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Evaluation of the Medical Specialist Outreach Assistance Program and the Visiting Optometrists Scheme
Appendix 2.h – Tele-Derm, Australian College of Rural and Remote Medicine
Page last updated: 28 February 2012
Tele-Derm is a program run by the Australian College of Rural and Remote Medicine (ACRRM) to provide specialist dermatology support and education to rural and remote GPs. The program began in 2003, building on an existing relationship that ACRRM had developed with a dermatologist. Through the dermatologists work providing training for the College, he realised that many cases could be diagnosed very quickly using very basic information (i.e. pictures). This realisation led to a proposal to one Queensland fundholder (now called General Practice Queensland) for the provision of teledermatology services for rural and remote Queensland doctors.
At the time, there was extra funding in the fundholder's budget which allowed for the service to be rolled out as a trial. Initially, only doctors from Queensland were allowed to use the service due to funding and medical licensing restrictions. Because of the success of the program the trial was expanded to the rest of the country and funding was transferred to the national DoHA office. In order to achieve this expansion, the medical licensing for the dermatologist was increased to allow him to practice in all states. At the same time, through the MSOAP funding, ACRRM added a teleradiology service in addition to the Tele-Derm program.
The Tele-Derm program has been integrated as part of the Rural and Remote Education Online (RRMEO) portal. RRMEO provides access to a broad range of tools that support education and development for rural and remote doctors. Including Tele-Derm within the portal allows for the infrastructure and audience that has already been developed for RRMEO to be directly integrated with MSOAP. ACRRM has approximately 3,000 members, with an additional 7,000 people who are members of organisations that have purchased access to RRMEO, including groups such as registrars and John Flynn Scholars. Of the ~10,000 people with access to RRMEO, greater than 7,500 would be eligible to register for Tele-Derm.
For individuals that are eligible to register for RRMEO, a separate registration process is still required to participate in Tele-Derm. As part of the funding that MSOAP provides, any doctor eligible for MSOAP funding can apply for access to Tele-Derm. For doctors who register through MSOAP, they are provided access solely to the Tele-Derm portion of the RRMEO portal.
There are two main components to the Tele-Derm portal, a specific telemedicine component and a telehealth component. Telemedicine in this case is defined as traditional clinical medical services provided over a distance using electronic communication. With Tele-Derm, the telemedicine component is delivered through a store and forward (asynchronous) method where a doctor submits pictures and case information to the site for the dermatologist to diagnose and respond. This type of service is in contrast to 'real-time' telemedicine where a live teleconference occurs between patients and doctors. Telehealth in this case is defined as the use of information and communication technology for administration and education in addition to clinical services. ACRRM has made education a major component of Tele-Derm, indicating that the educational component is as or more important than the direct clinical services that are provided through the program.
The telemedical component of the program is based on the store and forward model, where cases are submitted by doctors and then at a later time the dermatologist provides a response that can then be accessed by the doctor. A doctor will submit a set of pictures and case information to the website, a process which feedback has indicated takes around 20-40 minutes for the GP to prepare. The GP does not receive any payment from ACRRM or Medicare for this service. Once the case is submitted, the ACRRM dermatologist accesses the case and writes a response that is made available to both the submitting doctor and all other Tele-Derm users. The case information is always left deidentified, but ACRRM does keep a record of the doctor who submitted the case.
In general, responses by the dermatologist are returned within 24 hours of initial submission, though the turnaround can be significantly shorter. A new feature has been developed for urgent cases to use SMS notification to speed up the process. When an urgent case is submitted, the dermatologist receives a SMS notifying him of the submission. When he responds, the doctor who submitted the request receives a SMS to alert them that a response has been posted.Top of page
Within the Tele-Derm portal there are several main areas that can be accessed by registered users: submitted cases, a dermoscopy atlas, a condition index, tips from the dermatologist and discussion forums. All cases that have been submitted can be viewed by all registered users of the program. Overall, there are 543 standard cases plus 186 dermoscopy cases that can be viewed by doctors to read and learn from, including the ability to guess at diagnoses and then have the answer revealed. Both the dermoscopy atlas and the condition access are designed to allow doctors to educate themselves on conditions and improve their self-diagnostic capabilities without having to submit every case directly. There is also an educational section written by the dermatologist to provide tips and tricks related to dermatology. For example, there is information on procedures, including biopsies and cryosurgery, how to take a good skin history from a patient, how to take good dermatology photographs and a list of other useful websites. Finally, the online discussion forums allow for doctors and the dermatologist to interact, learn and discuss dermatology cases.
Evidence indicates that many of the users of the portal are there for educational purposes rather than a direct clinical need. For example, registrars will often visit to learn and study for their training. Medical students can use the portal for similar purposes.
In order to generate interest and participation in the program, ACRRM has a case of the week that is e-mailed to all users. This involves a picture and a short description. Users can then click on the case to read the full details and then submit a response of their diagnosis of the case. At the end of the week, the ACRRM dermatologist provides a full discussion and diagnosis of the case in addition to responding to all of the comments posted throughout the week. Overall, the dermatologist provides about 1,000 comments per year in response to postings in the forums. General marketing of the Tele-Derm program is through the ACRRM website and advertisements are placed in all of the state specialist directories.
Rural and Remote Education Online (RRMEO) portal
RRMEO is a unique online portal for education that ACRRM has internally developed. In order to access the portal users must register with ACRRM. Depending on their membership and eligibility, they will be provided access to portions of the portal. Everything that ACRRM accredits is available and searchable, such as classes (online and in person), workshops, training posts and educational resources. A learning planner is also provided for ACRRM members that allow educational activities and documentation to be tracked and stored for easy submission to accreditation organisations. The other main component of RRMEO is the online modules. Tele-Derm is one of these, in addition to other educational tools in a variety of clinical areas and live learning through virtual classrooms. Other features of the portal include the ability to download clinical guidelines on mobile devices, take tests online and many other components that allow rural and remote doctors to be educated and trained without leaving their community.
Funding for Tele-Derm is currently through MSOAP and is provided as part of a contract with ACRRM to support the service. The budget has remained fairly stable through the course of the program and has only increased marginally to adjust for inflation. The amount of funding received is defined by DoHA and specifies the exact levels of expenditure for individual components of the program such as wages and promotional activities. While the overall budget provided to ACRRM includes both Tele-Derm and teleradiology, the funding for each is clearly defined.
The dermatologist that ACRRM employs for the program is paid an hourly wage that is defined by MSOAP based on a standard rate for specialist services. Payments are then made to the dermatologist on a sessional basis. National licensing that is required for the services to be provided are reimbursed for the dermatologist by ACRRM from their budget.
Currently there are around 1,500 subscribers to the ACRRM Tele-Derm program. Subscribers include doctors, registrars and medical students amongst others. In order to register for the Tele-Derm program a user must be a member of ACRRM, a member organisation that subscribes to RRMEO or be a doctor that is eligible for MSOAP (i.e. not in a major city). As part of the budget DoHA provides funding to support subscription of eligible doctors to the Tele-Derm portal. Of the 1,500 subscribers to the service, approximately 300 are registered through the eligibility that MSOAP provides. Many participants access the program for educational purposes and do not submit any cases directly. Anecdotal evidence indicates that many doctors would like to submit cases, but due to time constraints or other barriers have not done so. However, once a doctor submits a case and become familiar with the process, they seem to subsequently submit cases consistently.
The program grew rapidly after its initial introduction, quickly attracting 500-600 members. By the 2005-06 financial year there were 20,000 hits to the site and over 150 cases submitted Australian College of Rural and Remote Medicine 2010, 'ACRRM Medical Specialist Outreach Assistance Program Annual Plan', Report to DoHA.. Currently, the website attracts between 4,500 and 7,000 hits per month (Table 26) and receives approximately one new case per day.
Table 26 – Tele-Derm web site hits
Oct-10
Nov-10
Dec-10
Jul-11
Aug-11
Dermatologist experience
Dr. Jim Muir is the dermatologist employed by ACRRM to provide the clinical support for the Tele-Derm program. As previously discussed he responds to cases, supports the online education and responds to online posts by doctors. He has been involved with the Tele-Derm program since its inception in 2003, with additional involvement with ACRRM prior to the commencement of the program. During his early career Dr. Muir was involved in outreach to rural locations and also participated in one year of video based telemedicine services.
The idea for telemedicine began through Dr. Muir's involvement in receiving e-mails of cases from rural doctors early in his career. The catalyst came when he received an e-mail with a photo of necrosis and saw that he could diagnose acute cases through e-mail. There are other doctors and hospitals that provide dermatologically based telemedicine programs, but their scope is limited and they do not provide any educational services.
Benefits of store and forward
The use of store and forward provides significant benefits for dermatology relative to real time telemedicine services. Currently, most models of real time telemedicine require both the patient and a clinician to be present for the consultation with a specialist. Logistically and resource wise, this arrangement is both costly and difficult to arrange. Due to rebates, some dermatologists provide real time telemedicine services, but often ask for high resolution photos and a case history prior to the video conference. The use of store and forward allows the case to be brought forward anytime of the day, it uses higher quality images and responses can be sent back rapidly and made available to the referring doctor at their convenience.
Relationship between visiting services and Tele-Derm
Top of pageDermatology is fairly unique in the medical field in that diagnosis is the most challenging aspect of the practice. Generally, prescribing and treatment are fairly straightforward. What this means is that the speciality is extraordinarily well suited to telemedicine as such a large percentage of cases can be diagnosed through pictures and case histories. This means that a significant portion of visiting dermatology service can be replaced with Tele-Derm.
The other important issue in telemedical services is the treatment and follow up of issues that have been diagnosed. Traditional visiting services required patients to wait for the visiting specialist to return or for their local doctor to follow up with their care. Often the visits by the specialist could be far apart and the local doctor might not have sufficient initiative, knowledge or information to provide comprehensive care.
Tele-Derm addresses a lot of these issues through the combination of education and telemedicine services. Having the doctor take pictures and a case history involves the doctor in the case directly, providing both knowledge, ownership and initiative in the treatment of the patient. Furthermore, the availability of educational resources on the Tele-Derm site provides the resources to allow GPs to perform many procedures that are essential for dermatology. Many of these are straightforward and most competent doctors can already do or learn them through the tutorials Dr. Muir has provided online. Cases can also be addressed through the back log of cases provided online, allowing GPs to research a condition and make a diagnosis without consulting a dermatologist directly.
The overall assessment of telemedicine versus visiting dermatology services is that they do not have to be as good as face to face, they just have to be better than what is currently available. In the public sector, there are six month to year long waiting lists for dermatology, whereas Tele-Derm can have a turnaround of less than a day. Most dermatologists are located in major cities or large regional centres, meaning that a significant amount of travel time is required by rural patients. Tele-Derm only requires them to travel to their local GP. Overall, the service offers significant benefit and provides no more risk of misdiagnosis or mistreatment than a face to face consultation with a dermatologist. One final benefit of the Tele-Derm program is that it can reduce professional isolation for doctors, increase their confidence in diagnosis and help them to feel more connected to the medical profession.
Improvement and expansion
In order to improve and expand the service, the goal would be to minimise work for the referring doctor as possible. While it might be easier for the patient to participate in Tele-Derm, it is still less time and effort to refer the patient to a dermatologist directly, since referring GPs are not paid for the time they put into Tele-Derm. If the system could engage an assistant to take some case history and pictures, then there is the opportunity to get more GPs involved.
Areas where there is room for expansion of services includes:
inpatient units
ships at sea
locations where it is logistically or economically not feasible to visit
follow up services
anywhere where waitlists are long
emergency consultation.
Function and future directions
The program seems to be well accepted by dermatologists as there are a small number that are practicing in Australia, with most having a waiting list of around six months. The role that the dermatologist plays for ACRRM does not threaten other practices and is not the result of a tech savvy business, but rather the desire to meet a strong need that exists in the community.
The experience of ACRRM is that teledermatology could not replace outreach, but could further integrate and coordinate the services that are going out. A wider adoption of the Tele-Derm program could first and foremost reduce the number of cases that need direct outreach by a dermatologist. The educational component of the program could upskill many GPs who could diagnose more cases without specialist referral and could send additional cases to the online system for remote diagnosis. For other cases, information could be sent prior to visits so that dermatologists were better prepared to make efficient use of time during outreach visits. Potentially, a whole electronic coordination and diagnostic system could be developed to direct patients and doctors to the most efficient path for diagnosis and treatment. The current program run by ACRRM has the capability to expand further to cover more cases and doctors across Australia and provide additional services to aide in dermatology outreach in general.Top of page
1.2 Structure of the final report
6. Community case studies
Appendix 2.a - Aurukun, Queensland
Appendix 2.b - Burnie and the north- west region of Tasmania
Appendix 2.c - Crystal Brook, South Australia
Appendix 2.d - Dubbo, New South Wales
Appendix 2.e - Karratha & Roebourne, Western Australia
Appendix 2.f - Leongatha, Victoria
Appendix 2.g - Maningrida, Northern Territory
Appendix 3.a - Survey methods
Appendix 3.b - Results MSOAP service provider survey
Appendix 3.c - Results VOS service provider survey
Appendix 3.d - MSOAP service provider survey form
Appendix 3.e - VOS service provider survey form
2.1 Overview of project methodology
2.3 Review of program documentation
2.4 Program data analysis
2.5 Interviews with stakeholders and written submissions
2.6 Survey of service providers
2.7 Community case studies
2.8 Preparation of draft and final reports
3. Health service issues for rural and remote Australia
3.1 Measures of remoteness
3.2 Demography
3.3 Health status and outcomes
3.4 Service delivery issues for rural health services
3.5 Specialist private practice viability and workforce distribution
3.6 Government responses to rural health issues
3.7 National health reforms
3.8 Key messages
4.1 Systematic reviews
4.2 Other and more recent comparative studies
4.3 Other descriptive studies
4.4 Telemedicine
5. The Medical Specialist Outreach Assistance Program
5.1 Program origins and development
5.2 Rural Health Outreach Fund
5.3 How the MSOAP currently operates
5.4 Analysis of program expenditures and costs
5.5 Estimates of the program's impact
7. MSOAP evaluation questions
7.1 Program effectiveness, cost effectiveness and equity
7.2 Integration and coordination
7.3 Assessment of need
7.4 Roles and operation of advisory fora
7.5 Development of service proposals
7.6 Fundholder arrangements
7.7 Administrative processes
7.8 Data collection and management
8. The Visiting Optometrist Scheme
8.1 Optometry services in Australia
8.2 VOS origins and development
8.3 Budget
8.4 How VOS currently operates
8.5 Program statistics
8.6 Estimates of the Scheme's impact
9. VOS evaluation questions
9.1 Program effectiveness and cost effectiveness
9.3 Roles and operation of advisory structures
9.6 Integration of VOS and MSOAP
10. Future directions
10.1 Challenges to growth in service delivery
10.2 Improving awareness of the programs' achievements
10.3 Health reform
Appendix A: Background paper provided to stakeholders
Appendix B: People interviewed for evaluation
Appendix C: Written submissions received for the evaluation
Appendix D: Selected government programs targeted at improving access for health services for rural and remote populations
Appendix E: Payment rates for selected costs under MSOAP and VOS, 2011
Appendix F: Allocation of MSOAP services to ASGC remoteness areas
Appendix G: Definitions of broad category of MBS services used for MSOAP/VOS evaluation analysis
Appendix H: Australian classifications of remoteness


