Dynamic squeezing in a single-mode boson field interacting with two-level system

Journal of Luminescence 101 (2003) 101–114

Dynamic squeezing in a single-mode boson field interactingwith two-level system

T. Sandua,*, V. Chihaiab, W.P. Kirka

aNanoFAB Center, University of Texas at Arlington, 500 S. Cooper Street, Arlington, TX 76012, USAbGZG Institute, G .ottingen University, Goldschmidtstr.1, G .ottingen 37077, Germany

Received 9 November 2001; received in revised form 13 May 2002; accepted 10 July 2002

Abstract

We studied the time evolution of a two-level electron system interacting with a single-mode bosonic field

(i.e. photons, phonons). We found that in the adiabatic limit (i.e. electron motion fast and boson motion slow) it is

possible to obtain a reduction in fluctuation for the position coordinate of boson and momentum coordinate as well,

depending on the adiabatic potential. If the system is on a lower adiabatic sheet we obtain a reduction in momentum

fluctuations. However, if the system is on an upper sheet the fluctuation in position coordinate (associated with the

bosonic field) is reduced. The maximum reduction is for a range of parameters given by g2=vE1; where g is the electron–

photon (phonon) coupling strength and 2v is the electron energy level splitting. Moreover, it is possible to generate

squeezed light at a frequency different from the pump frequency. The output frequency is mainly given by the curvature

of the adiabatic potential. The system is interesting because it can be implemented in nanoscale systems like a single or

double quantum well system interacting with a laser field of radiation. The degree of squeezing can be improved either

by time modulating the energy of the electron levels inside the quantum well or by modulating the laser field.

r 2002 Elsevier Science B.V. All rights reserved.

PACS: 42.50.d; 63.20.Kr

Keywords: Squeezed states; Coherent states; Spin-boson model; Electron–phonon interaction; Adiabatic coupling

1. Introduction

The study of squeezed states is interesting fortwo reasons. It represents a frontier of physics thatgives us new understanding of quantum mechanicsand also because of its potential application forreducing noise in instrumentation and commu-

nication [1]. Optical noise is an important criterionin the performance of optical systems. Forexample, the size of the fluctuations in theamplitude of light waves limits the ability tomeasure or communicate small amplitude signalswith a beam of light. In recent years, muchattention has been paid to the study and produc-tion of squeezed states of quantum oscillators (theuncertainty for one observable is reduced to avalue less than its ground state value and for theconjugate observable the uncertainty is raised).

*Corresponding author. Tel.: +817-272-7496; fax: +817-

272-7458.

E-mail address: [email protected] (T. Sandu).

0022-2313/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 2 - 2 3 1 3 ( 0 2 ) 0 0 3 9 3 - 9

Mainly, squeezed states of light were considered[1]. The squeezed states are encountered in other(e.g. material) quantum systems [2–4]. Squeezedwave packets may be generated by a suddenchange of oscillator frequency [5,6] or by a pulsedexcitation to another potential surface [7]. More-over, the non-linearity needed for the generationof squeezed states is typical of some systems suchas interacting electron–phonon systems [8] orvibronic ones [17]. Squeezing of molecular wavepackets has applications for investigating intra-molecular processes in real time. The success ofsuch investigation depends on the generation ofhighly localized wave packets [9,10].

We consider the interaction of a two-leveloscillator system as the simplest non-trivialHamiltonian model. In the basis of the two elec-tronic states, |1S and |2S, the Hamiltonian can bewritten as ð_ ¼ 1Þ

H ¼p2 þ o2q2

2s0 � gqsz � vsx; ð1Þ

where s0, sz, sx, are Pauli matrices and p and q arethe oscillator coordinates.

The essential parameters are o; g and v asso-ciated with frequency of the oscillator, electron–oscillator coupling strength and separation ofelectron levels, respectively. This model is calledeither the spin-boson model or the toy polaronmodel [11] and it is encountered in many problemsin chemistry, physics, and optics. Dynamic proper-ties of mixed valence systems generate electronictransfer from one part of the molecule to another,so that it can be conceived as a bistable molecule,which has a two-well potential separated by abarrier [12–14]. Such systems are coupled vibroni-cally and are in degenerate or quasi-degenerateelectronic states. In solid-state physics this modelcan describe situations such as an electronic pointdefect in a semiconductor [15] or the interaction ofa dipolar impurity with the crystal lattice [16]. TheHamiltonian is also isomorphic with the simplepseudo Jahn–Teller system [17] and two-levelsystem interacting with a single-mode radiationfield. Its simplification is the famous Jaynes–Cummings model that has played a very importantrole in our understanding of the interactionbetween radiation and matter in quantum optics

[18]. The ground state of the toy polaron showssqueezing of momentum fluctuation [8] in theadiabatic regime (region of the parameters:vb1; g > 1; and g2=vD1). This region is a transi-tion of electron from a localized state to adelocalized one. In general, the transition isaccompanied by anomalous position fluctuationsand squeezing of momentum. Moreover, it hasbeen suggested that squeezing is generated in aharmonic oscillator with time-dependent fre-quency [2–7]. Since the dynamics of the oscillatorwithin Eq. (1) is strongly anharmonic in theadiabatic regime (as we will see later on) and ifwe consider the anharmonicity as a change inpotential curvature, it is natural to look forsqueezing in the adiabatic region of parameters.

