Spontaneous emission of the non-Wiener type

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INTRODUCTION. Spontaneous atomic emission processes constitute the basis of quantum theory of the interaction of radi ation with matter, and the first steps in ...
ISSN 10637761, Journal of Experimental and Theoretical Physics, 2011, Vol. 113, No. 3, pp. 376–393. © Pleiades Publishing, Inc., 2011. Original Russian Text © A.M. Basharov, 2011, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 140, No. 3, pp. 431–449.

ATOMS, MOLECULES, OPTICS

Spontaneous Emission of the NonWiener Type A. M. Basharov Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia email: [email protected] Received December 6, 2010

Abstract—The spontaneous emission of a quantum particle and superradiation of an ensemble of identical quantum particles in a vacuum electromagnetic field with zero photon density are examined under the con ditions of significant Stark particle and field interaction. New fundamental effects are established: suppres sion of spontaneous emission by the Stark interaction, an additional “decay” shift in energy of the decaying level as a consequence of Stark interaction unrelated to the Lamb and Stark level shifts, excitation conserva tion phenomena in a sufficiently dense ensemble of identical particles and suppression of superradiaton in the decay of an ensemble of excited quantum particles of a certain density. The main equations describing the emission processes under conditions of significant Stark interaction are obtained in the effective Hamiltonian representation of quantum stochastic differential equations. It is proved that the Stark interaction between a single quantum particle and a broadband electromagnetic field is represented as a quantum Poisson process and the stochastic differential equations are of the nonWiener (generalized Langevin) type. From the exam ined case of spontaneous emission of a quantum particle, the main rules are formulated for studying open sys tems in the effective Hamiltonian representation. DOI: 10.1134/S1063776111080036

1. INTRODUCTION Spontaneous atomic emission processes constitute the basis of quantum theory of the interaction of radi ation with matter, and the first steps in studying them were taken as far back as 1927, when Dirac began to establish quantum theory [1]. These processes are also a simple example of the quantum dynamics of an open system, which is an atom interacting with the environ ment—a vacuum electromagnetic field. And this problem already has a long history, which goes back to the 1930 work by Weiskopf and Winger [2]. Another very important field of knowledge in which spontane ous emission processes play a determining role is the theory of continuous quantum measurements and quantum jumps. Measurement theory and the theory of open systems are closely interwoven in quantum theory, which became a subject of active discussion, apparently beginning with Davis’ works [3], and the processes of an atom’s dynamics in an external quan tum broadband (multimode) electromagnetic field are a very important fundamental example here. The problem of spontaneous emission of an atom is naturally represented as a problem of the dynamics of a quantum particle with two energy levels in an exter nal resonance broadband quantum electromagnetic field [4]. From the standpoint of interaction, a reso nance broadband external field causes direct single photon transitions between the quantum level of a par ticle (determining the natural spectral linewidth) and the energy shift of quantum levels [4, 5]. In an ordinary

atom, these effects are of different order in terms of the interaction (coupling) constant with the electromag netic field: transitions between atomic levels are described by the size of the first order of the smallness and are represented as the energy of interaction between an atom and the external field, where an atom’s level shift is described by the size of the second order coupling constant and consists of the Lamb shift [4] and the shift due to the highfrequency Stark effect [5]. In ordinary circumstances, the sizes of the Lamb and Stark shifts in atom levels in a vacuum field are negligibly small in comparison to the characteristic interaction energy of an atom and the external field and until now its effect on the dynamics of spontane ous emission of an excited atom has not been investi gated. However, a quantum twolevel particle success fully simulates not only a real isolated atom but also an atom or ion in a solidstate matrix as well—quite a variety of objects: from spin [6, 7] and excitons [7] to artificial emitters like a quantum dot [8] or an atomic– photon cluster [9]. In addition, such artificial media are being studied and designed as a photon crystal with features in the photon spectrum such that conditions can be created under which the Lamb and/or Stark level shifts and energy of interaction between the arti ficial emitter and the external broadband quantum field can be of the first order. Therefore, it is of undoubted interest to investigate the question of spon taneous emission of such a particle, since this funda

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SPONTANEOUS EMISSION OF THE NONWIENER TYPE

mental phenomenon will reflect on many other opti cal processes. In the paper, we solve the problem of spontaneous emission of a quantum particle in a resonance broad band electromagnetic field with allowance for the Stark of energy level shift in this field when the Lamb and/or Stark level shifts are comparable to the energy of interaction between the particle and the electro magnetic field. It is established that under such condi tions, the dynamics of a quantum particle is non Wiener or nonBrownian (opposite to the Wiener dynamics of an atom with the highfrequency Stark

377

ω E2

ω

E1

1

effect neglected). This leads to suppression of the relaxation of an excited state and an additional shift in the crossover frequency of the quantum particle during its spontaneous emission. For all intents and purposes, suppression of relaxation results from quantum inter ference of the real transition from the excited to the ground level and virtual transitions with return to the excited level (Fig. 1). It is these virtual transitions that result in the additional energy level shift from which a transition occurs to the ground state with emission of a photon. This additional shift of the excited level dif fers both from the ordinary Lamb level shift and the conventionally understood Stark level shift as the aver age value of the corresponding operator. This is because in the case of an electromagnetic field with zero photon density, this mean value is equal to zero and there is an additional level shift. In this case, such a shift of the excited energy level is called an emission Stark level shift. The paper also examines the collective spontane ous emission of an ensemble of identical quantum par ticles with allowance for the emission Stark level shift. In is shown that the collective interaction of the ensemble of identical quantum particles with a vac uum field intensifies the Stark interaction. It is estab lished that—in contrast to ordinary superradiation [10], in which with increasing number of excited atoms, the delay time is reduced; there is a decrease in duration; and the superradiation pulse intensity increases—a parameter region of the quantum parti cle exists in which the number of particles in the ensemble leads to the inverse effect—an increase in delay time and signal duration of collective spontane ous emission. Moreover, it has been discovered that a “critical number” of atoms in the ensemble exist at which (super)radiation from this ensemble is sup pressed; i.e., the system stabilizes in the excited state. 1

Using prefix “non” in definitions of “nonWiener,” “non Brownian,” and “nonLangevin” dynamics, the author empha sizes that additional important factors of another origin are taken into account rather than the opposition to the relevant terms so that the above terms can also be interpreted as “gener alized Wiener,” “generalized Brownian,” and “generalized Lan gevin” dynamics.

Fig. 1. Schematic depiction of a real transition from excited level E2 to the ground state E1 with quantum emis sion at frequency and virtual transitions without quantum emission with a return to the excited state. Transition E2 E1 is optically allowed.

This effect is a consequence of the discovered effect of suppression of relaxation by the Stark interaction and its intensification in the ensemble of collectively decaying identical quantum particles. In the paper, an apparatus is applied for represent ing the effective Hamiltonian of stochastic differential equations (SDEs) of a nonWiener type of quantum evolution to derive the kinetic equations and relax ation operator for analyzing the dynamics of a quan tum particle resonantly interacting with the electro magnetic field. Up until now, the application of similar (nonWiener or generalized Langevin) equations has been limited to a description of quantum jumps [11] and the technique of counting photons in continuous measurements [12, 13]; as well, the discussed sum mand in the Hamiltonian describing Stark energy level shifts has not been taken into account. To derive the kinetic equation and relaxation oper ator, various methods have been developed (see [14– 16]); however, the quantum SDE method [11–13, 15, 17–21] is not only more visually representative and simpler, but it also serves as an integral part of the mathematically correct description of a certain class of open systems. The efficiency of the Wienertype quan tum SDE method has been visually demonstrated in [10, 12, 13, 15, 18–25]. The paper focuses on the determining role of representing the effective Hamil tonian of the system for a noncontradictory analysis of the dynamics of such a system. In the known works on deriving kinetic equations for a twolevel atom in a resonance broadband quantum field by a Wiener type of SDE method, a Hamiltonian in the rotating wave approximation was used as the initial Hamiltonian [12, 13, 18, 19]. Such a Hamiltonian was obtained from the standard initial Hamiltonian of a twolevel particle neglecting part of the summands in the elec

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tric dipole operator that had rapidly oscillating multi pliers in the interaction representation. Application of this quantum SDE technique directly to the initial Hamiltonian gives a result that is unexpected and con tradictory to observation: the relaxation operator proves to be identically equal to zero [22], as if the two level atom is not experiencing any radiative decay! Thus, in the quantum SDE technique, the problem of the effective Hamiltonian distinctly arises—the main assumptions on the character of interaction of the open system with the environment should be applied not to any Hamiltonian (including both an arbitrarily general and “exact” initial Hamiltonian), but to the effective Hamiltonian. The necessary systematic prin ciple of obtaining such an effective Hamiltonian and the region of its applicability are formulated in this paper. Such an approach leads not only to simple sub stantiation of the summands of the rotating wave approximation, but also to the derivation of the main summand of this study describing the Stark level shift in the broadband quantum field and responsible for the nonWiener type of spontaneous atomic emission. In the paper, this approach also imposes certain restrictions on further use of kinetic equations and the relaxation operator in studies on the dynamics of open systems, which in many works are ignored, which leads to incorrect results. The paper is organized as follows. In Section 2, a unitary transformation of the Hamiltonian is applied jointly with perturbation theory to derive the effective Hamiltonian of the problem considered here. The conditions are discussed under which the Stark sum mand in the Hamiltonian becomes of the same order as the summand responsible for singlephoton transi tions between resonance quantum levels of a particle. In Section 3, the quantum SDE apparatus is derived and the role of representing the effective Hamiltonian is discussed in studying the dynamics of an open sys tem by the quantum SDE method. In Section 4, the nonWiener evolution operator of the considered problem is derived and on its basis, we formulate the kinetic equation and relaxation operator for the den sity matrix of the system in the case of a photonless quantum electromagnetic field. Section 5 discusses the features of nonWiener spontaneous emission of a singlequantum particle. Section 6 examines collec tive spontaneous emission of an ensemble of identical quantum particles. Section 7 discusses problems related to theoretical analysis of experimental situa tions in which this nonWiener type of spontaneous decay of an excited quantum particle studied is quite

realistic, but no one has succeeded yet in finding a rig orous solution to the problem. 2. REPRESENTATION OF THE EFFECTIVE HAMILTONIAN OF OPTICAL PROBLEMS We consider a stationary atom interacting in the electric dipole approximation with a resonance quan tum electromagnetic field. The initial Hamiltonian of such a system, H

