Polaritons in a Metal-Nanorod

Article pubs.acs.org/JPCA

Quantum Symmetry Breaking of Exciton/Polaritons in a MetalNanorod Plasmonic Array Svitlana Zaster,† Eric R. Bittner,*,† and Andrei Piryatinski*,‡ †

Department of Chemistry, University of Houston, Houston, Texas 77204, United States Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States

‡

ABSTRACT: We study the collective, superradiant behavior in the system of emitter-dressed Ag nanorods. Starting from the Drude model for the plasmon oscillations, we arrive at a semiempirical Hamiltonian describing the coupling between quantized surface plasmon modes and the quantum emitters that can be controlled by manipulating their geometry, spacing, and orientation. Further, identifying the lowest polariton mode as SP-states dressed by excitons in the vicinity of k = 0, we examine conditions allowing for the polariton quantum-phase transition. Though the system is formally a 1D array, we show that the polariton states of interest can undergo a quantum-phase transition to form a Bose condensate at finite temperatures for physically accessible parameter ranges.

1. INTRODUCTION Spontaneous emission is a powerful probe of the dynamics of a given system. For a set of isolated emitters, the physics of spontaneous emission is well understood and depends upon the internal details of the specific system. However, the collective emission from an ensemble of emitters is not so well understood and depends strongly upon the nature of the interactions between the emitters and their surrounding media. Superradiance occurs when the coupling between the emitters and radiation field is sufficiently high that entire system + field must be considered as a collective quantum state and can be described by the classic Dicke model.1 The Dicke model has been used to model a wide range of physical systems from BEC in optical cavities2−5 to Hawking radiation from black holes.6 A direct realization of the model using a cavity ring-laser was proposed by Dimer et al.7 Superradiance has also received renewed interest in the field of quantum optics7 and nanotechnologies.8−10 It also underlies the physics behind the formation of exciton/polariton Bose condensates in thin-film microcavities.4,11−25 Quantum plasmonics is a rapidly emerging field of nanotechnologies establishing control of light quantum properties via its interaction with nanoscale metallic structures showing surface-plasmon (SP) response.26 Typically the SP modes are excited via near-field interactions with quantum emitters such as atomic impurities or semiconductor nanostructures. The coupling between the quantum emitters and SP modes plays a key role in controlling the quantum properties of the photons. Previous theoretical investigations indicate that strong coupling between an ensemble of quantum emitters and a SP mode can result in superradient emission.27,28 Such calculations utilize a semiclassical approach and provide the correct trends in the emission line-shape with the number of emitters allowing for identification of the super- and subradiant © 2016 American Chemical Society

exciton-SP modes. However, a complete investigation of the cooperative thermodynamic properties leading to spontaneous symmetry breaking associated with the superradiant regime requires a fully quantum mechanical approach. To address this problem, we examine the on-set of collective excitonic states in one-dimensional periodic arrays of Ag nanorods dressed by dipole emitters as sketched in Figure 1. The near-field coupling between localized SP modes within the nanorods and the dipole emitters gives rise to a collective exciton-SP mode. Using a quantum mechanical model that

Figure 1. Sketch of a one-dimensional periodic structure of metal nanorods (blue spheroids) lying in the xz plane with period a along the x-axis. Angle θ between spheroid’s major axis and x-axis defines orientation of the SP-mode dipoles psp. Each nanorod is dressed by a quantum emitter (red disk) located a distance d above the nanorod and characterized by a transition dipole moment μqe held parallel to psp. The wavy curve represents a collective plasmon mode ψ̂ k due to near-field coupling between the nanorods, Jsp (eq 25). The emitters are coupled to the collective SP mode via λnk (eq 28). Special Issue: Ronnie Kosloff Festschrift Received: November 2, 2015 Published: February 23, 2016 3109

