Basic Concepts of Quantum Systems versus Classical Networks Pier Paolo Civalleri,1, ∗ † Marco Gilli and Michele Bonnin 1
Politecnico di Torino, Torino, Italy
SUMMARY The paper is intended to establish a conceptual bridge between Classical Network Theory and Quantum Mechanics. The concept of time–reversal symmetry is extended from Quantum Mechanics to Network Theory (in the state equations description), while concepts of passivity and losslessness are extended from Network Theory to Quantum Mechanics.
: nanocircuits; time reflection symmetry; passivity; losslessness; reflectance
1. INTRODUCTION There are several ways of correlating quantum systems and electrical networks. The most immediate one consists in looking at formal analogies based on the fact that state equations might have the same structure in both cases, although their physical interpretation should be entirely different. This way was followed in [1] for pure states described by a wave function and in [2] for mixtures (and obviously pure states) described by a density matrix in B representation. Alternatively one could think of describing the quantum system interacting with an external classical electromagnetic field by a set of relations involving only the latter — a network–like description at the ports which mediate the interaction between the quantum system and the external classical world. Finally one can try to extend some abstract properties which are proper to one field to the other and viceversa. This is the point of view adopted in this paper, where we show on the one hand how time reversal symmetry in Quantum Mechanics can be extended to Classical Systems in state–space representation, on the other how such concepts as passivity and losslessness can be transferred from Network Theory to Quantum Mechanics. Independently on the point of view one can assume, it must be recognized that a mapping of a quantum system on a classical network preserving the subsystems interconnections is plainly impossible, even in an approximate way, since the former must be described in a state space which is the tensor product of the state spaces of its components, while for the latter only their direct sum is needed. ∗
Correspondence to: Prof. Pier Paolo Civalleri, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy, Tel. +390115644066, Fax +390115644099 † e-mail:
[email protected]
2
,
Thus we are compelled to consider systems as a whole on both the quantum and the classical side, possibly open to the external world through some properly defined channels, that is n–ports. The n–port concept is probably the basic paradigm that can launch a bridge between quantum and classical systems. As we are concerned port quantities, they are amplitudes of matter or electromagnetic waves; in the last case, in which the field is usually treated classically, they interact with the quantum system by influencing some elements of the Hamiltonian matrix, i.e. some n–port parameters. Thus the n–port is nonlinear and, if the electromagnetic field is not constant, time–variant too.
2. ANTILINEAR AND ANTIUNITARY OPERATORS Time reversal symmetry in Quantum Mechanics is described by antiunitary operators. Since the formalism of the latter, a special case of antilinear operators, is not usually treated in elementary textbooks, we present in this section a short resume of their properties and of the practical rules to use them. The presentation is essentially based on [3], p. 632–681. An operator is said to be antilinear when it acts on kets according to the rule: ˆ 1 |1i + λ2 |2i) = λ∗1 A|1i ˆ + λ∗2 A|2i. ˆ A(λ
(1)
The algebraic operations involving Aˆ are defined consistently as follows: (i) Multiplication by a constant c: cAˆ = Aˆ c∗
(2)
(ii) Sum of two antilinear operators: same as for the linear ones. (iii) Products: (Aˆ 1 Aˆ 2 )|ui = Aˆ 1 (Aˆ 2 |ui)
(3)
The product of linear and antilinear operators is linear or antilinear according to the number of antilinear factors being even or odd. (iv) Inverse: same as for the linear ones. The action on bras is defined along the same lines as for linear operators. Consider the ˆ expression hv|(A|ui), where the parentheses indicate that operator Aˆ acts on ket |ui. Its complex conjugate is a linear form on |ui ∗ ˆ [hv|(A|ui)] = h f |ui
(4)
By definition bra h f | depends on hv| according to the rule ˆ h f | = hv|A.
(5)
∗ ˆ ˆ [hv|(A|ui)] = (hv|A)|ui
(6)
Thus we have the fundamental relation
3
from which it follows, for any complex constant λ, ∗ ˆ ˆ [hv|λ(A|ui)] = (hv|λA)|ui,
∗ ˆ ˆ [hv|λ(A|ui)] = λ∗ (hv|A)|ui
(7)
and therefore, since |ui is arbitrary, ˆ + λ∗2 (h2|A). ˆ (λ1 h1| + λ2 h2|)Aˆ = λ∗1 (h1|A)
(8)
Thus the action on bras is antilinear too. The adjoint A† of the antilinear operator Aˆ is defined as follows ˆ hv|(A|ui) = [(hu|Aˆ † )|vi]∗ = hu|(Aˆ† |vi).
(9)
An antiunitary operator Tˆ is an antilinear operator for which Tˆ −1 = Tˆ † .
(10)
Antiunitary transformations act on kets, bras, and linear operators according to the rules: |˜ui = Tˆ |ui
h˜v| = hv|Tˆ †
B˜ = Tˆ Bˆ Tˆ† .
(11)
Sesquilinear forms are transformed as follows ˜ ui = hv|Tˆ† (Tˆ Bˆ Tˆ† )Tˆ |ui = (hv|Tˆ† Tˆ Bˆ Tˆ† )Tˆ |ui = (hv| Bˆ Tˆ† )Tˆ |ui h˜v| B|˜ = [hv|( Bˆ Tˆ† Tˆ |ui)]∗
(12) ˆ ∗. = hv| B|ui
Hence for any complex constant c Tˆ cTˆ † = c∗
(13)
[ q, ˆ pˆ ] = j~ −→ [ q, ˜ p˜ ] = −j~.
(14)
and therefore
3. THE CONJUGATION OPERATOR To the coordinate representation {q} is associated a conjugation operator Kˆ q with the following properties: (i) For any |ui and |vi hv|(Kˆ q |ui) = [(hv|Kˆ q )|ui]∗
(15)
(ii) If |qi is a basis vector of {q}, then Kˆ q |qi = |qi
(16)
Eq. (15) shows that Kˆ q is antilinear. From eq. (16) it follows hq|Kˆ q† = hq|
(17)
4
,
ˆ = hq|qi or and therefore hq|Kˆ q† K|qi ˆ Kˆ q† Kˆ q = 1.
