Analysis of a Multimode Plasmonic Nanolaser with an

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Analysis of a Multi-Mode Plasmonic Nano-Laser With an Inhomogeneous Distribution of Molecular Emitters Yuan Zhang, and Klaus Mølmer J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b03918 • Publication Date (Web): 16 Jun 2017 Downloaded from http://pubs.acs.org on June 18, 2017

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The Journal of Physical Chemistry

Analysis of a Multi-mode Plasmonic Nano-laser with an Inhomogeneous Distribution of Molecular Emitters

Yuan Zhang ∗ and Klaus Mølmer ∗ Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark E-mail: [email protected]; [email protected]

Abstract We extend Lamb’s reduced density matrix laser theory to analyze the inhomogeneous molecular couplings and the mode-correlation in a plasmonic nano-laser consisting of a gold sphere and many dye molecules interacting with a driving optical field and with the quantized plasmon modes. The molecular inhomogeneity is accounted for by simulating their random distribution around the sphere. Our analysis shows that in order to obtain lasing we must employ a large number of strongly driven molecules to compensate strong damping of the plasmon modes. The compact molecular arrangement, however, can lead to molecular energy-shifts and thus reduces the excitation of the plasmon modes and ultimately suggests a maximum limit for the plasmon excitation in any specific system.

Introduction The interaction between metals and light has been investigated for more than a century with Maxwell’s electromagnetic theory. One essential insight obtained is that the electromagnetic (EM) 1

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field is enhanced and localized around metal nano-particles (MNP) and on the interfaces between metallic films and dielectrics 1 due to the excitation of surface plasmons involving collective oscillations of conductance electrons in the metal. The enhancement boosts the interaction between quantum emitters and the EM field 1–4 and thus leads to enhanced absorption, 5,6 emission 7,8 and Raman scattering. 9,10 This can be utilized to improve the sensibility of spectroscopic instruments 11 and the efficiency of LEDs 12,13 and solar cells. 14,15 The localization introduces EM modes with mode volume not limited by the wavelength of free-space light. 4,16 These modes can be excited if externally pumped quantum emitters are placed near MNPs or metallic films. Under suitable conditions, the energy loss of these modes can be even compensated and the system can achieve lasing. This phenomenon known as SPASER, was proposed by Bergman and Stockman 17 and verified firstly by Noginov and his co-worker 18 with an experiment involving a gold nano-sphere and many dye molecules. Since then many experimental demonstrations have been reported with structures like semiconductor wires 19–28 /squares 29,30 on metallic films, semiconductor pillars 31–36 /dots 37,38 /wires 39,40 inside metallic cavities as well as dye molecules in periodically arranged MNP arrays. 41–45 In order to theoretically describe these systems, one has to determine the lasing modes and consider how the gain medium transfers energy to these modes. The modes can be analyzed by solving Maxwell’s equations analytically 27,46 or numerically. 19–28,31–36,41–45,47 The energy transfer requires us to model the gain medium as a random spatial distribution of multi-level emitters. This multi-level model allows us to couple some levels with external driving fields or electron reservoirs to describe pumping mechanism and couple other levels with the lasing modes. Therefore, we have to deal with a complex theoretical problem involving many emitters, many levels and many modes. To reduce the complexity, semi-classical theories have been developed and utilized. In the theory proposed by Dridi and Schatz, 48,49 Maxwell’s equations for the field coupled with rate equations for four-level emitters are solved with a finite-difference time-domain method—a technique prevailing in electromagnetic simulations. 50 This approach has been successfully used to simulate the coupling of dye molecules with the local electromagnetic field in

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Enf (b) E ne

(a) Z COORDINATE (nm)

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photons

10 0

E driven

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m )

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0

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molecules

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(c)

Plasmon RDM ρμ ν Molecule-Plasmon (n) ρe μ , e ν Correlations: ( n) ( n) ~ ρf μ , g ν ρ gμ , f ν ~

(n)

ρgμ, g ν

~ ρ(gμn) , e ν −1 ( n) ~ ρ 1

e μ 1−1, g ν

ρ(n) f μ , e ν −1 1

ρ

(n) e μ 1−1, f ν

(n)

~ ρ(gμn) , e ν −1 ( n) ~ ρ

ρ(n) f μ , e ν −1

2

2

ρ

e μ 2−1, g ν

(n) e μ 2−1, f ν

(n)

ρf μ , f ν

~ ρ(gμn) , e ν −1 ~ ρ( n) 3

e μ 3−1, g ν

ρe μ −1, e ν −1

ρ(n) e μ −1, e ν −1

ρe μ −1, e ν −1

ρ(n) g μ −1, g ν − 1

ρ(n) g μ −1, g ν −1 ρμ −1 ν −1 (n) ρ x-mode f μ −1, f ν −1

(n) ρ g μ −1, g ν −1 ρμ −1 ν − 1 (n) y-mode ρf μ −1, f ν −1

1

1

ρ

1

1

(n) f μ 1−1, f ν1 −1

2

2

1

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ρe(n)μ −1, f ν 3

(n)

