On the ground state of a nonisolated two-state particle (*)
G. BENIVEGNA, A. MESSINAand E. PALADINO(**)INFM, Gruppo Nazionale del CNR and Centro Universitario del MURST
Istituto di Fisica, Università degli Studi di Palermo - Via Archirafi 36, 90123 Palermo, Italy
(ricevuto il 21 Ottobre 1996; approvato il 26 Novembre 1996)
Summary. — We derive some exact properties of the ground state of a two-state system coupled to a harmonic bath. The physical meaning of these properties as well as some implications are carefully pointed out.
PACS 03.65 – Quantum mechanics.
PACS 63.20.Mt – Phonon-defect interactions.
1. – Introduction and Hamiltonian model
The interaction of a two-state particle with bosonic external degrees of freedom is a general problem of lasting interest in many chemical and physical contexts. Very often, in fact, the treatment of various and very different complex phenomena may be fruitfully traced back (or mapped in) to the study of a truncated two-level localized nonisolated unit coupled to bosonic or fermionic excitations. Such an approach has been used to study many interesting problems such as phonon-assisted tunnelling of paraelectric or paraelastic defects in crystals [1, 2], tunnelling of molecules in low-temperature media [3-6], hopping of an electron between two different sites of a molecule (Holstein systems) [7, 8], and the interaction between vibrational modes and electron levels (Jahn-Teller systems) [9].
In this paper we consider the following Hamiltonian model [2, 10-13]:
H 4
!
i ˇvia†iai1!
i ei(ai1 a†i) sx1 ˇv0 2 sz, (1)where sj ( j 4x, y, z) are Pauli operators describing the particle degrees of freedom,
v0 denotes the bare tunnelling parameter, ai and ai† are Bose operators creating or
annihilating quanta of frequency vi. The positive parameter eiis the coupling constant
between the pseudospin and the i-th mode. Models very similar to (1) have been used to
(*) The authors of this paper have agreed to not receive the proofs for correction.
(**) Present address: Viale Andrea Doria 6, 90125 Catania, Italy. E-mail: PaladinoHalp1ct.ct.infn.it 905
study both the quantum dynamics of a two-state system coupled to an Ohmic bath [14-16] and to represent the macroscopic quantum tunnelling in Josephson junctions and SQUIDs [17]. Very recently, the first observation of resonant tunnelling between macroscopically distinct quantum levels has been reported [18].
Hamiltonian (1) encompasses many different physical situations. Unfortunately, an exact treatment of its eigensolution-problem in a generic point (]vi(, ]ei(, v0) of the microscopic parameter space seems to be hopelessly difficult. On the contrary, approximate analytical solutions may be found when the physical situation under scrutiny corresponds to points belonging to specific sectors of the parameter space, known as adiabatic and nonadiabatic regions [12, 13, 18-21]. The reason is that in such regions it is legitimate to single out in H a term which can be treated as a perturbation with respect to the others. Within this approach it has been shown that the structure of the ground state as well as the discreteness degree of the low-lying energy levels in the spectrum are qualitatively sensitive to the values of the parameters relative to the particular physical problem under investigation [12, 13, 22].
It is worth emphasising that this sensitivity is of experimental significance too in view of the fact that, at least for some systems described with this Hamiltonian model, an external control of the microscopic parameters appears possible [23].
In addition, we point out that the low-temperature dynamics of the system might display a great variety of different physical behaviours depending on the values assumed by the microscopic parameters appearing in H.
In this paper we prove some exact results concerning the ground state of a system described by Hamiltonian (1). A peculiar aspect of our treatment is that, circumventing the difficulties related to the eigenvalue problem resolution, it is valid in the whole range of microscopic relevant parameters. As a consequence, the results found in this paper are applicable to each physical system which can be modelled by (1).
In the next section we derive a property of the particle energy in the ground state of the system. In sect. 3 the existence of correlations between the two-state system and its environment is explored by studying the quantum covariance between a pair of appropriate coordinates relative to the two subsystems. Some physical implications as well as some conclusive remarks are presented in the last section.
