Correlated two-photon transport in a one-dimensional waveguide side-coupled to a nonlinear cavity
We investigate the transport properties of two photons inside a one-dimensional waveguide side-coupled to a single-mode nonlinear cavity. The cavity is filled with a nonlinear Kerr medium. Based on the Laplace transform method, we present an analytic solution for the quantum states of the two transmitted and reflected photons, which are initially prepared in a Lorentzian wave packet. The solution reveals how quantum correlation between the two photons emerge after the scattering by the nonlinear cavity. In particular, we show that the output wave function of the two photons in position space can be localized in relative coordinates, which is a feature that might be interpreted as a two-photon bound state in this waveguide-cavity system.
Creating quantum correlations among photons has been a subject of major interest for studying the foundations of quantum theory as well as applications in quantum information science. As direct interactions between photons in free space are extremely weak, generation of correlated photons generally requires nonlinear media. Electromagnetically induced transparency and photon blockade are mechanisms that have been exploited to achieve strongly interacting photons Schmidt1996 ; Deutsch1997 ; Harris1998 ; Kimble2005 . Recently, studies of two-photon scattering from a two-level system inside a one-dimensional (1D) waveguide have also reported various features of photon correlation Kojima2003 ; Fan2007 ; Roy20010 . For example, Shen and Fan Fan2007 have discovered the existence of two-photon bound states, and Roy Roy20010 has indicated an interesting application of the system as a few-photon optical diode. We also note that Shi and Sun Shi2009 have employed a formal scattering theory to study multi-photon transport in a 1D waveguide.
In this paper, we investigate the correlation properties of two photons in a 1D waveguide that is side-coupled to a nonlinear cavity filled with a Kerr medium (Fig. 1). The nonlinear cavity plays the role of a scatterer. It is worth noting that such a Kerr nonlinearity has also been employed in coupled cavity array systems for studying quantum phase transition Hartmann ; Greentree2006 ; Angelakis2007 ; Hartmann2010 ; Yamamoto2008 ; Tomadin2010 and nonclassical photon statistics Fazio2009 ; Savona2010 ; Ferretti2010 ; Bamba2010 . Here we will focus on the transport properties of two photons determined by the long time solution of the Schrödinger equation, assuming the initial photons are in wave packet forms. We will present an analytic solution based on the Laplace transform method, which has been applied to related photon-atom scattering problems tsoi . From the two-photon transmission and reflection amplitudes, we show how the two scattered photons can be correlated in frequency and position variables, with the latter revealing photon bunching and anti-bunching effects. Our solution also reveals a two-photon resonance condition when the incident photon energies match the cavity frequency shifted by the Kerr interaction. The behavior of transmission and reflection near the resonance will be discussed.
Ii Physical model
The physical model under investigation consists of an infinitely long 1D waveguide and a nonlinear cavity located at the origin (Fig. 1).
We consider a single-mode field in the cavity, which couples to right- and left-propagating fields of the waveguide via the side coupling Yariv2000 ; Fan2004 so that photons can tunnel between the waveguide and the nonlinear cavity. The Hamiltonian (with ) of the system is given by
Here and are annihilation and creation operators associated with the cavity mode with the resonance frequency . The second term in Eq. (1) describes the Kerr nonlinear interaction with strength . The Hamiltonian of free fields propagating in the waveguide is described by the third term, where () and () are, respectively, the annihilation (creation) operators for left- and right-propagating waves with wave number and frequency . These operators satisfy the commutation relations
Finally, the last term in the Hamiltonian (1) represents the coupling between the cavity and the waveguide, where is the tunneling strength.
For convenience, we introduce even- and odd-parity modes operators of the waveguide,
so that Hamiltonian (1) can be rewritten as
Here is introduced. We see that the interaction involves only even modes, and photons in the odd modes evolve freely in the waveguide. Therefore we shall focus on the calculation of the transport properties of the photons in even modes.
In the rotating frame with respect to , the Hamiltonian can be simplified to
where is the detuning. In this paper the dispersion relation for the modes in the waveguide is assumed to be linear, i.e., , and we will set the speed of light in the waveguide as .
Iii Single-photon transport
As a preparation for finding the solution for two-photon scattering, we first consider the single-photon problem Fanpapers ; Lukin-np ; Law2008 ; Sun ; Liao2009 . Note that the Kerr nonlinearity has zero effect for single photon states. The main purpose in this section is to present the single-photon transmission and reflection coefficients, which will appear in the two-photon solution later in the paper.