DOI: 10.1021/acs.jpca.5b10726 J. Phys. Chem. A 2016, 120, 3109−3116

Article

The Journal of Physical Chemistry A

considered in this paper. The amplitude of the metal nanoparticle SP dipole is given by

accounts for the explicit interactions between the quantum emitters and SP, we construct a Dicke Hamiltonian.1,3 In contrast to the conventional Dicke model whereby the quantum emitters are coupled to a single radiative cavity mode, our model couples the quantum emitters via near-field quasi-static Coulomb interactions. Within this model, the coupling between the emitters and the SP modes enhances the Coulomb couplings between emitters. Furthermore, this interaction can be controlled by manipulating the aspect ratio of the nanorods and their relative orientation. Because the transverse radiative potential is not included in the near-field regime, our model is free of issues raised in the literature regarding gauge invariance that leads to a violation of the Thomas−Reiche−Kuhn sum rule.29 By analyzing the thermodynamics of this collective system, we predict a symmetry breaking transition that leads a the collective, superradiant regime in a quasi-one-dimensional array. The paper organized as follows. In section 2 we derive a second-quantized SP model accounting for the local field interactions. In section 3, the Hamiltonian for the near-field coupled collective states of SP and quantum emitters and their mutual interactions is introduced. On the basis of this Hamiltonian, in section 4, we evaluate SP and exciton polariton modes and identify the lower-polariton mode as excitondressed SP states in the vicinity of k = 0. Further examination of the conditions for the symmetry breaking for this mode demonstrates the possibility of quantum-phase transition into the superradiant state at finite temperature. Section 5 concludes the paper.

psp =

⎛ ωspαsp ⎞1/2 ⎜ ⎟ ⎝ 2 ⎠

This depends on the SP energy, ωsp, and the nanoparticle polarizability, αsp, which we now derive. To describe the anisotropic SP response of the metal nanoparticles forming the array, we define each to be prolate spheroids and set the spheroid major axis to be oriented in the direction of the unit vector esp constrained to the (x, z)-plane. We further denote the spheroid major (minor) radius by a0 (b 0 ), the distance to the spheroid focal point as f0 = a0 2 − b0 2 and the inverse eccentricity as η0 = [1 − (b0/a0)2]−1/2. For large enough values of the aspect ratio a0/b0, one can consider the spheroidal nanoparticles as nanorods whose SP mode oscillating along the major axis couples strongly to the local electric fields whereas the contribution of the SP oscillations along the minor axis can be neglected. In this approximation, dipole moment of the spheroidal nanorod located at the nth site induced by the local electric field En can be represented as pn (ω) = α(ω)En

2. INTERACTING SP MODEL We shall begin by assuming that the metal nanoparticles interact via quasi-electrostatic interactions, giving rise to a collective surface-plasmon (SP) mode. Each nanoparticle is dressed by a single dipole emitter; these are then coupled both to each other via near-field interactions allowing an optical excitation to be exchanged between emitters and to the plasmon mode. For the sake of simplicity, we further assume that the metal nanoparticles, i.e., the nanorods, are strongly anisotropic and can be characterized by a single SP mode with the energy ωsp. This mode is polarized in the (x, z)-plane along the dimensionless unit vector esp, as illustrated in Figure 1. We also assume that the quantum emitter transition dipole is polarized in the same plane along the dimensionless unit vector eqe and parallel to the major-axis of the nanorod. This allows us to define the SP and the quantum emitter transition dipole operators as (1)

μn̂ = eqeμqe (σn̂+ + σn̂−)

(2)

(4)

where the nanorod polarizability α(ω) =

pn̂ = esppsp (ψn̂ † + ψn̂ )

(3)

4πε0f0 3 3

ϵ(ω) − 1 [Q 1(η0)/η0]ϵ(ω) − Q 1′(η0)

(5)

depends on the spheroid dielectric function ϵ(ω) and the geometric factors given in terms of a Legendre polynomial of the second kind, Q1(z) = (z/2) ln[(z + 1)/(z − 1)] − 1, and its derivative, Q1′ (z). A number of quantum electro-dynamics (QED) based second quantization approaches have been proposed to treat the SP modes in low-dimensional materials.31,32 Below, we apply one such procedure utilizing direct mapping on the Drude model33,34 adopted for confined geometry nanoparticles. To quantize the plasmon modes, the nanorods are described by the Drude dielectric function ϵ(ω) = ϵ∞ −

ωp 2 ω(ω + iγ )

