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Dynamics of many-body photon bound states in chiral waveguide QED
Sahand Mahmoodian Institute for Theoretical Physics, Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Appelstraße 2, 30167 Hannover, Germany Giuseppe Calajó ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain Darrick E. Chang ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain ICREA-Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain Klemens Hammerer Institute for Theoretical Physics, Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Appelstraße 2, 30167 Hannover, Germany Anders S. Sørensen Center for Hybrid Quantum Networks (Hy-Q), Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
We theoretically study the few- and many-body dynamics of photons in chiral waveguides. In particular, we examine pulse propagation through a system of N two-level systems chirally coupled to a waveguide. We show that the system supports correlated multi-photon bound states, which have a well-defined photon number n and propagate through the system with a group delay scaling as 1/n2. This has the interesting consequence that an incident coherent state pulse breaks up into different bound state components during propagation, which can become spatially separated at the output by photon number in a sufficiently long system. For sufficiently many photons and sufficiently short systems, we show that linear combinations of n-body bound states recover the well-known phenomenon of mean-field solitons in self-induced transparency. Our work thus covers the entire spectrum from few-photon quantum propagation to genuine quantum many-body (atom and photon) phenomena, and ultimately the quantum-to-classical transition. Finally, we demonstrate that the bound states can undergo elastic scattering with additional photons. Together, our results demonstrate that photon bound states are truly distinct physical objects emerging from the most elementary light-matter interaction between photons and two-level emitters. This opens the door to studying quantum many-body physics and soliton physics with photons in chiral waveguide QED.
Generating quantum many-body states of light remains one of the outstanding challenges of modern quantum optics Chang et al. (2014) . Such many-body states of light are of fundamental physical interest as they arise from non-equilibrium systems with strong interactions between light and matter. On the other hand, they also promise to form novel resources for quantum technologies, for example, in quantum enhanced metrology Paulisch et al. (2019) . The main obstacle in the pursuit of generating such many-body states has been the difficulty in developing one-dimensional systems with a sufficiently strong nonlinear response at the few-photon scale Chang et al. (2018) . Recently however, significant progress has been made in creating an ideal light–matter interface between atoms or artificial emitters coupled to a one-dimensional continuum of photons at optical Arcari et al. (2014); Goban et al. (2014); Lodahl et al. (2015) and microwave frequencies Bishop et al. (2009); Hoi et al. (2012); Mirhosseini et al. (2019) . Such an interface creates a highly nonlinear medium as photons propagating in a waveguide interact deterministically with atoms. Systems of this kind have been thus far used to propose or demonstrate the generation of states of photons with strong two- or three-body correlations Birnbaum et al. (2005); Faraon et al. (2008); Reinhard et al. (2012); Peyronel et al. (2012); Firstenberg et al. (2013); Javadi et al. (2015); Hamsen et al. (2017); De Santis et al. (2017); Liang et al. (2018); Mahmoodian et al. (2018); Stiesdal et al. (2018) .
Studies of photon correlations in these systems typically consider steady-state driving and photon correlations are subsequently measured in relative coordinates. On the other hand, here we show that pulse propagation through two-level systems (TLSs) strongly coupled to a waveguide leads to very distinct temporal features which reveal the underlying dynamics and has the potential to generate temporally ordered many-body states of light. In particular we theoretically consider the propagation of pulses of coherent and Fock states of light through a waveguide to which N TLSs are chirally coupled. We show that photons undergo a Wigner delay Wigner (1955); Bourgain et al. (2013) — a delay of the pulse center due the optical excitation transferring to the atom and back to the waveguide — which is dependent on the number of photons.
Figure 1: (a) N two-level atoms (blue circles) are chirally coupled to a waveguide with decay rate Γ and driven by an input Gaussian pulse, which can be a coherent or Fock state. The light pulse propagates with a correlation-number-dependent group velocity leading to an output state where one-, two- and three-photon bound states are spatially separated. (b) Schematic of the bound states propagation. When an n-photon bound state is scattered by an atom, it re-emits the absorbed photon with a stimulated emission rate Γn, which also coincides with the inverse of its width. Since only a single photon out of n is delayed by an amount 4/(Γn), the pulse preserves its shape but is delayed by τn=4/(Γn2).
Therefore, incident pulses break up into a state that is temporally ordered by its photon correlations. For example, as shown in Fig. 1(a), an incident coherent field can produce a pulse with three-photon correlations followed by two-photon correlations states followed by uncorrelated photons. The underlying physics that causes this time delay is examined by considering the many-body photonic scattering eigenstates of a TLS chirally coupled to a waveguide.
Of central importance are the class of bound eigenstates, where two or more photons propagate together. We show that the photon bound states propagate past each atom with a photon-number-dependent delay of τn=4/(n2Γ),where Γ is the decay rate into the waveguide, as can be understood in terms of absorption and stimulated emission of a photon as shown in Fig. 1(b). Since the interaction is chiral, the delay is also proportional to the number of emitters N in the waveguide. We also show that the pulse distortion encountered by an n-photon bound state scales as n−6. Pulses of higher order bound states therefore propagate with negligible distortion, i.e. much like a soliton. Indeed we show that the classical mean-field limit of a soliton solution emerges from a combination of photon bound states with n≫1. Going beyond mean field, we show that with a mesoscopic number of photons n but a large number of atoms N≫1, the soliton breaks apart into a temporally ordered many-body quantum state of light. In this limit mean-field theory fails and our full quantum many-body treatment provides a simple description of photon propagation in waveguide quantum electrodynamics (wQED).
In mean-field theories solitons are known to be highly stable object which are unaffected by external perturbations. Here we show that similar properties exist for few-photon bound states. We outline how one can conduct scattering experiments between photon bound states and individual photons. In the considered scattering experiments the bound state is deflected by the interaction but is otherwise unperturbed by it. Together the results obtained here demonstrates that the bound states should be considered as truly distinct physical entities emerging from the underlying light-matter interactions.
This manuscript is arranged as follows: in Section II we introduce the model for chiral wQED and outline various theoretical approaches for computing the dynamics through II.1 mean-field theory, II.2 the photon scattering eigenstates, and II.3 the matrix-product states (MPS). In Section III we compute how the input pulse propagates through the medium and compute the representation in terms of photon bound states. This is followed by Section IV which shows that the many-body photon bound states can be used to construct the mean-field soliton solutions obtained in self-induced transparency. Finally, in Section V we show that photon bound states can undergo elastic scattering with individual photons modifying the delay of the bound state but otherwise leaving it unaltered, much like classical solitons.
We consider a system of N two-level systems (TLS) chirally coupled coupled to a linearly dispersive one-dimensional bath of photons. The Hamiltonian for this system (ℏ=1) is
^H=−i∫dx^a†(x)∂∂x^a(x)+√ΓN∑i=1[^σ−i^a†(xi)+^σ+i^a(xi)], (1)
where all integrals are over R, σ±i are Pauli operators for the ith TLS, ^a(x) (^a†(x)) is a photon annihilation (creation) operator at position x, the group velocity is set to unity, and the energy has been renormalized to that of the TLS. The positions of the TLSs xi are arbitrary and only change the overall phase of the outgoing field. In this Hamiltonian the first term captures the free-propagation of photons in the waveguide with linear dispersion, while the next terms take into account the interaction between the photons and the emitters. In the absence of the interaction terms the photonic eigenstates of the Hamiltonian are plane waves with wavevector k and frequency ω=k. Equation (1) constitutes the typical scenario for chiral wQED Lodahl et al. (2017) . In this manuscript, we are interested in describing the propagation of a multi-photon input field through the system. This goal can be achieved by exactly computing the scattering eigenstates of Hamiltonian (1), as illustrated in Subsection II.2.
In addition to computing the eigenstates of (1), we introduce an equivalent formulation that is well-suited for using the MPS technique that we are going to introduce later in Subsection II.3. This approach is based on the observation that the transmitted field depends directly on the emitters evolution. This can be seen by formally integrating the Heisenberg equation for the field operator ^a(x,t), that provides, within Born-Markov approximation, the following generalized input-output relation for the transmitted field T. Caneva and Chang (2015); Manzoni et al. (2017)
^aout(t)=Ein(t)+i∑j√Γe−ik0xj^σ−j(t), (2)
where we have defined ^aout(t)=^a(x+N,t) as the output field measured right after the last atom. Note that, within this approach, we assume the input field Ein(t) to be a classical coherent field on resonance with the atomic transitions. With these assumptions the emitter dynamics driven by the input field is known to be described by a purely dissipative chiral master equation (ME) of the form Stannigel et al. (2012); Pichler et al. (2015) (see supplementary material (SM) for more information):
˙ρ=−iℏ(Heffρ−ρH†eff)+Γ∑ij^σ−iρ^σ+j. (3)
Heff=−iΓ2∑j^σ+j^σ−j+Hdrive−iΓ∑l>j^σ+l^σ−j, (4)
is the effective Hamiltonian, which provides the non-Hermitian collective evolution of the emitters, while the term Hdrive=−∑j√Γ[Ein(t)^σ+j+H.c.] gives the coupling of the emitters to the input field.
The combination of Eqs.(2) and (3) provide a full description of the photon propagation through the chiral medium. In particular the spin dynamics can be efficiently solved by making use of an MPS ansatz Verstraete et al. ; Schollwöck (2011) as recently described in Manzoni et al. (2017) . As will be shown in the following, this approach will allow us to fully explore the limit of many photons and large atomic arrays, a scenario that is challenging to simulate with standard numerical techniques.
ii.1 Mean-field Theory and Self-induced Transparency
Before considering the full many-body dynamics of the Hamiltonian (1), we consider the system within the mean-field limit. We present this mean-field limit to contrast its predictions with the full many-body theory presented below.
The first treatment of (1) within mean-field theory dates back to the work on self-induced transparency (SIT) McCall and Hahn (1967, 1969); Bullough et al. (1974) . In these early experiments, gasses of two-level atoms were excited by short intense laser pulses. Although the atoms are not ideally coupled to a single waveguide mode in such systems, the laser pulses are sufficiently short that decay channels to modes other than the laser mode can be neglected. Furthermore, both the weak coupling of the atoms to this mode and the high intensity of the laser means that one can consider the atoms as a spin continuum under the mean-field approximation where quantum correlations between the atoms and the light field can be neglected. Under these approximations, the equations of motion give the SIT equations