In the single-excitation subspace, an arbitrary state can be written as
where stands for the state with one photon in the cavity and no photon in the waveguide, and denotes the state with a vacuum cavity field and one photon in the th (even) mode of the waveguide. The time dependent variables and are the respective probability amplitudes.
By the Schrödinger equation , we have
By performing the Laplace transform defined by , Eq. (8) becomes
where and are the initial values of the probability amplitudes.
Assuming that initially the cavity is in the vacuum state and an incident single photon in the waveguide is prepared in a wave packet with a Lorentzian spectrum, the initial condition reads
where and are the detuning and spectral width of the photon, and is a normalization constant. The choice of in Eq. (10) has the advantage that analytic solutions can be obtained conveniently. In addition, by noting that corresponds to the monochromatic limit, an incident wave packet of a general form can be constructed by coherent superpositions of Lorentzian wave packets of various frequencies.
After some calculations, we obtain
Note that in obtaining Eq. (11), we have made the approximation: , where .
Taking the inverse Laplace transform of Eq. (11), in the long time limit, and , we have
Equation (12) shows that the scattering process results in a phase shift for a single photon with wave vector , where the phase shift is defined by .
In terms of the left- and right-propagation modes, if we assume a photon packet is incident onto the cavity from the left, then the initial state can be written as
In the long-time limit, the wave function becomes,
where the transmission and reflection amplitudes are defined as
A similar result has been obtained for the case that a single photon is scattered by a two-level system in a 1D waveguide Fanpapers , namely, the transmission amplitude is zero at the exact resonance. This effect was also reported in Ref. Yariv2000 for side coupling with a classical field.
Iv Correlated two-photon transport
iv.1 Equations of motion and solution
We now turn to the two-photon scattering problem. Since the total excitation number operator of the system is a conserved quantity, we can restrict the calculation to the two-excitation subspace. An arbitrary state in this subspace has the form:
where is the state of two photons in the nonlinear cavity and no photon in the waveguide, and is the state with one photon in the cavity and one photon with wave number in the waveguide. The last term represents the state with no photon in the cavity and two photons with wave numbers and in the waveguide. , , and denote the respective probability amplitudes.
By the Schrödinger equation, the probability amplitudes are governed by:
We assume that the two injected photons are initially prepared in a Lorentzian wave packet. The initial condition of the system reads,
with the normalization constant
Here and () are parameters defining the detunings and spectral widths of the two photons. Note that has been symmetrized in Eq. (19b) because of the bosonic character of photons.
We are interested in the asymptotic solution of in the long time limit. After a lengthy calculation (see Appendix A), we obtain for and ,
where and are defined in Eq. (13). The expression of is given by
From Eqs. (21) and (22), we notice that the term is a non-factorizable function of and , implying a correlation between the two output photons. has a numerator proportional to the strength of the Kerr nonlinearity in the cavity. In the case , Eq. (21) reduces to a simple expression , describing two independent scattered photons.
iv.2 Two-photon correlation in frequency variables
Let us express the results in terms of the left- and right-propagating modes. Assuming the two photons are injected from the left-hand side of the waveguide, then the initial wave function can be written as
According to Eq. (3) and the solution (21), we obtain the long-time wave function, up to an overall phase factor , as
Here and are, respectively, the two-photon transmission and two-photon reflection amplitudes, which correspond to the processes in which two photons with wave numbers and are transmitted into the right-propagation mode or reflected into the left-propagation mode. In addition, () relates to the process where the photon with wave number () is transmitted into the right-propagation mode and the photon with wave number () is reflected into the left-propagation mode.
We point out two interesting situations revealing the strong correlation of output photons in the frequency domain. The first situation is achieved by injecting two identical photons with and a narrow spectral width . This corresponds to the case when the peak frequency of the photons coincides with the resonant cavity. In this case the two photons are mainly reflected and uncorrelated [Fig. 2(a)], but if they are transmitted, they are strongly correlated [Fig. 2(b)]. This can be seen by the fact that at zero detuning, and hence the transmission of both photons is dominated by the term. In other words, the two-photon transmission near is almost entirely due to the nonlinearity in the cavity. Such a pair of transmitted photons is frequency correlated with the two-photon transmission probability concentrated along the line [Fig. 2(b)]. The uncertainty in the frequencies of individual transmitted photons is of the order of , whereas the uncertainty in the sum of the frequencies of both photons is of the order of . The smaller , the narrower is the distribution.
