A Trotter-Kato Theorem for Quantum Markov Limits

Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Hamiltonian generating the quantum Ito evolution.


Introduction
In the situation of regular perturbation theory, we typically have a Hamiltonian interaction of the form H = H 0 + H int with associated strongly continuous one-parameter unitarity groups U 0 (t) = e −itH0 (the free evolution) and U (t) = e −itH (the perturbed evolution), then we transform to the Dirac interaction picture by means of the unitary family V (t) = U 0 (−t) U (t). Although V (·) is strongly continuous, it does not form a one-parameter group but instead yields what is known as a left U 0 -cocycle: One obtains the interaction picture dynamical equation where Υ (t) = U 0 (t) * H int U 0 (t).
More generally, we may have a pair of unitary groups U (·) and U 0 (·) with Stone generators H and H 0 respectively, but where the domains of the generators are not dense. This is the situation of a singular perturbation. In this case we cannot expect the Dirac picture dynamical equation (2) to be anything but formal since the difference H int = H − H 0 is not densely defined.
Remarkably, the steps above can be reversed even for the situation of singular perturbations. If we assume at the outset a fixed free dynamics U 0 (·), with Stone generator H 0 , and a strongly continuous unitary left U 0 -cocycle V (·), then U (t) = U 0 (t) V (t) will then form a strongly continuous one-parameter unitary group with Stone generator H. In practice however the problem of reconstructing H from the prescribed H 0 and V (·) will be difficult.
In the situation of quantum stochastic evolutions introduced by Hudson and Parthasarathy [6], we have a strongly continuous adapted process V (·) satisfying a quantum stochastic differential equation (including Wiener and Poisson noise as special commutative cases) in place of (2), and the solution constitutes a cocycle with respect to the time-shift maps U 0 ≡ Θ (see below). Nevertheless, V (·) arises as the Dirac picture evolution for a singular perturbation of a unitary U (·) with some generator H with respect to the time-shift: it was a long standing problem to find an explicit form for H which was finally resolved by Gregoratti [4], see also [5].
The purpose of this paper is to approximate the singular perturbation arising in quantum stochastic evolution models by a sequence of regular perturbation models. That is, to construct a sequence of Hamiltonians yielding a regular perturbation V (k) (·) converging to a singular perturbation V (·) in some controlled way. We exploit the fact that the limit Hamiltonian is now known through the work of Chebotarev [1] and Gregoratti [4]. The strategy is to employ the Trotter-Kato theorem which guarantees strong uniform convergence of the unitaries once graph convergence of the Hamiltonians is established.

Quantum Stochastic Evolutions
The seminal work of Hudson and Parthasarathy [6] on quantum stochastic evolutions lead to explicit constructions of unitary adapted quantum stochastic processes V describing the the open dynamical evolution of a system with a singular Boson field environment. We fix the system Hilbert space h and model the environment as having n channels so that the underlying Fock space is F = Γ L 2 ([0, ∞)) . Here Γ (H) denotes the symmetric (boson) Fock space over a one-particle space H: we set the inner product as Ψ|Φ = ∞ m=0 1 m! Ψ m |Φ m and take the exponential vectors to be defined as (⊗ s denoting a symmetric tensor product) with test function f ∈ H. Here the one particle space is L 2 ([0, ∞), the space of complex-valued square-integrable functions on [0, ∞). We define the operators whereδ αβ is the Evans-Hudson delta defined to unity if α = β = 1 and zero otherwise. This may be written as In particular, we have the following theorem [6].
Theorem 1 There exists a unique solution V (·, ·) to the quantum stochastic differential equation with G αβ ∈ B (h). (We adopt the convention that we sum repeated Greek indices over the range 0, 1.) In particular, set V (t) = V (t, 0) then we have the quantum stochastic differential equation dV (t) = dG (t) V (t) which replaces the regular Dirac picture dynamical equation (2).
We refer to G = [G αβ ] ∈ B (h ⊕ h), as the coefficient matrix, and V as the left process generated by G. The conditions for the process V to be unitary are that G takes the form, with respect to the decomposition h ⊕ h, where S ∈ B (h)is a unitary, L ∈ B (h) and H ∈ B (h) is self-adjoint. We may write in more familiar notation [6] We denote the shift map on L 2 (R) by θ t , that is (θ t ) f (·) = f (· + t) and its second quantization as Θ t = I ⊗ Γ (θ t ). It then turns out that Θ * τ V (t, s) Θ τ = V (t + τ, s + τ ) and so V (t) = V (0, t) is a left unitary Θ-cocycle and that there must exist a self-adjoint operator H such that ) Here H will be a singular perturbation of generator of the shift, and its characterization was given by Gregoratti [4].

