Theory of non-Markovian dynamics in resonance fluorescence spectrum
We present a detailed theoretical study of non-Markovian dynamics in the fluorescence spectrum of a driven semiconductor quantum dot (QD), embedded in a cavity and coupled to a three-dimensional (3D) acoustic phonon reservoir. In particular, we investigate the effect of pure dephasing on one of the side-peaks of the Mollow-triplet spectrum, expressed in terms of the off-diagonal element of the reduced system operator. The QD is modeled as a two-level system with an excited state representing a single exciton, and ground state represents the absence of an exciton. Coupling to the radiative modes of the cavity is treated within usual Born-Markov approximation, whereas dot-phonon coupling is discussed within non-Markovian regime beyond Born approximation. Using an equation-of-motion technique, the dot-phonon coupling is solved exactly and found that the exact result coincides with that of obtained within Born approximation. Furthermore, a Markov approximation is carried out with respect to the phonon interaction and compared with the non-Markovian lineshape for different values of the phonon bath temperature. We have found that coupling to the phonons vanishes for a resonant pump laser. For a non-resonant pump, we have characterzied the effect of dot-laser detuning and temperature of the phonon bath on the lineshape. The sideband undergoes a distinct narrowing and aquires an asymmetric shape with increasing phonon bath temperature. We have explained this behavior using a dressed-state picture of the QD levels.
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The laws of quantum mechanics allow for quantum computersvon Neumann (1955); Benioff (1980); Feynman (1982), which are known to be significantly more powerful than classical computers. In a quantum computer, information is stored in quantum bits (qubits), rather than classical bitsDivincenzo (1997); Schumacher (1995). A single qubit represents a zero or one and is a two-level system with two energy levels which can be used to store and process the informationLoss and DiVincenzo (1998). An example of a two-level system, frequently used in quantum optics, is composed of the ground and excited states of an atom. Semiconductor QDs can be modeled as a two-level system with one exciton in the excited stateZrenner et al. (2002); Brunner et al. (2009); Strauß et al. (2016). Semiconductor QDs embedded inside a cavity has been a subject of intense research as a promising candidate for quantum computation and information tasks, as well as a source of single photonKuhlmann et al. (2015).
Recently, experiments on the cavity embedded QDs have been reported to show different spectral features of the Mollow-triplet fluorescence spectrumUlhaq et al. (2013); Cui and Raymer (2006); McCutcheon and Nazir (2010). In particular, a dot coupled to the acoustic phonon bath in the super-ohmic regime has shown modified spectral features as a function of the phonon bath temperature. More precisely, the triplet sideband is observed to show a systematic spectral sideband broadening for both resonant and off-resonant cases. This problem was studies experimentallyUlrich et al. (2011) and anlyzed theoreticallyRoy and Hughes (2012) in terms of the usual Born-Markov approximation. However, the pure dephasing process a 3D phonon bath is in the form a super-ohmic independent boson model (IBM), which is known to be highly non-MarkovianMcCutcheon and Nazir (2013); Weiler et al. (2012); Krummheuer et al. (2002); Vagov et al. (2014), and these results have been studied and discussed in terms of usual Born-Markov approximationHughes and Agarwal (2017); Kryuchkyan et al. (2017); Roy and Hughes (2012).
The system correlation function, for non-Markovian interactions (for e.g. nuclear spinsCoish and Loss (2004), phononsKaer et al. (2010)), decays with a typical time scale given by the correlation time which never dies-off to zero. In other words, the correlation time is non-zero and can be larger than the system decay time , which is the signature of a strongly history-dependent non-Markovian interaction.
Furthermore, the equation of motion for correlation function has an additional term known as irrelevant part, which is non-zero for non-Markovian interactions. This additional term vanishes for a Markov approximation, when the well-known QRT can be applied to find the system correlationSwain (1981). Recent theories [for e.g. Ref. Roy and Hughes, 2012] discussing fluorescence spectrum in solid-state systems rely on a history-independent Markov process and apply the usual QRT, giving rise to an exponential decay of the system correlation. Often, physical processes in solid-state systemsThanopulos et al. (2017); Toyli et al. (2016); McCutcheon (2016) are highly non-Markovian (history-dependent), and so the resulting spectrum using the QRT can no longer be used to describe their spectral properties.
In this paper, we analyze the effect of pure dephasing due to a 3D acoustic phonon bath on the fluorescence spectrum of a cavity coupled to a semiconductor QD. The associated emission spectrum can be directly related to a correlation function for system observables, which we evaluate beyond the Markov approximation using a Nakajima-Zwanzig GMEFick and Sauermann (1990); Swain (1981). Assuming a large band-width, coupling to the radiation modes of the cavity can be treated within Born-Markov approximation, which is relevant to cavity quantum electrodynamics (cavity-QED) experimentsMuller et al. (2007). The dot is represented by a two-level system with an exciton in the excited state coupled to a phonon reservoir, and can be modeled with the usual IBMMahan (1990).