The aim of this paper is to look for theproduction of squeezed states of a bosonic fieldin the adiabatic region of the parameters using atime-dependent variational approach and by directintegration of the time-dependent Schr .odingerequation. The model and its properties in theadiabatic limit are discussed in Section 2. Section 3is devoted to obtaining the equations of motion fordifferent trial functions with the time-dependentvariational principle (TDVP), which establishes aframework for the production of squeezed states.The use of TDVP allows us to quickly locate theHamiltonian parameters that are interesting fromthe point of view of squeezing. Also a scheme fornumerical integration of the time-dependent Schr-.odinger equation is presented. The discussion andconclusions are outlined in the last two sections. Itis shown that the mechanism for squeezing isbasically the change of the adiabatic potentialcurvature due to the electron–boson interaction.In addition, the inclusion of the higher-order non-linear terms describes with a high degree ofaccuracy the time evolution of Hamiltonian (1).

2. Theoretical model

2.1. Elimination of electron subsystem

Despite its simple structure Hamiltonian (1)cannot be diagonalized analytically. The Hamilto-nian, however, displays mirror symmetry. This

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114102

property allows us to separate the electron levelswith respect to the oscillator part and enables us tonumerically diagonalize the Hamiltonian [19,20].Analytical insights can be obtained in two limitingcases: large electronic separation and large elec-tron–boson coupling. Hamiltonian (1) has threeparts and three essential parameters: o; g; and v:We are going to work with dimensionless timeot-t: We will also set _ ¼ 1and o ¼ 1 withoutloss of generality. A systematic discussion ispossible just by choosing a hierarchy for theseparameters and performing suitable unitary trans-formations as follows. Let us consider the unitarytransformation [20]

U ¼ eS ¼ expðiLðqÞsyÞ ð2Þ

with LðqÞ a function of q; appropriately chosen, asshown below.

2.2. Large electronic separation (vb1; vbg)

We apply Eq. (2) to Hamiltonian (1) for L givenby the equation

tan ð2LðqÞÞ ¼ �v

gqð3Þ

and arrive at

H 0 ¼1

2ðp2 þ q2Þ þ

1

8

v2g2

ðv2 þ g2q2Þ2� sz

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ g2q2

p

þvg

4sy p

1

v2 þ g2q2þ

1

v2 þ g2q2p

� �: ð4Þ

For vb1; vbg the non-diagonal sy-term will besmall and it may be taken as a perturbation. Thus,the Hamiltonian is diagonal in the electronicsubspace and we have two separate effectivepotential sheets (sz ¼ 71) that are stronglyanharmonic. The second term in the Hamiltonianis sharply localized around q ¼ 0 and in this caseits contribution is negligible so, the lower/upperadiabatic sheet is given by

Vad ¼q2

28

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ g2q2

p: ð5Þ

The lower sheet may have either one minimum(if g2ov) or two minima and one maximum (ifg2 > v). The unitary transformation defined byEq. (3) transforms an initial state jF0S ¼ j1SwðqÞ;

where |1S is the first electronic state and wðqÞ is thevibrational wave function into

jF00S ¼ j1SwðqÞcosðLðqÞÞ � j2SwðqÞsinðLðqÞÞ: ð6Þ

Large coupling ðvb1; vbgÞ means

cosðLðqÞÞE� sinðLðqÞÞD1=ffiffiffi2

p: ð7Þ

Both electronic states will roughly have the sameoccupation probability, while the time evolutionwill be governed by anharmonic vibrationaldynamics generated by the adiabatic potential.The vibrational coordinate is moving in aneffective potential generated by averaging outthe motion of the electronic subsystem since theelectronic subsystem is moving faster than thevibrational subsystem (Fig. 1a). This justifies call-ing the effective potential sheets adiabatic poten-tial sheets. The non-diagonal term gives rise toelectronic transition between the two electronicsheets and it is small if vbg: It has a simpleapproximation given by

HND ¼g

2vpsy þ O

g

v

� �3

: ð8Þ

Therefore, neglecting HND we make an error ofthe order of magnitude of Oðg=vÞ:

2.3. Large coupling (gb1; gbv)

We may use the same transformation (2), whichgives rise to the same transformed Hamiltonian(4). Still, the sy term is small and is a perturbation.We have two effective potential sheets coupled bythe sy-term. We have two minima very wellrepresented by the first two terms in the expressionfor the effective potential sheets

V ¼q2

28

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ g2q2

pþ

1

8

v2g2

ðv2 þ g2q2Þ2: ð9Þ

The third term gives a sharply localized con-tribution around the origin. The time evolution ofthe states in the two sheets will be totally different.Suppose the initial state vector is the same as in thepreceding section. The transformed state vectorwill have the same expression as in Eq. (6).However, this time

cosðLðqÞÞE1; sinðLðqÞÞE0: ð10Þ

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114 103

The transformed state vector will remain in thesame electronic state and the dynamic evolution isstill governed by the anharmonic vibrationaldynamics of the transformed Hamiltonian. Thissuggests that the system remains trapped on oneelectronic level, this time the fast time evolution isgiven by the vibrational coordinate while theelectronic subsystem is the slow one (see Fig. 1b).We call this regime the anti-adiabatic regime, sinceusually, the adiabatic regime is when the electronmotion is the faster one, while the nuclei motion(i.e. vibrational) is the slow one.

