Time evolution of the one-dimensional Jaynes-Cummings-Hubbard Hamiltonian

The Jaynes-Cummings-Hubbard (JCH) system describes a network of single-mode photonic cavities connected via evanescent coupling. Each cavity contains a single two-level system which can be tuned in resonance with the cavity. Here, we explore the behavior of single excitations (where an excitation can be either photonic or atomic) in the linear JCH system, which describes a coupled cavity waveguide. We use direct, analytic diagonalization of the Hamiltonian to study cases where intercavity coupling is either uniform or varies parabolically along the chain. Both excitations located in a single cavity, as well as one excitation as a Gaussian pulse spread over many cavities, are investigated as initial states. We predict unusual behavior of this system in the time domain, including slower than expected propagation of the excitation and also splitting of the excitation into two distinct pulses, which travel at distinct speeds. In certain limits, we show that the JCH system mimics two Heisenberg spin chains.

Figure 1

(Color online) Visualizations of a 1D JCH system. (a) shows an example of a coupled cavity waveguide in a photonic crystal. A lattice of holes in the membrane provides variations in refractive index, hence trapping photons of a given frequency. Red spheres indicate two-level systems. These are placed at sites without holes, called defects, which form photonic cavities. (b) is a visualization of a portion of a JCH chain as logical elements. The bottom row of circles indicates the photonic cavities, linked together by the hopping rate $\kappa$. The top row of circles indicates the atoms, linked back to each photonic cavity by the coupling strength $\beta$ [defined in Eq. (1)].

Figure 2

(Color online) Space-time diagrams for evolution of an excitation $\left[|1⟩\otimes \left(|g,1⟩+|e,0⟩\right)/\sqrt{2}\right]$ along a chain of 100 JC cavities for three different parameter regimes. In each case, we plot the probability of occupation of a particular cavity (vertical axis) as a function of time (horizontal axis) for the system initialized in an even superposition of the atomic and photonic modes of the first cavity. The upper three plots show the population of the atomic components whereas the lower three plots are the photonic components. The ratio of cavity-cavity coupling to atom-photon coupling varies from left to right, with (a) and (b) $\kappa /\beta ={10}^{-3}$, (c) and (d) $\kappa /\beta =10$, and (e) and (f) $\kappa /\beta ={10}^3$. In plots (a), (b), and (f), dashed lines represent ${\mathcal{Q}}_{\wedge }$ as given in Eq. (15).

Figure 3

Dispersion of the wave packet, $\Delta Q$, at a fixed point in time, as a function of $\kappa /\beta$, with atom-cavity detuning $\Delta =0$ and number of cavities $N=100$. Solid (dashed) line shows $\Delta Q_{\text{atomic}}^{\text{JCH}}\left(\Delta Q_{\text{photonic}}^{\text{JCH}}\right)$. Dotted (dot-dashed) line shows $\Delta Q^{\text{Heis}}$ with $J=2\kappa \left(J=\kappa \right)$. The dispersion is measured at a time $T=N/4\kappa$, at which point the packet is “freely” evolving along the chain. This point is chosen such that the effects of the hard-wall boundaries can be ignored for both large and small $\kappa$. At the left of the plot, $\kappa /\beta ⪡1$ and the JCH chain mimics two identical Heisenberg spin chains. An example is shown of this localized behavior in Figs. 2a, 2b, where $\kappa /\beta ={10}^{-3}$. The corresponding spatial profile of the pulse at $T=N/4\kappa$ is shown in the left inset (solid line shows the photonic profile and dashed line shows the atomic profile, in this case they are coincident). At the right of the plot, $\kappa /\beta ⪢1$ and the photonic mode of the JCH chain mimics a Heisenberg spin chain, while the atomic mode does not propagate at all. An example of this type of localized behavior is shown in Figs. 2e, 2f, where $\kappa /\beta ={10}^3$, the corresponding profile of the pulse at $T=N/4\kappa$ is shown in the right inset. In the middle of the plot a traveling excitation shows a large amount of dispersion, an example of this delocalized behavior is shown in Figs. 2c, 2d, where $\kappa /\beta =10$ and the corresponding profile at $T=N/4\kappa$ is shown in the middle inset. The horizontal lines show the dispersion $\Delta Q^{\text{Heis}}$ of a Heisenberg spin chain with 100 spins after the initial state $|1⟩$ evolves until the front of the excitation is halfway (dotted line) and quarterway (dot-dashed line) along the excitation chain. The exact correspondence between the Heisenberg and JCH systems is seen in these asymptotic limits.

