Intraenvironmental Correlations in the Ground State of a Nonisolated Two-State Particle

(1)

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Intraenvironmental Correlations in the Ground State of a Nonisolated Two-State Particle

E. Paladino, G. Benivegna, A. Messina

To cite this version:

E. Paladino, G. Benivegna, A. Messina. Intraenvironmental Correlations in the Ground State of a Nonisolated Two-State Particle. Journal de Physique I, EDP Sciences, 1996, 6 (6), pp.783-791.

�10.1051/jp1:1996242�. �jpa-00247214�

(2)

Intraenvironmental Correlations in the Ground State of

_a

Nonisolated Two-State Particle

E. Paladino,

G.

Benivegna

and A. Messina

(*)

INFM,

Istituto di Fisica dell'Universith

degli

Studi di

Palermo, Gruppo

Nazionale del CNR and Centro Universitario del MURST Via Archirafi 36, 90123

Palermo, Italy

(Received

12 December 1995, revised 6

February

1996, accepted 29

February 1996)

PACS.63.20.Mt Phonon-defect interactions PACS.03.65.-w

Quantum

mechanics

Abstract. The existence of

entanglement

in the

ground

state of _a two-level particle

coupled

to _a bosonic environment is proved. The quantum covariances of pairs of simple dynamical

variables relative _to different

subsystems

_are

explicitly

shown to be bounded.

Physically

interpretable conditions for the _occurrence of weak intraenvironmental correlations _are

reported

and

discussed. The potentialities of _our treatment _are

briefly

put into evidence.

1. Introduction

Different

systems

^such as

paraelectric [1,2]

_or

paramagnetic

_[3] defects in _a

crystal, tunnelling

units in _a molecular _or

crystal

solid

[4-8],

_a two-level _atom in _a

cavity

field _[9] _or Jahn-Teller molecules

[10],

may be

investigated

^with ^the

help

of the _same Hamiltonian model. The _reason is that such

physical

or chemical _situations _can _be often

adequately represented

_as the interaction between _a localized

unit,

with _an effective finite dimensional Hilbert _space, and the

quantized

modes of _a bosonic field

[iii.

Of _course, the values of the

microscopic parameters appearing

in the Hamiltonian may differ

by

several orders of

magnitude passing

from _a

specific system

to another _one.

This circumstance sometimes

provides

the

possibility

of

singling

out in the Harniltonian

a dominant contribute which, in accordance with its nature, leads to

qualitatively

different

physical predictions

when _a

particular system

or another is considered.

Thus,

for

example,

the

structure of the

ground

state _as well _as whether the low

lying

_energy levels _can be

thought

as

well-isolated in the spectrum, are

quite

related _to the hierarchical

assumptions

made to treat the

problem [12,13].

^We ^remark

that,

at very low temperature, the

dynamical properties

of the

system might depend

_on such features of the

spectrum.

Moreover the existence of

degeneration

in the

ground

state may also be of relevance in the transition between weak and

strong coupling regimes [14-1ii

Most theoretical work _on this

system

is based _on

approximate approaches,

which unfortu-

nately,

far from the extreme

coupling regions,

turn out to be

inadequate.

^The

origin

of such

a failure _may be traced back to the

objective difficulty

in

guessing,

under

arbitrary coupling conditions,

the

existing entanglement

between the field and the localized unit and between the

(*)

Author for correspondence

(e-mail: messina@ipacuc.cuc.unipa.it)

©

Les

(ditions

_de

Physique

1996

(3)

784 JOURNAL DE

PHYSIQUE

relations, providing

_us with additional

conditions which _must be satisfied

by

_a trial state used in _a variational

approach,

should _at least make easier _a _more accurate construction of _an

approximate analytical

solution. In this paper we deduce and discuss _some _new exact results

concerning

the

ground

state of _a two-level unit

linearly coupled

to N

quantum

^harmonic oscillators.

We show that the

specific equilibrium

conditions which characterize the lowest _energy state 'of the

system

are at the

origin

of

peculiar

mutual correlations between the

subsystems

which

constitute the total

system.

