Fr\"olich-coupled qubits interacting with fermionic baths

We consider a macroscopic quantum system such as a qubit, interacting with a bath of fermions as in the Fr\"olich polaron model. The interaction Hamiltonian is thus linear in the macroscopic system variable, and bilinear in the fermions. Using the recently developed extension of Feynman-Vernon theory to non-harmonic baths we evaluate quadratic and the quartic terms in the influence action. We find that for this model the quartic term vanish by symmetry arguments. Although the influence of the bath on the system is of the same form as from bosonic harmonic oscillators up to effects to sixth order in the system-bath interaction, the temperature dependence is nevertheless rather different, unless rather contrived models are considered.


I. INTRODUCTION
The theory of open quantum systems has attracted increased attention in recent years, motivated by advances quantum information theory [1] and emerging quantum technologies [2,3]. For these to become practically useful in a broad range of applications a main roadblock to overcome is the strong tendency of large quantum systems to turn classical due to interactions with the rest of the world [4][5][6]. Open quantum systems encompass the various concepts and analytic and numerical techniques that have been developed to describe and estimate the development of a quantum system interacting with an environment [7,8].
A special place in open quantum system theory belongs to problems where a general system (the system of interest) interacts linearly with one or several baths of harmonic oscillators. One reason is that resistive elements in a small electrical circuit can be modeled as many LC elements in parallel, of which each one obeys the equation of a harmonic oscillator. At very low temperature as in quantum technology applications, these harmanic oscillators should be quantised [9]. A related reason is the number of physical environments (phonons, photons) that can also be directly described this way. In the Lagrangian formulation of quantum mechanics [10] the development of a wave function (unitary operator U ) is described by a path integral, while the development of a density matrix (quantum operation U · U † ) is described by two path integrals, one (forward path) for U and one (backward path) for U † . A third reason why harmonic oscillator baths are interesting is that the paths of such baths can be integrated out yielding the famous * eaurell@kth.se † jan.tuziemski@fysik.su.se; On leave from Department of Applied Physics and Mathematics, Gdansk University of Technology Feynman-Vernon theory [11]. The only trace of the bath (or baths) is then the Feynman-Vernon action, quadratic terms in the forward and backward paths. Nevertheless, most physical environments do only approximately or not at all consist of degrees of freedom that can be described as bosonic harmonic oscillators. Conduction band electrons in normal metals are for instance obviously fermions. Even if these fermions by themselves are free (and hence can be treated as fermionic harmonic oscillators), in the open quantum system context it is their interaction with the system of interest that counts. If that system is a macroscopic quantum variable such as a qubit, the simplest interaction that can be considered is quadratic in the fermionic variables and linear in the system of interest. As a term in an interaction Hamiltonain that is Xab where X is the macroscopic quantum variable of the system of interest and a and b are creation or destruction operators of the fermions. Interaction Hamiltonians of this type appear in the Frölich polaron model of the motion of a conduction electron in an ionic crystal [12,13]. In Feynman's variational treatment, one electron is modelled as a non-relativistic particle interacting with a bath of bosonic harmonic operators which are then integrated out. Here we are interested in the opposite case where one bosonic degree of freedom, i.e. the qubit, describes the system of interest, and we want to "integrate out" the fermions. One problem with such an approach is that fermionic functional integrals (Grassman integrals) are mathematically non-trivial objects. Another is that for the Frölichlike coupling both the bath Hamiltonian and the interaction are quadratic in the fermionic degrees of freedom; the result is hence two fermionic functional determinants depending on the forward and backward histories of the system of interest acting as external fields.
An approach to similar problems, used for a long time in condensed matter theory, is Keldysh techniques [14,15].
While essentially equivalent to Feynman-Vernon theory, Keldysh theory was developed for other applications, and encompassing from the start fermionic baths. The kernels of the quadratic terms in Feynman-Vernon theory can thus be identified with pair-wise bath correlation functions, in Keldysh theory referred to as "dressed non-equilibrium Greens functions". Here we will instead follow the recently developed extension of the Feynman-Vernon theory to non-harmonic baths [16]. One advantage of this approach is that it gives access also to terms in Feynman-Vernon influence functional higher than quadratic. Let us remark that from the functional integral point of view it is obvious that such terms must exist: while the bath can always be integrated out in principle, it is only for harmonic (bosonic or fermionic) baths that all the integrals are Gaussian and can be done in closed form.
