Letter pubs.acs.org/JPCL
Nonequilibrium Response of Nanosystems Coupled to Driven Quantum Baths Hermann Grabert,†,‡ Peter Nalbach,¶,§ Joscha Reichert,¶,∥ and Michael Thorwart*,¶,∥ †
Freiburg Institute for Advanced Studies (FRIAS), Universität Freiburg, Albertstraße 19, 79104 Freiburg, Germany Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany ¶ I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany § Westfälische Hochschule, Münsterstraße 265, 46397 Bocholt, Germany ∥ The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany ‡
S Supporting Information *
ABSTRACT: Commonly, nanosystems are characterized by their response to timedependent external fields in the presence of inevitable environmental fluctuations. The direct impact of the external driving on the environment is generally neglected. While this approach is satisfactory for macroscopic systems, on the nanoscale, an interaction of external fields with the environment is often unavoidable on principle. We extend the standard linear response theory of quantum dissipative systems to strongly driven baths. Significant modifications are found for two paradigm examples. First, we evaluate the polarizability of a molecule immersed in a strongly polarizable medium that responds to terahertz radiation. We find an increase of the molecular polarizability by about 30%. Second, we determine the response of a semiconductor quantum dot in close proximity to a metallic nanoparticle. Both are placed in a polarizable medium and exposed to electromagnetic irradiation. We show that the response of the quantum dot is qualitatively modified by the driven nanoparticle, including the generation of an additional channel of stimulated emission.
A
how the transition between the quantum and the classical world arises. Yet, the impact of driving forces on the environmental excitation modes is always neglected. Here, we generalize the theory of quantum dissipative response to include the driving of the environment. Sizable implications are shown to occur for the two paradigm examples. The model of the quantum dissipative harmonic oscillator is applied to the dynamic polarizability of a molecule dissolved in water,8−10 which is pumped in the terahertz range. The spinboson model applies to the optical absorption and emission of a quantum dot coupled to a metallic nanoparticle11 and subjected to electromagnetic radiation. In both cases, the driving of the bath leads to important signatures in the system’s response. Noteworthy, our approach applies likewise to all other quantum systems embedded in an environment causing linear dissipation (unpublished results). Hence, the present theory generalizes a generic textbook case ubiquitous in physics, chemistry, biochemistry, and engineering. A Molecule in a Strongly Terahertz-Driven Polarizable Medium. First, we consider a physical system that can be modeled by a dissipative quantum harmonic oscillator in a driven harmonic bath. In particular, we determine the contribution of a driven
fundamental scientific approach divides a complex physical system into a clearly identifiable “system” and a “bath” (or “environment”) surrounding the system of interest.1 An advanced formalism of system-bath models2−6 has laid the foundations for the modern theory of quantum dissipation. Time-dependent external fields are applied to record the system’s response.7 The quantum statistical description of dissipative quantum systems including their response to external forces is well established,4,5 yet the impact of the driving forces on the environmental modes is commonly neglected. This assumption is usually justified for macroscopic systems, however, for nanoscale systems, an interaction of the external fields with the environment is often unavoidable. In fact, the response of the system in its driven environment is implicitly recorded in experiments while usually being neglected