Abstract
A quantum system weakly coupled to a zerotemperature environment will relax, via spontaneous emission, to its groundstate. However, when the coupling to the environment is ultrastrong the groundstate is expected to become dressed with virtual excitations. This regime is difficult to capture with some traditional methods because of the explosion in the number of Matsubara frequencies, i.e., exponential terms in the freebath correlation function. To access this regime we generalize both the hierarchical equations of motion and pseudomode methods, taking into account this explosion using only a biexponential fitting function. We compare these methods to the reaction coordinate mapping, which helps show how these sometimes neglected Matsubara terms are important to regulate detailed balance and prevent the unphysical emission of virtual excitations. For the pseudomode method, we present a general proof of validity for the use of superficially unphysical Matsubaramodes, which mirror the mathematical essence of the Matsubara frequencies.
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Introduction
The spinboson model is a cornerstone of the theory of openquantum systems, and its elegance often belies its power to describe a wide range of phenomena^{1,2,3}. It not only allows us to understand the relationship between quantum dissipation and classical friction, but is a powerful model to study topics ranging from physical chemistry to quantum information. Practically speaking, a number of perturbative approaches and assumptions such as the Born–Markov and rotatingwave approximation (RWA) are usually employed to obtain tractable solutions^{2}. However, research areas, such as energy transport in photosynthetic systems^{4,5,6,7,8,9}, quantum thermodynamics^{10,11}, and the ultrastrong coupling regime in artificial lightmatter systems^{12,13,14,15,16,17}, have demanded the development of numerically exact methods to explore nonperturbative and nonMarkovian parameter regimes^{18,19,20}, which are out of reach of traditional approaches.
In the limit of a discrete environment consisting of a single bosonic mode, as arises in cavity QED (cQED)^{21}, the nonperturbative limit, in which the coupling is a significant fraction of the cavity frequency, is sometimes referred to as the ultrastrong coupling (USC) regime^{16,17}. This regime harbors a range of new physics, including higherorder coupling effects, the possibility to excite two atoms with one photon^{22}, the ability to prepare Bell and GHZ states in cQED^{23}, and the potential to generate a ground state which contains virtual excitations^{12,24,25,26}. In the latter, the excitations are called virtual because they are energetically trapped in the hybridized lightmatter ground state. A correct theoretical understanding of this trapping, such that unphysical emission from the ground state is avoided, was only developed recently^{27}. It is now understood that nonadiabatic external forces must be applied to transmute them into real, observable, excitations^{24,28,29,30,31}.
Numerical simulations^{32,33,34} have suggested that a similar phenomenon occurs when a spin, or twolevel system, is ultrastrongly coupled to a continuum environment (i.e., an infinite number of bosonic modes), as traditionally described with the spinboson model^{25}. This scenario is now becoming experimentally accessible in onedimensional transmission lines^{19,35}, and superconducting metamaterials^{36,37,38}. In addition, even at nonzero temperatures, it has been shown that virtual excitations can influence the efficiency of a photosynthetic quantum heat engine^{39}.
Methods and techniques are needed which are both quantitatively accurate and able to provide a qualitative understanding of the physics in this nonperturbative lowtemperature regime. However, many successful numerical methods used to study the spinboson model away from the normal perturbative limits, such as the hierarchical equations of motion (HEOM) technique^{40,41} and the pseudomode method^{42,43}, are limited in their ability to access low temperatures.
Here, we go beyond these restrictions and show how generalized versions of the HEOM and pseudomode methods can help us understand the nature of the ground state in the continuum, and explain how virtual excitations are trapped therein. In particular, in our generalization of the pseudomode method, the original continuum environment is replaced by a discrete set of modes^{42} which not only help to quantitatively describe the correct lowenergy nonperturbative physics but also help simplify the model to the essential mathematical elements needed to give a physical intuition about properties of the ground state. In addition, the methods we develop herein may also enable the exploration of virtual processes in quantum field theory, which are usually considered not physically accessible^{44}. They can also assist in the exploration of new physics and the development of applications in coupled lightmatter systems^{16,17}, and allow the modeling of complex lightharvesting processes in new parameter regimes^{9}.
