Ultrastrong coupling few-photon scattering theory

TL;DR

This paper develops a theoretical framework for analyzing single-photon scattering in a system where a two-level quantum emitter is ultrastrongly coupled to a one-dimensional waveguide, relevant for quantum optics and superconducting circuits.

Contribution

It introduces a combined polaron transformation and scattering theory approach to accurately compute photon scattering properties across a broad coupling range, including near the inelastic scattering regime.

Findings

Accurately predicts resonance frequencies, lineshapes, and linewidths for ultrastrong coupling.

Validates the method up to coupling strength α=0.3, close to the Toulouse point.

Models experimental results with superconducting circuits.

Abstract

We study the scattering of photons by a two-level system ultrastrongly coupled to a one-dimensional waveguide. Using a combination of the polaron transformation with scattering theory we can compute the one-photon scattering properties of the qubit for a broad range of coupling strengths, estimating resonance frequencies, lineshapes and linewidths. We validate numerically and analytically the accuracy of this technique up to $\alpha=0.3$, close to the Toulouse point $\alpha=1/2$, where inelastic scattering becomes relevant. These methods model recent experiments with superconducting circuits [P. Forn-D{\'\i}az et al., Nat. Phys. (2016)].

Figures

Equations

$$H=\frac{\Delta }{2}{\sigma }^z+\sum _k{\omega }_k{a}_k^{†}{a}_k+\frac{1}{\sqrt{L}}\sum _k{g}_k{\sigma }^x(a_k+a_k^{†}),$$

$$J(\omega )=\frac{2\pi }{L}\sum _k|g_k{|}^2\delta (\omega -{\omega }_k)\simeq \pi \alpha {\omega }^1{e}^{-\omega /{\omega }_c}.$$

$$H_{\text{p}}=\frac{\tilde{\Delta }}{2}{\sigma }^z{O}_{-\mathbf{f}}^{†}{O}_{\mathbf{f}}+\sum _k{\omega }_k{a}_k^{†}{a}_k+\sum _k\frac{G_k}{\sqrt{L}}{\sigma }^x(a_k+a_k^{†})+E_{\text{p}},$$

$$H_{\text{p}}^{(1)}$$

$$+\frac{2\sqrt{2}}{\sqrt{L}}\sum _k\tilde{\Delta }{f}_k(A_{+,k}^{†}{\sigma }^{-}+A_{+,k}{\sigma }^{+}).$$

$$r_k=\frac{1}{2}(s_k-1),\text{mbox}and{t}_k=\frac{1}{2}(s_k+1),$$

$$s_k=\frac{({\omega }_k-\tilde{\Delta })\tilde{\Delta }-({\omega }_k+\tilde{\Delta }){\Sigma }^{\ast }({\omega }_k)}{({\omega }_k-\tilde{\Delta })\tilde{\Delta }-({\omega }_k+\tilde{\Delta })\Sigma ({\omega }_k)}.$$

$${\delta }_L(\omega )=4{\tilde{\Delta }}^2{\int }_0^{\infty }\frac{dk}{2\pi }P\frac{f_k^2}{\omega -{\omega }_k},$$

$$r_k\simeq \frac{-i({\omega }_k+\tilde{\Delta })\frac{\Gamma }{2}}{({\omega }_k-\tilde{\Delta })\tilde{\Delta }+i({\omega }_k+\tilde{\Delta })\frac{\Gamma }{2}},$$

$$\Sigma (\omega )=\frac{2{\tilde{\Delta }}^2\alpha }{(\omega +\tilde{\Delta }{)}^2}\left[\omega \ln (\frac{\omega }{\tilde{\Delta }})-\omega -\tilde{\Delta }-i\pi \omega \right].$$

$$s_k=1+\frac{2iw}{1+2iw}[$$

$$+\ln w^2/(1+w^2)],$$

Full text

Waveguide quantum electrodynamics (QED) studies the interaction between propagating photons and quantum impurities in 1D environments. Reduced dimensionality empowers few-level systems with strong nonlinear features, and they become capable of fully reflecting individual photons Astafiev et al. (2010) or mediating a strong photon-photon interaction Hoi et al. (2013). In order for this to occur, the impurity –a two-level system or qubit— needs to be in the strong-coupling regime, where the spontaneous emission into the waveguide, $\Gamma$, dominates all other dissipation channels. This regime is achieved in experiments with superconducting circuits Astafiev et al. (2010); Hoi et al. (2013), neutral atoms Tiecke et al. (2014); Goban et al. (2015) and quantum dots in photonic crystals Arcari et al. (2014). In most experiments spontaneous emission is slower than the atom or photon oscillation frequencies, $\Gamma\ll\Delta,\omega$, allowing for a rotating-wave approximation (RWA) and theoretical predictions based on one- and few-photon wavefunctions Shen and Fan (2005, 2007), input-output theory Fan et al. (2010); Caneva et al. (2015) and path integral formalism Shi and Sun (2009); Shi et al. (2015).

