Space of dark states in Tavis-Cummings model Текст научной статьи по специальности «Физика»

ТЕОРЕТИЧЕСКИЕ ВОПРОСЫ ИНФОРМАТИКИ, ПРИКЛАДНОЙ МАТЕМАТИКИ, КОМПЬЮТЕРНЫХ НАУК И КОГНИТИВНО-ИНФОРМАЦИОННЫХ ТЕХНОЛОГИЙ / THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES

УДК 539.186.3

DOI: 10.25559/SITITO.15.201901.13-26

Space of Dark States in Tavis-Cummings Model

Yu. I. Ozhigov1,2

1 Lomonosov Moscow State University, Moscow, Russia

2 Institute of Physics and Technology of the Russian Academy of Sciences, Moscow, Russia ozhigov@cs.msu.su

The dark states of a group of two-level atoms in the Tavis-Cummings resonator with zero detuning are considered. In these states, atoms can not emit photons, despite having non-zero energy. They are stable and can serve as a controlled energy reservoir from which photons can be extracted by differentiated effects on atoms, for example, their spatial separation. Dark states are the simplest example of a subspace free of decoherence in the form of a photon flight, and therefore they are of interest to quantum computing. It is proved that a) the dimension of the subspace of dark states of atoms is the Catalan numbers, b) in the RWA approximation, any dark state is a linear combination of tensor products of singlet-type states and the ground states of individual atoms. For the exact model, in the case of the same force of interaction of atoms with the field, the same decomposition is true, and only singlets participate in the products and the dark states can neither emit a photon nor absorb it. The proof is based on the method of quantization of the amplitude of states of atomic ensembles, in which the roles of individual atoms are interchangeable. In such an ensemble there is a possibility of micro-causality: the trajectory of each quantum of amplitude can be uniquely assigned.

Keywords: Tavis-Cummings Model, Dark States, Decoherence Free Subspace.

Acknowledgements: The work is supported by the Russian Foundation for Basic Research within the scientific project No. 18-01-00695 a.

For citation: Ozhigov Yu.I. Space of Dark States in Tavis-Cummings Model. Sovremennye informa-cionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2019; 15(1):13-26. DOI: 10.25559/SITITO.15.201901.13-26

Abstract

G ®

Контент доступен под лицензией Creative Commons Attribution 4.0 License. The content is available under Creative Commons Attribution 4.0 License.

Modern Information Technologies and IT-Education

Пространство темных состояний в модели Тависа-Каммингса

Ю. И. Ожигов12

1 Московский государственный университет имени М.В. Ломоносова, г. Москва, Россия

2 «Физико-технологический институт имени К.А. Валиева» Российской академии наук, г. Москва, Россия

ozhigov@cs.msu.su

Аннотация

Рассматриваются темные состояния группы двух-уровневых атомов в резонаторе модели Тави-са-Каммингса с нулевой расстройкой. В этих состояниях атомы не могут испустить фотона, хотя обладают ненулевой энергией. Они устойчивы и могут служить управляемым энергетическим резервуаром, из которого можно извлечь фотоны путем дифференцированного воздействия на атомы, например, их пространственного разделения. Темные состояния — простейший пример подпространства, свободного от декогерентности в виде улета фотонов, и потому представляют интерес для квантовых вычислений. Доказано, что а) размерность подпространства темных состояний атомов есть числа Каталана, б) В RWA приближении любое темное состояние есть линейная комбинация тензорных произведений состояний синглетного типа и основных состояний отдельных атомов. Для точной модели в случае одинаковой силы взаимодействия атомов с полем справедливо то же разложение, причем в произведениях участвуют только сингле-ты и темные состояния не могут ни испустить фотона, ни поглотить его. Доказательство основано на методе квантования амплитуды состояний атомных ансамблей, в которых роли отдельных атомов взаимозаменяемы. В таком ансамбле имеется возможность микро-причинности: траекторию каждого кванта амплитуды можно определить однозначно.

Ключевые слова: модель Тависа-Каммингса, тёмное состояние, подпространство, свободное от декогерентности.

Благодарности: статья подготовлена при поддержке гранта Российского фонда фундаментальных исследований № 18-01-00695 а «Конечномерные модели квантовой электродинамики».

Для цитирования: Ожигов Ю. И. Пространство тёмных состояний в модели Тависа-Кам-мингса // Современные информационные технологии и ИТ-образование. 2019. Т. 15, № 1. С. 1326. DOI: 10.25559^1Т1Т0.15.201901.13-26

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019 ^ 2411-1473 sitito.cs.msu.ru

Introduction. Background

Interaction between light and matter described by quantum electrodynamics (QED) is the most fundamental force, and at the same time it represents the simplest illustration of the power of quantum theory (see [1],[2]) in its single-particle form, described by the Feynman diagrams. From a logical point of view, fully justified is quantum electrodynamics of a single charge, which can be renoral-ized by the theorem of Bogolubov and Parasuk (see [3] and also [4]).

For the many body quantum electrodynamics the corresctness rests not on the possibility to renormalize it but rather on the adequacy of the transition to tensor products of spaces of states that by default is considered an absolutely legal mathematical technique for systems of many bodies. This method never failed in cases where we could calculate the amplitude of the transition to the end, and gave predictions surprising on the accuracy. However, extrapolation of this technique to systems of many non-identical charges can not give any verifiable result due to the exponential growth of computational complexity with increasing number of charges. This led to the fundamental idea of a quantum computer ([5]), as a necessary tool for modeling complex multi-charge systems. A quantum computer with computational capabilities goes beyond the scope of the computational apparatus of physics accessible to us (fast quantum computation — see [11]), and therefore its very idea needs a particularly careful experimental verification and necessary refinements. The results of numerous experiments conducted since the early 1980s showed that it is hardly possible to build a quantum computer straightforwardly according to the original Feynman scheme ([5]) because of the decoherence phenomenon associated with the inability to isolate the quantum system from the medium (a review of approaches to open quantum systems, see the book [12]). Therefore, the problem of finding quantum states that would be isolated from the medium by its very form and would have sufficient flexibility to map all quantum states in general (a known attempt in this direction is a topological quantum computer, see [13]) has come to the forefront.

In this paper we study the simplest states of ensembles of two level atoms: dark states. It is proved that such states are exclusively superposition of tensor products of EPR singlets, e.g. states of the form |oi) - |l0). This means that optical darkness for two-level systems is closely related to the spin description: singlet states have zero total spin. Such a transparent connection exists only for two-level systems, that is, for spin 1/2.

Another aspect of the problem of quantum computers is overcoming the com — putational difficulties that inevitably arise when applying QED to the modeling of quantum computing. Quantum computation itself can be performed on the states of charged particles (spatial positions or spins), but the main source of decoherence is the interaction of charges with the field. Therefore, the simulation of a quantum computer must take place within the framework of QED, which is much more com— plicated than ordinary quantum mechanics, in which the field is manifested only in the form of a scalar potential.

Of particular importance are finite-dimensional models of QED, in which it is possible to reduce the complex states of the electromagnetic field to several qubits, meaning the presence or absence of a photon of a certain mode in a limited space — time region. The main of these models was proposed by Jaynes and Cummings for a two-level atom located in an optical Fabry-Perot resonator [6]), and

then was gen— eralized to ensembles of such atoms (the Tav-is-Cummings or Dick— see [7]) and on several cavities connected by an optical fiber (the Jaynes-Cummings-Hubbard model [14]). Within these models and their multiple options, it is possible to describe accu— rately the effects important for applications, for example, DAT (dephasing assisted transport— [15],[16]). On the basis of finite-dimensional models of QED it is possible to obtain nonlinear optical effects, which in principle opens door to construction of elementary gates for quantum computations (see [19]).The JCH model serves as an important generalization of the so-called continuous quantum walks ([17]) and can be used for their practical implementation.

The states of atoms with nonzero energy, in which they do not emit a photon are called dark states. Such states are not subject to decoherence because, even if they have a high energy of atomic excitations, they can stay in this state theoretically indefinitely for a long time without emitting photons. For two-level atoms, such states can be obtained in an optical cavity, for example, using the Stark-Zeeman effect ([18]).

It is possible to extract energy in the form of photons from an atomic system in a dark state by spatial separation of atoms, dephasing noise or other differentiated impacts to atoms. In this case, the resonator is needed only to obtain a dark state, the atomic system can be then removed from the cavity, while retaining the property of darkness, provided that we keept atoms together (for example, using optical tweezers).