(6)

where ϵ∞ is the dielectric constant due to the high frequency response of the metal ion core and the second term describes the response of the metal electron gas characterized by the bulk plasma frequency, ωp, and damping rate, γ. For the numerical calculations we use parameters entering Drude dielectric function obtained by Johnson and Christy of the SP response for bulk Ag.35 Specifically, we set the high frequency dielectric constant of the ion core to ϵ∞ = 3.7, the electron plasma frequency to ωp = 9.1 eV, and the damping constant to γ = 20 meV. The environment dielectric constant is set to unity. Substituting eq 6 into eq 5, we recast eq 4 to the following form

respectively. Here {ψ̂ n, ψ̂ †n} are boson operators for the SPs. The quantum emitters are approximated as two-level systems and described by SU(2) spin operators σ̂in with i = ±, z. The quantity, μqe, is the transition dipole matrix element between the ground and the excited states of each quantum emitter. Unless different values are specified, below we set quantum emitter transition dipole strengths to μqe = 75 D. Such a large value can be expected in epitaxially grown semiconductor quantum dots.30 However, molecular systems generally do not demonstrate such a large transition dipole and are not

⎡ ⎤ ωsp2 ⎥En(ω) pn (ω) = ⎢α∞ + αsp 2 ⎢⎣ ω(ω + iγ ) − ωsp ⎥⎦

(7)

where 3110


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The Journal of Physical Chemistry A 4πε0f0 3

α∞ =

ϵ∞ − 1 [Q 1(η0)/η0]ϵ∞ − Q 1′(η0)

3

dimensionless vector determining the direction of the local electric field induced by the spheroid polarization along its major axis is

(8)

⎡ ⎤ ⎛ r1′ + r2′ + 2f ⎞ 0⎟ ⎥ ′ ′ T(x ,y ,z) = ∇⎢z′ ln⎜⎜ − + r r 1 2 ⎟ ⎢⎣ ⎥⎦ ⎝ r1′ + r2′ − 2f0 ⎠

4πε0f0 3 [Q 1(η0)/η0] − Q 1′(η0)

αsp =

[Q 1(η0)/η0]ϵ∞ − Q 1′(η0)

3

(9)

are the higher frequency ion core and the low frequency electron gas polarizabilities, respectively. Note that the latter quantity serves as a multiplicative prefactor in the resonance response of the electron gas at the SP frequency of the nanorod ωsp =

where

[Q 1(η0)/η0]ϵ∞ − Q 1′(η0)

Transforming eq 7 to the time domain, t

∫−∞ dt′

pn (t ) = αspωsp2 ×e

−γ(t − t ′)/2

⎛ x′⎞ ⎛ sin θ − cos θ ⎞ x ⎟ ⎜ ⎟=⎜ ⎝ z′⎠ ⎝ cos θ sin θ ⎠ z

()

2

sin[ ωsp − γ /4 (t − t ′)] ωsp2 − γ 2/4

En(t ′)

one finds that the dipole moment of the nth site is a solution for the equations of motion of a driven harmonic oscillator (12)

N

En̂ = − ∑ [1 − δnm]

With this in mind, we introduce an effective SP coordinate, un, and conjugate momentum πn. By neglecting the damping term in eq 12, we arrive at the following effective SP Hamiltonian ⎛ π2 u 2⎞ = ∑ ⎜⎜ n + mspωsp 2 n ⎟⎟ − 2msp 2 ⎠ n=1 ⎝ N

HSP

m=1 N

n=1

ωspmsp 2

(ψn̂ † − ψn̂ )

1 (ψn̂ † + ψn̂ ) 2ωspmsp

un̂ =

N

3. COUPLED SP-EXCITON HAMILTONIAN Having defined our terms, we can now move on to define the total Hamiltonian describing the physical system. For this we extend eq 16 to read

(14)

Ĥ sp − x =

n=1

n=1

+

−

3 8πε0f0 3

(16)

ψk̂ = ψk̂ † =

n

k

1 N

∑ e−iknψn̂

1 N

∑ eiknψn̂ †

n

n

(23)

(24)

obey Bose commutation relation and describe collective SP modes of the nanorod array characterized by the wave vector k = −π, ..., +π. The second term describes coupling of the SP mode to quantum emitters with the ground-excited state transitions described by the spin operator σ̂±n with n = 1, 2, ..., N being the array site index. The last term, Ĥ XY, describes the quantum emitters and its explicit form will be specified below. Coupling between Neighboring Metal Nanorods. The first term in the Hamiltonian given by eq 22 is derived by