A main result of [16] is that higher-order Feynman-Vernon terms depend on cumulants of bath correlation functions. The first non-standard term in the extended Feynman-Vernon theory for the dynamics of the macroscopic system hence involves fourth-order cumulants of the correlation functions of the compound bath variables ab i.e. eighth-order fermionic correlations. Perhaps suprisingly we find that for the Frölich-coupled system these terms actually cancel in the influence function.
The paper is organized as follows.
In Section II we state the problem and make general remarks of what one can expect of the solution. In III we assume as a concrete example that the macroscopic variable is a qubit (a two-state system) coupled to the bath as in the spin-boson problem, and state more precisely the system-bath interaction we study in the rest of the paper. In Section IV we present the structure of the first term of the Feynamn-Vernon action and state that the second term in the expansion of the action vanishes in our model. Here we also sketch calculations of bath correlation functions of interest in our theory. In Section V the standard (second order) Feynamn-Vernon action of the considered model is compared to that of a harmonic bosonic bath. Appendices A, B contain summaries of technical details from [16], included for completeness. Appendix C presents the detailed argument that in the model considered the fourth order cumulant vanishes, and therefore there is no fourth order contribution to the generalized Feynman-Vernon action.

II. STATEMENT OF THE PROBLEM
Let us consider a system consisting of one macroscopic bosonic variable and a bath of free fermions as discussed above. That means a Hamiltonian where the first termĤ S is the Hamiltonian of the system. For a bosonic variable the evolution operator correspond-ing toĤ S can be written as a path integral where S[X] is the action of path X. The evolution operator acting on density matrices is similarly a double path integral over a "forward path" and a "backward path" where the slot marks where the initial density matrix is to be inserted. The free bath Hamiltonian in (1) iŝ where c † k (c k ) is the creation (destruction) of fermions, and the interaction Hamiltonian is of the type (below we will use a more specific model) Initially the bath and the system are assumed independent, and the bath is in thermal equilibrium at inverse temperature β. The evolution operator of the system is the quantum map (or quantum operator) given by where U = e − i (HS +HINT +HB )τ is the total evolution operator of the combined system and bath, ρ B (β) is the initial equilibrium density matrix of the bath, and · marks where to insert the initial density matrix of the system. Suppose that the evolution of the bath can also be written as a double path integral. If so the bath can be integrated out, so that we have The new term compared to (3) is the Feynman-Vernon influence functional, i.e. what remains after integrating out the bath paths while the system paths are held fixed. Although important general properties of the influence functional were stated in [11], in practice this formalism has mostly been used for when the baths are free bosons interacting linearly with a system. In that case all the path integrals over the baths are Gaussians, and F can be written as are two explicit quadratic functionals of the forward and backward system paths, usually known as Feynman-Vernon action.
On the other hand, it is not necessary to assume that the bath can be represented as path integrals. As reviewed in [17] and rederived in [16], the super-operator Φ in (6) can be computed perturbatively, and the terms translated back to a double path integral over the system. In this way one can identify the kernels in the actions S i [X, Y ] and S r [X, Y ] as being equilibrium pair correlations in the bath. Importantly this holds for any equilibrium bath. The price to pay if the bath is not harmonic is that there are higher-order terms that are respectively fourth, sixth etc order in the system variables X and Y .

III. A QUBIT COUPLED TO A FERMIONIC BATH AS IN SPIN-BOSON PROBLEM
For concreteness, and since this would be a main application to quantum information science, we now assume that the system of interest is a a qubit (a two-state system) governed by a system Hamiltonian The evolution operator e − i HS τ can be represented by inserting resolution of the identity between very small time increments ∆τ . The first term in (8) then only contributes if the state stays the same between two small time increments; that contribution is e ± i ε 2 ∆τ . The parameter ε is hence the level splitting. The second term in (8) on the other hand only contributes if the state changes over a small time increment, and the contribution is (±i ∆ 2 ∆τ ). The parameter ∆/2, which has dimension of a rate, is hence the tunelling element.