[∂∂t+∂∂x]a(x,t)=−i√Γσ−(x,t),∂∂tσ−(x,t)=i√Γσz(x,t)a(x,t),∂∂tσz(x,t)=4Im[a(x,t)σ∗−(x,t)]. (5)
Here, a(x,t)=⟨^a(x,t)⟩, ^σ−/z(x,t)=∑iσ−/zi(t)δ(x−xi), and σ−(x,t)=⟨^σ−(x,t)⟩, σz(x,t)=⟨^σz(x,t)⟩ are the expectation values of the spin operators in the continuum limit. These nonlinear equations of motion have SIT soliton solutions. Following the treatment in Bullough et al. (1974) , for a resonant pulse the field can be taken to be real, and one can map the equations of motion onto a nonlinear pendulum equation that can be solved exactly. This gives the fundamental soliton solution for the field
a(x,t)=√Γ¯n2sech[Γ¯n2(t−xV)], (6)
where ¯n is the number of photons in the field,(or more precisely, the total energy in the original SIT work), and V=¯n2Γ/(4ν) is the relative velocity in the SIT medium which depends on the field strength ¯n, the decay rate Γ, and the linear density of the gas ν.
An important feature of the solitonic solution (6) is that the integrated Rabi frequency Ω=2√Γ∫dta(x,t), which is proportional to the area under the pulse, is fixed by the relationship between the pulse amplitude and pulse width, and always evaluates to be 2π. This corresponds to a full Rabi cycle of complete excitation and subsequent deexcitation. The SIT soliton therefore can be physically interpreted as a rapid excitation and de-excitation of the atoms, which suppresses spontaneous emission of the excited state and thus makes the medium transparent.
Inspired by the apparent dependence of velocity on photon number, it is interesting to ask whether this property extends to the few-photon limit, thus enabling, e.g., photon-number separation at the output. A full quantum treatment is necessary to answer this question, which we turn to in the following sections.
ii.2 Many-body Scattering Eigenstates
In contrast to the mean-field treatment we now consider the full many-body eigenstates. Since the Hamiltonian (1) preserves the combined number of atomic and photonic excitations, eigenstates with different numbers of excitations decouple. By computing the eigenstates in the one- and two-excitation subspaces, one can generalize the result to an arbitrary number of excitations. This technique is often referred to as Bethe's ansatz Bethe (1931) and is used to diagonalize a class of one-dimensional many-body Hamiltonians Thacker (1981) . In particular, it has previously been used to diagonalize the Hamiltonian in Eq. (1) Rupasov and Yudson (1984) . Since we are interested in the state that emerges after interaction with the TLSs, we are interested in the scattering eigenstates. These are the photon states that interact with the TLSs and emerge unchanged apart from an overall transmission coefficient. The n-body scattering eigenstates have the form
|Sk⟩n=Ck,n,S1√n!∫dnx^a†(x)|0⟩n∏l<m[kl−km+iΓsgn(xl−xm)]n∏j=1eikjxj+↔, (7)
where Ck,n,S is a normalization constant which varies with wavevector k, excitation number n, and the type of eigenstates S, where S labels different states as explained below; ^a†(x)=^a†(x1)^a†(x2)…^a†(xn); dnx=dx1dx2…dxn; and ↔ indicates summing over all n! permutations of xi to symmetrize the wavefunction. The energy of the eigenstates is E=∑niki. Upon scattering off all N emitters, the eigenstates are multiplied by the eigenvalue tNk=∏nj=1tNkj, where tk=(k−iΓ/2)/(k+iΓ/2). Since, by assumptions, the system does not contain any dissipation and is chiral, all the transmission coefficients have |tk|=1. This means that the transmission coefficients simply multiply the eigenstate by a phase. For example a single photon on resonance undergoes a π-phase shift with t0=−1. Importantly, the phase that is imparted on the eigenstate varies with k, i.e., the TLSs introduce dispersion to the system. As we soon show, different types of eigenstates also accumulate different phases. We note that the output states is here described using the position coordinates x. This is equivalent to using a time variable t in (2) as we have set the group velocity to unity.
In addition to different eigenstates for different values of n, there are also different possible types of eigenstates within different excitation-number manifolds. In the single excitation subspace there is only one type of eigenstate and they are characterized by real wavenumbers k and are fully extended in space. For n=2, the wavenumbers k1 and k2 can either both be real which gives rise to a fully extended solution of the form (7). They can also assume complex values k1=E/2+iΓ/2 and k2=E/2−iΓ/2 which are called strings Thacker (1981) . These values give additional valid solutions. Since the wavevectors can form complex conjugate pairs, the eigenstates become localized in a relative coordinate while being fully extended within the center-of-mass coordinate. This localization is associated with the formation of photon bound states that manifest in the bunching of two photons which travel together during their propagation Shen and Fan (2007); Zheng et al. (2011) .
For n>2 the m-body string (m≤n) has the wavevector kj=K/m−i[m+1−2j]Γ/2 with j=1,2,…,m where K is the total energy of the m-string. In total, the n-photon manifold has p(n) string combinations where p(n) is the number of partitions of n. For example, for three photons, there is a completely extended scattering eigenstate, a completely bound eigenstate, and a hybrid state with two bound photons and one extended photon. The different string combinations give the different types of scattering eigenstates S by substituting the complex wavevectors into (7). The transmission coefficient is also obtained by substituting the complex wavevectors into the expression for tk.
For n photons, the n-string state is a fully localized bound state with energy E,
|BE⟩n=Cn,B√n!∫dnx^a†(x)|0⟩eiEn∑jxj−Γ2∑i<j|xi−xj|, (8)
where Cn,B=√Γn−1(n−1)!/(2πn). The transmission coefficient for the n-photon bound state is then,
tE,n=E−iΓn2/2E+iΓn2/2. (9)
Importantly, the phase of the transmission coefficient varies with n, i.e., the system has a photon-number-dependent dispersion.
With all the eigenstates at hand, the scattering matrix for interacting with all N emitters in the n-photon manifold can be formally written as
^SNn=∑S∑ktNk|Sk⟩nn⟨Sk|, (10)
where the sum over S is a sum over the different string combinations of the n-photon manifold. We note that the eigenstates are orthogonal, thus the scattering matrix for N emitters simply requires taking the eigenvalue to the N-th power. Since the number of string combinations increases as p(n), the number of terms in the sum increases exponentially for large n Rupasov and Yudson (1984); Shen and Shen (2015) . In this manuscript we compute the full output states for up to n=3 using this formalism. We note that a formalism exists where one does not have to sum over string combinations Yudson (1985) . Nevertheless the form used here is particularly insightful as it gives direct access to the number-dependent transmission coefficient which, as we show, plays a central role in understanding the many-body pulse propagation.
Figure 2: Transmitted power P(x) (solid curves) for (a) one- (b) two-, and (c) three-photon transport through N=20 TLSs for a resonant pulse with Γσ=3√2. In (b) and (c) dashed lines show the contributions from the different eigenstates. (d) Transmitted power and correlation functions for a coherent state with ¯n=0.5. Solid lines show photon scattering results truncating at three photons while dashed lines show MPS calculations performed by solving the ME with a Runge-Kutta algorithm and using maximum bond dimension Dmax=150. (e) Second-order correlation function G(2)(x1,x2) for the transmitted coherent state showing the two- and three-photon bound state contributions that are localized in the relative coordinate. (f) MPS calculation for input coherent state with ¯n=8, N=60 emitters and Γσ=2√2. Dashed vertical lines show Wigner delay τn=4N/(Γn2). With Pn(x) we indicate the contribution to the intensity coming from the emission of n photons within the space bin Δx=1. To perform this calculation we used a quantum trajectories MPS algorithm fixing the maximum bond dimension to Dmax=40.
ii.3 MPS Ansatz
In order to study the dynamics for stronger input pulses (n>3) than the one computed with the S-matrix formalism, we solve equations (2) and (3) using an MPS ansatz. Specifically, the system evolution can be solved either by directly solving the ME (3) Bienias et al. (2018) (method used for Fig. 2(d) and Fig. 3) or by using a quantum trajectories algorithm where the state of the system evolves under the effective Hamiltonian (4) and stochastically experiences quantum jumps Manzoni et al. (2017) (method used for Fig. 2(f)). In both cases an MPS representation is applied either to the quantum state or to a vectorized form of the density matrix. Here for simplicity we limit the discussion to the former while the latter is discussed in the supplementary material. The MPS ansatz consists of reshaping the generic quantum state |ϕ⟩=∑i1,..iNψi1,i2,..iN|i1,i2,..iN⟩ (with ij∈{g,e}) into a matrix product state of the form:
|ϕ⟩=∑i1,..iNAi1Ai2...AiN|i1,i2,..iN⟩, (11)
where, for each specific set of physical indices {i1,¯i2,..¯iN}, the product of the Aij matrices gives back the state coefficient ψi1,¯i2,..¯iN. Each matrix Aij has dimension Dj−1×Dj known as the bond dimension and finite edge boundary conditions are assumed by imposing D1=1 and DN=1. The bond dimension reflects the entanglement entropy. For instance, if Dj=1 for all j, the matrices Aij are scalars and the state (11) reduces to a product state with no entanglement. For arbitrary states the bond dimension grows exponentially with the number of particles. The advantage of the MPS ansatz is that, in many physical scenarios as the one considered here, the entanglement can grow slowly with the system size allowing an efficient description of the state in terms of a smaller bond dimension Schollwöck (2011) . An important figure of merit of the efficiency of the MPS ansatz is given by Dmax, the maximum bond dimension, used to represent the system during the entire time evolution (see SM for more information). We will make use of this quantity in Sec. IV to quantify the amount of many-body correlations present in the system.
Iii Many-body Pulse Propagation
We are interested in studying multi-photon propagation through the chirally coupled array. Here, we consider coherent and Fock input states with mode creation operator ^a†in=∫dxE(x)^a†(x), and we evaluate the transmitted field with the two methods described in the previous section. In particular, for the exact solution we compute the transmitted photon state using the eigenstates for up to three photons while for higher excitations we make use of the MPS ansatz. In Fig. 2 we consider the propagation of a Gaussian photonic mode with amplitude E(x)=eik0x−x2/(2σ2)/(√σπ1/4) where k0 is the detuning.
Figure 2 (a)-(c) shows the power P(x)=⟨^a†(x)^a(x)⟩ for one-, two-, and three-photon Fock states after propagating through N=20 TLSs. Here the x coordinate is chosen to be in the reference frame of the pulse propagating in the absence of emitters. In Fig. 2(b) we see that the two-photon bound state comes out earlier than the extended state. The two-photon bound state thus clearly propagates with a faster velocity than the extended state. The bound state also undergoes significantly less broadening and distortion. For the three-photon transport in Fig. 2(c) again the extended state is distorted and delayed, while the three-photon bound state has less significantly distortion and delay. In the three-photon manifold there is also a string combination that forms a hybrid state where one photon is completely extended while the other two are bound. The evolution of this state is as follows determined by the individual components of the state: the two bound photons propagate in a similar manner to the two-photon bound state in Fig. 2(b), while the extended photon propagates like a single photon. This separation of the propagation of the bound and extended parts of this state can be shown explicitly in the long pulse limit σΓ≫1 (see SM). This is significant because it implies that in order to understand the pulse evolution one does not have to understand the behaviour of all the different string combinations S. Rather, one can focus on simply understanding the behaviour of the bound states. On the other hand, for short pulses this separation is not completely true because one needs to include the effect of interactions between the components, see Sec. V.