We begin with an overview of our main results. We then introduce the spinboson model and freebath correlation functions, and provide an intuitive explanation of why omitting the apparently negligible Matsubara terms can have large consequences, even in the weak coupling regime. Next we demonstrate our correlation function fitting method for the HEOM, before turning to the pseudomode method and the reaction coordinate mapping to more transparently explain what happens when Matsubara terms are ignored in the ultrastrong coupling regime. Finally, we compare all three methods, with and without Matsubara contributions, and show their predictions for the dynamics and steadystate occupation of certain environment modes.
Results
Overview
We now summarize our main results in detail. As mentioned, using the HEOM technique in the lowtemperature limit is difficult^{45,46,47}, typically making this regime inaccessible. This is because the HEOM relies on a decomposition of the bath correlation function into a sum of exponentials. Unfortunately, due to the physical constraint disallowing Hamiltonians unbound from below (i.e., that the environment only consists of positive frequency modes), even a simple Lorentzian spectral density gives correlation functions which cannot be analytically decomposed into a finite sum. The same restriction has historically applied to the pseudomode method^{42,43}, as we will describe below.
To overcome this difficulty, we separate the correlation function into an analytical part, comprised of a finite number of exponentials, and the Matsubara part, given by an infinite sum of exponentials (the latter of which was neglected in other works studying the zerotemperature limit of the HEOM method^{48,49}). In the zerotemperature limit, we analytically integrate the infinite sum, and then fit it with a biexponential function. Fitting the total correlation function to exponentials for use with the HEOM has also been explored in refs. ^{45,46,47,50}, but our approach allows us to limit the fitting error^{51} to the Matsubara component, and gives us physical insight into the role of the different contributions to the correlation function. The fitting inevitably introduces some error in the system dynamics, which we analyze in detail in Supplementary Notes 3 and 4.
By comparing results with and without this Matsubara contribution, we find that the neglect of the Matsubara terms in both the HEOM formulation, and a generalized pseudomode method, induces a very specific error in the dynamics and steady state. This error corresponds to an unphysical system temperature, even at weak coupling, due to violation of detailed balance. Conversely, when including the Matsubara terms, detailed balance is restored, albeit with a finite error due to the fit. In the ultrastrong coupling regime, we find that, via comparison with the reaction coordinate method^{10,52,53,54} neglecting the Matsubara terms leads to an unphysical emission of photons from the ground state of the coupled lightmatter system (to which we will refer as our main example).
In generalizing the pseudomode method, which employs the fit of the Matsubara frequencies in the form of two additional zerofrequency Matsubara modes with nonHermitian coupling to the system, we find that it can exactly reproduce the full HEOM results for all parameter regimes. It can also be used to give meaning to the auxiliary density operators (ADOs) of the HEOM, indicating a strong relationship between the two approaches. To account for the unusual form of the Matsubara modes, we explicitly generalize the proof of validity of the pseudomode method^{42,43}. Our derivation shows that by combining the nonHermitian Hamiltonian together with what we call a pseudoSchrödinger equation, the Dyson equation for the reduced dynamics of the system is formally equivalent to one where the system is physically interacting with the original continuum environment.
The spinboson model
The iconic spinboson model considers a twolevel system (the spin, or qubit) in a bath of harmonic oscillators with the total systembath Hamiltonian given by (setting ħ = 1 throughout):
where ω_{q} is the qubit splitting, ω_{k} is the frequency of the kth bath mode, Δ is the tunneling matrix element, σ_{z(x)} are the Pauli matrices acting on the qubit. For later use, we define \(\bar \omega = (\omega _{\mathrm{q}}^2 + \Delta ^2)^{1/2}/2\), as the free qubit eigenfrequency. The kth mode of the bath, associated with annihilation operators b_{k}, interacts with the qubit via the operators \(\tilde X_k = g_k/\sqrt {2\omega _k} (b_k + b_k^\dagger )\) in terms of the couplings g_{k}, so that \(\tilde X = \mathop {\sum}\nolimits_k {\tilde X_k}\).