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Figure 1: Panel (a) illustrates our system, composed of a gold nano-sphere (10 nm radius) surrounded by a layer (12.5 nm inner- and 22.5 nm outer-radius) of 800 randomly distributed dye molecules with randomly oriented transition dipole moments; the driving field is polarized along z-axis. Panel Panel(b) (b)shows showsthe theeffective effectivethree-level three-level(E(E and EEnff)) molecules interacting cothe z-axis. ngg, ,EE nee and herently with the driving photons (the red arrow) and with the plasmons (the blue arrows) and experiencing dissipation (the vertical black arrows). Panel (c) shows how the reduced density ma trix elements of the plasmon modes ρµν with µ ≡ µx µy µz depend on the molecule-plasmon (n)

(n) correlations ρaµ,bν . The thin solid blue boxes indicate the correlations related to the plasmon x, y, and z modes (from left to right), and the colors of the arrows indicate the physical origin of the dependence, cf. the panel b. Approximate analytical expressions for the correlations are obtained to derive a closed set of equations for ρµν (for more details see text). 42,43 with randomly distributed experiments. However,emitters. those semi-classical theories evoke a mean-field approximation to re-

Most theories can quantum be viewed as effective the emitters are treated place the existing emitter-lasing mode correlation by descriptions, a product of where the emitters’ polarization and onlyfield in an average sense. They cannoreproduce characteristics measured in lasing experiments the amplitudes, and thus yield statisticalmain (quantum) information about the modes.beA cause complete the inhomogeneity of thetheory emitters becomestoirrelevant huge amounts of them are involved. more quantum laser is needed calculate ifthe plasmon number distribution and However, this may not be theUnfortunately, case in the plasmonic Because of thetreat strong the emitter-field correlations. so far, thenano-laser. existing quantum theories the inhomoemitters 51,52 and geneous subwavelength distribution of the in the of nano-laser, the spatial distribution of as identical two-level systems are near-field thus incapable dealing with randomly distributed

emitters can significantly affect inhomogeneous how they convertnear-field the pumping into the energy and emitters. Because of the strong in theenergy nano-laser, theplasmon spatial distribution

of emitters can significantly affect how they convert the pumping energy into the plasmon excita3



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tion and can even determine whether the systems can achieve lasing or not. To gain more insights into the physics involved, we analyze the inhomogeneity of the emitters with our quantum laser theory. In this article, we provide a systematic analysis of the inhomogeneity of molecular emitters in a nano-laser of Fig.1a, which resembles the one studied in ref 18. As the basis for our analysis, we will firstly describe our theoretical model in the following. To account for multi-molecules and multi-modes, we have extended significantly the density matrix laser theory of Lamb 53 for twolevel emitters and single lasing mode in our model. By following the procedure developed in refs 54 and 55, we first establish a reduced density operator equation for the entire system and then derive a quantum master equation for the reduced density matrix (RDM) of the plasmon modes. Our extended theory allows us to analyze how the molecular distribution affects the plasmon statistics and the molecule-induced mode-correlations. In the end, we summarize our findings and present an outlook for future work.

Theoretical Model As indicated by Fig.1a, we consider a random arrangement of molecular emitters separated by more than 2.5 nm from the surface of a gold nano-sphere of 10 nm radius. For less than 2.5 nm separation, the molecule-sphere Coulomb coupling is strong and overcomes the detuning of energy exchange process between the molecular emitters and the sphere multipole plasmon modes. Moreover, for less than 0.5 nm separation, the molecules may bind on the surface and electron tunneling may prevail. 56 These two processes deteriorate the lasing and should be avoided. For more than 2.5 nm separation, only the dipole plasmons obey the resonance condition and thus they dominate the energy exchange processes. In this case, the higher multipole plasmons still have minor influence on the system 57 but contribute only as an off-resonant reservoir to the excitedstate decay rate of the molecules. 58

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Reduced Density Operator Equation For a nano-sphere, there are three degenerate dipole plasmon modes with transition dipole moments pointing along three orthogonal axes in Cartesian coordinate system. Therefore, we can label them by j = x, y, z or j = 1, 2, 3. These modes can be described as quantum harmonic os+ cillators with Hamiltonian Hpl = ∑ j h¯ ω jC+ j C j , where C j and C j are creation and annihilation

operators and h¯ ω j = h¯ ωpl is their excitation energy. 59 We describe the molecules as three-level systems with the internal energy levels and transitions shown in Fig.1b. The molecular Hamiltoe nian reads He = ∑N n=1 ∑an Ena |an i han | where ground states |an = gn i, first |an = en i and second

|an = fn i excited states have the energies Eng ,Ene ,En f , respectively. 60 We assume that the plasmon

modes are resonant with the ground-to-first excited state transition, cf. the blue arrow in Fig.1b, ( jn) + e |g i he | and introduce the coupling Hamiltonian Vpl−e = h¯ ∑N v C + h.c. in the rotating n n j n=1 ge

wave approximation. If the molecules are very close to the surface of the metal nano-particle, the

electric field associated with the plasmons reduces with the distance exponentially. However, since the molecules are more than 2.5 nm from the surface for the reasons given above, the electric field decays only cubically with the distance, which is known as dipole approximation. In this case, the h i ( jn) (n) (n) coefficient h¯ vge = dge · d j − 3 dge · xˆ n d j · xˆ n / |Xn |3 is determined by the transition dipole (n)

moment dge of the molecules, d j = dpl e j of the plasmon modes as well as the distances Xn and

directional unit vectors xˆ n connecting the nth molecule and the sphere-center. We assume that a classical driving field is resonant with the ground-to-second excited state transition, cf. the red (n) iω0 t e |gn i h fn | + h.c. arrow in Fig.1b, and introduce the coupling Hamiltonian Ve (t) = h¯ ∑N n=1 vg f e (n)