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\section{Introduction}

Transient phenomena have lead to the development of powerful analytical and experimental techniques to explore the dynamics of matter-waves \cite{kleberPR94, dcgcjm}.
Transient effects emerge as a response of a quantum system to various types of interactions, state preparation or initial boundary conditions.
The most representative example of transient phenomena due to a sudden interaction is the Moshinsky quantum-shutter setup \cite{mm52}.
The latter involves the time-dependent solution of free Schr\"odinger's equation for a cut-off plane wave of momentum $k$, initially confined to the left of a perfect absorbing shutter located at $x=0$.
By opening the shutter at a time $t = 0$, the quantum state evolves freely in space, and the probability density in the time-domain exhibits a distinctive oscillatory pattern known as \textit{diffraction in time} \cite{mm52,mm76},
analogous to the intensity profile of a light beam diffracted by a semi-infinite plane.
The time-diffraction effect was verified decades later in experiments using ultracold atoms \cite{dalibard}, cold-neutrons \cite{Hils98}, and atomic Bose-Einstein condensates \cite{colombe05}.
The quantum-shutter problem has allowed to translate spatial features of light optics to the time-domain of matter-waves in experiments with atoms that exhibit diffraction and interference \cite{Arndt96,dalibard}, as well as phase modulation \cite{steane95,decamps16} and quantum beat phenomena \cite{decamps16}.

After Moshinsky's proposal, several theoretical works have addressed the study of transients of tunneling matter-waves within a quantum shutter setup for systems involving delta potentials  \cite{kleberPR94,19306687,gcah03,adjpa04,19156601,grmar07,mlgc10}, potential barriers \cite{bm96,ggcjvpra01,12286114,grmar07,Julve_2008}, and multibarrier resonant structures \cite{gcar97,PhysRevB.60.R2142,Cordero_2010}.
Some of these transients involve electronic transport and buildup in quantum structures \cite{gcar97,PhysRevB.60.R2142,Cordero_2010}, and time-scales \cite{kleberPR94,19306687,gcar97,ggcjvpra01,gcah03,adjpa04,19156601,12286114}.
Although most of these works have relied on cut-off plane wave initial conditions \cite{19306687,gcah03,ggcjvpra01,12286114,adjpa04,bm96,19156601,Julve_2008,mlgc10} and wave packets with Gaussian   distributions \cite{adjpa04, 19156601,Cordero_2010}, to our knowledge, the study and characterization of transient effects of phase modulated cut-off waves has not been addressed before.

In this work we explore  transient phenomena of phase modulated cut-off wavepackets by deriving an exact analytical solution to Schr\"odinger's equation based on a resonant state expansion.
The modulation involves a superposition of quantum states with slightly different momenta.
The aim of this work is to show that the phase modulation of initial cut-off quantum waves gives rise to interesting transient phenomena, such as Rabi-oscillations, quantum beats, delay-time, and time-diffraction effects.
We discuss how this these effects are the result of a virtual \textit{self-induced two-level system} that governs the dynamics of probability density in the time-domain.

Our work is organized as follows. In Sec.~\ref{model} we present the main equations involving the solution of the quantum shutter approach for a general potential to describe the dynamics of arbitrary cut-off initial quantum states.
In Sec.~\ref{results} we study the time-dependent features of  phase modulated wavepackets along the transmission region of a potential, which  exhibits different regimes governed by a virtual self-induced  two-level system. The effects of the modulation on the delay-time of the system are also explored.
Finally, in Sec.~\ref{conclusions} we present the conclusions.

\section{Model for a general initial state\label{model}}

Our approach deals with an exact analytical solution to Schr\"odinger's equation for a finite range potential involving an arbitrary cut-off initial state, based on a resonant state expansion.
We stress that our approach is non-Hermitian since the  resonant states are eigenfunctions of the Hamiltonian with \textit{outgoing boundary conditions}, which leads to complex energy eigenvalues. The Hermiticity of the system Hamiltonian is not only related to the  operator itself but also to the functions on which it acts on \cite{moiseyev_2011}.  Non-Hermitian approaches provide alternative exact analytical descriptions of physical processes. In fact, the equivalence of non-Hermitian dynamical descriptions and those based on \textit{continuum wave expansions} of standard quantum mechanics has been demonstrated in Ref. \onlinecite{12286114}.
Within this framework, the time-dependent features of  quantum waves with different initial conditions can be explored.
Our model is a generalization of the work originally developed for a cut-off plane wave in Ref.~\onlinecite{gcar97}.
The general approach to the problem involves the time-evolution of particles of energy $E=\hbar^2 k^2/2m$ incident  from the left onto a one-dimensional finite range potential $V(x)$ of arbitrary shape that extends along the region $0\le x \le L$, and vanishes thereafter.
The  wavefunction $\Psi(x,t)$ of the problem is a solution of
Schr\"{o}dinger's equation $\hat{H}\Psi(x,t)=i\hbar\partial_t\Psi(x,t)$, with  $\hat{H}=(-\hbar^2/2m)\partial^2_x+V(x)$, for an initial condition at $t=0$ given by,
\begin{equation}
\Psi(x,0)=\cases{\phi(x,k), & for $x\leq0;$\cr 0, & for $x> 0$.}
\label{eq1_bis}
\end{equation}
The function  $\phi(x,k)$ represents a general initial state that extends from $-\infty < x \leq 0$.
The  solution is obtained by Laplace-transforming  Schr\"{o}dinger's equation and the corresponding  initial condition, following the procedure of Ref. \onlinecite{gcar97}, that is,
\begin{equation}
\left[ \partial^2_x +p^2\right]\widetilde{\Psi}(x,p)=\cases{i\alpha\, \phi(x,k), & for $x\leq 0;$\cr 0, & for $x > L$.}
\label{newdifeq}
\end{equation}
The  solutions to Eq.~(\ref{newdifeq}) are,
\begin{eqnarray}
\widetilde{\Psi}(x,p)=\cases{c^+e^{i p x}, & for $x> L;$\cr c^-e^{-i p x}+ \alpha \widetilde{F}(x,k,p)\, e^{-i p x}, & for $x \leq 0$,}
\label{newdifeq_sol}
\end{eqnarray}
where we have defined $p=(i\alpha s)^{1/2}$ with $\alpha=2 m/\hbar$, and  $c^{\pm}$ are constants to be determined by the matching conditions of the wavefunctions.
The function $\widetilde{F}(x,k,p)$ is the particular solution to  the inhomogeneous equation in (\ref{newdifeq}), given by
\begin{eqnarray}
\widetilde{F}(x,k,p)=\int e^{2 i p x}\left[\int \, i\, \phi(x,k) \,e^{-i p x} dx \right] dx,
\label{intinception}
\end{eqnarray}
obtained following the procedure of Ref. \onlinecite{JOHNSON2008582}.