Ini

A

F

Int

= H +H +H ,

(1)

consists of the Hamiltonian of an isolated atom HA and the Hamiltonian of the electromagnetic field HF, and their interaction operator HInt, which have the form

∑ E |E 〉 〈E |,

A

H =

j

j

j

F

H =

Int

=

∑Γ

† ω ( bω

+ bω )

ω

∑d

kj |E k〉 〈E j|,

(2)

kj

∑ |E 〉 〈E | = 1, j

† ω bω ,

ω

j

H

∑ បωb

j

〈E j|E j〉 = δ jk ,

j

where quantum nondegenerate states |E j〉 of energy Ej characterize the atom, and dkj = 〈E k|d |E j〉 denotes the matrix elements of the dipole moment operator of the atom d = kj d kj |E k〉 〈E j| . We consider that the atomic levels are characterized by certain parity, such that 〈E k|d |E k〉 = 0. The photon annihilation and creation operators with frequency ω are given by the quantities † † bω and b ω : [bω, b ω' ] = δωω'. We ignore the recoil and degeneration effects and polarization features.



Wavevector |Ψ〉 of the atom + quantum electro magnetic field system satisfies the Schrödinger equa tion: iប ∂ |Ψ〉 = H |Ψ〉. ∂t Ini

(3)

Due to the unitary symmetry of quantum theory, we perform the following unitary transformation: ˜ 〉 = U |Ψ〉. |Ψ

(4)

The transition from vector |Ψ〉 to the new vector (4) is accompanied by a change in the Hamiltonian, ∂ † ˜ = UH Ini U † – iបU  H U , ∂t

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such that no description of the quantum system is given by the Schrödinger equation with the trans formed Hamiltonian (5): ∂ ˜ ˜ 〉. ˜ |Ψ iប  |Ψ 〉 = H ∂t

– iS



S = S,

(7)

2

– iS iS – i) [ S, O ] ( – i ) e Oe = O + ( +  [ S [ S, O ] ] 1! 2! 3

( –i ) +  [ S, [ S, [ S, O ] ] ] + …. 3! We expand transformed Hamiltonian (5) and S in a series over the interaction constant: (2)

S = S + S + …, ˜ = H ˜ (0) + H ˜ (1) + H ˜ ( 2 ) + …, H

(8)

where the upper index indicates the order of expansion over the coupling constant. Substituting (7), (8) into (5) with allowance for the Baker–Hausdorff formula and equating expressions of the same order of small ness, we obtain ˜ ( 0) = HA + HF , H

(9) (1)

˜ ( 1 ) = H Int – i [ S ( 1 ), H ˜ ( 0 ) ] + ប ∂S H , ∂t

(10)

(2)

In the absence of any resonances, effective Hamil tonian (5), (6), (8)–(11) is diagonal. The resonance conditions for interaction of the atom and the electro magnetic field bring effective Hamiltonian (5) and Schrödinger equation (6) to a closed system describing the interaction of the electromagnetic field with only the resonance quantum levels of the atom. The features of unitary transformation (4)–(8) in the case of interaction of quantum particles with clas sical electromagnetic fields are explained in mono graph [26]. In the quantum case, the method was used by the author in [9, 23–25] for conditions other than those considered here. The history of applying unitary transformation similar to (4)–(8) in nonlinear and quantum optics problems is considered in [27]. We make a principal assumption that the interac tion of an atom with a quantum electromagnetic field has a resonance character. According to Lax [28], the frequency spectrum of a broadband electromagnetic field splits into the total set of spectra of independent sources, each of which is in resonance with the corre sponding quantum transition of an atom. Further we will consider only one such source, central frequency ΩΓ, the spectrum of which is close to frequency ω21 of the optically allowed atomic transition |E 2〉 |E 1〉 (otherwise E2

3

E1) :

Ω Γ ≈ ω 21 ,

˜ ( 2 ) = – i [ S ( 1 ), H Int ] – i [ S ( 1 ), H ˜ (1) ] H 2 2 (2) ˜ (0) ∂S – i[S , H ] + ប , ∂t ….

˜ = lim H ˜ ( n ). H n→∞

in order to use the Baker–Hausdorff formula for arbi trary operator O:

(1)

tionary,” since it is a stationary point of the sequence of singletype unitary transformations ᑮ: ˜ (n + 1) = ᑮ(H ˜ ( n ) ), H ˜ = ᑮ(H ˜ ), H

(6)

We represent unitary operator U via the Hermitian operator: U = e ,

(11)

Expansion (8) and formulas (9)–(11), with the requirement of the absence of rapidly changing (in time) summands in the matrix elements of transformed

ω ij = ( E i – E j )/ប.

during the unitary transformation we proceed from the Schrödinger representation, then the matrix elements of the effective Hamiltonian should contain only “correct” rapidly changing multipliers, which are excluded in the transition to the interaction representation.

(12)

Before we derive the effective Hamiltonian corre sponding to resonance condition (12), we will discuss approaches used earlier in the context of subsequent application of the quantum SDE technique. If we sim ply limit ourselves in the initial Hamiltonian to only two resonance energy levels |E 1〉 and |E 2〉 , HInt HInt–TL (TL stands for twolevel), H

Int–TL

2

Hamiltonian (5) (in the interaction representation ), unambiguously determine unitary transformation (4)– (8), which are characterized as a transition to the effec tive Hamiltonian representation. The effective Hamil tonian representation, like the Heisenberg and Dirac representations, is closed in the sense that a repeat (or nth) unitary transformation ᑮ with the same properties determining it leaves the effective Hamiltonian “sta 2 If

379

=

∑Γ

† ω ( bω

+ b ω ) ( d 21 |E 2〉 〈E 1| + d 12 |E 2〉 〈E 1| ),

(13)

ω

then direct application of the quantum SDE technique with Hamiltonian (1) and interaction operator (13) gives an obviously incorrect result: the relaxation operator is identically equal to zero [22], as if the two level atom does not undergo any radiative decay. In works on calculating the relaxation operator [12, 3

The above requirements to the terms in the effective Hamilto nian unambiguously single out such a noise source from the broadband quantized electromagnetic field.

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13, 18, 19], it was proposed that the interaction oper ator be taken in the “initial Hamiltonian” in the rotat HInt–RF (RF stands ing wave approximation HInt for rotating frame): H

Int–RF

=

∑Γ



˜ (2) = H

(14)

×

† Γ ω b ω d 12 |E 1〉 〈E 2|,

The approach developed by the author [9, 23–25] on the basis of unitary transformation (4), (5) of the initial Hamiltonian assumes that as firstorder sum mands over the coupling constant with the electro magnetic field in the transformed Hamiltonian, it is necessary to take only those summands in the interac tion approximation that do not oscillate rapidly in time, namely:

1

k

k

∑ ∑ 2

Γω

ω

kj



Πk ( ν ) =

j

= –i

ω ω

ω

kj

2

d kj ⎛ 1 1    + ⎞ . ⎝ ប ω kj + ν ω kj – ν⎠

(15)

ω b ω d 21 |E 2〉 〈E 1|

∑Γ b ∑

2

d kj   |E k〉 〈E k|, ប ( ω kj – ω )

˜ ( 2 ) = H St + H Lamb , H

= H

Int–RF

∑Γ b ∑Γ † ω ω

St

H =

.

ω

×

This makes it possible to find operator S (1) from Eq. (10): S

(17)

k

In addition, the equation for determining S(2) follows from (11) and (17); however, the form of this operator will not be required here. It is important that operator S(2) determined in this way does not contain resonance denominators, so that expansion (8) is correct. The secondorder summand over coupling con stant (17) is represented in the form of two summands:

ω

(1)

k

where standard parameters of the theory of optical res onance processes are introduced [26]:

† ω b ω d 12 |E 1〉 〈E 2|

ω

∑Γ

ω'

k

the distinctive feature of which is the conservation of “excitation” in the system “twolevel atom + quan tized electromagnetic field.” As well, the remaining nonresonance atomic levels are not taken into account at all.