(18)
Thus Kˆ q is antiunitary. Again from eq. (16) we obtain ˆ Kˆ q2 = 1.
(19)
Hence Kˆ q is self–adjoint. The complex conjugate of a ket |ui is defined as the ket Kˆ q |ui. It follows that the complex conjugate of a bra hv| is the bra hv|Kˆ q . A representation {r} whose base vectors are such that Kˆ r |ri = |ri is called a real representation; the {q}–representation is real (eq. (16)). The complex conjugate of a linear operator Aˆ is defined as the operator Kˆ q Aˆ Kˆ q† . In particular, if ˆ Kˆ q Aˆ Kˆ q† = A, (20) the operator Aˆ is said to be real; if ˆ Kˆ q Aˆ Kˆ q† = −A,
(21)
the operator Aˆ is said to be imaginary. Complex conjugate operators in the {q} representation are represented by complex conjugate matrices: this is proved as follows. Let Aˆ be a linear operator represented as X Aˆ = ai j |qi ihq j |. (22) ij
If we take its complex conjugate, we obtain X X Kˆ q Aˆ Kˆ q† = a∗i j Kˆ q |qi ihq j |Kˆ q† = a∗i j |qi ihq j |. ij
(23)
ij
The definition above of complex conjugation is not the only possible. Actually any antiunitary operator whose square is 1ˆ could be provide an alternative definition of complex conjugation. According to our definition the position operator qˆ is real, the impulse operator pˆ is imaginary. In fact the eigenvalue equation for the operator qˆ reads q|qi ˆ = q|qi.
(24)
(Kˆ q qˆ Kˆ q† )Kˆ q |qi = qKˆ q |qi
(25)
(Kˆ q qˆ Kˆ q† )|qi = q|qi.
(26)
Taking the conjugate we get
or, according to the eq. (16) Thus from eqs. (24) and (26) we obtain Kˆ q qˆ Kˆ q† = qˆ that proves that the operator qˆ is real.
(27)
5
On the other hand eq. (14) gives Kˆ q pˆ Kˆ q† = − pˆ
(28)
which proves that the operator pˆ is imaginary. The eigenvalue equations for pˆ and its conjugate read p|pi ˆ = p|pi (29) and (Kˆ q pˆ Kˆ q† )Kˆ q |pi = pKˆ q |pi.
(30)
The latter equation can be rewritten as pˆ Kˆ q |pi = −pKˆ q |pi
(31)
Kˆ q |pi = | − pi.
(32)
from which we derive It is concluded that the {p} basis is imaginary with respect to complex conjugation Kˆ q . The reader can repeat for himself the above derivation by mutually exchanging symbols q and p wherever they occur. The antiunitary operator Kˆ p provides an alternative definition of complex conjugation, with respect to which pˆ is real, qˆ is imaginary and the {q} basis is not real. Time reversal symmetry (T–symmetry) is based on complex conjugation associated with the {q} basis. Parity and time reversal symmetry (PT–symmetry) is based on complex conjugation associated with the {p} basis. In the following we shall deal only with Tsymmetry. When the base is changed from {q} to say {ξ}, the conjugate of the image and the image of the conjugate do not coincide, unless the transformation matrix is real. Let (S ) be the transformation matrix from basis {q} to basis {ξ}. Then we have for any vector |ui (u)ξ = (S )(u)q
(33)
(u)∗ξ = (S )∗ (u)∗q
(34)
(v)ξ = (S )(u)∗q .
(35)
and while the image of (u)∗q is
The conjugate of the image, (u)∗ξ and the image of the conjugate, (v)ξ , do not coincide but for the case in which (S ) is real. In such a case it is immediate to verify that basis {ξ} is real.
4. TIME REVERSAL SYMMETRY Time reversal symmetry will be defined only for conservative systems, that is for systems whose Hamiltonian Hˆ does not depend on time. Such systems are not necessarily closed: they can interact with the outside world through physical fields that however must be constant in time. So the symmetry applies also to a restricted class of open systems.
6
, Time reversal symmetry is based on the operator Kˆ = Tˆ Kˆ q
(36)
ˆ In particular where Tˆ is a unitary operator whose choice depends on the Hamiltonian H. the choice will be different according to spin variables being or not present. The operator Kˆ can still be considered as a complex conjugation; however the existence of a real base depends on the choice of Tˆ . For a spinless Hamiltonian we choose Tˆ = 1 or
1. Spinless Hamiltonian.
Kˆ = Kˆ q .
(37)
Let the system, assumed to be in a pure state, be described by the S¨ equation j~
d ˆ |ψ(t)i = H|ψ(t)i. dt
(38)
Time reversal symmetry is obtained by simultaneously taking the complex conjugate of eq. (38) and reflecting time by t → −t. Eq. (38) gives j~
d ˆ ˆ K|ψ(−t)i = Kˆ Hˆ Kˆ † K|ψ(−t)i. dt
(39)
ˆ Thus the state vector K|ψ(−t)i satisfies the same equation as the state vector |ψ(t)i, i.e. the system is time–reversible, if and only if Kˆ Hˆ Kˆ † = Hˆ
(40)
that is Hˆ is real. The same result is obtained if the system is described by the V N equation j~
d ˆ ρ(t)]. ρ(t) ˆ = [H, dt
(41)
Taking the complex conjugate and changing t into −t we obtain j~
d ˆ ˆ K ρ(−t) ˆ Kˆ † = [Kˆ Hˆ Kˆ † , Kρ(−t) Kˆ † ]. dt
(42)
ˆ that is, Hˆ is Thus Kˆ ρ(−t) ˆ Kˆ † satisfies the same equation as ρ(t) ˆ if and only if Kˆ Hˆ Kˆ † = H, real. 2. General case. Time reversal inverts the spins. On the other hand transformation by Kˆ = Kˆ q inverts sˆy while leaves sˆx and sˆz unchanged. If however, after Kˆ q conjugation, one rotates by π the spins around the y-axis, sˆx and sˆz change their sign without affecting that of sˆy . Thus Kˆ = e−jπsy /~ Kˆ q (43) has the required property Kˆ sˆKˆ † = − sˆ.