(n)

in the rotating wave approximation. Here, the coefficient h¯ vg f = dg f · nE0 is determined by an(n)

other molecular transition dipole moment dg f and the driving field is specified by a frequency ω0 , a polarization vector n and an amplitude E0 . 61 Here, we consider continuous optical excitation and thus E0 is time-independent. The density operator ρˆ for the quantized plasmon modes and the molecular emitters obeys the

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following quantum master equation i ∂ ˆ , ρˆ = − Hpl + He +Vpl−e +Ve (t) , ρˆ − D [ρ] ∂t h¯

(1)

where the system dissipation is accounted for by the Lindblad terms: ˆ = (1/2) ∑ ku D [ρ] u

Lˆ u+ Lˆ u , ρˆ + − 2Lˆ u ρˆ Lˆ u+ .

(2)

The damping of the plasmon modes is included by terms with ku = γ j = γpl , Lˆ u = C j . The decay processes inside the molecules are included by terms with ku = ka→b , Lˆ u = |bn i han | for Ena > Enb (n)

for each molecule, cf. the black arrows in Fig.1b. For the sake of simplicity, we ignore molecular pure dephasing.

Plasmon Reduced Density Matrix Equation The main goal of our analysis is to determine the plasmon state populations (probabilities) and correlations as quantified by the plasmon reduced density matrix (RDM) with elements ρµν ≡ trS {ρˆ (t) |νi hµ|}, where trS denotes the trace over the system and |µi ≡ |µx i µy |µz i and |νi ≡ |νx i νy |νz i denote product states of the plasmon occupation number Fock states. From Eq.1, we (n)

(n)

(n)

observe that ρµν depends on the molecule-plasmon correlations ρeµ j −1,gν , ρgµ,eν j −1 and ρeµ,gν j +1 , (n)

(n)

ρgµ j +1,eν with a general definition ρaµ,bν ≡ trS {ρˆ (t) |bn i han | × |νi hµ|}, cf. Supporting Information. The equations for the correlations also follow from Eq.1. These equations result in dependence between the plasmon RDM and the correlations, shown in Fig.1c, which is caused by the couplings and the dissipation rates in the master equation 1. This dependence also indicates our procedure to solve those inter-dependent equations. Because both molecular and plasmonic dissipation rates contribute to the decay of the correlations, they must decay faster and thus may adiabatically 62 follow the plasmon RDM elements which are only affected by the plasmon damping rates. Because of the molecular dissipation, the correlations represented within the blue dashed box of Fig.1c 6


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depend on the correlations outside the box. Fortunately, they all can be expressed as functions of the plasmon RDM because of the symmetry hidden in the coupling Hamiltonians. Finally, we back substitute these expressions and obtain closed dynamic equations for the plasmon RDM, where the molecules contribute by several coefficients, cf. Eq. 74 in Supporting Information. The diagonal elements Pµ ≡ ρµ µ are the populations (the probabilities) of the plasmon number states |µi while the off-diagonal elements ρµν (µ 6= ν) represent the coherence of the plasmons. Here, we focus on the populations by solving the equations for those diagonal elements: # " 3 3 ∂ ( j) ( jk) Pµ = − ∑ γ j µ j + κ µ Pµ − ∑ ηµ Pµk −1 ∂t j=1 k=1 " # 3 3 ( j) ( jk) + ∑ γ j µ j + 1 + κµ j +1 Pµ j +1 − ∑ ηµ j +1 Pµ j +1µk −1 , j=1

( j)

(3)

k=1

( jn)

( jk)

e where the rates κµ ≡ − ∑N n=1 α µ µ and ηµ

( jkn)

e ≡ − ∑N n=1 βµ µ

include contributions from individ-

ual molecules, cf. Eqs. 75 and 76 in the Supporting Information. Here, µ j ± 1 for j = x indicates (µx ± 1, µy , µz ) and µ j + 1µk − 1 for j = x and k = y denotes (µx + 1, µy − 1, µz ). Since the former rates decrease the population of higher plasmon states but increase that of lower states, they can be interpreted as molecule-induced plasmon damping rates. Since the latter rates have the opposite effect on the population, they can be interpreted as molecule-induced plasmon pumping rates. The latter rates depend on two plasmon mode indices and thus account for correlation between different plasmon modes induced by the molecules. These rates can be considered as extended Einstein’s AB coefficients accounting for the multi-plasmon modes, the molecular pumping mechanism and the molecular inhomogeneity. For a system with single plasmon mode, these rates are reduced to κµ and ηµ . In the simplified version of Eq. 3, the term η1 P0 describes the rate of increasing population P1 and thus is related to spontaneous emission into that plasmon mode. The terms ηµ+1 Pµ for µ ≥ 1 describe the rates of increasing population Pµ+1 of the state with µ + 1 plasmon quanta and thus is related to stimulated emission. The terms κµ Pµ for µ ≥ 1 describe the rates of reducing population Pµ and thus is related to stimulated absorption. In steady-state the second line of Eq.3 is recovered if we replace µ j by µ j + 1 on the right side 7



