Single-photon superradiance and subradiance in helical collectives of quantum emitters

TL;DR

The paper addresses how a single photon interacts with infinite, continuously distributed emitters arranged on a line and on a helix to produce superradiant and subradiant states. By applying a Green's-function/open-quantum-system approach in the continuous limit, the authors derive closed-form eigenvalues for decay rates $\Gamma$ and collective Lamb shifts $E$, and they compare these results with infinite cylinders and discrete-line formalisms. Key contributions include explicit expressions for $\Gamma$ and $E$ on the infinite line and infinite helix, analysis of trapped states, and thermally averaged decay rates, as well as systematic comparisons across geometries and with realistic protein-fiber structures. The findings illuminate how geometric parameters like inverse pitch $\Omega$ and radius $r$ control radiative channels, offering design principles for superradiant devices and subradiant memories in biomaterial platforms, and they bridge different theoretical formalisms useful for quantum information processing in helical architectures. This work thus provides both fundamental insights into collective light-matter interactions and practical guidelines for engineering quantum optical phenomena in biological and biomimetic systems.

Abstract

Collective emission of light from distributions of two-level systems (TLSs) was first predicted in 1954 by Robert Dicke, who showed that when $N$ quantum emitters absorb photons, their collective radiative decay rate can be enhanced (superradiance) or suppressed (subradiance) relative to a single emitter. In this work, we derive novel analytical expressions for the collective decay rates and Lamb shifts for the interaction of a single photon with a continuous distribution of TLSs on an infinite line and an infinite helix. We compare these solutions to collectives of TLSs on a cylinder, finding limits in which the eigenvalues of structures of different dimensions are equal. We also compare our solution with arrangements where the emitter distribution is discrete rather than continuous, and when short- ($1/r^3$), intermediate- ($1/r^2$), and long-range ($1/r$) interaction terms are included. We find important differences between the discrete vector and continuous scalar emitter cases, which do not agree in the limit where discrete spacing goes to 0. The analytical solution for the helix is then used to make estimates of the maximally superradiant state, thermally averaged collective decay rate, and percentage of trapped states of quantum emitter architectures in protein fibers. Given the differences between our idealized infinite helix and the numerical model describing protein fibers, our analytical estimates show excellent agreement with the numerical results for sparse arrangements of emitters in protein fibers. Our work thus bridges the gap between different formalisms for superradiance, aids the engineering of devices which harness quantum optical effects for computing with superradiant error correction and subradiant memories, and motivates the discovery and creation of flexible platforms for quantum information processing using the intrinsic helical geometries of biomatter.

Figures (5)

