Quantum theoretical approach to a near-®eld optical system

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Quantum theoretical approach to a near-eld optical system

K . K O BA Y A S H I* & M. O H TS U *

*ERATO Ohtsu Localized Photon Project, Japan Science and Technology Corporation, 687-1

Tsuruma, Machida, Tokyo 194-0004, Japan

Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology,

4259 Nagatsuta-cho, Midori, Yokohama, Kanagawa 226-8502, Japan

Key words. Probe tipsample interaction, projection operators, quantum theory,

virtual photon model, Yukawa-type interaction.

Summary

The paper proposes a quantum theoretical formulation of

an optical near-eld system based on the projection-

operator method. Special attention is paid to a nanometric

probe tipquantum mechanical sample system, whoseinteractions are essential for describing such phenomena

as atom guidance and manipulation, or local excitation of a

single quantum dot. The relationship to the virtual photon

model an intuitive model is discussed, and the latter's

empirical assumption of the Yukawa-type interaction

between probe and sample is justied theoretically. Several

applications of the theory are briey outlined.

1. Introduction

Recent experimental study has revealed the possibility of

local excitation of a sample by means of an optical near eld

beyond the diffraction limit, as well as high-resolution imaging.

The uorescence lifetime of a single molecule, for example, has

been measured and its quenching behaviour has been discussed

in terms of the sampleprobe distance and the aperture size

of the probe (Trautmann & Ambrose, 1997).

Photoluminescence spectroscopy of a single quantum dot

at low temperature has been also investigated to obtain

information about electronic-level structures, the spin

relaxation mechanism, and the switching effect (Flack et al.,

1996; Matsumoto et al., 1998; Saiki & Ohtsu, 1998; Saiki

et al., 1998; Toda et al., 1998). Such experimental results

exemplify the need for quantum theoretical treatment of

optical near-eld problems. Several authors, in fact, havesemi-classically analysed near-eld spectroscopic problems

(Girard et al., 1995; Cho et al., 1996; Kobayashi, 1998),

formulating electronic structures in terms of quantum

mechanics and optical near elds in terms of classical

electromagnetics.

An atom, as a small sample, may be considered as an

ultimate candidate for nano/atom photonics. To this end,

atom reection by a near eld induced over a prism surface

has been experimentally veried (Balykin et al., 1988;

Aminoff et al., 1993; Landragin et al., 1996), and atoms

have been guided through a micrometre-sized hollow bre by

means of blue-detuned optical near eld (Ito et al., 1996).

Atomic deposition with nanometric precision using very-

low-energy atom deection or manipulation has also

been proposed (Ito et al., 1997; Ito et al., 1998), and

experimental verication is expected after technical chal-

lenges such as bre probe improvement (Mononobe et al.,

1998; Yatsui et al., 1998) and manipulation and detection

schemes (Hori, 1993; Ohtsu et al., 1993; Klimov & Letokov,

1995; Ito & Ohtsu, 1998).

Atomsubstrate interactions via a propagating light eld,

being a more basic physics issue, have been investigated

both theoretically and experimentally. Classical antennatheory can predict some kinds of atomic behaviour, but the

existence of pure cavity quantum electrodynamic (QED)

effects is well known (Hinds, 1994). The energy shift in the

ground state of an atom is one example in which quantum

theory is essential for explaining the phenomenon.

From the above observation it is natural to quantum-

theoretically formulate a near-eld optical system consist-

ing of an atom or molecule and an optical near-eld probe.

Such a formulation would be preferable if it could be related

to an intuitive model giving clear insights into the near-eld

optical system as well as scanning image characteristics. It

may be possible to use a semi-classical approach in order topredict the quantum-mechanical states of matter, but we

will employ a full quantum-mechanical approach, for the

following reasons. Experimental progress shows that an

optical near-eld probe tip reaches 150 nm in diameter

(Ohtsu, 1995), and it is not clear to deal with such a

microscopic tip and an atom or a molecule as two isolated

systems when they are coupled via a macroscopic bath

system (incident light, substrate, and/or bre probe). Under

Journal of Microscopy, Vol. 194, Pt 2/3, May/June 1999, pp. 249254.