∑ [1 − δnm]T([m−n]a ,0)pm̂ m=1 N

∑ (μm̂ ·T([m−n]a ,d))esp

̂ ∑ ∑ (σn̂+ + σn̂−)(λnkψk̂ † + λnk*ψk̂ ) + HXY (22)

N

m=1

1 2N

where the first term describes interaction free SP mode with the energies Ωsp(k). Field operators

with the dipole operator defined by eq 1 and the local electric field Ê̂ n(t) to be a superposition of the electric fields produced by the metal nanoparticles at sites m ≠ n and quantum emitters at all sites. Specifically, 3 En̂ = − 8πε0f0 3

∑ Ωsp(k)ψk̂ †ψk̂ k

(15)

N

∑ pn̂ ·En̂

(21)

Here rnm (Rnm) is the distance between metal nanoparticle on site n and a metal nanoparticle (quantum emitter) on site m, and rnm ̅ = rnm/rrm (R̅ nm = R nm/R rm).

(13)

results in the following quantized Hamiltonian (here and below ℏ = 1) Ĥ SP = ωsp ∑ ψn̂ †ψn̂ −

4πε0rnm3

4πε0R nm3

m=1

where psp = eu with e being electron charge. By writing out the equation of motion for un and comparing them with eq 12, one can easily identify the SP mass as msp = e2/αspωsp2. Applying second quantization to the SP momentum and coordinate πn̂ = i

pm̂ − 3 rnm ̅ ( rnm ̅ ·pm̂ )

μm̂ − 3R̅ nm(R̅ nm·μm̂ )

∑

−

N

∑ pn En

(20)

with θ being angle between the SP transition dipole and x-axis. Equations 17−20 for the local fields are derived using quasielectrostatic potential expansion in spheroidal coordinates36 and further retaining the dipole term. At distances, much larger that spheroidal aspect ratio, i.e., f 0 ≪ r1,2, eq 17 naturally acquires the following form of the field due to electric dipoles

(11)

pn̈ (t ) + γpṅ (t ) + ωsp2pn = αspωsp2En(t )

(19)

and the coordinates (x′, y, z′) associated with the spheroid frame of reference are related to the lab ones (x, y, z) via rotational transformation around the y-axis defined by the following equation

(10)

2

x′2 + y 2 + (z′ ± f0 )2

′ = r1,2

[Q 1(η0)/η0]ωp2

2

(18)

(17)

psp(ψ̂ †m

Here, p̂m = + ψ̂ m) is the scalar part of SP dipole operator defined in eq 1, a is the lattice period and d is the distance between metal nanorod and quantum emitter dressing it. The 3111


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Figure 2. (a) SP energy, ωsp, as a function of nanorod aspect ratio a0/b0. (b) Normalized near-field coupling between nanorods (eq 25) as a function of tilt-angle θ calculated for various aspect ratios a0/b0 and b0 = 2 nm. (c) For comparison, the normalized dipole−dipole coupling energy calculated for the same parameters according to eq 26 is provided.

of Jsp occurs at an angle close to the θ = 54.7°. For the cases shown here, the distance of separation between the centers of the nanorods is held fixed at a = 2a0 + δa and the end-to-end distance δa = 1 nm ensures that even at θ = 0 (linear aligned rods), the rods do not touch or overlap. For more closely packed rods (not shown in the plot), near-field effects become increasingly important and the crossover from negative to positive values of Jsp occurs at smaller angles. Comparison of the exact (eq 25) and dipole−dipole (eq 26) terms is provided in panels b and c of Figure 2. Our analysis indicates that the deviation from the dipole−dipole approximation is weak and only becomes significant for small angles and large aspect ratios. Exciton-SP Interaction Term. By performing the substitution of the expression for the local electric field produced by quantum emitters (i.e., the second term in eq 17) into the second term of the effective SP Hamiltonian (eq 16) and using the second-quantized representation for the corresponding dipole moments (eqs 1 and 2) as well as k-space representation for the SP field operators (eqs 23 and 24), we arrive at the second term of our Hamiltonian (eq 22) with the exciton-SP coupling parmaetr

starting with the effective SP Hamiltonian (eq 16) in which the second term is calculated using the local field coupling between the nearest neighbor nanorods (i.e., the first term in eq 17 with m = n ± 1) and the second quantized form of the SP dipole (eq 1). This results in the following near-field interaction energy between the nearest neighbor nanorods Jsp =