^We demonstrate that the

quantitative

and

for qualitative

under-

standing

^of _even a few

physical aspects

of the correlations which

get

established _among various

dynamical

variables of the combined system ⁱⁿ

stationary

conditions and in

particular

in its

ground

state,

might

offer _a constructive

point

of view useful to

single

out

interesting

features of the interaction into consideration. On the _one

hand,

in

fact,

the

knowledge

of such _as-

pects might provide suggestions

for

choosing,

in the _context of _a variational

approach,

_a trial

state flexible

enough

to

provide

_a

good analytical

solution for the

ground

state even when

any

perturbative

treatment of H fails. On the other hand when these correlations _are

phys- ically interpretable, they

_concur to achieve a more and _more realistic

picture

of the

ground

state nature. In addition their

knowledge might

be very useful to

develop

_a

_systematic

_way of

characterization of different

coupling regimes

_on

physical grounds [18].

An attractive aspect ^of our results is that

they

reveal the existence of

rigorous specific properties

which _are _common _to all the

physical _systems

modelled

by

the _same Hamiltonian.

This _means that _our

physical

conclusions _are valid _over the whole _range of characteristic

parameters appearing

in the Hamiltonian.

This _paper is

organized

_as follows. In the next Section _we will introduce the Hamiltonian model

remarking

_on its symmetry

properties.

In addition _we

present

^the mathematical

approach

_we

are

going

to _use in the _course of the _paper. The existence of intraenvironmental

ground

state correlations is

investigated

in Section

4,

whereas the

explicit

construction of bounds _on the

mean values of the interaction energy is contained in Section 5. In the

subsequent

Section 6 _we succeed in

characterizing

_a weak oscillator-oscillator

coupling regime

in terms of

microscopic parameters

^of the model. In the final Section _we summarize _our _mean

physical conclusions, mentioning possible applicative implications

of _our treatment _as well _as _some

speculations

for future work

regarding

this

interesting problem.

2. The Hamiltonian Model and the Mathematical

Approach

In this _paper _we

study

_a

physical system consisting

of _a _two-state

particle linearly interacting

with N quantum harmonic oscillators. We describe this situation with the

following

Hamilto- nian model:

H =

equivalent second-quantized

version _assumes the form:

H =

~j _huJ~o)o~

₊

~j

e~

(a~

⁺

a))

a~ +

~~°°

az

(2)

~ ~

2

The _energy

separation

between the levels of the

particle (spin

_or

pseudospin)

is

hula.

The aj

operators (j

= x, y,

z)

_are Pauli operators.

j(p~)

is the

position (conjugate momentum)

(4)

operator

^{of the} ^I-th ^oscillator

(mass

_m and elastic constant k~) ^whose quanta of

frequency

_uJ~

are created _or annihilated

by

the bosonic operators

a)

_and _a~

respectively.

We _assume _uJ~

#

_uJj

if I

# j.

The

positive coupling

constants _e~ and

Foi

_are related _as follows:

vifoi

=

@te~.

The linear

dependence

of the

operators

_q~, _p~ and

a),

a~ may be

expressed

_as:

fi ^1/2

q' "

~ °'~ (~'