3. Temporal evolution of a two-level electron

system interacting with a single mode boson field

Because there is no analytic solution of Eq. (1)to explore the time evolution of the Hamiltonian,we use first the time-dependent variational princi-ple for some trial functions [21–23] and then directnumerical integration of time-dependent Schro-dinger equation. For a trial function the optimalequations of motion for the given Hamiltonian (1)follow from ð_ ¼ 1Þ:

dZ t2

t1

L dt ¼ 0 ð11Þ

with L ¼ jCS� id= dtH jCS and the constraint/CjCS ¼ 1: When the above relation is applied togeneral state vectors free from artificial constraint,it is equivalent to the Schr .odinger equation.Choosing a necessarily limited set of trial func-tions, the time evolution equations that result maynot be in full agreement with the Schr .odingerequation. A clear level of the accuracy of theseequations does not yet exist. The equation ofmotion can be obtained in a straightforwardmanner as Euler–Lagrange equations.

Considering the electron–vibrational interac-tion, the first trial function is chosen to includethe displaced operator eigpsz and one photoncoherent or Glauber state [24]

jC1S ¼ eiðpq�bpÞe�igpsz j0Sðf1j1Sþ f2j2SÞ; ð12Þ

where p;b; f1; f2 are functions depending on time,|0S is the vacuum state for the quantum harmonicoscillator, |1S and |2S the electron levels. Thecoefficients f1 and f2 are complex valued functionswith squared modulus representing the probabilitythat the electron is on one level or the other.Subsequently, the dynamics of the functions p;b; f1and f2 obey the following equations:

’b ¼ p� 2igve�g2

ðf �1 f2 � f1 f �

2 Þ; ’p ¼ �b;

i ’f1 ¼ �gbf1 � ve�g2

f2; i ’f2 ¼ gbf2 � ve�g2

f1: ð13Þ

As a first remark, minimizing the functional/CjH jCS with the constraint /CjCS ¼ 1; wefind an approximation of the ground state energy:

E0 ¼1 � g2

2� ve�g2

: ð14Þ

0 2 4 6 8 10

-1.4

-1.0

-0.6

-0.2

0.2

0.6

1.0

1.4

<σz>

,<q>

(arb

itrar

y un

its)

time (a)

<σz>

<q>v=10;g=3.16

0 50 100-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

<σz>

,<q>

(arb

itrar

y un

its)

time(b)

<σz>

<q>v=0.2, g=2

Fig. 1. (a) Slow versus fast coordinate in the adiabatic limit.

The solid line is /szS; the average of sz and the dashed line is

/qS the average of q; v ¼ 10; g ¼ 3:16; (b) Slow versus fast

coordinate in the anti-adiabatic limit. The solid line is /szS; the

average of sz and the dashed line is /qS the average of q;v ¼ 0:2; g ¼ 2:

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114104

This value of E0 is the first approximation in v

(or g). For small g; this fact is more transparentwhen the electronic degree of freedom is elimi-nated [25]. A numerical exact solution presented inRef. [26] is based on Bargman representation [27],closely related to coherent states [24]. Secondly,the v factor is renormalized in Eq. (13) (theHolstein factor—renormalization of the electrontransfer integral) and the term �g2=2 is the smallpolaron energy [28].

In order to take into account the non-classicaleffects, the squeezed operator is introduced in thefollowing fashion [1,29–33]

jC2S ¼ eiðpq�bpÞe�igpszeirðpqþqpÞj0Sðf1j1Sþ f2j2SÞ;

ð15Þ

where r is still a real time-dependent variable andis a measure of the squeezing effect. The equationsof motion are

’b ¼ p� 2ivge�g2e4r

ðf �1 f2 � f1 f �

2 Þ; ’p ¼ �b;

i ’f1 ¼ �gbf1 � ve�g2e4r

f2; i ’f2 ¼ gbf2 � ve�g2e4r

f1;

2sinhð4rÞ þ 4vg2e4re�g2e4r

ðf �1 f2 � f1 f �

2 Þ ¼ 0: ð16Þ

The last equation in Eq. (16) is an integral ofmotion. It is easy to calculate the variances

Dp ¼e2rffiffiffi

2p ; Dq ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie�4r

2þ g2ð1 � ðjf1j2 � jf2j2Þ

2Þ

s:

ð17Þ

In addition, the minimum of /C2jH jC2S yieldsan approximation of the ground state [33]

E0 ¼coshð4rÞ

2�

g2

2� ve�g2e4r

; ð18Þ

where r is the solution of

2 sinhð4rÞ þ 4vg2e4re�g2e4r

¼ 0: ð19Þ

For v > 1 and g2=vE1 Eq. (19) gives a negativesolution for r which, according to Eq. (17),implies a reduction of momentum fluctuations(see Table 1).

