Fluctuations in quantum devices
Logical gates can be formalized by Boolean algebra whose elementary operations can be realized by devices that employ the interactions of macroscopic numbers of elementary excitations such as electrons, holes, photons etc. With increasing miniaturization to the nano scale and below, quantum fluc...
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Інститут фізики конденсованих систем НАН України
2004
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| Цитувати: | Fluctuations in quantum devices / H. Haken // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 527–537. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1190382017-06-04T03:02:31Z Fluctuations in quantum devices Haken, H. Logical gates can be formalized by Boolean algebra whose elementary operations can be realized by devices that employ the interactions of macroscopic numbers of elementary excitations such as electrons, holes, photons etc. With increasing miniaturization to the nano scale and below, quantum fluctuations become important and can no longer be ignored. Based on Heisenberg equations of motion for the creation and annihilation operators of elementary excitations, I determine the noise sources of composite quantum systems. Логічні фільтри можна формалізувати за допомогою булевої алгебри, елементарні операції якої можна реалізувати в квантових пристроях, де використовується взаємодія макроскопічної кількості елементарних збуджень (електрони, дірки, фотони і т.д.). З ростом мініатюризації системи до нанорозмірів, стають важливими квантові флуктуації, і ними вже не можна нехтувати. На основі рівняння Гейзенберга для операторів народження і знищення елементарних збуджень визначено характеристики джерел шуму в складних квантових пристроях. 2004 Article Fluctuations in quantum devices / H. Haken // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 527–537. — Бібліогр.: 4 назв. — англ. 1607-324X DOI:10.5488/CMP.7.3.527 PACS: 05.40.-a, 05.30.-d http://dspace.nbuv.gov.ua/handle/123456789/119038 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Logical gates can be formalized by Boolean algebra whose elementary operations
can be realized by devices that employ the interactions of macroscopic
numbers of elementary excitations such as electrons, holes, photons
etc. With increasing miniaturization to the nano scale and below, quantum
fluctuations become important and can no longer be ignored. Based
on Heisenberg equations of motion for the creation and annihilation operators
of elementary excitations, I determine the noise sources of composite
quantum systems. |
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Article |
| author |
Haken, H. |
| spellingShingle |
Haken, H. Fluctuations in quantum devices Condensed Matter Physics |
| author_facet |
Haken, H. |
| author_sort |
Haken, H. |
| title |
Fluctuations in quantum devices |
| title_short |
Fluctuations in quantum devices |
| title_full |
Fluctuations in quantum devices |
| title_fullStr |
Fluctuations in quantum devices |
| title_full_unstemmed |
Fluctuations in quantum devices |
| title_sort |
fluctuations in quantum devices |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119038 |
| citation_txt |
Fluctuations in quantum devices / H. Haken // Condensed Matter Physics. — 2004. — Т. 7, № 3(39). — С. 527–537. — Бібліогр.: 4 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hakenh fluctuationsinquantumdevices |
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2025-07-08T15:07:53Z |
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2025-07-08T15:07:53Z |
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1837091810452701184 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 3(39), pp. 527–537
Fluctuations in quantum devices
H.Haken
Institute for Theoretical Physics, Center of Synergetics,
University of Stuttgart,
Pfaffenwaldring 57/4, D–70550 Stuttgart, Germany
Received March 16, 2004
Logical gates can be formalized by Boolean algebra whose elementary op-
erations can be realized by devices that employ the interactions of macro-
scopic numbers of elementary excitations such as electrons, holes, pho-
tons etc. With increasing miniaturization to the nano scale and below, quan-
tum fluctuations become important and can no longer be ignored. Based
on Heisenberg equations of motion for the creation and annihilation opera-
tors of elementary excitations, I determine the noise sources of composite
quantum systems.
