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

1

Dynamic squeezing in a single-mode boson field interacting with

2-level system

T. Sandu a, *, V. Chihaia b, W. P. Kirk a a NanoFAB Center, University of Texas at Arlington, 500 S. Cooper Street, Arlington,

TX, 76012, USA b GZG Institute, Göttingen Univ.,Goldschmidtstr.1, Göttingen, 37077, Germany

PACS: 42.50d; 63.20.Kr

Keywords: squeezed states; coherent states; spin-boson model; electron-phonon

interaction; adiabatic coupling

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 1v

g2

≈ , 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.

DRAFT COPY

2

1. Introduction

The study of squeezed states is interesting for two reasons. It represents a frontier

of physics that gives us new understanding of quantum mechanics and also because of its

potential application for reducing noise in instrumentation and communication [1].

Optical noise is an important criterion in the performance of optical systems. For

example, the size of the fluctuations in the amplitude of light waves limits the ability to

measure or communicate small amplitude signals with a beam of light. In recent years,

much attention has been paid to the study and production of squeezed states of quantum

oscillators (the uncertainty for one observable is reduced to a value less than its ground state

value and for the conjugate observable the uncertainty is raised). Mainly, squeezed states of

light were considered [1]. The squeezed states are encountered in other (e.g. material)

quantum systems [2 - 4]. Squeezed wave packets may be generated by a sudden change of

oscillator frequency [5, 6] or by a pulsed excitation to another potential surface [7].

Moreover, the nonlinearity needed for the generation of squeezed states is typical of some

systems such as interacting electron-phonon systems [8] or vibronic ones [17]. Squeezing

of molecular wave packets has applications for investigating intramolecular processes in

real time. The success of such investigation depends on the generation of highly

localized wave packets [9, 10].

We consider the interaction of a two-level oscillator system as the simplest

nontrivial Hamiltonian model. In the basis of the two electronic states, 1> and 2>, the

Hamiltonian can be written as ( 1=ℏ ):

xz0

222

vgq2

qpH σσσω −−+= (1)

where σ0, σz, σx, are Pauli matrices and p and q are the oscillator coordinates.

The essential parameters are ω, g and v associated with frequency of the oscillator,

electron –oscillator coupling strength and separation of electron levels, respectively. This

model is called either the spin-boson model or the toy polaron model [11] and it is

encountered in many problems in chemistry, physics, and optics. Dynamic properties of

mixed valence systems generate electronic transfer 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 a barrier [12-14]. Such systems are coupled vibronically and are in

3

degenerate or quasi-degenerate electronic states. In solid-state physics this model can

describe situations such as an electronic point defect in a semiconductor [15] or the

interaction of a dipolar impurity with the crystal lattice [16]. The Hamiltonian is also

isomorphic with the simple pseudo Jahn-Teller system [17] and two-level system

interacting with a single-mode radiation field. Its simplification is the famous Jaynes-

Cummings model that has played a very important role in our understanding of the

interaction between radiation and matter in quantum optics [18]. The ground state of the

toy polaron shows squeezing of momentum fluctuation [8] in the adiabatic regime (region

of the parameters: v >>1 , g >1, and g2 /v ≅ 1). This region is a transition of electron from

a localized state to a delocalized one. In general, the transition is accompanied by

anomalous position fluctuations and squeezing of momentum. Moreover, it has been

suggested that squeezing is generated in a harmonic oscillator with time dependent

frequency [2-7]. Since the dynamics of the oscillator within (1) is strongly anharmonic in

the adiabatic regime (as we will see later on) and if we consider the anharmonicity as a

change in potential curvature, it is natural to look for squeezing in the adiabatic region of

parameters.

The aim of this paper is to look for the production of squeezed states of a bosonic

field in the adiabatic region of the parameters using a time-dependent variational

approach and by direct integration of the time-dependent Schrödinger equation. The

model and its properties in the adiabatic limit are discussed in Sec II. Sec. III gives a

brief summary of coherent states and squeezed states. Section IV is devoted to obtaining

the equations of motion for different trial functions with the time-dependent variational

principle (TDVP), which establishes a framework for the production of squeezed states.

