Dipole nanolasers: A study of their quantum properties

A. S. Rosenthal and Talal GhannamDepartment of Physics, Western Michigan University, Kalamazoo, Michigan 49008, USA

�Received 5 October 2008; published 23 April 2009�

In this paper we undertake a theoretical study of the quantum properties of a dipole nanolaser proposed byProtsenko et al. �Phys. Rev. A 71, 063812 �2005��. The dynamics of the system is studied in both density-matrix and quantum Langevin formulations, with attention directed to noise sources. The result of the analysisindicates that in the most directly accessible parameter regime, photons will be emitted in a broad spectraldistribution and the time delay between consecutive emissions of single photons exceeds the coherence time.Another parameter regime exists within which the emitted light more closely resembles conventional laserlight.

DOI: 10.1103/PhysRevA.79.043824 PACS number�s�: 42.55.Ah, 42.50.Ct

I. INTRODUCTION

During the past few years the need for small optical ele-ments to replace the current electronic ones has becomemore and more pressing. One essential element is a verysmall laser that can generate light efficiently and reliably.

In this paper we carry out a theoretical and computationalinvestigation of a dipole nanolaser �DNL�, a device pro-posed by Protsenko et al. �1�. The device under study con-sists of a two-level system �TLS� such as a quantum dot,whose electronic oscillations act, through their near fields, topump the dipole plasma modes of a metallic nanoparticle�abbreviated NP in this work�. Far-field radiation is gener-ated by coherent contributions of both the quantum dot andplasma oscillations. We describe this device in detail in thispaper. It is probably the smallest device that can conceivablybe produced with current or foreseeable technology that de-serves the name “laser.” A schematic drawing of the device isshown in Fig. 1.

Many different configurations of nanolasers have beenproposed in the literature such as photonic crystals �2�, sur-face polariton lasing �3�, quantum dots �4�, and nanowirelasers �5�. All these proposed nanolasers require a cavitywhich restricts their sizes to the wavelength of the light. TheDNL requires no such cavity and its size can therefore beconsiderably smaller, on the order of V�0.1 �m3.

In the paper introducing the idea of the DNL, Protsenko etal. �1� showed that the equations describing the plasmon os-cillations of the NP induced by the TLS map isomorphicallyonto the standard equations describing an electromagneticfield in a laser cavity produced by an active medium. The

analog of the active medium is the TLS and the analog of thelaser cavity field is the plasmon amplitude �6�. This workexamined a number of conditions necessary to produce plas-mon “lasing” and found that most of these were at, or near,the limits of what is currently technologically feasible. Leftuninvestigated were the properties of the light such a devicewould produce.

Despite the fact that it is radically different in size andoperational features from ordinary lasers, it will be shownthat the spectral width of the light emitted by a DNL isreduced with increasing amplitude, just as in a standard laser.Another prediction of this paper is that there are two princi-pal parameter regimes, one corresponding to the familiarcase of long “cavity” lifetimes �i.e., plasmon lifetime is longcompared to TLS lifetime� and the other to the opposite situ-ation where the TLS lifetime is long compared to that of theplasmon. It is the second parameter regime that is mostreadily attained with existing materials.

In Sec. II we establish the formalism and derive the equa-tions of motion of a DNL using the density-matrix andLangevin formalisms. In Sec. III we investigate thresholdconditions and steady-state operation. The core of this paperis contained in Sec. IV, where the spectral properties of thelight emitted by a DNL are investigated. Conclusions aregiven in Sec. VI.

II. EQUATIONS OF MOTION

The Hamiltonian is given by

FIG. 1. �Color online� The DNLconfiguration.

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H = ��0aH† aH +

��2

2�H,z + V�t� + Hres,0 + Hirrev. �1�

Here aH denotes the plasmon amplitude operator for a me-tallic nanoparticle; �H,� and �H,z denote transition and popu-lation operators for the TLS; V is the dipole-dipole interac-tion between TLS and NP; Hres,0 is the self-energy of thereservoirs to which the system is coupled; and Hirrev de-scribes coupling between these reservoirs and the DNL.