(44)
7
For the particular case of spin 1/2, eq. (43) becomes Kˆ = e−jπσˆ y /2 Kˆ q = −jσ ˆ y Kˆ q .
(45)
As an example, consider a particle of spin 1/2 in a magnetic field B. Its Hamiltonian reads1 −Bz Bx + jBy Hˆ = −µσˆ · B = −µ (46) Bx − jBy Bz
Under transformation (43) Hˆ changes its sign and therefore the system is not time reversible ([3], p. 675.) Obviously one could reestablish reversibility by changing B into −B; but since the magnetic field enters the definition of the system as a parameter, this would substantially create a different system. It is therefore better to consider two equal systems S1 and S2 merged in opposite magnetic fields. Their Hamiltonians are obviously opposite and turn the one into the other under time reversal. Such systems we will call time interreversible. We conclude this paragraph by remarking that Kˆ as defined in eq. (45) satisfies the equation ˆ Kˆ 2 = −1. (47) Thus there is no real basis associated with it as a complex conjugation operator. General Two-State Systems. A general two–level system is defined as one characterized by two states, that can generically be called up and down, and denoted as |2i and |1i. Such states could refer to two different configurations of a molecule, even not related to a coordinate system. In this case we define a complex conjugation operator Kˆ 0 with the property that Kˆ 0 |ii = |ii i = 1, 2. (48) Thus we can apply to such systems all the considerations developed in the first paragraph of this section.
5. TIME REVERSAL SYMMETRY IN BLOCH VARIABLES We shall limit our treatment to the case of two–level systems. Hermitian operators acting on vectors of two–dimensional H space H form in turn a linear vector space on the real field, since they form a class closed with respect to the sum of operators and the multiplication of an operator times a real constant. Such a vector space can be endowed with a inner product structure according to the definition ˆ Bi ˆ = Tr {Aˆ B} ˆ hA| 1
We define σ ˆx =
Ã
0 1 1 0
!
σ ˆy =
Ã
0 −j
j 0
!
(49) σ ˆz =
Ã
−1 0 0 1
!
.
Such definition assumes that the states are ordered in the sense in which energy increases. They differ from the usual conventions, but are convenient for our applications. See [4], p. 36.
8
,
and so becomes an H space that will be denoted as H(H, H). In such a space we introduce the hermitian canonic base represented by the linearly independent operators ˆ σ 1, ˆ x, σ ˆ y , and σ ˆ z . Any operator Aˆ ∈ H(H, H) can be represented as 1 1 1 1 Aˆ = A0 1ˆ + A1 σ ˆ x + A2 σ ˆ y + A3 σ ˆz 2 2 2 2
(50)
where A0 and the Ai ’s are real. Such components are called B variables. Thus the Hamiltonian and the density operators can be represented respectively as 1 1 1 1 Hˆ = H0 1ˆ + H1 σ ˆ x + H2 σ ˆ y + H3 σ ˆz 2 2 2 2
(51)
and
1 1 1 1 ρ(t) ˆ = 1ˆ + λ1 (t)σ ˆ x + λ2 (t)σ ˆ y + λ3 (t)σ ˆ z. 2 2 2 2 The V N equation (41) reads d~λ(t) = Ω~λ(t) dt
(52)
(53)
where ~λ = (λ1 , λ2 , λ3 )′ and 0 −ω0 ω2 0 −ω1 . Ω = ω0 −ω2 ω1 0
(54)
The elements of matrix Ω are related to the B components of the Hamiltonian by ω1 =
H1 ~
ω2 =
H2 ~
ω0 =
H3 . ~
(55)
All the above material is amply presented in [4], p. 176–180, from which we have also borrowed the notation. We shall discuss time reversal symmetry firstly for a spinless Hamiltonian.2 Taking the complex conjugate of eq. (52) and reversing time we find Kˆ 0 ρ(−t) ˆ Kˆ 0† = λ1 (−t)Kˆ 0 σ ˆ x Kˆ 0† + λ2 (−t)Kˆ 0 σ ˆ y Kˆ 0† + λ3 (−t)Kˆ 0 σ ˆ z Kˆ 0† .
(56)
The equation above can be rewritten as Kˆ 0 ρ(−t) ˆ Kˆ 0† = λ1 (−t)σ ˆ x + [−λ2 (−t)]σ ˆ y + λ3 (−t)σ ˆ z.
(57)
Thus the density operator B components undergo the transformation λ1 (t) → λ1 (−t) 2
λ2 (t) → −λ2 (−t)
λ3 (t) → λ3 (−t).
(58)
It is evident that σ ˆ x, σ ˆ y , and σ ˆ z have here only the meaning of base operators, not of physical spins.
The dynamical equation (53) can be explicitly written as λ1 (t) 0 −ω0 ω2 λ1 (t) d 0 −ω1 λ2 (t) λ2 (t) = ω0 dt λ3 (t) −ω2 ω1 0 λ3 (t)
and under time reversal becomes λ1 (−t) 0 −ω0 −ω2 λ1 (−t) d λ2 (−t) = ω0 0 −ω1 λ2 (−t) dt λ3 (−t) ω2 ω1 0 λ3 (−t)
.