• Figure 1: An infinite continuous line of quantum emitters exhibits either maximum decay-rate states, or trapped states with exactly zero decay rate. The blue curve is the plot of Eq. \ref{['eq:line_gamma']} as a function of $\kappa=k_z/k_0$ and the yellow curve is the plot of Eq. \ref{['eq:line_E']} as a function of $\kappa$. Trapped states occur for any $|\kappa|\geq1$, and the other states for $|\kappa|<1$ are all of equal maximal decay rate. The collective Lamb shift of the states at $\kappa=\pm1$ diverges to $-\infty$.
• Figure 2: Collective Lamb shifts and radiative decay rates for an infinite continuous helix of quantum emitters exhibit clear dependencies on the geometric parameters of inverse pitch ($\Omega = 2\pi/k_0 b$) and radius ($r = k_0R$), with the decay rates approaching those of the infinite continuous line in the limits $r\rightarrow0$ and $\Omega\rightarrow0$. The blue and orange curves in each plot correspond to Eqs. \ref{['eq:helix_eigenvalue_real_part']} and \ref{['eq:helix_eigenvalue_Lamb_shift']}, respectively, plotted as a function of $\kappa$ and evaluated at the displayed $\Omega$ and $r$ values. The black vertical dashed lines indicate asymptotes, while the gray horizontal dashed lines are reference grid lines at the vertical coordinate $1.0$. For the collective Lamb shifts, the infinite sum \ref{['eq:helix_eigenvalue_Lamb_shift']} is truncated to ten terms for plotting purposes. The top row shows fixed $\Omega=3$ and $r$ varying from $0.1$ to $10$ going left to right. The bottom row shows fixed $r=3$ and $\Omega$ varying from $0.1$ to $10$ going from left to right. In every plot, it is seen that $\kappa=1$ is where the maximum decay rate occurs. The condition \ref{['eq:trapped_states_condition']} for trapped states can also be seen in these plots. The decay rates in the two leftmost plots, $\Omega=3,r=0.1$ and $\Omega=0.1,r=3$, approach the decay rates of the infinite continuous line, which can be compared directly in Fig. \ref{['fig:infinite_line_vs_kz_k0']}.
• Figure 3: The thermally averaged collective decay rate $\langle\Gamma_\text{th}\rangle$ of an infinite helix of quantum emitters is consistently larger than that of an infinite cylinder, due to more eigenstates in the helices having high decay rates and large-magnitude, negative collective Lamb shifts. The vertical axis value of each point represents the value of the integral in Eq. \ref{['eq:thermal_Gamma_modified']} for the parameters specified by the $x$-axis and the legend, where we have set $\beta=1$. The dimensionless values $r\equiv k_0 R$ and $\Omega\equiv 2\pi/k_0 b$ are used on the $x$-axis, where $k_0$ is the excitation wavenumber, $b$ is the helical pitch, and $R$ is the radius. The blue (orange) points are for the infinite continuous helix, where $\Omega$ ($r$) is kept constant at $3$, and $r$ ($\Omega$) is given by the $x$-axis values. The green points are for the infinite cylinder solution from Ref. Svidzinsky2016. For the cylinder there is no $\Omega$ value, so $r$ is varied on the $x$-axis. The integral in Eq. \ref{['eq:thermal_Gamma_modified']} was approximated by discretizing $\kappa$ into intervals of length $\Delta\kappa=0.01$, such that $\kappa_i\in[0,0.01,\,...\,,4.99,5]$, and turning the integral into a sum.
• Figure 4: Collective Lamb shifts $E^\|$ and $E^\perp$ and radiative decay rates $\Gamma^\|$ and $\Gamma^\perp$ for a discrete line of quantum emitters modeled as transition dipole vectors in the limit of zero emitter spacing are distinct from those for a continuous line of scalar emitters. The collective Lamb shifts and decay rates from Ref. AAG2017, re-written here in Eqs. \ref{['eq:E_par']}-\ref{['eq:AAG_Gamma_perp']}, are plotted as a function of $k_z d/\pi$ for the infinite discrete line. Since the functions diverge when $d = 0$, the value $d=0.05$ was used for plotting purposes. The black vertical dashed lines indicate where $k_z=\pm k_0$. Note that there is a vertical asymptote for $E^{\perp}$ at $k_z=\pm k_0$, because at these values $E^{\perp}$ diverges to $-\infty$. These plots should be compared to Fig. \ref{['fig:infinite_line_vs_kz_k0']}, given by Eq. \ref{['eq:line_eigvals']}; though there are similarities, it can be seen that neither $E^\|$ nor $E^\perp$ converge to the collective Lamb shift of the infinite continuous line in the limit $d\rightarrow 0$. The same applies for $\Gamma^\|$ and $\Gamma^\perp$. This difference occurs because the interaction terms in the Hamiltonian from Ref. AAG2017 contain terms proportional to $1/r^2$ and $1/r^3$, which are not present in the Hamiltonian used in Fig. \ref{['fig:infinite_line_vs_kz_k0']}, and the emitters in the infinite continuous line from Fig. \ref{['fig:infinite_line_vs_kz_k0']} are scalar objects, with their polarization effects neglected.
• Figure 5: The infinite continuous helix provides a good approximation of the photophysics of densely spaced networks of molecular quantum emitters (tryptophans) in protein fibers. Panel a) shows, from left to right, a microtubule, an actin filament, and an amyloid fibril with the tryptophan amino acids highlighted in blue and red. Only the blue tryptophans are used to make the helical approximation of each structure. Panels b), c), and d) show only the blue tryptophans in the microtubule, actin filament, and amyloid fibril, respectively, from panel a). The best helical approximation of each tryptophan helix in panels b), c), and d) is overlaid in gray. The axes in panels b), c), and d) are all in units of nm. Panel e) shows the eigenspectrum of the infinite helix approximations of each structure for varying parameters $\Omega$ and $r$, but using the actual line density values $n_0$ from each protein fiber. The advantage of this data representation is that $\Gamma/\gamma$ can directly indicate which states are superradiant ($\Gamma/\gamma>1$) and subradiant ($\Gamma/\gamma<1$), rather than only trapped and non-trapped as in Fig. \ref{['fig:infinite_helix']}.