in the theoretical model. For quantum dissipative systems, two paradigms exist: a particle in a harmonic potential field coupled to an environment4,5 and the dissipative quantum mechanical two-state system (or spin-boson model).2,3,5 Both are well studied for the case of an undriven bath. The dissipative quantum harmonic oscillator is central since it is exactly solvable analytically. The spin-boson model serves as a keystone model for studying relaxation and dephasing of generic quantum systems coupled to quantum fluctuations with arbitrary coupling strength. Moreover, it allows the study of the fundamental question of © XXXX American Chemical Society
Received: March 29, 2016 Accepted: May 13, 2016
2015
DOI: 10.1021/acs.jpclett.6b00703 J. Phys. Chem. Lett. 2016, 7, 2015−2019
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The Journal of Physical Chemistry Letters
When the external field E is switched on at time t = 0, the timedependent field inside the sphere is
vibrational mode of a dielectric environment, specifically water, to the dynamic polarizability α(ω) of a polar molecule dissolved in water. In vacuum, the dynamic polarizability of a molecule is a sum of terms of the form9 α(ω) = (e2/M) f/ (Ω2−ω2), where Ω is the frequency of the vibrational transition and f the oscillator strength. We focus here on a single selected mode Ω and set f = 1. The molecule with dipole moment μ = eq may then be modeled by the Hamiltonian HS = p2/2M + MΩ2q2/2 of a particle with position q, momentum p, and mass M moving in a harmonic potential. A constant electric field E gives rise to an electric dipole moment μ = αE, where α = e2/ MΩ2 is the contribution of the mode Ω to the static polarizability. Then, we assume that the molecule is immersed in water, i.e., surrounded by a dielectric medium with a frequency-dependent complex dielectric function ε(ω) = ε′(ω) + iε″(ω). A standard model8−10 describes the dielectric by a continuum outside of a sphere of radius a surrounding the system as schematically shown in Figure 1. We add a time-dependent external electric
Esp(t ) = E(t ) +
∫0
t
⎛ 2μ(s) ⎞ ⎟ dsχ (t − s)⎜E(s) + ⎝ a3 ⎠
where we have introduced the temporal response function χ(t) −iωt = ∫∞ /2π. The field exerts a force eEsp(t) on the −∞dωχ(ω)e system where the last term in eq 3 gives rise to a frictional force. The expectation value ⟨μ(t)⟩ = e⟨q(t)⟩ of the dipole moment of the molecule embedded in a dielectric and driven by a timedependent external field E(t) obeys the equation of motion ⟨μ(̈ t )⟩ + Ω2⟨μ(t )⟩ =
e2 [E(t ) + M 2e 2 Ma3
−
∫t
∫t
t
dsχ (t − s)E(s)] 0
t
dsχ (t − s)⟨μ(s)⟩ 0
2
where we have used ⟨μ(t )⟩ = Re{α(ω),e polarizability
Figure 1. Sketch of a polar molecule dissolved in a polar solvent. The time-dependent electric field E(t) (red stripes) acts on both the solvent and the molecule with a molecular vibrational frequency Ω.
α(ω) =
field E(t) parallel to the dipole moment μ, which couples both to the dipole moment of the system and the dielectric medium. When the wavelength of the external field is much larger than the sphere radius a, we may disregard the spatial dependence of E. It is a matter of simple electrodynamics to determine for a given external field E(ω) and molecular dipole μ(ω) the additional field inside the sphere: 2μ(ω) ⎞ ε(ω) − 1 ⎛ ⎜E(ω) + ⎟ 2ε(ω) + 1 ⎝ a3 ⎠
ε(ω) − 1 = χ ′(ω) + iχ ″(ω) 2ε(ω) + 1
−iωt
e Ma3
=
α 2 Ω. a3
(5)
Via the relation
}, we obtain the molecular dynamic
1 + χ (ω) e2 2 ⎡ M Ω 1 − 2α3 χ (ω)⎤ − ω 2 ⎣ ⎦ a
(6)
Since χ(ω) is complex, the denominator of eq 6 gives the usual frequency shift and line broadening of the dissolved molecule, while the factor 1 + χ(ω) in the numerator modifies the response as a consequence of the direct coupling of the field to the surrounding dielectric. We note that the effect of the driving-induced bath force is included here on the level of linear response (or mean-field) theory appropiate to weakly driven systems with linear dissipation. Going beyond this is in principle possible, but becomes rather involved.12,13 To illustrate the magnitude of the effects due to the driven dielectric bath, we consider the collective vibrational mode of water in the THz range. The dielectric function of water in the range between 0.2 and 7 THz can effectively be described by15
