Lambda(Q^{\prime})={\displaystyle\int\limits_{Q^{\prime}}}\Psi^{\prime+}(x)\Psi^{\prime}(x)dE={\displaystyle\sum\limits_{\xi}}n(\xi),\
\xi:v(\xi)\in Q^{\prime};$

$n(\xi)$ are acting as eigen values of operator $\Lambda^{\prime}(Q)$. The
average amount of particles in $Q^{\prime}$:

$\langle
N\rangle(Q^{\prime},\Phi)=\left(\Phi(n),{\displaystyle\sum\limits_{\xi}}n(\xi)\Phi(n)\right)={\displaystyle\sum\limits_{\xi}}\langle
n(\xi)\rangle,\quad\xi:v(\xi)\in Q^{\prime};$

where $\langle n(\xi)\rangle$ is average particles amount in $v(\xi)$.

6.4. An Individual Particle as a Quantum Field subsystem. Let consider an
individual particle with WF $\psi(x)$ in terms of the second quantization as
subsystem $N=1$ of the system “QF”. We define it in the following way: we
observe a QF, and only the sessions of QF, that bring out $N=1$. We have:
$n=(0,0,\ldots 0,1,0,\ldots)$, $n_{i}=0,1$. The term “identical particles”
loses its meaning, and WF $\Phi(n)$ does not contain the permutation operator.
We find an average amount of particles in domain $Q\subset E:\langle
N\rangle(Q,\Phi)=\left(\Phi,\Lambda^{\prime}(Q)\Phi\right)$, using the finite
dimensional approximation given above. $\\{\psi(x,\xi)\\}$ — is the eigen
basis of the observed. To every $n$ corresponds the event $x(n)\in E$.
Identify the point $x(n)$ with the mark $\xi$ of the beam $v(\xi)$. We have:
WF $\Phi(n)=\psi(x(n))$ map the set of all $n$ into $U$. In agreement with (6)
should be put
$(\Phi_{1},\Phi_{2})={\displaystyle\sum\limits_{n}}\Phi_{1}(n)\Phi_{2}(n)w(\xi=x(n))$.
We have:

$\begin{array}[]{c}\|\Phi(n)\|^{2}=1;\langle
N\rangle(Q,\Phi)=\left(\Phi,\Lambda^{\prime}(Q)\Phi\right)={\displaystyle\sum\limits_{n}}\Phi^{2}(n)=\\\\[5.69054pt]
{\displaystyle\sum\limits_{\xi}}\psi^{2}(\xi)w(\xi)=P(Q^{\prime},\psi)\to{\displaystyle\int\limits_{Q}}\psi^{2}(\xi)d\xi,w(\xi)\to
0;\\\\[5.69054pt] n:x(n)\in Q,\xi:v(\xi)\in Q^{\prime}.\end{array}$

Thus, this function $\Phi(n)$ meets the definition of WF, and observing such
subsystem of QF gives the probability density (1) of the individual particle.

## 7\. About Relations of Uncertainties and Observation Accuracy Estimates

In non-relativistic QM the Heisenberg relations of uncertainties are the
effects of the statistical postulates described above. Within the framework of
traditional model of relativistic QP they no longer have this theoretical
basis due to the absence of the required law of the coordinates distribution.
Nevertheless, they are used in the same way, strictly speaking, now as an
independent postulate. In the model considered these relations obtain
substantiation. Therewith, while in non-relativistic QM the relation
coordinate-impulse and time-energy are deduced in different ways and have
different sense, [3] . 185 – 188, but herewith, they possess full formal and
semantic symmetry. Theoretical bottom threshold of uncertainty of coordinates:
$\triangle x>\triangle x_{\min}=\hbar c/\varepsilon$ ($\varepsilon$ — energy),
caused by these relations and absence of the QF effects (for photon this is an
order of its wave length), loses validity, because under the observation in
framework of our procedure QF effects are admissible. This theoretical bottom
threshold is replaced with the minimal practically acceptable value of
probability of the individual particle fixing under the observation of QF.

It is required to give a special interpretation of the uncertainties relation
time-speed-impulse $\mathbf{v}\triangle t\triangle p>\hbar$, and a derived
from it relation of impulse observation accuracy with duration of an
observation session $\triangle t\triangle p>\hbar/c\triangle t$. First of all,
does the concept “QP speed” make sense in the given model, and if yes, which
one?