+

ω' b ω'

∑ 2 ( Π ( ω ) + Π ( ω' ) ) |E 〉 〈E | +

∑Γ

† ω ω

ω

ω

˜ (1) = H

∑Γ b ∑Γ

ω b ω d 21 |E 2〉 〈E 1|

ω

+

oscillating multipliers in the interaction representa ˜ ( 2 ) in the form tion, we obtain H

d kj '   |E k〉 〈E j| + H.c., (16) ប ( ω jk + ω )

ω' b ω'

ω'

1

∑ 2 ( Π ( ω ) + Π ( ω' ) ) |E 〉 〈E |, k

k

k

k

k

H

Lamb

=

∑ ∑ 2

Γω

ω

kj

2

d kj   |E k〉 〈E k|. ប ( ω kj – ω )

Lamb

which includes both nonresonance atomic levels and which as a consequence determines the contribution of nonresonance atomic levels to the effective Hamil tonian. A prime signifies the absence of resonance denominators under the summation symbol in (16). ˜ ( 1 ) other summands Note that if in Hamiltonian H remain having multipliers that rapidly oscillate in time in the interaction representation, then operator S(1) would consist of summands with resonance denomi nators, which would contradict the idea of expanding transformed Hamiltonian (5) into series (8) over the coupling constant. Further, substituting the expressions for quantities ˜ ( 1 ) into formula (11) and leaving in the first (1) S and H two commutators only summands not having rapidly

One of these, H , is commonly called the Lamb summand, since precisely it describes the Lamb level shift. The other summand, HSt, we call the Stark inter action operator. It is analogous to the Stark level shift in a classical electromagnetic field with a strength of E = Ᏹe–iωt + c.c. [26]. From comparison of the obtained formulas with the expressions for the Stark shift presented in [26], we can propose a simple method for obtaining HSt in the case of quantum fields from the classical expressions by way of substitution:

∑Γ b ∑Γ † ω ω

2

Ᏹ Πk ( ω )

ω'

ω

×

ω' b ω'

1

∑ 2 ( Π ( ω ) + Π ( ω' ) ). k

k

k

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As a result, we take the sum of the first three sum mands in expansion (8) as the effective Hamiltonian and set ˜ = H ˜ (0) + H ˜ (1) + H ˜ (2) , H

ω

(18)

ωc

as well, system of equations (6) for the matrix elements describing the resonance levels and transitions between them turns out to be closed. For nonreso nance levels, the summands and operators (8) and (18) have diagonal form. We further introduce generating su(2) algebras

E2 ω

E1

1 R 3 =  |E 2〉 〈E 2| – 1 |E 1〉 〈E 1|, 2 2 R + = |E 2〉 〈E 1|,

R – = |E 1〉 〈E 2|

Fig. 2. Structure of resonance levels of an atom in the case of Raman resonance with an external broadband electro magnetic field and microresonator photon mode of fre quency ωc, ω – ωc ≈ (E2 – E1)/ប. Here, the real two quanta transfer from excited level E2 and emission of a photon ω compete with virtual transitions with return to excited level E2. Transition E2 E1 is optically forbid den. Such a situation takes place in the case of an atomic– photon cluster [9].

with the commutation relations [ R 3, R ± ] = ± R ± ,

[ R +, R – ] = 2R 3

and operator I

TL

= |E 2〉 〈E 2| + |E 1〉 〈E 1|,

commuting with all summands and operator (18). We rewrite (18) in the form ˜ = H Eff + 1 ( E ' + E ' )I TL + H Nonres , H 1 2 2 H

Eff

H

H

= H

TL

Int–Tr

TL

F

+H +H



† Γ ω b ω d 12 R –

+

ω

H

+H

Int–St

(19)

∑Γ

, +

=





Γω bω

ω



ω b ω d 21 R + ,

E '2 = E 2 +

H

Nonres

=

∑ ∑ ∑ ∑

d 2j   |E 2〉 〈E 2|, ប ( ω 2j – ω )

j

2 Γω

j

j ≠ 1, 2



2

k ≠ 1, 2 , j

d kj   |E k〉 〈E k|. ប ( ω kj – ω )

then the transformed Hamiltonian TL ˜˜ i ( E '1 + E '2 )I ⎞ ˜ H  H = exp ⎛   ⎝2 ⎠ ប TL ( E '1 + E '2 )I ⎞ 1 TL  –  ( E '1 + E '2 )I × exp ⎛ – j  ⎝ 2 ⎠ 2 ប

= H

2

∑ E |E 〉 〈E | + ∑ Γ b ∑ Γ j

2 Γω

TL ˜˜ ( E '1 + E '2 )I ⎞ ˜ |Ψ 〉 = exp ⎛ j   |Ψ〉, ⎝2 ⎠ ប

2

d 1j   |E 1〉 〈E 1|, ប ( ω 1j – ω )

2

j

k

ω'

Γω

ω

k

Here, the primes in the energy resonance levels E '1 and E '2 indicate inclusion into the mentioned energy quantities of the Lamb level shift.

Π 2 ( ω ) + Π 2 ( ω' ) – Π 1 ( ω ) – Π 1 ( ω' ) ⎫ +  R 3 ⎬, 2 ⎭

ω

k

If we now perform yet another unitary transforma tion,

Γ ω' b ω'

⎧ Π 1 ( ω ) + Π 1 ( ω' ) + Π 2 ( ω ) + Π 2 ( ω' ) TL × ⎨   I 4 ⎩

E 1' = E 1 +

∑ ω

ω

Int–St

k

k ≠ 1, 2

E '2 – E '1 ω '21 =  , ប

= បω '21 R 3 ,

=

Int–Tr

1

∑ 2 ( Π ( ω ) + Π ( ω' ) ) |E 〉 〈E |

×

† ω ω

j

ω

ω'

ω' b ω'

Eff

+H

Nonres

.

Thus, in problems in which only levels |E 1〉 and |E 2〉 are populated and there are no fields connecting them to other nonresonance levels, transformed Hamilto ˜˜ nian H reduces to effective Hamiltonian HEff and the

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Schrödinger equation, closed in regards to resonance levels, looks as follows: ˜˜ 〉 = H Eff |Ψ ˜˜ 〉. iប ∂ |Ψ ∂t

(20)

Operator U(t) satisfies the following Schrödinger equation: iប d U ( t ) = ( H dt

Int–Tr

(t) + H

Int–St

( t ) )U ( t ).

(22)

In the subspace of quantum states of a twolevel atom, operator ITL is unitary. Thus, the effective Hamiltonian of the problem HEff (19) is determined by generating su(2) algebras; further, the prime in the res onance transition frequency ω '21 indicating inclusion into the Lamb shift quantity will be omitted. We make the following remarks. 1. The summands in the interaction operator of an atom with the electromagnetic field are divided according to the following principle. Summand HInt–Tr describes real transitions between quantum levels Int–Tr 〈E 2|H |E 1〉 ≠ 0. Summand HInt–St is responsible only for shifts in energy levels of a twolevel atom and does Int–St |E 1〉 = 0. not describe real transitions, since 〈E 2|H

In the interaction representation,

2. In an ordinary atom, summands HInt–Tr and HInt–St are of different order over coupling constant Γ. How ever, they can be firstorder due to the anomalous smallness in the dipole moment quantity d21 for any reason. 3. In certain twoquanta and singlequantum reso nance processes of atomic interaction with electro magnetic fields, the effective interaction operator with such fields can also be represented as the sum HInt–Tr + HInt–St [9, 23–25, 27] (Fig. 2). As well, for twoquanta resonances, characteristic quantities HInt–Tr and HInt–St will be firstorder without additional requirements regarding dipole moment quantities [9, 23–26]. In the case of threequanta (or higher) resonance conditions, the characteristic quantity of the Stark level shift HInt–St, as a rule, substantially exceeds the quantity HInt–Tr.

and the solution to Eq. (22) can be represented as a series:

3. DESCRIPTION OF SPONTANEOUS EMISSION BY STOCHASTIC DIFFERENTIAL EQUATIONS IN THE EFFECTIVE HAMILTONIAN REPRESENTATION It is convenient (but not mandatory) to perform subsequent transformations in the interaction repre sentation. Then, the vector of state of an atom and ˜˜ electromagnetic field |Ψ ( t )〉 with the given initial state |Ψ 0〉 evolves according to the Schrödinger equation ˜˜ ˜˜ Int–Tr Int–St iប d |Ψ ( t )〉 = ( H (t) + H ( t ) ) |Ψ ( t )〉, dt

(21)

A F ˜˜ ˜˜ i ( H + H )t⎞ |Ψ |Ψ ( t )〉 = exp ⎛  〉, ⎝ ⎠ ប

H

×H H

U ( 0 ) = I.

A

Int–St

F

( H + H )t⎞ ( t ) = exp ⎛ i ⎝ ⎠ ប

Int–Tr

×H

A

F

– i ( H + H )t exp ⎛ ⎞ , ⎝ ⎠ ប A

F

i ( H + H )t ( t ) = exp ⎛ ⎞ ⎝ ⎠ ប

Int–St

A

F

i ( H + H )t⎞ , exp ⎛ – ⎝ ⎠ ប

t Int–Tr Int–St 1 U ( t ) = I +  ( H ( t' ) + H ( t' ) ) dt' iប

∫ 0

t t' Int–Tr Int–St 1 + 2 ( H ( t' ) + H ( t' ) ) ( iប ) 0 0

∫∫

× (H

Int–Tr

( t'' ) + H

Int–St

(23)

( t'' ) )dt'dt'' + …

t

← ⎛1 Int–Tr = T exp ⎜  ( H ( t' ) iប ⎝

∫ 0

+H

Int–St

⎞ ( t' ) )dt' ⎟ , ⎠



where T is the timesequencing parameter. We now discuss the requirements on the initial state of the system. We will assume that initially the state of A F the atom |Ψ 0 〉 and field |Ψ 0 〉 in no way correlate with each other; i.e., the initial state of the system is fac A F tored: |Ψ 0〉 = |Ψ 0 〉 ⊗ |Ψ 0 〉 . The field states corre sponding to different frequencies are also uncorrelated and characterized by the absence of photons, i.e., F



F

〈Ψ 0 |b ω b ω' |Ψ 0 〉 = 0,

the solution to which can be represented via the evolu tion operator U(t) (I is the unit operator): ˜˜ |Ψ ( t )〉 = U ( t ) |Ψ 0〉,

Int–Tr

F



F

〈Ψ 0 |b ω b ω' |Ψ 0 〉 = δ ( ω – ω' ), F

F

F





F

〈Ψ 0 |b ω b ω' |Ψ 0 〉 = 〈Ψ 0 |b ω b ω' |Ψ 0 〉 = 0.