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of the first line and this suggests a recursion relation for the populations. We obtain such a relation by setting the first line to zero:

Pµ =

( jk) 3 Pµk −1 ∑ j=1 ηµ . ( j) ∑3j=1 γ j µ j + κ µ

∑3k=1

(4)

Together with the normalization condition ∑µ Pµ = 1, the above relation can be utilized to easily calculate the populations according to the procedure outlined in Fig.S4 in Supporting Information. Although Pµx µy µz contains all the information about the three dipole plasmons, it is more intuitive to consider physical quantities related to one or two dipole plasmon. We calculate them by tracing out one mode to get Pµx µy = ∑µz Pµx µy µz , Pµy µz = ∑µx Pµx µy µz and Pµx µz = ∑µy Pµx µy µz (the joint population of two modes) or by tracing out two modes to get Pµx = ∑µy µz Pµx µy µz , Pµy = ∑µx µz Pµx µy µz and Pµz = ∑µx µy Pµx µy µz (the reduced population of one mode). We can also quantify the strength of plasmon excitation with the so-called plasmon mean numbers: N j ≡ ∑µ j µ j Pµ j and the plasmon statistics with the so-called (steady-state) second order correlation functions: (2) g j (0) ≡ ∑µ j µ j µ j − 1 Pµ j . To analyze how the individual molecules contribute to the plas(n)

(n)

mon excitation, we can calculate the population of individual molecular states: Pg ≡ ∑µ ρgµ,gµ , (n)

Pf

(n)

(n)

≡ ∑µ ρ f µ, f µ and Pe

(n)

≡ ∑µ ρeµ,eµ , which are actually determined by Pµx µy µz , cf. Eqs. 111,

112 and 113 in Supporting Information.

Results The above theoretical model allows us to consider many aspects of the plasmonic nano-laser. To gain a complete understanding, we have systematically investigated how the different parameters, for example, the strength of the external driven field, the decay rates of molecular levels, affect the system. With these understandings, in the present article, we would like to go one step further to study a more difficult problem, that is, how the molecular inhomogeneity may affect the system performance. From our theoretical model, we understand that the molecular inhomogeneity mainly

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originates from their positions and orientations of their transition dipole moments. For a realistic structure, the molecules are randomly distributed in three dimensions and they interact with the three plasmon modes simultaneously with different strengths, which complicates our analysis. In order to illustrate the physical insights, we will first of all consider a special configuration where all the molecules are located along the equator of the gold sphere, cf. Fig.2a. Actually, all the insights achieved are also valid for the realistic configurations discussed later on.

Configuration with Single Dipole Plasmon Mode ( jn) (n) For the configuration in Fig.2a, the molecule-plasmon coupling is reduced to h¯ vge = δ j,z ±dge dpl /Xn3

with positive (negative) sign for the molecules oriented along the positive (negative) z-axis. Ob-

viously, the dipole plasmon x- and y-mode are not involved and thus can be ignored in the following analysis. The coupling depends inversely cubically on the molecule-sphere center distance Xn = aMNP + dn , cf. the black squares in Fig.2b. Here, aMNP is the radius of the sphere and dn (n)

the distance to the sphere-surface. In contrast, the driving field coupling h¯ vg f of the molecules depends only on the molecular orientations but not the positions, cf. the red triangles in Fig.2b. If all the molecules have the same distance to the sphere-surface, they are equivalent and the resulting ideal system has been analyzed in details (unpublished results). We have obtained the optimal parameters of the system leading to the strongest plasmon excitation, cf. Table S1 in Supporting Information. These parameters will be used as reference parameters for the following simulations. To compensate the strong plasmon damping, the number of molecules coupled strongly with the plasmons is an essential parameter. It was demonstrated in the experiment 64 that the system properties like emission wavelength, intensity and pumping threshold strongly depend on the concentration (number) of the molecules. Here, we analyze this dependence from three aspects: density of molecules, spatial extension of molecular layer and molecular level-shift. As indicated by Fig.2b, the molecules close to the sphere couple strongly with the plasmons. Therefore, those molecules contribute more to the plasmon excitation than other molecules. By increasing the molecular density, we increase the number of molecules and thus the plasmon exci9