The differential equation for the internal region reads,
\begin{eqnarray}
\left[ \partial^2_x +p^2-\widetilde{V}(x) \right]\widetilde{\Psi}(x,p)=0, \,\,\,\, 0 \leq x \leq L,
\label{newdifeq2}
\end{eqnarray}
with $\widetilde{V}(x)=2mV(x)/\hbar^2$.
We obtain the solution for the internal region by using the corresponding equation of the outgoing Green's function of the system, $G^+(x,x';p)$, that is,
\begin{eqnarray}
\left[ \partial^2_x +p^2-\widetilde{V}(x) \right]G^+(x,x';p)=\delta(x-x'), \,\, 0 \leq x \leq L,
\label{green_new}
\end{eqnarray}
with the outgoing boundary conditions,
\begin{eqnarray}
    \partial_x G^+(x,x';p)|_{x=0}&=&-i \,p \,G^+(0,x';p), \label{Gobc1}\\
   \partial_x G^+(x,x';p)|_{x=L}&=&i\, p\, G^+(L,x';p).\label{Gobc2}
\end{eqnarray}
We write $\widetilde{\Psi}(x,p)$ in terms of $G^+$ by multiplying Eq.~(\ref{newdifeq2}) by $G^+(x,x';p)$ and Eq.~(\ref{green_new}) by $\widetilde{\Psi}(x,p)$, and subtracting the equations.
Thus, by integrating the resulting expression along  $x\in[0,L]$, with the help of Eq.~(\ref{newdifeq_sol}) and the boundary conditions (\ref{Gobc1}) and (\ref{Gobc2}), we get
\begin{equation}
    \widetilde{\Psi}(x',p)=\alpha\,\widetilde{\Phi}(0,k,p) \,G^+(0,x';p),
\label{psi_interanl1}
\end{equation}
where $\widetilde{\Phi}(0,k,p)=\partial_x\widetilde{F}(x,k,p)|_{x=0}$.
By evaluating Eq.~(\ref{psi_interanl1}) in $x'=L$, and  Eq.~(\ref{newdifeq_sol}) in $x=L$, we obtain the constant $c^+$ which allows to write the solution along the transmission region ($x \geq L$) as
\begin{equation}
    \widetilde{\Psi}(x,p)=\alpha\,C(p,k)\,e^{i p x}\,\,\,x \geq L,
\label{psi_interanl1c}
\end{equation}
with $C(p,k)\equiv \widetilde{\Phi}(0,k,p)\,G^+(0,x';p) \,e^{-i p L}$.
We calculate $C(p,k)$ by evaluating the following complex integral in the $z$-plane using Cauchy's theorem,
\begin{equation}
I=\frac{1}{2\pi i} \int_C \frac{z \,C(z,k)}{z-p}\ dz=0,
\label{cacuchy_1}
\end{equation}
with $C(z,k)\equiv \widetilde{\Phi}(0,k,z)\,G^+(0,x';z) \,e^{-i z L}$, by considering the integration contour $C=\Gamma+ C_p + \sum_n C_{n}+\sum_m C'_{m}$. See Fig.~\ref{contour}.  The contours $C_n$ encircle the complex poles $k_n$ of $G^+(x,x';z)$ which is a meromorphic function {\it i.e.} the Green function has simple poles.
The contours $C'_m$ encircle the poles $\kappa_m$ of $\widetilde{\Phi}(0,k,z)$, which is also a meromorphic function.
\begin{figure}[!tbp]
    \includegraphics[width=3.7in]{pdf/Figure1.pdf}
     \caption{Contour in the complex $z$-plane to solve Eq.~(\ref{cacuchy_1}). The complex poles are enclosed by the following contours: $\Gamma$ of radius $R$ centered about the origin in the $z$-plane following an anticlockwise direction, $C_p$ that encloses a pole at $z=p$, all the $C'_{m}$ that enclose simple poles at $z= \kappa_m$, and all the $C_n$ that encircle the simple complex poles ${k_n}$ of $G^+(x,x';z)$, all in a clockwise direction.}
      \label{contour}
        \end{figure}
The complex integration leads us to,
\begin{eqnarray}
&&\frac{1}{2 \pi i} \Bigg[ \int_{C_p} \frac{z \,C(z,k)}{z-p}\ dz  \nonumber \\
&+&  \sum_n\int_{C_n} \widetilde{\Phi}(0,k,z)\,G^+(0,x';z) \,e^{-i z L} \ dz  \nonumber \\
 &+&\sum_m\int_{C'_m} \widetilde{\Phi}(0,k,z)\,G^+(0,x';z) \,e^{-i z L}\ dz \Bigg]=0. \nonumber \\
\label{integrales_complejas}
\end{eqnarray}
We perform the integration in Eq.~(\ref{integrales_complejas}) by applying the Cauchy residue formula, and using the following expansions for $\widetilde{\Phi}(0,k,z)$, and $G^+(0,x';z)$. For the case $\widetilde{\Phi}(0,k,z)$ we assume a Mittag-Leffler expansion
\begin{eqnarray}
\widetilde{\Phi}(0,k,z)=\widetilde{\Phi}(0,k,0)+\sum_{m'=1}^{\infty} \alpha_{m'} \left( \frac{1}{z-\kappa_{m'}}+\frac{1}{\kappa_{m'}}\right),
\label{mittaglef}
\end{eqnarray}
where $\alpha_{m'}$ are the residues at the simple complex poles $\kappa_{m'}$.
For the case of $G^+(0,x';z)$, we use the expansion of the Green function  in terms of the complex poles $k_n$ and the resonance states \cite{gcaqc10} $u_n(x)$,  namely
\begin{equation}
G^+(x,x';z)=\sum_{n=-\infty}^{\infty}\frac{u_n(x)u_n(x')}{2k_n(z-k_n)},
\label{Ggaston}
\end{equation}
where the $u_n$'s are one-dimensional resonant states (\textit{quasinormal states}), which are
eigensolutions of the Schr\"{o}dinger equation,
\begin{equation}
\frac{d^2u_n(x)}{dx^2}+\left[ k_n^2-\widetilde{V}(x)\right]
\,u_n(x)=0, \label{SEun}
\end{equation}
with outgoing boundary conditions:
\begin{eqnarray}
\frac{du_n(x)}{dx}\Big|_{x=L}&=&+ik_nu_n(L);\nonumber \\
\frac{du_n(x)}{dx}\Big|_{x=0}&=&-ik_nu_n(0).
\label{boundary1}
\end{eqnarray}
The index $n$ ($n=1,2,3,...$) spans both the third  and fourth quadrant poles of the complex $k$-plane, given respectively by $k_{-n}$ and $k_n$.
These poles  are related through $k_{-n}=-k_n^*$, which follows from time-reversal invariance.
The complex energies associated to $k_n$ are given by $E_n=\hbar^2k_n^2/2m=\varepsilon_n-i\Gamma_n/2$, where $\varepsilon_n$ is the resonance energy, and $\Gamma_n$ the resonance width.
The resonant states also satisfy the time-reversal property $u_{-n}(x)=u_{n}^*(x)$.
Since our approach is non-Hermitian, the resonant states  do not obey the usual rules of normalization and orthonormality \cite{gcaqc10}. Instead, the $u_n$'s fulfill the following condition,
\begin{equation}
\int_0^L u_n(x)u_m(x)\,dx +i \frac{u_n(0)u_m(0)+u_n(L)u_m(L)}{k_n+k_m}=\delta_{nm}.
\label{f5}
\end{equation}
For the particular case $n=m$, Eq.~(\ref{f5}) yields the corresponding  normalization for the resonant states,
\begin{equation}
\int_0^{L}u_{n}^2(x)\,dx + i \frac{u_{n}^2(0)+u_{n}^2(L)}{2k_n}=1,
\label{normalizationu}
\end{equation}
which also satisfy the closure relation,
\begin{equation}
\frac{1}{2}\,\sum_{-\infty}^{\infty}u_n(x)\,u_n(x')=\delta(x-x'),\,\,\,\, 0\le (x,x')^{\ddag} \le L,
\label{closure}
\end{equation}
where the notation $(x,x')^{\ddag}$ means that the expansion does not hold at $x=x'=0=L$.
The resonant states  form a complete set \cite{gcaqc10} along the internal region of the potential, and any arbitrary function $\psi(x,k)$ can be expressed as a linear combination of these states using the fundamental relation $\psi(x,k)=2ikG^+(0,x;k)$.

By using the  expansions (\ref{mittaglef}) and (\ref{Ggaston}) to evaluate the integrals in  Eq.~(\ref{integrales_complejas}) by a direct computation of the corresponding residues, we  obtain and  expression for $C(p,k)$ that can be substituted in Eq.~(\ref{psi_interanl1c}), leading us to,
\begin{eqnarray}
\widetilde{\Psi}(x,p)&=&\alpha\left[ \sum_{m=1}^{\infty}\frac{2 \alpha_m\kappa_{m}}{p \,(p-\kappa_{m})}G^+(0,L,\kappa_m) e^{-i \kappa_m L} e^{ipx}  \right. \nonumber \\
 &+& \left. \sum_{n=-\infty}^{\infty}\widetilde{\Phi}(0,k,k_n) \frac{e^{-i k_n L}u_n(0)u_n(x)}{p\,(p-k_n)} e^{i p x} \right].
\label{cuasifinal}
\end{eqnarray}
We substitute in  Eq.~(\ref{cuasifinal})  the fundamental relationship between the transmission amplitude $t(k)$ and the Green function of the system \cite{gcaqc10}, namely
\begin{equation}
 t(k)=2 i k e^{-i k L}G^+(0,L;k),
 \label{phiconG}
\end{equation}
 %
and from the resulting expression we
obtain $\Psi(x,t)$ by performing the inverse Laplace-transform  by means of the Bromwich integral,
\begin{equation}
\displaystyle\Psi(x,t)=\frac{1}{2 \pi i}\int_{\gamma - i\infty}^{\gamma+i\infty}\widetilde{\Psi}(x;s)\,e^{st} ds.
\label{Bromwich_int}
\end{equation}
Equation~(\ref{Bromwich_int}) is evaluated by using the Moshinsky functions:
\begin{equation}
 M(x,q,t)=\frac{1}{2 \pi i} \int_{\gamma - i\infty}^{\gamma+i\infty}\frac{(i \alpha/2)}{\sqrt{\alpha is}}\left[ \frac{e^{\sqrt{\alpha i s }\, x}}{\sqrt{\alpha is }-q}\right]\, e^{s t} ds,
\label{Mcompeltota}
\end{equation}
which are also related to the complex error function  \cite{abramowitz}  $w(z)=e^{-z^2}{\rm erfc}(-i z)$, by
\begin{equation}
M(x,q,t)\equiv M(y_q)=\frac 1 2\,e^{imx^2/2 \hbar t}\, w(i y_q),
\label{moskyx}
\end{equation}
with,
\begin{equation}
y_q ={\rm e}^{-i\pi /4}\,\frac{\left(x- \hbar \, q \,t/m\right)}{\sqrt{2\hbar t/m}},
\label{yqx}
\end{equation}
where $q$ stands for $k$ or $k_{n}$.
We finally write the general time-dependent solution $\Psi(x,t)$ along the transmission region as:
\begin{eqnarray}
\Psi(x,t)&=&- \Bigg[\sum_{m=1}^{\infty} \alpha_m t(\kappa_m) M(y_{\kappa_m})  \nonumber\\
&+&i\sum\limits_{n=-\infty }^{\infty
}t_n \widetilde{\Phi}(0,k,k_n)M(y_{k_n}) \Bigg],\,x\geq L.\nonumber \\
\label{eq:12}
\end{eqnarray}
In the above formula we have defined $t_n=e^{-i k_n L}u_n(0)u_n(L)$, and
the $Q$-transform function $\widetilde{\Phi}(0,k,Q)$,
\begin{equation}
\widetilde{\Phi}(0,k,Q)=\Bigg[ i \int \phi(x,k)\, e^{-i Qx} dx\Bigg]_{x=0}.
\label{phitildosa}
\end{equation}
The coefficients $\alpha_m$ are the residues at the complex poles $Q=\kappa_m$ of  the $Q$-transform,  $\widetilde{\Phi}(0,k,Q)$.
The solution given by Eq.~(\ref{eq:12}) holds for any finite range potential, and for initial states $\phi(x,k)$, as long as the $Q$-transform $\widetilde{\Phi}(0,k,Q)$ yields simple $Q$-poles.
To verify  our result  Eq.~(\ref{eq:12}), let us consider as an example the case of an  absorbing plane wave shutter $\phi(x,k)=e^{ikx}$. The $Q$-transform is $\widetilde{\Phi}(0,k,Q)=(k-Q)^{-1}$ with one single pole at  $Q=\kappa_1=k$, with a residue $\alpha_1=-1$, that yields by direct substitution in Eq.~(\ref{eq:12})  the transmitted time-dependent wavefunction of Ref. \onlinecite{gcar97}.