Dark states have numerous uses. In particular, their role in the organization of inter-atomic interaction was considered in the work [26], for the control of solid-state spins — in work [22], for the control of macroscopic quantum systems— in work [27], one of the effects of the dark state in the light-harvesting complex can be found in the work [25]. Some methods for obtaining dark states in quantum dots can be read in papers [20], and also in [21]. The destruction of dark states by a magnetic field or modulated laser polarization was considered in [24]. In the works [8],[21],[10] singlet states are also considered as states with zero total spin forming the core of the decreasing operator, however, there is no detailed analysis of the structure of the subspace formed by them in these articles. The purpose of this paper is an explicit description of the of dark states. It follows from their definition that they form a subspace, which we will call dark subspace. We will be interested in the structure of this subspace and its dimension. The structure of dark states in the systems of kudits (d "two systems) is most thoroughly studied in the work [23]. In particular, for two-level systems in the work [23] it is proved that the dark states are precisely the stationary points of the tensor product of the groups SU(2). These stationary points are called in this work ''singlet states", since two-atom singlets of the EPR-pair type |oi^ - |l0) are invariant for this group. We shall prove that the dark states can be represented as a linear combination of products of simple singlets, that is, tensor products of EPR pairs. This fact justifies the term 'singlet state", having a chemical origin: singlet states of electron spins are pairing for atoms, that is, they make it possible to form a covalent bond. We consider Tavis-Cummings model, consisting of the optical cavity — the res— onator, and a group of identical two-level atoms inside it. The cavity length L = nc / a>c is equal to half the wavelength of a photon with a frequency , which differs from the frequency

of atomic transition ma by the small detuning 5 = mA-ma, || mA . A small detuning value provides a constructive interference of the

Modern Information Technologies and IT-Education

electric field of the photons inside the cavity and a long retention time of the photons of frequency m inside the cavity. In this case, we can write the Hamiltonian of the interaction of atoms and the field inside the cavity in the dipole approximation in the Jaynes-Tavis-Cummings form:

HTC = haAa+a + hma+ H,, H, =£gq(a+ + a)(a++ +aq), (1.1)

where + means conjugation, a+, a are field operators of creation — annihilation of photon, CT+,CT are raising and lowering operators of q-th atom, acting on its ground (o\ ^ and excited (i\ ^ states as

a |0) = 0, a 1 = |0) (here and below, by default, it is assumed

that the remaining state components are acted upon by the identity operator). Here the force of interaction of an individual atom q with the field gq = dEJhma/2e0L, Eq = sin(nxq / L) is the distribution of the photon field intensity along the resonator, x is the coordinate of the atom along the axis of the cavity, V is the effective cavity volume, d is the dipole moment of an atom, Eo is the electric JLji) (oi) +110)) constant. We suppose, for simplicity, that the detuning is '^ p

zero. The frequencies and strength of the interaction are always assumed to be nonzero.

We denote the part of the interaction of the Hamiltonian of the form ^ gq(a+aq + aa*+) by hrwa , and the other part of interaction

tion of matter and light it follows that the operator of emission of a photon in the RWA approximation is the action of the operator a+a, and for the exact model — of the operator a+(a+a+). Similarly, the photon absorption operator for the RWA approximation is , and for the exact model it coincides, to within an inversion of the field component, with the photon emission operator: a (a + a+). Therefore, the subspaces of dark and transparent states in the RWA approximation are the kernels of operators.& and correspondingly, and the invisible is the intersection of these sets. In the exact model the dark, transparent and invisible states are the same — the kernel of the operator a+a+.

So, the properties of darkness and transparency, taken separately from each other, depend on the applicability of RWA approximation to the considered model. The states and are dark and transparent only if it is applicable. If we refuse from the RWA approximation, these states will lose these properties. For example, the state becomes bright if the Hamiltonian has the form (1.1), since can go to a state with one photon of the form

Ë 9q (*+< + ) by H„o„RWA .

In the case of weak interaction g^ 1 we can leave only sum-

mands a+a aa+, conserving the energy, e.g. HRWA and the other

two, which do not conserve the energy h , we can omit (rotat' nonRWA

ing wave approximation RWA).

A state that can emit a photon is called a bright state. A state, which is not a bright will be thus dark (see, for example, [28]). A state that can not absorb a photon, we call transparent. A transparent dark state we call invisible. In other words: invisible is a state of atoms in the cavity, which can neither emit nor absorb a photon, e.g. the ensemble in this state does not interact with the field. A complete state of the system of atoms and the field has the form of a superposition of the basic states :|Y) = ^ hj j | ,

jp , ja

where the natural number j denotes the number of photons in the field, and the binary string j denotes the state of distinguishable atoms taken in a fixed order, so 0 and 1 denote the ground and excited states of the corresponding atom. Elements j , j, j of the string j = (j, j, j ), uniquely corresponding to atoms, we call qubits. A complete state of the system. Ibelongs to the tensor

I / gen

product.^ = ^ ® ^ of the state spaces of the field and states of atoms. In this article, we are only interested in processes with the emission of at most one photon, so the main object will be the atomic states having the form = j ,

which by default we call states, and the index a we omit.

If we assume RWA approximation, an example of a dark two atomic state can be: = |00), an example of a transparent = . We introduce the notation â = ~^aq ■ From the form of the interac-

T2

Throughout, we will identify the base state | j with the string of the binary expansion of the natural number j.

Let us consider an example of two-qubit states in the RWA approximation. First, let the interaction force of both atoms with the field be the same: g = g . We choose as the new basis the triplet and singlet states of the form

K>=100>.K>=I11).\t)=^2H+l01>)' Is)=^2H-

From them the singlet alone is invisible, and the triplet is neither dark nor transparent. Now suppose that g t g2, for example, atoms occupy different positions in the resonator. Then the state g2110_ g 101) (the atoms are numbered from left to right) will be dark, the state g |10 _ g2101) is transparent, and there will be no invisible states at all.

In the future, we consider the case of atoms with the same interaction energy with the field: g = g, i = 1,2,..., n, the detuning^-^ between the frequencies of atoms and the cavity is assumed to be zero, and we will consider only RWA approximation (unless explicitly stated otherwise), up to the last paragraph, where we consider the general case.

The weight (Hamming) v. of the basic state | j is the number of units in it. The ground state of the atoms | j is called equilibrium if its weight is half the number of all atoms. Equilibrium states, therefore, are possible only for systems with an even number of atoms. The superposition of equilibrium basis states is called the equilibrium state of atoms. A more general property of atomic states is linearity. The atomic state is linear if all its basic components have the same weight.

We will show that the invisibility property does not depend on the applicability of the RWA approximation, in particular, all invisible states are equilibrium.

Structure of the dark subspace

Let | j be the base state of the system of n qubits; we introduce the notation N = 2n — this is the dimension of the entire quantum state space of the n — qubit system. We denote by 1( j) the Hamming weight of this state, i.e. number of units in it; then the number of

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019 ISSN 2411-1473 sitito.cs.msu.ru

zeros in it is 0(j) = n -l(j). We define a binary relation on the basis states, denoted by Emissi0n(j, j'), which is true if and only if j' is obtained from j by replacing the single unit by zero. In other words, j' is obtained from j by the action of the decreasing operator j on one of the atoms in the excited state. In this case 1(j') = 1( j) -1.

The emission of a photon by an atomic system in a state j has the

form \o)P\j) ^ |i)J/>, ' (2.1)

where Emission(j, j').

For a basic state |j^ we call j' — family the set of basic states | j, such that Emissi0n(j, j') is true. In the other words, j' — family consists of basic states | j'), for which the transition of the form

(2.1) is the photon emission. j' — family we denote by [j'] and call the state | j^ its parent.

Note that two different families can have no more than one common member. Let us now consider an arbitrary atomic state

= j). From the definition of emission of a photon it fol-

j

lows that the state is dark if and only if the system of equations of the form

YK = 0, (2.2)

MJ ']

is satisfied for all j'= 0,1,...,2" -1. Note that it is sufficient to require that these equalities be satisfied only for j'= 0,1,...,2" -2, because the family -lj is empty: no state can pass to the basic state consisting of only excited atoms when the photon is emitted. We denote by b£ the set of basic n — qubit states j, such that 1(j) = k, and by ^ — the subspace spanned on b£. Then for any basic state j' its family completely belongs to b"(j,+1. Consequently, every dark state is a superposition of dark states belonging to subspaces ^, k = 0,1,...,n-1.We denote by d£ the subspace ^, consisting of dark states. Then = nKer(ct). We will always number the qubits from left to right, denoting by the symbol * the missing qubit, so that, for example, instead of 1 we write |o * 1).

The examples of states from d£ are the so called (n, k) -singlets: the states obtained by the tensor product of k samples of states of the form | o) | l) -11 10) , where 1 < p < q < n and n - 2k states of

the form |o) ,1" q" n. For n = 4, k = 2 (n, k) -singlets will be, for

example, the following states

(4.2) =(|0 *1*)-|1* 0 *))(*| 0 *1)-|*1* 0) ) = 10011) - 0110)-| 1001 + ln0^), (2.3) (4, 2) =(0)|1 -| 1)|0)) = 0101)-|0110)-11001) +|1010),

(4, 2) =(0**1) -| 1**0))(|*01*)-|*10*)) = 0011)-|0101 -| 101^ + ln0^).