Key words: quantum fluctuations, elementary excitations, Heisenberg
equations of motion, logical devices, noise in composite quantum systems
PACS: 05.40.-a, 05.30.-d
1. Introduction
Since my contribution is dedicated to the memory of my friend Yuri Klimon-
tovich, I wish to start with a few personal remarks. I met Yuri for the first time
in 1963 when I was invited to attend and to give a paper at the All Union Con-
ference on Solid State Physics in Moscow. This was my first visit to the Soviet
Union and I was overwhelmed by the warm reception I received there after such a
terrible war. I gave a lecture on laser theory that met special interest by Yuri. He
read my preprint and we had a very stimulating discussion on it. I was impressed
by the great warmth of Yuri and we became friends immediately. Later I had again
and again opportunities to meet Yuri at various conferences in the Soviet Union.
I especially remember a conference in Pushchino on the field of synergetics. I was
delighted to see the reception of our research and I realized how much contributions
had been done previously in Russia to this field. I was also very pleased that Yuri
and his colleagues initiated a permanent and very lively seminar on synergetics at
the Lomonossov University in Moscow. I tried of course to invite Yuri to my own
institute in Stuttgart because I had some funds there. For the well known reasons,
however, he was not able to come until the new situation occurred. Since then I
c© H.Haken 527
H.Haken
was very happy to have him several times in Stuttgart and we enjoyed both our
scientific discussions and life very much. I was deeply impressed by the profound
ideas of Yuri, especially in the field of non-equilibrium physics [1] that was a field
of our common interest and I profited again and again by his deep insights, e.g. on
his by now famous S-theorem. The last time I met Yuri was in Berlin at Werner
Ebeling’s institute just a few months before his untimely death. Yuri was such a
warm hearted and noble personality and I surely will not forget him.
My paper is motivated by the progress made in information technology and
here again especially in nanotechnology. So far information processing is based on
logical elements that use large numbers of elementary excitations such as electrons
so that quantum fluctuations can be ignored, at least in many cases. With increasing
miniaturization, however, the quantum domain becomes important. Therefore my
contribution tries to show how quantum fluctuations can be calculated in a unique
way for all kinds of elementary excitations and their interactions that may be used
in logical devices.
In my contribution I consider logical devices or, in other words, logical gates.
As is well known logical operations can be formalized by a Boolean algebra which
contains in particular the operations “and”, “or”, and the “exclusive or”. Their
corresponding truth tables can be realized by specific dynamical systems of which
we write down the typical equations
“and′′ :
dnr
dt
= gn1n2 , n > 0, (1)
“or′′ :
dnr
dt
= g(n1 + n2), n > 0, (2)
“exclusiveor′′ :
dx
dt
= −3yz − (x2 + y2 + z2)x, (3)
dy
dt
= −3xz − (x2 + y2 + z2)y, (4)
and
dz
dt
= −3xy − (x2 + y2 + z2)z. (5)
These equations must be solved by means of an initial value problem where the
transients provide the solutions to the corresponding truth tables. In the cases (1)
and (2) the reader may easily verify the validity of the corresponding truth tables
while with respect to the equations (3)–(5) I refer the reader to my book “Synergetic
computers and cognition” [2]. When we want to realize these abstract equations by
means of devices where e.g. the quantities n refer to currents or charge densities, we
must take care of their coupling to the surrounding which may be represented by
heatbaths. The elimination of the heatbath variables then leads to equations that
represent losses as well. Thus for instance equation (1) must be replaced by equation
dnr
dt
= gn1n2 − knr . (6)
With increasing miniaturization going to the nano scale and especially to still
smaller dimensions the processes must be treated from a quantum mechanical point
528
Fluctuations in quantum devices
of view. Thus the obvious question arises how to translate the above equations into
quantum mechanics. Here we must be very careful because there are different ways
depending on the experimental setup. In other words, we must take care of the
quantum mechanical preknowledge we have or, in other words, we must take care
of the way the initial states are prepared. Our considerations will apply basically
to all sorts of elementary excitations in solid state physics such as electrons, holes,
excitons, plasmons, polaritons, spin waves as well as to photons. In the first approach
I am proposing the replacement
nr, n1, n2 → quantum amplitudes b, a, c, (7)
where b, a, c will become quantum mechanical operators. In a more detailed descrip-
tion, these operators must be equipped with indices, e.g. wave-vectors. To bring out
the essentials, we drop such indices here. The interaction term in the case that will
give rise to equation (1) can be described by a Hamiltonian of the form
H = ~g(b+ac + c+a+b), (8)
where ~ is Planck’s constant divided by 2π and g is a coupling constant. The daggers
indicate creation operators. Operators without daggers are annihilation operators.