The use of TDVP allows us to quickly locate the Hamiltonian parameters that are

interesting from the point of view of squeezing. Also a scheme for numerical integration

of the time-dependent Schrödinger equation is presented. The discussion and conclusions

are outlined in the last two sections. It is shown that the mechanism for squeezing is

basically the change of the adiabatic potential curvature due to the electron-boson

interaction. In addition, the inclusion of the higher-order non-linear terms describes with

a high degree of accuracy the time evolution of the Hamiltonian (1).

4

2. Theoretical model

2.1. Elimination of electron subsystem

Despite its simple structure the Hamiltonian (1) cannot be diagonalized

analytically. The Hamiltonian, however, displays mirror symmetry. This property allows

us to separate the electron levels with respect to the oscillator part and enables us to

numerically diagonalize the Hamiltonian [19,20]. Analytical insights can be obtained in

two limiting cases: large electronic separation and large electron-boson coupling. The

Hamiltonian (1) has three parts and three essential parameters: ω, g and, v. We are going

to work with dimensionless time tt →ω . We will also set 1=ℏ and 1=ω without loss

of generality. A systematic discussion is possible just by choosing a hierarchy for these

parameters and performing suitable unitary transformations as follows. Let us consider

the unitary transformation [20]

( )( )yS qiexpeU σΛ== (2)

with Λ(q) a function of q, appropriately chosen, as shown below.

2.2. Large electronic separation (v>>1, v>>g)

We apply (2) to Hamiltonian (1) for Λ given by the equation

( )( ) ,gq

vqant −=Λ2 (3)

and arrive at

( ) ( )

++

++

++−+

++=

pqgv

1

qgv

1p

4

vg

qgvqgv

gv

8

1qp

2

1'H

222222y

222z2222

2222

σ

σ (4)

For v>>1, v>>g the non-diagonal yσ - term will be small and it may be taken as a

perturbation. Thus the Hamiltonian is diagonal in the electronic subspace and we have

two separate effective potential sheets ( 1z ±=σ ) that are strongly anharmonic. The

second term in the Hamiltonian is sharply localized around 0q = and in this case its

contribution is negligible so, the lower/upper adiabatic sheet is given by

5

2222

ad qgv2

qV += ∓ (5)

The lower sheet may have either one minimum (if vg2 < ) or two minima and one

maximum (if vg2 > ). The unitary transformation defined by (3) transforms an initial

state ( )q10 χΦ = ,where 1 is the first electronic state and ( )qχ is the vibrational

wave function into

( ) ( )( ) ( ) ( )( )qsinqqcosq ΛχΛχΦ 210 −=' (6)

Large coupling (v>>1, v>>g) means

( )( ) ( )( ) 21qsinqcos ≅−≈ ΛΛ (7)

Both electronic states will roughly have the same occupation probability, while the time

evolution will be governed by anharmonic vibrational dynamics generated by the

adiabatic potential. The vibrational coordinate is moving in an effective potential

generated by averaging out the motion of the electronic subsystem since the electronic

subsystem is moving faster than the vibrational subsystem (Fig. 1a). This justifies calling

the effective potential sheets adiabatic potential sheets. The non-diagonal term gives rise

to electronic transition between the two electronic sheets and it is small if v>>g. It has a

simple approximation given by

3

yND v

gOp

v2

gH

+= σ (8)

Therefore, neglecting HND we make an error of the order of magnitude of

v

gO .