These states will be linearly dependent, but any two of them are linearly independent and form a basis of which is easy to verify directly.

We note that for n = 2k all (n, k) — singlets are invisible without

RWA.

Theorem

dim D ) = max { - Ckn , 0}.

1 So defined distance

Any state from D is the linear combination of (n, k) — singlets Proof

At first we prove the point 1.

Since a state = is dark if and only if the system of equa-

j

tion (2.2) is satisfied, the belonging |Y)e D£ is equivalent to the satisfaction of the system consisting of all equalities of the form (2.2) for all j', such that 1(j') = k-1. If k = n, then dimD) = 0 and point 1 is satisfied; since it is sufficient to consider the case k < n._ Then to the different j' will correspond the different equations from Sn. Since the system has Ck variables and Ck-1 equations to prove point 1 it would suffice to show that all equations from Sn are independent.

Any permutation of # from the group Sn acts naturally on the set B = {0,1}" of all binary strings j of length n; the result of such action is denoted by # j. In particular, the substitution (a, b) e Sn acts as a transposition of two qubits with the numbers a and b of the given string. We will call such a transposition essential if it affected two qubits with the different values. Then those and only those transpositions that change the string on which they act will be essential.

Lemma 0. For any string j e B and any n eSn the string # j has the form (as ,bs )(as_1,bs_1 )..(a1,b1 )j, where all numbers al,a2,...,as, bl,b2,...,bsare different and s equals the double Hamming distance between j and #j.

Proof. Let s be minimal of such numbers that for some set of substitutions (aq,bq), q = 1,2,...s the string #j has the form (as,bs)(as_l,bs_l)...(aj,bj)j. We prove that all numbers al,a2,...,as, bl,b2,...,bs are different. Indeed, let it be wrong and some qubit is affected twice. Since always (a,b) = (b,a) and the substitutions of the

form (a, b) and (c, d) for the different a, b, c, d commute we can change the places of substitutions (aqso that two of them (aq,bq), (aq-l,bq, such that bq_1 = aq bcomes ajacent. Since s are minimal among the numbers of qubits aq, bq, aq-l, bq-lare exactly 3 different and we can assume that the numbers of qubits aq-t, aq, bq are different. The values of these qubits in the binary string j' = (aq-2, bq-2)(aq-3, bq-3)..(a1, bl) j we denote by a, b, c. Thanks to minimality of s we have a t b and we can assume that a = 0, b = 1. If c = 0, the substitution (aq, bq j is undue. If c = 1, then (aq, bq), (aq_t, bq_t)/ = (aq_t, b^j' and the condition of mnimality is violated again. Hence, all qubits participating in the considered substitutions have the different numbers and their values in each substitution are different as well. It involves that s is double Hamming distance between j and #j. Lemma 0 is proved. We define the natural metrics on the set as follows. The distance d(j, j') between basic states j, j' e B^j is defined as the half of Hamming distance between them that is by Lemma 0 is the minimal number of substitutions (permutations of a pair of qubits) in the transition from j to j'.'

Sequence of substitutions j0 ^ jl ^... jr we call correct if all passages jt ^ jM, i = 0,1,...,r-1 are essential substitutions and any qubit is affected in it no more than once. We fix the arbitrary j0 e B.

Lemma 1. Let j0, jl, j2,..., jr be a sequence of states from. If for any q = 0,1, ...,r-1

Modern Information Technologies and IT-Education

— through the number of substitutions are more convenient than Hamming because Hamming distance between elements of are always even.

d(j, j) = d((, j)-l, d((, jq) = 1, (2.4)

then there exists the correct sequence of substitutions of the form j0 ^ jj jr, in which substitutions are determined uniquely and vice versa, if there exists such correct sequence then for all q = 0,1,...,r -1 the equalities (2.4) are true.

Induction on r. The basis is evident. Step. Let Lemma 1 be true for r _1 and prove it for r. Let at first equations (2.4) be satisfied. By the induction hypothesis, there exists a correct sequence P of substitutions j ^... jr_!, and by d(jq+l, jq) = 1 the passage jr_i ^ jr — is a substitution as well. This substitution must change zero and one, because otherwise we would have the contradiction with the condition djjr-1, j0) = d(jr, j0)-l. Then, if this step violates the correctness, there is a qubit that participates twice in transpositions from j0 jr and it is affected just at the last step jr_1 ^ jr. But then we could reduce this sequence of substitutions, having received a contradiction with condition d(jq, j0) = d(jq+1, j0)-1. Indeed, without loss of generality we can assume that the sequence P moves units from qubits with numbers 1,2, ...,r _ 1 to the positions r, r +1, ...,2r -2 in random order, on which initially standed zeroes, and the last substitution jr_1 ^ jr moves the 2r _ 2 -th qubit either to the place r _1, or to the place 2r _1. In the first case the sequence P can be reduced to sharter since its result can be reached by the mobement of only r _ 2 qubits. In the second case we can reduce the sequence j0 ^ j jr, because it factually replaces only r _ 1 units by zeroes, and by Lemma 0 it means that d jjr_, j0) = d j jr, j0), which contradicts to the condition.

Let the sequence j0 jr be a correct sequence and by the

inductive hypothesis the equalities (2.4) are true for all q = 0,1,..., r-1. The second equality will be true because jr_ ^ j _ is a substitution. If the equality djjr-1, j0) = djjr, j0)-l is violated then the passage from j0 to jr can be fulfilled in less than r substitutions and Hamming distance between j0 and jr is less than 2r that contradicts to the correctness of the sequence j0 jr,

because in it each qubit is affected only once and the Hamming distance between j0 and jr is then 2r . Lemma 1 is proved. We define the partial order on , putting j < j2, if and only if there exists the correct sequence of substitutions of the j0 ^ jj j2. Then we can arrange all the states in Bn_1

at the nodes of the graph D, in the initial vertex of which is j0, and for any vertex j 'all vertices j lying above j' connected to j' by an edge satisfy the equalitiesd(j, j0) = d(j', j0)+1 and are obtained from j' by exactly one substitution. In this case, any monotonically increasing path on this graph will contain vertices in increasing order of djj, j0). The existence and uniqueness of such a graph D follows from Lemma 1. We enumerate tiers of this graph beginning with zero tier, consisting of only j0.

The basic states j' e B^, lying in the tier p, will be called the parents of rank p. The rank of such a parent is equal to the total number of qubit numbers that are equal to one in j0, and zero to j', that is, the Hamming distance between these vertices. We will denote the set of these qubit numbers by rem(j'). The rank of the state j e Bn is the minimal rank of the parent j' e B^ whose family contains j: j' e[j']. The rank of state j e B£ is denoted by r (j). Lemma 2. Let the parent j' e B^ have rank p. Then exactly p of its family members have rank p _1, the remaining n -k +1 - p have rank p.

Proof. We first we note that 0 < p < min {k -1, n - k +1} .It follows

from the definition of the rank of the elements B£ that the members of the family [j'] having rank p _ 1 are exactly the basic states j obtained from j' by replacing zero by a unit in some qubit from rem(j'). Then all other members of the family [j'] have rank p (see Figure 0). Lemma 2 is proved.

We note that, for example, for k = n, there is a unique family, whose parent has rank zero, and this family consists of exactly one member, in which all the qubits have the value one. The rank of this member will also be zero.

We define the amplitude values X° for all j e B£ depending on the

rank j as follows. Let p = r(j). We put

-(-'f-^p^ (2'5)

Q(n - k +' - s)

0 0 0 0 0 111

rank — 3

1 1

тапк - 2

1 1 1

^ ^ \i ^

111 0 0 10 0 0

J

Fig. 1. j' — parent of rank 3, obtained from j0 by the substitutions pointed in the upper part of the picture. Two members of its family have ranka 3, nd three members have rank 2: instead of substitution of unit instead of zero in any q -th

qubit from rem (jwe can omit the substitution with q -th qubit in the passage j0 ^ j' and so obtain instead of j' the new parent of the rank 2 for the member of family [ j']

The correctness of this equation follows from Lemma 2, which guarantees the absence of zeroes in the denominator. Indeed, since p < n -k + 1 the only possibility of appearance of such zeroes is the value s = p = n - k +1. But the total number of such states j e B£, for which p = n - k +1, by Lemma 2 equals zero. The equation (2.2) will not then be true for j ' = j0, because the sum of amplitude values for the members of family of rank zero by Lemma 2 is ((n -k + 1)/(n - k +1) = 1. For the members of family of nonzero rank p the equation (2.2) is satisfied. Really, in view of Lemma 2 in such a family there are exactly p members of rank p_1, and exactly n -k +1 -p of rank p. Substituting the amplitude values X° from (2.5) for p and p_1 we transform the equation (2.2) to the sum of numbers of the form

(-1)

^ (p-1)! p

(-1)

p !(n - к +1 - p)

= 0.