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In addition, F

F

F



F

〈Ψ 0 |b ω |Ψ 0 〉 = 〈Ψ 0 |b ω |Ψ 0 〉 = 0. This means that the electromagnetic field is a broad band electromagnetic field and can be considered as a thermostat with a central frequency ΩΓ = ω21. Until now, ordinary standard quantum theory pro cedures have been used. We now introduce new quan tities and make new principal assumptions. We deter mine the operators ∞

– i ( ω – ω 21 )t 1 b ( t ) =  dωe bω , 2π



–∞



† 1  dωe i ( ω – ω 21 )t b † , b ( t ) =  ω 2π



–∞

t

∫ dt'b ( t' ),

B(t) =

(26)

t †

B (t) =

0

∫ dt'b ( t' ), †

0

t

∫ dt'b ( t' )b ( t' ), †

Λ(t) =

0

as well, integration over ω is performed from –∞ to +∞ and not from 0 to +∞. This important assumption leads to the relations †

[ b ( t ), b ( t' ) ] = δ ( t – t' ),

=

Under Markov conditions, Eq. (22) is indetermi nate. This is clear if we consider in more detail the integrals encountered in formula (23). For instance, (23) contains integrals of the form t

∫ ϕ ( t' ) dB ( t' ), †

(27)

0

t2

where ϕ(t') is a certain operatorvalued function. Let, as usual,

∫ dt' ∫ dt''δ ( t' – t'' ) = min ( t , t ). 1

0

The performed approximations (28) and the adopted integration limits in determining quantities b(t) and b†(t) represent the Markov conditions for the interaction of an atom with the electromagnetic field—the dynamics of the broadband electromag netic field (24), (25) is determined by the field state at a given moment in time and does not depend on the formation prehistory of this state. They have been used in all previous derivations of the kinetic equation for an atom in a resonance field without allowance for the Stark summand in the interaction operator [18, 19].



[ B ( t 1 ), B ( t 2 ) ] t1

For simplicity, matrix element d21 was considered a real quantity. Without difficulty, its phase can be restored from comparison with the formula for HInt–Tr. The introduced parameters η1 and η3 are propor tional, respectively, to quantities Π2(ΩΓ) + Π1(ΩΓ) and Π2(ΩΓ) – Π1(ΩΓ), and ξ1 and ξ2 are proportional to Π1(ΩΓ) and Π2(ΩΓ). Without changing notation, here and further we consider all quantities, including time, dimensionless, omitting the Planck constant in the equation for the evolution operator. We restore the dimensions further in the section devoted to non Wiener spontaneous emission of a singlequantum particle.

[ B ( t ), B ( t' ) ] = t,



2

0

t

We further assume that coupling parameter Γω and Stark shift parameters Πk(ω) do not depend on fre quency ω: Γ ω = const = Γ ΩΓ , Π 1 ( ω ) = const = Π 1 ( Ω Γ ), Π 2 ( ω ) = const = Π 2 ( Ω Γ ).

Int–Tr

H



( tdt ) = χR + dB ( t ) + χR – dB ( t ),

Int–St

( t )dt = ( η 1 I + η 3 R 3 )dΛ ( t )

= ( ξ 1 |E 1〉 〈E 1| + ξ 2 |E 2〉 〈E 2| )dΛ ( t ), †







0

= lim

(28)

(29) (30)

dB ( t ) = B ( t + dt ) – B ( t ), dB ( t ) = B ( t + dt ) – B ( t ), dΛ ( t ) = Λ ( t + dt ) – Λ ( t ).

∫ ϕ ( t' ) dB ( t' ) (32)

N

The introduced quantities and assumptions (28) make it possible to write the interaction operators in the form H

383

∑ ϕ ( t' ) ( B ( t ) – B ( t †

i

N→∞ i=1



i

i – 1 ) ).

Here, points ti are such that 0 < t1 < t2 … < tN – 1 < t divide time interval (0, t) into N parts; as well, t0 = 0, tN = t, and t 'i is a certain point of the ith part of the indicated division: ti – 1 ≤ t '1 ≤ ti . With an increasing number of points in the divisions of interval (0, t) at N ∞, we also consider that the maximum size of intervals (ti – 1, ti) decreases to zero: max(ti – ti – 1) 0. We will consider that the meansquares limit is con sidered as the limit:

(31)

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It is easy to see that integrals of type (32) depend on the choice of point t 'i . For definiteness, we consider the expression t



N

∫ B ( t' ) dB ( t' ) = †

∑ B ( t' ) ( B ( t ) – B ( t †

lim

N→∞

0



i

i

i – 1 ) ).

i=1

Then, †





〈 B ( t 1 )B ( t 2 )〉 = 〈 [ B ( t 1 ), B ( t 2 ) ] + B ( t 2 )B ( t 1 )〉 F

F



= min ( t 1, t 2 ) + 〈Ψ 0 |B ( t 2 )B ( t 1 ) |Ψ 0 〉 = min ( t 1, t 2 ), N



N



∑ ( t' – t



B ( t 'i ) ( B ( t i ) – B ( t i – 1 ) ) =

i

i=1

i – 1 ).

i=1

We will consider that integrals of type (32) are taken in the Ito sense: t

N

∫ ϕ ( t' ) dB ( t' ) = †

0

lim

N→∞

As well, stochastic processes B(t), B†(t), and Λ(t) determine the quantum Wiener Q(t) and Poisson N(t) processes from the formulas [29, 30]

∑ ϕ(t

i – 1)(B





( t i ) – B ( t i – 1 ) ).

i=1

As well, quantities ϕ(t) are considered “nonanticipat ing”; i.e., in the statistical sense, they do not depend on the subsequent behavior of B(t), B†(t) for all future values t. This is expressed mathematically as the rela tions

Operators dB(t), dB†(t), and dΛ(t) are increments of annihilation and creation processes and the number of photons via which the Wiener and Poisson processes degenerate. In what follows, however, since this does not lead to misunderstandings, we will call B(t) and B†(t) (as well as dB(t) and dB†(t)) quantum Wiener processes and Λ(t) (and dΛ(t)) the quantum Poisson process. Processes Q(t) and N(t) further are not used anywhere. Relations (33) make it possible to now correctly redefine Eqs. (22). We consider the Ito differential of operator U(t) (22): dU ( t ) ≡ U ( t + dt ) – U ( t ). If we rewrite (23) in the form of products, Int

H ( tN – 1 ) U ( t ) = lim exp ⎛   ( t N – t N – 1 )⎞ … ⎝ ⎠ N→∞ i Int

H ( ti – 1 ) × exp ⎛   ( t i – t i – 1 )⎞ … ⎝ ⎠ i Int

H ( t0 ) × exp ⎛   ( t 1 – t 0 )⎞ , ⎝ i ⎠



[ ϕ ( t ), dB ( t ) ] = [ ϕ ( t ), dB ( t ) ] = [ ϕ ( t ), dΛ ( t ) ] = 0. We understand the Ito SDE to mean a relation of the form





t









where the stochastic integrals are understood in the Ito sense. Hudson and Parthasarathy [17] found that Ito dif ferentials (or increments) (31) in the case of the initial electromagnetic field state (24) and (25) satisfy the algebra †

dΛ ( t )dB ( t ) = dB ( t ), †

dB ( t )dΛ ( t ) = dB ( t ), dB ( t )dB ( t ) = dt, dΛ ( t )dB ( t ) = dΛ ( t )dt = dB ( t )dB ( t ) †







+ U ( t )dU ( t ) + dU ( t )dU ( t ).

t0

dΛ ( t )dΛ ( t ) = dΛ ( t ),

From this expression, the unitarity of the evolution operator and the correctness of the Ito differentiation rule are clear: †



(t) ,

d ( U ( t )U ( t ) ) = ( dU ( t ) )U ( t )

+ ε ( ϕ ( t ), t ) dΛ ( t ) + γ ( ϕ ( t ), t ) dt, t0

Int–St

+ ( η 1 I + η 3 R 3 )dΛ ( t ) ) ) – 1 }U ( t ).

t0

t

(t) + H



ϕ ( t ) – ϕ ( t 0 ) = α ( ϕ ( t ), t ) dB ( t ) + β ( ϕ ( t ), t ) dB ( t ) t0

Int–Tr

dU ( t ) = { exp ( – i ( χR + dB ( t ) + χR – dB ( t )

for which the following equality is justified: t

Int

H (t) = H

then with allowance for (30) and (31), we have



dϕ ( t ) = α ( ϕ ( t ), t )dB ( t ) + β ( ϕ ( t ), t )dB ( t ) + ε ( ϕ ( t ), t )dΛ ( t ) + γ ( ϕ ( t ), t )dt, t



Q ( t ) = B ( t ) + B ( t ), N ( t ) = Λ ( t ) + i ( B ( t ) – B ( t ) ).

= dB ( t )dΛ ( t ) = dB ( t )dt = dB ( t )dt = 0.