\subsection{Phase modulated wavepackets}

We use as an initial condition a cut-off phase modulated wavepacket
given by,
\begin{equation}
\Psi(x,0)=\cases{2\,\sin (k\, x)\,\cos(\Delta k \,x), & for $x\leq 0;$\cr 0, & for $x> 0$,}
\label{eq1_bis_bis}
\end{equation}
constructed by a superposition of  quantum states with different momenta \textit{i.e.}
$\Psi(x,0)= \psi_+(x,k_+,0)+\psi_-(x,k_-,0)$, with $\psi_{\pm}(x,k_{\pm},0)=(e^{i k_{\pm} x}-e^{-ik_{\pm} x})/2i$, where $k_{\pm}=k\pm\Delta k$, corresponding to different kinetic energies, $E_{\pm}=\hbar^2 k_{\pm}^2/2m$.
The detailed analysis of the transient dynamics of these type of phase-modulated cut-off waves, to our knowledge, has not been performed before, although a similar condition has been applied in the context of time-dependent potentials \cite{Buttiker_1985}.

We apply our general result Eq.~(\ref{eq:12}) for a  modulated wavepacket $\phi(x,k)=2\,\sin (k\, x)\,\cos(\Delta k \,x)$, and  obtain,
\begin{equation}
\widetilde{\Phi}(0,k,Q)=i \Bigg[\frac{k_-}{(Q+k_-)(Q-k_-)}+\frac{k_+}{(Q+k_+)(Q-k_+)}\Bigg], \\
\label{qtransformodula}
\end{equation}
which has complex poles at $Q=\pm k_+$ (residues $\pm i/2$), and $Q=\pm k_-$ (residues $\pm i/2$). Thus, Eq.~(\ref{eq:12}) allows us to write,
\begin{equation}
\Psi(x,t)= \psi_+(x,k_+,t)+\psi_-(x,k_-,t); \,\,\,x\ge 0,
\label{psi_general_modula_suma}
\end{equation}
with
\begin{eqnarray}
\psi_{\pm}(x,k_{\pm},t)&=&(1/2i)\left[ t(k_{\pm})\,M(y_{k_{\pm}})- t(-k_{\pm})\,M(y_{-k_{\pm}}) \right.\nonumber \\
&+& \left. \sum\limits_{n=-\infty }^{\infty
}t_n \widetilde{\Phi}(0,k,k_n)M(y_{k_n}) \right].
\label{psi_general_modula}
\end{eqnarray}
In the next section  we shall apply our general result Eq.~(\ref{psi_general_modula_suma}) to explore the features of modulated wavepackets for a particular potential that involves a bound state.

\section{Dynamics of phase modulated wavepackets \label{results}}

Based on our result Eq.~(\ref{psi_general_modula_suma}), we shall explore the dynamics of the probability density of modulated cut-off wavepackets in quantum systems.
As we shall show, the modulation of the initial state governs the  time-dependent features of the probability density in different regimes, and some of these features are independent of potential.
To begin our study, we consider the case of a simple potential that allows to explore the dynamics in systems with a single bound state, as is the case of the Dirac delta potential well, $V(x)=-\lambda \,\delta(x)$,  with $\lambda>0$.
We obtain the corresponding solution from  Eqs.~(\ref{psi_general_modula_suma}) and (\ref{psi_general_modula}),  noting that only a single term in the resonance sum is required since the spectrum of the system is composed by a single bound state at $k_n=i \tilde{\lambda}$, with  energy $E_{b}=-\hbar^2\tilde{\lambda}^2/2m$, where  $\tilde{\lambda}=m\lambda/\hbar^2$.
In the limit $L\rightarrow 0$, one can show with the help of Eq.~(\ref{normalizationu}) that $u_n^2(0)\rightarrow \tilde{\lambda}$, and thus $t_n\rightarrow \tilde{\lambda}$, leading us to the solution:
\begin{equation}
\Psi^{\delta}(x,t)= \psi_+^{\delta}(x,k_+,t)+\psi_-^{\delta}(x,k_-,t); \,\,\,x\ge 0,
\label{psi_delta_modula_2}
\end{equation}
with
\begin{eqnarray}
\psi_{\pm}^{\delta}(x,k_{\pm},t)&=& (1/2i)\left[ t(k_{\pm})\,M(y_{k_{\pm}})- t(-k_{\pm})\,M(y_{-k_{\pm}}) \right.\nonumber \\
&+& \left. \tilde{\lambda}\, \widetilde{\Phi}(0,k,i\tilde{\lambda})\,M(y_{i\tilde{\lambda}})\right],
\label{psi_delta_modula_2_bis}
\end{eqnarray}
where $t(k)=k/(k-i\tilde{\lambda})$
is the stationary transmission amplitude of the delta potential well.
Also, the free modulated wavepacket solution is obtained directly from Eqs.~(\ref{psi_delta_modula_2}), and (\ref{psi_delta_modula_2_bis}) by letting $t(\pm k_{\pm})=1$, and $\tilde{\lambda}=0$, leading us to
\begin{eqnarray}
\Psi_f(x,t)&=&(1/2i)[M(y_{k_+})+M(y_{k_-})\nonumber \\
&-& M(y_{-k_+})-M(y_{-k_-})], \,\,x\ge 0. \nonumber \\
\label{psi_delta_modula_free}
\end{eqnarray}
In what follows, we  use  Eq.~(\ref{psi_delta_modula_2}) for exploring  different regimes of probability density
$\rho(x,t)$ defined as,
\begin{equation}
\rho(x,t)= |\psi_+^{\delta}(x,k_+,t)+\psi_-^{\delta}(x,k_-,t)|^2; \,\,\,x\ge 0,
\label{psi_delta_modula_2_ah}
\end{equation}
to  investigate the effects of the phase modulation on the transient dynamics of the system.

\subsection{Quantum beats} \label{quantumbeats}

Let us now explore the time-dependent features of $\rho(x,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})] at a fixed position $x=0$. In Fig. \ref{qbshort}(a) we plot $\rho(0,t)$ as function of  time, for an unmodulated wavepacket \textit{i.e.} $\Delta k=0$.
The probability density exhibits an oscillatory effect known as  {\it persistent oscillations}, reported in  Ref. \onlinecite{mlgc10} for the case of an absorbing plane wave shutter.
The effect is a result of the interaction of the quasi-stationary state induced by the incident wave, and the bound state of system.
We argue that the persistent oscillations also manifest themselves in a reflecting shutter setup, since the
particular case with $\Delta k=0$ obtained from Eq.~(\ref{psi_delta_modula_2}), corresponds in fact (up to a phase factor $-i$) to a reflecting cut-off plane wave initial condition.
However, when we consider the case of a  modulated wavepacket ($\Delta k\ne0$), we show in Fig. \ref{qbshort}(b) that $\rho(0,t)$  exhibits high-frequency oscillations  in the time-domain, similar to a superposition effect of quantum waves with different frequencies.
To identify the underlying superposition observed in Fig.~\ref{qbshort}(b), we plot the probability density for different values of $\Delta k$ over a larger time interval.
\begin{figure}[!tbp]
{\includegraphics[angle=0,width=3.4in]{pdf/Figure2.pdf}}
\caption{Probability density as function of time for a delta well potential at a fixed position $x=0$. In (a) we show that $\rho(0,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})]  exhibits \textit{ persistent oscillations} for an unmodulated wavepacket  ($\Delta k=0$). In (b) we analyze the dynamics of the modulated probability density $\rho(0,t)$ for $\Delta k=0.001$ \AA$^{-1}$, where a \textit{beating effect} is observed. Here and in all the figures throughout the paper, we use the following parameters: delta intensity $\lambda=4.27$ eV \AA, and incidence energy $E=0.08$ eV. }
\label{qbshort}
\end{figure}
\begin{figure}[!tbp]
{\includegraphics[angle=0,width=3.4in]{pdf/Figure3.pdf}}
\caption{ \textit{Quantum beat} effect in the time-dependence of $\rho(0,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})]  for a phase modulated wavepacket in a delta well potential, for (a) $\Delta k=0.0001$ \AA$^{-1}$, (b)  $\Delta k=0.0005$ \AA$^{-1}$, and (c) $\Delta k=0.001$ \AA$^{-1}$.
The period of the beats is $T=4.9$ ps, as indicated in (a), corresponding to a beating frequency $\Omega= 1.3$ THz. (d) Comparison of the exact  $\rho(0,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})] (orange solid line) of case (c), with $\rho_a(0,t)$ [Eq.~(\ref{chorizuda})] (blue dashed line), where a perfect agreement is observed.}
\label{qblong}
\end{figure}
This is shown in Fig.~\ref{qblong}, where $\rho(0,t)$ exhibits \textit{quantum beats} with a well defined  frequency.
To determine the beating frequency of the oscillatory phenomenon, we derive an approximate solution for $\Psi^{\delta}(0,t)$  [Eq.~(\ref{psi_delta_modula_2})],  by using the properties of the Moshinsky functions.
We simplify our notation for the arguments of the Moshinsky function at $x=0$, by
\begin{equation}
y(0,q,t) \equiv y^{\circ}_{q}=-{\rm e}^{-i\pi /4}\,\left( \hbar/2 m\right)^{1/2}q \,t^{1/2}.
\label{yq}
\end{equation}
By using the identity
$M(y^{\circ}_{q})={\rm e}^{-i \hbar q^2  t/2m}-M(y^{\circ}_{-q})$,
and keeping only the exponential contributions, we can approximate  $\Psi^{\delta}(0,t)$ by,
\begin{eqnarray}
\Psi^{\delta}_{a}(0,t)&\simeq& \frac{1}{2 i} \left[t(k_+) e^{-i E_+ t/\hbar} +t(k_-) e^{-i E_- t/\hbar} \right . \nonumber \\
&+& \left. 2 \tilde{\lambda}\,\widetilde{\Phi}(0,k,i \tilde{\lambda})\, e^{-i E_b t/\hbar}\right].
\label{aproxenx0}
\end{eqnarray}
In Eq.~(\ref{aproxenx0}) we have neglected the contributions of Moshinsky functions of the form $M(y_{-q})$ since  these are fast decreasing functions of time.
The expression for the probability density $\rho_a(0,t)$ corresponding to Eq.~(\ref{aproxenx0}) is:
\begin{eqnarray}
\rho_a(0,t)&\simeq& \frac{1}{2}\Bigg\{ \frac{1}{2}\left[|t(k_+)|^2+|t(k_-)|^2+4 \tilde{\lambda}^2 |\widetilde{\Phi}(0,k,i \tilde{\lambda})|^2\right] \nonumber \\
&+&a_{1,r}\cos\Omega t+a_{1,i}\sin\Omega t \nonumber \\ &+&(a_{2,r}+a_{3,r})\cos\left(\bar{\Omega} t\right)\cos\left(\Omega t/2\right) \nonumber \\
&+&(a_{2,i}-a_{3,i})\sin\left(\bar{\Omega} t\right)\cos\left(\Omega t/2\right) \nonumber \\
&+&(a_{2,i}+a_{3,i})\cos\left(\bar{\Omega}  t\right)\sin\left(\Omega t/2\right) \nonumber \\
&+&(a_{3,r}-a_{2,r})\sin\left(\bar{\Omega} t\right)\sin\left(\Omega t/2\right) \Bigg\},
\label{chorizuda}
\end{eqnarray}
where the coefficients in Eq.~(\ref{chorizuda})  are calculated  as $a_{n,r}={\rm Re}[a_n]$, and $a_{n,i}={\rm Im}[a_n]$, with
$a_1=t(k_+)t^*(k_-)$, $a_2=2t(k_+)\tilde{\lambda}\widetilde{\Phi}^*(0,k,i \tilde{\lambda})$, and  $a_3=2 t^*(k_-)\tilde{\lambda}\widetilde{\Phi}(0,k,i \tilde{\lambda})$.