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Figure 2: with single dipole plasmon Panel (a): a gold properties. nano-spherePanel (the green Figure 2: Configuration Systems with single dipole plasmon modemode. and their steady-state (a): a filled circle) with 240 randomly distributed molecules in a layer (r = 12.5 innerand r = out in gold nano-sphere (the green filled circle) with 240 randomly distributed molecules in a layer (r22.5 in = nm outer-radius); the=black crosses indicate the molecular dipole moment along 12.5 inner- and rout 22.5dots nm and outer-radius); the black dots and transition crosses indicate the molecular the z-axis;dipole the driving field is polarized the positive z-axis. Panel (b): the thepositive couplingz-axis. with transition moment along the z-axis;along the driving field is polarized along the dipole plasmon (thewith blackthe squares) and with (the the driving field (the redwith triangles) for molecules Panel (b): the coupling dipole plasmon black squares) and the driving field (the with different distances d to the sphere-surface. Panel (c): plasmon number state population n with different distances dn to the sphere-surface. Panel (c): plasmon red triangles) for molecules P for systems with increasing numberwith of molecules to 220 in N steps of 20 (from µ e from 20 z number state population Pµz for systems increasingNnumber of molecules e from 20 to 220 the left to right curve); the inset shows plasmon mean number N (the black squares with z in steps of 20 (from the left to right curve); the inset shows plasmon mean number Nz fitted (the black (2) −4 N 2 ) and the g (0)-functions (2) octagons fitted with −1.54 +fitted 8.26 × 10−2 Ne + + 1.54 × 10 (the eNe + 1.54 ×z 10−4 Ne2 ) and the squares with −1.54 8.26 × 10−2 gzred (0)-functions (the red (n) (n) (n) −0.02N e −0.02N 0.74e + 0.98) versus Ne . ePanel (d):versus population of molecular states of Pa molecular versus dnstates ; Pg P (the octagons fitted with 0.74e + 0.98) Ne . Panel (d): population a (n) (n) (n) (n) (n) black squares and curves), P (the red up-triangles and curves), P (the green down-triangles e versus dn ; Pg (the black squares and curves), Pe (the red up-triangles f and curves), Pf (the green and curves); the arrows on the right indicate the increase of N . Panel with different e down-triangles and curves); the arrows on the right indicate the increase(e,f): of Nsystems systems e . Panel (e,f): widths W = r − r of the molecular layer (fixed molecular density); panel e shows out z (the in with different widths W = rout − rin of the molecular layer (fixed molecular density); panel eNshows −0.72W 2 black fitted with + 18.49) and gzand (0)-function (the red fittedfitted with N black squares fitted−19.79e with −19.79e−0.72W + 18.49) g2z (0)-function (thecircles red circles z (thesquares (n) (n) (n) −0.28W (n) (n) d with W from 1 nm to 100 nm; the 0.50e0.50e−0.28W + 1.06) ; panel f shows Pg , PPeg(n)and versus n with + 1.06) ; panel f shows , Pe Pf and Pf versus dn with W from 1 nm to 100 arrows indicate the increase of W . Physical parameters are specified 1 in Supplemental nm; the arrows indicate the increase of W . The strength E0 and energy hin ¯ ωTable 0 of the external field are Material. fixed. All the parameters are specified in Table S1 in Supporting Information.

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tation. This is clearly demonstrated in Fig.2c by the increased populations Pµz of higher plasmon excited states and the increased plasmon mean number Nz (the black dots and curve in the inset) with increasing number of molecules Ne from 10 to 240. For larger Ne , Pµz resemble Poisson distributions indicating the formation of a coherent state and Nz is much larger than unity (maximal value 25). Following the explanation of Eq. 3, we know that the spontaneous emission is proportional to Pµ=0 while the stimulated emission and absorption are proportional to Pµ≥1 . Fig.2c shows that Pµ=0 reduces and Pµ≥1 increases gradually when Ne increases, which indicates that the spontaneous emission is overtaken by the stimulated emission. All these results indicate that the (2)

systems are lasing. This conclusion is further confirmed by the gz (0)-function, cf. the red dots and curve in the inset of Fig.2c, which approaches unity for large Ne , i.e. the Poisson limit. The fluctuation of the dots around the curves in the inset is caused by different molecular distribution in each simulation and may thus characterize fluctuations encountered in experiments. To understand why the increasing molecular density can increase the plasmon excitation, we (n)

analyze the state population for every molecule Pa , cf. Fig.2d. Here, the different molecular indices n are displayed with the molecule-sphere surface distance and the indices a = g, e, f indicate the three molecular states. First, we notice that the molecule-plasmon coupling Vpl−e leads to reversible processes (the spontaneous emission, the stimulated emission and absorption of the plasmons) since it enters into our master equation as a coherent coupling, cf. Eq.1. These pro(n)

cesses tend to balance the population of the molecular excited states Pe (n)

(n)

and ground states Pg . (n)

This leads to the reduced Pe , cf. the red curves and arrow, and the increased Pg , cf. the black curves and arrow, with increasing Ne . In addition, because the reduced coupling with increasing (n)

distance dn (cf. Fig.2b) reduces the rate of these processes, the Pe

(n)

with increasing dn . The population of the higher excited state Pf

(n)

increase and Pg

decrease

is mainly determined by the

strong decay rate from this state to the middle excited state and thus is always smaller than the other populations. In the following, we consider the effect of the spatial extent (width W ) of the molecular layer, cf. Fig.2e, which can be also studied in experiments like refs. 18,64 by precisely controlling

11



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Page 12 of 25

the synthesis time of the molecular layer. In this case, the molecules added far away from the sphere-surface contribute less to the plasmon excitation because of the reduced molecule-plasmon (2)

coupling, cf. Fig.2b. As a result, the plasmon mean number Nz and the gz (0)-function saturate for large W as displayed by Fig.2e. In addition, we find that the data points are close to the fitted curve for small W but fluctuate a lot for large W . This can be easily understood with the (n)

change of the molecular state population Pa , cf. Fig. 2f. When W increases from 1 nm to 30 (n)

(n)

(n)

nm, Pg , Pe , Pf

change dramatically since all molecules contribute to the plasmon excitation.