9
(59)
(60)
The form of the equation does not change if and only if ω2 = 0, i.e., taking into account eq. (55), H2 = 0. But eq. (51) shows that this is just the condition that Hˆ be real. Thus eq. (59) assumes the form λ1 (t) 0 −ω0 0 λ1 (t) d (61) λ2 (t) = ω0 0 −ω1 λ2 (t) . dt λ3 (t) 0 ω1 0 λ3 (t)
The case of a particle of spin 1/2 in a magnetic field can be treated as an interreversibility problem. Systems S1 and S2 are characterized by matrix Ω of eq. (54) and its negative (that coincides with its transpose), so that the equations change the one into the other as time is reversed.
6. TIME REVERSAL IN CLASSICAL NETWORKS We assume an autonomous network N composed of a finite number of ideal elements. It can be analyzed by writing down all element equations and the constraints in the form of a set of independent equations expressing Kirchhoff’s voltage and current laws (tableau equations). Since time reversal leaves unchanged the signs of voltages while reverses those of currents, we can easily classify the elements as time reversible or irreversible. 1. Capacitors, inductors, ideal transformers, current controlled current sources, and voltage controlled voltage sources are time reversible. 2. Resistors, gyrators, voltage controlled current sources, and current controlled voltage sources are time irreversible. However pairs of elements of the same kind with opposite parameter signs are interreversible. The Kirchhoff’s laws, on their side, are invariant to the change of t into −t. It follows that a pair of networks, one of which is obtained from the other by leaving unchanged the parameters of the elements of type 1 and changing the signs to those of the elements of type 2, are time interreversible. In particular, networks containing only elements of type 1 are time reversible, networks containing only elements of type 2 are time interreversible with their opposites.3 3
We denote as opposite of a network that obtained by changing the signs of all of its elements.
10
, The energy stored in the network has the form w=
1X 1X 2 Ch u2h + Lh ih 2 2
(62)
where the sums are extended to all capacitive and to all inductive branches respectively. If there are loops of capacitors or cut-sets of inductors, some voltages and some currents will be expressed as linear combinations of a set of independent voltages and of a set of independent currents respectively. This will cause the energy to be expressible as the sum of two nonnegative definite quadratic forms in the independent voltages and the independent currents respectively. Each of these quadratic forms can be brought to a sum of squares by a suitable congruence, so that the energy assumes the form 1 w(t) = [x1′ (t)x1 (t) + x2′ (t)x2 (t)]. 2
(63)
Observe that subvector x1 (t) only depends on the capacitor voltages, while subvector x2 (t) only depends on the inductor currents. Thus under time reversal the two subvectors transform as follows x1 (t) → x1 (−t) x2 (t) → −x2 (−t). (64) Mixing constraints may occur in presence of gyrators, current controlled voltage sources and voltage controlled current sources. In such cases usually a state variable simply changes its status leaving one of the subvectors to enter the other. For example, if we have a capacitor connected at the output of a current controlled voltage source of transresistance rm , its energy becomes 1 1 w = Cv22 = Crm2 i21 2 2
(65)
so that the corresponding term migrates from x1 to x2 (possibly adding to a preexisting term, if current i1 flows through an inductor.) In conclusion we can regard the network as a nondynamic n–port, whose nC ports are terminated on unit capacitors and nL = n − nC are terminated on unit capacitors. Thus the network equations can be written as dx = Ax(t) dt
(66)
or in partitioned form d dt
x1 (t) x2 (t)
A11 A12 = A21 A22
x1 (t) x2 (t)
.
(67)
The state matrix A is the negative of the hybrid matrix of the nondynamic n–port having as independent variables the independent voltages and the independent currents. By applying to eq. (67)the transformation of eq. (64) we find A12 x1 (−t) d x1 (−t) −A11 . (68) = dt x (−t) A x2 (−t) 2 21 −A22
11
A network is time–reversible if and only if the reversed time state satisfies the same equation as the original one, i.e. if and only if A11 = −A11 = 0 and A22 = −A22 = 0, and the state matrix has the form 0 A12 . (69) A = A21 0 Two networks, denoted by indices α and β, are interreversible if and only if Aα,11 Aα,12 Aα = Aα,21 Aα,22
−Aβ,11 Aβ,12 = A = β Aβ,21 −Aβ,22
.
(70)
Conservative Networks. The conservation of the stored energy implies that dw dx′ (t) dx(t) = x(t) + x′ (t) = x′ (t)(A′ + A)x(t) = 0 dt dt dt
(71)
A′ + A = 0
(72)
and therefore or A11 = −A′11
A12 = −A′21
A21 = −A′12
A22 = −A′22 .
(73)
Conservative networks may be reciprocal or not. Conditions for reciprocity are ([5], p. 73) A11 = A′11
A12 = −A′21
A21 = −A′12
A22 = A′22 .
(74)
Thus a lossless reciprocal network is characterized by a state matrix A =
0 −A′12
A12 0
(75)
and therefore is time reversible. If the requirement of reciprocity is released (that is, if gyrators are admitted in addition to capacitors and inductors), A11 and A22 need not to be zero. In this case we consider the transpose network (obtained by reversing all gyrator polarities) described by the state matrix ′ A11 −A′21 . AT = (76) ′ ′ −A12 A22
Since AT is conservative such being A, it can be written as −A11 A12 AT = A21 −A22
.
Hence, due to eq. (70), two transpose conservative networks are time interreversible.