Received 6 December 1998; accepted 9 February 1999

q 1999 The Royal Microscopical Society 249

Correspondence to: K. Kobayashi. Tel: 81 42 788 6036; fax: 81 42 788

6031; e-mail: [email protected]


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this condition, the states of the total system should be

different from those of the two isolated systems, and it is

very important to consistently handle the interaction

between the two from the quantum-mechanical viewpoint,

although the existing theoretical approaches have neglected

to do so from the outset. Moreover there have been no attempts

to give any intuitive model from microscopic approaches.

We pay special attention to a nanometric probe tip

quantum mechanical sample system whose interactions areessential for describing such phenomena as those in atom

deection or manipulation experiments using optical near

elds. The purpose of this paper is to propose a unied and

systematic formulation of optical near-eld problems in order

to discuss the quantum-mechanical observables of such a

system. We will then discuss the relationship to the virtual

photon model (Hori, 1993; Kobayashi & Ohtsu, 1998; Ohtsu

& Hori, 1998) an intuitive model based on experimental

results which assumes the Yukawa-type interaction between

a probe tip and sample as an elementary process. Taking

advantage of the characteristics of the virtual photon model,

we will then describe a potential application of the theory.

The paper is organized as follows. The next section

outlines the method on which our approach is based. In

Section 3, Yukawa-type interaction between sample and

probe tip, which was empirically assumed in the virtual

photon model, is derived as an effective sampleprobe

interaction. One example of a case in which the virtual

photon model can offer physical insight is discussed in

Section 4, and we also give an example of the application of

our formulation on the basis of the virtual photon model.

Finally, Section 5 offers some concluding remarks.

2. Method of projection operatorsWe will employ the method of projection operators and

discuss the near-eld optical system, since we have to

describe the difference between coupled and isolated

quantum-mechanical states in a unied and systematic

way. It is very appropriate to extract an arbitrary number of

degrees of freedom from the many degrees of freedom of the

system, and to renormalize the effects of the other degrees of

freedom. Even if we are, for example, not able to obtain all of

the exact states for the total system, we can derive the exact

effective interaction in terms of a few states for the isolated

system, as will be shown below (see Eqs.(1011)).

First, let us briey outline the method of static projectionoperators (Hyuga & Ohtsubo, 1978; Kobayashi & Kohmura,

1984; Fulde, 1995). Let the total Hamiltonian H for a near-

eld optical system such as a near-eld optical microscope

and spectroscope be

H H0 V; 1

H0 HA HB Hbath; 2

where the Hamiltonians HA and HB describe the states of the

probe tip and sample, respectively, as isolated quantum-

mechanical systems. The Hamiltonian Hbath represents the

states of light and innite matter (substrate or bre probe),

while V denotes the interaction between the system A (or B)

and the bath system (Fig. 1). Steady states of H and H0 are

assumed in the following discussion, and the eigenvalues

and eigenstates ofH are written as El and jWli, respectively,

i.e.HjWli H0 VjWli EljWli: 3

Dening the static projection operators P and Q 1 P in

the usual manner as

jW1l i PjWli; jW

2l i QjWli; 4

P2

P; PQ QP 0; 5

we divide the eigenstates jWli into two groups, jW( 1)l i in P-

space and jW( 2)l i in Q-space. It is possible to formally express

QjW( 2)l i by PjW( 1)l i as

QjW

2

l i JEl

H0

1

QVPjW

1

l i; 6

J

1 El H01

Q V1

; 7

and the eigenstates jWli for the total Hamiltonian can then

be expressed in terms of the eigenstates in P-space as

jWli P QijWli JPjW1l i: 8

Since both of the states satisfy the normalization condition,

we can rewrite this as

jWli JPPJJP

1=2jW

1l i: 9

We thus obtain an effective operator Oeff in the P-space,

which is equivalent to an arbitrary operator O in the fullspace, corresponding to physical observables (Kobayashi &

Kohmura, 1984):

hWmj OjWli hW1m j OeffjW

1l i; 10

Oeff PJJP

1=2PJ

OJP PJJP1=2: 11

From the above Eqs (10 11), it follows that relevant

quantities such as the probe tipsample interaction and

transition dipole moments can be exactly expressed by using

a small number of bases in P-space after renormalizing the

effects from Q-space. This is one of the advantages of the

method, and is desirable from the computational viewpoint.