Therefore, Nz increases and the molecular inhomogeneity has less influence on the plasmon ex(n)

(n)

(n)

citation. However, when W increases further, Pg , Pe , Pf

change less and now the molecules

distributed in the region near to the sphere will significantly affect Nz . In addition, we calculated the threshold strength of the external pumping field for the systems studied here, cf. the Fig. S1a in Supporting Information. The threshold strength is defined as the strength at which the plasmon mean number equals unity. We saw that the threshold decreases from about 7 × 107 V/m for W = 5

nm to about 2.1 × 107 V/m for W = 40 nm and then maintains the same value for much larger W . The initial threshold reduction is due to the added molecules contributing to the plasmon excitation while the saturation of the threshold is due to the weak coupling of the remote molecules and the plasmon mode. The compact molecular arrangement around the sphere implies that the molecules may directly interact with each other through dipolar interaction. For not too high concentration, electron transfer between molecules can be ignored but direct energy exchange between excited molecular dipoles can lead to energy-shifts (inhomogeneous broadening). In principle, such effects can be accurately described by directly incorporating the inter-molecular energy exchange into the system Hamiltonian in the master equation 1. However, here, we follow an easier, phenomenological way to account for such effects by introducing random energy shift δ E (n) to individual molecules with n √ o a Gaussian distribution, 63 cf. Fig. 3a, p (δ E) = 1/ 2πσ exp − (δ E)2 / 2σ 2 (with stan(n)

dard deviation σ ). It means that the transition energies are modified as h¯ ωeg = h¯ ωeg + δ E (n) and (n)

h¯ ω f g = h¯ ω f g + δ E (n) , compared to the values in Table S1 in Supporting Information. 12


Page 13 of 25

(a) 0.7 50 0.6 40 0.5 30 0.4 0.3 20 0.2 10 0.1 0200 150 100 50 0 50 100 150 200 0.0 ENERGY SHIFT (meV)

20 15

0.15

10

0.10

50

0.8 POPULATION

DISTRIBUTION (x100)

1.14 1.12 1.10 1.08 1.06 1.04 1.02

0.6 0.4 0.2 0.0 2

4


12

(d)

1.0

(c)

0.8

20 40 60 80 100 DEVIATION (meV)

0.6 0.4 0.2

0.05 0.00 0

(b)

1.0

POPULATION

0.20

gx(2)(0)FUNCTION

30 25

MEAN NUMBER

NUMBER OF MOLECULES

60

POPULATION

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

10

15 20 25 30 35 PLASMON STATES

40

45

0.0 2

4


12

Figure 3: of molecular energy-shift δ E (n)δfor with Ne = 250 molecules. Panelwith (a): Figure 3: Effect Influence of molecular energy-shift E (n)a system on steady-state properties of a system (n) (n) histogram of δ E as well as the Gaussian distribution with the deviation σ =50 meV. Panel (b): Ne = 250 molecules. Panel (a): histogram of δ E as well as the Gaussian distribution with the (n) (n) the sphere-surface, P(n) (the populationσ of molecular states versus the distance g deviation =50 meV. Panel (b):Papopulation of molecular molecular states Pa toversus the molecular distance (n) (n) (n) (n) black and curve), red squares up-triangles curve), down-triangles and to the squares sphere-surface, Pg Pe(the(the black and and curve), Pe P(the redgreen up-triangles and curve), f (the (n) curve); the random populations fitted by polynomial function. Panel (c): plasmon state population Pf (the green down-triangles and curve); the random populations fitted by polynomial function. Pµz with from 0 meV toP100with meVincreasing in steps ofσ10from meV0 (from the100 right to left curve); Panel (c):increasing plasmon σ state population meV to meV in steps of the 10 µz (2) inset:(from plasmon number Nz (the and curve) gz (0)-function dots and meV the mean right to left curve); theblack inset:dots plasmon meanand number Nz (the black(the dotsred and curve) (2)Panel (d): fitted populations of molecular states for increasing σ indicated by the arrows. curve). and gz (0)-function (the red dots and curve). Panel (d): fitted populations of molecular states 7 V/m. Physical parameters are specified in Theincreasing strength ofσthe drivingby field E0 = 9 × 10strength Table for indicated theisarrows. The of the driving field is E0 = 9 × 107 V/m. 1 in Supplemental Material. Physical parameters are specified in Table S1 in Supporting Information.

12 13



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Page 14 of 25

The consequence of energy-shifts is to perturb the perfect resonant condition assumed previously for the molecular pumping and the molecule-plasmon energy transfer. This is reflected (n)

by the irregular change of the state populations Pa

for the molecules at similar distances to the

sphere-surface, cf. the dots in Fig.3b. However, since the majority of molecules has no or small (n)

energy-shift as shown in Fig.3a, the populations Pa

in Fig.3b still roughly follow the same trend

observed in Fig.2d, cf. the solid lines. The broadening of the transition energies is also reflected in the shift of the plasmon state population Pµz to lower states, a reduced plasmon mean number (2)