(77)
12
,
Nonconservative Networks. The time reversibility conditions A11 = 0 and A22 = 0 can be satisfied with A12 = A′21 . Such a network is nonconservative and nonreciprocal. It is immediately seen that all the eigenvalues appear in opposite pairs, as λ–roots of the eigenvalue equation det(λ2 1 − A12 A′12 ) = 0. (78) Consider, as a simple example, a circuit composed of an inductor of inductance L and a capacitor of capacitance C connected at the ports of a negative impedance converter. Assuming that at both ports of the latter the current flows out from the positive terminals, we get the equations di(t) L = u(t) dt (79) du(t) = i(t) C dt According to our previous definition the system is time reversible, since mappings t → −t, u(t) → u(−t), and i(t) → −i(−t) transform the previous equations into themselves. Here the presence of the negative impedance converter guarantees that the network is nonreciprocal and active (thus nonconservative.) Although the stored energy is not conserved, the Lagrangian function 1 L = [x2′ (t)x2 (t) − x1′ (t)x1 (t)] (80) 2 (i.e. the difference between the magnetic and the electric energy) does not change with time. Note that for a network including negative impedance converters the definition of stored energy is somewhat ambiguous; if for example the negative impedance converter loaded by the capacitor of capacitance C is simply replaced by a negative capacitor of capacitance −C, the stored energy would precisely coincide with the previously defined Lagrangian function and thus the network would become conservative! In the literature (see e.g. [6], p. 3,11, [7], p. 79–81) a conservative system is usually defined as one on which only forces deriving from a potential act, which is equivalent to state that the total energy remains constant in time. The example above suggests that such a statement should be supplemented with that the system be isolated. As a conclusion, time reversibility in classical dynamics is by no means limited to conservative systems but can occasionally extend to active ones. This opens interesting questions about open quantum systems, inasmuch as they be in contact with energy sources.
7. TWO–LEVEL OPEN QUANTUM SYSTEMS We discuss the concepts of passivity and losslessness for the simple case of a two–level quantum system. The extension to the case in which the states are n does not require new concepts, but only an heavier formalism. The case in which the states are infinite in number is not treated here. We consider a two–level quantum system interacting with a thermal bath under the assumptions and the approximations illustrated in [4], p. 227–268.
13
Its Hamiltonian operator can be represented in the energy representation by the matrix E1 V . (81) (H) = ∗ V E2
where V is a small perturbing term describing a reversible interaction with the outside world. Then its eigenvalues, if second order terms are neglected, are still E1 and E2 . Its state can be represented accordingly by the density matrix ρ11 ρ12 . (82) (ρ) = ∗ ρ12 ρ22 The interaction with the bath can be described in terms of either the transition frequencies between the states W12 (2 → 1) and W21 (1 → 2) and the adiabatic transition ad within the states,4 or the longitudinal and the transverse relaxation times frequency W12 T 1 and T 2 . Such quantities are related by the equations ([4], p. 257–258) 1 = W12 + W21 , T1
1 1 ad = (W12 + W21 ) + W12 T2 2
(83)
so that the relaxation times satisfy the equality ([4], p. 258) 2 1 1 = + ∗. T2 T1 τ
(84)
where
1 ad = 2W12 . τ∗ Thus the V N equations of the open system read ([4], p. 256.) ¢ j¡ dρ11 = − Vρ∗12 − V ∗ ρ12 − W21 ρ11 + W12 ρ22 dt ~ ¢ j¡ dρ22 = − V ∗ ρ12 − Vρ∗12 + W21 ρ11 − W12 ρ22 dt ~ " # dρ12 j 1 ad = [(E2 − E1 )ρ12 + V(ρ11 − ρ22 )] − (W12 + W21 ) + W12 + j∆ω12 ρ12 . dt ~ 2
(85)
(86)
In absence of external fields (V = 0) the system reaches the equilibrium state ρ11 =
W12 , W12 + W21
ρ22 =
W21 , W12 + W21
ρ12 = 0
(87)
corresponding to the thermal equilibrium of populations and to a complete decoherence. Alternatively the system can be described in the B space of operators. Then the Hamiltonian operator is represented as 1 1 1 1 Hˆ = H0 1ˆ + H1 σ ˆ 1 + H2 σ ˆ 2 + H3 σ ˆ3 2 2 2 2 4
(88)
Such a frequency is related to the elastic collisions between particles, that do not change the energy but destroy the coherence. Obviously it is symmetric in the indices 1 and 2.
14
,
where H0 = E 1 + E 2 ,
H1 = Re V,
H2 = Im V,
H3 = −(E2 − E1 ) + ~∆ω12
(89)
and the density operator as 1 1 1 1 ρˆ = 1ˆ + λ1 σ ˆ 1 + λ2 σ ˆ 2 + λ3 σ ˆ3 2 2 2 2 or in matrix form
1 1 − λ3 λ1 + jλ2 (ρ) = 2 λ − jλ 1 + λ3 1 2
.
(90)
(91)
Using B variables eqs. (86) become Im V E2 − E1 1 − + ∆ω12 − λ T2 ~ ~ 1 1 Re V d E2 − E1 − ∆ω12 − − λ2 = ~ T2 ~ dt Re V 1 Im V λ3 − − ~ ~ T1
0 λ1 0 λ2 + . 1 W12 − W21 λ3 − T 1 W12 + W21 (92) Again, in absence of external fields, the system reaches the equilibrium state λ1 = 0,
λ2 = 0,
λ3 = −
W12 − W21 , W12 + W21
(93)
which is easily seen to coincide with that described by eqs. (87). The interaction energy assumes a particularly simple form if we assume that the system possesses some kind of electric dipole moment µˆ represented in the energy basis as ([4], p. 267) 0 µ (µ) = (94) µ 0 on which acts a weak parallel electric field E. Then we have V = −µE
(95)
and eqs. (86) and (92) simplify accordingly. Population reversal cannot occur if V is constant, but becomes possible if it is time– dependent. In particular, under the assumption of eq. (95), it can be obtained by parametric pulse excitation, i.e. by choosing E(t) =
~π δ(t) µ
(96)
(see [4], p. 262.) The approximations involved in eqs. (86) and (92) should be carefully considered. Firstly the diagonal form of the relaxation matrix5 implies that the Hamiltonian be diagonal (see the derivation in [4], § 3.3.) The transformation from the diagonal form of the 5
The relaxation matrix is defined as diag{1/T 2 , 1/T 2 , 1/T 1 }.