The effect of the bath can be considerably simplified when the initial state of the environmental modes is Gaussian, and in a product state with the system (the qubit). Specifically, we assume the bath to be in a thermal state at a temperature T. In this case, the influence of the environment is contained in the twotime correlation function \(C(t) = \langle \tilde X(t)\tilde X(0)\rangle\). The correlation function of the free bath, when it is not in contact with the system, can be written (in the continuum limit) as,
Here, \(J(\omega ) = \pi \mathop {\sum}\nolimits_k {g_k^2} /2\omega _k\delta (\omega  \omega _k)\) is the spectral density, which parameterizes the coupling coefficients g_{k}, and β = 1/k_{B}T is the inverse temperature. Throughout this article, we focus on the following underdamped Brownian motion spectral density,
which is characterized by a resonance frequency ω_{0}, a width γ, and a strength λ. A spectral density of this form is a convenient basis, in which one can represent a range of other spectral densities^{55,56}.
In the underdamped limit (γ < 2ω_{0}), it is convenient to decompose the correlation function, for Eq. (3) in Eq. (2), as C(t) = C_{0}(t) + M(t), where
in terms of \(C_0^{\mathrm{R}} = \coth \left[ {\beta (\Omega + i\Gamma )/2} \right]\exp (i\Omega t) + {\mathrm{H}}{\mathrm{.c}}{\mathrm{.}}\) (where H.c. denotes Hermitian conjugation) and \(C_0^{\mathrm{I}} = e^{  i\Omega t}  e^{i\Omega t}\), and
with the definitions Γ = γ/2, \(\Omega ^2 = \omega _0^2  \Gamma ^2\), and \(\epsilon _k = 2\pi k/\beta\) (k ∈ ℕ) for the Matsubara frequencies.
Intuitively, the C_{0}(t) part of the correlation function characterizes the resonant part of the bath, with a shifted resonant frequency Ω and decay rate γ/2. On the other hand, the M(t) part of the correlation function seems to have a less transparent description: it has no resonances but infinite subcontributions which decay at rates equal to \(\epsilon _k\) (hence we will name it the Matsubara correlation). One way to explore its meaning is to study what happens to the qubit dynamics after imposing C(t) → C_{0}(t), i.e., completely neglecting it. Note that this will induce an error even at zero temperature (β → ∞) due to the competition between the factor β^{−1} and the Matsubara frequencies approaching the continuum.
To proceed with our intuitive analysis, it is worth considering the Fourier transform of the correlation function, i.e., the power spectrum \(S(\omega ) = {\int}_{  \infty }^\infty {\mathrm{d}} t\,C(t)e^{i\omega t} = J(\omega )[1 + \coth \,(\beta \omega /2)]\). From this expression, it is possible to check that the powerspectrum encodes the symmetry condition
When the coupling to the environmental degrees of freedom is small compared with the qubit eigenfrequency \(\bar \omega = (\omega _{\mathrm{q}}^2 + \Delta ^2)^{1/2}/2\), the effect of the bath can be studied perturbatively (for example by using the Fermi golden rule). In this case, the qubit will absorb (relax) energy from (into) the environment at rates proportional to \(S(  \bar \omega )\) (\(S(\bar \omega )\)) so that Eq. (6) encodes the physical meaning of the detailed balance condition. As a consequence, by neglecting the Matsubara correlations, we are then going to break this balance^{57,58,59}. Nevertheless, the qubit will still reach an equilibrium thermal state at the effective temperature
where \(S_0(\omega ) = {\int}_{  \infty }^\infty {\mathrm{d}} t\,C_0(t)e^{i\omega t}\). The relation between β_{eff} and the actual temperature β intuitively quantifies the effect of the Matsubara correlations when the coupling to the environment is very weak.
On the other hand, when the coupling with the environment starts to be a significant fraction of the system eigenfrequency, hybridization effects between the system and the bath become relevant. As it will be shown in a later section, the Matsubara correlations are essential to be able to correctly model both the nonMarkovian and the equilibrium properties in this parameter regime (and which, in this case, were encoded in the detailed balance condition). We first describe the HEOM, and how the Matsubara term can be included, even at zero temperature, with a fitting approach.
The hierarchical equations of motion
The HEOM method can in principle describe the exact behavior of the system in contact with a bosonic environment, without approximations. The derivation can be found in refs. ^{40,41,48}, and the general procedure can be described as follows. Using the Gaussian properties of the free bath, one can write down a formally exact timeordered integral for the reduced state of the system (or equivalently, a pathintegral representation). This is difficult to solve directly. However, by assuming that the free bath correlation functions can be written as a sum of exponentials, one can take repeated time derivatives to construct an exact series of coupled equations describing the physical density matrix, and auxiliary ones encoding the correlations between system and environment. These can be truncated at a level that gives convergent results.