A qualitative treatment of the dynamics ofEq. (13) is straightforward. After some algebraicmanipulations, it is easily observed that, duringthe motion, /CjH jCS and jf1j2 þ jf2j2 (energy andprobability) are conserved. To demonstrate com-

plete integrability of Eq. (13), it is necessary tofind the third integral of motion [34]. Theseconserved quantities lead to the conclusion thatb2 þ p2 and f1f �

2 þ f �1 f2 are bounded

jf1f �2 þ f �

1 f2jpjf1j2 þ jf2j2; ð20Þ

b2 þ p2p2E þ g2 � 1 þ 2v expð�g2Þðjf1j2 þ jf2j

2Þ;

ð21Þ

where E is the energy. The Eqs. (13) have only onestationary point

p ¼ 0; b ¼ 0; f1 ¼ 0; f2 ¼ 0: ð22Þ

Linear analysis around the stationary pointgives a matrix that has only imaginary eigenvalues.Therefore, the stationary point is elliptic. Scalingp;b; f1; f2 and gðp-p=k; b-b=k; f1-f1=k; f2-f2=k; g-kg; k real factor), Eq. (13) can be studiedaround the stationary point [35]. Thus, Eq. (13)has oscillatory behavior. This fact justifies thename ‘‘bistable molecule’’ or ‘‘toy’’ polaron. Thesame conclusions can be achieved for Eq. (16).

Numerical integration of the time-dependentSchrodinger equation was performed by using asymplectic integrator of order two [36]. Thepurpose of the integration scheme was to find afactorization for the evolution operator

Uðt; t þ dtÞ ¼ e�iðTþV Þ dt ð23Þ

Table 1

Comparison of ground state energy calculated with Eqs. (14)

and (18)

Ground

state

energy,

Eq. (18)

Ground

state

energy,

Eq. (14)

exp ð4rÞ r

g2 ¼ 4 �2.053 �1.555 0.2178 �0.3810

v ¼ 3

g2 ¼ 4 �2.514 �1.573 0.1745 �0.4365

v ¼ 3

g2 ¼ 4 �3.040 �1.592 0.1489 �0.4761

v ¼ 3

The values considering squeezing show that the ground state

energy is lowered significantly.

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114 105

into factors easier to handle. More explicitly itmeans the following decomposition can be made:

e�tðAþBÞ ¼ e�t1Ae�t2Be�t3A?e�tnA þ Oðtsþ1Þ: ð24Þ

Here, the parameters tj are proportional to t andthey are determined so that the product (24)expresses the sth order approximant. The simplestdecomposition is

e�tðAþBÞ ¼ e�tAe�tB þ Oðt2Þ ð25Þ

and is the first order in t. The second-orderfactorization applied to Eq. (23) is

e�i TþVð Þ dt ¼ e�ið1=2ÞV dte�iT dte�ið1=2ÞV dt þ Oðdt3Þ:

ð26Þ

According to this scheme we separated theaction of the kinetic part from the action ofpotential energy part. The kinetic part T may nowact on the Fourier transform of the wave function(in the Fourier transformed space—momentumspace—the momentum operator has a multiplica-tive form). The full propagation is given as

Cðt þ dtÞEe�iV dt=2FT½e�iT dtFT�1½e�iV dt=2CðtÞ;

ð27Þ

where FT and FT�1 are direct and inverse Fouriertransform, respectively. We checked the numericalvalidity of Eq. (26) for limiting, solvable cases ofHamiltonian (1) like-v ¼ 0 and/or g ¼ 0 and theresults reproduce the analytic solutions.

4. Discussion

Our study was focused on three sets of initialconditions as follows:

(A) The system is in the ground state of theoscillator and in the symmetric combinationof those two states;

(B) The system is in the ground state of theoscillator and in the anti-symmetric combi-nation of those two states;

(C) The system is in the ground state of theoscillator and on the first electronic state.

For this selection the symmetric combination(A) is associated with the lower potential energy

sheet and anti-symmetric combination (B) isassociated with upper potential energy sheet. Thecase (C) is essentially on the lower sheet, at themiddle of the distance between the bottoms ofthese potential energy sheets.

Numerical integration of the Schr .odingerequation (Eq. (27)) indicates squeezing in theadiabatic region only, more precisely, in theregion of transition from one minimum totwo minima for the lower sheet (g2=vE1). Insolid-state physics this is associated with localiza-tion/delocalization transition from a trappedelectron to quasifree one, which is usually accom-panied by an anomalous position fluctuation[8,15,26]. This manifests as squeezed states formomentum fluctuations revealed for the case (A).We can see a large amount of squeezing for twoof the sets of parameters v ¼ 4; g ¼ 2 and v ¼10; g ¼ 3:16 in Fig. 2. The squeezing for theother quadrature (q) is presented in the sameregion of parameters, but for case (B) wherethe electron is on the upper adiabatic (effective)potential energy (Fig. 3). In the case (C) the systemreaches momentum squeezed states to a lesserdegree than case (A) (Fig. 4). We continuedby doing a systematic study for g2=vE1: In thefirst case we kept v constant, v ¼ 10; andwe looked for the quadrature fluctuations forall three situations (A)–(C). The calculations weremade for g ¼ 1; 3; 3:16; 3:5; and 4. The resultsare shown in Figs. 5 and 6. For a better view of theresults we made temporal averages for ðDpÞ2 andðDqÞ2noted as /ðDpÞ2St and /ðDqÞ2St; respec-tively. As can be seen, the maximum productionof squeezing is when g2=vo1; i.e. the region withjust one minimum for the lower adiabatic poten-tial. The amount of squeezing is much less forthe case (C). Similar behavior is for another setof parameters: g ¼ 2 and v ¼ 3; 3:5; 4; 5; 6 (Fig. 7).Interesting is the case when g ¼ 1; where thereis a tiny squeezing and when the system is preparedaccording to (C) it changes its state of squeezingfrom one quadrature to the other and back(see Fig. 8). Large coupling region ðgbv; g > 1Þshows large fluctuations in both quadratures,mainly due to the fact that electron dynamics takeplace between those two deep and relatively faraway minima of the effective potential (for

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114106

example, the distance between this minima isroughly 2g).