15
Hamiltonian to that of eq. (81) is represented in the B space by a rotation through the Euler angles α and β (see [3], p. 524, for their definition), which generates nondiagonal elements in the relaxation matrix. If and only if the coupling between the states induced by the reversible interaction with the outside world, represented by the interaction energy V, is small with respect to the energy gap E2 − E1 , the rotations are small of the same order, and the nondiagonal elements of the relaxation matrix become small of the second order and hence negligible.
8. PASSIVITY AND LOSSLESSNESS Passivity and losslessness are concepts proper of Classical Network Theory. We shall show how they can be extended to Quantum Mechanics preserving their physical content, although not necessarily the details of formalism. It is convenient to introduce at the outset some terminology. An abstract n–port is one described by a set of n independent equations between the 2n port quantities (voltages and currents or incident and reflected waves.) A concrete n–port is a specific structure realizing a given set of port equations. Thus an abstract n–port is the class of equivalence of all concrete n–ports having the same port equations ([5], p. 3–4). An abstract n–port is said to be passive if the energy flowed into its ports from time τ = −∞ to any instant t, for all admissible excitations, is nonnegative [8] : Z t p(τ) dτ ≥ 0 ∀t (97) −∞
where p(τ) is the instantaneous power absorbed at time τ. An abstract n–port is said to be lossless if it is passive and moreover all the energy flowed into its ports over the time interval −∞ < τ < +∞ is eventually flowed out again: Z +∞ p(τ) dτ = 0. (98) −∞
Such definitions are essential part of an axiomatic approach to Network Theory. A concrete n–port is said to be passive if the energy flowed into its ports in the time interval (−∞, t) equals the energy irreversibly transformed into heat in such a time interval plus the energy stored at time t, or, equivalently, if the instantaneous power absorbed at any instant equals the sum of the dissipated power and the rate of increase of stored energy: Z Z t
t
pd (τ) dτ + w(t)
p(τ) dτ =
(99)
−∞
−∞
or
dw (100) dt where pd is the dissipated power and w the stored energy. Since the dissipated power and the stored energy (electric and magnetic) are both nonnegative, a concrete passive n–port is abstractly passive.6 The existence of integrals in eq. (99) implies that w(−∞) = 0, p = pd +
6
The reverse implication is clearly not true.
16
,
so that integrating eq. (100) one precisely recovers eq. (99). Moreover it must be noted that the assumption pd ≥ 0 implies that the n–port be at an higher temperature than the environment, which is the usual situation of electrical devices; this has some relevance when the concept of passivity is extended to quantum open systems. A concrete n–port is said to be lossless if eqs. (99) and (100) are satisfied with pd = 0, ∀τ, t. In other words a lossless n–port is one which is devoid of dissipation and delivers no power to the environment. We shall extend these concepts to Quantum Mechanics in the simple case of a two– level system. The generalization to a n-level systems poses no new conceptual problems, but simply requires a more elaborate formalism. Instead of looking for a detailed description of the parametric interaction between the system and the outside world, we shall simply identify the quantum correspondent of eq. (100). To this aim, we firstly calculate the average stored energy from eqs. (81) and (82): hEi = Tr{Hˆ ρ} ˆ = E1 ρ11 + E2 ρ22 + Vρ∗12 + V ∗ ρ12
(101)
and hence the energy rate of change from eqs. (86) d dρ11 dρ22 d hEi = E1 + E2 + (Vρ∗12 + V ∗ ρ12 ) dt dt dt dt d j (E2 − E1 )(Vρ∗12 − V ∗ ρ12 ) + (Vρ∗12 + V ∗ ρ12 ) + (E2 − E1 )(W21 ρ11 − W12 ρ22 ). = ~ dt (102) By reordering the terms of the above equation, and multiplying them times the number of particles in the system, N, we get j d [N(Vρ∗12 + V ∗ ρ12 )] + N(E2 − E1 )(Vρ∗12 − V ∗ ρ12 ) dt ~ d = N(E2 − E1 )(W12 ρ22 − W21 ρ11 ) + hNEi. dt
(103)
The first term on the right–hand side can be understood as follows. The fraction of particles in the excited state, Nρ22 , times the probability of transition to the ground state per unit time due to the interaction with the bath, W12 , gives the average number of particles that decay per unit time and the average number of photons released to the bath. This quantity multiplied by E2 − E1 gives the energy delivered to the bath per unit time, i.e., the dissipated power. The second term on the right–hand side is the rate of increase of the stored energy. Thus the term in the left–hand side must represent the input power; this interpretation is corroborated by the fact that the term vanishes with the interaction energy V. Thus eq. (103) represents eq. (100) in the case of a two–level quantum system. There is an important point to observe. In Network Theory it is assumed pd ≥ 0 for all admissible inputs and for all t, while in Quantum Mechanics the sign of pd depends on the instantaneous values of the populations. In other words, in the classical framework the thermal bath can only absorb energy while in the quantum context it can occasionally release some amount of it to the system. This is due to the fact that in the first case the temperature of the system is higher than that of the environment, while in the second is equal, since the system is in thermal equilibrium with the bath. The symmetry between the two cases is restored if the bath temperature is assumed to be zero: in such a case, in
17
fact, we have W21 = 0 and therefore pd ≥ 0. However, in equilibrium ρ22 is zero too, and to have transitions from state 2 to state 1, the former must be continuously fed with new particles, that can occupy the higher energy level only if their temperature is greater than zero. The assumption of zero temperature is physically reasonable as an approximation since at room temperature we have kT ≈ 10−20 J. Also the concept of losslessness requires a deeper insight. Suppose that the contact between the system and the bath is interrupted and moreover no elastic collisions take place within the system: then W12 = W21 = Wad = 0, ∆ω12 = 0 and eq. (92) reduces to E2 − E1 Im V 0 − λ 1 ~ ~ d E − E1 Re V λ2 = 2 0 − dt ~ ~ Re V λ3 Im V 0 − ~ ~
λ1 λ2 . λ3
(104)
In such a case the system is lossless in the usual sense and its time evolution, when the external excitation is turned off, is unitary. But consider now the case in which 1/T 1 = 0 and 2/T 2 = 1/τ∗ (see eq. (84)). Eq. (86) then becomes E2 − E1 Im V ad −W12 − + ∆ω12 λ1 ~ ~ d E2 − E1 Re V ad λ2 = − ∆ω12 −W12 − dt ~ ~ Im V Re V λ3 0 − ~ ~
λ1 λ2 . λ3
(105)
Then with no external excitation λ1 and λ2 decay exponentially while λ3 remains constant. Comparing with eq. (91), it is seen that the populations of the two energy levels do not change, while their coherence decreases to zero. The system is still lossless, but, due to elastic collisions, undergoes a progressive decoherence. This fact seems not to have a counterpart in classical networks, probably due to the absence of the concept of coherence in classical context. The two cases of losslessness give rise to unitary and non–unitary free evolutions. Finally, we would like to point out that conservative systems are a particular case of lossless ones, corresponding to the condition V = const.