2. – The ground-state particle energy: an exact property
When the particle-environment coupling constants ]ei( appearing in model (1) do
not vanish, the ground state Ngb of H describes the existence of entanglement between the small and the large subsystems. In particular, this means that the two-level system energy, ˇv0
2 sz, undergoes parameter-dependent fluctuations around an expectation
value likewise sensitive to the values of the parameters appearing in (1).
For weak coupling the mean value of sz on Ngb differs very little from 21. On the
other hand, in the opposite regime the probability of finding the two-state particle in its bare excited state tends to become equal to that of finding it in its bare ground state. In this section we shall prove that whatever the values of the positive parameters ]vi(, ]ei(, v0are, the following inequality:
agNszNgb E 0 ,
(2)
Let us denote by Egthe lowest energy eigenvalue of H. By definition we thus have Eg4 agNHNgb 4
!
i 41 N ˇviagNai†aiNgb 1!
i 41 N eiagN(ai1 a†i) sxNgb 1 ˇv0 2 agNszNgb . (3)Suppose now, by absurd, that
agNszNgb D 0
(4)
and consider the following auxiliary state:
Ncb 4 sxNgb .
(5)
This vector is normalized as Ngb is, because sx is unitary. The evaluation of the mean
value of H on Ncb gives acNHNcb 4
!
i 41 N ˇviagNai†aiNgb 1!
i 41 N eiagN(ai1 a†i) sxNgb 2 ˇv0 2 agNszNgb . (6)Equations (3) and (6) immediately yield
acNHNcb2Eg4 2 agNˇv0szNgb E 0 .
(7)
This inequality contradicts the fact that Egis the lowest energy level of the system and,
being a direct consequence of the assumption (4), it implies that agNszNgb G 0 .
(8)
In order to complete the demonstration of the inequality (2) we still have to exclude the possibility that the mean value of the energy of the two-state particle might be null when the energy of the global system assumes its lowest value.
With this aim let us observe that if there existed a set of positive values of the
parameters of our model (1) for which agNszNgb 4 0 , then we would have Eg4 agNH 2
ˇv0
2 szNgb. In turn, this implies either that the fundamental levels of H and
g
H 2ˇv02 sz
h
coincide, or that Eg is higher than the lowest energy level ofg
H 2ˇv02 sz
h
. It is not difficult to convince oneself that such latter possibility cannotoccur.
As far as the former one, we may argue that it is false too, noticing that the lowest energy eigenstates of
g
H 2ˇv02 sz
h
are not stationary states of H as it should be.Thus we arrive at the conclusion that the condition agNszNgb 4 0 can occur if and
only if v04 0 , whereas, otherwise, the expectation value of the two-level system
energy is negative according to inequality (2).
Albeit many efforts have been made to construct a physical characterization of how the structure of the ground state of the Hamiltonian (1) is modified in the passage from the weak- to the strong-coupling regime [8, 10, 11, 13, 20, 21], the modalities of such a transition are still to be clarified under many aspects. In the framework of this open problem it is now under our active consideration the possibility of using a mathematical
approach analogous to that used in this paper to study how agNszNgb varies from its
minimum value 21 to its upper bound 0 as a function of an appropriate coupling strength increasing from 0 to infinity.
3. – Particle-environment ground-state correlations
It has been recently shown that the ground state of a nonisolated two-state particle linearly coupled to a harmonic-oscillator bath may be labelled with the help of the conservative operator P 4exp [ipN] where
N 4
!
i 41 N a†iai1 sz 2 1 1 2 (9)is the total number operator.
By exploiting this ground-state symmetry classification it is possible to show that the expectation value on Ngb of sxand of (a†i1 ai), for any i, vanish [24].
On the other hand, in the lowest energy state of the particle-environment system peculiar mutual correlations exist between the small and the large subsystems of the total system [25].
In order to investigate more in detail this point we consider the quantum covariance
between the two-level system coordinate sx and the i-th bath oscillator coordinate
(a†i1 ai).
By definition [26] we have (10) Covg
(s
x, (ai1 ai†))
44 agN(ai1 a†i) sxNgb 2 agN(ai1 a†i) NgbagNsxNgb 4 agN(ai1 a†i) sxNgb .