The problem then lies in parameterizing the correlation functions of a given physical bath with a sum of exponentials. In practice, one can either fit^{46,47,50} the correlation functions directly with exponentials or fit the spectral density using a sum of overdamped (DrudeLorentz) or underdamped Brownian motion spectral densities^{7,55,56}. However, for the latter, as one might expect from the discussion so far, the Matsubara frequencies in Eq. (5) become increasingly important at low temperatures. These frequencies, in the HEOM, are numerically challenging to take into account due to the increasing number of auxiliary density operators^{45,60} (though using an alternative Padé decomposition with the HEOM has been explored as a way to optimally capture the influence of these terms^{61}).
In the zerotemperature (β → ∞) limit, the Matsubara frequencies \(\epsilon _k = 2{\mathrm{\pi}} k/\beta\) approach a continuum, i.e., 2π/β → dx → 0 for 2πk/β → x. As a consequence, we can represent the Matsubara correlation in Eq. (5) as the integral
However, this integral representation does not give a direct solution in exponential form. Using a fitting procedure, we have found that we can capture the influence of these terms with a biexponential function,
where c_{m} and μ_{i} are real (for the choice of Matsubara decomposition we use here). Adding more exponential terms increases the accuracy of the fit only marginally for the parameter ranges we study here. In addition, each exponent leads to a large numerical overhead with the HEOM method, thus one would like to keep the number of exponents to a minimum. In Fig. 1, we give an example of the fitting of the correlation function.
Given the above decomposition, we can finally write the full equations of motion. However, a fully generic formulation of the HEOM^{46} treats the real and imaginary parts of the correlation function separately, which turns the (for β = ∞) single nonMatsubara exponent in Eq. (4) into four exponents. It is more numerically convenient to reduce these to two exponents, following ref. ^{48}, by defining (again, only for β = ∞ for notational simplicity) the new parameters c_{3} = λ^{2}(1 − i)/4Ω, c_{4} = λ^{2} (1 + i)/4Ω, μ_{3} = −iΩ + Γ, and μ_{4} = iΩ + Γ. Meanwhile, as described above, the Matsubara terms are entirely real, and given by Eq. (9).
In the HEOM itself, we denote the physical and auxiliary density matrices as \(\rho _{\bar n}\) where \(\bar n = [n_1,n_2,..,n_K]\), (where here K = 4), is a multiindex composed of nonnegative integers n_{k}. The physical density matrix of the system, traced over the environment, is given by \(\rho _{\bar 0} = \rho _{[0,0,...,0]} \equiv {\mathrm{Tr}}_E(\rho _T)\). Any other index denotes an auxiliary density operator which encodes the correlations between system and environment, as we will discuss later. We use \(\rho _{\bar n_{k^ \pm }}\) to denote a higherorder ADO, which differs from \(\rho _{\bar n}\) in the kth index by ±1. For instance, \(\rho _{0_{2^ + }} = \rho _{[0,1,0,...,0]}\). The equations of motion given by HEOM can be compactly written as
where \({\cal{L}}\rho = [H_s,\rho ]\) and the \({\cal{L}}_k^ \pm\) are Liouville space operators, depending on the spinbath coupling operator and the exponential decomposition of the correlation function^{40,41} given by \({\cal{L}}_k^  \rho _{\bar n_{k^  }} = n_k(c_k^R[Q,\rho _{\bar n_{k^  }}] + c_k^I\{ Q,\rho _{n_{k^  }}\} )\) and \({\cal{L}}_k^ + \rho _{\bar n_{k^ + }} = [Q,\rho _{n_{k^ + }}]\). Note again that this is not a generic construction^{46}, but is specific for the choice of decomposition of correlation functions we use here.
Environment as a discrete set of modes
Before discussing results predicted by the HEOM, it is useful to consider two complementary methods based on discrete decompositions of the environment. The idea that the behavior of an infinite continuum environment can be described by a finite set of discrete modes arises in both the methodology of pseudomodes^{42,43,62,63,64} and the socalled reaction coordinate mapping^{52,53,54}. The former is based on the identification of frequencies in the correlation functions which are then assigned to a set of unphysical pseudomodes^{42,43}. In contrast, the reaction coordinate (RC) method is instead based on a formal mapping of the full Hamiltonian environment Hamiltonian to a single reaction coordinate and a residual (perturbative) environment.