In order to facilitate my approach I will consider in the present case a subspace
spanned by the vectors
a+c+φ0 and b+φ0 , (9)
where φ0 is the vacuum state. In order to come as close as possible to the description
in the terminology of a dynamical system I consider Heisenberg equations of motion.
As is well known, for any operator Ω the Heisenberg equations can be derived by
means of the Hamiltonian in the form
dΩ
dt
=
i
~
[H, Ω] , (10)
where the square bracket indicates the commutator
[H, Ω] = HΩ − ΩH. (11)
Thus the Heisenberg equation for the annihilation operator b reads
db
dt
= i
[
g(b+ac + c+a+b), b
]
(12)
or because of commutation relations
db
dt
= igac
[
b+, b
]
. (13)
Using the explicit commutation relation
[
b+, b
]
= −1 (14)
529
H.Haken
we eventually arrive at
db
dt
= −igac. (15)
Similarly the Heisenberg equations for the operator product ac read
d(ac)
dt
= ig
[
(b+ac + c+a+b, ac
]
(16)
or
d(ac)
dt
= igb
[
c+a+, ac
]
= igb(c+a+ac − acc+a+) (17)
which because of
aa+ − a+a = 1 (18)
can be cast into the form
d(ac)
dt
= igb(c+a+ac − (1 + a+a)(1 + c+c)) (19)
or in short
d(ac)
dt
= −igb(1 + a+a + c+c). (20)
Having in mind that the operators act in the subspace (9) the equation (20) reduces
to
d(ac)
dt
= −igb. (21)
By a similar approach we obtain the relations
d(bac)
dt
= ig
[
b+ac + c+a+b, bac =
]
=
= igac
[
b+, b
]
+ igb
[
c+a+, ac
]
=
= −igac − igb in subspace! (22)
So far we have not taken into account the interaction of our quantum system with
heatbaths. Having eliminated them (which can be done in a model like fashion
explicitly) we may take care of their effect using the damping constants so that the
above equations are translated into
db
dt
= −igac − κb, (23)
d(ac)
dt
= −igb − γac (24)
and
d(bac)
dt
= −igac − igb − γ′(bac), (25)
respectively. However, this phenomenological approach has a great drawback be-
cause, as one may easily show, it violates quantum mechanical consistency. As has
530
Fluctuations in quantum devices
been shown especially in the context of laser physics this violation of quantum me-
chanical consistency can be remedied by the introduction of quantum noise operators
Γ so that for instance equation (23) is transformed into
db
dt
= −igac − κb + Γb(t). (26)
For reasons that will become obvious herein below I reformulate the correspond-
ing quantum mechanical equations using projection operators Pij according to the
scheme:
b → P 1
01 ; ac = P 2
0011 ; bac → P 1
01P
2
0011 . (27)
This transforms the equations (23) and (24) into
dP 1
01
dt
= −igP 2
0011 − κP 1
01 + Γ1
01 (28)
and
dP 2
0011
dt
= −igP 1
01 − γP 2
0011 + Γ2
0011 , (29)
respectively, and using the notation
P 1
01P
2
0011 = P 0
000,111 (30)
we can cast (25) into the form
dP 0
000,111
dt
= −igP 2
0011 − igP 1
01 − γ′P 0
000,111 + Γ0
000,111 . (31)
The obvious question arises in which way we may determine the fluctuating forces
Γ1
01, Γ2
0011, Γ0
000,111. Before we turn our attention to the solution of this task, we
discuss a second interpretation. Namely the initial state may also be prepared using