iii.1 Evolution of bound states
To better quantify the difference in propagation observed above, let us compute the pulse evolution in the center-of-mass coordinate. This can be done by using the form of the n-photon bound state and its transmission coefficient given in Eqs. (8) and (9). Within the n-photon manifold the projection of an input Gaussian state and the bound state is n⟨BE|in⟩=cne−(E−nk0)2σ2/(2n), where cn is a constant in E. Here it is convenient to use Jacobi coordinates xc=∑njxj/n, xJi=∑ij=1xj/i−xi+1, where i∈{1,2,…,n−1}. The resulting bound-state contribution is then
|out⟩n,bound=cn√n!∫dnx^a†(x)|0⟩e−Γ2g(xJ)×∫dEtNE,neiExc−E2σ2/(2n), (12)
where g(xJ)=∑i<j|xi−xj| written in Jacobi coordinates. The center-of-mass evolution is clearly determined by the second integral. This now has the standard form of Gaussian pulse propagation through a linear dispersive medium. Defining tNE,n≡eiNϕ(E), the first to third derivatives of ϕ(E) give the delay, broadening, and distortion that the pulse undergoes per emitter respectively. The delay per emitter is τn(k0)=Γ/(k20+n2Γ2/4). This is largest for a resonant pulse
τn=4/(n2Γ). (13)
This gives the Wigner delay imparted on an n-photon bound state by a single emitter in wQED: the photons propagate with a number-dependent velocity. By taking higher-order derivatives we also compute the pulse broadening b(k0)=−32k0Γ/[n(4k20+n2Γ2)2], which is zero on resonance. The third-order pulse distortion term on resonance is d=−32/(n6Γ3). The pulse distortion is thus drastically reduced for higher-order bound states. This indicates that many-photon bound states suffer negligible pulse distortion while propagating through the array of nonlinear and dispersive atoms.
In order to verify that indeed the physics of the bound states dominates the wave-packet evolution we also compute the evolution of a coherent input pulse as shown in Figure 2(d). Here, the pulse width Γσ=3√2 is the same as in Figs. 2(a)-(c), while the average photon number in the pulse is ¯n=0.5. We compute the output both by truncating the coherent state to three photons, and solving exactly using Eq. (10), or by solving Eqs. (2) and(3) with the MPS algorithm. The evolution of the bound states is seen in the position of the peaks in the power distribution ⟨^a†(x)^a(x)⟩ as well as in the difference between the power and the n-th order correlation functions G(n)(x)=⟨[^a†(x)]n[^a(x)]n⟩. The localized nature of the bound states in the relative coordinates is shown in Fig. 2(e) where we computed the two-point second order correlation function G(2)(x1,x2)=⟨^a†(x1)^a†(x2)^a(x1)^a(x2)⟩. While exact analytical calculations become unfeasible for ¯n≫1, the validity of our arguments and the importance of the bound states can still be seen in MPS simulations. For example, in Fig. 2(f), we calculate the transmitted power for an input coherent state ¯n=8 and N=60. Here, for this particular system length, the photon-number-dependent Wigner delay clearly manifests itself as separate peaks for up to six-photon bound states.
This low-distortion propagation of the bound states can be intuitively explained by returning to the simple schematic shown in Fig. 1(b). When a multi-photon bound state propagates trough the atomic array one of the photons in the packet can be absorbed and re-emitted by the atom. This process occurs on a time scale ruled by the inverse of the photon-number dependent stimulated emission rate that coincides with the bound-state packet width, Δtn∼1/(Γn), allowing the pulse to preserve its shape. This continuous absorption and re-emission of photons during the bound-state propagation leads to a time delay of one out of n photons by an amount 4/Γn, leading to the group delay in Eq. (13).
Finally, while these results obtained in the quantum regime were initially inspired by the classical regime of SIT, we now establish a concrete relationship between the SIT solitons and quantum bound states.
Iv Connection to the SIT soliton
In the previous sections we have shown how the Hamiltonian (1) leads to the SIT solitonic solutions in the mean-field limit, and that the full quantum mechanical treatment of this Hamiltonian predicts correlation-ordered photon propagation. In this section we aim to bridge the gap between these two regimes: first we show that indeed the many-body theory reduces to the mean-field result in the limit of large photon number. Secondly, we derive the quantum corrections to the mean-field results which become relevant when both the number of photons n and the number of emitters N are large. Finally, we push the numerical simulations to the many-photon limit to verify the theoretical predictions.
Let us consider a wave packet composed of a linear combination of many-body bound states. This is expressed with the ansatz
|ψ⟩=∑n∫dEcn(E)|BE⟩n. (14)
We later show that this is the expected form of the state for a high-power coherent input with a large spectral width. Such a state can be probed by measuring either the field ⟨ψ|^a(x)|ψ⟩ or the m-th normally ordered observable, which we consider in the center-of-mass coordinate ⟨ψ|(^a†(x))m(^a(x))m|ψ⟩. We compute these observables in the limit where the average photon number is large ¯n≫1, the pulses are spectrally broad σ≪1/Γ and the order of the correlation function is much less than the photon number m≪n. These give (see SM for full calculations),
⟨ψ|^a(x)|ψ⟩ =¯n√Γ2sech(¯nΓx2), (15)
⟨ψ|(^a(x)†)m(^a(x))m|ψ⟩ =[¯n√Γ2sech(¯nΓx2)]2m. (16)
where ¯n is the average number of photons in the pulse. These observables reproduce the fundamental soliton solution of SIT McCall and Hahn (1969); Bullough et al. (1974); Yulin et al. (2018) given in Eq. (6). Self-induced transparency is thus the classical limit of the photon bound state when the photon number becomes large, or conversely photon bound state are simply the quantum limit of a soliton, a quantum soliton. Just like the SIT solitons, the integrated Rabi frequency of the pulse is Ω=2√Γ∫dx⟨^a(x)⟩=2π, i.e., the intense pulse of light rapidly excites and de-excites the emitters resulting in a 2π Rabi oscillation. Since we consider high photon numbers n≫1, the effects of pulse distortion due to higher-order dispersion are negligible and interaction with the emitters simply maps x→x+4N/(¯n2Γ).
Figure 3: (a) Maximum bond dimensions Dmax of the MPS calculations required to obtain a tolerance less than tol∼10−4 versus input power ¯n for different TLS number N. (v) Output power for N=20 atoms for different strength of a coherent input pulse with solitonic shape. The results are computed by directly solving the ME (3) (solid lines) with the MPS algorithm where we fixed Dmax=150. In the second and the third panel we included the results given by Eq. 17 (dashed lines) and in the third the mean-field solitonic ansatz(dotted line).
iv.1 Beyond mean-field theory
As a lowest-order approximation, the variation in photon number n making up the pulse (14) can be ignored, and the state will simply propagate at a reduced speed dictated by the mean photon number. This leads to the SIT soliton prediction, that simply maps x→x+4N/(¯n2Γ). However, for a coherent state, the uncertainty in the photon number scales as Δn∼√¯n, which leads to a gradual broadening of the pulse due to the different photon number components accumulating different time delays. This can become significant for a sufficiently large number of emitters N. This broadening is not captured by the mean-field theory. The breakdown of the mean-field theory therefore occurs when the difference in the delay of the ¯n and the ¯n+Δn photon bound states becomes on the order of the of the width of the ¯n photon bound state. This gives (τ¯n+Δn−τn)Δt(n)∼N/n3/2. This means that when N/n3/2≳1 the mean-field theory breaks down even for large input power. This provides the boundary between mean-field theory and genuine quantum many-body dynamics.
When ¯n≫1 and N≳n3/2, the Fock-state-dependent delay can be included in the expression for the field and the power to give
⟨^a(x)⟩=e−|α|2∑nα2nn!n√Γ2sech[nΓ2(x+4Nn2Γ)],⟨^a†(x)^a(x)⟩=e−|α|2∑nα2nn!n2Γ4sech[nΓ2(x+4Nn2Γ)]2, (17)
with equivalent expressions for higher-order correlation functions. We note that similar expressions exist for the bound state of the nonlinear Schrödinger equation with an attractive interaction Lai and Haus (1989a, b) .
Equations (17) provide a simple description of the observables of a quantum many-body state of light. In order to investigate the full transition from the multi-photon bound-state propagation to the formation of the SIT solitons we make use again of the master equation simulation. In Fig. 3 we plot the transmitted power of input pulses with solitonic amplitude Ein(x)=¯n√Γ2sech(¯nΓx2) for different amplitude strength ¯n. In the first box we see again the formation of the bound states peaks which tend to reduce their time delay as the input power is increased. For intermediate input pulses (second box) the bound states with a large number of photons get more populated, and they accumulate toward a single peak as the difference in delay times for large photon numbers become less distinguishable. In this regime the transmitted power starts to be well-described by Eq.(16). For even higher input power (third box) the individual bound states are no longer recognizable and a solitonic pulse well-described by mean-field theory emerges. These results show how the SIT solitons emerge from a superposition of photon bound states that, in the limit of few photons, can be indeed interpreted as quantum solitons. On the other hand it is important to emphasize the difference in physical effects. While the SIT solitons can be fully described by a mean-field semi-classical treatment, the formation of distinct bound-state peaks is characterized by a highly-correlated state of light and represents the break down of the mean-field solution due to quantum effects.
Within the MPS ansatz, one natural way to characterize the amount of correlations in the system is to allow the maximum truncated bond dimension Dmax to vary, and to record the value Dthmax(Na,¯n) at which the truncation error exceeds some acceptable threshold value (see SM). In Fig. 3, we show this quantity as a function of the solitonic input pulse strength ¯n. We see that the breakdown of the mean-field description occurs in the regime where bound state formation occurs and amount of correlation in the system is high (large values of Dmax), while in the limit of large ¯n the bond dimension tends to shrink approaching the mean-field limit (Dthmax=1).
V Soliton Interactions
Figure 4: Evolution of the state |in⟩ (see text) demonstrating the interaction between a two-photon bound state centered at a1=−10/Γ and a single photon centered at a2=15/Γ. The output observables (a) ⟨^a†(x)^a(x)⟩, (b) G(2)(x), and (c) G(3)(x) as are plotted versus the number of emitters N.
So far we have characterized the propagation of light through ideal media and show that it can be understood in terms of the photon bound state. To fully characterize and understand these objects it is important to also investigate their interactions and robustness to disturbances. To this end, we now make a preliminary investigation of a scattering experiment between a single photon and a two-photon bound state. We consider the input state |in⟩=C∫d3xa†(x)|0⟩ϕ(x1,x2,x3) with C being a constant and
ϕ(x1,x2,x3)=e−(x1−a2)2/(2σ2)e−(x2+x3−a1)2/(4σ2)×e−Γ/2|x3−x2|+↔, (18)