Pseudomodes model
As shown in the seminal work of Garraway^{42} (and recently confirmed and generalized in ref. ^{43}), as long as the free correlation function of a discrete set of modes accurately reproduces the correlation function of the full bath, their effect on a given system should be identical, a concept that recalls in spirit Baudrillard: “The simulacrum is never that which conceals the truth—it is the truth which conceals that there is none. The simulacrum is true.”^{65}.
From the discussion so far, and the generalized proof in ref. ^{43}, it is evident that we can capture the full correlation function of the free environment, Eq. (2), with a single underdamped mode for the nonMatsubara part Eq. (4), and two additional modes, from the fitting procedure Eq. (9), which capture the Matsubara frequency contributions Eq. (5). By construction, at zero temperature, the resulting dynamics of the system coupled to these effective modes should obey the total Hamiltonian,
Here, ζ_{1} = Ω, \(\Omega = \sqrt {\omega _0^2  \Gamma ^2}\), ζ_{2} = ζ_{3} = 0, \(\lambda _1 = \lambda /\sqrt {2\Omega }\), \(\lambda _2 = \sqrt {c_1}\), \(\lambda _3 = \sqrt {c_2}\) (where c_{1} and c_{2} are the coefficients of the fitted Matsubara terms in Eq. (9), and ζ_{2} = ζ_{3} = 0 because Eq. (9) contains no oscillating components).
The damping of each pseudomode is simply described by a Lindbladian with the corresponding loss rate,
where \({\cal{G}}_1 = {\mathrm{\Gamma }}\), \({\cal{G}}_2 = \mu _1\), \({\cal{G}}_3 = \mu _2\).
Note that the couplings λ_{2} and λ_{3} between the pseudomodes associated with the Matsubara terms and the system are complex (since c_{1} and c_{2} are required to be negative), and thus the above Hamiltonian is strangely nonHermitian^{66}. This situation is not immediately covered by the general proof in ref. ^{43}. We extend that proof in Supplementary Note 6, and show that, to properly take into account the negative c_{1} and c_{2}, the dynamics of the system has to be computed by solving the following equation of motion for the density matrix ρ (which, throughout this article, will be referred to as the pseudoShrödinger equation for simplicity)
where \(D[\rho ] = \sum_{i = 1}^3 {D_i} [a_i]\). The adjective “pseudo” not only refers to the pseudomodes in question but also to the fact that, when H_{pm} is nonHermitian, we are purposely not taking the Hermitian conjugate when H_{pm} acts on the right of ρ.
While we refer to Supplementary Note 6 for a detailed justification, given the nonHermitian nature of the Hamiltonian in Eq. (11), it is worth presenting here a sketch of the proof.
Following a parallel strategy to the one presented in ref. ^{43}, it is possible to show that the dynamics of observables in the system+pseudomodes space (obtained by solving the pseudoShrödinger equation above), is equivalent to a reduced pseudounitary dynamics, in which each pseudomode is coupled to a bosonic environment under a rotatingwave approximation and with a constant spectral density (defined for both positive and negative frequencies).
As mentioned, the prefix pseudo refers to the fact that the Hermitian conjugate is never taken when considering equation of motion for the density matrix. From this auxiliary model, the reduced system’s dynamics can be obtained through a Dyson equation. When the pseudomodes and their environments are in an initial Gaussian state, this equation is fully specified by the twotime correlation function of the coupling operator \(\sum_{i = 1}^3 {\lambda _i} (a_i + a_i^\dagger )\).
The advantage of considering a nonHermitian Hamiltonian together with a pseudoSchrödinger equation in this derivation is that, by doing so, the Dyson equation for the reduced dynamics of the system is formally equivalent to one where the system is physically interacting with a single environment via a Hermitian coupling operator characterized by the same correlation function C_{0}(t) + M_{biexp}(t). This completes the proof.