Figure 4

(Color online) Evolution of a Heisenberg spin chain Hamiltonian [Eq. (14)] with $J=1$. (a) Evolution of $|1⟩$ under uniform coupling. Note that at first, the excitation travels fairly neatly, but both within a single pass, as well as during the reflection, the excitation pulse spreads out. Note the faint lines which are parallel but displaced from the wave front. These lines are due to the nature of the Heisenberg Hamiltonian. (b) Evolution of $|1⟩$ under parabolic coupling. Note how the evolution is smooth and repetitive. The excitation travels sinusoidally from one end of the line of spins to the other end and back again, without spreading out. (c) Evolution of a Gaussian pulse as in Eq. (26), with $Q_c=50$, $s=10$, and $k=\pi /2$, with uniform coupling. Note how the evolution is dispersion free.

Figure 5

Evolution of a Heisenberg spin chain Hamiltonian [Eq. (14)], with $J=1$. Solid lines display the expectation value of the position $⟨Q⟩$ and dashed lines show the dispersion $\Delta Q$. (a) Evolution of $|1⟩$ under uniform coupling. $\Delta Q$ increases with time. This indicates increasing dispersion as time progresses. (b) Evolution of $|1⟩$ under parabolic coupling. Here $\Delta Q$ does not increase with time, as such this indicates dispersion-free evolution. (c) Evolution of a Gaussian pulse as in Eq. (26), with $Q_c=50$, $s=10$, and $k=\pi /2$, with uniform coupling. As in (b), there is no overall increase in dispersion. As such, this pulse is also dispersion free.

Figure 6

(Color online) Space-time diagrams for evolution of an excitation $\left(|1⟩\otimes \left(|g,1⟩+|e,0⟩\right)/\sqrt{2}\right)$ along a chain of 100 JC cavities with parabolic intercavity coupling profile. The upper plots show the population of the atomic components whereas the lower plots are the photonic components. The ratio of cavity-cavity coupling to atom-photon coupling varies from left to right, with (a) and (b) $\kappa /\beta ={10}^{-4}$, (c) and (d) $\kappa /\beta ={10}^0$, and (e) and (f) $\kappa /\beta ={10}^3$. In plots (a), (b), and (f), the white dashed lines represent $⟨Q\left(t\right)⟩$, as given in Eq. (29). The effect of the parabolic coupling is to constrain the pulse, resulting in well-defined and reversible dispersion.

Figure 7

Dispersion of the wave packet, $\Delta Q$, at a fixed point in time for a JCH system with parabolic coupling. Solid (dashed) line shows $\Delta Q_{\text{atomic}}^{\text{JCH}}\left(\Delta Q_{\text{photonic}}^{\text{JCH}}\right)$. Dotted (dot-dashed) line shows $\Delta Q^{\text{Heis}}$ with $J=2\kappa \left(J=\kappa \right)$. The parameter ranges are identical to Fig. 3, although the range of the vertical axis is different. Insets show the pulse profile at three values of $\kappa /\beta$. Solid (dashed) line represents the photonic (atomic) profile. The result is that, while the dispersion is considerably larger than the uniform coupling case, in this case, the pulse is more “well behaved.” More precisely, the pulse has a Gaussian profile away from the boundaries and its evolution is completely reversible. In the extreme limits for $\kappa$, we have again perfect agreement with that expected from the appropriate spin chain model.