i.e., a product state composed of a two-photon bound state and a single photon which are centered at a1 and a2 respectively with a1<a2. Figure 4 shows the evolution of the pulse delay for this state as it propagates through the ensemble, i.e. for different N. Here since a1<a2 and the Wigner delay is larger for single photons τ1>τ2, the bound-state photons catch up to and overtake the single photon. In this process the photons interact when the two parts overlap. The interaction causes a change in the Wigner delays which is seen as kinks in the lines in Figs. 4(a)-(b). The region of interaction is highlighted by the third-order correlations shown in Fig.4(c). After the interaction N≳10 the lines in Fig. 4 (a) continue with the same slopes as before the interaction, signifying that there is still a two-photon bound state and an unbound photon. The collision between the bound state and the free photon is thus elastic and the bound state is stable against external influence. Potentially, the interaction identified here could be used to realize passive photon–photon gates for quantum computing.
Vi Conclusion
In conclusion we have shown that chiral wQED platforms provide a highly nonlinear medium suited for exploring nonlinear optics at the quantum level. From the most elementary light-matter interaction, the interaction between photons and two-level systems, emerges correlated photonic states. These bound states are truly distinct physical objects with their own dispersion relation and are stable against external influence. Our work provides a clear recipe for how these features can potentially be observed in experiments. In the limit where the number of photons is high the bound state approach approach the well-known soliton solution of SIT. Our full quantum description on the other hand covers the entire spectrum from few-photon quantum propagation to genuine quantum many-body (atom and photon) phenomena, and ultimately the quantum-to-classical transition. In particular the analysis highlights how the mean-field solution with weak quantum correlations break down through a region of maximal quantum correlations, until finally resulting in a state with weaker correlations. This phenomena can be used to generate quantum many-body states of light with strong position-dependent photon–photon correlations. With strong light–matter interaction available in current circuit QED platforms, the physics presented here can be observed with current setups provided they are modified to exhibit a chiral interaction.
Vii Acknowledgements
S.M. and K.H. acknowledge funding from DFG through CRC 1227 DQ-mat, projects A05 and A06, and "Niedersächsisches Vorab" through the "Quantum-and Nano-Metrology (QUANOMET)". G.C. and D.E.C. acknowledge support from ERC Starting Grant FOQAL, MINECO Severo Ochoa Grant SEV-2015-0522, CERCAProgramme/Generalitat de Catalunya, Fundacio PrivadaCellex, and Plan Nacional Grant ALIQS, funded by MCIU, AEI, and FEDER. A.S.S. acknowledge support from the Danish National Research Foundation (Center of Excellence Hy-Q).
Chang et al. (2014) D. E. Chang, V. Vuletić, and M. D. Lukin, Nat. Photonics 8, 685 (2014).
Paulisch et al. (2019) V. Paulisch, M. Perarnau-Llobet, A. González-Tudela, and J. I. Cirac, Phys. Rev. A 99, 043807 (2019).
Chang et al. (2018) D. E. Chang, J. S. Douglas, A. González-Tudela, C.-L. Hung, and H. J. Kimble, Rev. Mod. Phys. 90, 031002 (2018).
Arcari et al. (2014) M. Arcari, I. Söllner, A. Javadi, S. L. Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, Phys. Rev. Lett. 113, 093603 (2014).
Goban et al. (2014) A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, Nat. Commun. 5, 3808 (2014).
Lodahl et al. (2015) P. Lodahl, S. Mahmoodian, and S. Stobbe, Rev. Mod. Phys. 87, 347 (2015).
Bishop et al. (2009) L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M. Girvin, and R. J. Schoelkopf, Nature Physics 5, 105 (2009).
Hoi et al. (2012) I.-C. Hoi, T. Palomaki, J. Lindkvist, G. Johansson, P. Delsing, and C. M. Wilson, Phys. Rev. Lett. 108, 263601 (2012).
Mirhosseini et al. (2019) M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. Dieterle, A. J. Keller, A. Asenjo-Garcia, D. E. Chang, and O. Painter, Nature 569, 692 (2019).
Birnbaum et al. (2005) K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005).
Faraon et al. (2008) A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučkovic, Nat. Phys. 4, 859 (2008).
Reinhard et al. (2012) A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, and E. L. H. andA. Imamoğlu, Nat. Photonics 6, 93 (2012).
Peyronel et al. (2012) T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, Nature 488, 57 (2012).
Firstenberg et al. (2013) O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletić, Nature 502, 71 (2013).
Javadi et al. (2015) A. Javadi, I. Söllner, M. Arcari, S. Lindskov Hansen, L. Midolo, S. Mahmoodian, G. Kiršanskė, T. Pregnolato, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, Nat. Commun. 6, 8655 (2015).
Hamsen et al. (2017) C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, Phys. Rev. Lett. 118, 133604 (2017).
De Santis et al. (2017) L. De Santis, C. Antón, B. Reznychenko, N. Somaschi, G. Coppola, J. Senellart, C. Gómez, A. Lemaître, I. Sagnes, A. G. White, L. Lanco, A. Aufféves, and P. Senellart, Nat. Nanotechnol. 12, 663 (2017).
Liang et al. (2018) Q.-Y. Liang, A. V. Venkatramani, S. H. Cantu, T. L. Nicholson, M. J. Gullans, A. V. Gorshkov, J. D. Thompson, C. Chin, M. D. Lukin, and V. Vuletić, Science 359, 783 (2018).
Mahmoodian et al. (2018) S. Mahmoodian, M. Čepulkovskis, S. Das, P. Lodahl, K. Hammerer, and A. S. Sørensen, Phys. Rev. Lett. 121, 143601 (2018).
Stiesdal et al. (2018) N. Stiesdal, J. Kumlin, K. Kleinbeck, P. Lunt, C. Braun, A. Paris-Mandoki, C. Tresp, H. P. Büchler, and S. Hofferberth, Phys. Rev. Lett. 121, 103601 (2018).
Wigner (1955) E. P. Wigner, Phys. Rev. 98, 145 (1955).
Bourgain et al. (2013) R. Bourgain, J. Pellegrino, S. Jennewein, Y. R. P. Sortais, and A. Browaeys, Opt. Lett. 38, 1963 (2013).
Lodahl et al. (2017) P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Nature 541, 473 (2017).
T. Caneva and Chang (2015) T. S. J. S. D. J. I. C. T. Caneva, M. T. Manzoni and D. E. Chang, New J. Phys. 17, 113001 (2015).
Manzoni et al. (2017) M. T. Manzoni, D. E. Chang, and J. S. Douglas, Nat. Commun. 8, 1743 (2017).
Stannigel et al. (2012) K. Stannigel, P. Rabl, and P. Zoller, New J. Phys. 14, 063014 (2012).
Pichler et al. (2015) H. Pichler, T. Ramos, A. J. Daley, and P. Zoller, Phys. Rev. A 91, 042116 (2015).
(28) F. Verstraete, V. Murg, and J. I. Cirac, .
Schollwöck (2011) U. Schollwöck, Ann. Phys. 326, 96 (2011).
McCall and Hahn (1967) S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967).
McCall and Hahn (1969) S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 (1969).
Bullough et al. (1974) R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, Opto-electronics 6, 121 (1974).
Bethe (1931) H. A. Bethe, Z. Phys. 71, 205 (1931).
Thacker (1981) H. B. Thacker, Rev. Mod. Phys. 53, 253 (1981).
Rupasov and Yudson (1984) V. I. Rupasov and V. I. Yudson, Sov. Phys. JETP 59, 478 (1984).
Shen and Fan (2007) J.-T. Shen and S. Fan, Phys. Rev. A 76, 062709 (2007).
Zheng et al. (2011) H. Zheng, J. Gauthier, and H. U. Baranger, Phys. Rev. Lett. 107, 223601 (2011).
Shen and Shen (2015) Y. Shen and J.-T. Shen, Phys. Rev. A 92, 033803 (2015).
Yudson (1985) V. I. Yudson, Sov. Phys. JETP 88, 1043 (1985).
Bienias et al. (2018) P. Bienias, J. Douglas, A. Paris-Mandoki, P. Titum, I. Mirgorodskiy, C. Tresp, E. Zeuthen, M. J. Gullans, M. Manzoni, S. Hofferberth, D. Chang, and A. Gorshkov, arXiv:1807.07586 (2018).
Yulin et al. (2018) A. V. Yulin, I. V. Iorsh, and I. A. Shelykh, New Journal of Physics 20, 053065 (2018).
Lai and Haus (1989a) Y. Lai and H. A. Haus, Phys. Rev. A 40, 844 (1989a).
Lai and Haus (1989b) Y. Lai and H. A. Haus, Phys. Rev. A 40, 854 (1989b).
Bateman (1954) H. Bateman, Tables of Integral Transforms, Vol. I (McGaw-Hill Book Company, 1954) p. 30.
Chang et al. (2012) D. E. Chang, L. Jiang, A. V. Gorshkov, and H. J. Kimble, New J. Phys. 14, 063003 (2012).
Gonzalez-Tudela et al. (2011) A. Gonzalez-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, Phys. Rev. Lett. 106, 020501 (2011).
Supplementary Material: Dynamics of many-body photon bound states in chiral waveguide QED
S1 One-, Two- and Three-photon Transport using Scattering Eigenstates
In this section we detail how we compute the ouput photon states when N emitters are impinged upon by one, two, or three photons. We also combine these scattering calculations to consider scattering of a coherent state with up to three photons.
s1.1 Single-Photon Transport
In the few-photon transport calculations we we consider Gaussian input states. The normalized single-photon input state has the form
|in⟩1=∫dx^a†(x)|0⟩1√σπ14eik0xe−x2/(2σ2) (S1)
with its Fourier representation
|in⟩1=∫dk^a†(k)|0⟩√σπ1/4e−(k−k0)2σ2/2. (S2)
The creation operators ^a†(k) are eigenstates of the single-photon scattering operator. After N scattering events, we have ^a†(k)→tNk^a†(k), with the transmission coefficient
tk=k−iΓ/2k+iΓ/2. (S3)
Throughout this document we consider an ideal 1D reservoir without losses, and we also set vg=1. The scattered state is thus
|out⟩1=√σ2π1π1/4∫dx^a†(x)|0⟩∫dktNke−(k−k0)2σ2/2eikx. (S4)
This is computed numerically. Most of the integrals that appear in the few-photon transport calculations are Fourier integrals, which can be efficiently computed using Fast Fourier Transforms.
s1.2 Two-Photon Transport
The two-photon scattering matrix has two types of eigenstates, extended states |WE,Δ⟩ and bound states |BE⟩ Shen and Fan (2007) . The scattering matrix has the form
^SN22=12∫dEdΔtNE2+ΔtNE2−Δ|WE,Δ⟩2⟨WE,Δ|+∫dEtNE,2|BE⟩2⟨BE|, (S5)
where all integrals range from −∞ to ∞. The two-photon scattering eigenstates can be obtained from Bethe's ansatz in the main text (7). In a position-space representation they are
|WE,Δ⟩2=1√2∫dx1dx2^a†(x1)^a†(x2)|0⟩WE,Δ(xc,x)|BE⟩2=1√2∫dx1dx2^a†(x1)^a†(x2)|0⟩BE(xc,x). (S6)
WE,Δ(xc,x)=1√4Δ2+Γ2√22πeiExc[2Δcos(Δx)−Γsgn(x)sin(Δx)],BE(xc,x)=√Γ4πeiExce−Γ2|x|, (S7)
tE,2=E−2iΓE+2iΓ, (S8)
and xc=x1+x22, x=x1−x2, E=k+p is a two-photon detuning, and Δ=k−p2 is a difference in photon energies, where k and p are photon detunings.
The input state for two photons is
|in⟩2=1√2∫dx1dx2^a†(x1)^a†(x2)|0⟩1σ√πe−x21/2σ2e−x22/2σ2eik0(x1+x2)=1√2∫dx1dx2^a†(x1)^a†(x2)|0⟩1σ√πe−x2c/σ2e−x2/4σ2e2ik0xc. (S9)
Projecting this on the two-photon states gives
2⟨BE|in⟩2=√Γσe−(E−2k0)2σ2/4eΓ2σ2/4Erfc(Γσ2) (S10)
2⟨WE,Δ|in⟩2=4σπ√21√4Δ2+Γ2e−(E−2k0)2σ2/4[√πΔe−Δ2σ2−ΓDF(Δσ)], (S11)
where Erfc=1−Erf, where Erf is the error function, and DF is Dawson's Function.
The output state is then written in terms of two integrals given by the bound-state and extended-state contributions
|out⟩2=1√2∫dx1dx2^a†(x1)^a†(x2)|0⟩[ψB(xc,x)+ψW(xc,x)], (S12)
ψB(xc,x)=√Γσ√Γ4πeΓ2σ2/4Erfc(Γσ2)e−Γ2|x|∫dEtNE,2e−(E−2k0)2σ2/4eiExc, (S13)
ψW(xc,x)=σπ2∫dEdΔtNE2+ΔtNE2−Δe−(E−2k0)2σ2/4eiExc14Δ2+Γ2[√πΔe−Δ2σ2−ΓDF(Δσ)]×[2Δcos(Δx)−Γsgn(x)sin(Δx)]. (S14)
The integrals in these two functions are computed numerically.
s1.3 Three-photon transport
Using the same rules rules we construct the three-photon scattering matrix for N emitters
^SN33=13!∫d3ktNk|Wk⟩3⟨Wk|+12∫dKdktNktNK,2|HK,k⟩3⟨HK,k|+∫dEtNE,3|BE⟩3⟨BE|, (S15)
tE,3=tE/3−iΓtE/3tE/3+iΓ=E−9iΓ/2E+9iΓ/2. (S16)
The eigenstates are
|Wk⟩3=1√3!∫dx1dx2dx3^a†(x1)^a†(x2)^a†(x3)|0⟩Wk,3(x1,x2,x3)|HK,k⟩3=1√3!∫dx1dx2dx3^a†(x1)^a†(x2)^a†(x3)|0⟩HK,k,3(x1,x2,x3)|BE⟩3=1√3!∫dx1dx2dx3^a†(x1)^a†(x2)^a†(x3)|0⟩BE,3(x1,x2,x3) (S17)
with the normalized real-space representations,
Wk,3(x1,x2,x3)=1√3!(2π)3∏3m<n[(km−kn)2+Γ2]{3∏m<n[km−kn−iΓsgn(xn−xm)]3∏j=1eikjxj}+↔,HK,k,3(x1,x2,x3)=−iΓ2π√3Γ[(K2−k)2+Γ24][(K2−k)2+9Γ24]θ(x2−x1)[K2−k−iΓ2−iΓsgn(x3−x1)]×[K2−k+iΓ2−iΓsgn(x3−x2)]eiK2(x1+x2)e−Γ2(x2−x1)eikx3+↔,BE,3(x1,x2,x3)=Γ√3πeiE(x1+x2+x3)/3e−Γ/2(|x1−x2|+|x1−x3|+|x2−x3|). (S18)
The three-photon Gaussian input state is
|in⟩3=1√3!∫dx1dx2dx3^a†(x1)^a†(x2)^a†(x3)|0⟩1π3/4σ3/2eik0(x1+x2+x3)e−x21/(2σ2)e−x22/(2σ2)e−x23/(2σ2). (S19)
Both the projection of the input state on the scattering eigenstates and the subsequent integrals in (S15) are performed numerically.
s1.4 Output states and observables
We now have one-, two-, and three-photon output states which we express as
|out⟩1=∫dx1^a†(x1)|0⟩ψ1(x1),|out⟩2=1√2∫dx1dx2^a†(x1)^a†(x2)|0⟩ψ2(x1,x2),|out⟩3=1√3!∫dx1dx2dx3^a†(x1)^a†(x2)^a†(x3)|0⟩ψ3(x1,x2,x3). (S20)
These are also used to compute the output for coherent states
We are interested in computing the normally ordered expectation values and correlations ⟨^a†(x)^a(x)⟩, G(2)(xc,x)=⟨^a†(xc)^a†(xc+x)^a(xc+x)^a(xc)⟩, and G(3)(xc,x,y)=⟨^a