(33)

Further expanding the exponent in the series and using the Hudson–Parthasarathy algebra (33), we obtain the following Ito equations for the evolution operator: dU ( t ) = A 0 dtU ( t ) + A + dB ( t )U ( t ) †

+ A – dB ( t )U ( t ) + A Λ d ( t )U ( t ), †











dU ( t ) = U ( t )A 0 dt + U ( t )dB ( t )A + †







+ U ( t )dB ( t )A – + U ( t )dΛ ( t )A Λ , 2

A0 = χ R+

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exp ( – i ( η 1 I + η 3 R 3 ) ) – 1 + i ( η 1 I + η 3 R 3 ) ×    R – , 2 ( η1 I + η3 R3 )

η 1 I + η 3 R 3 – sin ( η 1 I + η 3 R 3 ) 2 – χ R +    R– 2 ( η1 I + η3 R3 )

exp ( – i ( η 1 I + η 3 R 3 ) ) – 1 A – =   ηR – , η1 I + η3 R3

1 – cos ( η I + η 3 R 3 ) 2 – iχ R + 1 R – . 2 ( η1 I + η3 R3 )

exp ( – i ( η 1 I + η 3 R 3 ) ) – 1 A + = χR +   , η1 I + η3 R3 A Λ = exp ( – i ( η 1 I + η 3 R 3 ) ) – 1. Here we understand the operators exp ( – i ( η 1 I + η 3 R 3 ) ) – 1 + i ( η 1 I + η 3 R 3 )    , 2 ( η1 I + η3 R3 ) exp ( – i ( η 1 I + η 3 P 3 ) ) – 1   η1 I + η3 R3 to mean Taylor series of the corresponding functions from x: η1I + η3R3 x with subsequent inverse sub stitution x η1I + η3R3. In the absence of Stark interaction η1 = η3 = 0, Eq. (34) describes the already studied case corre sponding to the Wiener relaxation of an atom. It is determined by the presence of only summands pro portional to dt and increments of the quantum Wiener process dB(t) and dB†(t), as well as the absence of sum mands proportional to the increment of the quantum Poisson process dΛ(t). The Wiener equations, like in the classical case, are determined only by Wiener pro cesses [19]. The dependence of the evolution operator on [30] is a feature of nonWiener process (synonyms: nonBrownian process, generalized Langevin pro cess) [30]. The general analytical structure of such equations was studied in [30]. Equation (34) for the evolution operator generates the stochastic equation for the wavefunction of the system, cos ( η 1 I + η 3 R 3 ) – 1 2 d |Ψ ( t )〉 = – iχ R + ⎛ i    2 ⎝ ( η1 I + η3 R3 ) η 1 I + η 3 R 3 – sin ( η 1 I + η 3 R 3 ) ⎞ –    R – dt |Ψ ( t )〉 2 ⎠ ( η1 I + η3 R3 ) ⎧ cos ( η 1 I + η 3 R 3 ) – 1 sin ( η 1 I + η 3 R 3 ) ⎫ + χ ⎨   – i  ⎬ η1 I + η3 R3 η1 I + η3 R3 ⎭ ⎩

(35)



× R – dB ( t ) |Ψ ( t )〉 with the effective nonHermitian interaction Hamil tonian

385

At η1 = η3 = 0, these expressions passed to the known expressions [16]. The Schrödinger SDE is convenient for numerical simulation of being “trajectory” of the quantum sys tem; however, we will construct the subsequent analy sis of nonWiener spontaneous emission from the equation for the system density matrix, which we will obtain and solve in later sections of the paper. In concluding this section, we emphasized the role of the effective Hamiltonian representation in the quantum SDE technique. In passing to the effective Hamiltonian representation, there is also a change in the vector describing the initial state of the system. Therefore, conditions (24) and (25) are imposed pre cisely on the transformed vectors of states. Otherwise, like in [22], a trivial result follows: in such a field, a quantum system does not evolve, which does not cor respond to reality and is not a consequence of ignoring the Poisson process. As well, using the obtained equa tions, we cannot, as usual, analyze situations in which a new separation of fast and slow subsystems occurs over any parameter. For instance, we do not need to study how the equations change with increasing detuning of central frequency ΩΓ of the spectrum of the electromagnetic field from frequency ω21 of the resonance transition, i.e., dependence on parameter Δ = ΩΓ – ω21. This parameter is absent in the above mentioned equations in which ω21 ≈ ΩΓ, but it is easy to recover it, which will lead to substitution of opera − iΔt tors R± in Eqs. (34) and (35): R± R± e + . Further, an equation will be obtained for the density matrix of the quantum particle, which will already not depend on parameter Δ. However, this does not mean that the equations will be justified for arbitrary values of parameter Δ. At large values of parameter Δ, for instance, in comparison to the spectral linewidth of the resonance transition obtained at Δ = 0, the above mentioned equations become incorrect and it is nec essary in this case to obtain a new effective Hamilto nian according to the method explained in the previ ous section and then switch to the Markov approxima tion and derive new SDEs. These equations will already differ from the results obtained from the abovementioned Eqs. (34) and (35) primarily in dif ferent values of parameters appearing in them. In addition, the SDE will be determined by a new noise source corresponding to another spectral region of the broadband electromagnetic field. Note that the described limiting situation can be achieved in such media as a photon crystal with peculiarities in the elec tromagnetic wave spectrum.

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This is similar in other examples of open quantum systems: SDEs and the equations determined by them for the density matrix of an open system with fast and slow subsystems will differ depending on the method by which they are obtained. It is necessary at first to separate the fast and slow subsystems in the open sys tem and obtain for them the effective Hamiltonian, SDEs, and equations for the density matrix. If we do not first separate the fast and slow subsystems in the open system and obtain the effective Hamiltonian of the problem, SDEs, and equations for the density matrix, as though it were impossible to perform such division, then further, from the obtained kinetic equa tions, we should not attempt to study the limiting case in which fast and slow subsystems are separated that are not taken into account in deriving the effective Hamiltonian. And the obtained equations will differ from those obtained based on the effective Hamilto nian with allowance for separation of fast and slow subsystems in the open system. It is important that the equations obtained by the various abovementioned methods will differ either fundamentally or by the value of the input coefficients. However, there will always be such differences. In the context of optical problems, attention to this was paid for the first time in [24] in the case of the dispersion limit for atoms in a microresonator with losses at the mirrors. And since, in connection with the general Lindblad result [28] regarding the “Lindblad” type of kinetic equations (see also Eq. (38) below), study of various problems begins with formulating the kinetic equations in the Lindblad form and their subsequent analysis, some times, including also the limiting cases of separating the fast and slow subsystems, it should be admitted that such studies are not quite correct. The author mentions certain examples of such studies in [27]. 4. KINETIC EQUATIONS FOR THE ATOMIC DENSITY MATRIX AND THE RELAXATION OPERATOR









+ ρ ( t )A 0 dt + ρ ( t )dB ( t )A + + ρ ( t )dB ( t )A – †





+ ρ ( t )dΛ ( t )A Λ + A + dB ( t )ρ ( t )dB ( t )A + †



+ A + dB ( t )ρ ( t )dB ( t )A – + A + dB ( t )ρ ( t )dΛ ( t )A Λ †









+ A – dB ( t )ρ ( t )dB ( t )A + + A – dB ( t )ρ ( t )dB ( t )A – †







+ A – dB ( t )ρ ( t )dΛ ( t )A Λ + A Λ dΛ ( t )ρ ( t )dB ( t )A + †



+ A Λ dΛ ( t )ρ ( t )dB ( t )A – + A Λ dΛ ( t )ρ ( t )dΛ ( t )A Λ . The kinetic equation for the density matrix ρTL(t) = TrFρ(t) for a twolevel subsystem following from the equation for ρ(t) after its averaging over the electro magnetic field states with allowance for (33) and the relations †

Tr F ( ρ ( t )dB ( t ) ) = Tr F ( ρ ( t )dB ( t ) ) = Tr F ( ρ ( t )dΛ ( t ) ) = 0 already has a quite simple form: TL

TL

TL





TL

dρ ( t ) = A 0 dtρ ( t ) + ρ ( t )A 0 dt + A – ρ ( t )A – dt or TL cos ( η 1 I + η 3 R 3 ) – 1 2 dρ    = χ R + ⎛  2 ⎝ dt ( η1 I + η3 R3 )

η 1 I + η 3 R 3 – sin ( η 1 I + η 3 R 3 ) ⎞ TL + i    R– ρ 2 ⎠ ( η1 I + η3 R3 ) cos ( η 1 I + η 3 R 3 ) – 1 2 TL + χ ρ R + ⎛   ⎝ ( η I + η R )2 1 3 3 η 1 I + η 3 R 3 – sin ( η 1 I + η 3 R 3 ) ⎞ – i    R– 2 ⎠ ( η1 I + η3 R3 )

(36)

The equation for the density matrix of the system, †

ρ ( t ) = U ( t ) |Ψ 0〉 〈Ψ 0|U ( t ) is also generated by Eq. (34) for the evolution operator and is derived, as usual, in the stochastic equation technique by calculating the increment: †

dρ ( t ) = ρ ( t + dt ) – ρ ( t ) = dU ( t ) |Ψ 0〉 〈Ψ 0|U ( t ) †



+ U ( t ) |Ψ 0〉 〈Ψ 0|dU ( t ) + dU ( t ) |Ψ 0〉 〈Ψ 0|dU ( t ). The equation for the density matrix ρ(t) of the entire system, obtained using Eq. (34) and the Hudson– Parthasarathy algebra, is quite cumbersome: dρ ( t ) = A 0 dtρ ( t ) + A + dB ( t )ρ ( t ) †

+ A – dB ( t )ρ ( t ) + A Λ dΛ ( t )ρ ( t )

2 ⎧ cos ( η 1 I + η 3 R 3 ) + χ ⎨   ⎩ η1 I + η3 R3

sin ( η 1 I + η 3 R 3 ) ⎫ TL – i   ⎬R – ρ R + η1 I + η3 R3 ⎭ ⎧ cos ( η 1 I + η 3 R 3 ) – 1 sin ( η 1 I + η 3 R 3 ) ⎫ × ⎨   + i  ⎬. η1 I + η3 R3 η1 I + η3 R3 ⎭ ⎩ This form (36) of the equation for the density matrix of a single emitter is convenient for subsequent generalization to the case of emission of an ensemble of identical emitters. To analyze the effect of nonreso nance levels, Eq. (36) can also be written in a form fol