Nz and an increased gz (0)-function with increasing deviation σ of the energy-shift from 0 meV to 100 meV, cf. Fig.3c. These features can be understood by analyzing the contribution of indi(n)

vidual molecules through their state populations Pa . As shown in Fig.3d, the populations of the (n)

molecular middle excited states Pe

(n)

increase while those of the ground states Pg

decrease with

increasing σ . These results reflect that the molecules are on average less affected by the plasmons and thus contribute less to the plasmon excitation. In addition, we calculated the threshold strength of the external pumping field for the systems studied here, cf. Fig. S1b in the Supporting Information. We saw that the threshold increases from about 3 × 107 V/m to 9 × 107 V/m with increasing deviation of the molecular energy-shift from 0 to 100 meV. The threshold increase is caused by the off-resonant condition of the molecules to the external field and the plasmon, which makes the optical pumping and the subsequent energy transfer to the plasmon less efficient. The features described above indicate that the lasing performance is strongly affected by the inhomogeneous molecular energy-shift. Finally, we point out that the standard deviation σ characterizes energyshifts due to intra-molecular interactions and should hence depend on the molecular concentration.

Configuration with Two and Three Dipole Plasmon Modes (n)

We now consider the more complex situation where the molecular transition dipole moments dge (n)

and dg f orient randomly in the x-y plane, cf. Fig.4a. In this case, the molecules couple with the dipole plasmon x− and y−mode simultaneously with random strength, cf. the upper panel of Fig.4b. In addition, the driving field coupling becomes also random as shown in the lower 14


Page 15 of 25

Y COORDINATE (nm)

10 0 10 20

20

10

0

10

X COORDINATE (nm)

20

COUPLING (meV) COUPLING (meV)

(a)

20

30 20 10 0 10 20 30 30 20 10 0 10 20 30 2

(b)

4


0.34 0.30 0.26 0.22 0.19 0.15 0.11 0.07 0.04 0.00 50 40 30 E 20 OD 10 X M

POPULATION (x100)

0.8 0.6

0.4 0.2 0.0 2

12

(d)

(c)

1.0

POPULATION

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4


12

0 10 Y 20 MOD30 E 40 50 0

Figure dipole plasmon modes. the configuration for N Figure4:4:Configuration Systems withwith twotwo dipole plasmon modes andPanel their(a): steady-state properties. Panel (a): e = 500 molecules is similar a except that the dipole moments orient randomly the configuration fortoNFig.2 molecules is molecular similar to transition Fig.2a except that the molecular transition e = 500 in the xy-plane and the driving field is along the positive x-axis. Panel (b): the couplings of dipole moments orient randomly in the xy-plane and the driving field is along the positive x-axis. (n) (n) molecules distances d toatthe sphere-surface; shows the the coupling Panel (b): atthedifferent couplings of molecules different distances the d upper to thepanel sphere-surface; upper with plasmon x-modewith (the the black squares), and with plasmon y-mode (the the red plasmon triangles); part the shows the coupling plasmon x-mode (the the black squares), and with ythe lower shows the the coupling the driving field. Panel population of molecular mode (thepanel red triangles); lower with part shows the coupling with(c): the the driving field. Panel (c): the (n) (n) (n) (n) (n) (n) (n) states Pa versus d (n) ; Pg states (the P black squares), upper triangles), population of molecular d (n) ;PePg (the (thered black squares), Pe P(the red green upperdown triana versus f (the (n) triangles); the curves are exponentially fits to the average population as function of distance. Panel gles), Pf (the green down triangles); the curves are exponentially fits to the average population as (d): joint of population plasmon Fock states. parameters are specified in Table 1 in µx µy of function distance.PPanel (d): joint population PµxPhysical µy of plasmon Fock states. Physical parameters Supplemental Material. are specified in Table S1 in Supporting Information.

14 15



1000

(b)

(a)

800 600

400 200 0 30

0.27 0.24 0.21 0.18 0.15 0.12 0.09 0.06 0.03 0.00 50 40 30 DE 20 O 10 Y M

POPULATION (x100)

NUMBER OF MOLECULES

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 25

20

0 10 20 10 COUPLING (meV)

30

0 10 X 20 MOD30 E 40 50 0

Figure 5: Configuration Systems shown in Fig. 1a with three dipole plasmon modes andPanel their (a): steady-state shown in Fig. 1a with three dipole plasmon modes. stacked properties. Panel (a): stacked histogram of the molecule-plasmon coupling; blue, red and orange histogram of the molecule-plasmon coupling; blue, red and orange bars are for x,y,z-mode respecbars for x,y,z-mode respectively. Panel (b):a the jointwith population Pµx µy forPhysical a systemparameters with 800 tively.arePanel (b): the joint population P system 800 molecules. µx µy for molecules. parameters are specified the Table S1 in Supporting Information. are specifiedPhysical the Table 1 in Supplemental Material. panel of Fig.4b because it also depends oninthe orientations. This implies that theBasically, molecules at ing deviation σ of molecular energy-shifts Fig.1(d,e) in Supplemental Material. they similar distance to thelike sphere-surface experience different couplings and this leads to the random show similar features those in Fig.2c and Fig.3c respectively. (n)

population states Pa as displayed in However, the maximum of Finally, of let molecular us turn to the realistic configuration of Fig.4c. Fig.1a. In this case,because the randomly distributed the molecule-plasmon couplingcouple decreases increasing distance sphere-surface d (n) , the molecules in three dimensions with with the three plasmon modesto in the a similar pattern, cf. Fig.5a (n)