The resultant fluorescence spectrum has three components due to dressing of the levels by a pump laserMollow (1969); Cohen-Tannoudji and Grynberg (2004), and we project the system into a dressed state basis which allows us to characterize the three components of triplet separatelyHopfmann et al. (2017); Laucht et al. (2016). We have solved the dot-phonon coupling using an exact approach beyond Born approximation, and the exact result coincides with that of obtained within Born approximation. Phonon coupling gives rise to a frequency dependent frequent-shift and dephasing, which bring non-Lorentzian features in the fluorescence spectrum. Frequency-shift and dephasing due to phonon interaction are strongly temperature-dependent and vanish for the lower temperatures. We found that in the dressed-state basis levels of interest are coupled asymmetrically to the phonons and have vanishing dephasing and frequency-shift for a resonant dot-laser frequency. We have also observed that the sideband undergoes a distinct narrowing and becomes asymmetric with increasing temperature, which is explained using the dressed-states energy levels.
This paper is organized as follows: In Sec. II, we start with discussing the setup and establish the formula for resonance fluorescence spectrum of a general two-level system. In Sec. III, we introduce the model Hamiltonian for a driven cavity-QED two-level system interacting with a phonon bath. In Sec. IV, we discuss and derive the exact form of Nakajima-Zwanzig GME for the dynamics of the reduced density matrix and correlation function. We obtain the expressions for the lineshape functions in both Markovian and non-Markovian regimes. In Sec. V, we present our results and discuss the plots in different parametric regimes. In Sec. VI, we conclude with a discussion and summary of the results. Other technical details are discussed in the Appendixes.
Ii Fluorescence spectrum
We start with the model Hamiltonian of a general two-level system interacting with radiation modes of the electromagnetic field, which can be written as system (), field (), and interaction () in terms of the standard Jaynes-Cummings model within a rotating-wave approximation (RWA):
where and are the raising and lowering operators between excited state and ground state in the Hilbert space of the system and , are the annihilation and creation operators in the Hilbert space of a set of electromagnetic modes coupled to the system. Coupling to the radiation modes is given by a coupling constant to the mode of frequency . For simplicity of the units, we have also used . Here, we adopt the well-known theory of gedanken spectrum analyzer given in Ref. [Scully and Zubairy, 1997], and assume that the radiation field emitted by the system is detected by a two-level atom (detector) with a transition frequency , which is prepared in its ground state initially, see Fig. 1.
The Hamiltonian for the detector is given by
and coupling between detector atom and the radiation field, included in the Hamiltonian , is given by the Hamiltonian
where is the coupling of detector to a field mode . Therefore, the Hamiltonian for entire system including system and detector, as well as coupling to the radiation modes can be written as:
According to Wiener-Khintchine theorem, fluorescence spectrum, in the stationary regime and in the interaction picture with respect to the detector, is given by Fourier transform of the correlation functionScully and Zubairy (1997)
where is the detector dipole matrix element with positive-frequency part of the electric field is defined by
and the negative-frequency part of the electric field is . The quantity is the electric field per photon and is the effective volume of a cubic cavity resonator. We rewrite the correlation function, in Eq. (7), in terms of the system operators using the methods described in Refs. Scully and Zubairy, 1997; Cohen-Tannoudji and Grynberg, 2004:
here is the detector response function [discussed in Appendix A]. The average , in Eq. (9), is given with respect to the stationary density matrix , where . Using the cyclic property of trace, fluorescence spectrum can be written as
here operator is defined as
and . Since and are operators in the system Hilbert space and , the evolution of is determined by the Hamiltonian of emitting system and radiation field, , in the absence of detector and can be computed without using the well-known QRTSwain (1981).
We consider a driven two-level cavity-QED system with an excited state representing a single exciton, and a ground state with no exciton. The QD interacts with the cavity photons and a phonon reservoir which is coupled to the excited state , as shown in Fig. 1. The Hamiltonian for the total system reads,
where is the transition frequency of the two-level system and is the frequency of laser field. The photon (phonon) modes are represented by bosonic fields with frequencies () with creation and annihilation operators () and (), respectively. The system-photon (system-phonon) coupling strength is given by (), and (Rabi frequency) is the coupling between the two-level system and laser field. The system operators are denoted by where and .