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lowing from the second form of Stark interaction operator (3): TL

dρ = 2 R  χ + dt



k = 1, 2

cos ξ k – 1 ξ k – sin ξ k⎞ ⎛   + i   2 2 ⎝ ⎠ ξ ξ k

× |E k〉 〈E k|R – ρ 2

TL

+ χ ρ R+



k = 1, 2

k

TL

cos ξ k – 1 ξ k – sin ξ k⎞ ⎛   – i   2 2 ⎝ ⎠ ξ ξ k

k

(37)

× |E k〉 〈E k|R – +χ

2



k = 1, 2

×

cos ξ k – 1 sin ξ k⎞ TL ⎛   – i  |E k〉 〈E k|R – ρ R + ⎝ ⎠ ξk ξk



k' = 1, 2

cos ξ k' – 1 sin ξ k'⎞ ⎛   + i  |E k'〉 〈E k'|. ⎝ ξ k' ξ k' ⎠

Now it is possible to take into consideration in Eq. (37) the presence in (17) of nonresonance states |E k〉 , k ≠ 1, 2. As well, like before, we consider that there is only one quantum field resonant to the tran sition |E 2〉 |E 1〉 . According to formula (17) and the abovementioned procedure of deriving the kinetic equations, in (37), we must sum over k and k' to reach all levels; as well, we need to replace the den sity matrix of the twolevel system ρTL by the density matrix of a quantum particle ρA. Due to the presence in (37) of operators R±, we find that the nonreso nance states have no effect on the dynamics of reso nance levels; otherwise, it is like via parameters ξ1 and ξ2, and

which follows from the effective Hamiltonian repre sentation, is justified both for the Wiener case [28] and for the nonWiener case considered. This can be for mulated in another way as follows: the nonresonance levels of a quantum particle are insensitive to the Stark interaction of resonance levels with a broadband quantum field with zero photon density. 5. SUPPRESSION OF SPONTANEOUS EMISSION AND THE INITIAL SHIFT OF THE DECAYING LEVEL UNDER NONWIENER DYNAMICS OF A SINGLEQUANTUM PARTICLE Before writing Eq. (37) for the resonance matrix elements of the density matrix of a single quantum particle, we again transform (37) to clearly see its Lindblad [31] form: TL ξ 1 – sin ξ 1⎞ 2 cos ξ 1 – 1 dρ  + i    = χ ⎛  2 2 ⎝ ⎠ dt ξ1 ξ1

× |E 2〉 〈E 2|ρ

This result is important in the following aspect. In the case of classical fields [26], the Stark shift of each level consists of the Stark shifts of all fields used in the problem. In the quantum case with zero photon den sity, the Stark level shift is generally equal to zero due to relations (24), (25), whence F

〈Ψ 0 |

∑Γ b ∑Γ † ω ω

1

∑ 2 ( Π ( ω ) + Π ( ω' ) ) |E 〉 〈E |Ψ 〉 = 0. k

k

k

k

F 0

k

However, as is seen from the following section, the Stark interaction operator causes a new additional energy shift of the decaying level. And nonresonance levels do not have this additional shift. Partitioning of the broadband quantum electromagnetic field in the case of zero photon density into independent sources,

(38)

2 cos ξ 1 – 1 TL – 2χ   |E 1〉 〈E 2|ρ |E 2〉 〈E 1| 2 ξ1 TL

St–Tr



TL

TL



TL



= i[ρ , H ] – L Lρ – ρ L L + 2Lρ L , in complete accordance with the general theory of quantum SDEs [30]. Here, the operator H

St–Tr

2 ξ 1 – sin ξ 1 = – χ   |E 2〉 〈E 2| 2 ξ1

describes the additional shift of the decaying level mandatory for nonWiener decay of the upper level of a quantum particle, and 1 – cos ξ L = χ 1 |E 1〉 〈E 2| ξ1 is the Lindblad generator. The additional shift 2 ξ 1 – sin ξ 1 – χ   2 ξ1

ω'

ω

×

ω' b ω'

TL

ξ 1 – sin ξ 1⎞ 2 TL cos ξ 1 – 1 + χ ρ ⎛   – i   |E 2〉 〈E 2| 2 2 ⎝ ⎠ ξ1 ξ1

A

dρ 〈E k|  |E k〉 = 0 for k ≠ 1, 2. dt

387

of the decaying level |E 2〉 differs from the Lamb shift and results precisely from the process of its decay. It also differs from the conventionally understood Stark shift, represented in the effective Hamiltonian by the ˜ ( 2 ) and equal to zero in the case of zero summand H photon density. We will call the shift of the decaying level a decay Stark shift. The full picture of the non

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BASHAROV 2 Π 2 ( ω 21 ) – Π 1 ( ω 21 ) η 3 = χ  , 2 ( d 12 ) /បω 21

Wiener dynamics of a twolevel particle is given by the equation for the matrix elements: TL

dρ 22 2 1 – cos ξ 1 TL ρ 22 ,  = – 2χ  2 dt ξ1 TL

dρ 11 2 1 – cos ξ 1 TL ρ 22 ,  = 2χ  2 dt ξ1

(39)

TL

dρ 21 ξ 1 – sin ξ 1⎞ TL 2 1 – cos ξ 1  – i   ρ 21 .  = – χ ⎛  2 2 ⎝ ⎠ dt ξ1 ξ1 Thus, the Stark summand HInt–St(t) in interaction operator leads to renormalization of the constants of ordinary radiative decay into an additional shift in the resonance atomic transition frequency. Add any value of the parameter of the Stark inter action of the ground level ξ1, the Stark interaction with a vacuum electromagnetic field suppresses radiative transitions in a single emitter, decreasing the radiative decay constant. As well, it is as if the Poisson process stabilizes the excited atom. This is distinctly clear from the simplified equation for the case of small values of parameter ξ1: TL

2

ξ dρ 22 TL 2  = – χ ⎛ 1 – 1 ⎞ ρ 22 , ⎝ ⎠ dt 12 TL

2

ξ dρ 11 TL 2  = χ ⎛ 1 – 1 ⎞ ρ 22 , ⎝ dt 12⎠ TL dρ 21

2 ξ1

ξ TL 1 2  = –  χ ⎛ 1 –  + i 1⎞ ρ 21 . dt 2 ⎝ 12 3⎠ Note that the Stark level shift can be considered as the result of virtual transitions with absorption and emission of a virtual quantum leading to a return back to this level. In a quantum case, the transition to inter fere with the real transition from an excited state to a lower level. As a result, the total transition velocity to a lower level decreases. In a hypothetical situation, at a sufficient value of the Stark shift ξ1 2π, a particle participate only in virtual transitions and a transition to the ground state does not occur; i.e., the Stark shift freezes the particle at the excited level. Recall that the aforementioned refers to the case of zero photon den sity of a vacuum electromagnetic field. In the absence of direct quantum transitions between energy levels (χ = 0), the Poisson process in an photonless electromagnetic field in no way reflects on the state of an atom. We now write Eq. (36) for dimensional quantities. If we choose frequency ω21 as the characteristic fre –1

quency and ω 21 , then –1

χ = Γ ΩΓ ប d 12 ,

2 Π 2 ( ω 21 ) + Π 1 ( ω 21 ) η 1 = χ   , 2 2 ( d 12 ) /បω 21

(40)

2 Π 1 ( ω 21 ) 2 Π 2 ( ω 21 ) ξ 1 = χ  , ξ 2 = χ  . 2 2 ( d 12 ) /បω 21 ( d 12 ) /បω 21 Here, d12 is a real quantity, and in Eqs. (36)–(38), to obtain a dimensional equation it is necessary to per form the substitution t ω21t. Recall that due to the considered resonance ΩΓ = ω21, and with the chosen method of bringing it to dimensional form, the follow ing inequality should also be satisfied,

χ2 Ⰶ 1, so that expansions (8) make sense. In the case of an atomic–photon cluster, the pro cess of direct singlephoton transition between reso nance levels is described by a summand of the same order as the Stark interaction. Instead of H

Int–Tr

=

∑Γ

ω b ω d 21 R +

+ H.c.

ω

we have the operator [9] H

Int–Tr

= – gR + c



∑Γ

ω b ω Π 21 ( Ω Γ )

+ H.c.,

ω

which is determined by the characteristic parameter of the twoquanta transition operator Π 21 ( Ω Γ ) =

 ⎛  + ⎞ . ∑  ប ⎝ω + Ω ω –Ω ⎠ d 2j d j1

1

j2

j

1

Γ

j1

Γ

This parameter, generally speaking, is of the same order as quantities Π1(ΩΓ) and Π2(ΩΓ). Here, g is the coupling constant of an atom with photons of the microresonator mode described by the creation c† and annihilation c operators, as well as by frequency ωc. Together with the central frequency of the broadband quantum electromagnetic field, microresonator pho tons are in Raman resonance ΩΓ – ωc = ω21 with the optically forbidden atomic transition |E 2〉 |E 1〉 [9]. Quantity g, generally speaking, has the same order as quantity Γ ΩΓ Δω c , where Δωc is the spectral width of the microresonator mode, such that –1

2

–1

χ = gΓ ΩΓ Π 21 ( Ω Γ )ប ∼ Γ ΩΓ Π 21 ( Ω Γ )ប Δω c , Π k ( Ω Γ )Ω Γ 2 Π k ( Ω Γ )បΩ Γ 2 ∼ χ  . ξ k = χ ប  2 Π 21 ( Ω Γ )Δω c g ( Π 21 ( Ω Γ ) )

(41)

Since Δωc Ⰶ ΩΓ, at χ2 Ⰶ 1, parameters ξk (and ηk) can be on the order of unity and owing to the number of atoms making up the atomic–photon cluster, the Stark shift intensifies. The case of smallness of param eters ξk is considered in [9]. Thus, in the case of singlequantum resonance of a single particle, special conditions are needed such that

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parameters ξk are all of the order of unity and the effects of suppressing relaxation and the additional energy shift of the excited level are noticeable. Mean while, for collective spontaneous emission under con ditions of both singlequantum and twoquanta reso nance, parameters ξk can be on the order of unity as a consequence of the additional proportionality to the number of particles, and the described effects can be substantial; as well, the spontaneous emission is of nonWiener type. Note that in the model situation, when η1 = 0, the Hamiltonian of the problem degenerates only via gen erators of the su(2) algebra, which can be represented in the form of the model Hamiltonian









H = បω 21 R 3 + បωb ω b ω dω + Γ ω b ω D 12 ( ω )R – dω



+ Γ ω b ω D 21 ( ω )R + dω



389

I 1.2 2

3

1 0.8

4 0.4 0 0

2

4

6

8

10 t

Fig. 3. Normalized superradiation intensity profile as a function of dimensionless time for a different number of quantum particles: 13 (1), 17 (2), 21 (3), 25 (4). η1 = 0.2, η3 = 0. The maximum intensity for the case of 13 particles corresponds to unity of superradiation intensity.