We have

V�t� = − E� · �̂2 = ��int��−�t�a†�t� + �+�t�a�t�� ,

�int =1

4��m��0 · �2 − 3��0 · r����2 · r��

Rq3 � ,

Hres,0 = ��j

� jAj†Aj + Cj

†Cj ,

Hirrev = ��j

� j�Aja† + Aj

†a� + � j,2�Cj�+ + �−Cj†� . �2�

�int is the coupling strength; �0 are �2 are the dipole matrixelements for NP and TLS, respectively. The A’s represent thecombined effects of electromagnetic radiation and NP ther-mal phonons, while the C’s represent the combined effects ofelectromagnetic radiation, thermal noise, and electronicpump noise contributing to decay and dephasing of the TLS.The decay constant describes the rate of spontaneous emis-sion for the NP and also includes damping due to othersources such as Ohmic losses, while 2 describes decay ofthe TLS induced by the C reservoir.

The final slowly varying operator equations of motiontake the form

d

dtA = −

2A�t� − i�int− + FA�t� ,

d

dt−�t� = −

c

2−�t� + i�intzA + ei��−�2�tFC,−�t� ,

d

dtz = − pz + 2i�int�−A† − +A� − 2 + FC,z�t� +

�p

2,

�3�

where the operators denoted by upper case letters are theslowly varying terms of the a and � operators. The noiseoperators FA�t�, FC,−�t�, and FC,z�t� have bilinear expectationvalues given in the Appendix. �p represents the pump rate ofthe TLS. p=c+�p /2 and c=2�2n̄C+1�. n̄C is the averagenumber of quanta in the C reservoir.

III. THRESHOLD AND STEADY-STATE CONDITIONS

We can find the threshold conditions for the system bymaking the semiclassical approximation of factorizing ex-pectation values. This is equivalent to assuming that the NPplasmons are in a coherent state.

The steady-state conditions are found to be

z0 � �p

2− 2�

p + 4GN0=

2G, �4�

A0 ��� 1

2 �p

2− 2� −

p

4G� , �5�

−0 � i2�int

Cz0A0. �6�

In the above equations, a zero subscript denotes a steady-state value. The pumping threshold condition is given by thefollowing:

�p �2C + 4G2

2G − � , �7�

where G=2�int2 /C. N0�A0

�A0 is the plasmon number.It can be easily shown that requiring the threshold value

of z0 to be less than 1 implies

�2 c

4. �8�

Equation �8� is the lasing condition. All the above conditions,though phrased in slightly different language, are consistentwith the work of Protsenko et al. �1�.

IV. INTENSITY SPECTRUM

In determining the intensity spectrum we will take twoapproaches. In the first we follow the usual laser model inwhich the phase decay of the TLS coherence is large: 2�. In the second approach, corresponding more closely toconventional materials and technology, we assume the oppo-site condition �2.

A. Case 1

When 2�, both � and z decay rapidly compared tothe plasmon field and can be assumed to follow this field ontime scales long compared to 2

−1. This allows us to make an

adiabatic approximation in Eq. �3�, setting ̇��t�= ̇z�t�=0,from which it follows that

− =2i�

czA +

2

cFc,−�t�, + = −

2i�

czA

† +2

cFc,+�t� ,

z =2i�

p�−A† − +A� +

�

p+ Fz�t�, where � =

�p

2− 2.

This leads to the following equation for the plasmon ampli-tude:

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dA

dt= �−

2+

2�2�� + Fz�t��cpMK

�A −i8�3Fc,+�t�

pc2MK

AA

−2i�

KcFc,−�t� + Fa�t� , �9�

where M =1+ �4�2 /cp�A†A and K=1+ �4�2 /cpM�A†A.To calculate the intensity spectrum we assume that we are

far above threshold, that we can neglect the fluctuations inthe amplitude inasmuch as this quantity is constrained tofluctuate about its steady-state value, while the phase canchange freely �7�. We therefore consider only the time de-pendence in the phase. The plasmon field operator A�t� canbe written classically as A�t�=A0e−i�, where A0 is the steady-state value of A given in Eq. �5� and � is the phase diffusionconstant. The intensity is then given by the following expres-sion:

�A�t�†A�0�� = N0e−�1/2�����t��2� = N0e−�D���� = N0e−�t,

�10�

where

�D���� =1

2lim

�t→0

�����t��2��t

.

The spectral width is found to be

� =1

4N0 �n̄a +

1

2� +

�2

c� . �11�

We see that the spectral width of the intensity spectrum � forthe NP is inversely proportional to the plasmon number N0 asknown from standard laser theory, and contains a term in-versely proportional to c. This latter term represents thedecrease in NP lifetime due its coupling to a decaying TLS.This means that as the decay rate from the excited states ofthe TLS increases the spectral width of the NP becomes nar-rower, which is counterintuitive. To understand this behaviorwe go back to equations of motion �3�. The width of thespectrum of the NP arises from spontaneous decay of the NP.This decay can happen in many different ways. The NP plas-mons are coupled to the NP reservoir and can spontaneouslydecay into it, or the decay may be due to the coupling to theTLS: the plasmons can send a virtual photon to the TLS,which in turn decays to the surrounding reservoir, or the TLScould return the virtual photon back to the NP, which in turndecays back into its reservoir, and so on. These processeslead to the broadening of the NP spectrum.