the occupation numbers and we may consider an experimental setup in which only
occupation numbers of elementary excitations are concerned. Thus our goal is to
translate the equation
dnr
dt
= gn1n2 (32)
correspondingly which we first consider without damping. In our model we also
assume that
d(n1n2)
dt
= gnr (33)
holds. From (33) and (32) we easily derive
d(nrn1n2)
dt
= gn2
1n
2
2 + n2
rg. (34)
Introducing losses we may supplement the equations correspondingly so that for
instance we obtain
dnr
dt
= gn1n2 − κnr (35)
531
H.Haken
and
d(n1n2)
dt
= gnr − γn1n2 . (36)
When we choose κ = γ = g, (35) and (36) lead to a conservation law. In order to
translate these equations into quantum mechanics we introduce projection operators
according to
nr → P 1
11 , n1n2 → P 2
11,11 (37)
so that we now have to consider the equations
dP 1
11
dt
= gP 2
11,11 − κP 1
11 + Γ1
11 , (38)
dP 2
11,11
dt
= gP 1
11 − γP 2
11,11 + Γ2
11,11 , (39)
and
d
(
P 1
11P
2
11,11
)
dt
= gP 2
11,11P
2
11,11 + gP 1
11P
1
11 − (κ + γ)P 1
11P
2
11,11 + Γ0
111,111 . (40)
Making use of the property of projection operators (see also below for details)
P 1
11P
1
11 = P 1
11 (41)
and of
P 1
11P
22
11,11 = P 0
111,111 (42)
we may cast (40)into the form
dP 0
111,111
dt
= gP 2
11,11 + gP 1
11 − (κ + γ)P 0
111,111 + Γ0
111,111 . (43)
Again it remains our task to determine the fluctuating forces Γ explicitly.
2. Haken-Weidlich theorem [3,4]
In the following we use the fundamental quantum mechanical property of pro-
jection operators
PijPlk = δjlPik . (44)
We denote quantum statistical averages by square brackets. We assume that for in-
stance phenomenologically or partly phenomenologically and partly from first prin-
ciples the following averaged equations are given
d 〈Pij〉
dt
=
∑
kl
〈Mij,klPkl〉, (45)
where the elements M do not depend on P , but may depend on variables of other
quantum systems. As one may show the solutions to (45) do not obey the quantum
532
Fluctuations in quantum devices
mechanical consistency relations (44). To restore quantum mechanical consistency
we introduce the equations
dPij
dt
=
∑
kl
MijklPkl + Γij(t) . (46)
We assume that the averages vanish
〈Γij(t)〉 = 0 (47)
and that the fluctuating forces are δ-correlated in time
〈Γij(t)Γkl(t
′)〉 = Gij,klδ(t − t′). (48)
This is the only assumption to be made in the present context. In many cases it
is fulfilled if for instance the reservoirs are broadband or the relaxation time of the
fluctuating forces is short compared to that of all other processes in the system.
The Haken-Weidlich theorem states that the strength of the fluctuating forces is
uniquely determined by
Gij,kl =
∑
mn
〈(δjkMil,mn − δnlMij,mk − δmiMkl,jn) Pmn〉. (49)
As a comparison with the previous section shows, M may be decomposed into
Mij,kl = M
(1)
ij,kl + M
(2)
ij,kl , (50)
where
M (1) stems from
i
~
[H, Pij] . (51)
As can be shown, the terms M (1) cancel each other so that it is sufficient to determine
the strengths of the fluctuating forces by using M (2) instead of M in (49).