(1)
This includes the applied field E(ω), the response of the dielectric medium to the applied field, and the reaction of the dielectric medium to the system dipole μ(ω). This latter term is the well-known Onsager reaction field and is responsible for the dissipative motion of the system. The middle term, arising from driving the environmental modes, is denoted as the cavity field in the theory of solvation.12−14 Its contribution to the effective polarizability has been determined by recursively calculating higher-order hyperpolarizabilities12 or by numerical selfconsistent reaction-field simulations,13 while we provide in the following analytical results on the basis of linear response theory. In other circumstances of driven quantum systems, such a driven-bath term is usually disregarded.5,9,10 As seen from eq 1, the response of the dielectric medium is characterized by the complex frequency-dependent susceptibility: χ (ω) =
(4)
It describes a harmonic motion with vibrational frequency Ω driven by the field given in eq 3. For a periodic external field E(t ) = Re,e−iωt switched on at t0 → − ∞, the solution of this equation follows as ⎫ ⎧ ⎪ 1 + χ (ω) e2 ⎪ −iωt ⟨μ(t )⟩ = Re⎨ ,e ⎬ 2 α 2 2 M ⎪ Ω ⎡1 − 3 χ (ω)⎤ − ω ⎪ ⎦ ⎩ ⎣ ⎭ a
Esp(ω) = E(ω) +
(3)
ε(ω) =
As Δε1 Δε2 + + 2 + ε∞ 1 − iωτ1 1 − iωτ2 ωs − ω 2 − iωγs (7)
with parameters fitted to experimental data: Δε1 = 73.9, τ1 = 8.76 ps, Δε2 = 1.56, τ2 = 0.224 ps, As = 35.1 × (2π)2THz2, ωs = 5.3 × (2π) THz, γs = 5.4 × (2π) THz, ε∞ = 2.34. This model describes the slow and fast Debye dielectric modes and the intermolecular stretching vibrational mode around 5 THz. Figure 2a) shows the real and imaginary parts of ε(ω) together with the contribution of the third term in eq 7 marked as εs(ω). Also, the resulting susceptibility χ(ω) is shown. In Figure 2 b), we show the dynamic polarizability α(ω) = α′(ω) + iα″(ω) for a test molecule with a typical vibrational frequency Ω = 4.5 × (2π) THz. In particular, we compare the cases when the
(2) 2016
DOI: 10.1021/acs.jpclett.6b00703 J. Phys. Chem. Lett. 2016, 7, 2015−2019
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The Journal of Physical Chemistry Letters
element. The Pauli matrices σi(i = 1,2,3) describe a pseudospin in the two-dimensional Hilbert space of the SQD exciton. For the sake of simplicity, we assume that the dipole moment of the dot is along the z-axis connecting the MNP and the SQD (see Figure 3), and we apply an external time-dependent electric field E(t) along the same axis. We also focus on the case R ≫ as,am, where the electric fields caused by the SQD and the MNP in the surrounding medium are of dipolar form. These assumptions are not necessary16 but simplify the analysis without affecting the conclusions. Three spatial regions with different dielectric functions arise. From an evaluation of the electrostatic potential within the SQD, within the MNP and in the surrounding medium, and obeying the proper boundary conditions, one finds for the field Es at the SQD Es = (1 + χs )E + 2χs Figure 2. (a) Dielectric function ε(ω) = ε′(ω) + iε″(ω) of the water solvent (top: real part; bottom: imaginary part) with the intermolecular stretching mode at ωs = 2π × 5.3 THz,15 which contributes εs(ω). Moreover, we show the susceptibility χ(ω) = χ′(ω) + iχ″(ω). (b) Dynamic polarizability α(ω) = α′(ω) + iα″(ω) for a test molecule with a vibrational frequency Ω = 4.5 × (2π) THz for the undriven (blue) and driven (red) bath (top: real part, bottom: imaginary part).