Equation �3� implies that the NP operator A�t� is propor-tional to dipole moment of the TLS −:

d

dtA = −

2A�t� − i�int− − + FA�t� .

But in the adiabatic approximation − itself is proportionalto the inverse of c �larger c implies more population in theground state and therefore decreased transition dipole mo-ment�. Thus when c increases, the dipole moment of theTLS − decreases, which in turn reduces the coupling be-tween the TLS and the NP, which in turn decreases the line-width of the NP.

B. Case 2

If �2 we cannot eliminate the time derivatives of the’s, but we can eliminate the time derivative of the plasmonfield amplitude A�t�. This results in a set of nonlinear equa-tions which can be solved numerically but not analytically.To increase insight into the properties of the solution, wesuppose below that the population inversion of the TLS z isconstant. This can be justified if pumping of the TLS is suf-ficiently fast. Fixing the inversion allows us to solve for thespectrum exactly using the same phase diffusion technique asabove. The spectral width is found to be

� =1

8N0�2n̄a + 1� . �12�

By comparing to Eq. �11� we find that this spectrum is thesame as the first terms in the spectrum of the first case. Thedifference is that the noise term coming from the coupling tothe TLS is missing because the plasmon lifetime is so muchshorter than the TLS lifetime.

V. RESULTS AND DISCUSSION

Assuming the TLS is a quantum dot, the dipole momentof the TLS is on the order of �2=ed= �1.6�10−19��10�10−9��10−27 C m. The induced dipole moment of the NPis given by

�0 = − 2�2� a

Rq�3 �r − �m

��r + 2�m�� . �13�

At the plasmon resonance frequency, the quantity in bracketsdiverges. For silver NPs in a silicon matrix, this happens at afrequency �c of about 4.275�1015 Hz. To ensure the over-lapping of the NP and TLS near fields and that the dipoleapproximation is still valid, a restriction on the distance be-tween the two elements is required. This restriction can begiven by the following equation �see Ref. �1��:

rcr − �a + r2� � max�2r2,2a� , �14�

where a and r2 are the diameters of the NP and the TLS,respectively. The distance between the two elements Rqshould be equal or bigger than the critical value rcr.

The spontaneous decay of the TLS can be calculated asfollows �8�:

2 =�3�2

2

3��0�c3 + �7.6 � 108 s−1 K−1�T , �15�

where T is the operating temperature. In this work, T lies inthe range of 30–100 K so that radiative damping of the TLSdominates nonradiative damping. We calculated a value of on the order of 1011 s−1.

The NP decay time can be taken to be around 1014 s−1

�9�, and it can be approximated by the following equation:

=2

3�0

2 k3

��0+

�02

��T0. �16�

k is the wave vector corresponding to radiation in the sur-rounding material, and �T0 is the polarizability of the nano-particle and is given by �10�

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�T0 =4

3�a3��ip +

��rp − 1�2

�ip� . �17�

The values of the permittivities are calculated by interpolat-ing data taken from �11�.

The decay constant for the TLS and the NP given by theabove equations have the following values: �1014 s−1 and2�1011 s−1. It is clear that the condition for adiabatic ap-proximation 2� used in case 1, the standard laser theory,is not valid for silver particles in a silicon matrix. Othersimple materials such as gold have optical properties that donot change this conclusion.

A. Intensity spectrum

The photon intensity spectrum is formally defined as

S1��,r� = limT→�

1

2��

r/c

T+r/c

d�ei���Etot− �r,t�Etot

+ �r,t + ��� .

�18�

Using the parameters above, this spectrum can be evaluatedto be

S1��� =A0

2

2��INP��� + ITLS���L2� 1

�� − �0�2 + �2� ,

�19�

where

INP = � �02

4��0�2c2r��0 �

r

r� �

r

r�2

,

ITLS = � �02

4��0�2c2r��2 �

r

r� �

r

r�2

.

The I’s are the intensities of light emitted from the uncoupledTLS and NP separately, and

L =2��

cp + 8�2N0=

2

��20�

is a constant, computed from the solution to the equations ofmotion, depending on the strength of the coupling of thesystem and on the decay of the NP �i.e., �.