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text:
Q: Display the number of times a form has been submitted I have a wordpress website, on each post I have a html submission form that emails user info to me.
1) I need to be able to display how many times people have submitted the form (i.e. "xx people have submitted this form")
2) I need to store that number in a database and be able to call it on pages that display multiple posts on one page (for instance a search results page) - so that when there are several posts on display, you can still see "xx people have submitted this form" on each individual post.
Hope that makes sense. I know it's complex and my php skills aren't incredible so any code snippets would be greatly appreciated.
A screenshot of what I'm after can be found here: SCREENSHOT OF WEBSITE - you can see the form is under "Express Interest" and is just a submit button (the data is in some hidden fields)
Thanks, Sam

A: Easy,
Create a column for submitted numbers.
Every time form successfully submitted add +1 to that column.

A: You should really think about an auto-increment id column in your database.
autoincr
As an alternative you can write the number into a file with php...

redpajama_set_name: RedPajamaStackExchange


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text:
Zobaczyć na własne oczy (tytuł oryginalny: Të shoh në sy) – albański film fabularny z roku 1986 w reżyserii Mevlana Shanaja.

Opis fabuły
Akcja filmu rozgrywa się w niewielkim przedsiębiorstwie w północnej Albanii. Diana jest tam lekarką, a Gjergji ekonomistą. Oboje są idealistami, czym narażają się na śmieszność wśród załogi. Przed przybyciem Gjergjego, Diana była już zaręczona. Uczucie, którym darzy młodego ekonomistę nie jest dobrze widziane w jej środowisku.

Obsada
 Arben Imami jako Gjergji Morava
 Elvira Diamanti jako Diana
 Sotiraq Bratko jako Nesti
 Agim Shuke jako wiceprzewodniczący
 Shpresa Bërdëllima jako żona wiceprzewodniczącego
 Birçe Hasko jako magazynier Andrea
 Mimika Luca jako matka Gjergjego
 Stavri Shkurti jako dyrektor
 Karafil Shena jako kierowca
 Ketrin Gjonça
 Aurela Hyseni
 Gëzim Rudi

Bibliografia

 Archiwum filmu albańskiego

Albańskie filmy obyczajowe
Albańskie filmy telewizyjne
Albańskie filmy z 1986 roku
Filmy w reżyserii Mevlana Shanaja

redpajama_set_name: RedPajamaWikipedia


It is that time of year to hold myself accountable to the progress or, lack of, made in 2012. On the completed front I did lose 20 pounds and read a book a week. While I did purchase a guitar in 2012 it has seen little use so fail on the music playing front. For travel I did about 108 days on the road this year with visits to Finland, England, and France so I'll give myself a 1/2 point on travel but with two personal trips already locked in for 2013 I'm off to a strong start.
Health: Get down to my college weight (we are getting close), Exercise 3 times a week, Continued no Soda, Less Alcohol, and eat a whole hell of a lot better. I'd say I'm 40% on my way to becoming a vegetarian at my current rate of dietary change.
Four Great Personal travel adventures in 2013. I have two planned and two more to lock in.
Listen better. I spend too much time talking. I learn a lot more when I listen with my mind as well as my ears.
Continue reading a Book a week. I read a ton of business books in 2012 and I look forward to delving into a lot more history and fiction in 2013.
I look forward to seeing you in 2013 for a salad. Until then may your flights be short and life's adventures rich.


This profile was current as of 1984, when he was part of the on-campus Kellogg community.
Eduardo Viola is professor of international relations at University of Brasilia and coordinator of the Brazilian Research Network on International Relations and Climate Change. He has published on issues of globalization and governance, democracy and democratization in South America, and the global and South American politics of climate change. Viola has been visiting professor at several international universities, including the University of Colorado Boulder, University of Amsterdam, and Stanford University. A former Kellogg Institute visiting fellow, he holds a PhD from the Universidade de São Paulo.