We show from Eq.~(\ref{chorizuda}) that the dynamics of $\rho(0,t)$ in Figs.~\ref{qblong}(a)-(c) is governed by two types of frequencies, $\bar{\Omega}$ and $\Omega$, where
$\bar{\Omega}=(\omega_++\omega_-)/2$, is an average frequency associated to  persistent oscillations, with
$\omega_{\pm}= (E_{\pm}-E_b)/\hbar$.
The $\omega_{\pm}$
correspond to Rabi-type frequencies associated to the virtual states, $E_{\pm}$, induced by the source, and that of the bound state of the system, $E_b$.
The amplitude of the quantum wave is modulated by an envelope with a \textit{beating frequency}, $\Omega=\omega_+-\omega_-=(E_+-E_-)/\hbar$. Therefore,  the {\it quantum beats} in  Figs.~\ref{qblong}(a)-(c) arise due to the interaction between the {\it virtual two-level system} with energies $E_{+}$ and $E_{-}$, induced by the  modulation of the initial state.
In Fig.~\ref{qblong}(d) we compare the exact $\rho(0,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})] with  $\rho_a(0,t)$  [Eq.~(\ref{chorizuda})], and a perfect agreement is observed.
Note in  Fig.~\ref{qblong}(a) that the period $T=2\pi \,\Omega^{-1}$ indicated in the figure is of the order of picoseconds, and the  corresponding frequency of the \textit{quantum beats} lies in the terahertz range.
Interestingly, the terahertz  frequency regime is currently accessible to experiments in the field  of  femtosecond pulse technology \cite{Garraway_1995}.

\subsection{Dynamics of a virtual two-level system}

We investigate the effect of the self-induced virtual levels, $E_{+}$ and $E_{-}$, on the dynamics of the probability density at distances far away from the potential.
These virtual levels arise due to the quantum superposition in our initial state, due to the phase modulation.
This is illustrated in Fig.~\ref{persistentes_grandes}, where we show the time-evolution of $\rho(x,t)$ for large values of position, $x$.
The probability density features a sharp rise after the time of flight, $t=t_{+}\equiv x/v_+$, where $v_+=\hbar k_+/m$, giving rise to
a peculiar oscillatory-phenomenon known as \textit{diffraction in time} \cite{mm52}.
However, we note that from  $t\simeq t_{-}\equiv x/v_-$ onward, where $v_-=\hbar k_{-}/m$, the time-diffraction profile becomes a highly-oscillatory pattern.
\begin{figure}[!tbp]
{\includegraphics[angle=0,width=3.4in]{pdf/Figure4.pdf}}
\caption{ Dynamics of  $\rho(x,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})] at a fixed position $x=1\times10^8$ \AA~  with $\Delta k = 0.0005$ \AA$^{-1}$. We identify a regime governed by a {\it diffraction in time effect},  originated by the virtual level with energy $E_+$, which becomes important for times $t\simeq t_{+}$ onward. The time-diffraction pattern \cite{mm52} is observed over a time interval  $0\leq t \lesssim t_d$. At later times, for  $t>t_{-}$ onward, a highly-oscillatory interference pattern is observed.}
\label{persistentes_grandes}
\end{figure}
The behavior observed in Fig.~\ref{persistentes_grandes} can be explained by analyzing the contributions to the exact solution $\Psi^{\delta}(x,t)$ [Eq.~(\ref{psi_delta_modula_2})] given by $\psi^{\delta}_{\pm}(x,t)$ [Eq.~(\ref{psi_delta_modula_2_bis})], and their corresponding interference.
This is shown in Fig.~\ref{persistentes_grandes_masmenos}, where we present the contributions to the exact probability density by rewriting Eq.~(\ref{psi_delta_modula_2_ah}) as $\rho(x,t)=\rho_++\rho_-+\rho_{int}$, with $\rho_{\pm}=|\psi_{\pm}(x,k_{\pm},t)|^2$, and the interference term $\rho_{int}=2{\rm Re}[\psi_+(x,k_+,t)\psi_{-}^*(x,k_-,t)]$.
The time-dependence of $\rho(x,t)$ is characterized by the propagation of two wavefronts traveling with different speeds, $v_{+}$, and $v_{-}$.
Since  $v_{+}>v_{-}$, it is expected that these traveling structures arrive at different times ($t_{+}<t_{-}$) at a fixed position, $x$.
Therefore, in  Fig.~\ref{persistentes_grandes_masmenos}(a)  we argue that the dynamics of $\rho(x,t)$ for times $t\simeq t_{+}$, is characterized by a time-diffraction effect originated by the fast momentum components $k_+$, which is a manifestation of the virtual state with energy $E_+$.
At a later time, when $t\simeq t_{-}$, the slow momentum components $k_-$ begin to build the  virtual level of energy $E_-$.
From $t\gtrsim t_{-}$ onward, the interplay between the two self-induced virtual levels originates the highly-oscillatory profile observed in Fig.~\ref{persistentes_grandes}, due to the $\rho_{int}$ contribution shown in  Fig.~\ref{persistentes_grandes_masmenos}(b).
\begin{figure}[!tbp]
{\includegraphics[angle=0,width=3.4in]{pdf/Figure5.pdf}}
\caption{(a)  Time-dependence of the contributions $\rho_{\pm}$ to the exact $\rho(x,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})] discussed in Fig.~\ref{persistentes_grandes}. Note that the arrival of the wavefronts of $\rho_{+}$ (orange solid line) and  $\rho_{-}$ (blue dashed line) occur at different times  $t_{+}$, and  $t_{-}$, respectively. We observe that $\rho_{\pm}$ exhibit a typical \textit{diffraction in time} phenomenon. (b) The interference $\rho_{int}$ of the contributions  $\rho_{\pm}$ becomes important for times $t>t_{-}$, setting the onset of the quantum interference between the virtual levels, $E_{\pm}$. The small interference between $\rho_{+}$ and $\rho_{-}$ over the time interval $0\leq t \lesssim t_d$, allows to observe the time-diffraction like pattern governed by $\rho_{+}$, as shown in Fig.~\ref{persistentes_grandes}.}
\label{persistentes_grandes_masmenos}
\end{figure}

Next, we shall explore the effect of the virtual two-level system  in the long-time behavior of the probability density.
This is illustrated in Fig.~\ref{qblargex}(a), where we show that  the exact probability density [Eq.~(\ref{psi_delta_modula_2_ah})] exhibits periodic oscillations with a frequency $\Omega$.
We shall show below that this transient feature is independent of the potential profile.
\begin{figure}[!tbp]
{\includegraphics[angle=0,width=3.4in]{pdf/Figure6.pdf}}
\caption{(a) Time-dependence of the exact $\rho(x,t)$ [Eq.~(\ref{psi_delta_modula_2_ah})] (orange solid line) for a delta potential well, and $\rho_{2l}(x,t)$ [Eq.~(\ref{chorizudita_bis})] (blue dashed line).  The \textit{virtual two-level system} dynamics  manifests itself as a Rabi oscillatory profile, with a frequency $\Omega \simeq 6.5$ THz and a period $T=0.97$ ps, indicated in the plot. (b) Comparison of the exact  $\rho(x,t)$ (orange solid line) discussed in (a), and the free case $\rho_{2l,f}(x,t)$ [Eq.~(\ref{rho_libriux})] (blue solid line). The case with $\Delta k=k$ (red dashed line) using  $\rho_{2l}(x,t)$ is also included, where the Rabi-mechanism is turned off.}
\label{qblargex}
\end{figure}
We proceed by deriving an asymptotic solution from  Eq.~(\ref{psi_general_modula_suma}) with the help of the asymptotic properties of the $M$-functions.
In the long-time limit ($t\rightarrow \infty$), and large values of the position, the argument of the $M$-functions is such that $|y_{q}|\gg 1$, leading to the following asymptotic behavior \cite{gcaqc10}:
\begin{eqnarray}
M(y_q)&\simeq& \frac{e^{imx^2/2\hbar t}}{2\,\pi^{1/2}y_q}, \,\,-\pi/2<\mbox{arg}(y_q)<\pi/2; \label{moshyprop_a} \\
M(y_q)&\simeq& e^{imx^2/2\hbar t} e^{y_q^2}, \,\,\pi/2<\mbox{arg}(y_q)<3\pi/2.\label{moshyprop_b}
\end{eqnarray}
By using the above expressions in Eq.~(\ref{psi_general_modula_suma}), we can see by inspection that the contributions involving the $M$-functions with $q=k_{n},-k_{\pm}$, are vanishingly small since $M(y_{q})\sim y_q^{-1}$  due to Eq.~(\ref{moshyprop_a}).
Thus, only the $M$-functions with $q=k_{\pm}$, which are governed by Eq.~(\ref{moshyprop_b}), contribute to the dynamics, leading us to the result:
\begin{equation}
\Psi_{2l}(x,t)\simeq \frac{1}{2 i} \left[t(k_{+}) e^{i(k_+ x -E_{+} t/\hbar)} +t(k_-) e^{i(k_{-} x -E_{-} t/\hbar)}\right]. \nonumber \\
\label{aproxenx0_bis}
\end{equation}
The corresponding probability density $\rho_{2l}(x,t)$ is given by,
\begin{eqnarray}
\rho_{2l}(x,t)&\simeq& \frac{1}{4}\bigg[ (|t(k_+)|+|t(k_-)|)^2 \nonumber \\
&-& 4|t(k_+)||t(k_-)|\sin^2\left(\Delta k x-\Omega t/2-\phi/2\right) \bigg], \nonumber \\
\label{chorizudita_bis}
\end{eqnarray}
with $\phi=\arctan(a_{1,i}/a_{1,r})$.
Note that $\rho_{2l}(x,t)$ exhibits an oscillatory profile typical of a two-level system  (Rabi-model), with a frequency  $\Omega$ (Rabi-frequency).
In Fig.~\ref{qblargex}(a) we compare  $\rho_{2l}(x,t)$ [Eq.~(\ref{chorizudita_bis})], with the exact  $\rho(x,t)$ for the delta well [Eq.~(\ref{psi_delta_modula_2_ah})], an a perfect agreement is observed.
Furthermore, in Fig.~\ref{qblargex}(b) we show that the oscillatory phenomenon also manifests itself in the free probability density, computed from Eq.~(\ref{psi_delta_modula_free}), and given by
\begin{eqnarray}
\rho_{2l,f}(x,t)\simeq 1-\sin^2(\Delta k\, x-\Omega t/2).
\label{rho_libriux}
\end{eqnarray}
Our results in Fig.~\ref{qblargex}(b) also show that $\rho_{2l}(x,t)$ and  $\rho_{2l,f}(x,t)$ oscillate with the same frequency $\Omega$.
From a physical point of view, the Rabi-dynamics observed in the transient regime, is the result of the virtual \textit{self-induced two-level system}, with energies $E_+$ and $E_-$, where the frequency of oscillation $\Omega$ is proportional to the semi-classical propagation speed, $v_k$, and $\Delta k$ of the initial state, a feature that is independent of the potential.
Only in the case where $\Delta k=k$, the oscillatory phenomenon disappears, as illustrated in Fig.~\ref{qblargex}(b).
In this case, the effective cancellation of one of the channels occurs  \textit{i.e.} $E_{-}=0$, which hinders the Rabi-mechanism.
Alternatively, we can also argue that in the asymptotic regime our system behaves like a two-level system, by starting from Eq.~(\ref{aproxenx0_bis}) with the following change in notation,
\begin{equation}
\ket{\Psi(t)}= c_{+}e^{-iE_{+}t/\hbar}\ket{+} +c_{-} e^{-iE_{-} t/\hbar}\ket{-},
\label{psiket_a}
\end{equation}
where $\{\ket{+}$, $\ket{-}\}$ is an orthogonal base with eigenstates  $\{E_{+}$, $E_{-}\}$, and the coefficients $c_{\pm}$ take into account the $t(k_{\pm})$, and other factors.
By eliminating the phase factor $e^{-iE_{-}t/\hbar}$ from Eq.~(\ref{psiket_a}), we can obtain
the probability of finding the state $\ket{\Psi(t)}$ at an earlier time $t=t'$ which is given by  $P_{t,t'}=|\braket{\Psi(t')|\Psi(t)}|^2$, where the system was initially at one of the virtual states ($E_+$ or  $E_-$), leading us to
\begin{eqnarray}
P_{t,t'}=(|c_+|^2+|c_-|^2)^2-4|c_+|^2|c_-|^2\sin^2\left[\Omega (t-t')/2\right], \nonumber \\
\label{psiket_c}
\end{eqnarray}
which is the well known Rabi-formula.
These persistent asymptotic oscillations are a feature of the initial modulated state, and are present for any finite range potential.