Q(n - к +1-s) П(п - к +1 - s)

Fulfillment of the equation (2.2) for any family of nonzero rank and its violation for a family of zero rank with the chosen values of variables proves that the equation (2.2) for j' = j0 does not depend on other equations of this kind. Since j0 e is arbitrary, all the equations in (2.2) are independent, as required.

s-U

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019 ISSN 2411-1473 sitito.cs.msu.ru

THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES

The point 1 of the Theorem is proved.

We note that from this point it follows that every state invisible in the RWA approximation is an equilibrium state. Indeed, if the state is dark, then 2k " n, because otherwise the dimension of the dark subspace is zero. On the other hand, if the state is transparent, then when zeros are replaced by ones and vice versa, it becomes dark, and we have 2k > n, whence n = 2k.

where e L(p, <l) for s e{0,1, -1}. The application of anti-symmetrization to such states gives zero. Antisymmetrization applied to the states from Lpq, gives their doubling. If |®) e Ker(Anpq), then, since, according to what has been proved, the orthogonal component of the state vanishes by antisymmetriza-tion, and the straight component — doubles, we have e (Lp ^. Lemma 3 is proved.

We introduce the projector Vp on the subspace Lp in a natural

way:

V = - Y

p, q 2 ^ I

2 ker(p, q)

k)(.

Sp,q ® k

(2.8)

Fig. 2. Structure of the singlet state. The tensor product includes all pairs of qubits connected by any arc, so that the values of the qubits are selected either as shown in the figure or in the opposite way. The sign of the pair is positive, if 1precedes 0 (as indicated in the figure), and negative otherwise

We now prove item 2. Any (n, fc) -singlet can be represented, up to a permutation of qubits, in the following non-normalized form, where the factors of the form |0) are omitted (the number of such factors is n - 2k):

1= |(1*...**...* 0 — 0 *...**...* 1)(* 1... **.. .0 *—*0.. .**.. .1*) (2.6)

(*...* 10 *...* — *...* 01*... which is schematically depicted in Figure 1.

The linear span of the set A is denoted by L(A), the orthogonal complement to the subspace L is denoted by L1, the cardinality of an arbitrary set A is denoted by |A|.

Let p, q be a pair of numbers of qubits, p t q. Consider the two qubit space l(p, q), generated by the qubits with numbers p and q, and introduce the following notation for singlet and triplet states in this space:

Sp,q = | 0 pi 1 q -I 1 pi 0 q = \ 0 p 11 q 1 pi 0 q ^ ,q =| 0 pi ^ q ^q = 1 p 1 q ^

The first is a singlet, the other three are triplet states. These states form an orthogonal basis in l (p, q).

Consider an arbitrary state | Y) e L(B^ and let (p, q) Y) denote the state obtained from by permuting the qubits p and q. We introduce the antisymmetrization procedure for the state — by the equality

V) = \V)-(p, q.

We note that if was dark then Anpq will be dark as well for all p, q.

By r (p, q) we denote the set of basic states |r) of the set of all atoms but two: p and q. We denote by Lpq the subspace H, consisting of states of the form spq 0, where e L(r(p, q). These subspaces in general case are not orthogonal for the different pairs p, q .2 Lemma 3.

For p t q and | e Lpq the following equalities take place:

Im (Anp,q) = Lpq, Ker (Anp,q) = Up^^ | <F) = 21 <F).

Proof. By the definition, antisymmetrization on p, q always gives a

state belonging to Lpq. We have: Lp consists of the states of the

form

C K) + tpq H + tpq lO-

Lemma 3 can then be written in an equivalent form as the following

Corollary:

Corollary.

AnP q = 2 Vp ,q.

A state |D) s D£, k > 0 we call singular if it is orthogonal to all (n, k) = singlets.

To prove part 2 of the theorem, it suffices to show that the singular state must be zero. For this we need a number of additional facts concerning the subspace D£ of the dark states. Lemma 4.

For k > 0 Dn c L | U Lp

Proof.

In this Lemma it is necessary to represent any dark state in the form of a sum of states, in each of which a certain two-qubit singlet presents as a tensor factor. The difficulty here is that singlets are not orthogonal, and two such states may overlap. Therefore, in order to prove this Lemma, we need to consider in more detail the trajectories of individual small portions of the amplitude before they are completely calcelled by virtual emission of a photon. The action of the group Sn on qubits as their transpositions can be naturally extended to the operators on the whole space of quantum states H, namely: on the basic states of atoms the transposition an e Sn acts straightforwardly to the atomic component and leaves the field component unchanged and

nlLxjP,i\ip)\i)=^h.;\J*)n\ J).

jp,j jp,j

For the Hamiltonian aH, acting on the whole space of states H we denote by GH the subgroup Sn, consisting of all transpositions t of atomic qubits, such that [H,t] = 0. Let A c{0,1,...,2"-lj be subset of basic states of n — qubit atomic system. Its linear span L (A) we call connected with respect to H, if for all two states , | j} e A there exists the transposition of qubits t e GH, such that t (i) = j. In this case for any basic state of photons the subspace j^j®L(A) we call connected with respect to H as well. The state Yj 0 of n — qubit system we call connected with respect to H, if it belongs to a connected subspace with respect to H; in this case the state of the whole system of the field and atoms of the form \jp)we call connected with respect to H as well. Proposition.

If 1^) = I j is connected with respect to H, then any two col-

j

umns of the

It is easy to show that dot product of two states from D^ 2 which are tensor products of EPR-singlets is always some degree of two.

Modern Information Technologies and IT-Education

matrix H with numbers J1 = (jp, jt), J2 = (jp, j2) and with arbitrary equal field component | j^j, such that and Xj2 are nonzero, differ from each other only by permuting the elements. Indeed, for such basic states j1 and j2, according to the definition of the H -connection, there exists t & Gh, such that j2 =r(j1). Columns with numbers J1, J2 consist of the amplitudes of the states H|Jj) and H\J2), respectively. From the commutation condition, we have tH| Jj = Hr\ J) = H| J2), and this just means that the column J2 is obtained from the column J1 by permuting elements induced by t . The Proposition is proved.

Example. We consider Tavis-Cummings Hamiltonian HRWA with zero detuning for n atoms interacting identically with the field. Then GH = Snthat can be verified straightforwardly: for the random transposition t = (p, q) of two atomic qubits and a basic state of the whole system atoms and field = | jp^|j) the coinsidence of states tH\Jand Ht | J follows from the equality of forces of interaction between atoms and field. It means that any transposition of atomic qubits commutes with Hamiltonian. Let Tin k be the linear span of such basic states, in which atomic parts have energy kaho (contain ka unitsa), and photonic part is a| kp^ where

ka, kp are natural numbers. Then wi-Q k will be connected with respect to HRWA.

Our goal is to show that if the state of the whole system of atoms and field is connected with respect to the Hamiltonian H, then the amplitudes of all the basis states in can be broken up into small portions — amplitude quanta, so that for each quantum its trajectory will be uniquely determined under the action of the Hamiltonian H on a small time interval, in particular, it will be uniquely determined, with which exactly other quantum of amplitude it will cancel when summing the amplitudes to obtain the subsequent state in unitary evolution exp (-iH t/ti).

Let |^) = | jj) be an arbitrary connected with respect to

j

H state of the whole system. In what follows we will use the notations , | j) and b for designation of basic states of the whole system of atoms and field, if the opposite is not written directly. We introduce the important concept of an amplitude quantum as a simple formalization of the transformation of a small portion of the amplitude in evolution on a small time interval when passing between different basis states.

Let T = {+1,-1, + i, -i} be a set of 4 elements, called amplitude types: real positive, real negative, and analogous imaginary. The product of types is determined in a natural way: as a product of numbers. A quantum of amplitude of the size e > 0 is a train of the form

K = (s, id, \bm), \bfln), tm, tfln) (2.9)

where \ b„), \ b^) are two different basic states of the system of atoms and photons, id is a unique identification number that distinguishes this quantum among all others, tn

form b

^ K)

is called a state transition, t

tßn e T. Transition of the

Transitions of states and types of amplitude quanta actually indicate how this state should change over time, and their choice depends on the choice of the Hamiltonian; the quantum size of the amplitude indicates the accuracy of the discrete approximation of the action of the Hamiltonian using amplitude quanta. The set 0 of amplitude quanta of the size e is called quantization of the amplitude if the following condition is fulfilled: Q. In the set 0 there is no such amplitude quanta k1 and k2 , that their state transitions are the same, tn (kj ) = tin (k2) and wherein tf,n (k) = -tfin (k2), and also there are no such quanta of amplitude K and K2, that sin (k) = sin (k2) and tin (k) = -tin (k2) . The condition Q means that in the transition described by the symbol " ^" the final value of the amplitude quantum can not be cancelled with the final value of a similar amplitude quantum. We introduce the notation d( j) = {K: sin (k) = jj .If |j), |i) are basic states, tt, tj e T are types, d is quantization of the amplitude, we introduce the notation

Ke (i, j, t,, tj) = {k e 0 (j), tn (k) = tj, tjn (k) = ti, sfln (k) = i}. For any complex z, we define its relation to the type t eT in the natural way: [z ] = \Re (z)|, if t = +1 and Re (z )> 0, or t = -1 and Re(z)< 0; [z] = |/m (z), if t = +i and Im (z)> 0 , or t = -i and Im (z)< 0; [z] = 0 in all other cases.