distance-dependent averaged Pa show a similar behavior as in Fig.2d. The co-existence of the (see also the coupling of individual molecule in Fig.2a in Supplemental Material). This implies xis directly shown the joint plasmon stateinpopulation for ais system with thatand all -y themode plasmon modes will be by excited by the molecules similar wayPµand reflected by x µy this N 500populations molecules, Pcf. Here, bettera visualize Pµx µ(10, is shown as a smooth the joint , Pµy µz and Pµxto peak around for a system with Nsurface. e= µx µyFig.4d. µz with e = 800 y , it10) The population a maximum µx =in25Supplemental and µy = 25, which indicates that the both plasmon molecules, cf. has Fig.5b (see alsoaround Fig.2(b,c) Material). In addition, we also find modes are excited to thePsame In addition, havethe also analyzed the plasmon state the increased population , Pµz of higher plasmonwe states, increased plasmon mean numµx , Pµystrength. (2)

(2)

(2) (2) populations thethe plasmon mean numbers Nx , gN(2) the gx (0)andincreasing gy (0)-function ber Nx , Ny , NP well reduced gx (0) , gy (0) (0)-functions with numberfor of µy , as zy and z µas x,P

systems with molecular density (number of molecules) in Fig.S2a-c and with increasing molecules Ne increasing ( cf. Fig.2(d,e,f) in Supplemental Material). deviation σ to ofachieve molecular Fig.S2d-fper inmode, Supporting Information. Basically, they In order the energy-shifts same plasmoninexcitation we must double (triple) the number show similar in features like those Fig.2cmodes and Fig.3c respectively. of molecules the case with twoin(three) compared to the single mode case. Incidentally, our Finally, us turn to the realistic of Fig.1a. In this thesignificant randomly asymmetry distributed results showletthat the polarization of configuration the driving field alone does notcase, cause molecules inexcitation three dimensions couple with the three plasmon modes in a similar pattern, cf. Fig.5a between the of the three plasmon modes. (see also the coupling of individual molecules in Fig.S3a in Supporting Information). This implies that all the plasmon modes will be excited by the molecules in a similar way and this is reflected by

Conclusions

the joint populations Pµx µy , Pµy µz and Pµx µz with a peak around (10, 10) for a system with Ne = 800 In summary,cf.weFig.5b have developed quantum in laser theory based on reducedIn density matrix molecules, (see also aFig.S3b,c Supporting Information). addition, we equations also find and applied it to a plasmonic nano-laser consisting of a gold nano-sphere and many dye molecules.

16 15


Page 17 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the increased population Pµx , Pµy , Pµz of higher plasmon states, the increased plasmon mean num(2)

(2)

(2)

ber Nx , Ny , Nz as well as the reduced gx (0) , gy (0) , gz (0)-functions with increasing number of molecules Ne ( cf. Fig.S3d-f in Supporting Information). In addition, we calculated the threshold strength of the external pumping field for systems of 240 molecules in a fixed layer of 10 nm width. Since these systems assume the three configurations shown in Fig. 2, 4, 5, respectively, the molecules couple actually with single, double and triple plasmon modes respectively. The pumping threshold is about 3.9 × 107 V/m for the system with

triple plasmon modes. It is reduced to about 2.7 × 107 V/m for the system with double plasmons

and is further reduced to 1.5 × 107 V/m for the system with a single plasmon mode. Here, we have

doubled the molecule-plasmon coupling in the latter case to ensure a comparable coupling strength in all the systems.

Conclusions In summary, we have developed a quantum laser theory based on reduced density matrix equations and applied it to a plasmonic nano-laser consisting of a gold nano-sphere and many dye molecules. Our study reveals that the molecular inhomogeneity and the multi-plasmon modes make strong molecular pumping necessary to compensate strong plasmon damping and to achieve lasing. By increasing the molecular density, the plasmon excitation increases, but molecular energy-shifts due to inter-molecular interaction may ultimately reduce the plasmon excitation. This may constitute a fundamental limitation for the further miniaturization of the plasmonic nano-laser. In this article, we modeled the molecular emitters as three-level systems. However, the procedure illustrated can be readily applied to the emitters with arbitrary level structure, which will be necessary to study the influence of other intrinsic processes of the emitters on the laser performance. For example, by introducing more intermediate molecular vibrational levels, in principle, we can study how the intra-molecular vibrational energy redistribution and the temperature of the environment affect the system performance. This extended theory may be utilized to analyze the

17



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experiments, 43 where the varying excitation energy of lattice plasmons due to the surrounding material affects the dye molecules used by picking up the molecular energy levels resonant to the plasmons. This study will not only provide more insights into the interplay of the plasmons and the gain material but may also suggest how to optimize the system performance.

Supporting Information Supporting Information includes: Simulation Parameters, Supplemental Results, Detailed Derivation of Equations.

Acknowledgments Y. Z. and K. M. acknowledge Volkhard May for several illuminating discussions. This work was supported by Villum Foundation (Y. Z. and K. M.).

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Analysis of a Multimode Plasmonic Nano-laser with an Inhomogeneous Distribution of Molecular Emitters Figure 6: TOC Graphic

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Analysis of a Multimode Plasmonic Nanolaser with an

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