3. Effective sampleprobe interaction and virtual photon

model

On the basis of the above formalism, we now discuss the

virtual photon model (Hori, 1993; Kobayashi & Ohtsu,

1998; Ohtsu & Hori, 1998) a unique and intuitive

model in particular, its empirical assumption regarding

250 K . K O B A Y A S H I A N D M . O H T S U

q 1999 The Royal Microscopical Society, Journal of Microscopy, 194, 249254


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the probesample interaction in a fundamental process. The

model assumes virtual photons with nonzero mass (meff)

tunnelling between nanometric objects, similar to electron

tunnelling in scanning tunnelling microscopy (STM). Thismeans that the Yukawa-type interaction is responsible for

the interaction between a nanometric probe tip and sample.

In other words, the Yukawa potential (emeff t/t) is assumed

to be generated near the nanometric object (source) as a

result of the interaction between incident light and the

object, for both of the illumination and collection modes in

near-eld optical microscopy and spectroscopy. Owing to the

assumption, it can describe optical near elds localized on

the material surface without solving the Maxwell equation,

and a novel scheme for manipulating atoms by using the

optical near eld was successfully proposed and analysed

(Hori, 1993; Ohtsu et al., 1993).

Let us now adopt more specic Hamiltonians as

components of H0 in order to derive an explicit form of

the effective sampleprobe interaction. First we choose a

Hamiltonian for Ha (a A, B),

Ha

n

Enaenacna; 12

Ena "

2k2a

2mc

"2

2me

np

a

2or

"2

2me

np

b

2; 13

where the creation and annihilation operators are denoted

as cna and cna, and the relevant energy levels are designated

as En(a), and electric dipole transitions between those levels

are allowed. Note that this Hamiltonian corresponds to one

in which electrons with effective mass me and wave number

ka are conned in an innite well potential of size a(b). The

ground and excited states will be designated as jAi, jA*i,jBi, jB*i, for later use.

Next, let us describe Hbath in terms of excitons (B, B ),

photons (a, a), and their interactions. Excitons with

energy "Q represent an innite crystal lattice with

site number j. Photons with a wave vector k~ G~and energy

"q interact with the excitons as m~jD~ (r~j) in a multipolar

QED Hamiltonian (Craig & Thirunamachandran, 1984),

where m~and D~(r~j) are the dipole operator of matter and the

displacement eld operator at site j, respectively. As is well

known (Hopeld, 1958; Knoester & Mukamel, 1989;

Juzeliunas & Andrews, 1994), this Hamiltonian can be

diagonalized in terms of exciton-polariton modes (energy

"qk~n, creation/annihilation operator yk~v, yk~n as

Hbath

j

3i1

"Q Bji Bji ~k;~G

2l1

"q~k~Ca

~k~G;la~k~G;l

i

~mj~D ~rj 14

~k;n

"q~kny

~kny~kn: 15

Here the indices i and l label the three Cartesian

axes and the transverse polarization of light, respectively.The k~ summation extends over the rst Brillouin zone,

and G~ runs over the reciprocal lattice. The summation ofu

runs over all branches of exciton-polaritons. We will use the

ground state of Hbath, j0i as one component of the P-space.