Figure 8

(Color online) Space-time diagrams for evolution of a Gaussian pulse along a chain of 100 JC cavities with a uniform intercavity coupling profile. The upper plots show the population of the atomic components whereas the lower plots are the photonic components. The ratio of cavity-cavity coupling to atom-photon coupling varies from left to right with (a) and (b) $\kappa /\beta ={10}^{-2}$, (c) and (d) $\kappa /\beta ={10}^0$, and (e) and (f) $\kappa /\beta ={10}^3$. In plots (a), (b), and (f), the white dashed lines represent ${\mathcal{Q}}_{\wedge }^{\text{Gaussian}}$, as is given in Eq. (28). The effect of the Gaussian pulse is to choose a narrow momentum distribution, which allows the excitation to propagate freely in the system.

Figure 9

Dispersion of a Gaussian wave packet, $\Delta Q$, at a fixed point in time, as a function of $\kappa /\beta$ at $\Delta =0$. The dispersion is measured at time $T=N/8\kappa$, at which point the packet is freely evolving along the chain. The initial state of the Gaussian is given by Eq. (27), with center of the pulse at $Q_c=N/2$, width of the pulse $s=N/10$, wave number $k=\pi /2$, and number of cavities $N=100$. The width of the pulse is small enough such that the boundaries do not have an effect on $\Delta Q$ for the value of $T=N/8\kappa$ in the limits $\kappa /\beta ⪡1$ and $\kappa /\beta ⪢1$. At the left of the plot, $\kappa /\beta ⪡1$ and the JCH chain mimics two identical Heisenberg spin chains. An example is shown of this localized behavior in Figs. 8a, 8b, where $\kappa /\beta ={10}^{-2}$. The corresponding spatial profile of the pulse is shown in left inset (solid line shows the photonic profile and dashed line shows the atomic profile; in this case they are coincident). At the right of the plot $\kappa /\beta ⪢1$ and the photonic mode of the JCH chain mimics a Heisenberg spin chain, while the atomic mode does not propagate at all. An example of this type of localized behavior is shown in Figs. 8e, 8f, where $\kappa /\beta ={10}^3$. The corresponding profile of the pulse at $T=N/8\kappa$ is shown in the right inset. In the middle of the plot, a traveling excitation shows a large amount of dispersion. An example of this delocalized behavior is shown in Figs. 8c, 8d, where $\kappa /\beta =10$, and the corresponding profile at $T=N/8\kappa$ is shown in the middle inset. Horizontal line shows the dispersion $\Delta Q^{\text{Heis}}$ of a freely evolving pulse [this value does not change, provided the pulse is not interacting with the boundary, as is evidenced by Fig. 5c].

Figure 10

The pulse profile for various points in time as a function of cavity number for evolution of a Heisenberg spin chain. (a) Evolution of $|1⟩$ with uniform coupling between spins. (b) Evolution of $|1⟩$ with parabolic coupling between spins. (c) Evolution of a Gaussian pulse as given in Eq. (26) with uniform coupling between spins. In the first uniform case, the profile is initially given by a Kronecker delta function, but later spreads into a function with one primary peak and a number of smaller trailing peaks. In the parabolic case, at each end of the chain, the pulse is given by a Kronecker delta function, while in the middle of the chain, the pulse approximates a Gaussian. In the uniform coupling case with an initial Gaussian pulse, it evolves along the chain with fixed profile, and at the ends of the chain interferes and changes direction.

Figure 11

(Color online) These plots show the evolution of [(a) and (b)] the state $|1⟩\otimes \left(|e,0⟩+|g,1⟩\right)$ and (c) a Gaussian pulse in a one-dimensional JCH system consisting of 100 cavities, with $\kappa =\beta$ and $\Delta /\beta ={10}^3$. Note the time scales are different in each case. (a) Evolution in a uniform chain, where the dashed lines are a triangle wave giving approximate evolution of the wave front, as given in Eq. (15). (b) Evolution in a parabolically coupled chain, where the dashed lines are as given in Eq. (29). (c) Evolution in a uniformly coupled chain, with an initial Gaussian pulse, as given in Eq. (27). Dashed lines show the expectation value of position as given in Eq. (28).