The paths in X and Y in the path integral in (3) are nothing but a way to represent e − i HS τ and e i HS τ , and are hence piece-wise constant, equal to ±1. Before continuing we note a clash of conventions: X and Y are in the literature on open quantum systems used to refer to the history of a system variable which is intergrated over. In our case these are the histories (forward and backward) of a representation ofσ z , and X is also used for the system part of the interaction Hamiltonian. This is the convention we follow. In the quantum information literature X and Y instead refer to the operatorsσ x and σ y , while the operatorσ z is written Z. We do not follow this convention. Now, it is convenient to include the contributions from the level splitting in the actions in (3), and the contributions from the tunelling elements in the path measures DX and DY . If so DX and DY are nothing but the path probabilities of (classical) Poisson point processes, except that the jump rates are purely imaginary. That is, we can interpret X as s i , n, t 1 , . . . , t n where s i is the initial state (up or down), n is the number of jumps and t 1 < t 2 < . . . are the jump times. The purely imaginary path measures are then The advantage of the above is that it can accomodate also a coupling to a bath when that coupling is proportional to σ z . When the bath is composed of bosonic harmonic oscillators this is the spin-boson problem; the above path integral was developed by Leggett and collaborators for that problem in [18]. For our problem we will consider the interaction Hamiltonian is where X (σ z ) is the system part of the interaction, c † l , c † k (c k , c l ) are the creation (destruction) operators of two fermions, and g kl is a coupling constant. Due to the anticommutation rules for fermions we can set g kl = −g lk . For the following sections it is convenient to introduce an interaction representation based on (4) and (8). Bath destcruction operators transform as where ω i ≡ ǫ i / , and bath creation operators as c † Explicit form of the transformed system operator X in (10) is not relevant for further considerations. In this representation the interaction Hamiltonian is We also assume that the bath is initially in a thermal

IV. THE GENERALIZED FEYNMAN-VERNON ACTION TERMS
In most cases discussed in literature Feynman-Vernon action involves only term of the second order i.e. the one consisting of two variables of the system corresponding to forward and backward paths. This is because usually the environment can be modeled as a bath of free bosons linearly coupled to a system and, in order to trace out environmental degrees of freedom, one needs to perform a Gaussian integral, what results in the Feynman-Vernon influence functional that involves quadratic functionals of forward and backward paths. However, this is not always the case and for other type of baths and couplings, e.g. the fermionic bath considered here, higher order terms in the action appear. A systematic way of dealing with such situations was formulated in [16] and is summarized in Appendix A and B. In that approach the total Feynman-Vernon action is expressed as a sum of terms of different order (i.e. involving different number of system variables). In the Appendix B we show that expression for the the usual quadratic Feynman-Vernon action can be rewritten such that .] is a functional over paths of the system, which explicit form is given by Eq. (B5), X s and Y s are forward and backward path of the system evaluated at time s, and C(t 1 , t 2 ) is the bath correlation function. For the problem considered here it reads cos is thermal expectation value of fermionic operators. Derivation of the above result relies on two simple facts. The first that, in general, consecutive action terms depend on the following bath correlation functions The above is non-zero only if the number of all fermion indices k's and l's is even. For thee quadratic action term we find that the only non-zero contribution is from which Eq. (13) immediately follows. The next term in the action expansion involves fourth-order cumulant function multiplying appropriately time-ordered combination of four super-operators of the qubit. We show in Appendix C that for our model all those terms vanish and hence there is no fourth order contribution to the Feynman-Vernon action.

V. PHYSICAL ANALYSIS OF THE QUADRATIC ACTION TERMS
In this section we analyze the quadratic term of the action. Our aim is to compare it to the action for a harmonic bosonic bath that is linear coupled to a system. The easiest way of doing this is by rewriting the Feynman-Vernon action with the help of imaginary and real parts of the kernel k I (t) and k R (t) respectively. The general expression reads where X t , Y t correspond to forward and backward path of a system operator. For the fermionic bath considered here the kernels k I (t) and k R (t) are respectively: whereas for bosonic baths coupled linearly to the system (see e.g. [8]): (17) Let us now discuss differences and similarities between those expressions. Imaginary kernels modify action and hence describe dissipation. For harmonic bosonic baths imaginary kernel is temperature independent, what is not the case for the model consider here. However, we can consider two temperature regimes with simpler behavior. In the low temperature regime (β ≫ 1) the fermionic kernel resembles the bosonic one k I (s − u) ≈ 2i k,l g 2 kl sin [(ω k + ω l ) (s − u)], with frequency of a bosonic mode replaced by sum of frequencies of interacting fermions. In the opposite regime i.e. high temperatures (β ≪ 1) the dissipation kernel vanishes. On the other hand, the real kernel introduces noise and is responsible for decoherence process. In the bosonic case decoherence strength increases with temperature. For the fermionic model in the low temperature limit (β ≫ 1) the real kernel k R (s − u) ≈ −2 k,l g 2 kl cos [(ω k + ω l ) (s − u)] is similar to the bosonic one. However, magnitude of the fermionic kernel does not grow with temperature: The high temperature limit (β ≪ 1) of the real kernel reads As we can see, for the low temperatures the fermionic bath behaves similarly to the bosonic one and the differences between them are most important in the high temperature regime.