The spectral width � is given by Eq. �11� for case 1,

� =1

4N0 �n̄a +

1

2� +

�2

c�

and by Eq. �12� for case 2,

� =�2n̄a + 1�

8N0 .

The system most closely resembling a typical laser wouldhave parameters appropriate to case: 2�. While this doesnot correspond to metal NPs in a silicon matrix, it mightconceivably be engineered by using kinds of materials dif-ferent from those we have been considering. We choosesome arbitrary numbers for and 2. For acceptable pumprates and for n̄c=0, we plot the spectrum in Fig. 2 for case 1with the following parameters: n̄a=0 and n̄c=0, �=1013 s−1, Rq=50 nm, �p=1013 s−1, a=7 nm, 2=1013 s−1, and =1012 s−1.

FIG. 2. �Color online� Spectrum of light emitted from theTLS-NP system vertical axis in units of �V s /m�2 and the electricfields are evaluated 1 �m from the TLS-NP system.

FIG. 3. �Color online� The reduced spectral width of the lightemitted by the system for case 1 when nc=12. Units are same asthose in Fig. 2.

FIG. 4. �Color online� The relation of the plasmon number N0 tothe lifetime of the NP in units of inverse seconds.

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Here the number of plasmons excited, N0, is 5, and thelight emitted from NP is much bigger than that from theTLS, INP� ITLSL2, in agreement with the work of Protsenkoet al. �1�. The linewidth is about the same as for the NP, ��=1012. The factor 1 /N0 tends to decrease the width,while the factor �2 /c tends to increase it. The net result islittle change.

By increasing the pumping of the TLS, we see a reductionin the linewidth. Also increasing the noise quanta of the TLSwill reduce the linewidth even more. For example, increasingthe number of noise quanta around the TLS to n̄c=12 willreduce the width almost 100 times ��1010 as shown in Fig.3.

In this case the number of plasmons excited is about 45,which helps reducing the width along with c. The rate ofphoton emission is �N0=1010�45�1012 photons s−1. Wenote that corrections to the dipole approximation for standardmaterials cannot bring us into case 1 because those correc-tions not only increase �0 but also greatly increase ��0

2.The differences between case 1 and case 2 have to do with

two effects: the increased value of the plasmon width �al-most entirely radiative� and the corresponding decrease inplasmon number, as shown in Fig. 4. The pumping also de-pends sensitively on as seen in Fig. 5.

For the second case, using parameters suitable to metalnanoparticles such as silver and gold in a silicon matrix, weobtain the typical distribution shown in Fig. 6. As in case 1,the width can be reduced by increasing the pumping. But to

reduce the width 100 times below =1014 s−1 the pumpingneeds to be increased greatly, into the optical range �p=1015 s−1 or higher, and the stability of the system at suchhigh pump values would need to be addressed, even if thesevalues could be technologically attained. Again, as for thefirst case, increasing c will decrease the linewidth. This isnot obvious from Eq. �12� as c is not explicitly presentthere, but it is still there implicitly in N0, as seen in Fig. 7.

An increased value of c requires, of course, an increasedpump rate to exceed the threshold condition. We plot in Fig.8 a typical spectrum for c�1012 s−1.

In case 2 the number of plasmons is much smaller thanfor case 1, on the order of 0.6. Additionally, the light emittedcomes roughly equally from the TLS and NP, INP� ITLSL2, indisagreement with �1�. In this case, photons are emitted atthe rate �N0=1013�0.6�1013 photons s−1.

We also realize that the maximum of S1 for case 2 is onthe order of 10−26 V2 s2 /m2, which is 107 times smaller thanthe intensity produced in the first case. This is due mainly tothe fact that in case 1 the number of plasmons excited, N0, isabout 100 times bigger in case 1 than in case 2.

B. Coherence length

The coherence length is defined by lc=�2 /��, where��=�2� /c, with � as the linewidth. For our silver-in-siliconparameters:

FIG. 5. �Color online� The dependence of pumping �s−1� on thedecays of the NP, �s−1�.

FIG. 6. �Color online� The spectral width of the light emitted bythe system for regime of the following numbers: n̄a=0 and n̄c=0,�0=10−26, Rq=50 nm, �p=1013, a=7 nm, and =1014 s−1. Herethe spectral width actually increases above : ��4�1014 s−1.