2.3. Large coupling (g>>1, g>>v)

We may use the same transformation (2), which gives rise to the same

transformed Hamiltonian (4). Still, the yσ term is small and is a perturbation. We have

two effective potential sheets coupled by the yσ -term. We have two minima very well

represented by the first two terms in the expression for the effective potential sheets

( )2222

22222

2

qgv

gv

8

1qgv

2

qV

+++= ∓ (9)

6

The third term gives a sharply localized contribution around the origin. The time

evolution of the states in the two sheets will be totally different. Suppose the initial state

vector is the same as in the preceding section. The transformed state vector will have the

same expression as in (6). However, this time

( )( ) ( )( ) 0qsin;1qcos ≈≈ ΛΛ (10)

The transformed state vector will remain in the same electronic state and the dynamic

evolution is still governed by the anharmonic vibrational dynamics of the transformed

Hamiltonian. This suggests that the system remains trapped on one electronic level, this

time the fast time evolution is given by the vibrational coordinate while the electronic

subsystem is the slow one (see Fig. 1b). We call this regime the anti-adiabatic regime,

since, usually, the adiabatic regime is when the electron motion 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 (1) to explore the time evolution of the

Hamiltonian, we use first the time-dependent variational principle for some trial functions

[21 - 23] and then direct numerical integration of time-dependent Schrodinger equation.

For a trial function the optimal equations of motion for the given Hamiltonian (1) follow

from (ℏ = 1):

0t

tLdtδ

2

1

=∫ , (11)

with id

L Hdt

= Ψ − Ψ and the constraint 1=ΨΨ . When the above relation is

applied to general state vectors free from artificial constraint, it is equivalent to the

Schrödinger equation. Choosing a necessarily limited set of trial functions, the time

evolution equations that result may not be in full agreement with the Schrödinger

equation. A clear level of the accuracy of these equations does not yet exist. The

equation of motion can be obtained in a straightforward manner as Euler-Lagrange

equations.

7

Considering the electron-vibrational interaction, the first trial function is chosen to

include the displaced operator zpgie σ and one photon coherent or Glauber state [24]

( ) ( )2f1f0eeΨ 21pgipqi

1z += − σβπ

. (12)

where π , β, f1 , f2 are functions depending on time,0 is the vacuum state for the

quantum harmonic oscillator, 1 and 2 the electron levels. The coefficients f1 and f2

are complex valued functions with squared modulus representing the probability that the

electron is on one level or the other. Subsequently, the dynamics of the functions π, β, f1

and f2 obey the following equations:

( )

−=−−=

−=−−=••

•−

•

.fevfgfi;fevfgfi

βπ;ffff evgi2πβ

1-

222-

11

*212

*1

g

22

2

gg ββ (13)

As a first remark, minimizing the functional >< ΨΨ H with the constraint <ΨΨ >=1,

we find an approximation of the ground state energy:

2g-2

0 ve2

g1E −−= . (14)

This value of E0 is the first approximation in v (or g). For small g, this fact is more

transparent when the electronic degree of freedom is eliminated [25]. A numerical exact

solution presented in [26] is based on Bargman representation [27], closely related to

coherent states [24]. Secondly, the v factor is renormalized in (13) (the Holstein factor-

renormalization of the electron transfer integral) and the term -g2/2 is the small polaron

energy [28].

In order to take into account the non-classical effects, the squeezed operator is

introduced in the following fashion [1, 29-33]

( ) ( ) ( )2f1f0eeeΨ 21pqqpripigpβqπi

2z += +−− σ , (15)

where r is still a real time-dependent variable and is a measure of the squeezing effect.

The equations of motion are:

( )

( )

2 4

2 4 2 4

2 4

1 2 1 2

1 2 2 11 2

2 41 2 1 2

2

2 sinh 4 4 0 .

r

r r

r

e

- e - e

er

g

g g

g

β π ivge f f f f ; π β

i f gβf ve f ; i f gβf ve f

( r) vg e e f f f f

• •− ∗ ∗

• •

− ∗ ∗

= − − = − = − − = −

⋅ + ⋅ + =

(16)

8

The last equation in (16) is an integral of motion. It is easy to calculate the variances:

( )

−−+==

− 22

2

2

12

r4r2

ff1g2

eq;

2

e∆p ∆ . (17)

In addition, the minimum of 22 ΨHΨ yields an approximation of the ground state[33]:

( ) r42e-2

0gve

2

g

2

r4coshE −−= , (18)

where r is the solution of

( ) 0eevg4r4sinh2r42er42 g =+ − . (19)

For v >1and g2 /v ≈ 1 equation (19) gives a negative solution for r which, according to

(17), implies a reduction of momentum fluctuations (see Table 1).