+ Γ ω Γ ω' b ω b ω' Π ( ω, ω' )R 3 dω dω' with certain parameters D12(ω) and Π(ω, ω'). The use of such a model Hamiltonian is convenient in a variety of studies. As well, the dimensionless equations for the matrix elements of the density matrix obtained in the Markov approximation coincide with Eqs. (39), in which quantity ξ1 is replaced by a certain dimension less parameter η determined by the quantity Π(ω, ω') [32]. However, in the approach that uses the indicated model Hamiltonian, it is impossible to determine which level shifts are caused by the Stark interaction. It is also impossible to judge the role of nonresonance levels and their behavior in the nonWiener decay of an excited level. Further, we continue to use dimensionless quanti ties and equations. 6. NONWIENER SUPERRADIATION OF AN ENSEMBLE OF IDENTICAL QUANTUM PARTICLES Equation (36) is easily generalized to the case of interaction with a broadband quantum electromag netic field with zero photon density of an ensemble of identical particles concentrated in a bulk with a char acteristic size smaller than the resonance wavelength. It is commonly accepted to talk about spontaneous emis sion of an ensemble of excited atoms as superradiation [10]. If, as is usual in superradiation theory, we ignore the dipole–dipole interaction of identical particles and small summands similar in nature that appear as a con sequence of the unitary transformation in the effective Hamiltonian, then it is easy to obtain the following kinetic equation for density matrix ρN of the ensemble from N = 2r identical particles in field (24), (25): N cos ( η 1 NI + η 3 R 3 ) – 1 2 dρ    = χ R + ⎛  2 ⎝ dt ( η 1 NI + η 3 R 3 )

η 1 NI + η 3 R 3 – sin ( η 1 NI + η 3 R 3 ) ⎞ N + i   R – ρ ⎠ ( η 1 NI + η 3 R 3 ) cos ( η 1 NI + η 3 R 3 ) – 1 2 N + χ ρ R + ⎛   ⎝ ( η NI + η R ) 2 1 3 3 η 1 NI + η 3 R 3 – sin ( η 1 NI + η 3 R 3 ) ⎞ – i    R– 2 ⎠ ( η 1 NI + η 3 R 3 )

(42)

2 cos ( η 1 NI + η 3 R 3 ) – 1 – i sin ( η 1 NI + η 3 R 3 ) + χ   η 1 NI + η 3 R 3 N

× R– ρ R+ cos ( η 1 NI + η 3 R 3 ) – 1 + i sin ( η 1 NI + η 3 R 3 ) ×  . η 1 NI + η 3 R 3 Here it is assumed that the initial state of particles is symmetrical relative to their rearrangements, and operators R± and R3 realize the (N + 1)dimensional representation of the su(2) algebra. The states of the emitters are represented by eigen vectors |m〉 of operator R3: R3 |m〉 = m |m〉 , –r ≤ m ≤ r. The state of the ensemble from completely excited quantum particles is given by vector |r〉 , and the state of the ensemble, all particles of which are in the ground state, is given by vector |– r〉 . In studying super radiation, we consider that at the initial moment in time t = 0, all quantum particles of the ensemble are A excited, such that |Ψ 0 〉 = |r〉 . Equation (42) includes quantities η1NI and η3R3, which results in an important consequence of elective decay: in an ensemble of identical particles, the Stark shift intensifies. In the model case, when the perimeters of Stark level shifts are equal, Π2(ω21) = Π1(ω21); i.e., when η3 = 0, the nonWiener superradiation is described by

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1.2 0.8

2

(b)

(c)

ω

ω

ω

3

1 4

0.4

0

(a)

1

2

3

4

5 t

Fig. 4. Normalized superradiation intensity profile of an ensemble of 13 particles as a function of dimensionless times for a different Stark interaction value: η3 = 0.5 (1), 0.6 (2) 0.7 (3), 0.9 (4). η3 = 0.05. The maximum intensity in the case of η3 = 0.5 corresponds to unity of superradia tion intensity.

the same formulas as conventional (Wiener) superra diation, but with a renormalized constant χ, which is clear from the equations for diagonal matrix elements:

E2

E2

E2

ω

ω

ω

ωcl

ωc

E1

E1

ω E1

Fig. 5. Schematic depiction of real transitions to ground level E1 with quantum emission of frequency ω and of vir tual transitions without quantum emission with return to excited level E2 in the case of twophoton resonances. Bold solid and dotted arrows denote quanta of the microresona tor mode of frequency ωc and a classical electromagnetic field of frequency ωcl. Transition E2 E1 is optically forbidden.

N

dρ mm 1 – cos ( η 1 N ) N 2  ρ mm  = – 2χ g mm – 1  2 dt ( η1 N ) 1 – cos ( η 1 N ) N 2 + 2χ g m + 1m   ρ m + 1m + 1 , 2 ( η1 N ) g mm – 1 = 〈m|R + |m – 1〉 〈m – 1|R – |m〉 = ( r + m ) ( r – m + 1 ). Therefore, in the approximation of a large number of excitations N Ⰷ 1 [10], we have 2 2 I ( t ) ≈ γ ⋅ 1 N sinh γ ⋅ 1 N ( t – t D ) , 4 2

1 – cos ( η 1 N ) 2 γ = 2χ αបω 21  , 2 ( η1 N ) –1

(43)

t D = ( γN ) ln ( γN ). In contrast to the case of ordinary superradiation, the delay time and pulse duration of nonWiener superradiation depends nonmonotonically on the number of particles. If in ordinary superradiation with an increasing number of particles the delay time and pulse duration decrease, than in nonWiener superra diation, an increase in them can be observed due to the dependence γ in formula (43) on the nonWiener fac tor (1 – cos(η1N))/(η1N)2. As well, there exists a “crit ical number” Ncr of excited quantum particles η1Ncr = 2π from the ensemble of which (super)radiation is suppressed. This effect is a consequence of the effect of relaxation being suppressed by the Stark interaction. As well, since the size of the Stark energy level shift of an ensemble of identical particles is proportional to the

number of particles in the ensemble, the increase in the number of atoms leads to an effective increase in the Stark interaction. As the end result, all of this is related to the increase in the effective dipole moment of a sym metrized ensemble of identical quantum particles. In the general case, we have N

dρ mm 2  = – 2χ g mm – 1 dt 1 – cos ( η 1 N + η 3 ( m – 1 ) ) N ×    ρ mm 2 ( η1 N + η3 ( m – 1 ) ) 1 – cos ( η N + η 3 m ) N 2 + 2χ g m + 1m 1 ρ m + 1m + 1 . 2 ( η1 N + η3 m ) The mean intensity I (t) of superradiation is propor tional to the decrease in the energy of the ensemble: N N I ( t ) = – α d Tr ( H ρ ), dt N where H = បω21R3 and the geometric coefficient α is introduced. Then, N/2

I(t) = α



N

2

បω 21 2χ G mm – 1 ρ mm

m = – N/2 N/2

≡ α'



N

G mm – 1 ρ mm ,

m = – N/1

1 – cos ( η 1 N + η 3 ( m – 1 ) ) G mm – 1 = g mm – 1   . 2 ( η1 N + η3 ( m – 1 ) )

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ω E2

391

These new regular patterns of collective spontane ous emission of an ensemble of identical quantum par ticles can serve as the starting point for further, more detailed study in which dipole–dipole interaction of identical quantum particles, their different mutual locations, etc., should be taken into account.

ω

7. CONCLUSIONS E2' Dipoledipole interaction E1' Spontaneously emitting atom

E1 Atom of another sort

Fig. 6. Twoquanta processes with spontaneous quantum emission ω with transition from the excited to the ground level E2 E1 in one particle during its interaction with an atom from the surroundings and it’s excitation to level E '2 . Transition E2 E1 is optically forbidden, and E '2

E '1 is optically allowed.

During emission, an exciting ensemble of quantum particles subsequently passes from a fully inverted state |r〉 to a state |m〉 , m = r – 1, r – 2, …. If η1 ≥ 0, η3 > 0 and the number of particles in the ensemble is such that there exists a number mcr, –r < mcr < r at which η1N + η3(mcr – 1) ≈ 2π, effect of “termination” of the emission process and stabilization of the excited ensemble in relation to collective decay is possible, since the transition velocity G mcr mcr – 1 from the state |m cr〉 to a lower energy level |m cr – 1〉 becomes equal to zero. Figure 3 shows the time profiles of the intensity of collective spontaneous emission intensity of an excited ensemble of identical quantum particles for different numbers of particles in the ensemble. Clearly, at the chosen parameter values, with an increasing number of particles, the delay time and emission pulse width increase in opposition to the case of ordinary superradiation. Figure 4 shows the time profiles of the collective spontaneous emission intensity of an excited ensemble of identical quantum particles for different values of parameter η3 in a region far from the spontaneous decay suppression mode. With increasing Stark inter action described by parameter η3, the delay time and superradiation pulse width increase and an insignifi cant increase in intensity is also observed.