The second situation of interest is two-photon resonance occurring when the sum of energies of the two incident photons equals to the energy of a cavity containing two photons, i.e., . In this case the photons can jointly enter the cavity. We show in Figs. 2(c) and 2(d) an example with , where the frequency correlation appears more effectively in the reflected amplitude , since (). This is shown in the narrow distribution in Fig. 2(d). The transmission part [Fig. 2(c)], although they carry most of the probabilities, are almost uncorrelated.
iv.3 Two-photon correlation in position variables
We now discuss the spatial features of the output photons. For simplicity, but without loss of generality, we consider the monochromatic limit of incident photons. The two-photon transmission amplitude projected in position space reads (see Appendix B)
Here we have defined and for the center-of-mass and relative coordinates respectively, and and . We note that Eq. (26) is a product of the center-of-mass wave function and the relative wave function , with describing the center-of-mass motion of the two transmitted photons. The second term of is a function localized around with a width . We remark that a similar feature was reported in Ref. Fan2007 in a photon-atom scattering problem, where the exponential decaying function is connected to the existence of photon bound states.
To reveal spatial correlations, we take so that the first term of can be suppressed. This is shown in Fig. 3(a) for various values of . Note that is proportional to the joint probability of photons with a separation , therefore the decaying feature corresponds to photon bunching. In particular, the joint probability of having both transmitted photons at the same position increases with increasing , but it saturates when [Fig. 3(b)].
For the two-photon reflection amplitude in position space, we carry out a similar calculation and obtain,
At , the second term causes a dip in at [Fig. 4(a)], which is a signature of photon antibunching as the reflected photons repel each other. As increases, the joint probability of having both reflected photons at the same position decreases [Fig. 4(b)], which is in contrast to the transmitted part.
Finally we describe the effects of two-photon resonance around discussed in the previous section. For simplicity we again consider the case here. In Fig. 5, we illustrate the dependence of the relative two-photon wave function on . The effect of two-photon resonance is most apparent in Fig. 5(a), where the reflected two-photon wave function is strongly localized around when . Away from the resonance, the reflected two-photon wave function exhibits an oscillatory pattern in [Fig. 5(c)], which is controlled by the two-photon detuning . We also plot the transmitted two-photon wave function in Figs. 5(b) and 5(d), in which similar oscillatory patterns are observed.
In conclusion, we have presented an analytic solution of two-photon scattering inside a one-dimensional waveguide that is side-coupled to a Kerr-type nonlinear cavity. The system provides a scheme to realize correlated two-photon transport. The Kerr nonlinearity is found to correlate photons in frequency variables such that is a constant, which is a constraint because of the energy conservation. In position space, we have shown that the Kerr nonlinearity can cause the two photons ‘stick’ together with an average separation distance of the order of . We may interpret the result as a two-photon bound state, because of the exponential decaying shape of relative wave function. However, because of the interference with single photon processes described by the first term in Eq. (25), features of photon correlation may only be observed efficiently in certain directions. Finally, we note that recent studies of a related topic have considered using a single atom as a scatterer Fan2007 ; sunnew . However, in view of recent progress in achieving a giant Kerr nonlinearity Schmidt1996 ; Deutsch1997 ; Harris1998 , our work suggests that a nonlinear cavity may be an alternative way of regarding the correlated two-photon transport problem.
Acknowledgements.This work is supported by the Research Grants Council of Hong Kong, Special Administrative Region of China (Project No. 401408).
Appendix A Solution of Eq. (18) by the Laplace transform method
In this appendix, we give a detailed derivation for the solution of Eq. (18) which governs the transport of two photons in a waveguide. We use the Laplace transform method to solve these equations. Under the initial condition (19), Eq. (18) becomes
where we have made the approximation .
The solution of Eq. (31), by inspection, takes the form:
Then from Eq. (30c) we obtain the following expression
Until now, we have obtained the expression for . Then we can get the expression for the probability amplitude by performing the inverse Laplace transform of . In particular, since we are interested in the output state of the two photons, here we present only the long-time solution of as
where and have been defined in Eq. (13), and the expression for the correlation term is
Appendix B Derivation of two-photon output state in position space
In this appendix, we derive the wave function of the two-photon output state (LABEL:longtsolu) in position space. For the two-photon transmission process, the corresponding wave function in position space can be written as
In Eq. (37), symmetrization of the two photons has been taken into account by introducing . According to the initial condition given in Eq. (19b), we can get the expression for the independent transport part as
where we introduce the center-of-mass coordinator , the relative coordinator , the total momentum , and the relative momentum . is the Heaviside step function and is defined in Eq. (16). Note that here we have taken the approximation under the assumption of .
According to Eq. (36), the Fourier transform of the correlation part can be written as
for . The Fourier transform of the correlation part can be obtained as