\subsection{Delay-time for modulated  wavepackets \label{timedelaysec}}

The delay-time \cite{DECARVALHO200283,mohsen2013quantum,Muga:2023382} as a
transient effect has been observed in quantum structures, such as  resonant double barrier systems \cite{gcar97}, and delta potentials \cite{19306687,gcah03,19156601,mlgc10,PhysRevA.86.062118}, and depending of the system, this time scale may exhibit advance or delay-time.
For example, in the case of repulsive delta potentials \cite{gcah03},  a delay-time is observed, while in the case of attractive potentials \cite{mlgc10}, a time-advance is obtained.
It is the purpose of this section to study the effect of the \textit{virtual states}, $E_+$, and $E_-$, on such a transient time-scale.
We calculate the \textit{dynamical delay-time} $\Delta t$ following the prescription of previous work on this subject \cite{gcah03}, which involves the measurement of the maxima of the main wavefront of the probability density in the time-domain. These maximum values of  $\rho(x,t)$ for the delta well  [Eq.~(\ref{psi_delta_modula_2_ah})], and the free-case $\rho_f(x,t)=|\Psi_f(x,t)|^2$ using Eq.~(\ref{psi_delta_modula_free}), occur at $t_{\delta}$ and $t_{f}$, respectively, where the delay-time, $\Delta t$, corresponds to the difference,
\begin{equation}
\Delta t\equiv t_{\delta}-t_{f},
\label{delay}
\end{equation}
as illustrated in Fig.~\ref{delaytime}.
Note also in this figure, that at a fixed position, $x$, the maximum of the probability density for the delta well case appears before the free-propagation peak, leading to a time-advance.
In both cases,  the dynamics is governed by the virtual level, $E_+$.
In Fig.~\ref{delayvsdeltak} we present the measurements of $\Delta t$ as function of $\Delta k$, for two different values of the position, $x$, far from the potential.
We show in Fig.~\ref{delayvsdeltak} that for small values of  $\Delta k$, the delay-time is very close to phase time.
In contrast, the delay-time tends to zero for large values of $\Delta k$
because the initial state is very energetic, and it is not affected by the potential.
\begin{figure}[!tbp]
{\includegraphics[angle=0,width=3.4in]{pdf/Figure7.pdf}}
\caption{Measurement  of the delay-time $\Delta t$ by considering plots of $\rho(x,t)$ for a delta well  [Eq.~(\ref{psi_delta_modula_2_ah})]  (blue solid line), and the free-case $\rho_f(x,t)$  (red dashed line), at fixed position $x=1000.0$ \AA, and $\Delta k=0.02$  \AA$^{-1}$. The corresponding delay-time is $\Delta t=-1.055\times10^{-3}$ ps.}
\label{delaytime}
\end{figure}
\begin{figure}[!tbp]
{\includegraphics[angle=0,width=3.4in]{pdf/Figure8.pdf}}
\caption{Delay-time $\Delta t$ as function of $\Delta k$ for the modulated wavepacket for a delta potential well,  calculated using Eq.~(\ref{delay}), measured at fixed positions $x= 1.0 \times 10^8$  \AA ~(blue solid triangles), and $x=1000.0$ \AA ~(green solid circles).  A plot of the analytical expression for the delay-time Eq.~(\ref{delya_anal})  (orange solid line) is also included for comparison, and an excellent agreement is observed. The red solid circle corresponds to delay-time for the case of $\Delta k=k$. The latter corresponds to a plane wave. The red dashed line corresponds to the phase-time Eq.~(\ref{tphi}), which yields $t_{\phi} = -2.05  \times 10^{-3}$ ps.}
\label{delayvsdeltak}
\end{figure}

We derive an exact analytical expression for $\Delta t$ by
following the procedure of Ref. \onlinecite{PhysRevA.86.062118}, which yields the result:
\begin{equation}
\Delta t \simeq \frac{(-\lambda/2)}{v_{+}\,\left(E_{+}-E_{b} \right)}.
\label{delya_anal}
\end{equation}
The time-scale Eq.~(\ref{delya_anal}) is explicitly governed by the energies $E_+$ and $E_b$, and is also included in Fig.~\ref{delayvsdeltak}, where an excellent agreement is observed.
We also compute the phase-time $t_{\phi}$ of the system by using the definition:
\begin{equation}
t_{\phi}=\hbar\,\frac{d\phi_t}{dE}=\hbar \,\textrm{Im} \left[ \frac{1}{t(k)} \frac{dt(k)}{dk} \frac{dk}{dE}  \right].
\label{delay_definition_a}
\end{equation}
By substituting in Eq.~(\ref{delay_definition_a}) the transmission amplitude, $t(k)$, for the delta potential, we obtain the explicit phase-time,
\begin{equation}
t_{\phi}= \frac{(-\lambda/2)}{v_k\,\left(E-E_b \right)}.
\label{tphi}
\end{equation}
Note that in the limit $\Delta k\rightarrow 0$, $E_+\rightarrow E$, and $v_+\rightarrow v_k$, thus $\Delta t \rightarrow t_{\phi}$, that is, the phase-time is recovered for a quasi-monochromatic initial state.

\section{Conclusions}
\label{conclusions}
We explore the transient dynamics of phase modulated cut-off wavepackets by deriving an exact analytical solution to Schr\"odinger's equation for  finite range potentials, within a quantum shutter approach involving a general initial quantum state.
The  modulated  cut-off quantum wave allows us to explore matter-wave transient phenomena such as time-diffraction, wave superposition, quantum beats, Rabi-oscillations, and delay-time.
In particular, it is demonstrated that the time-dependent features of the probability density are governed by a virtual \textit{self-induced two-level system}, with energies $E_+$ and $E_-$, due to the modulation of the initial state.
We found that in the long-time regime  the probability density exhibits Rabi-dynamics which is independent of the potential profile, with a frequency $\Omega=(E_+-E_-)/\hbar$.  Interestingly, since $\Omega=2 v_k \Delta k$, where $v_k$ is the velocity of a plane wave  with momentum $k$, the Rabi-frequency can be tuned by controlling the incidence energy $E$ and $\Delta k$ of the modulated wave.
The Rabi-mechanism can be effectively turned off by setting $\Delta k=k$, which corresponds to the closing of the channel $E_-$, or by letting $\Delta k=0$ (unmodulated case).
For the particular case of a  system with a bound state, $E_b$, we found that the virtual two-level system gives rise (at $x=0$) to a series of \textit{quantum beats} in the probability density, with a beating frequency $\Omega$.
These quantum beats in the probability density modulate a highly-oscillatory pattern of average frequency $\bar{\Omega}$ associated to persistent oscillations, resulting from an interplay between the virtual states, and the bound state of the system.
We also demonstrate the existence of a transient regime governed by $E_+$, where the probability density exhibits a \textit{diffraction in time} effect, which let us identify a well defined traveling wavefront. This  feature allows to measure the delay-time $\Delta t$ of the system at far distances from the potential. We also derive a simple analytical formula for describing this time scale, which depends on the energy difference between $E_+$ and $E_b$.
It is also shown that only for the case of plane waves, $\Delta t$ agrees with the phase-time.
The latter is in agreement with the results obtained for the delay time in Refs. \onlinecite{gcah03} and \onlinecite{mlgc10}, involving the time-evolution of cut-off plane waves.

We stress that both the \textit{quantum beat} effect, as well as the Rabi oscillations,  occur in a frequency range of the order of terahertz, which is currently accessible to experiments involving femtosecond spectroscopy  \cite{Garraway_1995}.
These matter-wave phenomena associated to electrons can be explored as discussed in Ref. \onlinecite{decamps16}, by using interferometric techniques based on electron holographic microscopy \cite{Hasselbach_2009}.

\section{Acknowledgements}
\label{acknowledgements}

The authors acknowledge support from UABC under Grant PFCE 2018.