+
4as3am3 6
R
μs as3
−
2am3 R3
(1 + χs )χm E
⎡ μ⎤ (1 + χs )χm ⎢χs E − (1 + χs ) s3 ⎥ as ⎦ ⎣
(8)
where χm = (ε0 − εm)/(2ε0 + εm), χs = (ε0 − εs)/(2ε0 + εs), and where we have suppressed the frequency-dependence of all quantities. Equation 8 is valid apart from terms of higher order in a3s /R3, a3m/R3 and when the external field and the two dipole moments are oriented (anti)parallel (see Figure 3). The first term in eq 8 gives the externally applied field at the SQD, and the second term is the reaction field of the surrounding medium to the dipolar field of the SQD. The third term represents the field at the SQD caused by the polarization of the MNP induced by the applied field E. Finally, the fourth and fifth terms describe the effect of the additional polarization of the MNP induced by the dipolar field of the SQD, which includes a part already present in the absence of driving and a contribution induced by the external field. Using μs = μσ1, the result in eq 8 may be written as Es(ω) = E(ω) + χ(ω)σ1(ω) + λ(ω)E(ω), with the functions
external electric field drives only the molecule, and when it drives both the molecule and the surrounding water. We find a significant enhancement of the real part of the polarizability by up to 33% and of the imaginary part by 34%. The significant enhancement of the response is solely due to the additional enhancement of the electric field generated by the polar solvent, which itself is polarized by the external field. A Semiconductor Quantum Dot Close to a Metallic Nanoparticle in an Irradiated Medium. As the second paradigm, we consider a model of a quantum two-level system coupled to a driven harmonic bath. Such a model describes, e.g., a spherical semiconductor quantum dot (SQD) of radius as and dielectric function εs(ω), which is embedded in a medium with dielectric function ε0(ω) and which contains at distance R from the SQD a metallic nanoparticle (MNP) of radius am and dielectric function εm(ω).11 A sketch is shown in Figure 3. We focus on
χ=
⎤ 2as3am3 2μ ⎡ ⎢ (1 + χs )2 χm ⎥ χ − s 3 6 as ⎣ R ⎦
(9)
and λ = χs −
⎛ 2a 3 ⎞ (1 + χs )χm ⎜1 − 3s χs ⎟ R R ⎠ ⎝
2am3 3
(10)
Here, χ(ω) describes the reaction field giving rise to the frictional influence of the environment, while λ(ω) arises from the driving of the environmental modes by the applied field. We study the interesting case when the coupling of the SQD to the MNP dominates the two-state dynamics. To this end, ε0 and εs are assumed constant. The dielectric function of the MNP is taken of Drude-Sommerfeld form εm(ω) = 1 − ω2p/(ω2 + iΓω), where ωp is the plasma frequency and τD = 1/Γ is the relaxation time. From eq 9, we obtain the absorptive part of χ(ω) as
Figure 3. Spherical SQD in close proximity to a MNP immersed in a medium with dielectric constant ε0. The system is driven by a timedependent electric field indicated by the red arrows.
χ ″(ω) = −
two excitonic states of the SQD. The Hamiltonian of a two1 level SQD is Hs = 2 ℏω0σ3 − μs Es , where ℏω0 is the energy difference between the ground state (σ3 = −1) and the exciton state (σ3 = 1). Moreover, μs = μσ1 is the electric dipole moment of the SQD and μ is the interband optical transition matrix
=κ
4μam3 R6
(1 + χs )2 χm″ (ω) Γωr2ω
(ω 2 − ωr2)2 + Γ 2ω 2
(11)
where 2017
DOI: 10.1021/acs.jpclett.6b00703 J. Phys. Chem. Lett. 2016, 7, 2015−2019
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The Journal of Physical Chemistry Letters κ=
2233ε03μam3 2
6
(2ε0 + εs) (2ε0 + 1)R
states of the SQD and induces two fundamental resonances. The resonance at the larger frequency ω2,max is dominated by the MNP. Interestingly enough, the driven bath modes cause a decrease of the height of response peak 1 down to 43% of its value for the undriven case, while at peak 2 the response is enhanced by a factor of ∼16 (real part) or of ∼4 (imaginary part). Moreover, the SQD, which only absorbs light under conventional direct driving, also emits light when the driven MNP plasma mode is taken into account. This qualitative change in the response of the SQD in the presence of a pronounced peak in the spectral density of driven bath modes can be directly used to identify the underlying mechanism in an experiment. The key prerequisite for the qualitatively different response is that a well pronounced bath mode exists to which the external drive couples. Such a bath mode is generated by the presence of a single metallic nanoparticle. Driven bath modes occur quite frequently in physics, chemistry, and adjacent fields, and the generalized response theory presented here can likewise be applied to a large variety of other systems (unpublished results) which will be the subject of future work. In any case, the concept of driven quantum baths provides the unifying theoretical framework to determine the dynamical response of quantum systems to externally driven environments. Another interesting example in which the cooperative response of many environmental degrees of freedom enhances the dynamical response of a central system to a globally applied external driving field is surface enhanced Raman scattering (SERS).22,23 Surface plasmons are excited and strongly couple to molecules adsorbed on the surface.