Superconducting circuits are waveguide QED systems where the qubit-photon coupling can match the qubit and photon energies, $\Gamma\sim\Delta,\omega$. This so called ultrastrong-coupling regime (USC) causes the breakdown of RWA predictions, the excitation of qubit-photon entangled ground states Sanchez-Burillo et al. (2014), extremely broadband interactions Peropadre et al. (2013), and a phase transition into the localization regime Leggett et al. (1987). The USC was first demonstrated in resonators Niemczyk et al. (2010); Forn-Díaz et al. (2010), where it admits an analytic description Braak (2011). More recently, the USC regime has been explored using propagating photons in microwave guides Forn-Díaz et al. (2016) and studying the resonance spectrum of the qubit in the transmission line. This new generation of experiments opens a very challenging theoretical problem: the integration of USC in the waveguide QED framework, moving beyond the study of dissipation Leggett et al. (1987), to photon-qubit scattering and interactions.

While this question has been addressed using numerical methods such as Matrix Product States or MPS Peropadre et al. (2013); Díaz-Camacho et al. (2016), the Numerical Renormalization Group Bera et al. (2016), in this work we develop fully analytic predictions for the photon-qubit interaction, which are accurate for a broad range of the USC regime. Our starting point is the spin-boson model for a waveguide of length $L$

[TABLE]

We do not work with this Hamiltonian, but build a transformed one $H_{\text{p}}=U_{\text{p}}^{\dagger}HU_{\text{p}}$ using an optimized polaron transformation $U_{\text{p}}$ that eliminates most of the qubit-photon entanglement in the ground state Díaz-Camacho et al. (2016). The new Hamiltonian $H_{\text{p}}$ can be manipulated and combined with scattering theory Shi et al. (2015) to predict the dynamics of few-photon wavepackets. We show results for the superconducting Ohmic spin-boson model, where the spectral function is linear up to a cutoff $\omega_{c}$

[TABLE]

Interestingly, we recover cutoff independent predictions for the resonance and linewidth of elastic single-photon scattering in the USC regime. These results are validated with analytics at the Toulouse point Guinea et al. (1985); Toulouse (1969) at $\alpha=1/2$ and also with moderate-size matrix-product state (MPS) numerical simulations of the qubit spontaneous emission. Both methods attest the qualitative $(\alpha\in[0.3,0.5])$ and even quantitative $(\alpha\in[0,0.3])$ accuracy of our techniques in modeling new and state-of-the-art experiments such as the single-photon scattering with tuneable coupling qubits by Forn-Díaz et al. Forn-Díaz et al. (2016). This work opens the door to studying multi-photon scattering in more complex experiments with transmons or $\Lambda$-level schemes, or the development of accurate models for photon mediated interactions in open waveguides, which would have important applications in the quantum simulations of Ising-like Hamiltonians Kurcz et al. (2014) and annealing.

Model setup.—

Our starting point is the Hamiltonian (1) that models the interaction between a two-level system and a photonic waveguide with periodic boundary conditions. The Pauli matrices $\sigma^{x,z}$ are defined in the qubit basis $\ket{e}$ and $\ket{g}$ for excited and ground states. The qubit couples to photons with momenta $k$, with anihilation (creation) operators $a_{k}$ $(a_{k}^{\dagger})$. We will conduct analytic calculations with a linear dispersion $\omega_{k}=c|k|$ and couplings $g_{k}=\sqrt{\pi\alpha c\omega_{k}/2}e^{-\omega_{k}/2\omega_{c}}$ that reproduce the Ohmic spectral function (2), and numerics with a hard cut-off $\omega_{k}=\omega_{c}\sqrt{[1-\cos(k)]/2}$, couplings $g_{k}=\sqrt{\pi\alpha c\omega_{k}/2}$ and $L$ equal to the number of modes.