We call 0— shift of the state the state = , where

for every basic

tt= (¿1 W) = e X tm (*).

(2.10)

transition. Let's choose the identification numbers so that if they coincide, all other attributes of the quantum also coincide, that is, the identification number uniquely determines the quantum of amplitude. There must be an infinite number of quanta with any set of attributes, except for the identification number. Thus, we will identify the amplitude quantum with its identification number, without further specifying this. We introduce the notation:

tin (K) = tin, tfln (K) = tfln, Sn (K) = bn, sfin (K) = bfl„

Quantization of amplitude 0 actually specifies the transition

We fix the dimension dim (i) of the state space, and we will make estimates (from above) of the positive quantities: the time and size of the quantum of amplitude to within an order of magnitude, assuming all the constants to depend only on independent constants: dim (i )and on the minimum and maximum absolute values of the elements of the Hamiltonian H. In this case, the term strict order will mean an estimate from above as well as from below by positive numbers that depend only on independent constants. We show that for the state connected with respect to H and for any however small s > 0 there exists S > 0 of strict order e and quantization of the amplitude 0 with the size of strict order e2 such that 0 approximates the state with error e and the state of the form SH\ Y) with the same error is approximated by 0 -shift. Then, passing to the Tavis-Cummings Hamiltonian, we fix the error of our approximation to zero: s ^ 0, so that the overwhelming (for s ^ 0 ) number of amplitude quanta is cancelled with each other, giving in the limit the state from I . .

LI U Lp

V

Lemma 4.1.

Let be a state of the whole system of atoms and field connected with respect to H. Then for any number s> 0 there exists the amplitude quantization 0 of the size e of the order e2, the number ej, of the order e and the number c of the stricked 1, such that the following conditions are satisfied: for any basic state j

z i-z i+¿Ii i-I и -m

<e

(2.11)

Where

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019 ISSN 2411-1473 sitito.cs.msu.ru

THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES

_R+={k: K€d(j), tln (k) = +1}, R_={k: Ked(j), tm (K) = -1}, I+={k: Ked( j), tln (K) = +i}, I_={k: ks6( j), tln (k) = -^ and for any basic states and any types tJ,ti eT the following

inequality takes place ' ' (2.12)

f \

e 1

\yKEKt(( ,j,t, 'j ) J

- - [( il *>< i| H j 1

<£l.

Proof. The meaning of the point 1) is that the quantization of the amplitude gives a good approximation of the amplitudes of the state | Y); the meaning of the point 2) is that this quantization d in the realization of transitions for all quantums of the size e for each gives an approximation with an error of the order e of the state cH| Y) (see Lemma 4.2 Further).

Let there be given a state connected with respect to H| Y) = X^j I j

and a number s> 0. For |j with nonzero Xj ^ 0 let

xi ={j~ sign-re(g + g + ■•• + £) + signimi(g+g+ ... + g)

(2.13)

JsNj

X. is the best approximation of the M,

We introduce the notation v = j + — this is the number of

occurrences of e in any column of the the expansion of the matrix (2.14). The definition of connectivity involves that for any j = 0,1,2,..., N -1, such that Xj ^ 0 one of numbers (i |H| j}, i = 0,1,2,..., N-1 is nonzero, hence for the sufficiently small e the number v will be nonzero as well and for the sufficiently small e this number will be of the order 1/e . We denote by Zt j the set of occurences of the letter e in the right side of the expression (2.14), Zj = [J Z{ j . Then the number of elements in the set Zj is v .

We take the lesser value of amplitude quantum: e = s/v . We substitute in expression (2.13) instead of each occurrence of e its formal expansion of the form s = e + e +... + e, having obtained a decomposition of the amplitudes of the initial state into smaller numbers:

V v v (2.16)

Xj jM signre(e + e + .+ + e + e + <; + .. +e + ... + e + e +... + e) +

Mj

+ signimi (e + e +. + + e + e + + +. . , + e + . ,. + e + e +.. . + e)

where signreeMj + signim.„^.j amplitude Xj with precision e; Mj,Nj are the natural numbers. Thus, the point 1) of the Lemma will be almost fulfilled, only without determining the final states |i) and finite types tt, which depend on the Hamiltonian.

We approximate each element of the Hamiltonian in the same way as we approx— imated the amplitudes of the initial state: (i |H| j) *+(g + g + ... + e) ± i (g+g + ... + g) (2.14)

where Rij, Ii} are the natural numbers; real and imaginary parts — with accuracy e each, and the signs before the real and imaginary parts are chosen proceeding from the fact that this approximation should be as accurate as possible for the selected e. Amplitudes of the resultant state H| T) are obtained by multiplying all possible expressions (2.13) with all possible expressions

(2.14):

(i|H| j) «(signreMjE + i signimNje)(±Rije ±i IijE). (2.15)

Each occurrence of the expression e2 in the amplitudes of the resultant state after the parentheses are opened on the right side of

(2.15) will be obtained by multiplying a certain occurrence of e in the right part of (2.13) by a certain occurrence of e in the right part of (2.14). The problem is that the same occurrence of e in (2.13) corresponds not to one but several occurrences of e2 to the result, and therefore we can not associate the amplitude quanta directly with occurrences of e in (2.13).

How many occurrences of e2 in the amplitudes of the state H| Y) correspond to one occurrence of e in the approximation of the amplitude Xj j|of the state |Y) ? This number, the multiplicity of the given occurrence of e, is equal to j + Ii ^. These numbers

can be different for an arbitrary Hamiltonian H and states |Y). However, since | Y) is connected with respect to H, by virtue of the Proposition, the columns of the matrix with different numbers j for nonzero Xj will differ only by permuting the elements, therefore the numbers j + Ii ^ for different j will be the same.

Let , W,

be the sets of occurrences of the letter e into

the right side of the expression (2.16), marked with upper braces. Each of these sets has v elements, as in the defined above sets Zj. Hence we can build for each such set Wj one-to-one mapping of the form % : WJ ^ Zj. For each occurrence of e in (2.13) we natirally define its descendants— the occurrences of e in (2.16); descendants for each occurrence will be v .

To each pair of the form (wJs, % (wJs), where wJs e WJ, we put in correspondence the state and the type transition naturally. Namely, the state transition will be j ^ i for such i, that £ (wJs) e Zt j; the type transition tn ^ tfln is defined so that tn is the type of the occurrence3 wJs, and the type tfln is the multiplication of the type tn by the type of occurrence % (w^. The sets Wj do not intersect for the different pairs j, s, therefore we consider the domain of definition of the function | all occurrences of e in the right side of (2.16) (see Figure 2).

We associate each occurrence of e in the expression (2.16) with a unique identifier and determine its amplitude quantum so that: a) the initial state and initial type of this quantum correspond to this occurrence; and b) the transition and types for a given quantum correspond to the mapping | in the sense defined above. The condition Q is satisfied, since there are no cancelling terms in the expression for the matrix element (2.14). Therefore, we determined the quantization of the amplitude.

Then the point 1 of Lemma 4.1 will be fulfilled by the initial choice of the partition (2.13). In view of our definition of the function |, the amplitude distribution in the |0Y) state will be proportional to the amplitude distribution in the state cH | Y) for any constant c > 0. In fact, we are talking about the choice of the time value t = c in the action of the operator tH on the initial state. In order to determine the value of c necessary for the fulfillment of the point 2, we calculate the contribution of each occurrence of l e in the right side of equation a(2.15) and compare it with the deposit of the corresponding letter e in |0Y).

3 The type of an occurrence is also defined naturally, after opening parentheses, for example, for the occurrence ... ie ... its type i .

Modern Information Technologies and IT-Education

22

ТЕОРЕТИЧЕСКИЕ ВОПРОСЫ ИНФОРМАТИКИ, ПРИКЛАДНОЙ МАТЕМАТИКИ, КОМПЬЮТЕРНЫХ НАУК И КОГНИТИВНО-ИНФОРМАЦИОННЫХ ТЕХНОЛОГИЙ

В

Fig. 3. A. Multiplication of the state vector by the matrix H . The deposit of each occurrence of e is multiplied by e . B. 6 — shift of the initial state. The size of amplitude quantum e has the order e2 We fix some type transition tn ^ tfln and some state transition sin ^ sfln. We call an occurrence of e2 in the result of opening parentheses in (2.15) corresponding to these transitions if j = sin, i = sfln, and this occurrence is obtained by the multiplication of the occurrence of e of the type tnn in the first multiplier of the right side of (2.15) by the occurrence of e in the second multiplier of the type t', so that tint' = tfln. Each of such occurrence of e2 corresponds to unique quantum of amplitude of the size e from the amplitude quantization defined above through the function which has the same state anf type transitions: this quantum corresponds to the occurrence of e that are mapped by the one-to-one correspondence | into the initial occurrence of e2. Hence the target value of c we can find from the proportion e2/l = e/c, whence, taking e =s/v, we obtain c = 1/vs, that has the order 1. Since the accuracy of the approximation of the final state by 6 — shift coincides in order of magnitude with e, we obtain the inequality (2.12). Lemma 4.1 is proved. Lemma 4.1 straightforwardly gives

Corollary

In the conditions of Lemma 4.1. || |№F)-cH| Y) || has the order e.