In this section we wish to prove that in the ground state of the system sxand (a†i1 ai)
are correlated in such a way that the following inequality holds: Covg
(s
x, (ai1 ai†))
E 0 .(11)
To demonstrate this result, once more, we refer to eq. (3), assuming, by absurd, that agN(ai1 a†i) sxNgb D 0 .
(12)
Let us introduce the following auxiliary state:
NWb 4 cos (pa†iai) Ngb .
(13)
Since sin (pa†
iai) is the null operator, then
cos (pa†
iai) 4exp [ipa†iai] 4exp [2ipa†iai] 4cos [2pa†ia] .
This means that cos (pa†
iai) is unitary and Hermitian, so that the auxiliary state NWb is
normalized as Ngb is. Another property of NWb is that it is linearly independent from Ngb. Let us see this last point more in detail. It is easy to convince oneself that Ngb may be chosen real. This means that fixing an arbitrary real basis in the Hilbert space S of the system, the coefficients of the expression of Ngb in such a basis are real. From this property it follows that, if NWb were linearly dependent on Ngb, then it would be
cos (pa†iai) Ngb. Thus Ngb cannot simultaneously have nonvanishing projections on even
and odd eigenstates of ai†ai. Of course, this condition is not compatible with the fact
that Ngb is an eigenstate of H in view of the linearity of the coupling term in (1).
Exploiting the anticommutation properties of the operators cos (pa†
iai) and ai or
a†i, we easily get
aWNHNWb2Eg4 2 2 eiagN(ai1 ai†) sxNgb E 0
(14)
as a consequence of the hypothesis (12). Since the inequality (14) is not compatible with the definition of Ngb, we must conclude that
agN(ai1 a†i) sxNgb G 0 .
(15)
In order to show that condition (15) is indeed true in the stronger form expressed by (11), let us suppose, by absurd, the existence of a suitable set of positive values of the parameters such that the following equation is satisfied:
agN(ai1 a†i) sxNgb 4 0 .
(16)
Then we may represent Egas follows:
Eg4 agNHNgb 4 agNˇvia†iai1 H
(i) (N 21)Ngb , (17)
where H(N 21)(i) is given by
H(N 21)(i) 4
!
s 41 N 8 ˇvsa†sas1!
s 41 N 8 es(as1 as†) sx1 ˇv0 2 sz (18) with!
s 41 N 8 4!
s 41 i 21 1!
s 4i11 N . (19)If we denote by NgN 21b a ground state of H
(i)
(N 21), then a ground state of ˇvia†iai1
H(N 21)(i) has the form NgN 21b N0b. From eq. (17) we easily infer that Egmust be either
equal to or larger than the expectation value of H(N 21)(i) on the state NgN 21b. Noting that
aOiNagN 21NHNgN 21b NOib 4 aOiNagN 21Nˇvia†iai1 H
(i)
(N 21)NgN 21b NOib ,
(20)
we deduce, in view of (16), that Eg cannot exceed the lowest eigenvalue of H
(i) (N 21). It should therefore come out that
Eg4 agNHNgb 4 aOiNagN 21Nˇvia†iai1 H
(i)
(N 21)NgN 21b NOib
(21)
which, in turn, should imply
HNgN 21b NOib 4EgNgN 21b NOib
(22)
which is evidently false. In fact, due to the presence in H of a term linear in aiand a†i,
the state appearing in the l.h.s. of eq. (22) is linearly independent from that appearing in the r.h.s. of the same equation. Summing up, for every positive value of the parameters and for all the field modes, we succeeded in proving condition (11). It is possible to give a physically transparent interpretation of such a condition.
Writing explicitly the second time derivative of the coordinate operator (a†
i1 ai)
relative to the i-th oscillator in the bath, it is immediate to verify that the dynamics of such an oscillator is influenced by the actions of two “forces”. The first is proportional to (a†
i1 ai), thus representing the elastic force of the same oscillator. The second one,
proportional to the particle coordinate sx, is the force exerted on the i-th oscillator by
the two-level system.
These identifications enhance the conceptual interest for inequality (11), allowing to read it as the existence of correlations between the internal and external forces on each environmental oscillator in the ground state of the global system. In other words, condition (11) singles out a particular physical aspect pertinent to the field and the two-state unit subsystems when the energy of the global system takes its lowest possible value.