VI. DISCUSSION
In this paper we addressed the model of a macroscopic quantum variable such as a qubit interacting with a fermionic bath. The coupling between the qubit and the bath is quadratic in fermionic operators, and the bath is initially in a thermal state. To investigate this system we employed the extension of the Feynman-Vernon influence functional technique that allows to systematically study higher order contributions to the Feynamn-Vernon action that arise from system-bath interaction being non-linear with respect to bath operators. We explicitly computed the second order contribution to the Feynman-Vernon action. While this is the standard term having the same functional form also in the case of bosonic harmonic baths, the dependence on temperature will in general be different for a fermionic bath with two-fermion coupling. We identified one regime where nevertheless the fermionic environment mimics a bosonic one. Finally, we showed (details in appendix) that the fourth order terms in the generalized Feynman-Vernon influence action vanish for the model considered. The first non-zero corrections to Feynman-Vernon or Keldysh theory are hence of sixth order in the system-bath interaction coefficient. Here we briefly sketch how the cumulant expansion can be used to express influence of the bath on the system. Firstly we summarize necessary notation from [16]. That paper employs the super-operator approach to find dynamics of the system interacting with the environment: A map governing evolution of system operators is obtained by tracing out bath degrees of freedom from the formal solution of the full (i.e. including system and the bath) Liouvillevon Neumann equation. A crucial step in performing the trace is evaluation of multi-time superoperator correlation functions (correlation functions with indices) in the bath, which are defined in terms of ordinary bath correlation by where Q b (t i ) are time evolved bath operators from the interaction part of the Hamiltonian (in interaction picture) and ρ B (t 0 ) is the initial state of the bath. As the starting point was Liouvillevon Neumann equation, one needs to include indecies d 1 , · · · , d n to time-order the operators in two groups, one (d i = " < ") acting from the right on the bath density matrix in ascending time order, and other other (d i = " > ") acting from the left in descending time order.
Let us recall that, for the ordinary operator correlation functions, successive orders of cumulants (cluster expansion) are defined inductively as The only difference between standard and super-operator correlation functions is that the latter need to be time ordered of time as determined by the indices d 1 , . . . , d N . Once this is done one can write a general cumulant expansion as C d1,···dN (t 1 , · · · , t N ) = (all possible groupings) (groups of one time) where N can be even or odd, and where the times on the right-hand side are inserted after the re-ordering. All odd order cumulants vanish for a bath where the Hamiltonian is an even function (as in our case) and the second order cumulant is the same as the second order correlation function. The first non-trivial cumulant is then where we have retained the super-operator notation on the right-hand side.
All cumulants beyond G 2 vanish for correlation functions of (classical) Gaussian processes [19]. This also holds as for operator correlation functions of harmonic bosonic baths, because in the path integral language these are all determined by Gaussian integrals. Alternatively, all higher-order operator correlation functions are in a bath of free bosons by Wick theorem given by combinations of pairwise operator correlation functions, which give same expressions as the cumulants used here. For free fermions all higher-order correlation functions are also given in terms of pair-wise combinations of pair-wise correlation functions, but with signs, and therefore different from the cumulants used here.

Appendix B: Generalized Feynman-Vernon actions
This Appendix summarizes the derivation of the generalized Feynman-Vernon action from [16] and relates it to the cluster expansion. The multi-time super-operator function in (A1) multiplies super-operator representation of the system operator. The connection to the path integral formulation is established in the following way: For indices d i = " < " the super-operators correspond to forward paths X i , whereas for indices d i = " > " to a backward paths Y i (with a negative sign). A general bath correlation function is represented as in Eq. (A2) then a relevant series summation is performed. As a result, one obtains a reduced system propagator of the form (7), where contributions to the generalized Feynman-Vernon action S (n,m) contain n number of X and m number Y as The last term in the above expression is the cumulant of the operator correlation function with an appropriate time ordering (first times for the backward path in reverse chronological order, then times for the forward path in chronological order). The term corresponding to m + n = 2 is the standard quadratic Feynman-Vernon action as given by Eq. (13).