+

°) ₍₃₎

p~ = I

~~~°~)

^~~~

(a) a~). (4)

2

It is

immediately

verified that the

following

transformation

j

# -tXj

&) #

-O)

a~

₌ _-a~ ^I =

1,.. ,N (5)

ay

# -Oy

iz

" Oz

is _a

symmetry

of H. The canonical

change

of variables

expressed by (5)

is

generated by

the N mode

parity

operator

P exP

~r _al

a~ +

i~

⁺

ill ⁽⁶⁾

which

generalizes

_a similar one-mode operator

previously

^introduced

[17]. Transforming

H with the

help

of the operator

[17,19]

N

T = exp

Ii

^~

(a~ 1) ~j a)a~ (7)

~

i=1

yields k

=

TtHT

_as follows

k

=

~j (huJ~a)o~

^{+ e~}

(o)

⁺

a~)

⁺

~~°°

az

fl

cos

(gra)a~) ⁽⁸⁾

~

2

Thus the search of the _common

eigenstates

^of ^H ^and

P,

_appears to be

equivalent

to the

diagonalization

of the

following purely

bosonic Hamiltonians

(W

=

+1)

where W is the

eigenvalue

of P.

As _we have said in the introduction _we do _not attempt to solve the

eigenvalue problem

of H.

Our main

goal

is to shed

light

into the nature of correlations manifested

by

the

system

in its

ground

state. To reach this

objective

we look for relations

involving

the

quantum

covariance

Covw(A,B)

of

simple pairs

of Hermitian

operators

^related to different

subsystems,

^defined

as

[20]

:

C°~W A,

B ^~~~

(~fii

AB

~fii (~fii _IA ~fii (~fii

B

~fii o)

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786 JOURNAL DE

PHYSIQUE

I N°6

where

j~fif)

is _a

generic eigenstate

^of H with _a definite

parity.

The

computation

tool for

obtaining

such relations is very

simple.

From the well-known

expression

^{of the} ^time derivative of _any operator ^A [21]

/t

~

~t

^~ ^~

^'

^~~~~

it is

straightforward

to

get that,

when A is _a time

independent operator,

(1fisllH,Ajjqfi~)

= o

(12)

where _{j~fis) is} _a

generic stationary

state of the

system. Choosing appropriately

the

operator A,

we may

give equation (12)

the form of _a "balance

equation"

between the

expectation

values of

simple

operators. ^This

approach

has

just

^been

presented

in reference

[17]

ⁱⁿ ^connection ^with

a

simpler

Hamiltonian model. Here _we

generalize

^and ^extend

it, confirming

in this _way _our

expectation

about the

flexibility

and

potentiality

of this method.

3. Ground State Intraenvironmental Correlations

The Hamiltonian model

(1)

does not contain intraenvironmental

coupling

terms. There is however _no doubt _on the existence of

entanglement

_among ^different

degrees

of freedom of the

combined

system

in its lowest energy ^state. We may

qualitatively

understand the _occurrence of correlations _among the environmental oscillators _as

arising

from their _common

coupling

with the _same

subsystem (particle).

In this _paper we shall derive and discuss _some

simple

relations which make

physically transparent

^the role

played by

each

single oscillator-particle

interaction _on the appearance of _a direct mutual influence between distinct coordinates

belong- ing

to the

(pseudo-spin)-environment _system.

To establish these relations _we make _use of the mathematical

approach

illustrated in the

preceding

section. Iii order to

exploit constructively equation (12),

_we seek

products

of two

simple operators

relative to two

arbitrarily prefixed

environment modes. A direct

inspection

of H

suggests

^the ^choice _q~pj and qjpi.

Since

qr(pr)

is

unitarily mapped

into

-qr(-pr) by P,

the

expectation

values of

q~qj(p~pj),

taken _on _a

parity

^defined

stationary state,

^coincide ^with ^the ^covariance ^between ^the two op-

erators.

Identifying

A with q~pj and qjpi ⁱⁿ

equation (12),

_we

get

two

independent

relations

from which it is easy ^to obtain

FQ~(1b71~j

Ox

j~§I) j

[jj~'l~ ^'~~°~

'~i~~

~~~~

_~~~

(~~~

~~

w2Fo~(~fiT~jjj(~fii)~-

WI FOJ

(~li _'~'°~

_'l~~'~~l

(14)

covw (pi,

_pj ₌ _m ^?

(wj Wf)

We _now _prove that in the

ground

state

jg)

^of ^H ^the two covariances

expressed by equations (13)

and

(14)

cannot

simultaneously

vanish. To this end it is

enough

to show

that,

when the energy

of the

system

assumes its lowest

possible value,

then

necessarily

(glqra~lgl # o,

_r

=

1, 2,..

,

N.

(is)

Let _us suppose,

by absurd,

the existence of _a suitable set of

positive

values of the

parameters (uJ~), (e~),

_uJ0 such that the _mean value of _qra~ _on

jg)

^vanish.

Then,

in view of

equation (3),

we

get

(gj(ar

₊

o))a~jg)

= 0.

(16)

(6)

Then _we may represent

Eg

as follows:

~g

H)j_~j

^then a

ground

state of

huJra)ar

₊

H)j_~j

has the form

jgN-1)10r).

From

equation (17)

_we

easily

infer that

Eg

must be either

equal

to or

larger

than the

expectation

value of

H)j/_~j

on the _state

jgN-I). Noting

that

(°ri(~N-liHi~N-I)i°r)

"

(On(~N-lii~"r°)°r

₊

H)l/-I))i~N-I)i°r) (20)

we

deduce,

in view of

equation (17),

that

Eg

cannot exceed the lowest

eigenvalue

of

H)j_~~.

It should therefore _come out that

Eg

₌

(gjHjg)

=

(0rj(gN-il(huJra)or

₊

H)~/_~j)jgN-1)10r) ⁽²¹⁾

which,

in

turn,

^should

imply

HlgN-ill°ri

₌

EglgN-ill°ri (22)

which is

evidently

false.