The corollary means that we can assign to each quantum of the amplitude its own history, that is, to assign to it the portion of the amplitude in the state cH| Y), which is in the natural sense the descendant of a given quantum. In particular, we can say that two quanta of amplitude cancel each other when 6 shift, if their descendants cancel each other Now we can prove Lemma 4.

Choose a number k e {1, 2,..., N -1} and |D)e D£. We consider the subspace 7îÇo, defined above. The state |0 |D) e Ô" o will be connected with respect to Hamiltonian H = HRWA -ktiml, because Dn c Oc", all states from B£ are obtained from each other by permutations of atomic qubits and all such permutations commute with Hamiltonian H (see the example to the Proposition above). Then H coincides with the operator a+ct+ aâ+ on the subspace O = ® O, e.g. the dark states from D£ are the atomic parts of the states from the kernel of H, limited onO. Since all atoms interact in the same way with light, we can assume that all nonzero elements of H are the same, and changing the time scale — that they are equal to one.

We apply Lemma 4 to the Hamiltonian H and the initial state |Y) = | 0 |D) s Ker (ct)| . For the arbitrary s> 0 we obtain the approximation of the state cH|Y) with the accuracy of the order e by 6 — shift for that amplitude quantization 6 with the quantum of

the size e of the order e2 whose existence is asserted in Lemma 4.1. We have cH| Y) = 0. Further in the transition |0 ^ we omit the photonic part.

The Corollary from Lemma 4.1 means that we can expand the amplitudes Xj j|y) of the initial state into the sum of the terms ±(i )e so that each occurrence of such a term in the expansion of the amplitude of any basic state | j in the state | Y) there will correspond exactly one term of the form ±(i )e in the expansion of the amplitude of some basis state |i) to the resulting state |0Y), this correspondence will be one-to-one, and the transition ^ will be the emission of a photon, that is, the atomic part state |i) will be obtained from the atomic part | j by replacing one unit with zero. We combine some occurrences of e in the amplitudes of the decomposition of the resultant state into mutually cancelling pairs: ±(i )e corresponding to one basic state. Then the corresponding terms of the initial state will be EPR singlets, since the pair of initial basic states | j belongs to the same family, because of the Q property of quantization of amplitudes, they are different, and their amplitudes are opposite. Since the difference between |0Y) and cH| Y) = 0 (c , of course, depends on e) converges to zero for s^ 0 by (2.12), the fraction of the cancelling quanta can be made arbitrarily close to unity as e decreases.

The sum of such pairs of states will belong to a set of the form Lp q , since such a cancellation means the presence of one singlet in the expansion of the basis states. Since there is a fixed number of basic states, letting s ^ 0, we get a sequence of linear combinations of states from Lp that converges to some such combination, which is the desired representation of |D). Lemma 4 is proved. Let |D0) be a singular state. By Lemma 4, we have

|A> = 1 <

(2.17)

where |Dp^ are the states of n-2 qubits.

Each summand of this sum belongs to the subspace Lp . The difficulty is that we can not say that |Dp ^ are dark states, that is, the emission of a photon by atoms in any of these states can be compensated by the emission of a photon by an atom whose state belongs to another |Dp, , where p' ^ p or q' ^ q.

We will overcome this difficulty with the help of an antisymmetriza-tion operation. We put \D'p ^ = Anp . Then \D'p for any p T q will be singular, since the darkness and orthogonality of the singlet is preserved under permutation of atoms and subtraction. We show that there is nonzero among all possible states \D'p ^. Indeed, let all such states be zero. Then, by Lemma 3, for any pair p * q |_D0) e Lpq, and, the state |D0) belongs to the orthogonal complement of the linear span of all Lp . But in this case it is zero, since it belongs to this linear span by virtue of (2.17).

Thus, among \d'„ „) there is a nonzero; let it correspond to the pair

i pq>

p = 1, q = 2: D A. This state is singular, and it belongs to L12, that is, it has the form s12 0|_Dj). Then |Dj is also a singular state of n-2 qubits. Indeed, |Dj is a dark one, since it was obtained by splitting one Sj 2 singlet from the dark state. If it is not singular, then it would have a nonzero projection onto the linear span of (n -2, k) singlets obtained by the removing of the first two qubits from the main space. But then multiplying it by one singlet would also have a non-zero projection already on the linear span of (n, fc) singlets, which contradicts the singularity of |Dt' .

Thus, is a singular state of n - 2 qubits. We apply the same ar-

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019 ISSN 2411-1473 sitito.cs.msu.ru

guments to it as to |D0), getting singular |D2) from n-4 qubits, etc. In the end, we get a singular Dk singlet, which contradicts the definition of the singularity. The Theorem is proved. Note that if in the RWA approximation the state is dark, but not invisible, then n/2 > k and in each component of its singlet decomposition there are zero tensor factors of the form |0.. For an invisible state there are no such zero components, that is, only singlets are present.

So, we see that the dark states in the exact Tavis-Cummings model coincide with the invisible states for this model in the RWA approximation. Indeed, the latter, as follows from the Theorem, are linear combinations of the tensor products of the EPR singlet |0l)-| 10, and each such singlet itself will be dark in the exact Tavis-Cum-mings model, as is easily seen directly, applying the Hamiltonian HTC to such an EPR pair. This explains the advantage of the term "dark states": it covers not only those that do not emit light, but also do not absorb light.

The algebraic definition of a dark state for two-level atoms is as follows: J±|y) = 0, where J± is an increasing and decreasing operator. It is proved in the paper [23] that this is equivalent to the fulfillment of the inequality U| Y) = | Y) for any operator U e SU(2) (such states |Y) in this work are called "singlet"). Applying our Theorem, we find that the stationary points of the group U, U eSU(2) are exactly linear combinations of tensor EPR-sin-glet products, which means the equivalence of the definition of darkness in [23] and our definition of darkness for an exact model. The work [23] contains a similar algebraic characteristic of the dark states of d — level atoms is also given for d > 2; an explicit description of such states is an interesting problem.

Almost dark states

Consider the state \aD) = |11)-|00of two identical two-level atoms that is not dark, but represents an example of an almost dark state. At low frequencies o, this state will persist for a long time, not emitting a photon. Indeed, in the exact Tavis-Cummings model, the transition to the ground state with the emission of a photon for this state can occur in two ways: either the photon is emitted by an excited atom or it arises together with the excitation of another atom in the ground state. It is not difficult to see that the amplitudes of these processes are opposite.

This, however, does not mean that the emission of a photon is impossible at all. The matter is that the excited state 1 and the basic |0 evolve differently: the phase of the excited state changes faster than the ground state, since coa > 0. Therefore, the states resulting from the emission or production of a photon will differ slightly in phase and there will be no complete cancellation of the amplitudes. This almost dark state differs from the singlet state: in the latter, both transitions are completely equal in both RWA and in the exact model. But if oa is very small compared to g/ti (the limit of strong interaction, opposite to RWA), then an almost dark state will be at rest for a long time and will not emit a photon. The tensor product of simple EPR singlets and states of the form \aD), and linear combinations of such states will also remain unchanged long for small o. Is it true that such linear combinations exhaust all states that have the property of almost darkness, that is, of arbitrarily long conservation for small o ? This question is still open.

Some generalizations

First, assuming, as before, the equality of forces of interaction with the field of all atoms, we give up the RWA approximation, and consider the case of the exact solution. The set of dark states for the exact Hamiltonian is Ker (ct + ct+) = Ker (<r)n Ker (ct+) , since a lowers the Hamming weight of the basic states, and a+ increases it. Given that the replacement of the zeros to ones and vice versa subspaces of Ker (<r)and Ker (ct+) are moving one to another, and singlet only changes the sign, and applying to Ker (<r)and Ker (cf+j item 2 of the Theorem, we get that the dark state for the exact Hamiltonian are linear combination of (2k, k)— singlets. These states will be also invisible. In particular, dark states will exist only for ensembles with an even number of atoms.