Instead of (1), we implement approximations on the equivalent model $H_{\text{p}}=U_{\text{p}}^{\dagger}HU_{\text{p}}$ after a polaron transformation $U_{\text{p}}=\exp[-\sigma^{x}\sum_{k}f_{k}(a_{k}^{\dagger}-a_{k})/\sqrt{L}]$, which disentangles the bosonic and qubit states

[TABLE]

The renormalized qubit energy $\tilde{\Delta}=\Delta e^{-2\sum_{k}|f_{k}|^{2}/L}$ appears with the operators $O_{\mathbf{f}}=\exp(2\sigma^{x}\sum_{k}f_{k}a_{k}/\sqrt{L})$ from normal ordering. The Silbey-Harris prescription Silbey and Harris (1984); Harris and Silbey (1985) optimizes $f_{k}=g_{k}/(\omega_{k}+\tilde{\Delta}),$ reducing the effective coupling $G_{k}=\tilde{\Delta}f_{k}$, and making the ground state of $H_{\text{p}}$ as close to $\ket{g}\ket{0}$ as possible.

Despite the highly nonlocal term $\sigma^{z}O_{-\mathbf{f}}^{\dagger}O_{\mathbf{f}}$, it is possible to diagonalize $H_{\text{p}}$ using MPS ansatz García-Ripoll (2006); Verstraete et al. (2008); Orús (2014), a variational estimate of the ground state wavefunction $\ket{\psi}=\sum_{s,\mathbf{n}}\mathrm{tr}\left[A^{s}_{0}A^{n_{1}}_{1}\cdots A^{n_{N}}_{N}\right]\ket{s,n_{1}\ldots n_{N}},$ where $A_{i}^{x}\in\mathbb{C}^{\xi\times\xi}$ are different matrices labeled by physical degrees of freedom: the qubit states, $s\in\{g,e\}$, or the photon occupation numbers $n_{i}$ of the associated momenta $k_{i}$. Numerical optimizations with the hard cut-off model $\omega_{c}/\Delta=3,6$ and $9$ show less than $2\%$ qubit excitation probability and a negligible amount of photons below the Toulouse point $\alpha=1/2$ [cf. Fig. 1]. Interestingly, most qubit-photon entanglement is removed by the polaron transformation and the MPS converges with a small bond dimension $\xi\ll 20$ (small matrices), and a small cut-off $n_{i}\leq 4$, significantly improving over earlier simulations with $H$ Peropadre et al. (2013).

Excitation conserving polaron.—

Inspired by the simplicity of the ground state, we will now assume that the low-energy dynamics of $H_{\text{p}}$ admits also a simple description as quasiparticles on a close-to-vacuum state. For that we select the single-particle section Hamiltonian

[TABLE]

This model introduces annihilation operators $A_{\pm,k>0}=(a_{k}\pm a_{-k})/\sqrt{2}$ for the symmetric and anti-symmetric modes, and is restricted to work with a single excitation, $N=\sigma^{+}\sigma^{-}+\sum_{k}a^{\dagger}_{k}a_{k}=1$, which is sufficient for single photon emission and scattering. Note the spin-dependent potential, $V_{\text{local}}=-4\sigma^{z}\tilde{\Delta}\sum_{kp}f_{k}f_{p}A^{\dagger}_{+,p}A_{+,k}/L$, essential to capture the whole dynamics

We have compared both models using MPS simulations of a low-energy problem in which an excited qubit relaxes into a vacuum of photons, $\ket{\psi(0)}=\ket{e}\ket{0}$. As shown in Fig. 2a, the wavefunction at all times $\ket{\psi(t)}$ remains in the single excitation sector up to $\alpha\sim 0.3$. Moreover, Figs. 2b-c demonstrate an excellent agreement between the RWA (4) and the Silbey-Harris Hamiltonian (3), not only in the qubit dynamics, but also in the emission spectrum. The quantitative disagreement is largely accounted for by (i) the additional dressing of localized photons by the qubit [cf. Fig. 1] and (ii) inleastic three- or more-photon contributions above $\alpha=0.35$ Goldstein et al. (2013).