It follows from Eq. (15) that photons and innite matter

may be a collection of harmonic oscillators. In this

representation, a near-eld optical probe tip interacts with

a nanometric sample via exciton-polariton mediation as


Fig. 1. Schematic illustration of how to compose a total Hamiltonian

H

for a near-eld optical system from three components, HA, H

B,

and Hbath. (a) The states of the probe tip, the sample, and the

bath system, as each isolated system, are described by the Hamilto-

nian HA, HB, and Hbath, respectively. Depending on the sample, HBmay contain the centre of mass motion as well as internal degrees

of freedom. Atom de ection system by the optical near eld

described by the total Hamiltonian H as an example. Incident

and deected atoms have the momenta "k~and "k~ H, respectively.

Q U A N T U M T H E O R E T I C A L A P P R O A C H T O N E A R - F I E L D O P T I C A L S Y S T E M 251


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follows:

V BaA

~ma~D~ra

BaA

~kn

i"V0

p

2p

"

rK

H~kn

ay~kn KH~kn

ay~kn

ba ba;

16

where KH

k~n(a) and K

H

*k~n(a) are the coupling coefcients

between the probe (sample) and exciton-polaritons, and baand ba are the creation and annihilation operators leading

to electric dipole transitions of the sampleprobe system,

relating to Eq. (12). It is worth reminding ourselves that

there is no direct interaction between sample and probe tip

in this Hamiltonian.

As explained in the previous section, one can exactly

express the relevant interaction by using a small number of

bases in the P-space, after renormalizing the effects from the

Q-space. We then choose a combination of the ve states,

jAi, jA*i, jBi, jB*i and j0i, as the P-space bases illustrated in

Fig. 2, and set jW~(1)l i jAi jB*i j0i, jW(1)m i jA*i jBi j0i, in

order to evaluate the effective sampleprobe interaction Veff(AB) from Eqs (10) and (11):

VeffAB W1m j VeffjW

1l

W

1m jPJ

JP

1=2PJ

VJP PJJP1=2jW1l

h i17

Noting that the operator J satises the equation

J; H0P E H0JP JE H0P

VJP JP VJP; 18

let us expand J as J P Sn1 g(n)

J(n), and obtain pertur-

bative solutions of J. We can then determine Veff (AB)

perturbatively in the order of V. Since there are no

contributions in the lowest order of V because PVP 0,

the nonzero contributions to Veff (AB) come from the second

order of V as

VeffAB hA

jhBjh0j

P VQ E0p E0Q

1 VP

P VE0p E0Q

1Q VP

jAijB

ij0i 19

4p2p3

n

d3k K

H

~knAKH

~knBq~kn Q0B

KH

~knBKH

~knAq~kn Q0A

4 5:

20

Here the summation of k~ is replaced by integrating k~ , and

H0 in the denominators with respect to the corresponding

eigenvalues. The excitation energies, "Q0(A) and "Q0(B),

are

"Q0A 3"2

2me

p

a

2; "Q0B

3"2

2me

p

b

2; 21

from Eq. (13). We can similarly show how hW( 1)l jVeffjW(1)m i

contribute to the effective sampleprobe interaction

Veff(BA).

On the basis of the dispersion relation of an exciton-

polariton (effective mass: mp, one of the solutions of the

dispersion relation: qk ) obtained from Eq. (15), we make the

following assumptions, which would be appropriate for

actual near-eld experiments:

i q~kn qk "

2mpk

2,

"

2mpk

2; qk < Q0;

ii KH~kn , const exp i~k~r:

Dening the ratio g (3mp/me)1/2 for exciton-polaritons

and electrons in the probe tip (sample), we apply theseassumptions to Eq. (20), and obtain the explicit function

form of

VeffAB VeffBA ~expigpt=b

t

expgpt=a

t

expigpt=a

t

expgpt=b

t;

t j~tA ~tBj: 22

This gives the effective sampleprobe interaction derived

from microscopic consideration, and the second and fourth

terms are the Yukawa function, which was empirically

assumed in the virtual photon model. The formula alsoshows that optical near eld is effectively localized near the

sampleprobe system, and that the decay length is

proportional to the inverse of the probe (sample) size and

consistent with the experimental results (Saiki et al., 1996).