3psp2 8πε0f0 3

(esp·T(a ,0)) (25)

where f0 = a0 2 − b0 2 is the distance to the focal point of the spheroid measured from its center and vector T(x,z) given by eqs 18−20. a is the distance between centers of nearest neighbor nanorods. In the limit f 0 ≪ r1,2 describing the nanorod separation significantly exceeding their size, eq 25 recovers the spherical dipole−dipole interaction term Jspdd =

psp2 4πε0a3

(1 − 3(esp·ex)2 )

(26)

Here ex is a dimensionless unit vector denoting orientation of the x-axis. Note that this expression directly follows from the first term in the asymptotic expression of the local field given by eq 21. Furthermore, neglecting the coupling terms between the next to the nearest neighbor nanoparticles and higher, we diagonalized the SP terms by performing its Fourier transformation. This results in the first term in eq 22 with the SP energy dispersion Ωsp(k) = ωsp + 2Jsp cos(k)

λnk = e−ink [λ 0 + 2λ1 cos k + i2λ 2 sin k] = e−ink Λk

(28)

This term depends on the interaction energy between n-th emitter the nanorod it dresses and the nearest neighbor nanorods given by λ0 =

(27)

where k = −π, ..., −π. According to eq 27 the plasmonic band is determined by the values of ωsp and Jsp. Figure 2a shows the variation of the surface plasmon energy ωsp with increasing nanorod aspect ratio a0/b0, where a0 (b0) is the spheroid major (minor) radius. By simply varying the major radius, one can tune this frequency over a wide range spanning the near-infrared through ultraviolet spectrum. In Figure 2b, we show the variation in the near-field coupling between emitters (eq 25) for various nanorod aspect ratios a0/b0. The position of the SP band edge within the Brillouin zone depends on the sign of Jsp. According to Figure 2b, the sign of Jsp can be varied from negative to positive by changing the polarization of the SP mode from the ex to ez directions. The crossover between negative and positive values

λ1,2 =

3psp μqe 8πε0f0 3

(eqe ·T(0,d)) (29)

3psp μqe 16πε0f0 3

[(eqe ·T(−a ,d)) ± (eqe ·T(a ,d))] (30)

respectively. Here, a is distance between the centers of nearest neighbor nanords and d is distance between the centers of a nanorod and the quantum emitter sitting on top of it. Provided f 0 ≪ r1,2, the coupling constants given by eqs 29 and 30 acquire the following dipole−dipole interaction form λ 0dd = 3112

psp μqe 4πε0d3

[(esp·eqe) − 3(esp·ey)(eqe ·ey)]

(31)


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Figure 3. Squared absolute value of the exciton-SP coupling, |Λk|2, calculated for various values of the nanorod aspect ratio a0/b0, width b0, and tiltangle θ at k = 0 in units of the SP frequency, ωsp. The distance between each quantum emitter and the nanorod it dresses is set to d = 3 nm.

Ĥ XY. However, all raising/lowering operators belonging to different sites commute, [σ̂αi , σ̂βj ] = 0 for i ≠ j = 1, ..., N, with N being number of quantum emitters and α, β = x, y, z. The Pauli principle requires that they anticommute between sites. Consequently, the σ̂i ± = (σ̂xi ± σ̂yi )/√2 operators as defined above describe neither fermions nor bosons and are inconvenient for describing a many-body system. This can be resolved defining a new set of operators (Holstein−Primakoff transformation)

psp μqe

λ1dd =

2

2 3/2

4πε0(a + d )

× [(esp·eqe) − 3 cos2 α(esp·ex)(eqe ·ex) − 3 sin 2 α(esp·ey)(eqe ·ey)] 3psp μqe

λ 2dd =

8πε0(a 2 + d 2)3/2

(31)

sin 2α

× [(esp·ex)(eqe ·ey) + (esp·ey)(eqe ·ex)]