Now, on the contrary, we assume that the RWA approximation is true, but the forces of interaction of atoms with the field a are

-iq

different positive real num bers. \

Now dark subspace is Ker I ^ gq^q I. Let s e B£ be a binary train,

in which zeroes stand on the positions Sj, s2,..., sk. We introduce the notations rs = n gq.

qefa sг, ,.,sk}

It follows from the definition of Hamiltonian and numbers rs that

the atomic state | Y) = ~^Xj | j is dark if and only if the following

j

system of equations:

£ = 0, (4.1)

is satisfied for all j' = 0,1,..., 2" -1, which is connected with the system (2.2) naturally: X0 is a solution of (4.1) if and only if XS = XHrs is a solution of (4.1).

The point 1 of the Theorem is then satisfied because the dimension of the dark subspace does not depend on gq, the point 2 will be also true if only instead of singlet we always consider the "distributed singlet": two qubit state of the form |s12) = g1 |0jl2) - g2 |lj02). Such a state is obtained from the singlet by adiabatic change of coordinates of atoms inside the cavity (for example, by optical tweezers), so that the coefficient gq depends on the coordinate of q — th atom (see the first paragraph).

In this case dark states will not be transparent already when n = 2, because transparent will be anti-singlet of the form |(s= g2|0jl2)-gJljOj). The transparency does not thus connected with the stability of the state in the time in contrast with the darkness, which guarantees such a stability. By the same reason in the case of exact Hamiltonian and the different forces of interaction there is no dark states even for n = 2.

Conclusion

An explicit form of the dark states of an ensemble with an even number of identical two-level atoms in the framework of the Tav-is-Cummings model was studied. At the same force of interaction of atoms with light atomic ensembles in these states do not interact at all with the mode of the cavity, and therefore — theoretically — remain unchanged even when the ensemble of atoms is extracted from the resonator. Spatial separation of the dark ensemble or thermal dephasing immediately leads to the emission of photons. Dark states can be used to protect quantum computing, as energy storage, and so on.

Modern Information Technologies and IT-Education

The dimension of the dark subspaces is equal to the Catalan numbers. An explicit form of their structure is established: dark states are linear combinations of tensor products of EPR singlet states. Subject to the applicability of the RWA approximation, the dark property is maintained at the vacuum state of the cavity field in the case of adiabatic dilution of atoms, in which the force of interaction with light becomes different. However, such ensembles will interact with light if the state of the field in the cavity is not vacuum. The search for further applications of dark states and methods for obtaining them is a task for further research. Almost dark states, which are a linear combination of triplets, were also considered; they interact very weakly with light at small values of the excitation energy of atoms, which can be realized, for example, for Rydberg States. Classification of almost dark states as well as dark states in systems of d -level atoms at d > 2 represent separate problems. In proving the key result of the paper — point 2 of the Theorem, the method of amplitude quanta was developed — small portions of the amplitude of basis states, the trajectory of which can be determined in advance in the course of evolution. This method assumes the passage to the limit, but allows us to prove the algebraic property of dark states. It can be of interest for studying the physics of quantum computers and their scalability.

References

[1] Feynman R.P. QED: The Strange Theory of Light and Matter Princeton University Press, 1985. (In Eng.)

[2] Feynman R.P. Theory of Fundamental Processes. Addison Wesley, 1961. (In Eng.)

[3] Bogoliubov N.N., Parasiuk O.S. On the theory of multiplication of causal singular functions. Doklady of the Academy of Sciences of the USSR = The Reports of the Academy of Science. 1955; 100(1):25-28. (In Russ.)

[4] Anikin S.A., Zavyalov O.I., Polivanov M.K. Simple proof of the Bogolyubov - Parasyuk theorem. Theoretical and Mathematical Physics. 1973; 17(2):1082-1088. (In Eng.) DOI: 10.1007/BF01037256

[5] Feynman R.P. Simulating Physics with Computers. International Journal of Theoretical Physics. 1982; 21(6-7):467-488. (In Eng.) DOI: 10.1007/BF02650179

[6] Jaynes E.T., Cummings F.W. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proceedings of the IEEE. 1963; 51(1):89-109. (In Eng.) DOI: 10.1109/PROC.1963.1664

[7] Dicke R.H. Coherence in Spontaneous Radiation Processes. Physical Review. 1954; 93(1):99-110. (In Eng.) DOI: 10.1103/PhysRev.93.99

[8] Cohen E., Hansen T., Itzhaki N. From Entanglement Witness to Generalized Catalan Numbers. Scientific Reports. 2016; 6(30232). (In Eng.) DOI: 10.1038/srep30232

[9] Toth G. Entanglement Witnesses in Spin Models. Physical Review A. 2005; 71(1):010301(R). (In Eng.) DOI: 10.1103/ PhysRevA.71.010301

[10] Cirac J.I., Ekert A.K., Macchiavello C. Optimal Purification of Single Qubits. Physical Review Letters. 1999; 82(21):4344-4347. (In Eng.) DOI: 10.1103/PhysRevLett.82.4344

[11] Grover L.K. A fast quantum mechanical algorithm for database search. Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC), May 1996. Melville, NY. 2006; 810:212-219. Available at: https://arxiv

org/abs/quant-ph/9605043 (accessed 17.02.2019). (In Eng.)

[12] Breuer H., Petruccione F. The Theory of Open Quantum Systems. Oxford, 2007. (In Eng.) DOI: 10.1093/acprof:o so/9780199213900.001.0001

[13] Freedman M., Kitaev A., Larsen M., Wang Z. Topological quantum computation. Bulletin of the American Mathematical Society. 2003; 40(1):31-38. (In Eng.) DOI: 10.1090/S0273-0979-02-00964-3

[14] Angelakis D.G., Santos M.F., Bose S. Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays. Physical Review A. 2007; 76(03):031805(R). (In Eng.) DOI: 10.1103/PhysRevA.76.031805

[15] Huelga S., Plenio M. Vibration, Quanta and Biology Contemporary Physics. 2013; 54(4):181-207. (In Eng.) DOI: 10.1080/00405000.2013.829687

[16] Plenio M.B., Huelga S.F. Dephasing assisted transport: Quantum networks and biomolecules. New Journal of Physics. 2008; 10:113019. (In Eng.) DOI: 10.1088/13672630/10/11/113019

[17] Ambainis A. Quantum walks and their algorithmic applications. International Journal of Quantum Information. 2003; 1(4):507-518. (In Eng.) DOI: 10.1142/ S0219749903000383

[18] Ozhigov Y.I. Dark states of atomic ensembles: properties and preparation. Proc. SPIE10224, International Conference on Micro— and Nano-Electronics 2016, vol. 10224, id. 102242Y, 8 pp. (In Eng.) DOI: 10.1117/12.2264516

[19] Azuma H. Quantum Computation with the Jaynes-Cummings Model. Progress of Theoretical Physics. 2011; 126(3):369-385. (In Eng.) DOI: 10.1143/PTP.126.369

[20] Pöltl C., Emary C., Brandes T. Spin entangled two-particle dark state in quantum transport through coupled quantum dots. Physical Review B. 2013; 87(4):045416. (In Eng.) DOI: 10.1103/PhysRevB.87.045416

[21] Tanamoto T., Ono K., Nori F. Steady-State Solution for Dark States Using a Three-Level System in Coupled Quantum Dots. Japanese Journal of Applied Physics. 2012; 51(2):2BJ07. (In Eng.) DOI: 10.1143/jjap.51.02bj07

[22] Hansom J., Schulte C., Le Gall C., Matthiesen C., Clarke E., Hugues M., Taylor J.M., Atatüre M. Environment-assisted quantum control of a solid-state spin via coherent dark states. Nature Physics. 2014; 10:725-730. (In Eng.) DOI: 10.1038/nphys3077

[23] Kok P., Nemoto K., Munro W.J. Properties of multi-partite dark states. 2002. Available at: https://arxiv.org/abs/ quant-ph/0201138 (accessed 17.02.2019). (In Eng.)

[24] Berkeland D.J., Boshier M.G. Destabilization of dark states and optical spectroscopy in Zeeman-degenerate atomic systems. 2001. Available at: http://arxiv.org/pdf/quant-ph/0111018v1.pdf (accessed 17.02.2019). (In Eng.)

[25] Ferretti M., Hendrikx R., Romero E., Southall J., Cogdell R. J., Novoderezhkin V.I., Scholes G.D., van Grondelle R. Dark States in the Light-Harvesting complex 2 Revealed by Two-dimensional Electronic Spectroscopy. Scientific Reports. 2016; 6:20834. (In Eng.) DOI: 10.1038/srep20834

[26] André A., Duan L.-M., Lukin M.D. Coherent Atom Interactions Mediated by Dark-State Polaritons. Physical Review Letters. 2002; 88(24):243602. (In Eng.) DOI: 10.1103/ PhysRevLett.88.243602

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019 ISSN 2411-1473 sitito.cs.msu.ru

THEORETICAL QUESTIONS OF COMPUTER SCIENCE, COMPUTATIONAL MATHEMATICS, COMPUTER SCIENCE AND COGNITIVE INFORMATION TECHNOLOGIES

[27] Lee E.S., Geckeler C., Heurich J., Gupta A., Cheong K.-I., Secrest S., Meystre P. Dark states of dressed Bose-Einstein condensates. Physical Review A. 1999; 60(5):4006-4011. (In Eng.) DOI: 10.1103/PhysRevA.60.4006

[28] Fink J.M., Bianchetti R., Baur M., Göppl M., Steffen L., Filipp S., Leek P.J., Blais A., Wallraff A. Dressed Collective Qubit States and the Tavis-Cummings Model in Circuit QED. Physical Review Letters. 2009; 103(8):083601. (In Eng.) DOI: 10.1103/PhysRevLett.103.083601

Submitted 17.02.2019; revised 25.03.2019; published online 19.04.2019.