A qualitative treatment of the dynamics of (13) is straightforward. After some

algebraic manipulations, it is easily observed that, during the motion, >< ΨHΨ and

2

2

2

1 ff + (energy and probability) are conserved. To demonstrate complete integra-

bility of (13), it is necessary to find the third integral of motion [34]. These conserved

quantities lead to the conclusion that 22 πβ + and 2*

1*21 ffff + are bounded:

2

2

2

12*

1*21 ffffff +≤+ , (20)

( )( )2

2

2

12222 ffgexpv21gE2 +−+−+≤+ πβ , (21)

where E is the energy. The equations (13) have only one stationary point:

0f;0f;0;0 21 ==== βπ (22)

Linear analysis around the stationary point gives a matrix that has only imaginary

eigenvalues. Therefore the stationary point is elliptic. Scaling π, β, f1, f2 and g (π →π/k,

β→β/k, f1 → f1/k, f2 → f2 /k, g →kg, k real factor), eq. (13) can be studied around the

stationary point [35]. Thus eq. (13) has oscillatory behavior. This fact justifies the name

“bistable molecule” or “toy” polaron. The same conclusions can be achieved for (16).

Numerical integration of the time-dependent Schrodinger equation was performed

by using a symplectic integrator of order 2 [36]. The purpose of the integration scheme

was to find a factorization for the evolution operator

( ) ( ) tett,tU VTi δδ +−=+ (23)

9

into factors easier to handle. More explicitly it means the following decomposition can

be made:

( ) ( )1sAtAtBtAtBAt t0eeeee n321 +−−−−+− += … . (24)

Here the parameters tj are proportional to t and they are determined so that the product

(24) expresses the sth order approximant. The simplest decomposition is

( ) ( )2tBtABAt t0eee += −−+− (25)

and is the first order in t. The second order factorization applied to (23) is

( )

+=

−−

−+− 3

V2

1i

iTtV

2

1i

VTi t0t

eteete δδ

δδ

δ (26)

According to this scheme we separated the action of the kinetic part from the action of

potential energy part. The kinetic part T may now act on the Fourier transform of the

wave function (in the Fourier transformed space -momentum space- the momentum

operator has a multiplicative form). The full propagation is given as

( ) ( )

≈+ −−−−

teFTeFTett2/tiV1tiT2/tiV ΨδΨ δδδ (27)

where FT and FT -1 are direct and inverse Fourier transform, respectively. We checked

the numerical validity of (26) for limiting, solvable cases of Hamiltonian (1) like v = 0

and/or g = 0 and the results reproduce the analytic solutions.

4. Discussion

Our study was focused on three sets of initial conditions as follows

(A) - the system is in the ground state of the oscillator and in the symmetric combination

of those two states;

(B) – the system is in the ground state of the oscillator and in the anti-symmetric

combination of those two states;

(C) - the system is in the ground state of the oscillator 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) is associated with upper potential

energy sheet. The case (C) is essentially on the lower sheet, at the middle of the distance

between the bottoms of these potential energy sheets.

10

Numerical integration of the Schrödinger equation (eq. (27)) indicates squeezing

in the adiabatic region only, more precisely, in the region of transition from one minimum

to two minima for the lower sheet ( 1v

g2

≈ ). In solid-state physics this is associated with

localization/delocalization transition from a trapped electron to quasifree one, which is

usually accompanied by an anomalous position fluctuation [8, 15, 26]. This manifests as

squeezed states for momentum fluctuations revealed for the case (A). We can see a large

amount of squeezing for two of the sets of parameters v=4, g=2 and v=10, g=3.16 in Fig.