Numerical calculations were also performed forEq. (16). We differentiated the last equation ofEq. (16) to transform it into ordinary differentialequations (ODE). Then it was numerically inte-grated by a fourth order Runge–Kutta algorithm.Considering that the system was prepared in theground state of the oscillator and with the electronlocated on state |1S, the initial conditions are: p ¼0; b ¼ �g; f1 ¼ 1; f2 ¼ 0 and r ¼ 0: We found thesame qualitative pattern: maximum of squeezing isobtained for g2=vo1 (Fig. 9). The dynamics ismuch more regular than the dynamics shown by

direct integration. We believe that the agreementwith the true dynamics is just for a short time withrespect to the oscillator frequency in our case withrespect to 1ðot51Þ: For this reason, we usedEq. (16) to locate the interesting region of squeez-ing. The integration of Eq. (16) is much faster thanintegration of the time-dependent Schr .odingerequation.

If the system with Hamiltonian described byEq. (1) is a two-level system interacting with anelectromagnetic field, the output field has changedfrequency (‘‘dressed’’ bosons). The dynamics canbe described using adiabatic potentials given byEq. (5). Using a Taylor expansion around the

0 400.30

0.35

0.40

0.45

0.50

0.55

0.60

(∆q)

2

(a)

v=5; g=2asym

0 20

20

400.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

(∆q)

2

(b)

v=10; g=3asym

time

time

Fig. 3. (a) Time evolution of ðDqÞ2; the dispersion of q squared.

As long as the value is below 0.5 we get squeezing. v ¼ 5; g ¼ 2

and the initial conditions are the case (B); (b) Same as (a) but

v ¼ 10; g ¼ 3.

0 20 400.1

0.2

0.3

0.4

0.5

0.6

0.7

(∆p)

2

(a)

v=4; g=2sym

0 20 400.1

0.2

0.3

0.4

0.5

0.6

(∆p)

2

time(b)

v=10; g=3.16sym

time

Fig. 2. (a) Time evolution of ðDpÞ2; the dispersion of p squared.

As long as the value is below 0.5 we get squeezing. v ¼ 4; g ¼ 2

and the initial conditions are the case (A); (b) Same as (a) but

v ¼ 10; g ¼ 3:16.

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114 107

origin, one can get ðg2=vo1Þ

Vad ¼ 8v þ 18g2

v

� �q27

1

8

g4

v3q4 þ O

g

v

� �6

: ð28Þ

In the first approximation the frequency of theeffective dynamics reads

o2eff ¼ 18

g2

v; ð29Þ

where upper sign is for the lower sheet and lowersign for the upper sheet. When g2=v > 1 we havetwo minima and the effective potential is given by

o2eff ¼ 1 �

v2

g4: ð30Þ

Accordingly, we have to look at this effectivedynamics to account for squeezing of the outputfield. Defining the conjugate quadratures in theusual way as

X1 ¼ qffiffiffiffiffiffiffiffioeff

p; X2 ¼ p=

ffiffiffiffiffiffiffiffioeff

pð31Þ

the condition for squeezing is

DX1o1ffiffiffi2

p ; DX2o1ffiffiffi2

p : ð32Þ

The results are shown in Figs. 10–12. Thedynamics look more mixed and the features getreversed. We get a much larger degree of squeezingfor X1; associated with the q coordinate, in the caseof symmetric initial condition (A), while the anti-

symmetric case (B) shows squeezing for X2: Againthe squeezing is much stronger for g2=vo1: Theresults can be explained by squeezing produced bya sudden change of frequency [5] plus thesqueezing produced by the quartic anharmonicityinduced by the adiabatic potential. In Figs. 13 and14 we have numerically integrated the Schr .odingerequation for Hamiltonian (1), with potentialenergy given up to fourth order by Eq. (28) andharmonic oscillator with frequency given byEq. (29). The initial conditions were the groundstate of the unperturbed harmonic oscillator of

1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.5

1.0

1.5

2.0

<(∆

p)2 > t

g

sym

< > = time ave.t

1.0 1.5 2.0 2.5 3.0 3.5 4.00.2

0.3

0.4

0.5

0.6

0.7

<(∆

q)2 > t

g

v=10< >

t=time average

v=10

asym

(a)

(b)

Fig. 5. (a) /ðDpÞ2St; the temporal averages of ðDpÞ2. The initial

conditions are those of case (A), v ¼ 10; (b) /ðDqÞ2St; the

temporal averages of ðDqÞ2. The initial conditions are those of

case (B), v ¼ 10:

0 20 40

0.3

0.4

0.5

0.6

0.7

0.8

(∆p)

2

case C

time

v=10; g=3.16

Fig. 4. Time evolution of ðDpÞ2; the dispersion of p squared. As

long as the value is below 0.5 we get squeezing. v ¼ 10; g ¼3:162 and the initial conditions are the case (C).