3. Composite quantum systems
In the following I want to generalize the Haken-Weidlich theorem to composite
quantum systems. This requires the introduction of the appropriate multiplication
rules of projection operators. First we adopt the already known rule
P 1
ijP
1
kl = δjkP
1
il , (52)
where the upper index lrefers to the specific subsystem. Similarly we have
P 2
ijP
2
kl = δjkP
2
il . (53)
However, what is new is the relation for the composite system given by
P 1
ijP
2
kl = P 0
ik,jl . (54)
533
H.Haken
From (52)–(54) we may deduce the following multiplication rules:
P 1
ijP
0
i′k′,j′l′ ≡ P 1
ijP
1
i′,j′P
2
k′l′ = δji′P
0
ik′,j′l′ , (55)
P 2
ijP
0
i′k′,j′l′ = P 2
ijP
1
i′j′P
2
k′l′ = δjk′P 0
i′i,j′l′ , (56)
P 0
ij,klP
0
i′j′,k′l′ = P 1
ikP
2
jlP
1
i′k′P
2
j′l′ = δki′δljP
0
ijk′l′ . (57)
For the following we need a concise notation so that we introduce the following
abbreviation:
P K
i
∼
, j
∼
for K = 1, i
∼
= 1, j
∼
= j,
K = 2, i
∼
= 1, j
∼
= j,
K = 0, i
∼
= i1i2, j
∼
= j1j2 . (58)
Having used it we can cast the relations (52)–(54) in the concise form
P K
i
∼
, j
∼
P L
i′
∼
,j′
∼
= hKLV
i
∼
, j
∼
;i′
∼
,j′
∼
;i′′
∼
,j′′
∼
· P V
i′′
∼
j′′
∼
, (59)
where for instance
(K, L) : 1, 1, h111
ij,i′j′,i′′j′′ = δji′δi′′iδj′j′′ (60)
holds. The basic idea is now similar to that of section 4. We assume that the averaged
equations
d
dt
〈
P K
i
∼
, j
∼
〉
=
〈
∑
k
∼
,l,
∼
L
MKL
i
∼
j
∼
,k
∼
l
∼
P L
k
∼
l
∼
〉
, (61)
that may be either based on Hamiltonians and on phenomenologically added incoher-
ent terms or containing only incoherent terms. We want to convert these equations
into quantum mechanically consistent equations by adding fluctuating forces
d
dt
P K
i
∼
, j
∼
=
∑
k
∼
,l,
∼
L
MKL
i
∼
j
∼
,k
∼
l
∼
P L
k
∼
l
∼
+ ΓL
i
∼
j
∼
. (62)
We lump the projection operators together to a state vector
A =
P (1)
P (2)
P (0)
(63)
that has to obey equations that we write in the form
dA
dt
= MA + Γ. (64)
534
Fluctuations in quantum devices
The formal solution of (64) reads
A =
∫ t
G(t, τ)Γ(τ)dτ + Ah , (65)
where G is the Green’s function with the property
G(t, t) = E (66)
and Ah a solution to the homogeneous equation (64). We consider
〈
ÃBA
〉
=
〈(∫ t
Γ̃(τ)G̃(t, τ)dτ + Ãh
)
B
(∫ t
G(t, τ ′)Γ(τ ′)dτ ′ + Ah
)〉
, (67)
where the tilde refers to the transposed matrix or transposed vector
Ã, B̃, etc. (68)
We have introduced a matrix B in the form
B =
B11 B12 B10
B21 B22 B20
B01 B02 B00
, (69)
where each submatrix is further labelled by means of indices,
I = i
∼
, j
∼
, (70)
where eventually we will choose only one non-vanishing element. Because of (59) we
obtain for the l. h. side of (64)
〈
P K
I BKL
IJ P L
J
〉
= BKL
IJ
〈
P K
I P L
J
〉
= BKL
IJ hKLV
I,I′,I′′
〈
P V
I′′
〉
. (71)
Taking the derivative with respect of time, we then obtain for the l. h. side of (46)
the relation
l.h.s
d
dt
: BKL
IJ hKLV
I,I′,I′′
∑
J,W
〈
MV W
I′′J P W
J
〉
. (72)
After differentiation the r. h. side (67) contains the terms
r.h.s
d
dt
:
〈
Γ̃(t)B
(∫ t
0
G(t, τ)Γ(τ)dτ + Ah
)〉
, (73)
〈(∫ t
0
Γ̃(τ)G̃(t, τ)dτ + Ãh
)
BΓ(τ)
〉
, (74)
〈
ÃM̃BA
〉
, (75)
〈
ÃB̃MA
〉
. (76)
535
H.Haken
A simple analysis and using a single element of B transforms (73) into