4. – Physical implications and conclusive remarks
The results obtained in the previous sections may be exploited to deduce a physical
consequence concerning the behaviour of the lowest energy eigenvalue Eg of H in the
parameter space V, defined by the conditions viD 0 , eiD 0 for any i, and v0D 0. Taking into account that the normalization of Ngb is valid in each point of V, it is easy to verify the following orthogonality condition:
agNg 8zb 40
(23)
between Ngb and the real unnormalized state Ng 8zb which is the partial derivative of the
ground state Ngb with respect to some parameter here generically denoted by z. Since Ngb is a function of all the parameters appearing in (1), we make evident such a dependence writing
Eg4 agNHNgb 4 Eg(]vi(, ]ei(, v0) . (24)
The differential of Egmay be formally cast in the form
dEg4
!
i ¯Eg ¯vi dvi1!
i ¯Eg ¯ei dei1 ¯Eg ¯v0 dv0. (25) Considering that ¯Eg ¯z 4 ¯ ¯zagNHNgb 4 ag 8z NHNgb 1 agN ¯H ¯z Ngb 1 agNHNg 8zb (26)and taking into account the condition (23), we immediately get ¯Eg
¯z 4 agN
¯H
¯z Ngb .
(27)
This result is embodied in the well-known Feynman-Hellmann theorem. Since H
contains linearly all the microscopic parameters of the model, the expression of dEg
becomes dEg4
!
i ˇagNa†iaiNgb dvi1!
i agN(a † i1 ai) sxNgb dei1 ˇ 2agNszNgb dv0. (28)This equation says that the variation of Eg correspondent to an infinitesimal
displacement in V, is related to the mean values of the operators a†iai, (a†i1 ai) sx
and szon the ground state Ngb.
The mean value of a†
iai on Ngb is, of course, positive in V, for any i. In fact the
condition agNa†
iaiNgb 4 0, implying necessarily that Ngb is an eigenvector of the
operator a†
iai, is incompatible, in V, with Ngb being the ground state of H. At the same
time the inequalities deduced in the previous sections may be read in this context saying that the other first partial derivatives of the function Eg(]vi(, ]ei(, v0) cannot vanish at any point of V.
This implies that, whatever the physical system under consideration is, it turns out that
d Egc0 .
(29)
In words this means that the first differential of Egcannot be null in correspondence to
arbitrary increments of the microscopic parameters of the system appearing in eq. (1). From inequalities (2) and (11) and from the fact that agNa†iaiNgb is positive in V for
any i, we may draw moreover the conclusion that the hypersurface representing the function Eg(]vi(, ]ei(, v0) in V does not possess saddle points. Inequality (2) in fact says that Eg, taken as a function of v0 alone, decreases as v0 increases. At the same time inequality (11) means that Eg is a decreasing function of each coupling constant ei.
We wish to conclude adding some remarks on the content of this paper. We have presented exact proofs of some properties possessed by the ground state of a Hamiltonian model widely used in the literature.
Our results are expressed both by inequalities (2) and (11) and by their implications on Egdiscussed in this section and described by eq. (29).
Distinctive features of our treatment are both the fact that it has been developed without introducing any restrictive condition on the parameters and the circumstance that it has not been based on a preliminary explicit knowledge of the parameter dependence of the lowest energy eigenvalue and eigenvector of H. From a mathematical point of view, the derivation of our results is attractive in itself because our approach, with the help of the two auxiliary states introduced by eqs. (5) and (13), is entirely based on general properties of the lowest energy level. We wish to point out the inherent physical interest of inequality (11) in the sense that it permits to disclose general aspects of the correlation existing between the two-state particle and its bosonic environment in the fundamental state of the combined system.
We believe that the results presented in this paper may be of some help in the framework of a variational approach to the problem of the ground state of H. This is especially true when any perturbative treatment of this problem turns out to be inadequate.
* * *
We express our gratitude to Drs. M. PALMA, R. PASSANTE and G. SALAMONE for
carefully reading our manuscript. Partial financial support by the CRRNSM-Regione Sicilia and CNR is acknowledged.
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