FIG. 7. �Color online� Dependence of the mean plasmon num-ber on effective decay of the TLS c. n̄a=0 and n̄c=12, �0

=10−26 C m, Rq=50 nm, a=7 nm, and =1014 s−1.

FIG. 8. �Color online� The spectral width of the light emitted bythe system. Here the spectral width is less than : ��1013 s−1.n̄a=0, and n̄c=12, �0=10−26 C m, Rq=50 nm, �p=5�1015 s−1,a=7 nm, and =1014 s−1.

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�� =�10−6�21014

3 � 108 � 10−6 ⇒ lc � 10−6 m = 1 �m.

Since photons are being emitted at the rate of1011 photons s−1 and the duration of emission is on averageonly 10−14 s, the average separation between photons isabout 3 mm. We conclude that there is no coherence fromone emission to another.

C. Second-order coherence

The second-order degree of coherence measures the de-gree of fluctuation in intensities, of the sort that would bemeasured in a Hanbury-Brown-Twiss-type experiment.

The second-order coherence spectrum S2�� ,r�, defined by

S2��,r� = limT→�

1

2��

r/c

T+r/c

dt1�r/c

T+r/c

dt2ei��t1−t2�

��Etot− �r,t1�Etot

− �r,t2�Etot+ �r,t2�Etot

+ �r,t1�� , �21�

is given for the DNL by

S2��,r� =A0

4

2��INP + ITLSL2�2 1

�� − �0�2 + 4�2� �22�

and is shown in Fig. 9.In general, the narrower the spectrum is, the fewer the

fluctuations are and the larger the second-order coherences.Increasing the pumping more will decrease the width as thisincreases the number of plasmons excited, N0, which in turn,as mentioned before, decreases the linewidth. This is shownin Fig. 10.

As it is obvious from Eq. �22� the first-order and second-order coherences share almost the same Lorentzian line-width. Reduction in the spectral width in the intensity willautomatically lead to reduction in the width of second-ordercoherence. The second-order coherence is a strictly decreas-ing function of time and implies classical-type light.

VI. CONCLUSION

It is clear that the dipole nanolaser does not produce any-thing like ordinary laser light. In fact, what it produces are

single photons that can lie anywhere in a broad spectral re-gion. These photons are weakly correlated with one another�correlation length�1 �m� and are emitted in a two-lobeddipole pattern. The advantage of using the dipole nanolaserover simply a pumped TLS is the strong mutual interactionof TLS and NP, which greatly increases the radiant efficiencyof the system �L2�10 for case 2�. The price one pays for thisenhanced efficiency �at least when operating with metal NPsin a silicon matrix� is a broad spectrum. The photons areemitted in short time intervals, the average duration of aradiation process being 1 /�, and the interval between suc-cessive emissions is about �� / P�1 /�.

Another result of this work is the prediction of a narrow-�regime of parameters as in case 1. Should these parametersbe realizable by existing or new materials and technology,the dipole nanolaser would produce light of a width similarto that of conventional multimode lasers.

A spectral width of 1012 s−1 hardly resembles ordinarylaser light and a spectral width of 1014 s−1 even less so. Inthe latter case we are not speaking about a continuouslyemitted beam but rather a series of single-photon pulsessimilar to that emitted by the TLS acting alone. This is remi-niscent of the device described by Bergman and Stockman�3�, in which modes confined to a metallic surface are gen-erated as femtosecond pulses. Even the lower limit of1012 s−1 implies significant temporal structure in the emittedlight. We cannot predict the nature of this structure withoutgoing beyond the factorization approximation.

APPENDIX

Observables are computed using the following represen-tations for averages of bilinear products of noise operators:

�Fa†�t�Fa�t��� = n̄A

2��t − t�� ,

�Fa�t�Fa†�t��� = �n̄A + 1�

2��t − t�� ,

FIG. 9. �Color online� Second-order coherence for the situation��1013 s−1, �0=10−26 C m, Rq=50 nm, �p=5�1015 s−1, a=7 nm, and =1014 s−1. FIG. 10. �Color online� The relation of the plasmon number N0

on pumping of the TLS �s−1�.

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�F+�t�F−�t��� = �1

22n̄C +

�p

8�1 − ��z�t������t − t�� ,

�F−�t�F+�t��� = �1

22�n̄C + 1� +

�p

8���z�t�� − 1����t − t�� ,

�Fz�t�Fz�t��� = p + �2 −�p

2���z�t�����t − t�� ,

where n̄A and n̄C are the average numbers of quanta in the Aand C reservoirs, respectively.

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