2. The squeezing for the other quadrature (q) is presented in the same region of

parameters, but for case (B) where the electron is on the upper adiabatic (effective)

potential energy (Fig. 3). In the case (C) the system reaches momentum squeezed states

to a lesser degree than case (A) (Fig. 4). We continued by doing a systematic study for

1v

g2

≈ . In the first case we kept v constant, v=10, and we looked for the quadrature

fluctuations for all three situations (A), (B), and (C). The calculations were made for

g=1, 3, 3.16, 3.5, and 4. The results are shown in Figs. 5 and 6. For a better view of the

results we made temporal averages for ( )2p∆ and ( )2q∆ noted as ( ) t2p >< ∆ and

( ) t2q >< ∆ , respectively. As can be seen, the maximum production of squeezing is when

1v

g2

< , i.e. the region with just one minimum for the lower adiabatic potential. The

amount of squeezing is much less for the case (C). Similar behavior is for another set of

parameters: g=2 and v=3, 3.5, 4, 5, 6 (Fig. 7). Interesting is the case when g=1, where

there is a tiny squeezing and when the system is prepared according to (C) it changes its

state of squeezing from one quadrature to the other and back (see Fig. 8). Large coupling

region (g>>v, g>1) shows large fluctuations in both quadratures, mainly due to the fact

that electron dynamics take place between those two deep and relatively far away minima

of the effective potential (for example, the distance between this minima is roughly 2g).

Numerical calculations were also performed for (16). We differentiated the last

equation of (16) to transform it into ordinary differential equations (ODE). Then it was

numerically integrated by a fourth order Runge-Kutta algorithm. Considering that the

system was prepared in the ground state of the oscillator and with the electron located on

11

state 1 , the initial conditions are: π =0; β = -g; f1 = 1; f2 = 0 and r = 0. We found the

same qualitative pattern: maximum of squeezing is obtained for 1v

g2

< (Fig. 9). The

dynamics is much more regular than the dynamics shown by direct integration. We

believe that the agreement with the true dynamics is just for a short time with respect to

the oscillator frequency in our case with respect to 1( 1t <<ω ). For this reason, we used

(16) to locate the interesting region of squeezing. The integration of (16) is much faster

than integration of the time-dependent Schrödinger equation.

If the system with Hamiltonian described by eq. (1) is a 2-level system interacting

with an electromagnetic field, the output field has changed frequency (“dressed” bosons).

The dynamics can be described using adiabatic potentials given by eq. (5). Using a

Taylor expansion around the origin, one can get ( 1v

g2

< )

64

3

42

2

ad v

g0q

v

g

8

1q

v

g1vV

+±

+= ∓∓ . (28)

In the first approximation the frequency of the effective dynamics reads

v

g1

22eff ∓=ω , (29)

where upper sign is for the lower sheet and lower sign for the upper sheet. When 1v

g2

>

we have two minima and the effective potential is given by

4

22eff

g

v1−=ω . (30)

Accordingly, we have to look at this effective dynamics to account for squeezing of the

output field. Defining the conjugate quadratures in the usual way as

eff2eff1 pXqX ωω == , (31)

the condition for squeezing is

2

1X

2

1X 21 << ∆∆ , . (32)

The results are shown in Figs. (10-12). The dynamics look more mixed and the features

get reversed. We get a much larger degree of squeezing for 1X , associated with the q

12

coordinate, in the case of symmetric initial condition (A), while the anti-symmetric case

(B) shows squeezing for 2X . Again the squeezing is much stronger for 1v

g2

< . The

results can be explained by squeezing produced by a sudden change of frequency [5] plus

the squeezing produced by the quartic anharmonicity induced by the adiabatic potential.