Equations (13)

and

(14)

and the

supplementary

condition

(IS)

have _a clear

physical meaning rigorously proving that,

in the

ground

state of _a nonisolated two-state

particle,

the presence of intraenvironmental correlations is _a _common

aspect

^of all

coupling regimes

and that their _occurrence is _a direct consequence of the interaction of each oscillator with the two-level

system.

^It ^is ⁱⁿ ^fact immediate to _see from

equations (13)

and

(14)

that

when the

coupling

between _a

given

environment

degree

of freedom and the

pseudospin

_may

be

neglected,

then both the covariances

vanish,

whatever the

intensity

of

coupling

of the other oscillators with the two-level

system

îs. Ît îs

interesting

to look for weak intraenvironment- correlation conditions without any

assumption

_on the

particle-environment coupling regime.

To this end _we focus _our attention _on the _mean value of the interaction energy in the

ground

state of the

system.

4. Lower and

Upper

Bounds _on the

Expectation

Value of the Ground State Particle-

Oscillator Interaction

Energy

In this section _we wish to demonstrate that the interaction energy between the two-state

particle

and each environmental oscillator satisfies _a double

inequality

in the

ground

state of the

system.

To this end let's introduce the

following auxiliary

states:

l~Ji)

=

Tilg)

₊

CDS(7r°lai)lg) (23)

1~22) ⁼

T21g)

+

exPinilai al)a~ilg) ₁₂₄₎

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788 JOURNAL DE

PHYSIQUE

I N°6

where1J~

=

~~ and I is _a

prefixed

index between 1 and N.

huJ~

Since the

operators Ti

^and

T2

_are

unitary,

_{j~2i) and} _j~22) _are normalized like

jg)

^is. ^Moreover both

j~2i)

^and j~22) ^are

linearly independent

from

jg). Considering

^that:

T)a~Ti

" -o~

(25)

T/o~T2

_" _ai+q~ax

(26)

it is not difficult to show that

T)HTI

= H

2e~(ai

+

a) _)a~ ₍₂₇₎

~~~

_~ _{~ ~~}_~°~ _~

_°~ _~°~

_~ _~

^~~

_~

_~'~

_~~~

°~ ~~~~

where

az

₌

T)azT2.

Exploiting equations (27)

and

(28)

_we

easily _get:

(~JilHl~2i) (glHlg)

₌

-2611gl(ai

+

_on

_jg) _represents

also the covariance between the I-th oscillator coordinate and that relative to the

pseudo-spin. Inequality (31)

has the remarkable

physical meaning that,

what-

ever the

coupling strength,

the

assumption

of _a

positive

covariance between these coordinates is not

compatible

with the

system being

in its

ground

state.

5. Intraenvironmental

Weak-Entanglement

Conditions

With the

help

of

equation (31)

_we _may _now deduce the main result of this paper: the existence of _an _upper bound for the absolute value of the covariance between

pairs

of operators relative to two oscillators in the

particle

environment.

It is convenient to write down

equations (13)

and

(14)

in adimensional form:

~°~(t~'

+ t~~,t~J ⁺

°~)

~~~~~~~~~~~ ^~

~~~()2

~/~~~~~~~~

^~

^~~~~~~ _()

j 1

6~ldjjgj(Oj

+

tkj)Oxlgl ~J~°'19'(°'

+

°))°~'91 coy(iia~ «]),i(aj _aj)) (33)

= 2

h(u~j uJi)

inserting equations (3)

and

(4)

into the two

expressions (13)

and

(14). Taking advantage

from the double

inequality (31) conveniently

written _as

I(gl(a~

+

al)axlg)I I ()

+

@) (34)

(8)

it is

possible

to convince oneself that

jcov(a~

+

a),

aj ⁺

aj _)j

< ~~°°

[eijuJ~j (3~ij

⁺

1)

+ e~juJ~i(3~i~ +

1)] (35) jAuJj

jcov(I(a~ at ), I(aj aj))j

< ~~°°

[e~juJ~j

(3~ij

+

1)

+ ej~uJ~j(3~i~

+1)] (36)

~ l/~1°1

between the r-th oscillator and the two-level

particle [2,14,15].

It is _common in the literature to

classify

three different

coupling regimes:

weak,

intermediate and

strong,

ⁱⁿ accordance with values of _~ir such that: _~ir <

I,

_~ir m

1,

~ir » 1

respectively.

In the

preceding section,

_we have

pointed

out

that,

in accordance with _our

physical expectations,

^when an environmental mode is

weakly coupled

with the

pseudospin,

its

entanglement

with any other mode is small. An

interesting question

is thus whether _a condition of weak

ground

state intraenvironmental

correlation,

_may be

compatible

with the existence of

particle-oscillator

intermediate _or

strong coupling.

From

equations (35)

and

(36), assuming

~T~ ^SQS ~Tj ^SQS~T > 1

(41)

we obtain

that,

when

~~

~

~~j~