Single-photon scattering estimates.—

The MPS suggest that we can work with $H_{\text{p}}^{(1)}$ up to $\alpha\simeq 0.3$. To get a feeling for this value, let us cut the waveguide to make a $\lambda/2$ resonator which is resonant with the qubit. We will find a qubit-cavity coupling $g_{\text{cav}}=\sqrt{\alpha}\Delta$. Thus, values of $\alpha=0.3$ correspond to $g_{\text{cav}}\simeq 0.55\Delta$ inside a cavity: a coupling so strong, that the bandwidth of photons is comparable to the qubit energy $\Gamma/\tilde{\Delta}\simeq 1$ [cf. Fig. 2c], the so called ultrastrong coupling regime.

Our goal is to analyse scattering in the USC regime, developing formulas that can be used to model experiments Forn-Díaz et al. (2016). We will apply scattering theory to $H_{\text{p}}^{(1)}$, focusing on the low power regime of at most one photon. The reflection and transmission coefficients

[TABLE]

are constructed from the chiral phase shift $|s_{k}|=1$ experienced by photons in the $A_{k,+}$ modes. These can be computed using scattering formalism Shi et al. (2015) or Lippmann-Schwinger theory

[TABLE]

The self-energy $\Sigma(\omega)=\delta_{L}(\omega)-i\Gamma(\omega)/2$ contains a Lamb-shift

[TABLE]

and a decay rate $\Gamma(\omega)=4\tilde{\Delta}^{2}f_{k_{0}}^{2}\left|\partial\omega_{k}/\partial k\right|_{k=k_{0}}^{-1}$ given by the solution $k_{0}\equiv k_{0}(\omega)$ of $\omega_{k_{0}}=\omega$. The reflectivity $R_{k}=\left|r_{k}\right|^{2}$ and transmissivity $T_{k}=\left|t_{k}\right|^{2}$ determined by Eq. (6), satisfy $R+T=1$ and provide a concrete prediction for the lineshape of a single-photon scattering experiment, for all dispersion relations and frequency dependent couplings. When the dynamics of the spontaneous emission is slower than that of the qubit and photons, i.e., $|\Gamma|\ll\tilde{\Delta},\omega_{k}$, the Markov approximation reflects into a negligible potential $V_{\mathrm{local}}$ and Lamb-shift $\delta_{L}\sim 0$ and a uniform $\Gamma(\omega)\sim\Gamma$. We recover the usual formula

[TABLE]

predicting total reflection $R=1$ for resonant photons $\omega_{k}\rightarrow\tilde{\Delta}$, and displaying the usual Lorentzian profile from scattering experiments in the strong coupling regime with superconducting circuits Astafiev et al. (2010); Hoi et al. (2013) or quantum dots Arcari et al. (2014). For USC, however, the self-energy and the local potential $V_{\mathrm{local}}$ induce significant distortions and asymmetries in the lineshapes, as expected from both earlier numerics Peropadre et al. (2013) and experiments Forn-Díaz et al. (2016).

Open transmission line.-

We particularize the predictions to the Ohmic coupling of a superconducting qubit with a transmission line. In the limit of large cut-off $\omega_{c}$, we may approximate $\tilde{\Delta}=\Delta\left({e\Delta}/{\omega_{c}}\right)^{\alpha/(1-\alpha)}$, with varying prefactors depending on the details of the model. However, independent of the renormalization scheme, our scattering estimates lead to the same self-energy

[TABLE]

This prediction does not involve the cut-off: this information is implicit in the value of $\tilde{\Delta}$, which the optimal transformation uses to regularize the couplings in both the infrarred and ultraviolet limits.

The formula above has three important consequences: (i) The lineshape profiles are asymmetric for even moderate values of $\alpha\simeq 0.1$ [See Figs. 3a-b]. (ii) As shown in Fig. 3a, the scattering resonance $\omega_{\text{reson}}$, defined as the frequency of maximum reflection, does not necessarily match the value $\tilde{\Delta}$. This is due to a very large Lamb shift, of the same order of magnitude as the spontaneous emission rate itself. (ii) The linewidth $\Gamma$ depends on both $\alpha$ and $\omega$, and it is not possible to calibrate the interaction strength using the formula $\Gamma/\tilde{\Delta}\simeq\pi\alpha$, from the non-USC regime [cf. Fig. 3c]. In other words, while it is true that we can distinguish the USC regime by the condition that $\Gamma/\pi$ be comparable to $\omega_{\text{reson}}$, a calibration of $\alpha$ demands the mathematical modelization of the line shapes.