9. THE TWO–LEVEL QUANTUM SYSTEM AS A LINEAR ONE–PORT Under suitable approximations the two–level quantum system can be considered as a linear one–port and its reflectance calculated. We suppose that the quantum system is driven by an electromagnetic wave impinging upon it, that the effect of the magnetic field B is negligible, and moreover that the wavelengths of the harmonic components of such a wave are all large compared with the dimensions of the system. Under such assumptions the interaction energy can be represented as in eq. (95) ([9], p. 176). Moreover, for the sake of simplicity, we assume that the bath is at zero temperature, so that W21 = 0.
18
, Thus, according to eq. (102), the dissipated power has the expression pd = N(E2 − E1 )W12 ρ22
(106)
The density ρ22 at the thermodynamic equilibrium is zero. Hence dissipation is entirely due to the nonzero density owed to the action of the electric field. Since the problem is inherently nonlinear and time–variant, some approximation must be introduced at the outset to allow its replacement by a linear time–invariant one. We assume that the electric field, E, is harmonic of angular frequency ω 1 E = EM cos ωt = EM (ejωt + e−jωt ). 2
(107)
Moreover we assume that EM is small, so that, when the field is applied to the system from time zero to some T (that will be chosen large enough), the density of the ground state is not appreciably reduced: ρ11 (T ) ≈ 1 and hence ρ22 ≪ 1. Let ρii , i = 1, 2, be represented as a superposition of pure state densities X X ρii = ciν c∗iν = |ciν |2 . (108) ν
ν
Each amplitude ciν , neglecting the coupling with the bath, evolves in time according to the S¨ equation. In the interaction picture we have −jω0 t c 0 Ve c 1ν 1ν d . (109) j~ = dt c c2ν V ∗ ejω0 t 0 2ν
We integrate the second of eqs. (109), taking into account eq. (107), over the time interval (0, T ). The result of the integration is " # µEM T sin[(ω − ω0 )T/2] −j(ω−ω0 )T/2 sin[(ω + ω0 )T/2] j(ω+ω0 )T/2 c2ν (T ) = j e + e c1ν (0). 2~ (ω − ω0 )T/2 (ω + ω0 )T/2 (110) Multiplying eq. (110) by its complex conjugate, summing over ν, and taking into account that ρ11 (0) ≈ 1, we get " 1 µ µT ¶2 sin2 [(ω − ω0 )T/2] sin2 [(ω + ω0 )T/2] + ρ22 (T ) = 2 ~ [(ω − ω0 )T/2]2 [(ω + ω0 )T/2]2 (111) # 2 EM sin[(ω − ω0 )T/2] sin[(ω + ω0 )T/2] +2 · cos ωT (ω − ω0 )T/2 (ω + ω0 )T/2 2 (see [10], p. 9–13). The action of the bath is now taken into consideration by putting together eqs. (106) and (111) to yield µ µT ¶2 1 " sin2 [(ω − ω )T/2] sin2 [(ω + ω )T/2] 0 0 + pd (T ) = N(E2 − E1 )W12 ~ 2ǫ0 cσ [(ω − ω0 )T/2]2 [(ω + ω0 )T/2]2 # 1 sin[(ω − ω0 )T/2] sin[(ω + ω0 )T/2] · cos ωT ǫ0 cE2M σ +2 (ω − ω0 )T/2 (ω + ω0 )T/2 2 (112)
19
where ǫ0 is the electric permeability of the vacuum, c the velocity of light, and σ is the surface illuminated by the field. Now 1 pi = ǫ0 c E2M σ 2
(113)
is just the incident power of the electromagnetic wave. Let now the one–port be represented in D form. Let a1 (t) and b2 (t) be the normalized incident wave at port 1 and the normalized reflected wave at port 2. With the harmonic excitation a1 (t) = A1 ejωt (114) we obtain at time T b2 (T ) =
Z
T
def
s21 (τ)e−jωτ dτ · A1 ejωT = s21 (T, ω)A1 ejωT
(115)
|b2 (T )|2 = |s21 (T, ω)|2 |A1 |2 .
(116)
0
and consequently For the excitation of eq. (107) the above equation becomes h |b2 (T )|2 = 41 |s21 (T, ω)|2 + |s∗21 (T, ω)|2 + |s21 (T, ω)|2 ej2(ωT +φ(ω)) i +|s∗21 (T, ω)|2 e−j2(ωT +φ(ω)) |A1 |2 .