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114108

Eq. (1) (i.e. with o ¼ 1). One can see that thesolution given by the system with potential energyas in Eq. (28) up to the fourth-order approximates,quite well the full solution. This approximation isbetter for the upper potential sheet since thequartic perturbation has a smaller weight withrespect to the harmonic part of the potentialenergy. Suppose, we initially have a harmonicoscillator in the ground state with frequency o ¼ 1and suddenly change the frequency to o0 ¼ 1 þ D;D real and jDjo1: For tp0

ðDqÞ2 ¼ ðDpÞ2 ¼1

2: ð33Þ

For tX0 we then obtain

ðDqÞ2 ¼1

4ð1 þ DÞð1 þ DÞð1 þ cosð2ð1 þ DÞtÞÞ½

þ1

1 þ Dð1 � cosð2ð1 þ DÞtÞÞ

;

ðDpÞ2 ¼ð1 þ DÞ

4

1

1 þ Dð1 þ cosð2ð1 þ DÞtÞÞ

þð1 þ DÞð1 � cosð2ð1 þ DÞtÞÞ:

ð34Þ

The system shows squeezing for the quadraturesgiven by Eq. (31) with o ¼ 1 þ D: However, wemay easily check that ðDqÞ2p1=2 for D > 0 andðDpÞ2p1=2 for Do0: The squeezing is strongly

dependent on frequency change, the bigger thechange, the stronger the squeezing. This explainswhy there is smaller squeezing for g2=v > 1 and forthe strong coupling limit, since the effectivefrequency for these cases is given by Eq. (30),which is close to 1. On the other hand these are incomplete agreement with previous studies that theground state is actually a squeezed state [8,29,30–33]. The adiabatic potentials induce a renor-malization and subsequently a change of fre-quency for the boson field, which require asqueezed ground state. Let us recall that thesqueezed states are the unitary transformations

3.0 3.5 4.0 4.5 5.0 5.5 6.00.2

0.3

0.4

0.5

0.6

0.7

<(p)

2 > t

v

symg=2< >

t=time average

3.0 3.5 4.0 4.5 5.0 5.5 6.00.00

0.25

0.50

0.75

1.00

<(q)

2 > t

v

asymg=2< >

t=time average

∆∆

(a)

(b)

Fig. 7. (a) /ðDpÞ2St; the temporal averages of ðDpÞ2. The initial

conditions are those of case (A), g ¼ 2; (b) /ðDqÞ2St the

temporal averages of ðDqÞ2. The initial conditions are those of

case (B), g ¼ 2:

1.0 1.5 2.0 2.5 3.0 3.5 4.00.00

0.25

0.50

0.75

1.00

1.25

<(∆ p

)2 > t

g

case C v=10< >

t=time average

Fig. 6. /ðDpÞ2St; the temporal averages of ðDpÞ2. The initial

conditions are those of case (C); v ¼ 10:

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114 109

that transform the ground state of an oscillatorwith frequency o into the ground state of anoscillator with frequency o0 [1]

j0So0 ¼ UþS j0So; ð35Þ

where UþS is the Hermitian conjugate of the

operator given by (in the second quantization)

US ¼ eirðpqþqpÞ ¼ erðaa�aþaþÞ ð36Þ

and r is

r ¼ lno0

o

� �: ð37Þ

Thus, to move to the new ground state we haveto act by a unitary transformation given byEq. (36) with r given by Eq. (37). As we can nowsee, the new ground state is squeezed in momen-tum if o0oo; i.e. the system is on a lower adiabaticsheet, whereas for o0 > o the new state shows thesqueezing in position coordinate, i.e. the system ison the upper adiabatic sheet. The analysis can bemoved further to include the anharmonicityinduced by these adiabatic potentials. Contribu-tions given by adding the quartic terms (Eq. (28))to this analysis is to strengthen the squeezed statesin both cases. The positive quartic perturbationwill induce a squeezing in momentum, while thenegative perturbation will induce squeezing inposition.

It is worthwhile to mention that the systemgiven by Hamiltonian (1) can be implemented in ascheme including a two-level system generated bya single or double quantum well system ininteraction with a laser field. The input radiationcan be the light beam from a quantum cascadelaser with a wavelength of 4.5 mm [37] and theoutput target radiation with 8.4 mm wavelength[38]. The energy corresponding to 4.5 mm is275.5 meV and is considered the reference energyin Hamiltonian (1). We look for a value of4 for the parameter v. The energy separation be-tween electron levels is 2v ¼ 2:2 eV. A 2.83 nm

0 20 40 60 80 1000.45

0.46

0.47

0.48

0.49

0.50

0.51

0.52

0.53

0.54

0.55

(∆q)

2

(b)

0 20 40 60 80 1000.45

0.46

0.47

0.48

0.49

0.50

0.51

0.52

0.53

0.54

0.55

(∆p)

2

(a) time

time

case Cv=10; g=1

case Cv=10; g=1

Fig. 8. (a) Time evolution of ðDpÞ2. The initial conditions are

those of case (C), g ¼ 1; v ¼ 10: We can see a small amount of

squeezing; (b) Time evolution of ðDqÞ2. The initial conditions

are those of case (C), g ¼ 1; v ¼ 10:

3.0 3.5 4.0 4.5 5.0

0.1

0.3

0.5

0.7

<(∆

p)2 > t

v

g=2;