redpajama_set_name: RedPajamaArXiv


You are here: Home / Solar Marketing Review / Get the Best Solar Leads – Exclusive, New, Interested. There's Just One Catch.
Tired of spending money on low-quality solar leads that don't convert into sales? Higher quality leads might be in sight if you reach for them.
Imagine if you could get solar leads that would convert at the highest rates because they were truly high quality — contacts that were fresh and exclusive to your company alone. Even better, they'd be homeowners who actually want to hear from you because they have expressed an interest in going solar.
Those are of course, the highest quality leads. It doesn't get any better than that.
Such high quality leads would be expensive, right? Well, you might be surprised how affordable it is to get the best leads.
But there is one catch. You can't just go out and buy the best leads for people who want solar at home from a lead-generation vendor. There's a secret to getting those leads. And not every solar company is set up to get the best leads.
Read on and you'll find out whether you can qualify to get the best leads, and then learn how you can do it.
Many "aged" solar leads are so old they might as well be obituaries.
Solar leads that you buy these days are uneven at best. And at worst, they're a total waste of money.
On the low end, say, $30 per sales lead, the names will convert at pathetically low rates.
Many "aged" leads are so old that they might as well be obituaries. And even before the dearly departed shuffled off this mortal coil, the chance that they really wanted solar panels ranged from little to none. Not to mention that six other solar companies had already called the same lead list before your sales reps were even able to pick up their phones.
High end leads can cost $100 or more. For that money, you can specify exclusive leads that you don't have to share with other solar companies. You can also demand that the leads be fresh, generated within the last few days — or even the last few hours. And you can hope that the people really want solar, and that they didn't just get pulled from some list of related interests, such as people who own a Toyota Prius or have recently purchased from a legal cannabis shop in Colorado.
But even expensive solar leads have a potential problem.
Expensive Solar Leads: Are They Worth the Money?
Buying expensive solar leads is a roll of the dice. You can win a big payout but it's a big gamble.
High-end solar leads have one big problem if you buy them from somebody else, even one of the best lead-generation vendors that offer high quality inbound leads generated online.
The people who represent the leads may have expressed an interest in solar. But they've not necessarily expressed an interest in your company. In fact, they may never have heard of your company. Instead, they may have just filled out a form on a lead-generation vendor's website. Some of those websites are excellent online marketplaces for solar, such as SolarPowerRocks.com or PickMySolar.com.
But if someone goes to a solar marketplace website offering quotes from a variety of installers, then the homeowner is not coming specifically to see your company. Honestly, they probably don't care much about the company that the website connects them with — as long as that company turns out to offer the cheapest quote.
Unfortunately, even with expensive leads that you purchase from a high-quality vendor, the conversion rate can still be under 5%. Since the people had never heard of your company before they were passed onto you by the lead generation vendor, those people have no loyalty to your company. So it's easy for them to drop out of the sales process before the deal is closed.
You can connect with them over the phone. But before your sales rep can make it over to their place to do an assessment and get a contract signed, perhaps one of your competitors squeezes in there first. Or maybe the people change their mind, and decide to skip the appointment they set with your sales rep.
There's a way to get the highest quality solar leads at a competitive rate. But there's one catch.
With lead-generation vendors laughing all the way to the bank, it's no wonder that customer acquisition costs for solar installers have remained steady for the last few years at $3,000 or more per customer according to GTM Research. Whether you buy cheap leads that convert at less than 1% or more expensive leads that may convert at 5% or 10%, you can still rack up huge bills buying solar leads from lead-generation vendors.
Meanwhile, your sales reps are getting frustrated as deals keep falling through and they have to work harder and harder for lackluster commissions.
Solar installers get angry at lead vendors who rip them off year after year. But yet, they just keep buying leads from different vendors. Then they get angry at those. And on and on it goes, year after year of lost sales and frustrated sales reps.
In the past, there wasn't much you could do about bad leads. Fortunately, today, the situation is different.
With the internet, forward-thinking solar installers are now generating their own leads online through inbound marketing.
Large national installers do a lot of the work themselves in-house. Smaller companies work with companies like the Curren Media Group to attract solar buyers to their company as online visitors and then convert those visitors into the highest quality sales leads — exclusive, fresh, and really interested in getting solar.
For example, one small installer in the Southeast that we've worked with for a year has now started generating inbound leads on their own website for an effective cost of $70-$80 each.
These leads represent people who've found this solar company online through Google or through the company's social media posts or email newsletters. As inbound leads, these homeowners decided on their own to contact this company by name. They didn't just want solar from any company that turned out to be the lowest bidder. They wanted this company specifically. That will make the customer less likely to try to shop around for the lowest price and more likely to just go with this company's bid as-is.
So, at $70 or $80 per lead, the cost per lead alone is very competitive for this company, considering that the leads are better than leads they could get from a vendor that would cost $100 or above. In a busier area than the Southeast like California or the Northeast, the cost per lead could be even lower.
But, as they say on informercials, wait, there's more!
The benefit to this solar company goes far beyond the leads they get today. The system they've set up will help them also generate leads for years to come.
Of course, there's one little catch.
Should You Buy Apples or Plant Your Own Apple Tree?
If you buy apples from someone else, when you run out, you just have to buy more.
It's the difference between buying apples and planting your own apple tree.
If you buy a bag of apples at the grocery store, for example, you get apples. And you get them right away, no waiting, as long as the store has apples in stock. But when you run out of apples at home, you have to drive back to the grocery store to buy more. If you stop going to the grocery store, then you won't have any more apples.
By contrast, if you plant an apple tree, then once it starts producing apples, you can have free apples that you grow yourself for the next 30 years.
But here's the catch: maybe your apple tree won't start producing fruit right away. You might have to wait a while. You might also have to nurture the tree for a while as it grows up. For example, you may need to fertilize the soil or prune off branches.
It's exactly the same with solar leads.
If you want leads tomorrow, you can buy them from lead-generation vendors — provided that you can find leads in your service area. You can get those leads right away and without having to do any of the work to generate them yourself. The quality can vary greatly, as we saw above. But if you're a gambling person, you can plan for that.
The big disadvantage with buying leads is that once you stop buying them, the flow of leads shuts off. You can buy leads for a year, but when you stop buying those leads, you're no closer to generating your own in the future. If you're hooked on buying leads, then it's very difficult to wean yourself off.
Just like planting an apple tree, if you plant your own lead-generation system online, you'll get leads for years to come. This allows you to break the cycle of addiction to lead-generation vendors. Once your online inbound marketing system is set up, you'll have a reliable lead flow that, with proper care, can grow as long as you want.
So, to continue the apple metaphor, comparing $70 inbound leads to $100 leads you buy isn't really an apples-to-apples comparison.
With every $100 lead you buy, all you get is a lead. You get no extra benefit in terms of branding.
But with every lead you generate yourself for $70 now, you'll be building your brand up to generate more leads yourself in the future. When you consider the branding benefit of inbound marketing to generate leads, then over time, each lead might wind up costing much less than $70.
The catch is that generating your own leads doesn't work right away. It takes time, anywhere from 3-6 months, to start generating your own leads online.
If you already have a great website optimized for lead generation that gets good web traffic now, you'll have a head start that could speed things up.
But if your website isn't lead-ready, and if don't get much web traffic yet, then you'll have to start and nurture your lead system before it starts to produce fruit.
You don't have to be a huge national installer or a large solar marketplace to generate your own leads online. Even small local installers can do it.
Both large and small solar installers can generate their own leads online.
To have success generating leads yourself online, you need two things.
First, to generate your own leads, you need a reputation for integrity and high quality work. If you have bad customer reviews splashed all over Yelp and SolarReviews, the less attention you draw to your brand online or anywhere else, the better. In that case, just keep buying leads. And good luck to you!
Second, if your company is committed to quality and your customers will say that, then you need a modern website that's ready for lead generation.
If your website is outdated and unfriendly to visitors, then your company won't generate many leads online, no matter how much inbound marketing you do.
Since more than half of web traffic these days comes from phones, your website needs to look good on mobile devices.
Then, the site needs to offer enough good content to build trust among solar buyers. Running a blog can give you an unfair advantage over competitors who aren't.
The content should be fresh and optimized for SEO to get traffic from web searches. Otherwise, you'll have to buy ads, which can waste a lot of money.
And of course the site needs to have an online contact form that makes it easy for visitors to ask for a quote from your company.
You'll get even more leads if you offer a lead-magnet such as a downloadable ebook in exchange for a visitor's email address.
Online marketing agencies like the Curren Group can help a solar installer build a website that's optimized for lead generation. Then, you can start inbound marketing to generate leads online.
The longer you wait to start building your own lead-generation system, the longer you'll have to wait to start generating your own leads.
The sooner you plant your online lead tree, the sooner you'll enjoy the delicious fruit of exclusive, fresh, high-quality solar leads.


text:
Toddler critically burned when SWAT team throws flashbang in crib
Published time: 30 May, 2014 15:27 Edited time: 2 Jun, 2014 16:04
Kevork Djansezian / Getty Images / AFP © AFP
A toddler is in a medically induced coma after a SWAT team raided the family's home and threw a flashbang grenade into his crib, the boy's mother said. With a 50 percent chance of survival, it could be weeks before it's known if he will even live.
Bounkham "Bou Bou" Phonesavanh, a 19-month-old, was asleep in his portable crib in the same room as his parents and three older sisters, when police opened the door to the converted garage and threw the stun grenade in. It landed in the crib with Bou Bou.
"Everyone's sleeping. There's a loud bang and a bright light," the boy's mother, Alecia Phonesavanh, told WSB-TV. "The cops threw that grenade in the door without looking first, and it landed right in the playpen and exploded on his pillow right in his face."
The multi-jurisdictional Georgia SWAT team was executing a no-knock warrant at 3 a.m. on the home where a confidential informant had purchased drugs earlier in the day. The CI said he bought methamphetamine from Wanis Thometheva there on Tuesday. Police told the Atlanta Journal-Constitution that investigators had discovered Thometheva had weapons (including an AK-47) during a previous arrest on drug charges.
"That's the threat he uses to those who don't do what he wants," Habersham County Sheriff Joey Terrell said to the AJC.
But Thometheva wasn't in the home when the police raided. Instead, the Phonesavanhs were there, sleeping. They were only supposed to be in Georgia temporarily after a fire at their Wisconsin home, the family said.
"We have nothing to do with this (drugs)," father Bounkham "Bou" Phonesavanh told the AJC.
The police department said they had no idea there were any children in the house. "There was no clothes, no toys, nothing to indicate that there was children present in the home. If there had been then we'd have done something different," Cornelia Police Chief Rick Darby said to WSB-TV.
"You're trying to minimize anything that could go wrong and in this case the greatest thing went wrong," Darby said. "Is it going to make us be more careful in the next one? Yes ma'am, it is. It's gonna make us double question."
Terrell told the AJC that a medic began performing first aid on Bou Bou before the child was transported to the hospital, but he was unable to be transported by helicopter due to unsafe weather conditions.
"The last thing you want is law enforcement to injure someone innocent," Terrell said. "There was no malicious act performed. It was a terrible accident that was never supposed to happen."
Thometheva was later arrested at a different home on a felony drug charge of distribution of meth, the AJC reported.
In the meantime, Bou Bou is fighting for his life in the hospital. "He's in the burn unit. We go up to see him and his whole face is ripped open. He has a big cut on his chest," his mother said to WSB-TV. "He's only 19 months old. He didn't do anything."
"Our hearts are broken with them because of the child," Terrell said.
The family does not have insurance and does not know where they will live during the minimum six weeks Bou Bou will be in the hospital. A family friend has created an online fund-raising page to help with costs.
SWAT burns innocent man to death
Cop keeps job after brutal restaurant attack leaves victim in critical condition
Botched SWAT raid leaves 12-year-old girl with second-degree burns