In addition, we can infer the so-called resonance condition

that a similar size gives maximum coupling between sample

and probe tip, as can be determined by a simple estimation

based on the static dipole model.

Fig. 2. P- and Q-space spanned by bases each of which corresponds

to an eigenstate of HA, HB, and Hbath. The P-space is depicted by

hatching.




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4. Practical implications of the virtual photon model and

application of our formulation

Since the empirical assumption of the virtual photon model

was justied on the basis of microscopic theory in the

previous section, we can take advantage of the character-

istics of the model to develop an application of our

formulation. We have, so far, emphasized the quantum-

mechanical aspects of the optical near-eld systems. The

model also predicts the fundamental performance of near-

eld optical microscopy much faster than the conventional

approach in computation time, that is, the intensity

distribution in the near-eld region of a subwavelength

metallic circular aperture irradiated by the propagating

light. The resolution of the microscope was also clearly

explained by the model. In addition, we can easily handle

more realistic probesample systems, and discuss the

dependence of detected signals on the probe shape, or on

the taper angle v (see Fig. 3). Let the probe be a set of

dielectric spheres of different diameters that approximates a

conical shape. From Eq. (22), the Yukawa potential is

generated between two arbitrary points on the surfaces ofthe probe sphere pi and sample sphere sj. Thus pickup signal

I can be written as

I~probe

i

samplej

dj

dr1

pi

dt2Vefft12

2; t12 j~t1 ~t2j:

23

If the so-called resonance condition is satised, the pickup

signal becomes proportional to exp[const/sin(v/2)].

Turning to the quantum-mechanical system again, we

discuss the possibility of applying the formulation to atom

deection and manipulation with a sharpened bre probe.

These techniques are unique and will be essential forcarrying an atom to a desired point on a substrate with high

spatial accuracy far beyond the diffraction limit. As shown

in Fig. 1, suppose that an incident atom with momentum

"k~ is deected into momentum "k~ H by an optical near eld

Veff. Then the differential cross-section or angular distribu-

tion of the deected atom is given by

dj

dQ~

M

2p"2~k Hh Veff ~k i

2 24

where M is the mass of the atom. This kind of evaluation

will be required in the design of atom guidance and

manipulation systems using the optical near eld, as well asin the discussion of basic experimental results.

One implication related to cavity QED problems is that the

transition dipole moment of an atom or a molecule m~0 is

changed by the inuence of the probe tip and optical near

eld, that is, the difference between the isolated and coupled

states. Perturbative evaluation of Eqs (10) and (11) with

O m~ shows that the renormalized transition dipole

moment m~eff can be categorized into two classes: one directly

renormalized into m~0, and the other not renormalized into m~0,

owing to transitions between higher energy levels. It is

interesting to estimate these effects numerically.

5. Conclusion

We have developed a microscopic theory for an optical near-eld system, using a quantum-mechanical technique

known as the projection-operator method. It was shown

that the Yukawa-type interaction is involved in effective

probe tipsample interaction, indicating that the empirical

assumption of the virtual photon model is justied. The key

points are that a probe tip exists near the sample, and that

the electron energies in the probe tip or sample are inversely

proportional to the square of its size, owing to the


Fig. 3. Fibre probe tip with taper angle v as a collection of dielectric

spheres with different radii. The i-th and j-th spheres of the probe

and sample are denoted as pi and sj, respectively. The probe sample

separation is shown as d.

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connement effect. We discussed a few applications of the

theory, taking advantage of the characteristics of the virtual

photon model. In future, we need to examine in detail the

assumptions and approximations used, as well as what kind

of normal modes should be included for optical near-eld

problems, in order to make the formulation more rigid.

Acknowledgements

The authors gratefully acknowledge fruitful conversations

with H. Hori of Yamanashi University and H. Ito of Tokyo

Institute of Technology. K.K. also appreciates valuable

suggestions from T. Kohmura of University of Tsukuba.

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