< >t=time average

variational time-dependent

Fig. 9. The temporal average of ðDpÞ2 in the time-dependent

variational scheme. The initial conditions are those of case (C);

g ¼ 2:

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114110

Al0.35Ga0.65As quantum well sandwiched betweenAlAs layers gives about 2.2 eV energy separationbetween the top most heavy hole subband and thelowest electron conduction subband (Fig. 15). Thevalue of g is 1.689 according to Eq. (29). For anyIII–V compound a widely accepted value forinterband momentum matrix element is given bythe relation 2p2=m0E23 eV: Assuming a unitaryoverlap of the electron and hole envelope wavefunction, the value of 1.689 for g is obtained by anelectromagnetic field with strength around 106 V/cm. The light is polarized in order to preventintraband coupling, i.e. the high-frequency electric

field lies in the layer plane. The advantages of thisscheme is that we can tailor parameters such asenergy splitting and strength of coupling with alarge degree of freedom. Also the system is quitesimple to implement. A large degree of squeezingmight be obtained by modulating the energysplitting (given by parameter v) and the electron–photon coupling (given by parameter g) byapplying an AC bias across the quantum wellsystem [2–7]. The applied electric field shifts theelectron and hole wave functions in oppositedirection, modulating the overlap integral andthus the electron–photon coupling g. In this way,

0 2 4 6 8 10 12 14 16 18 200.3

0.5

0.7

0.9

(X∆

1 )2

asymv=5, g=2

0 2 4 6 8 10 12 14 16 18 200.3

0.5

0.7

0.9

(X

2 )2

asymv=5, g=2

time

∆

(a)

(b) time

Fig. 11. (a) The temporal evolution of ðDX1Þ2. The initial

conditions are those of case (B), g ¼ 2; v ¼ 5; (b) The temporal

evolution of ðDX2Þ2. The initial conditions are those of case (B),

g ¼ 2; v ¼ 5:

0 10 20 30 40 50 60 70 80 90 100

0.1

0.3

0.5

0.7

0.9

(∆X

1 )2

sym

0 5 10 15 20 25 30 35 400.0

0.5

1.0

1.5

2.0

2.5

(∆X

2 )2

sym

(a)

(b)

time

v=10, g=3

v=10,g=3

time

Fig. 10. (a) The temporal evolution of ðDX1Þ2. The initial

conditions are those of case (A), g ¼ 3; v ¼ 10; (b) The temporal

evolution of ðDX2Þ2. The initial conditions are those of case (A),

g ¼ 3; v ¼ 10:

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114 111

the effective frequency is dependent on time andadds a new source of squeezing (see Eq. (29)). Onthe other hand one might modulate the laser pulse,i.e. modulating the coupling g and, according tothe same Eq. (29), we produce a frequency changein the system and consequently, another source ofsqueezing. One shortcoming of this scheme is thatit produces squeezed light at a different frequencythan that of the pump.

6. Concluding remarks

In summary, we studied the temporal evolutionof a two-level system interacting with a boson fieldin the adiabatic and anti-adiabatic limit. In the

adiabatic limit, the system shows a reduction ofmomentum or position for the bosonic subsystemdepending on what adiabatic potential the systemis in. Moreover, the frequency of the boson fieldchanges according to the curvature of the adia-batic potential. In the new variables the systemshows squeezing. The mechanism of squeezing iswell described by a sudden change of frequency.The system presents potential applications togenerate squeezed light in nanostructures likequantum wells.

0 2 4 6 8 10

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

<q

2 >

0 2 4 6 8 10 12 14

0.1

0.2

0.3

0.4

0.5

0.6

0.7

<p

2 >

exact solution approx. sol. oscillator

sym, v=5, g=2

exact solution approx. sol. oscillator

sym, v=5, g=2

time(a)

(b) time

Fig. 13. (a,b) /p2S and /q2S for the Hamiltonian (1) (solid

line), the oscillator with the potential energy given by Eq. (28)

(dashed line) and, the harmonic oscillator with the frequency

given by Eq. (29) (dotted line). The initial conditions are case

(A); v ¼ 2; g ¼ 5:

1 2.5 3 3.50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

<(∆X

1 )2 > t

g

3 5 60.4

0.5

0.6

0.7

0.8

<(∆ X

2 )2 > t

v

sym v=10

(a)

(b)

asym g=2

Fig. 12. (a) /ðDX1Þ2St; the temporal averages of ðDX1Þ

2. The

initial conditions are those of case (A), v ¼ 10; (b) /ðDX2Þ2St;

the temporal averages of ðDX2Þ2. The initial conditions are

those of case (B), g ¼ 2:

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114112

Acknowledgements

One of the authors (T.S.) is grateful to Dr. K.Clark for his comments on the manuscript. Thiswork was supported in part by NASA GrantNCC3-516, and by the Texas Advanced Technol-ogy under Grant No. 003594-00326-1999.

References

[1] M.O. Scully, M.S. Zubary, Quantum Optics, Cambridge

University Press, Cambridge, 1996: see also issues on

squeezing and non-classical light, Appl. Phys. B 55(3)

(1992); J. Mod. Opt. 34(6) (1987); J. Mod. Opt. 34(7)

(1987).

[2] C.O. Lo, Phys. Rev. A 45 (1992) 5262.