† σ̂iz = 2bî bî − 1

(32)

where sin α = d/(a2 + d2)1/2, and a set of unit vectors ex, ey, and ez determines orientation of x, y, and z axes (Figure 1). Alternatively, eqs 31 and 32 can be obtained from eqs 16 and 21. In Figure 3 we show the absolute value of the exciton-SP coupling, |Λk|2, calculated according to eqs 28−30 for various values of the nanorod aspect ratio a0/b0, width b0, and tilt-angle θ at k = 0 in units of the SP frequency, ωsp. The results indicate that the highest couplings are achieved with thin (b0 = 0.5 nm) nanorods of aspect ratios between 10 and 15 and alignment defined by θ = 60°. The use of approximate eqs 31 and 32 (not shown) does not bring significant deviations from the results in Figure 3. Quantum Emitter Hamiltonian. We now focus on the last term in eq 22 describing the quantum emitters and their coupling to each other resulting in the delocalized exciton states. Our exciton model for this is identical to the so-called XY model describing a linear chain of spins coupled to a transverse magnetic field with the Hamiltonian ̂ = HXY

ωqe 2

∑ σn̂ z + n

Jqe 2

∑ (σn̂+σn̂−+ 1 + σn̂++ 1σn̂−) n

Jqe =

4πε0a3

(1 − 3(eqe ·ex)2 )

σ̂i+ =

† 2 (1 − bî bî )1/2 bi

σ̂i− =

2 bî (1 − bî bî )1/2

†

(36)

†

(37)

where the bî ’s obey the canonical Bose algebra [bî , b†ĵ ] = δij. This transformation preserves spin operator commutation relation [σ̂+i , σ̂−j ] = 2δijσ̂zi and can be used to describe the propagation of the quantum emitter excitons using the identity 1 2

N

∑ (σn̂+σn̂−+ 1 + σn̂+σn̂−+ 1) n=1 N

=

†

†

∑ (bn̂ bn̂ + 1 + bn̂ + 1bn̂ ) + 6(1/N ) n=1

(38)

We introduce a small 6(1/N ) error associated with the terms operator terms higher than the second order where N is the number of quantum emitters describing the effects associated with deviation from Bose statistics. It is also important to note that the quantum emitter Hamiltonian explicitly preserves the total number of bosons (excitations) in the system viz. N̂ b = ∑nb†n̂ bn̂ and [Ĥ XY, N̂ b] = 0. As a result, we can write total spin operator Ŝz = ∑iσ̂zi = 2N̂ b − N, and observe that each spinless excitation created by b†î carries an Szωqe/2 = ωqe quantum of energy. The quantum-emitter terms can now be brought into diagonal form by defining the transformation

(33)

Here, ωqe is the two-level system transition energy. The nearest neighbor interaction between the quantum emitters can be well represented by the dipole−dipole interaction energy μqe 2

(35)

Ĥ x =

(34)

†

∑ bî Hijbĵ ij

where μqe is the quantum emitter transition dipole and a is the distance between neighboring emitters. Because the model corresponds to the free-propagation of excitons on a lattice, one is tempted to immediately diagonalize

(39)

where Hij = ωqeδij + Jqe (δi , j + 1 + δj , i + 1) 3113

(40) DOI: 10.1021/acs.jpca.5b10726 J. Phys. Chem. A 2016, 120, 3109−3116

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Figure 4. Noninteracting SP, Ωsp, and exciton Ωk dispersion curves calculated according to eqs 27 and 44, respectively. The following geometric parameters for the array are adopted in the calculations a0 = 10 nm, b0 = 1 nm, θ = 0, and a = 20 nm. (b) Upper, Ω+, and lower, Ω−, polariton branches (eq 52) formed after exciton-SP interaction, Λk (eqs 28−30) is taken into account for the parameters adopted in (a).

Introducing the Fourier expansion of the exciton operators bn̂ = †

bn̂ =

1 N

∑ eiknbk̂

1 N

∑ e−iknbk̂

(41)

k

bk̂ = −vk Ψ+̂ (k) + uk Ψ̂−(k)

(48)

(42)

k

uk =

vk =

†

∑ ωx(k)bk̂ bk̂

(43)

k

Ω+(k) − Ωsp(k) (Ω+(k) − Ωsp(k))2 + |Λk|2

Ωx(k) = ωqe + 2Jqe cos(k)

H±̂ =

(44)

Λk (Ω+(k) − Ωsp(k))2 + |Λk|2

(50)