Yuri I. Ozhigov, Professor, Department of Supercomputers and Quantum Informatics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University (1, Leninskie gory, Moscow 119991, Russia); Leading Researcher, Institute of Physics and Technology of the Russian Academy of Sciences (36, Build. 1, Nakhimovsky Av., Moscow 117218, Russia), Dr. Sci. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-4957-9063, ozhigov@

[12]

[13]

[14]

[15]

[16]

The author has read and approved the final manuscript. [17]

Список использованных источников

[1] Feynman R.P. QED: The Strange Theory of Light and Matter [18] Princeton University Press, 1985. 158 p.

[2] Feynman R.P. Theory of Fundamental Processes. Addison Wesley, 1961. 172 p.

[3] Боголюбов Н.Н., Парасюк О.С. К теории умножения при- [19] чинных сингулярных функций // Доклады АН СССР 1955. Т. 100, № 1. С. 25-28.

[4] Аникин С.А., Завьялов О.И., Поливанов М.К. Одно простое [20] доказательство теоремы Боголюбова - Парасюка // Теоретическая и математическая физика. 1973. Т. 17, № 2. С. 189-198. URL: http://www.mathnet.ru/links/6f777bfcbe1f0 8ec7c3a70af66b8be55/tmf3938.pdf (дата обращения: [21] 17.02.2019).

[5] Feynman R.P. Simulating Physics with Computers // International Journal of Theoretical Physics. 1982. Vol. 21, Issue 6-7. Pp. 467-488. DOI: 10.1007/BF02650179 [22]

[6] Jaynes E.T., Cummings F.W. Comparison of quantum and semiclassical radiation theories with application to the beam maser // Proceedings of the IEEE. 1963. Vol. 51, Issue 1. Pp. 89-109. DOI: 10.1109/PR0C.1963.1664

[7] Dicke R.H. Coherence in Spontaneous Radiation Processes [23] // Physical Review. 1954. Vol. 93, Issue 1. Pp. 99-110. DOI: 10.1103/PhysRev.93.99

[8] Cohen E., Hansen T., Itzhaki N. From Entanglement Witness [24] to Generalized Catalan Numbers // Scientific Reports. 2016.

Vol. 6, id. 30232. DOI: 10.1038/srep30232

[9] Toth G. Entanglement Witnesses in Spin Models // Physical Review A. 2005. Vol. 71. Issue 1, id. 010301(R). DOI: [25] 10.1103/PhysRevA.71.010301

[10] Cirac J.I., Ekert A.K., Macchiavello C. Optimal Purification of Single Qubits // Physical Review Letters. 1999. Vol. 82, Issue21.Pp. 4344-4347.DOI: 10.1103/PhysRevLett.82.4344

[11] Grover L.K. A fast quantum mechanical algorithm for [26]

database search // Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (STOC), May 1996. Melville, NY, 2006. Vol. 810. Pp. 212-219. URL: https:// arxiv.org/abs/quant-ph/9605043 (дата обращения: 17.02.2019).

Breuer H., Petruccione F. The Theory of Open Quantum Systems. Oxford, 2007. 636 p. DOI: 10.1093/acprof:o so/9780199213900.001.0001

Freedman M., Kitaev A., Larsen M., Wang Z. Topological quantum computation // Bulletin of the American Mathematical Society. 2003. Vol. 40, Issue 1. Pp. 31-38. DOI: 10.1090/S0273-0979-02-00964-3

Angelakis D.G., Santos M.F., Bose S. Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays // Physical Review A. 2007. Vol. 76, Issue 03, id. 031805(R). DOI: 10.1103/PhysRevA.76.031805 Huelga S., Plenio M. Vibration, Quanta and Biology // Contemporary Physics. 2013. Vol. 54, Issue 4. Pp. 181-207. DOI: 10.1080/00405000.2013.829687 Plenio M.B., Huelga S.F. Dephasing assisted transport: Quantum networks and biomolecules // New Journal of Physics. 2008. Vol. 10, id. 113019. DOI: 10.1088/13672630/10/11/113019

Ambainis A. Quantum walks and their algorithmic applications // International Journal of Quantum Information. 2003. Vol. 1, Issue 4. Pp. 507-518. DOI: 10.1142/S0219749903000383

Ozhigov Y.I. Dark states of atomic ensembles: properties and

preparation // Proc. SPIE 10224, International Conference

on Micro- and Nano-Electronics 2016. Vol. 10224, id.

102242Y, 8 pp. DOI: 10.1117/12.2264516

Azuma H. Quantum Computation with the Jaynes-Cummings

Model // Progress of Theoretical Physics. 2011. Vol. 126,

Issue 3. Pp. 369-385. DOI: 10.1143/PTP.126.369

Poltl C., Emary C., Brandes T. Spin entangled two-particle

dark state in quantum transport through coupled quantum

dots // Physical Review B. 2013. Vol. 87, issue 4, id. 045416.

DOI: 10.1103/PhysRevB.87.045416

Tanamoto T., Ono K., Nori F. Steady-State Solution for Dark States Using a Three-Level System in Coupled Quantum Dots // Japanese Journal of Applied Physics. 2012. Vol. 51, issue 2, id. 2BJ07. DOI: 10.1143/jjap.51.02bj07 Hansom J., Schulte C., Le Gall C., Matthiesen C., Clarke E., Hugues M., Taylor J.M., Atature M. Environment-assisted quantum control of a solid-state spin via coherent dark states // Nature Physics. 2014. Vol. 10. Pp. 725-730. DOI: 10.1038/nphys3077

Kok P., Nemoto K., Munro W.J. Properties of multi-partite dark states. 2002. URL: https://arxiv.org/abs/quant-ph/0201138 (дата обращения: 17.02.2019). Berkeland D.J., Boshier M.G. Destabilization of dark states and optical spectroscopy in Zeeman-degenerate atomic systems. 2001. URL: http://arxiv.org/pdf/quant-ph/0111018v1.pdf (дата обращения: 17.02.2019). Ferretti M., Hendrikx R., Romero E., Southall J., Cogdell R. J., Novoderezhkin V.I., Scholes G.D., van Grondelle R. Dark States in the Light-Harvesting complex 2 Revealed by Two-dimensional Electronic Spectroscopy // Scientific Reports. 2016. Vol. 6, id. 20834. DOI: 10.1038/srep20834 André A., Duan L.-M., Lukin M.D. Coherent Atom Interactions

Modern Information Technologies and IT-Education

Mediated by Dark-State Polaritons // Physical Review Letters. 2002. Vol. 88, Issue 24. Pp. 243602. DOI: 10.1103/ PhysRevLett.88.243602

[27] Lee E.S., Geckeler C., Heurich J., Gupta A., Cheong K.-I., Secrest S., Meystre P. Dark states of dressed Bose-Einstein condensates // Physical Review A. 1999. Vol. 60, Issue 5. Pp. 4006-4011. DOI: 10.1103/PhysRevA.60.4006

[28] Fink J.M., Bianchetti R., Baur M., Göppl M., Steffen L., Filipp S., Leek P.J., Blais A., Wallraff A. Dressed Collective Qubit States and the Tavis-Cummings Model in Circuit QED // Physical Review Letters. 2009. Vol. 103, issue 8, id. 083601. DOI: 10.1103/PhysRevLett.103.083601

Поступила 17.02.2019; принята к публикации 25.03.2019; опубликована онлайн 19.04.2019.

|об авторе:|

Ожигов Юрий Игоревич, профессор, кафедра суперкомпьютеров и квантовой информатики, факультет вычислительной математики и кибернетики, Московский государственный университет имени М.В. Ломоносова (119991, Россия, г. Москва, ГСП-1, Ленинские горы, д. 1); ведущий научный сотрудник, «Физико-технологический институт имени К.А. Валиева» Российской академии наук (117218, Россия, г. Москва, Нахимовский проспект, д. 36, корп. 1), доктор физико-математических наук, ORCID: http://orcid.org/0000-0002-4957-9063, ozhigov@cs.msu.su

Автор прочитал и одобрил окончательный вариант рукописи.

Современные информационные технологии и ИТ-образование

Том 15, № 1. 2019 ISSN 2411-1473 sitito.cs.msu.ru