The explicit time dependence in [Eq. (12)] can be removed by going to a rotating frame and applying a RWA. We perform the RWA on both the driving term and the system-photon coupling term, in which we neglect the rapidly-oscillating terms and keeping only the time-independent part. The resulting RWA Hamiltonian in the rotated frame is then
where and are the detunings of atomic and cavity frequencies from the laser pump frequency . In general, operators in the rest frame are transformed to the rotating frame according to the following relations:
Due to presence of the intense laser field the two bare states are strongly coupled to each other and give rise to two dressed states. The dressed states can be written, in terms of the bare states, as:
where and with the mixing angle is given by , where is the dressed Rabi frequency.
Furthermore, following closely the discussing on IBM in Ref. [Mahan, 1990], we apply another transformation by using a canonical transformation , where is an anti-hermitian operator, to write the total Hamiltonian in terms of the free and perturbed parts as, after transforming it to the dressed-state basis
where the free part is
and the perturbed part is given by
where , and . The polaron transformation introduces a frequency shift as: , where is the polaron shift. We have also performed a Schrieffer-Wolff transformation on the above Hamiltonian to get rid of the energy exchange term ( lifetime process) due to phonons, see Appendix B. The separation into pure dephasing and transition terms is determined by the form of system coupling, i.e. dephasing terms contain the diagonal coupling and transition terms contain the off-diagonal couplings. Accordingly, fluorescence spectrum, in Eq. (10), can be written in Laplace domain using the dressed-state representation as a three-peak spectrum [Ref. Cohen-Tannoudji and Grynberg, 2004]:
where is the probe detuning, we have used the relations , and . The polaron transformation does not affect the fluorescence spectrum in above expression as acts on the phonon mode Hilbert space and commutes with the system operators. It can be seen that the first term in Eq. (27) gives the central peak whereas last two terms give rise to satellite peaks of the Mollow-triplet, shown in Fig. 2. We are interested in the influence of pure dephasing due to phonon interaction, which only affects the off-diagonal elements of the system operator. From here and what follows, we will study the effect of pure dephasing on the fluorescence spectrum, with a particular focus on one of the Mollow-triplet sidebands (Stokes line). Alternatively, when the width of each side peak , [see Eq. (63), below] is small compared to the peak separation , we approximate the spectrum near the side peak centered at by, see Fig. 2
In order to compute the spectrum, we will evaluate the dynamics of matrix element in the dressed-state representation with Hamiltonian in the polaron frame, given by Eq. (18).
iii.2 Initial conditions
The radiation and phonon modes are decoupled from the system for times (where is a time in the distant past), and prepared independently in the states described by density matrices , and , respectively. The interactions (photons and phonons) are switched on at this time , and the state of entire system is described by the full density matrix :
where initial density matrix for photon is described by vacuum of the cavity modes
and phonon modes are described by a canonical ensemble at temperature :
We recall the operator , where
is analogous to the density matrix operator with a modified initial condition, given by
where stationary density matrix and hence account for conditions that accumulate between the system and interactions in the time interval . We choose an initial condition when exciton is in excited state given by , and evolves in the presence of pump laser. We switch on the detector at and subsequently calculate the dynamics of for , with initial condition given by the steady state density matrix accumulated between the time interval .
Iv Generalized Master Equation
We are interested in the dynamics of reduced system operator, after tracing over variables of photon and phonon modes; . To study the dynamics of reduced system operator , we introduce a projection superoperator , defined by its action on an operator: .
Both operators, and , follow the same von-Neumann equation, and can be written in form of exact Nakajima-Zwanzig GMESwain (1981), using
where is the self-energy superoperator
and is the full Liouvillian superoperator, defined as and . We have used the properties of projection operator: and
also introducing its complement . We can derive an equation of motion for analogous to the equation for [Eq. (34)]. However, an additional term appears because , and we have assumed that the full density matrix operator is not separable for all times i.e. . The resulting motion equation for is then,
where is defined in Eq. (35) and the last term in the above equation contains , i.e. the irrelevant part of is non-zero. When the radiation and phonon modes are described by Eqs. (30) and (31), the projection operator follows some useful identities
where and are reduced self-energy and irrelevant part superoperators, respectively, and is the reduced system operator. Comparing Eqs. (42) and (43), we found that the first two terms are identical but there is an additional irrelevant part, , is present in the equation for . This term vanishes in a Markov approximation and usual QRT can be applied to find the system correlationSwain (1981). In addition to this, we have also assumed that the full density matrix is not separable for all times which is another reason that QRT is no longer valid in the present case. Furthermore, for a non-Markovian equation the irrelevant term is non-zero and usual QRT can not be used to compute the system correlation in this casede Vega and Alonso (2008); Lax (2000); Jin et al. (2016). The equation for off-diagonal matrix element, , is coupled to both diagonal and off-diagonal elements of the reduced system operator, and can be written in the form
with a non-zero irrelevant part matrix element expressed as (see Appendix C)
The self-energy superoperator can be decomposed into three terms as
where the first and second terms in above expression are pure dephasing ( pure dephasing time) processes due to photon and phonon couplings, respectively, and the third term gives rise to transition ( lifetime) due to radiative modes coupling. We treat the dot-cavity coupling self-energy within Born-Markov approximation to second-order in perturbation Liouvillain , see Appendix D.1. Applying the continuum of modes for cavity given by Lorentzian density of state [Eq. (127)], we find the self-energy matrix elements in their Laplace domian, defined as ,
where is the detuning of cavity from laser pump frequency and is cavity bandwidth. Here, Born approximation is justified by finding that the higher order terms in reduced self-energy are suppressed by a small parameter . Similarly for phonon modes, applying a continuum of modes [see Appendix D.2] for the deformation potential coupling mechanismKrummheuer et al. (2002) , where is the phonon coupling parameter in the units of and is the phonon cut-off frequency, we obtain
here is Bose function. In above expression, the dot-phonon interaction self-energy is evaluated using exact approach within non-Markovian limit and it is found that Born approximation is exact in this case, see Appendix D.3. Furthermore, the reduced self-energy matrix elements couple to the populations can as well be written within usual Born approximation
and similarly for the coherence ,
Equation for the coherence in Eq. (46) contains both diagonal and off-diagonal elements of the self-energy superoperator, some of them are fast moving compare to others. In next section, we will perform a secular approximationCohen-Tannoudji and Grynberg (2004) to get rid of the fast oscillating terms.