Accrington Stanley Community Trust and the clubs Academy Department are teaming up to offer young footballers a unique opportunity to train with the academy and their coaches!
The first Elite Soccer School of 2016 will run during the February half term. Mon 15th Feb-Fri 19th Feb and is open to young footballers aged 7-12 who currently play grass-roots football. Academy coaches will be delivering five days of sessions aimed at helping youngsters develop all areas of their game and also provide them with the opportunity to show off their skills in front of Accrington Stanley's recruitment scouts.
"You usually have to write in or be scouted in your team to get a trial but these camps with the trust give our academy recruitment scouts a chance to come and see players, it's also a chance for you to work with our academy coaching staff and train alongside our signed academy players. I do hope you take this unique opportunity and we hope to see you on trial or in an academy team with the club soon" David Bailey Academy Head of Recruitment & Talent ID.
The course will be based at Accrington Academy High School on their state of the art 3G pitch and will run from 9am-3pm, children are asked to bring a packed lunch and appropriate clothing for all weather. Places can be booked through the Community Trust website by Clicking here.
For more information please contact us on 01254 304071.


text:
MND Group
Oil & Gas Production
Public Contracts and Electronic Auctions
Registration Form for Suppliers
Promotion of Sports
Promotion of Culture
What are some of the analyses performed on a routine basis?
Main Well Control Department
Geological Characteristics of Exploration Areas
Oil and Gas Prospecting
MND conducts hydrocarbon exploration activities within 5 exploration licenses over a total area of 2,200 km2 in South Moravia.
Due to their geological character, all of these exploration areas are suitable for the formation of hydrocarbon deposits.
The Vienna Basin, consisting of a long strip of land, extending from the borders with Slovakia and Austria to the town of Moravský Písek, is historically the oldest of these. In terms of its regional-geological setting, the area is made up of a massive sedimentary pool, comprising a geological formation that extends to our country from Austria and Slovakia. The most promising hydrocarbon deposits are located in Sarmatian and Badenian sandstones at a depth of 600 to 3,000 m.
Currently the most important exploration area held by MND are the slopes of the Bohemian Massif. They are situated north-west of the Vienna Basin and extend nearly as far as to Brno. It is a region surrounded by the Bohemian Massif from the north-west and the Western Carpathians from the south-east. The two major exploration areas are Nesvačilka and Vranovice. Commercially viable hydrocarbon deposits are most likely to occur in Jurassic and Paleogene rock formations, lithologically sandstones, conglomerates and carbonates buried at depths of 1,000 to 4,000 meters.
MND a.s.
Úprkova 807/6
695 01 HODONÍN
info@mnd.cz
https://www.facebook.com/MNDczhttps://www.youtube.com/user/MNDcz/https://www.linkedin.com/company/mnd-a-s-
www.mndgroup.eu
Gas from the Source
www.mnd.cz
Copyright © 2018 MND

redpajama_set_name: RedPajamaCommonCrawl


Destiny is an online persistent world first-person shooter video game in a Si-Fi world. It was written by Joseph Staten, Dave Mongan and Joshua Rubin developed by Bungie and publishing it on a ten-year publishing deal with Activision. The game was released on 9th September 2014 on gaming consoles like PlayStation 3, PlayStation 4, Xbox 360, and Xbox One.
Have an active Bungie.net account if you don't, create one and Enter the cheat codes at "Bungie's Code Redemption Page". Your PlayStation Network ID and/or Xbox Live Username should be connected to the account to get your rewards. Enter the codes in the "Enter Your Code" field.
Refresh the page after every code entered for the next code. Now when you start the game, the cheat code rewards will be added. At your bottom of the screen you will see extra content that means its getting unlocked.
You can check for available Shaders for new armor color schemes and new Emblems for player nameplate background also you can read the Grimoire Cards from Bungie.net for extra story from the Game.
*Remember- Emblems and Shaders can be picked up at the Special Orders Vendor in the Tower, but Some items requires exploring the world of Destiny so they wont be available immediately.
Want an item that you don't have or missing? Just sign into your profile, now go to user profile page, then click on the "codes" button, click "Use Now" button you will see next to each item.


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Microsoft aims to become more gender-inclusive with first Hack for Her Summit
Though technology has been a largely male sector for decades, Hack for Her looks to be Microsoft's latest move towards integrating both genders into the field. The first summit was held on January 12, and more information has come out regarding the summit and its goals, opening the gate for more events like it to be hosted. Below is a highlights video from the recent summit which highlights the effort's moves thus far as well as its goals.
Previously, we've known that adult women are a prime demographic for developers, with casual and mobile games leading the charge. We've also seen #ILookLikeAnEngineer speed across Twitter from various women-led teams at Microsoft who worked to develop products from Cortana to HoloLens. The women already in the technology industry is an amazing move forward from where we've been previously, but the Code for Her Summit looks to increase the number even further.
As executive sponsor of Hack for Her, as well as executive vice president of Business Development at Microsoft, Peggy Johnson feels strongly about this new move for inclusive development: "Building products for everyone starts with gender-inclusive design." Johnson has also partnered with design experts from various fields of study to educate and aid companies in gender-inclusive development methods.
As Microsoft wrote in their official blog post about Hack for Her, "Our goal is to grow the Hack for Her community by growing awareness of its opportunities and benefits, invite participants to host hands-on hackathons and education summits, and create tools to apply the gender lens." Code for Her also offers the chance for companies to take advantage of the $18 trillion market that women make up globally and work to improve products to benefit more people in general.
As Microsoft notes, you can find the effort on Twitter at @HackForHer, as well as information on co-hosting or co-sponsoring an event by emailing [email protected]
Discover the evolution of working with Office 365 and WeWork
Kantar: Windows Phone continues to decline in global markets


With changes in Windows 10's patching options introduced in the Windows 10 1607 ("Anniversary") release, Microsoft has published a concise summary of the patching options and how they affect different editions of Windows 10, for those without SCCM or third-party patching technologies such as Shavlik.
Microsoft has also announced changes to the servicing model for Windows 7 and Windows 8.1, by introducing monthly rollups instead of individual packages from October 2016. This will result in us having fewer updates to manage, but each update now fixes many problems.
One of Microsoft's challenges in producing reliable patches is the vast number of combinations of patches that can be deployed by its customers. This has a direct impact on testing and, ultimately, patch quality. While the change might not have an immediate effect on patch quality, it will make Microsoft's testing job dramatically simpler and the end result should be a better patching experience.