## 8\. Material for the experiment

It provided by the new properties of RQP and QF being predicated here: the
probability density formulas for various types of RQP coordinates and formulas
for distributions of amounts of particles into space-time for QF. The simplest
variant to check the latter: to locate WF $\Phi$ at eigenbasis of 4-impulse,
to calculate the corresponding theoretical $\langle N\rangle(Q,\Phi)$ and
comparing it with directly measured $\langle N\rangle(Q,\Phi)$. A significant
new feature of the model observation: nonconditionality of the measure from
the moment of time $t$. For a time $T$ the repeated reactions of particles
with instrument of observation are possible, and respectively — nonuniqueness
value $y$ when measurement. The latter should be eliminated in any way. Unlike
textbook views, an individual RQP observation under procedures described are
not burdened with problem of preventing effects of QF. The particular interest
represent the photon coordinates observation. From one side, it is ideally
provided the uniqueness of the collision with the particle-device as far as it
disappears under the reaction. From the other side the validity of (1) for it
is stipulated with the additional postulate, which is alternate to
electrodynamic principle of gradient invariance. And an experiment will
determine the alternative: either the given principle is invalid here, and the
(1) is valid, or our postulate of photon is invalid, and it’s coordinate
distribution is not determined.

## 9\. Conclusions

When the statistical part of RQP has been included in logic of the SR
sufficiently fully, there appears, despite common opinion, a theoretical
possibility to preserve informative proper-ties of the WF in the extent
comparable to non-relativistic QM. There appears a theoretical op-portunity to
observe spatial-time coordinates of the RQP, and the presentation of the
probability density $g(x)=\psi(x)^{2}$ satisfying all necessary requirements
gets defined. Here WF $\psi(x)$ maps space-time $E$ into Euclidean space $U$,
characteristic for each type of particles: boson, real and complex, scalar and
vector, including photon, and electron. In application to photon this formula
of the density is conditioned by an additional assumption, that is alternative
to the electro-dynamic principle of gradient invariance. Density $g(x)$ in a
simplest case is formally alike the density of the probability of non-
relativistic particle’s coordinates, but has other content: other observation
conditions, other properties, other interpretation of density. Observation
sessions are not caused by moment $t$, i.e. time is excluded from the
observation process as a parameter, but is present in the argument $x$ and the
observable value $y=x$. Density now is relativistic invariant. Instead of
stochastic dance of the particle, described by the probability flow with
spatial density $f(t,\mathbf{x})=|\psi(t,\mathbf{x})|^{2}$, we have
probabilistic distribution of particle’s appearance in space-time with density
$g(x)$, not resulted, in general, to motion in the space. In a simplest case
the idea of such distribution of coordinates was considered in [4], and [5] is
a short version of this paper. The interpretation of distributions of energy-
impulse of RQP is also clarified here. Relations of uncertainty of Heisenberg
are the yield of the postulates of non-relativistic QM. In the frame-work of
traditional model of RQP they no longer have this theoretical rationale, as
there is no law of coordinates’ distribution, necessary for that. These
relations obtain the justification in the model considered. An observation
procedure, filtering effects of QF, have been proposed and justified.
Respective restrictions of the RQP observation precision, caused by these
effects, are abandoned. As applied to the QF, this new constructs are
transformed to new characteristics of the particles distribution in space-
time, which complete distribution throughout impulses. The operators of these
distributions and the invariant relativistic description for free QF have been
obtained.

Abbreviations: Quantum Particle — QP; Wave Function — WF; quantum mechanics —
QM; relativistic QP — RQP; Special Relativity — SR; Hermitian Quadratic Form —
HQF; Wave Equation — WE; Relativistic QM — RQM.

## References

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  * [2] P.A.M. Dirak. Prinicples of Quantum Mechanics. Oxford, Clarendon Press, 1958.
  * [3] L.D. Landau, E.M. Lifshits. Theoreticheska’a Physica, T. III, Kvantova’a Mechanika. Phys-MathGis, Moscow, 1963, 702 p. (in Russian).
  * [4] V.F. Krotov. // Quadratic Parametrical Families of the Measures and Basics of Quantum Me-chanics. International Journal of Computing Anticipatory Systems, 2006, v. 17, p. 77 – 92.
  * [5] V.F. Krotov. O statisticheskih svoistvah volnovoj functsii svobodnoj relativistscoj chastitsy. // Trudy Instituta Problem Upravleni’a RAN, v. XXVIII, Moskva, 2008, p. 5 – 13 (in Russian).