iv.1 Secular approximation
The secular approximation consists in neglecting the fast oscillating terms in Markov equation-of-motion, and the equation for [Eq. (46)] is decoupled from populations and coherence within a secular approximationCohen-Tannoudji and Grynberg (2004). Here, we consider a general equation of motion for the operator , without irrelevant part matrix elements:
and would like to perform the secular approximation in order to get rid of the fast oscillating terms. To this end, introducing a rotating frame
where is the total frequency shift given implicitly by the expression
In the weak coupling regime, , the frequency shift to the leading order in
Here, the purpose of introducing a rotating frame is to get rid of all oscillating parts from and hence obtain an equation for
where . Due to presence of the oscillatory exponential in the last two terms in RHS of the above equation, we decompose it into slow-varying and fast-oscillating parts as:
where S and F stand for slow and fast, respectively. Carrying out Markov approximation on slow term by replacing: , , and finally extending the upper limit of integration to infinity, we obtain
Condition for validity of Markov approximation: decays on a time scale which is the decay time of and given by the relationFick and Sauermann (1990)
and the condition for validity of Markov approximation:
which leads to a similar condition . For large and due to presence of the highly oscillatory exponential , the effects of terms will eventually average out to the smaller values compared to . Therefore, in secular approximation when
we neglect the fast oscillating terms in Markovian approximation since this term will oscillate fast and average out to a smaller value compared to the slow term. On substituting for from Eq. (54) and for , , we get an explicit condition for the validity of secular approximation as: . In the similar manner, we can also neglect the contribution form and from Eq. (46). Going back to lab frame and within secular approximation, we obtain the expression in Laplace transform:
The irrelevant part matrix elements, due to photon and phonon , both are identically zero under a Markov approximation. On substituting for from Eq. (68) in the expression (28), we obtain an expression for one-peak Markovian spectrum
Here, we have substituted the expression for the self-energy matrix elements given by Eqs. (52) and (53). Expression for the lineshape in Eq. (68) is Markovian with respect to both photon and phonon interactions. However, we are interested in the non-Markovian regime with respect to phonon coupling and an equation for always contains an extra small term due to non-Markovian interaction, known as irrelevant part matrix element. Assuming that the irrelevant part is associated with a smallness in the present problem and in order to get the further insight, we will estimate the typical size of its contribution and find a regime where non-Markovian correction is dominant compared to its irrelevant part contribution. Equation for in Laplace domain with its irrelevant part matrix element can be written as,
The smallness of irrelevant part compared to non-Markovian self-energy due to phonon interaction can be justified by the inequality given by
and we only keep the contribution from self-energy matrix element. Furthermore, expanding Eq. (72) in the powers of self-energy and ignoring the higher order terms, we have
and assuming that irrelevant part gives rise to a small contribution and comparing it with self-energy contribution leads to the following inequality,
The one-peak spectrum is centered around , with a width mostly dominated by Markovian decay rate , estimating the size of above inequality around , and it is found that irrelevant part matrix element is always suppressed by a small parameter, , compared to self-energy matrix element contribution, also see Appendix C. Following the above discussion, we neglect the irrelevant part from the equation for and after going back to lab frame we have an equation written in Laplace domain,
Above expression is Markovian with respect to photon coupling but non-Markovian in terms of phonon interaction, where Markovian frequency shift () and decay rate () are given by