The distinguishing physical features of nonWiener relaxation are the competition and superposition of real transitions with quantum emission and virtual transitions as a result of Stark interaction without quantum emission with a return to the excited level. As a result, the Stark interaction stabilizes the excited quantum particle, suppressing spontaneous emission in a singlequantum particle; in an ensemble of iden tical particles, weak Stark interaction is intensified by the collective process of atoms interacting with the vacuum field, and it suppresses collective spontaneous emission as well as superradiation. The most suitable problems for experimentally studying nonWiener spontaneous emission are undoubtedly collective (onequantum, twoquanta) spontaneous emission and relaxation processes in artificial emitters, e.g., of atomic–photon cluster type [9]. In the case of a single atomic–photon cluster, the necessary intensity of the Stark interaction is provided by a sufficient number of atoms in the cluster. In comparison to the case of one quantum decay studied here, twoquanta relaxation processes are complicated for studying, e.g., twopho ton decay of an excited state in which the atom emits two photons (see Fig. 5a). The author knows of no rig orous theoretical analysis to date of such processes using only the one hypothesis on the Markov behavior of the interaction of atoms with a broadband electro magnetic field. The reason for this is partly due to the absence of a suitable mathematical description of two quanta decay with emission of two photons using quantum SDEs. Analysis of twophoton decay under significant Stark interaction (Fig. 1) using methods [14–16], alternatives to the quantum SDE method, encounters significant computational difficulties. The fact of the matter is that in methods [14–16], to obtain the kinetic equation of a twophoton emitting quan tum particle, it is necessary to add up the entire infinite series mandatory in the final analysis of the essence of Stark interaction, like for the Poisson process. In par ticular, this is indicative of the presence in the kinetic equations obtained in this paper, e.g., in Eq. (38), of 2 the nonWiener factor (1 – cosξ1)/ ξ 1 . To date, all kinetic equations derived by alternative methods [14– 16] have used only the first nondisappearing summand of the aforementioned infinite series. Therefore, the known works on collective spontaneous twophoton decay in a broadband vacuum have a limited region of applicability in which the Stark interaction can be ignored. In the quantum SDE technique, analogous

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summation, as was demonstrated in this study, occurs automatically owing to the Hudson–Parthasarathy algebra, namely, the radiative transitions with deexci tation of only one photon. The following cases can be referred to spontaneous emission with deexcitation of a single photon with the necessity of allowing for strong Stark interaction in an ensemble of identical photons. (1) The case of an optically allowed transition (Fig. 1). (2) Twoquanta relaxation processes in which only one quantum is also emitted. As well, the other quantum participating in the twoquanta resonance transition is taken from either the external additional coherent field (frequen cies ωcl, Fig. 5b), from the microresonator photon or phonon mode (frequencies ωc and a small spectral width, Figs. 2, 5c), or owing to the interaction with surrounding atoms of the medium (Fig. 6). Note that at present, schemes analogous to those presented in Figs. 2 and 5c have been seriously discussed as practi cal schemes of atomic–phonon interfaces of quantum computing devices [33]. Twoquanta processes with emission of a single quantum and participation of microresonator mode photons (Figs. 2, 5c) are described based on the notion of an atomic–photon cluster [9, 34]. In [9], the author had to limit himself to a narrow parameter region in which it was possible to ignore the Stark shift with a broadband quantum field (in order to avoid examining nonWiener processes) and only Wiener decay of an atomic–photon cluster was considered. The results of this paper are also directly applicable for describing nonWiener decay of a singly excited atomic–photon cluster. The case of nonWiener decay of a multiply excited atomic–photon cluster can be considered using the approach developed in the paper; however, here, the features of an atomic–photon cluster will also manifest themselves in connection with its description by a thirdorder polynomial algebra. The case presented in Fig. 5c has been examined by the author in [23], in which the Stark shift is reduced to a “strong” Wiener process proportional to the amplitude of the coherent field, and to a weak Poisson process, which was ignored. However, in such condi tions, the Poisson process can be seen in the case of collective interaction with the vacuum field of an atomic ensemble. There is no quantum SDE apparatus for analyzing the case of using a broadband electromagnetic field with a nonzero photon density: it is necessary to obtain a generalization of the Hudson–Parthasarathy algebra to the case of nonzero photon density. However, there are still insurmountable mathematical difficulties in generalizing the Hudson–Parthasarathy algebra. The case of spontaneous emission of an atom in media with peculiarities in the photon spectrum due to additional interaction with the atoms of the medium (Fig. 6) was examined by the author in [25]. However, there, the problem was also solved in a narrow param

eter region in which the Stark interaction was ignored because at that time he had no knowledge of how to take this into account. Nevertheless, in problems sim ilar to [25], the Stark interaction can also be quite sig nificant for an ensemble of admixture atoms. In concluding, we emphasize that the Stark inter action is in a certain sense universal, since it represents a secondorder field effect independently of resonance conditions and is intensified in an ensemble of a cer tain number of identical particles. There are quite var ious physical conditions of spontaneous quantum par ticle emission and an ensemble of identical particles in which precisely a nonWiener type of spontaneous emission should be determining, and the approach described in this paper may be applicable for their description in the case of zero photon density of an electromagnetic vacuum. ACKNOWLEDGMENTS The author thanks V.P. Belavkin for his remarks on [30] and on the terminology used therein and A.M. Chebotarev for discussions of mathematical questions connected to the Poisson and Levy pro cesses. The work was financed in part by the Russian Foundation for Basic Research (project no. 0902 00503a). REFERENCES 1. P. A. M. Dirac, Proc. R. Soc. London, Ser. A 114, 710 (1927). 2. V. Weisskopf and E. Wigner, Z. Phys.: Condens. Matter 63, 54 (1930); V. Weisskopf and E. Wigner, Z. Phys.: Condens. Matter 65, 18 (1931). 3. E. B. Davies, Quantum Theory of Open Systems (Aca demic, London, 1976). 4. L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms (Willey, New York, 1975). 5. V. S. Butylkin, A. E. Kaplan, Yu. G. Khronopulo, and E. I. Yakubovich, Resonant Nonlinear Interactions of Light with Matter (Nauka, Moscow, 1977; Springer, Berlin, 1989). 6. F. Bloch, Phys. Rev. 70, 460 (1946). 7. Y. Yamamoto and A. Imamoglu, Mesoscopic Quantum Optics (Wiley, New York, 1999). 8. D. Loss and D. P. DiVincenzo, Phys. Rev. A: At., Mol., Opt. Phys. 57, 120 (1998). 9. A. M. Basharov, JETP 110 (6), 951 (2010). 10. M. G. Benedict, A. M. Ermolaev, V. A. Malyshev, I. V. Sokolov, and E. D. Trifonov, SuperRadiance: Mul tiatomic Coherent Emission (Institute of Physics, Bristol and Philadelphia, 1996). 11. H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993). 12. A. Barchielli, Phys. Rev. A: At., Mol., Opt. Phys. 34, 1642 (1986). 13. A. Barchielli and V. P. Belavkin, J. Phys. A: Math. Gen. 24, 1495 (1991).

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SPONTANEOUS EMISSION OF THE NONWIENER TYPE 14. K. Blum, Density Matrix Theory and Applications (Ple num, New York, 1981; Mir, Moscow, 1983). 15. H.P. Breuer and F. Petruccione, Theory of Open Quan tum Systems (Oxford University Press, Oxford, 2002). 16. V. E. Tarasov, Quantum Mechanics of NonHamiltonian and Dissipative Systems (Elsevier, Amsterdam, 2008). 17. R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys. 93, 301 (1984). 18. C. W. Gardiner and M. J. Collet, Phys. Rev. A: At., Mol., Opt. Phys. 31, 3761 (1985). 19. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, 2000). 20. A. M. Chebotarev, Lectures on Quantum Probability (Sociedad Mathematica Mexicana, Mexico, United Mexican States, 2000). 21. A. S. Kholevo, Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravleniya 83, 31 (1991). 22. W. J. Munro and C. W. Gardiner, Phys. Rev. A: At., Mol., Opt. Phys. 53, 2633 (1996). 23. A. M. Basharov, Sov. Phys. JETP 75 (4), 611 (1992).

393

24. A. M. Basharov, V. N. Gorbachev, and A. A. Rodich kina, Phys. Rev. A: At., Mol., Opt. Phys. 74, 042313 (2006). 25. A. M. Basharov, JETP 89 (6), 1063 (1999). 26. A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves (Kluwer, Dordrecht, The Netherlands, 1999). 27. A. M. Basharov, Teor. Fiz. (Samara) 9, 7 (2008). 28. M. Lax, Phys. Rev. 145, 110 (1966). 29. V. P. Belavkin, Usp. Mat. Nauk 47, 47 (1992). 30. V. P. Belavkin, Theor. Math. Phys. 110 (1), 35 (1997). 31. G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 32. A. M. Basharov, Phys. Lett. A 375, 784 (2011); http://arxiv.org/abs/1101.3288. 33. K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev. Mod. Phys. 82, 1041 (2010). 34. A. M. Basharov, Bull. Russ. Acad. Sci.: Phys. 75 (2), 161 (2011).

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