To summarize, the reduced system dynamics computed from Eq. (13) is equivalent to that of the original spinboson model, Eq. (1), under the assumption (or, in our case, approximation, due to the fitting procedure used to capture the Matsubara terms) that the correlation in Eq. (2) has the form
Remarkably, we will see in a later section that this pseudomode model precisely reproduces the results of the HEOM model, both when the Matsubara frequencies (modes) are neglected, and when they are included, and it also allows for an interpretation of the auxiliary density matrices in the HEOM. In addition, the latter suggests that the HEOM can be derived, in some cases, from the pseudomode model itself (akin to the dissipaton model introduced by Yan^{67}). It is also interesting to note that like the HEOM^{68}, and unlike a normal Lindblad master equation, Eq. (13) does not guarantee complete positivity because of the nonHermitian couplings. In Supplementary Note 8, we discuss this in detail, and provide criteria for guaranteeing complete positivity in terms of the parameters in the fit.
We finish this section with a brief note on the effect of neglecting the Matsubara correlations, i.e., in considering the approximation \(C(t) \mapsto C_0(t)\). In this case, only a single pseudomode is needed, i.e., i = 1 in Eqs. (11) and (12). Alternatively, as we show in Supplementary Note 7, this single pseudomode can be understood as mediating the interaction between the system and a residual bath of bosonic modes (with annihilation operator f_{k} and frequency \(\omega _k \prime\)) with the Hamiltonian
where the couplings \(g_\alpha \prime\) describing the interaction with the residual environment are characterized by the spectral density J_{Mats}(ω) = γΩ and defined for both positive and negative frequencies. This system has an interesting relation to another technique used to model the spinboson model: the reaction coordinate mapping.
Reaction coordinate (RC) mapping
Returning to the full spinboson Hamiltonian, in the reaction coordinate approach a unitary transformation maps the environment to a singlemode reaction coordinate and a residual bath. As discussed in^{10,54,69}, for the underdamped Brownian motion spectral density the new Hamiltonian is
where the residual bath, described by operators d_{k}, with frequencies \(\omega _k {\prime\prime}\) and couplings \(g_k {\prime\prime}\), has an Ohmic spectral density J_{res}(ω) = γω. Importantly, this Hamiltonian is still “exact”, and properties of the RC mode are related to the original environment^{54} via
Using this new degree of freedom, for small γ, such that a Born–Markovsecular approximation for the residual bath is valid, one can derive a new master equation which describes the dynamics of the system coupled to the reaction coordinate, and which preserves detailed balance by definition [see Supplementary Eq. (3) in Supplementary Note 1].
As can be seen by direct comparison, the Hamiltonians in Eqs. (15) and (16) are related by a rotatingwave approximation (RWA) and a Markov approximation in the coupling between the reaction coordinate and its residual environment (and a renormalization of the RC frequency). In Supplementary Note 1, we present an alternative intuitive argument showing why applying a RWA and Markov approximation leads to a correlation function without Matsubara terms. Subsequently, deriving a master equation for the residual environment under these conditions leads to one which does not conserve detailed balance as explicitly shown in Supplementary Eq. (7).
Overall this suggests that the Matsubara frequencies play two roles: first of all, they restore detailed balance, both on the level of the system (in the weak coupling regime, as expected), and also on the level of the system and RC mode (in the strong coupling and narrowbath regime). Secondly, beyond the weak coupling and narrowbath regime, they describe the nonnegligible influence of “background” modes in the environment not captured by the reaction coordinate itself (e.g., strong correlations with the residual bath).
Virtual excitations in the ground state
Before discussing the strong coupling limit, we first consider the weak coupling case and illustrate how neglecting the Matsubara terms leads to an artificial temperature. In Fig. 2, we plot the probability for the qubit to be excited in the steady state ρ_{11}(t) = 〈1ρ(t → ∞)1〉 (where 1〉 is the excited state of the free qubit Hamiltonian) as a function of qubit frequency \(\bar \omega\). We immediately see that both HEOM and pseudomode approaches give a steadystate population identical to that suggested by Eq. (7) when the Matsubara terms are neglected. Similarly, by fitting the Matsubara terms, and introducing them into both methods, we find that the population drops close to zero, as expected. The residual error in the fit produces a deviation from the expected β = ∞ ↔ ρ_{11}(t → ∞) = 0, which becomes large as the qubit frequency becomes small, and hence more sensitive to the residual effective temperature.