Renaming the variables so that times are always ordered s > u and rewriting the resulting expression in terms of sum and difference of system paths χ s = X s + Y s and ξ s = X s − Y s gives (B4) We want to simplify the above expression with regard to the correlation function and shift all time re-orderings to the system operators. Therefore we rewrite it as can be moved to the left and, if time ordering is preserved, the sign of this expression remains the same. The possible pairings for k's are where we introduced a symbolic notation: The first and second square bracket groups operators with k and k ′ respectively and number inside bracket denote indices of re-ordered times s i . To get the total operator acting on ρ B in Eq. (C4) one needs to multiply the k, k; operators with l, l ′ operators and include appropriate combination of coupling constants. In fact there are just two possible pre-factors: g 2 kl g 2 k ′ l ′ for terms of a form [ab] k [cd] k ′ [ab] l [cd] l ′ and g kl g kl ′ g k ′ l g k ′ l ′ for all others. It will prove convenient to write the resulting expression in the following form The above expressions are written using re-ordered times s 1 , s 2 , s 3 , s 4 . Assuming that superoperator indices d 1 , d 2 , d 3 , d 4 are fixed we relate re-ordered times to unconstrained times in the following way where indices a, b, c, d are related to values in1, 2, 3, 4 through a permutation where indices d, t indicate dependence of the permutation on the superoperator ordering d i and unconstrained times t j . Then we have that e.g. [12] if the order of the operators is the same as (a, b), and if the order of the operators is the opposite of (a, b). Additionally we will need the sign of a permutation S(m → n), then we can rewrite the first bracket in Eq. (C9) as The above expression is a sum that goes over 3 permutations of (abcd) where the first is the identity, S(abcd → abcd) = 1. It can be extended to the sum over all 24 permutations of (abcd) The same argument applies to the l, l ′ terms in the second bracket of Eq. (C9) so we can rewrite both brackets as Let us consider product of two terms from the two sums We need to consider the following cases: 1. The same pairing i.e.
• xy = x ′ y ′ and zw = z ′ w ′ • xy = z ′ w ′ and zw = x ′ y ′ Those terms are of the same structure as the ones in Eq. (C8). 2. Different pairing e.g. 1 64 S(1234 → xyzw)S d,k (x, y)S d,k ′ (z, w)S(1234 → xzyw)S d,l (x, z)S d,l ′ (y, w). The overall expression is summed over indices d i and integrated over times t j . Consider therefore the following change of variables with rest of them unchanged. Now we will analyze how such a change affects the sign of the considered term (it is useful to bring in the dependence of t and t ′ ). We have We compare the above to the effect of permuting indices x ↔ y.
and all the above same relations hold for k ′ , l, l ′ . Therefore we found that all terms where the pairing is not the same cancel pairwise. As a result the only non-zero term, from Eq. (C8) and (C9) is Performing the trace yields the following result k,l,k ′ ,l ′ g 2 kl g 2 k ′ l ′ ( 12 k 34 k ′ 12 l 34 l ′ + 13 k 24 k ′ 13 l 24 l ′ + 14 k 23 k ′ 14 l 23 l ′ ) , where ab k is thermal expectation value and we restored the sum over bath degrees of freedom.
2. Case II: k1 = k2 = k3 = k4, l i ′ = l j ′ -quartic pair-wise couplings Using the operator notation introduced in the previous case we find that the relevant expression is This can be evaluated into g 2 kl g 2 kl ′ ( 12 k 34 k − 13 k 24 k + 14 k 23 k ) ( 12 k 34 l ′ − 13 k 24 l ′ + 14 k 23 l ′ ) , where ab k is thermal expectation value. The above expression can be simplified using exactly the same discussion as in the previous Subsection. The reason for this is that it does not involve indices k's and l's but only time orderings and signs of permutations. Therefore we find that the final expression is k,l,l ′ g 2 kl g 2 kl ′ ( 12 k 34 k 12 l 34 l ′ + 13 k 24 k 13 l 24 l ′ + 14 k 23 k 14 l 23 l ′ ) , where we restored the sum over bath degrees of freedom.
Here again we can apply reasoning from Subsection (C 1), so finally we have k,l g 4 kl ( 12 k 34 k 12 l 34 l + 13 k 24 k 13 l 24 l + 14 k 23 k 14 l 23 l ) ,
Compering the above to the results of Subsections C 1, C 2, C 3 we find that line (C29) equals to Eq. (C22), line (C30) equals to Eq. (C25) and line (C31) equals to Eq. (C28). As a result, in our model the fourth order cumulant vanishes.