(117)
where φ(ω) is the phase of s21 (jω). For any ω, we consider a frequency interval centered at ω whose width is of the order of 2π/T and average both sides of eq. (117) over it (coarse graining). The result is 1 (118) |b2 (T )|2 = |s21 (T, ω)|2 |A1 |2 . 2 where for the sake of simplicity we have used the same symbol for |b2 |2 (T ) (which obviously also depends on ω) before and after the averaging. Since the transmittance is Z T Z ∞ −jωτ s21 (jω) = lim s21 (τ)e dτ = s21 (τ)e−jωτ dτ (119) T →∞
0
0
it is clear that its squared modulus can be calculated as the ratio of power debited to the load resistor at time T and the incident power at port 1. By applying the considerations above to eq. (112), we obtain (in the approximation of a large T ): µ µT ¶2 1 " sin2 [(ω − ω )T/2] sin2 [(ω + ω )T/2] 0 0 + |s21 (jω)|2 = N(E2 − E1 )W12 2 ~ 2ǫ0 cσ [(ω − ω0 )T/2] [(ω + ω0 )T/2]2 # sin[(ω − ω0 )T/2] sin[(ω + ω0 )T/2] +2 · cos ωT . (ω − ω0 )T/2 (ω + ω0 )T/2 (120) Passivity implies that |s21 (jω)|2 ≤ 1. For the given values of the physical constants, this puts an upper bound to the pumping time T (eq. (120)). When this is fixed, the condition ρ22 ≪ 1 imposes an upper bound to EM (eq. (111)). Since we must satisfy two conditions
20
,
and we have at disposal two parameters, we can always choose T in such a way as to have |s21 (jω0 )|2 = 1 and EM so that ρ22 ≪ 1. Whether or not the maximum allowable value of T complies with the condition that s21 (T, jω) ≈ s21 (jω) depends on the values of the other parameters appearing in eq. (120); a failure would mean that the approximations are not appropriate. Once |s21 (jω)|2 is given, the squared modulus of the reflectance is simply |s11 (jω)|2 = 1 − |s21 (jω)|2
(121)
due to the losslessness of the D two–port. Finally the complex reflectance s11 (jω) and the complex transmittance s21 (jω) are determined as functions of the angular frequency by calculating their minimum phases from the logarithms of their moduli by H transform. A particularly convenient way of carrying out this last step consists in using the W–L transform, whose application is briefly sketched in the next section. It must be noted that in most treatments eq. (111), and consequently eqs. (112) and (120), are substantially simplified as only the first term inside the square brackets is taken into consideration. This is perfectly allowable on the point of view of numerical calculations as long as we are interested in the performance near resonance, because there the contributions of the other two terms are negligible. But such an assumption destroys the evenness of the moduli of the scattering parameters, thus the oddness of their phases, and therefore makes the application of the concepts of Classical Network Theory impossible.
10. THE SCATTERING PARAMETER CONSTRUCTION BY WIENER–LEE TRANSFORM Consider three complex frequency planes, p = σ + jω, w = u + jv, and θ = γ + jδ. We define the following analytic functions mapping the second onto the first and the third onto the second w−1 p=Ω , w = eθ (122) w+1 where Ω is a constant introduced for dimensional purposes. The compound function θ → p is θ (123) p = Ω tanh . 2 The imaginary axis of the θ–plane is transformed onto the unit circle of the w–plane, and this one onto the imaginary axis of the p–plane: δ δ w = ejδ → p = Ω tanh j = jΩ tan = jω 2 2
(124)
The real axis of the θ–plane maps onto that of the w–plane and this onto that of the p– plane. Thus both transformations and their resultant are real. Finally the exterior and the interior of the unit circle of the w–plane are mapped onto the right and left half–planes of the p–plane. Let F(p) be a real function of p analytic in the right half–plane and continuous in the closure. The first of mappings (122) allows to define a new real function G(w) of w such
21
that G(w) = F(p). The function G(w) is analytic outside the unit circle and continuous in the closure; hence it can be expanded in T series around the point at the infinity ∞ X cn G(w) = wn n=0
(125)
where all coefficients cn are real due to the reality of G(w). Continuity in the closure allows to represent the function G(w) on the unit circle as G(ejδ ) =
∞ X
cn e−jnδ .
(126)
n=0
From eq. (126) it is immediately found Re G(δ) =
∞ X
Im G(δ) = −
cn cos nδ,
∞ X
cn sin nδ
(127)
n=1
n=0
that evidently form an H transform pair. Let now s(p) = s˜(ejδ ) represent either functions s11 (p) or s21 (p) and let ln s˜(ejδ ) = ln | s˜(ejδ )| + jφ(δ)
(128)
Since the modulus is known on the unit circle (eqs. (120) and (121)), its logarithm can be expanded in a F series there. Since the function is even in ω, and because of the reality of the correspondences also in δ, the series will include only cosines, ∞ X
jδ
ln |s(e )| =
cn cos nδ.
(129)
n=0
Then, according to eq. (127), the phase will be expressed as φ(δ) = −
∞ X
cn sin nδ.
(130)
n=0
Note that it is possible to avoid the frequency distortion introduced by the mapping ω = Ω tan
δ 2
(131)
by choosing Ω much larger than the upper bound of the frequency band of interest. In such an instance tan δ ≪ 1 and, by using tan x ≈ x, eqs. (129) and (130) become ∞ X
Ã
2n ln |s(e )| = cn cos ω Ω n=0 jδ
and
∞ X
!
! 2n ω . φ(δ) = − cn sin Ω n=0 Ã
(132)
(133)
22
, 11. CONCLUSIONS
We have shown that the concept of time reversibility, as presented in Quantum Mechanics, can be extended in a profitable way to classical autonomous linear systems, and that, on the other hand, some concepts proper of Classical Network Theory, like passivity and losslessness, provide useful insights in the performance of quantum systems. In particular the calculation of the reflectance of a two–state quantum system driven by an electromagnetic wave shows the potentialities of the network paradigm. Conceptual interrelations between the two fields appear to be of great importance in the perspective of the development of nanotechnologies.
ACKNOWLEDGEMENTS
This research was partially supported by “Ministero dell’Istruzione, dell’Universit`a e della Ricerca (MIUR)”, Rome, Italy, under FIRB Project no. RBAU01LRKJ.
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