As we increase the coupling, the detailed balance condition in terms of the bare qubit Hamiltonian and the original bath temperature in Eq. (7) is no longer a good measure of the fit. This is because in the ultrastrong coupling regime, when the effect of the environment on the system is nonperturbative, socalled virtual excitations can become important^{25}. For example, in this scenario, the hybridized system–environment ground state (which in principle should be the steadystate at zero temperature) contains a finite population of photons which cannot be directly observed (or emitted into other modes or environments).
In our treatment of the ultrastrong coupling regime, we find that the Matsubara terms are crucial to obtain the correct photon population in a single collective mode, and trap that population. In order to show this, we first consider the RC picture where the collective bath coordinates are defined in terms of a single mode^{54}, as per Eq. (17). The RC mapping gives a very clear picture of the dominant influence of the environment in terms of the collective RC mode such that any virtual or real photon population of the collective mode is given by the expectation of the number operator \(\left\langle {a^\dagger a} \right\rangle\) (though this does not directly correspond to the original bathmode occupation).
Can a similar quantity be extracted from the HEOM? It has been shown^{70,71} that higherorder moments of the total bath coupling operator can be extracted from certain combinations of auxiliary density operators returned by the HEOM. Similarly, for a single undamped mode, ref. ^{72} showed that the population is given by the secondlevel auxiliary density matrix. In our case, we can extract populations that correspond to precisely those of the pseudomodes (see Supplementary Note 2). For example, the occupation of the first pseudomode is given by
It is clear then that the ADOs and the pseudomodes bear a close relationship.
As we can see in Fig. 3 (starting from the initial condition of a zerotemperature environment, and the qubit in the ground state of the free system Hamiltonian), in the absence of the Matsubara terms, the population of the excited state of the twolevel system (see Supplementary Fig. 1), and the population of the a_{1} mode predicted by the HEOM from Eq. (18) match closely that of the RC model with the approximation of the RWA for the RCresidual bath coupling and a flatresidualbath approximation [described by Supplementary Eq. (7)]. In this case, the population increases to a steady state which can be ascribed to the artificial nonequilibrium situation induced by neglecting the Matsubara correlation. In the RC model without Matsubara contributions, since the state ρ(t) of the qubit and RC mode evolves through the Lindblad equation shown in Supplementary Eq. (7), the rate of energy dissipation into the residual environment is given, in terms of the bare annihilation operator a, by
i.e., proportional to the average photons in the steady state. However, we know that this emission is unphysical, as it both violates detailed balance and energy conservation.
In contrast, the addition of the Matsubara terms to the HEOM, the addition of the Matsubara modes to the pseudomode model, and the corresponding removal of the unphysical assumptions in the RC model, results in dynamics in all three cases which tend toward a steady state which is close to the ground state of the coupled systemRC Hamiltonian. In this case, the HEOM and pseudomode model match exactly, while the RC model gives a qualitative agreement. This trend is one of our primary results: the addition of Matsubara terms to the HEOM (or equivalently Matsubara modes to the pseudomode model) restores detailed balance in terms of the coupled systemRC Hamiltonian (up to a residual error from the fit) and traps photons in an effective ground state, as confirmed by the RC model. In this case, the state ρ(t) of the qubit and RC mode evolves through the Lindblad equation shown in Supplementary Eqs. (1) and (3) characterized by jump operators between eigenstates. As a consequence, since the steady state is the ground state, there is no steadystate energy dissipation [see Supplementary Eq. (4)] into the residual bath.
As γ is increased, cf. Fig. 4, we see a deviation between HEOM and RC models (see also Supplementary Fig. 1 for a comparison of system populations, and Supplementary Note 5 for a discussion of the steady state as a function of coupling strength). For strong coupling and broad baths, the Matsubara terms become more relevant, as does the error arising from the fitting procedure. In Supplementary Note 4, we perform an error analysis which suggests that the difference between the RC results and the HEOM results exceed potential errors arising from the fit. Thus, we primarily ascribe this difference to the breakdown of the perturbative approximation for the residual bath in the RC model, which becomes more pronounced as γ is increased.