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Christian belief systems
Competing theories of eschatology,
end times, and millennialism
Eschatology is a Christian term that means the study of the end of history from a religious perspective. Probably more obscure theological text has been written on this topic than on any other belief in Christendom.
The Bible contains many prophecies about the future. The Christian Scriptures (New Testament) in particular talks extensively about the return of Jesus Christ to this earth. 1 This is usually called his "second coming," or "parousia." The Gospel of Matthew, Chapter 24, is devoted to this topic, as is much of the book of Revelation, and 1 Thessalonians 4:16-18. A literal interpretation of the Bible shows that four important events are predicted:
The Millennium: Revelation describes an important interval lasting for 1000 years when Christ rules. 1 This is a golden era; a time of universal peace. The concept was first proposed by the ancient Persian religion of Zoroastrianism.
The Tribulation: This is a 7 year interval when a world religious-political leader called the Antichrist takes power.
Armageddon is a terrible war provoked by the Antichrist. Most people on earth will die. God's anger, hatred, and wrath are poured out over mankind and the earth at this time. A series of violent events are predicted in Daniel 9, Matthew 24, and Revelation 4-19.
The Rapture: 1 Thessalonians 4:16-18 describes a miraculous event when Christ will descend from the heavens towards the Earth. Many conservative Protestants believe that faithful "born again" Christians who have previously died will be resurrected, rise from their graves, and ascend to meet Jesus in the sky. Immediately afterwards, "born again" Christians who had not died will also ascend into the air. They will abandon cars, airplanes, factory jobs, homes, families, friends etc. Since the vast majority of humans are not "born again," most people will remain behind on earth. More details on the rapture.
Unfortunately, this and various other Biblical passages predicting the future are ambiguous. The events themselves are open to many interpretations. There is no clear indication of either their timing or sequence. Some Christians believe that "millennium" does not mean a time interval of exactly 1,000 years. Rather it refers to a long interval of time. Some Christians interpret events mentioned in the Christian Scriptures as descriptions of real happenings in our future; others interpret them symbolically and/or as events that have already occurred millennia ago.
This leaves the passages open to many conflicting beliefs about the end times. A lot of intra-denominational and inter-denominational strife has resulted from disagreements about end time prophecy. For example, the Roman Catholic Church and most mainline and liberal denominations do not have expectation that a Rapture will occur in the way anticipated by many fundamentalist and other evangelical faith groups.
End times beliefs among Protestants:
Various Protestant denominations and para-church organizations promote one of six main systems of prophecy concerning the "end times"
Historical Premillennialism: This belief was held by a large percentage of Christians "during the first three centuries of the Christian era, and is found in the works of Papias, Irenaeus, Justin Martyr, Tertullian, Hippolytus, Methodius, Commodianus, and Lactanitus." 2 The Antichrist first appears on earth and the seven year Tribulation begins. Next comes the Rapture. Christ and his Church return to earth to rule for a Millennium. The faithful will spend eternity in the New Jerusalem. It is a gigantic cubical structure, some 1,380 miles in height, width and depth, which will have descended to Earth. New Jerusalem is a.k.a. Celestial city, City of God, Heavenly Jerusalem, Holy city, Shining City on a Hill, Tabernacle of God, Zion, etc. The forces of evil will have been conquered. The faithful will live during this thousand years of peace in Jerusalem, while occupying spiritual bodies. After this period, all people are judged.
Christianity became the official religion of Rome in the fourth century CE. Premillennialism was declared a heresy at the Council of Ephesus (431). Amillennialism soon became the prevailing doctrine of the Roman Catholic Church and premillennialism was suppressed.
Dispensational Premillennialism: (a.k.a. Dispensationalism) Premillennialism, declared a heresy in ancient times, was reintroduced circa 1830. Most people credit John N. Darby with its resurrection. He was a minister of the Church of Ireland, a denomination in the Anglican communion, and the founder of the Plymouth Brethren. However, author Dave MacPherson claims that British pastor Edward Irving was the actual person responsible, and that a conspiracy was organized to give Darby the credit. 14 Premillenialism received general acceptance by most Fundamentalists and other Evangelical Christians after the publishing of the Scofield Reference Bible in 1909. As in Historic Premillennialism, the Tribulation is believed to precede the second coming of Christ, and the subsequent establishment of the millennial kingdom -- a thousand-year golden age on Earth. The Final Judgment follows the millennium. But, theologians are divided over the timing of the Rapture. Many Premillennialists search world events and signs in the heavens for some indication of the Tribulation, which they anticipate will arrive at any time.
All of the theories that have been proposed about the timing of the Rapture appear to contradict some passages in the Bible. Current beliefs include:
Pre-Tribulation Rapture: (or "pre-trib") The Rapture happens just before the Tribulation, so that believers will not have to experience any of its disruption and pain. The main difficulties with pre-trib are contained in the Olivet Prophecy of Jesus. In Matthew 24, Mark 13, and Luke 21, Jesus describes the terrible destruction and loss of life of the tribulation period. The disaster is believed to be so intense that no human (Christian or non-Christian alike) would remain alive, except that God shortens the duration of the disaster for the sake of the believers. Jesus then continues by describing his return towards earth immediately after the terrible devastation. From this passage, it seems clear that the rapture will follow the Tribulation. The supporters of the "pre-trib" position suggest that Jesus will have a total of three comings: the first during the first century CE; the second at the start of the tribulation, and a third at the end of the tribulation.
Post-tribulation Rapture: (or "post-trib") The faithful experience the full horrors of the entire Tribulation and are raptured only at the end of the 7 years. The main problem with this theory is that there are many Bible passages which state that Christ's return will be at a time that cannot be predicted. But the Tribulation period starts with the arrival of the Antichrist and an interval of peace. Precisely 42 months later, a sudden shift occurs, a peace treaty is broken, and devastation begins. These would be well defined dates that would allow an accurate prediction of the end of the Tribulation. There are other weaknesses to this theory. 7
Mid-Tribulation Rapture: (or "mid-trib") The Rapture happens hafl-way through the seven year tribulation -- that is: 42 months into the Tribulation. Up to that time, the Antichrist brings peace to the world. After 42 months, events take a sudden turn for the worse. Some supporters of the "mid-trib" position suggest that there will be many mini-raptures.
Pre-wrath Rapture: This is a new theory, promoted by Marvin Rosenthal, former director of Friends of Israel, and others. Their view teaches that the church must experience most of the Tribulation, and then be raptured towards the end of the Tribulation period.
Partial Rapture: This theory teaches that the faithful born-again believers are raptured just before the Tribulation. Newly born again believers are are raptured during or at the end of the Tribulation.
The latter three theories contain some of the weaknesses of pre-trib and post-trib.
All of the Premillennialist beliefs teach that the Tribulation is followed by 1000 years of peace when all live under the authority of Christ. Afterwards, in a brief, final battle, Satan is permanently conquered.
Dispensational Premillennialism contains an internal conflict. Its advocates generally believe "that the moral conditions of the world and the church are destined get increasingly worse. When they get almost unbearably bad, the Lord Jesus will return in the clouds to 'rapture' the living saints up to heaven." 11 However, they tend to be very outspoken and active in their opposition to many behaviors that they consider to be extremely sinful: abortion access, equal rights for homosexuals, same-sex marriage, pre-marital sex, adultery, sex education in schools, access to physician assisted suicide, the use of embryonic stem cells in healing, etc. By their opposition to these "hot" religiously controversial topics, they are slowing down their adoption, thus delaying Jesus' return to earth, the rapture and the 1,000 year millennium.
Amillennialism: (Also sometimes referred to as nonmillennialism, nunc-millennialism, or realized millennialism). Amillennialists believe that the millennium is not an actual physical realm on Earth. They do not believe that it will last 1,000 years. Rather it began at the time of Pentecost (circa 30 CE) and is currently active in the world today through the presence of the heavenly reign of Christ, the Bible, the Holy Spirit, and the activities of Christian faith groups. Both good and evil will continue in the world during this time. Lawlessness, a falling away from the Church, and persecution of Christians will increase in magnitude. Finally, the current Church Age will end suddenly at Christ's second coming. A type of Rapture will happen when Christ returns: believers will rise to meet Jesus in the sky. All will then shortly thereafter return to Earth. The Day of Judgment will then occur. Events described in The Olivet Discourse (Matthew 24, Mark 13, and Luke 21) and in most of the book of Revelation are seen as occurrences which have already happened, or which are symbolic in nature and not to be taken literally. The Antichrist is looked upon figuratively and not as a real person.
This belief was held by many leaders of the early Christian church during the first and second centuries CE. Simultaneously, other leaders -- perhaps the majority -- taught a version of premillennialism that is very different from today's dispensational premillennialism. St. Augustine (354 - 430 CE), often called the "Father of Amillennialism" was largely responsible for the establishment of amillennialism as the formal church belief. It remained the generally accepted system throughout Christianity until after the Reformation in the 16th century. Many Christian denominations -- including the Anglican Communion, Disciples of Christ, Lutheran, Orthodox, Reformed, Roman Catholic, and some Baptists continue to teach Amillennialism.
Postmillennialism: (Also known as "Christian Reconstruction", "Kingdom Now Theology" and "Dominion Theology.") This belief arose during the early 19th century CE. According to author Loraine Boettner, Postmillennialism involves "that view of last things which holds that the kingdom of God is now being extended in the world through the preaching of the gospel and the saving work of the Holy Spirit, that the world eventually is to be Christianized, and that the return of Christ will occur at the close of a long period of righteousness and peace, commonly called the millennium." 12 The theory is based on the perception of a gradual movement towards social perfection. They predicted that a massive religious revival, spiritual awakening and purification would occur. The entire human race would be converted to Christianity, including the Jews. A millennium of peace and righteousness follows. After the millennium, Jesus returns to earth, resurrects the dead believers, and conducts the last judgment. The Rapture and Tribulation are largely ignored. This belief is being actively promoted today by the Chalcedon Foundation and other groups within the Christian Reconstruction movement whose goal is to convert the U.S. government and those of other countries to a theocracy -- a Christian version of Iran, with the loss of essentially all personal freedoms.
Preterism is a belief that the events prophesized in the New Testament have already happened. The great war of Armageddon in the book of Revelation occurred in the late 60's and early 70's CE when the temple in Jerusalem was destroyed, many Jews were killed and the rest were driven from Palestine. When Jesus talked about the end of the world, he did not mean that the physical world would be no more. He taught that the old worldview held by various contemporary Jewish groups was coming to an end, to be replaced by a new concept, the Kingdom of God. Thus, all of the major elements in the book of Revelation (Tribulation, Armageddon, Rapture, etc.) actually took place in the first century CE. 8
No Millennialism: Most skeptics and liberal Christian theologians largely interpret the contents of the books of Daniel and Revelation as having no meaningful prophetic information for our future. Many regard Revelation as being composed of visions, hallucinations or nightmares of the author, of little meaning for Christians today. Some believe that the purpose of the book of Revelation was to stiffen resolve in the early Christian movement so that members could withstand persecution by the Roman Empire. Thus, its purpose was to predict persecutions and other events that were to happened to the early Christian church. They also reject the apparent prophecies in the Book of Daniel. They believe that Daniel was written early in the 2nd century BCE, long after most of the events had actually happened. It was history recorded, not their future prophesized.
End times beliefs among Roman Catholics:
Roman Catholics generally follow the teachings of Augustine and the Protestant reformers, and accept Amillennialism. However, they do not generally use the term. They anticipate Jesus coming to Earth and gathering the Church together. But they generally do not use the term "rapture" either According to Catholic Answers, "The Church has rejected the premillennial position, sometimes called 'millenarianism' (see the Catechism of the Catholic Church 676). In the 1940s the Holy Office judged that premillennialism 'cannot safely be taught,' though the Church has not dogmatically defined this issue." 13
One source is receiving a lot of attention recently. St. Malachy (1094-1148 CE) was the bishop of Armagh in what is now Northern Ireland. He wrote a book titled "Prophesies of the Popes." It descrbes a vision that he allegedly received concerning the 112 popes who would follow his contemporary -- Pope Celestine II -- who died in 1144. The book stored in the Vatican archives, from where it was rediscovered and published in 1595 CE. Some believe that the book describes the present pope -- Pope Francis -- as the last pope before the destruction of Rome, the end of the Church, and the return of Jesus Christ to Earth. 15,16
An Amazon book description of a modern-day book "Petrus Romanus: The Final Pope Is Here," states:
"According to the prophecy, the next pope (following Benedict XVI) is to be the final pontiff, Petrus Romanus or Peter the Roman. The idea by some Catholics that the next pope on St. Malachy's list heralds the beginning of "great apostasy" followed by "great tribulation" sets the stage for the imminent unfolding of apocalyptic events, something many non-Catholics would agree with. This would give rise to a false prophet, who according to the book of Revelation leads the world's religious communities into embracing a political leader known as Antichrist. In recent history, several Catholic priests--some deceased now--have been surprisingly outspoken on what they have seen as this inevitable danger rising from within the ranks of Catholicism as a result of secret satanic "Illuminati-Masonic" influences. These priests claim secret knowledge of an multinational power elite and occult hierarchy operating behind supranatural and global political machinations. Among this secret society are sinister false Catholic infiltrators who understand that, as the Roman Catholic Church represents one-sixth of the world's population and over half of all Christians, it is indispensable for controlling future global elements in matters of church and state and the fulfillment of a diabolical plan they call "Alta Vendita," which is set to assume control of the papacy and to help the False Prophet deceive the world's faithful (including Catholics) into worshipping Antichrist. 16
Unfortunately for the conspiracy theorists who promote these beliefs, the 112th pope has just been elected, and his name is not Petrus Romanus. It is Pope Francis.
Related essays on this web site:
The millennium and the end of the world
Will the end of the world happen in our immediate future?
Revelation 20:1 to 7 from the Christian Scriptures (New Testament)
Lonnie Kent York, "History of Millennialism" at: http://www.restorent.com/
Mike Rouse & James Underwood, "Post Tribulation Rapture - Defined and Refined" at: http://208.128.2.3/
Grover Gunn, III, "Postmillennialism Today" at: http://www.wavefront.com/
Rev. Ralph A. Smith, "The Biblical and Theological Issues", at: http://www.berith.org/
Catholic Answers, Inc., "Are You Pre, Mid, or Post?", at: http://www.catholic.com/
The Bible Prophecy and the Rapture Report lists some weaknesses of the post-trib belief. See: http://www.serve.com/
The Preterist Archive is at: http://www.preteristarchive.com
FAS Millennia Monitor at: http://www.fas.org/ This website has many links to Premillennialist, Postmillennialist and Amilennialist resources.
The Chalcedon Foundation's web site is at: http://www.chalcedon.edu/
P.A. Smith, "Jerry Falwell's eschatological schizophrenia," WorldNetDaily™, at: http://www.worldnetdaily.com/
Loraine Boettner, "The Millennium," P&R Press, (1992). Read reviews or order this book safely from Amazon.com online book store.
"The Rapture," Catholic Answers, at: http://www.catholic.com/
Dave MacPherson, "The Rapture Plot," Millennium Press, (2000). Read reviews or order this book safely from Amazon.com online book store
Ken Carpenter, " The Last Pope?," DVD, WND Films, (2013). Read reviews or order this video safely from Amazon.com online book store
Thomas Horn, Chris D. Putnam, "Petrus Romanus: The Final Pope Is Here,"
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Rutgers Men's and Women's Lacrosse Host Doubleheader
By: Scott Logan
PISCATAWAY, NJ—Rutgers lacrosse fans will be treated to a double dose of action this weekend, as both the men's and women's teams will play at High Point Solutions Stadium on February 27.
The men's team has been the most pleasant surprise among Rutgers' spring sports. The Scarlet Knights come into the weekend contest at 2-0.
It is a small sample size, but the team has impressed in those two games, and have even received votes in the most recent Inside Lacrosse poll.
A dominant win over Fairfield could be enough to propel the Scarlet Knights into next week's rankings.
Rutgers began its season with a 19-11 win over St. John's. The Scarlet Knights were unphased by the freezing temperatures that day, as the offense came out hot. Senior captain Scott Bieda scored five goals and four assists, earning him Big Ten and NCAA Offensive Player of the Week honors.
Redshirt freshman attacker Adam Charalambides began his Scarlet Knights career with a bang against the Red Storm, scoring a team-high six goals and was recognized as the Big Ten Freshman of the Week.
Last Saturday, Rutgers went on the road for a much tougher test, taking on Army in West Point. It was a revenge game of sorts for the Scarlet Knights, who dropped a close 9-7 game to #11 Army in Piscataway last season.
In last week's matchup, fans were treated to an exhilarating, back-and-forth matchup. Army held an 11-10 lead in the fourth quarter, but with just over five minutes left in the contest, Rutgers went on a 3-0 run to secure the 13-11 victory.
Six of Rutgers' 13 goals came from Charalambides, who was named Big Ten Freshman of the Week for a second week in a row.
Fairfield heads to Piscataway with an 0-2 record. The Stags dropped a pair of home contests to Richmond (11-10) and Bucknell (13-8).
For what it's worth, Richmond received votes in the latest Inside Lacrosse poll, and Bucknell is listed at #20, so Fairfield has a couple of quality losses on their schedule.
Before the men take the field at approximately 3:30pm, the women will look to even up their record at 2-2 when they host Delaware at noon.
The women's team will attempt to rebound from a 14-2 blowout loss to #15 Johns Hopkins last Sunday in front of the home crowd.
While the Scarlet Knights took it on the chin in their last contest and sit at just 1-2 on the season, there is still a lot of optimism about this team. They opened the season with an impressive road victory, downing Villanova 14-8 on February 12.
They concluded a brief two-game road trip in Long Branch when they faced interstate foe Monmouth. Unfortunately, Monmouth jumped out an early 7-0 lead, and the Scarlet Knights never recovered. Despite the 12-10 loss, Rutgers' ladies showed a lot of fight, going on a 10-5 run to finish the game.
The loss to Johns Hopkins did not have many positives to point out, but there rarely are when going against a perrenial lacrosse powerhouse like the Blue Jays.
Delaware presents another challenge for Rutgers, as the Blue Hens come into Saturday's contest at 3-1. While they have wins over Georgetown, George Washington, and La Salle, perhaps Delaware's most impressive showing came in their one loss.
The Blue Hens went to Philadelphia to visit UPenn, then ranked 13th in the nation. Delaware was unable to pull off the upset, but went to toe-to-toe with one of the country's best teams, losing a close 12-10 contest.
A win over Delaware on Saturday would be big for the Scarlet Knights, who look to put last year's 2-16 record in the rearview mirror.
Scott Logan