In Figs. 13 and 14 we have numerically integrated the Schrödinger equation for

Hamiltonian (1), with potential energy given up to fourth order by eq. (28) and harmonic

oscillator with frequency given by eq. (29). The initial conditions were the ground state

of the unperturbed harmonic oscillator of eq. (1) (i.e. with 1=ω ). One can see that the

solution given by the system with potential energy as in eq. (28) up to the fourth order

approximates quite well the full solution. This approximation is better for the upper

potential sheet since the quartic perturbation has a smaller weight with respect to the

harmonic part of the potential energy. Suppose we initially have a harmonic oscillator in

the ground state with frequency 1=ω and suddenly change the frequency to ∆ω += 1' ,

∆ real and 1<∆ . For 0t ≤

( ) ( )2

1pq 22 == ∆∆ . (33)

For 0t ≥ we then obtain

( ) ( ) ( ) ( )( )( ) ( )( )( )

( ) ( ) ( )( )( ) ( ) ( )( )( )

+−+++++

+=

+−+

+++++

=

t12cos11t12cos11

1

4

1p

t12cos11

1t12cos11

14

1q

2

2

∆∆∆∆

∆∆

∆∆

∆∆∆

∆ (34)

The system shows squeezing for the quadratures given by eq. (31) with ∆ω += 1 .

However, we may easily check that ( )2

1q 2 ≤∆ for 0>∆ and ( )

2

1p 2 ≤∆ for 0<∆ . The

squeezing is strongly dependent on frequency change, the bigger the change, the stronger

the squeezing. This explains why there is smaller squeezing for 1v

g2

> and for the

strong coupling limit, since the effective frequency for these cases is given by eq. (30),

which is close to 1. On the other hand these are in complete agreement with previous

studies that the ground state is actually a squeezed state [8, 29, 30-33]. The adiabatic

potentials induce a renormalization and subsequently a change of frequency for the boson

13

field, which require a squeezed ground state. Let’s recall that the squeezed states are the

unitary transformations that transform the ground state of an oscillator with frequency ω

into the ground state of an oscillator with frequency 'ω [1]

ωω0

sU0

'

+= (35)

where +s

U is the Hermitian conjugate of the operator given by

( ) ( )++−+ == aaaarqppqir ee

sU (in the second quantization), (36)

and r is

=ωω'

lnr (37)

Thus to move to the new ground state we have to act by a unitary transformation given by

eq. (36) with r given by eq. (37). As we can now see, the new ground state is squeezed in

momentum if ωω <' , i.e. the system is on a lower adiabatic sheet, whereas for ωω >' the

new state shows the squeezing in position coordinate, i.e. the system is on the upper

adiabatic sheet. The analysis can be moved further to include the anharmonicity induced

by these adiabatic potentials. Contributions given by adding the quartic terms (eq. (28))

to this analysis is to strengthen the squeezed states in both cases. The positive quartic

perturbation will induce a squeezing in momentum, while the negative perturbation will

induce squeezing in position.

It is worthwhile to mention that the system given by Hamiltonian (1) can be

implemented in a scheme including a 2-level system generated by a single or double

quantum well system in interaction with a laser field. The input radiation can be the light

beam from a quantum cascade laser with a wavelength of 4.5 µm [37] and the output

target radiation with 8.4 µm wavelength [38]. The energy corresponding to 4.5 µm is

275.5 meV and is considered the reference energy in Hamiltonian (1). We look for a

value of 4 for the parameter v. The energy separation between electron levels is 2v = 2.2

eV. A 2.83 nm Al0.35Ga0.65As quantum well sandwiched between AlAs layers gives

about 2.2 eV energy separation between the top most heavy hole subband and the lowest

electron conduction subband (Fig. 15). The value of g is 1.689 according to eq. (29). For

any III-V compound a widely accepted value for interband momentum matrix element is

14

given by the relation 23mp2 02 ≈ eV. Assuming a unitary overlap of the electron and

hole envelope wave function, the value of 1.689 for g is obtained by an electromagnetic

field with strength around 106 V/cm. The light is polarized in order to prevent intraband

coupling, i.e. the high frequency electric field lies in the layer plane. The advantages of

this scheme is that we can tailor parameters such as energy splitting and strength of

coupling with a large degree of freedom. Also the system is quite simple to implement.