∑ {Ω+(k) Ψ+̂ (k)† Ψ+̂ (k) + Ω−(k) Ψ̂−(k)† Ψ̂−(k)} k

Similar to SP (eq 27), the position of the quantum emitter band edge within the Brillouin zone depends on the sign of Jqe. Because we keep orientation of the quantum emitter and SP transition dipoles parallel, the sign of Jqe has an angular dependence similar to that shown for the SP in Figure 2c. Pulling everything together, our model Hamiltonian (eq 22) takes the form of a series of independent boson terms

(51)

where the Bose operators Ψ̂+(k) and Ψ̂−(k) naturally describe the quasiparticles associated with the upper, Ω+(k), and lower, Ω−(k), polariton branches, respectively. The associated polariton energies are Ω±(k) =

†

∑ {Ωsp(k)ψk̂ †ψk̂ + Ωx(k)bk̂ bk̂

Ω ph(k) + Ωsp(k) 2

k

±

†

+ (bk̂ + bk̂ )(Λ*k ψk̂ † + Λkψk̂ )}

(45)

Note that this is similar to the Dicke model for the coupling between a photon cavity mode and an ensemble of spins that has been widely used in atomic and molecular physics. However, in this case, each cavity (i.e., SP) mode, ψ̂ k, is coupled to a single exciton mode bk̂ rather than to an ensemble of independent two-state atoms.

†

∑ {Ωsp(k)ψk̂ †ψk̂ + Ωx(k)bk̂ bk̂ k †

+ (Λ*k ψk̂ †bk̂ + Λkbk̂ ψk̂ )}

⎛ Ω ph(k) − Ωsp(k) ⎞2 ⎜⎜ ⎟⎟ + |Λk|2 2 ⎝ ⎠

(52)

In Figure 4a, we plot noninteracting SP, Ωsp(k), and exciton, Ωk(k), dispersion curves. Turning on interaction Λk splits the curves at the crossing points into polariton branches Ω± as shown in panel b. These are analogous to the upper and lower polariton branches except that both the SP and excitonic modes are described by cos(k) dispersion terms. Comparing panels b with a, one can conclude that in the vicinity of k = 0 the upper polariton branch, Ω+(k), represents the exciton states dressed by SP and the lower one, Ω−(k), the SP states dressed by excitons. The state dressing shows up as the energy shift of the Ω±(k=0) compared to the energies of Ωsp(k=0) and Ωx(k=0). The polariton Rabi splitting is Ω+(k) − Ω−(k) = 100 meV. It values is 4 times larger than typical plasmon damping rate γ = 20 meV in silver,37 resulting in distinguishable exciton-lake and SP-like branches. At this point, we focus on the SP-like, Ω−(k), polariton branch and ask whether or not a 1D system described by a cos(k) dispersion can undergo a quantum-phase transition to

4. QUANTUM SYMMETRY BREAKING Having defined our system by the Hamiltonian given in eq 45, we diagonalize this Hamiltonian. For this purpose we adopt the rotating wave approximation (RWA) ĤRWA =

(49)

and result in the following transformed Hamiltonian,

with the energy dispersion

Ĥ SP =

(47)

The unitary transformation matrix elements eliminating the desired term now read

†

where k = −π, ..., −π, and bk̂ ’s are canonical boson operators describing delocalized exciton states with momentum k, we thus arrive at the diagonal exciton Hamiltonian Ĥ x =

ψ̂k = uk Ψ+̂ (k) + vk Ψ̂−(k)

(46)

To eliminate the exciton-SP coupling, we adopt the following polariton transformation for the SP and photon operators: 3114


Article


the fractional population in the excited states will be less than that of the ground state, implying that under these conditions, the population will collapse to the lowest wave-vector state with decreasing temperature. In Figure 5 we present a phase diagram showing the critical λ vs β|J| as determined by the values of λ and g = β|J| where Nex in

form a Bose−Einstein condensate (BEC). In general, the BEC transition is forbidden by symmetry for 1- and 2-dimensional systems. The reason for this can be traced to a nonremovable singularity at k = 0 when the number density is integrated over all momentum states and is the basis of the Mermin−Wagner theorem. However, for interacting systems and systems in potential wells, BEC is possible even in 1 and 2 dimensions. To determine whether or not such transitions are possible in our system, we consider the average population number for a weakly interacting Bose ensemble and compare the total ground state population to the total excited population at finite temperature. Let us introduce partition function ̂

̂

Ξ = Tr[e−β(H − μN )]