One might attribute the difference to the fact that the RC model does not take into account the frequency shift that we see in Eq. (4). However, phenomenologically solving for the ground state of the system coupled to an RC mode with renormalized frequency Ω actually predicts a larger population (shown by the red dotdashed line in Fig. 4) than the normal systemRC ground state due to the decreased frequency of the nonMatsubara mode^{25}. In addition, the predicted population is also larger than the full HEOM/pseudomode results, which suggests that, as γ is increased, the correlations between the system and the pseudomodes associated with the Matsubara frequencies become stronger, and actually reduce the population in the nonMatsubara pseudomode^{25}. However, without the RC model to guide us with a physical interpretation in this limit, it becomes difficult to associate the populations of the Matsubara modes to real physical modes, collective or otherwise^{53,73,74,75,76}. In fact, as described earlier, since their contribution to the correlation functions of the bath is negative in the parameter regimes we consider here, in the pseudomode model their coupling to the system is nonHermitian, accentuating their nature as simulacra.
Discussion
We have analyzed the dynamics and steadystate properties of the zerotemperature spinboson model in the strong and ultrastrong coupling regime using three different techniques. We showed that the Matsubara terms, taken into account with a fitting procedure in the HEOM and pseudomode methods, restore detailed balance (albeit up to a residual error), even in the ultrastrong coupling regime. This was validated by a comparison with the reaction coordinate method, which also indicates that the Matsubara terms are important for the correct “trapping” of virtual excitations in the collective ground state.
Simultaneously, we showed that a pseudomode model can exactly capture the same dynamics as the HEOM, and can take into account negative contributions to the correlation functions, like the Matsubara frequencies, via a pseudoSchrödinger equation.
Our results also elucidate the relationships and differences between the three methods employed herein, particularly the strong relationship between the pseudomode treatment and the HEOM.
Future work includes generalizing to arbitrary spectral densities for systems such as superconducting qubits coupled to transmission lines (with potentially structured environments^{77}), and photosynthetic complexes^{4,5,6,7,8,9}. In addition, in the broadbath limit, it may be possible to assign direct physical meaning to the ADOs of the HEOM, and the Matsubara modes of the pseudomode method, by comparison with bosonicchain mappings of the environment^{53,73,74,75,76,78}, in the same way the RC mapping guides us in this work. This might allow, for example, inspection of spatial dependencies of the photon population, as revealed by other methods^{32,33}. We hope that these insights can help toward a better understanding of ultrastrong coupling at zero temperature in continuum systems, and emphasize the impact of the positive frequency nature of many physical environments (and the resulting appearance of Matsubara frequencies).
Data availability
Dataset sharing is not applicable to this article as no data sets were generated or analyzed during this study.
Code availability
The numerical code used to generate most of the figures in this work is available at https://doi.org/10.5281/zenodo.3294068 and https://github.com/pyquantum/matsubara, and is documented at https://matsubara.readthedocs.io/. It is available to use under the MIT license, and uses the opensource library QuTiP^{79,80}
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Acknowledgements
We would like to thank Stephen Hughes for helpful suggestions on the pseudomode approach, and Ken Funo, David Zueco, Simone De Liberato, HuanYu Ku, and Fabrizio Minganti for feedback and comments. We would also like to thank Fabio Mascherpa for bringing to our attention their recent work^{78}. S.A. was supported by the RIKEN IPA program. N.L. acknowledges support from JST PRESTO, Grant no. JPMJPR18GC. N.L. and F.N. acknowledge support from the RIKENAIST Joint Research Fund and the Sir John Templeton Foundation. F.N. is partly supported by the MURI Center for Dynamic MagnetoOptics via the Air Force Office of Scientific Research (AFOSR) (FA95501410040), Army Research Office (ARO) (Grant no. W911NF1810358), Asian Office of Aerospace Research and Development (AOARD) (Grant no. FA23861814045), Japan Science and Technology Agency (JST) (the QLEAP program, and CREST Grant no. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (JSPSRFBR Grant no. 175250023, JSPSFWO Grant no. VS.059.18N). M.C. acknowledges support from NSAF no. U1730449.
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N.L. and S.A. contributed equally to the development of the numerical simulations. M.C. performed analytical calculations, and developed the proof of the pseudomode method. N.L. conceived the project. N.L and F.N. supervised the research. All authors discussed the results and contributed to writing the paper.
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Lambert, N., Ahmed, S., Cirio, M. et al. Modelling the ultrastrongly coupled spinboson model with unphysical modes. Nat Commun 10, 3721 (2019). https://doi.org/10.1038/s41467019116561
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DOI: https://doi.org/10.1038/s41467019116561
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