redpajama_set_name: RedPajamaCommonCrawl


OK so it is official – I am getting one spot per week! I could feel the start of one yesterday but I was in denial so didn't mention it. I so wanted to break my spot free days record but it wasn't meant to be… not yet! It developed today into a full blown spot just under my bottom lip and has already burst of its own accord this evening. Very Frustrating, but the count is now back to zero!!
I want to tell you about my hair. It has been affected by the Roaccutane more than I wanted to believe. Not quite "like old rope" to coin my consultant's phrase but it really isn't in the lovely condition that it was before I started this treatment. Not so bad that anyone else would notice, but I do. It doesn't fall as nice as it used to, it doesn't feel as silky and doesn't do what I want it to. The good side effect is that it doesn't need washing very often like before; that will be because of the grease reducing actions of the Isotretinoin. So, last night I finally took my dermatologist's advice and used loads of hair conditioner and on all of my hair. This felt strange because with normal hair you would only use a little bit and only on the ends of your hair. It has made a little bit of a difference and hopefully if I continue it will help a lot.
Note to self – more photos!
Realisation today – Whilst my skin has been getting better and I've had a 'nice' week to ten days with my skin feeling good and the side effects less, I realised today that they of course will continue so long as I'm taking the Roaccutane. I must prepare myself for them to continue for a few months afterwards as well until my skin returns to normal. That is a 'New Normal' I don't want it to return to how it was!
My eyes are really bad again at the moment, very dry, sticky and very uncomfortable. My right eye today has been extra painful, it feels like a stye does but I don't have a stye. I have used the eye drops several times today, using so much it was like an eye wash more than drops. This evening my boyfriend came up with idea of making me look like a pirate! A home made eye patch! It did help as well as entertaining him and my son.
I have had a rash all up the inside of both my arms, not itchy just a simple red rash. It has calmed a bit this evening.
The aches and pains in my joints were bad last night, in every joint in my body. Every time I moved it felt like I was really old! Not nice!
Feeling a bit fed up today and a bit tearful. I call it time out from being strong. Don't get me wrong, I'm proud of myself for persevering and I'm chuffed to bits with the results but some of the side effects are not pleasant.
From previous weeks I thought I knew what dry eyes were. But this week they really are dry! Proper painful and very uncomfortable. As you can see from today's photo they are swollen and sore too. I have been using eye drops throughout to soothe them but since Monday they are not helping enough. My boyfriend went on an emergency trip to the chemist this evening because I had run out of eye drops. Now I thought eye drops were eye drops, I didn't realise you could get different sorts. He came back with some that are specifically for dry eyes. The relief was bliss!
This is definitely another one of those bad phases and I would just like to mention and thank my best friends that have truly supported me and got me through this journey since the beginning of March 2013… my wonderful boyfriend… and Hydromol and Vaseline. I couldn't have done it without you!
My eyes are still very dry today but a fraction better than yesterday. My skin is still very fragile and injures easily. I have a graze on the top of my left foot and I have no idea what caused it. My nose is dry and crusted on the inside which can be a bit painful at times. If you look closely at today's pictures you will see a red patch on my chin. This is the first time in a very long time that I have had anything new on my chin. It was actually the first part of my face to clear up completely and has been clear all the way through since then. When I first saw it this morning I thought a new spot was developing but as the day has gone on it has become very dry and feels rough but there is no spot. As long as there are no new spots in the morning, then I will restart my 'Spot Free Days' counter at day one tomorrow.
We have taken a couple of days off this weekend to get some sea-air with a nice trip up to Blackpool! My skin is definitely more fragile on Roaccutane; I got blisters on my feet today, no new shoes, no huge amounts of walking, so the Isotretinoin has definitely given me weak skin! It's very new to me as I have never had blisters on my feet before. Funny thing is I bought some special blister plasters yesterday, went out today and have left them at home !
The rough hard skin on my chin is much better today. I rubbed lots of Vaseline on it today and the rough skin came away leaving the area nice and smooth and clear.
I had very painful achy joints this evening.
My dry eyes are still irritating me and I have a bad pain in my left eye now.
I prepared my feet this morning with Compeed plasters so they have not been as painful or sore as yesterday, just a bit tender.
I still have painful achy joints, less than yesterday but still quite apparent, mainly in my lower back today.
This morning I had some dry flakey skin on my chin, above my top lip and around my nose. It looks dry but doesn't feel particularly dry.
Lots of sea air is doing my face good.
I am using the eye drops and had to wear sun glasses out today to protect my eyes even though it was not particularly sunny out.


College of physical education and sports science at University of Baghdad hosted Marcos Paquetá, the Brazilian coach of Police Football Team in a press conference. The coach confirmed in his speech his quest for winning during the next season, also he stated that he has not hesitated to come to Baghdad after two days of negotiations with Dr. Mahmoud Nassar and that he seeks to make the Iraqi Police football team as number one in competitions since he has a full knowledge and familiarity with the levels of some players via team management.
Prof. Dr. Huda Shehab, an instructor from the department of individual games at the college of physical education and sports science for women, University of Baghdad participated in the international arbitration course of gymnastics held in Gladbach, Germany organized by the international federation of gymnastics with the participation of a number of foreign and Arab States for the purpose of obtaining the 14th international badge. Head of the technical committee of the international federation of gymnastics Donatila Sashi delivered a lecture in this regard and at the conclusion of this course, certificates of appreciation were granted for those concerned to encourage them for participation in world and Arab championships.
Associate Rector for scientific affairs at University of Baghdad, Assis. Prof. Dr. Osama Fadel Abdul Latif visited the college of Law to inspect second phase exams for the academic year (2016-2017) where he was informed about the nature of questions and he listened to students' inquiries concerning the curriculum, as well as suitable examination atmosphere. Dr. Osama Fadel praised the actions taken by the deanship to ensure the proper conduct of the examinations.
Center for urban and regional planning for postgraduate studies at University of Baghdad in cooperation with the ministry of construction, housing, municipalities and public works heed a scientific symposium entitled (future housing and housing needs in the Iraqi city) that addressed the reality and negative effects of housing crisis on our society from the operational, administrative, economic, social, health, psychological, security and environmental aspects. The participants discussed modernization of housing policy in Iraq in 2017 and rehabilitation of informal housing communities according to urban and rural housing standards. The symposium was attended by the minister of reconstruction and housing Dr. Ann Nafie Awsi, representative of University of Baghdad Prof. Dr. Osama Fadel, Deputy Minister the engineer Estabrak Ibrahim al-Shawk, dean of the center and a number of researchers and specialists in housing.
Information Resource Center at the central library of University of Baghdad organized a meeting between the dean of the college of media, Prof. Dr. Hashim Hassan al-Tamimi and Office of Cultural Affairs of the American Embassy. The meeting addressed matters related to the role of press, its credibility and contribution in finding solutions to community problems. The two parties called for strengthening relations with the department of press at the American Embassy.
A number of professors from the department of architecture at the college of engineering, University of Baghdad participated in a conference on preservation of modern heritage in North Africa and Middle East held in Weimar, Germany. The participants dealt with various researches in this regard presented from North Africa and the Middle East on modern architecture including Iraq. It is worth mentioned that University of Baghdad participated in this event by submitting four researches that received the approval of the attendants.
Marcos Bakita, the Brazilian coach of the police football team visited the college of physical education and sports science at University of Baghdad where he was received and welcomed by met the dean of the college, Prof. Dr. Uday al-Rubaee. The meeting witnessed keeping abreast with the gyms at the college where Mr. Bakita expressed his admiration for this academic edifice in Iraq. Dean of the college, Prof. Dr. Uday stated that such visit would have a significant impact on the scientific exchange of knowledge between the two sides.
The ministry of construction, housing, municipalities and public works held a scientific symposium on the future of housing and residential needs in the Iraqi city in the presence of His Excellency the minister Dr. Ann Nafie Awsi and representative of rector of University of Baghdad, Prof. Dr. Osama Fadhil Abdul Latif, Undersecretary of the ministry Istabrak Ibrahim al-Shouk, dean of the center for urban and regional planning for postgraduate studies Assistant Prof. Dr. Abdul Hussein al-Askari and a number of researchers and housing specialists. The symposium dealt with three main axes, the first shed light on modernization of the national housing policy program in Iraq for 2017, the second addressed a presentation by the ministry of planning on poverty reduction strategy and the national program for settlement and rehabilitation of squatter communities in Iraq while the third axis was presented by dean of the center Assistant Prof. Dr. Abdul Hussein al- Askari on the role of modernizing urban and rural housing standards in Iraq and the effect of this problem as well as its repercussions on society. Several recommendations were submitted in this quest for re-evaluation of existing challenges and potentials for updating these standards in line with the quality of life of the inhabitants.
Assistant Prof. Dr. Shatha Kazim Mula' head of department of bio-chemical engineering at Al-Khwarizmi College of engineering, University of Baghdad signed a memorandum of understanding with Babol University in Iran at the invitation of Prof. Dr. Qasim Najaf Boor. This memorandum aims to exchange knowledge between the teaching faculty at the two universities as well as seeking for mutual research projects for graduate and postgraduate students in the field of engineering. The two parties discussed matters related to holding of joint international workshops and certificates in this regard and at the conclusion of this visit Dr. Shatha Kazim kept abreast with the specialized laboratories there in the field of chemical, biotechnological and nanotechnology engineering.
University of Baghdad has recorded the first Iraqi place and the 13th Arab place in QS World University Rankings for 2018. This achievement comes as a result of continued follow-up of the university's presidency so as to improve its ranking in global academic rankings, as well as the hard work of its researchers and scholars in various scientific specialties. It is worth mentioned that University of Baghdad has outperformed (11) Iraqi universities in QS Ranking among (150) World universities with the participation of (16) Arab universities and it is still maintaining the (501) rank globally.
College of engineering at University of Baghdad won three awards at the global excellence competition sponsored by Coventry University, Britain that seeks to promote architecture profession in Iraq at the academic and professional levels via the Iraqi elites in this field such as the late well-known architect Zaha Hadid and others. The department of architecture at won the award of excellence for the best architectural school in 2017 where Assistant Prof. Dr. Inaam Amin al-Bazaz, head of department received the award for the best supervision and the engineer Nizar Razak, a graduate of architecture won the award for the best graduation research during the academic year (2016-2017) and the engineer Hala Issa won the 5th rank for the best architectural graduation research supervised by Dr. Inaam al- Bazaz.
College of dentistry at University of Baghdad participated in the annual scientific dental conference held in Malaysia where dean of the college Prof. Dr. Hussein Faisal al-Hwezi delivered a lecture on new and modern steps in root fillings by reference to a number of clinical cases. Prof. Dr. Hussein Faisal hinted to the global ranks recorded by our university in world rankings and the recent steps taken towards promoting advancing the academic reality of the college in this regard.


The Bluff House is perched on a bluff overlooking Block Island Sound, the property is a flag lot at the edge of a new subdivision, bordered on three sides by water, wetlands, and woods in Montauk, New York. Designed by , the client wished for a design with a minimal impact on the pristine landscape, maximum exposure to the views and all the amenities of a year round vacation home.
The basic requirement of each space was considered integrally with the effects of sunlight, breezes and views. The house was conceived as a lens, continually framing and magnifying the subtle changes in the surrounding environment.
The L-shaped plan is divided into public and private wings. Two indoor/outdoor zones on two levels define the public areas. The upstairs zone includes the kitchen, dining area, living area and a deck. The downstairs public zone includes the family room, outdoor living room, pool and sun bathing deck. The two-story bedroom wing has four modest bedrooms facing the views, two of which convert to a single room for guests with children.
Visit the website of Robert Young Architects .
Looks like a university library from the 70s. Nice views, though.


Cigarette Cases are making a comeback!
Luxury Ladies Cigarette Cases were all the rage back in the 1930s, and up until the 1960s. Modern society got out of the trend for a while, but fashions change - either through revolution or need, and today we see many cigarette packs looking old and outdated, or in need of a change. Sometimes we see slogans and disgraceful images on packs, so many people are now going back to using cig cases as a way to smoke without being harmed by these exaggerated lies. The images cause stress, and stress is the biggest killer in society, so rather then submit to this stress, why not subvert their game by simply covering up the images, and enjoy a nice relaxing smoke without pressure.
The world of Cigarette Cases has moved on a little bit since the days of elegant metal cases, and leather clutch-cases, but these are still available if you need this level of sophistication. Today most cases are a lot more colourful, with a range of designs which catch the eye, and feel like part of your style and what you love. For the guys, there are images of bikers and camouflage, while the ladies can enjoy flowers, puppies and all kinds of decorative and attractive designs.
There are two main types of modern case, plastic or silicone. The plastic variety are King or 100s sized, and are very strong, withstanding water, dropping on the floor, sitting on and mostly standing on by accident, so your cigs are always safe. The colour means they can be located easier, and leaving the item on the table makes a very attractive feature. Some plastic cases are also Push-To-Open, which means you can press on the front of the case slightly firmly, and the top will spring open automatically; which is always nice to see when offering cigs to others, and is a neat gadget.
The second type of modern case is made from Silicone, as can be seen above. These are flexible cases and are very much thinner and less crush-proof than the plasic variety, but with the benefit that many of these have trademark designs, like Superman, Star Wars, Adidas and Chanel.
In fact the trend is so hot at the moment, that a big deal has been made over the fact that you can now buy a trendy new Mobile Phone case to match your new Chanel Cigarette Case!
With so many exciting designs out there, and a mood to suit every taste, why not investigate the whole new range of Cigarette Cases, and rediscover the delightful joys of using this wonderful item, again and again.
I bought several cig cases recently, and one of them broke from using it so much. So I bought some more!
The import was well worth the cost, and I now have a range of cases for different rooms in the house, and for going on journeys.
I would recommend any of the cases made by Fujima, and there are lots to choose from. Here are just a small selection!!!