A large degree of squeezing might be obtained by modulating the energy splitting (given

by parameter v) and the electron- photon coupling (given by parameter g) by applying an

ac bias across the quantum well system [2-7]. The applied electric field shifts the

electron and hole wave functions in opposite direction, modulating the overlap integral

and thus the electron-photon coupling g. In this way the effective frequency is dependent

on time and adds a new source of squeezing (see eq. (29)). On the other hand one might

modulate the laser pulse, i.e. modulating the coupling g and, according to the same

equation (29), we produce a frequency change in the system and consequently, another

source of squeezing. One shortcoming of this scheme is that it produces squeezed light at

a different frequency than that of the pump.

6. Concluding remarks

In summary, we studied the temporal evolution of a two-level system interacting

with a boson field in the adiabatic and anti-adiabatic limit. In the adiabatic limit, the

system shows a reduction of momentum or position for the bosonic subsystem depending

on what adiabatic potential the system is in. Moreover, the frequency of the boson field

changes according to the curvature of the adiabatic potential. In the new variables the

system shows squeezing. The mechanism of squeezing is well described by a sudden

change of frequency. The system presents potential applications to generate squeezed

light in nanostructures like quantum wells.

Acknowledgements

One of the authors (T. S.) is grateful to Dr. K. Clark for his comments on the

manuscript. This work was supported in part by NASA grant NCC3-516, and by the

Texas Advanced Technology under grant No. 003594-00326-1999

15

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16

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17

Figure Captions

Fig. 1.(a) Slow versus fast coordinate in the adiabatic limit. The solid line is <σz> , the

average of σz and the dashed line is <q> the average of q; v=10, g=3.16;

(b) Slow versus fast coordinate in the anti-adiabatic limit. The solid line is <σz> , the

average of σz and the dashed line is <q> the average of q; v=0.2, g=2.

Fig 2. (a) Time evolution of (∆p)2, the dispersion of p squared. As long as the value is

below 0.5 we get squeezing; v=4, g=2 and the initial conditions are the case (A).

(b) Same as (a) but v=10, g=3.16.

Fig 3.(a) Time evolution of (∆q)2 , the dispersion of q squared. As long as the value is

below 0.5 we get squeezing. v=5, g=2 and the initial conditions are the case (B).

(b) Same as (a) but v=10, g=3.

Fig 4. Time evolution of (∆p)2 , the dispersion of p squared. As long as the value is below

0.5 we get squeezing. v=10, g=3.162 and the initial conditions are the case (C).

Fig. 5. (a) <(∆p)2 > t ,the temporal averages of (∆p)2. The initial conditions are those of

case (A); v=10.

(b) <(∆q)2 > t ,the temporal averages of (∆q)2. The initial conditions are those of case (B);

v=10.

18

Fig. 6. <(∆p)2 > t ,the temporal averages of (∆p)2. The initial conditions are those of case

(C); v=10.

Fig. 7. (a) <(∆p)2 > t ,the temporal averages of (∆p)2. The initial conditions are those of

case (A); g=2.

(b) <(∆q)2 > t ,the temporal averages of (∆q)2. The initial conditions are those of case (B);

g=2.

Fig.8. (a) Time evolution of (∆p)2. The initial conditions are those of case (C); g=1,

v=10. We can see a small amount of squeezing.

(b) Time evolution of (∆q)2. The initial conditions are those of case (C); g=1, v=10.

Fig.9. The temporal average of (∆p)2 in the time-dependent variational scheme. The

initial conditions are those of case (C); g=2

Fig. 10. (a) The temporal evolution of (∆X1)2. The initial conditions are those of case (A);

g=3, v=10

(b) The temporal evolution of (∆X2)2. The initial conditions are those of case (A); g=3,

v=10.

Fig. 11. (a) The temporal evolution of (∆X1)2. The initial conditions are those of case (B);

g=2, v=5.

19

(b) The temporal evolution of (∆X2)2. The initial conditions are those of case (B); g=2,

v=5.

Fig. 12. (a) <(∆X1)2 > t , the temporal averages of (∆X1)

2. The initial conditions are those

of case (A); v=10;

(b) <(∆X2)2 > t , the temporal averages of (∆X2)

2. The initial conditions are those of case

(B); g=2.