BKL
IJ
1
2
GKL
IJ (77)
and the same expression results from (74). The expression (76) can easily be trans-
formed into
BKL
IJ
〈
P K
I
∑
NU
MLU
JN P U
N
〉
= BKL
IJ
∑
NU
hKUV
INJ ′
〈
MLU
JN P V
J ′
〉
. (78)
Similarly the expression 〈
(MA)T
BKL
IJ P L
J
〉
, (79)
that stems from (75) can be transformed into
BKL
IJ
∑
I′,V ′
〈
MKV ′
IJ ′ P W
J ′′
〉
hV ′LW
J ′JJ ′′ . (80)
Collecting all expressions and choosing only one matrix element B with its specific
indices, we obtain our final result
GKL
IJ = hKLV
I,I′,I′′′
∑
JW
〈
MV W
I′′J P W
J
〉
−
∑
NU
hKUV
INJ ′
〈
MLU
JN P V
J ′
〉
−
∑
J ′,V ′
hV ′LW
J ′JJ ′′
〈
MKV ′
IJ ′ P W
J ′′
〉
.
(81)
This is the desired extension of the theorem of section 4 to a composite quantum
system.
4. Conclusion and outlook
Some general remarks about the applicability of our above formalism may be in
order. The projection operators correspond, at least in general, to physical observ-
ables, such as occupation numbers, (complex) amplitudes etc. Using them, we may
calculate correlation functions of the form
〈
P V
I,J(t)P V ′
I′,J ′(t′)
〉
(82)
or 〈(
P V
I,J(t) −
〈
P V
I,J(t)
〉)(
P V ′
I′,J ′(t′) −
〈
P V ′
I′,J ′(t′)
〉)〉
. (83)
In particular, the latter form (83) enables us to determine the contribution of the
fluctuations. We may thus determine the error made by a quantum device, e.g.
by a logical gate. The formalism is rather general in that it does not only apply
to elementary excitations, but also to general collective states provided that they
can be characterized by quantum numbers and that their generalized Heisenberg
equations are known.
A final remark should be made. As our above formalism reveals, Hamiltonian
quantum systems are noise free. Only when they are coupled to reservoirs that cause
incoherent processes, fluctuations become manifest.
536
Fluctuations in quantum devices
References
1. Klimontovich Yu. L. Statistical Theory of Open Systems. Kluwer Academic Publishers,
1995
2. Haken H. Synergetic Computers and Cognition, 2nd ed. Berlin, Springer, 2004, (to
appear).
3. Haken H., Weidlich W. // Z. Physik, 1966, vol. 189, p. 1.
4. Haken H. Laser Theory. Encyclopedia of Physics XXV/2c. Berlin, Springer, 1970.
Флуктуації в квантових пристроях
Г.Хакен
Інститут теоретичної фізики,
Центр синергетики університету Штутгарта,
Пфаффенвальдрінг, 57/4, D-70550, Штутгарт, Німеччина
Отримано 16 березня 2004 р.
Логічні фільтри можна формалізувати за допомогою булевої алгеб-
ри, елементарні операції якої можна реалізувати в квантових при-
строях, де використовується взаємодія макроскопічної кількості еле-
ментарних збуджень (електрони, дірки, фотони і т.д.). З ростом мі-
ніатюризації системи до нанорозмірів, стають важливими квантові
флуктуації, і ними вже не можна нехтувати. На основі рівняння Гей-
зенберга для операторів народження і знищення елементарних збу-
джень визначено характеристики джерел шуму в складних кванто-
вих пристроях.
Ключові слова: квантові флуктуації, елементарні збудження,
рівняння руху Гейзенберга, логічні пристрої, шуми в складних
квантових системах
PACS: 05.40.-a, 05.30.-d
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