~~~~

then both covariances

(35)

^and

(36)

^become ^much ^smaller ^that

unity.

This

physically

_means

that when

inequalities (41)

^and

(42)

_are

simultaneously satisfied,

the

specific equilibrium

_con-

ditions which

get

established in the

correspondent ground

state of the

particle-environment system

are

compatible

with the existence of weak

entanglement

between the two oscillators under consideration. In other words the crossed

coupling

condition

expressed by inequal- ity (42), conjugated

with the

assumption

on the

coupling strength expressed by equation (41),

legitimate neglecting

_as first

approximation

the presence of mutual influence between the two oscillators induced

by

their _common interaction with the two-level

particle.

We

point

out

that,

in accordance with the

weak-entanglement

condition

(42),

values of _~i

larger

^than

unity

_may be

admitted, provided

that

jAuJj

is

appropriately greater

than _uJo.

6.

Summary

and

Concluding

Remarks

In the

preceding

^sections we have

presented

_new

physically transparent

^results

concerning

_some aspects ^of ^the intraenvironmental correlations

existing

ⁱⁿ ^the

ground

state of _a two-state particle

linearly coupled

to _a bosonic environment. Our

treatment,

_as

usual,

^simulates such _an

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790 JOURNAL DE

PHYSIQUE

I N°6

environment

introducing

_a set of quantum harmonic oscillators without however

assuming

_a

specific dispersion

law in the

correspondent, widely used,

Hamiltonian model

(I).

At the _same

time _we do not fix _a

particular oscillator-dependence

of the

coupling

constants. In this _way _our results possess

general validity

and

consequently

_may be

applied

is several different

physical

contexts.

Studying

the

quantum

covariances between the coordinates of _any _two environmental

oscillators,

_as well _as between their relative

conjugate

_momenta, in _an

arbitrary parity-defined stationary state,

we have found _a

quantitative

connection between the _appearance of entan-

glement

in the environment and the direct

oscillator-particle coupling.

^The circumstance that this last

quantity satisfies,

in the

ground

state of the total

system,

the double

inequality (31),

deduced in this article for the first

time, represents

^the

key

idea for

constructing

_upper bounds

on the absolute values of both oscillator-oscillator covariances. The mathematical

dependence

of such _upper bounds _on the

microscopic

model

parameters appearing

in the

Hamiltonian,

has

provided

_us with the

possibility

of

characterizing

a

weak-entanglement

intraenvironmental

regime

in _terms of

frequencies

and

coupling

constants. It is remarkable that such _a

regime

_may

occur, in the

ground

state of the

system,

even for

moderately large coupling strength.

It is worthwhile that all the

physical

conclusions reached in this _paper _are based _on _an exact mathematical

approach

which

strategically

circumvents the

knowledge

of

analytical explicit

form of the

ground

state of the

system.

^We ^deem ^that ^this

approach

_may be furthermore

exploited.

In

particular

_we refer to the fact that in this _paper _we have focused _our attention

on two

specific

choices of the

operators

to be inserted into

equation (12).

Other

possibilities

may be of _course considered

eventually leading

to relations like

equations (13)

and

(14).

We

believe that the

knowledge

of additional _exact and

independent

relations between _mean values of

simple operators possessing

_a clear

physical meaning might

be of _some aid in

addressing

intuition toward _a well-based choice of _a

ground

state trial ket in the framework of _a variational

approach.

Acknowledgments

We wish to express our

gratitude

to Dr. G. M.

Palma,

Dr. R. Passante and Dr. G. Salamone for

carefully reading

_our

manuscript.

Partial financial

support by

the CRRNSM-REGIONE

SICILIA and CNR is

acknowledged.

References

[1] Shore H-B- and Sander

L-M-, Phys.

Rev. B12

(1975) 1546;

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