[3] X. Ma, W. Rhodes, Phys. Rev. A 39 (1989) 1941.

[4] G.S. Agarwal, S.A. Kumar, Phys. Rev. Lett. 67 (1991)

3665.

[5] J. Jansky, Y.Y. Yushin, Phys. Rev. A 39 (1989) 5445.

[6] J. Jansky, T. Kobayashi, An.V. Vinogradov, Opt. Com-

mun. 76 (1990) 30.

[7] J. Jansky, P. Adam, An.V. Vinogradov, T. Kobayashi,

Spectrochim. Acta 48A (1992) 31.

[8] D. Feinberg, E. Ciuchi, E. de Pasquale, Int. J. Mod. Phys.

B 4 (1990) 1317.

[9] A.H. Zewail, Science 242 (1988) 1645.

[10] G. Alber, P. Zoller, Phys. Rep. 199 (1991) 231.

[11] T. Holstein, Ann. Phys. 8 (1959) 343.

[12] S.B. Piepho, E.R. Krausz, P.N. Schatz, J. Am. Chem. Soc.

100 (1978) 2996 (see also K. Prassides (Ed.), Mixed

Valence Systems: Applications in Chemistry, Physics and

Biology, NATO ASI Series, Kluwer Academic Publishers,

Dordrecht, 1991).

[13] J.H. Ammeter, Nouv. J. Chim. 4 (1980) 631.

[14] O. Kahn, J.P. Launay, Chemtronics 3 (1988) 140.

[15] Y. Toyozawa, in: C.G. Kuper, G.D. Whitfield (Eds.),

Polarons, Excitons, Oliver and Boyd, Edinburgh, 1963.

[16] T.L. Estle, Phys. Rev. 176 (1968) 1056.

[17] I.B. Bersuker, V.Z. Polinger, Vibronic Interactions in

Molecules and Crystals, Springer, Berlin, 1989, p. 69.

[18] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89.

[19] R.L. Fulton, M.J. Gouterman, Chem. Phys. 53 (1961) 105.

[20] M. Wagner, J. Phys. A 17 (1984) 3409.

[21] P. Langhof, S.T. Epstein, M. Karplus, Rev. Mod. Phys. A

44 (1972) 602.

40 45 50 55 60 65 70

-2.0

-1.5

0.5

1.0AlAsAl Ga As

e1

hh2lh1hh1

2ν

nm

0.35 0.65AlAs

Ene

rgy

(eV

)

Fig. 15. Energy levels of 2.83 nm quantum well; e1 is the first

conduction subband, hh1 and hh2 are the first and second heavy

hole subbands, and lh1 is the first light hole subband.

0 2 4 6 8 10 12 140.2

0.3

0.4

0.5

0.6

0.7

<q

2 >

0 2 4 6 8 100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

<p2 >

exact solution approx. sol. oscillator

asym, v=5,g=2

exact solution approx.sol. oscillator

asym, v=5, g=2

time(a)

(b) time

Fig. 14. (a,b) /p2S and /q2S for the Hamiltonian (1) (solid

line), the oscillator with the potential energy given by Eq. (28)

(dashed line) and, the harmonic oscillator with the frequency

given by Eq. (29) (dotted line). The initial conditions are case

(B); v ¼ 2; g ¼ 5:

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114 113

[22] R. Steib, J.L. Schoendorff, H.J. Korsch, P. Reineker,

J. Lumin. 76 & 77 (1998) 595.

[23] J.K. Klauder, Ann. Phys. 11 (1960) 123.

[24] R.J. Glauber, Phys. Rev. 131 (1963) 2766.

[25] H.B. Shore, L.M. Sander, Phys. Rev. B 7 (1973) 4537;

P. Gosar, Physica A 85 (1976) 374.

[26] S. Weber, H. B .uttner, Solid State Commun. 56 (1985) 395.

[27] V. Bargman, Commun. Pure Appl. Math. 14 (1962) 187.

[28] J.B. Allen, R. Silbey, Chem. Phys. 43 (1979) 341.

[29] H. Zeng, Solid State Commun. 65 (1988) 731.

[30] Xian-Feng Peng, Phys. Stat. Sol. B 217 (2000) 887.

[31] X. Su, H. Zheng, Int. J. Mod. Phys. B 12 (1998) 2225.

[32] H. Morawitz, P. Reineker, V.Z. Kresin, J. Lumin. 76 & 77

(1998) 567.

[33] Xiao-feng Pang, J. Phys. Chem. Solids 62 (2001) 491.

[34] V.I. Arnold, Mathematical Methods of Classical

Mechanics, Springer, New York, Heidelberg, Berlin, 1978.

[35] R. Seydel, From Equilibrium to Chaos, Elsevier,

New York, Amsterdam, London, 1988.

[36] S.A. Chin, Phys. Lett. A 226 (1997) 344.

[37] J. Faist, F. Capasso, C. Sirtori, D.L. Sivco, A.L.

Hutchinson, A.Y. Cho, Appl. Phys. Lett. 66 (1995) 538.

[38] C. Sirtori, J. Faist, F. Capasso, D.L. Sivco, A.L.

Hutchinson, A.Y. Cho, Appl. Phys. Lett. 66 (1995) 3242.

T. Sandu et al. / Journal of Luminescence 101 (2003) 101–114114