Toulouse point, $\alpha=1/2$.—

While the above scattering formulas work for a broad range of couplings, $\alpha\in[0,0.3]$, it is interesting to study the source of deviations for very strong interactions, up to the phase transition into the Kondo regime. Fortunately, the spin-boson model with an Ohmic spectrum admits an analytical solution at the Toulouse point $\alpha=1/2$, which already has been used to study scattering properties Goldstein et al. (2013).

The basic idea is to realize that at $\alpha=1/2$, working with a polaron displacement $f_{k}^{\prime}=-g_{k}/\omega_{k}$ we can cancel completely the linear coupling terms, $G_{k}=0$, and map the spin and $A_{+}$ modes to the density fluctuations a fermionic bath, $A^{\dagger}_{q,+}=\sqrt{\frac{2\pi}{Lq}}\sum_{k}c_{k+q}^{\dagger}c_{k}$. The resulting model can be diagonalized, giving as ground state $\ket{\mathrm{GS}}=\ket{0}_{-}\ket{\text{FS}}_{f}$ a product of the vacuum $\ket{0}_{-}$ of the anti-symmetric modes and the Fermi sea $\ket{\text{FS}}_{f}$ of the new fermions. To compute the scattering matrix, we express the asymptotic state with one incoming photon $\ket{\phi_{\mathrm{in}}}=a_{k}^{\dagger}U_{\mathrm{p}}\ket{0}_{-}\ket{\text{FS}}_{f}$ using fermionic operators, and compute the out-going asymptotic state $\ket{\phi_{\mathrm{out}}}=\lim_{T\rightarrow\infty}e^{-iHT}\ket{\phi_{\mathrm{in}}}$ using the Green function approach in the infinite line limit $L\rightarrow\infty$. The outgoing wavefunction contains both a single photon (elastic) component, as well as multiphoton (inelastic) contributions Goldstein et al. (2013). The single-photon reflection and transmission (5) derive from

[TABLE]

with $w=\pi\Delta^{2}/(4\omega_{c}k)$. The value $s_{k}$ is no longer a phase and as a result, $P_{1}=|R_{k}|+|T_{k}|<1$. The true dynamics deviates from the polaron-RWA predictions because of multi-photon processes, which, as already shown in Ref. Goldstein et al. (2013), are dominated by three-photon corrections $P_{3}=|\braket{0}{\prod_{i=1}^{3}A_{+,p_{i}}^{\dagger}}{\phi_{\mathrm{out}}}/\sqrt{3!}|^{2}$ with momenta $p_{1}+p_{2}+p_{3}=k$. In Figs. 4a-b we show the elastic transmission and reflection coefficients for both the exact Toulouse wavefunction and the polaron-RWA approximation. Considering that $\alpha=1/2$ is a very large interaction ($g_{\text{cav}}\simeq 0.71\Delta$ in the cavity), we find a very good qualitative and almost quantitative agreement between both methods in the elastic sector.

Summary and discussion.—

In this work we have derived analytical estimates of the lineshapes and resonance frequencies for single-photon scattering in the USC regime, $\alpha\in[0,0.3]$ in the spin boson model —or $g/\omega\in[0,55\%]$ if we would cut the same line to shape a cavity—. These estimates are supported by strong numerical evidence that the static and dynamic properties of the spin-boson model can be well approximated by a RWA version of the polaron transformation for this range of couplings. Our predictions apply to experiments with superconducting qubits in open transmission lines Forn-Díaz et al. (2016) and represent an important milestone in the integration of the USC regime in waveguide QED theory. The techniques presented in this work can be immediately extended to other dispersion relations and coupling strengths, including, for instance, USC scattering in photonic bandgaps and cavity arrays with bound states Sanchez-Burillo et al. (2014). It is also possible to account for radiative losses, heating and dephasing, depending on the qubit nature and its energy gap: in all cases the formulas generalize with the change $\Sigma(\omega)\to\Sigma(\omega)-i\Gamma_{\varphi}$, where $\Gamma_{\varphi}$ includes all additional dephasing sources. We expect that the ideas put forward in this work will stimulate and simplify future experiments with superconducting circuits in the USC regime, as well as help in the development of a complete theory for USC scattering and effective interactions in multi-qubit setups.

Acknowledgements.

JJGR acknowledges support from MINECO/FEDER Project FIS2015-70856-P and CAM PRICYT Project QUITEMAD+ S2013/ICE-2801. This work was funded by the European Union Integrated project Simulators and Interfaces with Quantum Systems (SIQS).