A while back I went to the Gosh A/W event and fell in love with many of their products. I think Gosh is a great highstreet brand (although it can be a little over priced) and I am a great fan of their nail polishes. I don't know too much about this polish, I'm not sure if it is out already although I am assuming it is limited edition. I love the colour Spot On! Pink its a mid toned pink leaning slightly on the red side.
This polish was a great to apply; I like brush, it applied really easily as well as quickly. This colour needed just two coats and dried really quickly. As an extra bonus I really like the packaging I think the little stars make it super cute. The only thing that confuses me slightly is that I see this colour as being quite summery and not something I would expect to see in an Autumn/Winter Collection- either way great polish!
What colours have you tried from Gosh?
The brush on both sides of the mascara is normal size average mascara brush which I was quite happy to see, the size makes the mascara easy and quick to apply.
I was left with white lashes which didn't look much thicker and only a little longer….
My lashes look much thicker and longer however I was still disappointed. Before apply mascara I always curl my lashes however this really didn't hold the curl at all, making my lashes almost instantly straight! Although it did add volume my lashes were crispy and flaky which always gets really uncomfortable towards the end of the day.
Overall, this mascara deliver what it claims to, thicker lashes however for me the fact it doesn't hold a curl means my eyes don't look as open as I like and the dryness of the products gets uncomfortable once it begins to flake.
Have you tried any Gosh mascaras?


Hello Badgers! My name is Nathan C. Chick and I am a 4th year Business Communication student here at Brock. I have been involved with BUSU for 3 years and have held 5 different BUSU jobs, ranging from a cook at your favorite Bar on-campus, to currently working as a Marketing Coordinator for you, the students. Informing the student body of all BUSU events and programs has been my job for the past 3 years and I wish to keep the ball rolling. I have developed a passion for this organization that can be matched by none. From helping organize and execute your favorite O-Week and Frost week events, to talking with students in the halls each and every day, I have been and will continue to be here for the students to make your experience at Brock the best it can be. BUSU is a second home to hundreds of student employees who work hard to provide Brock with the services necessary in everyday life. My goal for VPSS includes carrying over current successful programs within the Student Services department, such as the GB Rise and Grind Promotion and Wellness Wednesdays. Furthermore, with your support I wish to make the 2017-18 school year one that will renovate Union Station for customers and Isaac's events, Restructure BUSU jobs to create more opportunities for students, as well as provide GB services at the Marilyn I. Walker school of Fine and Performing Arts. These initiatives are both practical and timely for the next school year and keep the VPSS focus on making each day a little better for the most important people at Brock, you the students.
Thank you for your time and consideration, let us work together to improve the 2017-18 school year for all.
To the best of your understanding what is the role and responsibilities of the Vice President of Student Services?
"I've been working for BUSU for three now. I've held five different jobs and these are positions that reign from working in Union Station, the kitchen at Skybar, Isaac's, interning for two years on the marketing street team and working one year in the marketing office. The one thing I've learned throughout that time and experience is that it changes every year. There's the core values of the VPSS role, including student involvement, making sure large events like O-Week, Frosh Week and Wellness Week are carried out to the best of its ability to have the students involved. Making sure the time year at Brock isn't all class and work, that it includes the fun bonding experience. Also being heavily involved with clubs. Making sure clubs are running efficiently and effectively. As well as students support for athletics through Isaacs Army traditionally. That being said, it's inherent in the successful candidate and their values, and their passion. If I were to be the successful candidate for next year, we'd still maintain an emphasis on the large events. But shifting the focus to each and everyday. Making the small things better.
Why have you decided to run for the position of VPSS?
As I've mentioned, I've been working for BUSU for three years – this has been a long time coming I think. Ever since I started flipping burgers in Skybar kitchen, I wanted to do this, so I've been grinding for the last three years, learning every aspect to BUSU. There's still a lot more to learn, obviously no one can understand every aspect of the organization because it's so large. I inherently ran this year because I think it was time to change the focus of the VPSS role, and to give students a successful candidate that will come into the position with the knowledge and experience that's going to take the VPSS roles and BUSU next year from a good year to a great year.
The first one would be freeing up the Isaac's Army budget. By no means do I want to get rid of Isaac's Army, I think its a fantastic organization. However, the Isaac's Army coordinator position is eating up $25,000 of the student budget, that's approximately $15,000 paying the coordinator and approximately $10,000 running events, swag and so forth. Making sure we can open up that $25,000 to distribute towards other events, to make weekly events better.
Renovating Union Station is my second point. It's time for a renovation – it's been awhile since Union Station has had a face lift. Especially the tables and chairs. The tables are wobbly and the chairs aren't that comfortable – and students pay a lot of money. Union Station is one of the cheapest places to eat, so we have to make sure it's the most comfortable too. Also, I would like to build a new stage that is wheelchair accessible. There's a gentlemen in a wheelchair that comes to Isaac's every Thursday, and can't access the stage – that's unacceptable. To people that think it will be too expensive, in no way to I want to tear down the building and build a new one. All this is attainable, so it won't take to long and we have budgets for that.


Ira Paulo Pozon, 2018 AsiaGlobal Fellow, addresses the common misconceptions about lobbying and discusses the qualities of an ideal lobbyist.
We usually see them on American TV shows and movies. The savvy lobbyists representing industry interests speaking to regulators and legislators in an attempt to influence public policy. They speak of vast sums of money legally sent to government officials to sway their positions on certain public issues.
"Thomas Jefferson Johnson: Terry, tell me something. With all this money coming in from both sides, how does anything ever get done?
This is, of course, a comedic exaggeration of what lobbying is all about, though the practice does indeed involve significant sums of money. Through the years, the practice of lobbying has gained a bad reputation, one where it is perceived as some form of legalized bribery.
In a calendar quarter in 2015, the top 10 lobby groups in Washington spent $64 million influencing the Executive and Legislative branches.
Lobbying is not only legal, but a Constitutionally protected right, at least in the US. It is contained in the First Amendment right to "petition the Government for a redress of grievances".
Lobbying is so much more than a legalized means for funneling money. It is a means to gain or use widespread support on issues and interests to influence decisions. For example, the Tech Lobby represents the needs and interests of technology companies and one issue was trying to lower corporate tax rates. The American Association of Retired Persons (AARP) is an NGO and lobbyist focusing on retiree issues such as healthcare, social security, and age-discrimination.
Effectively, a lobbyist, in my eyes, is an advocate. The lobbyist carries the banner of the issue, sector, industry, or group, and acts as their voice to policymakers. Lobbyists do so much more than what they are perceived of, they gain the sectoral sentiment on a more grassroots level, conduct the research, negotiate with key individuals, and advocate for their causes.
I consider myself a lobbyist of sorts. Not one with access to large sums, but rather one that has the know-how and the network to best advocate for certain changes. For example, in this new digital age, we are faced with breakthrough technological advancements that will surely change so much within the status quo. Consider the Fourth Industrial Revolution (4IR), automation, artificial intelligence (AI) , the gig or sharing economy, and blockchain, to name only a few. Each of these phenomena either has changed or is about to change the way we do things. We now rely on ride-hailing or ride-sharing applications more than ever, for instance, and yet numerous governments and legal jurisdictions have trouble regulating this $100-billion industry.
Consider the more technical aspects, and one can already foresee how difficult it would be for industries to even speak on the same language as the policymakers. One tends to recall the humorous exchanges when a Senate Committee grilled Facebook's Mark Zuckerberg for hours, when clearly there were misunderstandings about how the technology worked in the first place.
Fact of the matter is, as I have been saying for quite a while now, technology is going to hit us faster than ever before, and we are not ready. Governments have difficulty even considering how to regulate blockchain, while labor and employment regulators have yet to fully see how AI will affect the workforce, and how educators and institutions should already be looking on how to provide the youth with the right skills for the age of AI.
That is, with all humility, where I come in. As a lawyer, consultant, and tech advocate, I find myself as the ideal lobbyist, one that acts as advocate, but also one that is a coach or educator of sorts, to help sectors in society, whether private or public. I help clients in understanding and preparing their businesses – whether in financial, education, training, service – in identifying the challenges they will face and the opportunities they can seek, through these technological changes, and assisting them in getting the point across through to the government on the needs and issues of the industry.
That is my version of a lobbyist — an advocate, industry expert, educator, and trumpet, one that gets things done for the benefit of the client, the industry, and society at large.
This article first appeared in The Manila Times on January 03, 2019.


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