(53)

where μ is the chemical potential and N̂ is the number of bosons in the system. We shall assume at this point that the system is open and that the total number of bosons in the system is allowed to fluctuate about some average value. Writing the free energy as − βA = log Ξ, we compute the occupation numbers for the lower polariton curve as ⎛ ∂A ⎞ N=⎜ ⎟ ⎝ ∂μ ⎠T , V π

λ + 1−λ

λe−β Ω−(k)

∑

1 − λe−β Ω−(k)

k≠0

eq 58 is less than N0 from eq 57. A low chemical potential corresponds to larger values of fugacity. For small β|J|, the fractional occupation of the ground state is small and Nex > N0 for all values of λ < 1. Decreasing the temperature or increasing the coupling gives rise to a critical condition where Nex < N0 at some λ < 1, indicating that at this condition, symmetry breaking must occur as the excited state population begins to collapse to the ground state. We thus determine the phase diagram by plotting the loci of where Nex = N0, given λ and β|J|. To the right of the curve, corresponding to low-temperature or large J, is the condensate/superradiant regime. To the left is the normal regime at high temperature or small J. Our theory predicts that the system is expected to become superradiant for all cases where β|J| ≳ 13. In terms of the physical parameters, |J| characterizes the band shape of the lower Ω−(k) branch. Taking the system described in Figure 4 as a typical parametric case, with |J|∼ 150 meV, the superradiant regime occurs when T ≲ 134 K. Even if our estimate of the coupling is off by even a few orders of magnitude in the coupling strength, the transition would be accessible at any reasonable temperature.

(55)

1 − λe−β Ω−(k)

k =−π

=

(54)

λe−β Ω−(k)

∑

=

Figure 5. Phase diagram for ideal Bose-gas in a 1D band. Here we take the scaled temperature β|J| as an order parameter and λc as the critical fugacity.

= N0 + Nex

(56) (57)

Here, we define the fugacity (absolute activity) λ ∈ [0, 1), such that the energy origin is at the bottom of the Ω−(k) dispersion curve and β = 1/kBT and the sum is taken over the first Brillouin zone. The first term thus represents the number of particles in the lowest energy state whereas the second sum reflects the number of particles in the excited states. BEC occurs when at finite temperature and λ < 1 the ground state population diverges logarithmically while the excited state population remains finite. To a good approximation, close to k = 0, Ω−(k) ≈ |J|(1 − cos(k))/2 = |J| sin2(k/2). We take the sum over k to an integral and expand the integrand in a geometric series. The resulting integrals can be performed exactly producing an expression for the number particles in the excited states, viz. Nex = 2

∫0

2

λe−g sin (k /2)

π

+

dk

2

1 − λe−g sin (k /2)

∞

=

5. CONCLUSION We propose a model for interacting collective SP modes and exciton states in an array of metal nanorods dressed by quantum emitters. Starting with a second-quantized picture of the anisotropic SP response, we construct a model Hamiltonian and explicitly compute the coupling required parameters. The final model can be solved exactly, resulting in upper and lower polariton dispersion curves. Moreover, we provide a proof of principle demonstration that the model admits a BEC transition to create a condensate of SP-excitonic states. Our work suggests a set of material parameters and nanorod alignments that can be optimized to achieve this condition under currently achievable laboratory and synthetic conditions.38

∑ λn[2 ∫

π +

2

dk e−gnsin (k /2)]

0

n=1 ∞

= 2π ∑ λ ne−gn /2I0(gn/2) n=1

(58)

where g = β|J| and I0(x) is the modified Bessel function of the first kind. The final sum can be evaluated numerically and rapidly converges for a given finite value of β|J| > 0 and 0 ≤ λ < 1. Formally, this is a number density because the integration is over only the first Brillouin zone. At some critical value of λ and finite temperature (holding the total number N0 + Nex fixed),

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AUTHOR INFORMATION

Corresponding Authors

*E. R. Bittner. E-mail: [email protected]. 3115


Article

The Journal of Physical Chemistry A *A. Piryatinski. E-mail: [email protected].

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work at the University of Houston was funded in part by the National Science Foundation (CHE-1362006) and the Robert A. Welch Foundation (E-1337). The work at Los Alamos National Lab was funded by the Los Alamos Directed Research and Development (LDRD) funds.

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Polaritons in a Metal-Nanorod

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