(12)
The absorptive part χ″(ω) of the environmental susceptibility represents an effective spectral density of a bosonic environment of the effective spin- 1 . It has a structured Ohmic form 2 with a Breit−Wigner resonance with resonance frequency ωr = ωp/ 2ε0 + 1 . The quantum dynamics is illustrated in terms of the mean dipole moment Q(t) = ⟨μs(t)⟩ = Re{Q(ω)e−iωt} and has been studied previously17−19 for a structured (undriven) environment. We assume a monochromatic externally applied field E(t ) = Re{,e−iωt }. The driven bath modes, which have not been taken into account so far, can be incorporated into previous approaches in terms of the effective field Eeff (t ) = Re{Λ(ω),e−iωt }
(13)
where Λ(ω) = 1 + λ(ω). The linear response of the SQD to the monochromatic external field follows from Q (ω) = Φ0(ω)Eeff (ω) = Φ(ω), , with the linear susceptibility Φ(ω) of the quantum dot. The susceptibility Φ0(ω) in the absence of driving of the bath modes has been evaluated previously in closed form19 for a structured spectral density with small κ, any value of Γ and zero temperature. Combining the explicit result for Φ0(ω) of Gan et al.19 with eq 10, we readily obtain the zero temperature linear susceptibility Φ(ω) = λ(ω)Φ0(ω) of an SQD in the presence of a driven MNP (see Supporting Information). To be specific, we show in Figures 4 a) and b) results for the frequency-dependent susceptibility of a CdSe nanocrystal
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.6b00703. Explicit equations for the linear susceptibility of the semiconductor quantum dot embedded in a driven dielectric medium and in the presence of a metallic nanoparticle in close vicinity (PDF)
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■ Figure 4. Dielectric susceptibility of the SQD with peak 1 (a) and peak 2 (b) (top: real part, bottom: imaginary part) for the undriven (blue, Λ(ω) = 1) and the driven (red) environment for R = 20 nm. Frequencies are shifted by the renormalized resonance frequencies ω1,max ≈ ω0 for peak 1 and ω2,max ≈ ωr for peak 2.
ACKNOWLEDGMENTS J.R., P.N., and M.T. acknowledge financial support of The Hamburg Centre for Ultrafast Imaging (CUI) with the German Excellence Initiative supported by the Deutsche Forschungsgemeinschaft. M.T. acknowledges the kind hospitality of the FRIAS during an extended research stay during the summer of 2014.
forming a two-level SQD coupled to a gold MNP at a distance of R = 20 nm. Appropriate parameters for the gold MNP are ℏωp = 8.5 eV and τD = 14 fs.20 Further, the MNP radius is taken11 as am = 7.5 nm. For the SQD we choose parameters11,21 appropriate for CdSe nanocrystals, with radius as = 0.65 nm and an excitonic energy gap of ℏω0 = 2.5 eV. Its dielectric constant is εs = 6.0. Finally, we fix the dielectric constant of the medium to ε0 = 1. The coupling to the MNP splits the two quantum
(1) Bogoliubov, N. N. On Some Statistical Methods in Mathematical Physics; Ukrainian SSR Academy of Sciences: Kiev, Ukraine, 1945 (in Russian). (2) Caldeira, A. O.; Leggett, A. J. Influence of Dissipation on Quantum Tunneling in Macroscopic Systems. Phys. Rev. Lett. 1981, 46, 211−214. (3) Leggett, A. J.; Chakravarty, S.; Dorsey, A. T.; Fisher, M. P. A.; Garg, A.; Zwerger, W. Dynamics of the Dissipative Two-State System.
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