Fig. 13. <p2> and <q2> 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 (29)(dotted line). The initial conditions are case (A); v=2,g=5.

Fig. 14. <p2> and <q2> 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 (29)(dotted line). The initial conditions are case (B); v=2, g=5.

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.

20

Tables

Table 1. Comparison of ground state energy calculated with eq. (14) and (18). The values

considering squeezing show that the ground state energy is lowered significantly.

21

Fig. 1

0 2 4 6 8 10

-1.4

-1.0

-0.6

-0.2

0.2

0.6

1.0

1.4

< σσ σσz>

,<q>

(arb

itra

ry u

nits

)

time

(a) <σσσσ

z>

<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

< σσ σσz>

,<q>

(arb

itrar

y u

nits

)

time

(b) <σσσσz

> <q>

v=0.2, g=2

22

Fig. 2

0 20 400.1

0.2

0.3

0.4

0.5

0.6

0.7

( ∆∆ ∆∆p)

2

time

(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

23

Fig. 3

0 20 400.30

0.35

0.40

0.45

0.50

0.55

0.60

( ∆∆ ∆∆q)

2

time

(a)v=5; g=2asym

0 20 400.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

( ∆∆ ∆∆q)

2

time

(b)v=10; g=3asym

24

Fig. 4

0 20 40

0.3

0.4

0.5

0.6

0.7

0.8

( ∆∆ ∆∆p)

2

time

case Cv=10; g=3.16

25

Fig. 5

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

(a) symv=10< >

t=time ave.

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

(b) asymv=10< >

t=time average

26

Fig. 6

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

27

Fig. 7

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

(a) 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

(b) asymg=2< >

t=time average

28

Fig. 8

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

time

(b)case Cv=10; g=1

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

time

(a)case Cv=10; g=1

29

Fig. 9

3.0 3.5 4.0 4.5 5.0

0.1

0.3

0.5

0.7

<(∆∆ ∆∆p

)2 > t

v

g=2;variational time-dependent< >

t=time average

30

Fig. 10

0 10 20 30 40 50 60 70 80 90 100

0.1

0.3

0.5

0.7

0.9

( ∆∆ ∆∆X

1 )2

time

(a) symv=10, g=3

0 5 10 15 20 25 30 35 400.0

0.5

1.0

1.5

2.0

2.5

( ∆∆ ∆∆X

2 )2

time

(b) symv=10,g=3

31

Fig. 11

0 2 4 6 8 10 12 14 16 18 200.3

0.5

0.7

0.9

( ∆∆ ∆∆X

1 )2

time

(a) 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

time

(b) asymv=5, g=2

32

Fig. 12

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

(a) sym v=10

3 5 60.4

0.5

0.6

0.7

0.8

<(∆∆ ∆∆X

2 )2 > t

v

(b)asym g=2

33

Fig. 13

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

<q2 >

time

exact solution approx. sol. oscillator

sym, v=5, g=2

0 2 4 6 8 10 12 14

0.1

0.2

0.3

0.4

0.5

0.6

0.7

<p2 >

time

exact solution approx. sol. oscillator

sym, v=5, g=2

34

Fig. 14

0 2 4 6 8 10 12 140.2

0.3

0.4

0.5

0.6

0.7

<q2 >

time

exact solution approx. sol. oscillator

asym, v=5, g=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 >

time

exact solution approx. sol. oscillator

asym, v=5, g=2

35

40 45 50 55 60 65 70

-2.0

-1.5

0.5

1.0AlAsAl

0.35Ga

0.65AsAlAs

e1

hh2lh1hh1

2 ν

Ene

rgy

(eV

)

nm

Fig. 15

36

Ground State

Energy, eq. (18)

Ground State

Energy, eq(14)

exp(4r) r

g2=4, v=3 -2.053 -1.555 0.2178 -0.3810

g2=4, v=4 -2.514 -1.573 0.1745 -0.4365

g2=4, v=5 -3.040 -1.592 0.1489 -0.4761

Table 1