Physical Motivation
As a precursor to and motivation for further approximations, we fix on a simple model of a quantum mechanical system S coupled to a boson field reservoir R. In the Markov approximation we assume that the auto-correlation time of the field processes vanishes in the limit: this includes weak coupling (van Hove) and low density limits. The Hilbert space for the field is the Fock space F R = Γ H 1 R with one-particle space H 1 R = L 2 R 3 taken as the momentum space. It is convenient to write annihilation operators formally as where a p , a * p ′ = δ (p − p ′ ). In particular, let us fix a collection of functions g i ∈ L 2 R 3 , i = 1, · · · , n, and set where ω = ω (p) is a given function (determining the dispersion relation for the free quanta) and k is a dimensionless parameter rescaling time. We have the commutation relations The limit k → ∞ leads to singular commutation relations, and it is convenient to introduce smeared fields in which case we have the two-point function (and define an operator C by) For convenience, let us assume that γ ij = δ ij . Then the A (ϕ, k) are smeared versions of the annihilators on Γ C n ⊗ L 2 (R) .
The limit k ↑ ∞ corresponds to the smeared field becoming singular and this leads to a quantum Markovian approximation. The formulation of such models was first given and treated in a systematic way by Accardi, Frigero and Lu who developed a set of powerful quantum functional central limit theorems including the weak coupling [7] and low density [8] regimes. Theorem 2 is an extension of these which includes both quantum diffusion and jump terms [2,3].
with Ω R the Fock vacuum of F R . The solution to the equation exists and we have the limit Note that is the complex damping coefficient (PV denotes the Cauchy principal value of the improper integral), and by rescaling the g i 's if necessary we may always assume that γ ij = δ ij without loss of generality. The proof of theorem is given in [2] and requires a development and a uniform estimation of the Dyson series expansion.
In the case n = 1 considered in this paper, the triple (S, L, H) from (4) obtained through (5) is where κ + = 1 2 + iσ is now a complex scalar. Our objective is reappraise Theorem 2, where we will prove a related result by an alternative technique. Using the Trotter-Kato theorem, we will establish a stronger mode of convergence (uniformly on compact intervals of time and strongly in the Hilbert space) by means of a graph convergence of the Hamiltonians. The new approach has the advantage of been simpler and is likely to be more readily extended to other cases, for instance a continuum of input channels as originally treated in [6], which cannot be treated by the perturbative techniques used in the proof of Theorem 2.
In this paper we restrict to the case of a single input channel (n = 1) and will return to the general case in a subsequent publication.

Trotter-Kato Theorems for Quantum Stochastic Limits
Our main results will employ the Trotter-Kato theorem, which we recall next in a particularly convenient form.
Theorem 3 (Trotter-Kato) Let H be a Hilbert space and let U (k) (·) and U (·) be strongly continuous one-parameter groups of unitaries on H with Stone generators H (k) and H, respectively. Let D be a core for H. The following are equivalent 2. For all 0 ≤ T < ∞ and all f ∈ H we have The theorem yields a strong uniform convergence if we can establish graph convergence of the Hamiltonians. We now present the Trotter-Kato theorems for the class of problems that interest us, treating the first and second quantized problems in sequence.
Note that κ + and κ − are complex conjugate: Let h be a Hilbert space and let E be a bounded self-adjoint operator on h. We consider the following family of operators on L 2 (R; h) where W 1,2 (X; h), X ⊆ R, denotes the Sobolev space of h-valued functions square integrable on X with square integrable weak derivatives on X. It follows easily that H (k) is self-adjoint for every k > 0. We define a unitary operator on h by We can define an operator H on L 2 (R; h) by It follows easily that H is self-adjoint. We define the following (strongly continuous) one-parameter groups of unitaries on L 2 (R; h) We would like to prove the following theorem.
Theorem 5 Let 0 ≤ T < ∞. We have the following We prove Theorem 5 at the end of this subsection. From the Trotter-Kato Theorem 3, it suffices to find, for every f ∈ Dom(H), a sequence f (k) ∈ Dom(H (k) ) that satisfies condition (i) of the Theorem.
Proof. Note that the C 1 -function h is Lipschitz on the support of η, that is, there exists a positive constant C such that where supp(η) denotes the support of η. Taking the limit for y to 0 + gives |h(x)| ≤ C|x|, x, y ∈ supp(η).
We can define M := max x∈(0,∞) |η(x)| and let L be a number to the right of the support of η. Now we have Lemma 8 If f is in Dom(H) ∩ C ∞ (R\{0}), and f (k) is given by Definition 6, then we have Proof. Note that the first limit follows immediately from a standard result on approximations by convolutions, see e.g. [9,Thm. 2.16]. For the second limit, note that where ∂ ac f denotes the absolutely continuous part of the distributional derivative of f . Using [9, Thm. 2.16] once more, we find that That is, all we need to show is that with rate 1 k . Using the boundary condition for f , we therefore find that with rate 1 k . Note that the L 2 -norm of g (k) grows with rate √ k, so that the limit in Eq. (11) follows. This completes the proof of the Lemma. Proof.
[of Theorem 5] The Theorem follows from a combination of the results in Theorem 3 and Lemma 8 and the fact that Dom(H) ∩ C ∞ (R\{0}) is a core for H. The latter follows from [9, Thm. 7.6].

A Second Quantized Model
Let E αβ be bounded operators on h such that E † αβ = E βα for α, β ∈ {0, 1}. Consider the following family of operators on h ⊗ F H (k) = idΓ(∂) + E 11 A † (g (k) )A(g (k) ) + E 10 A † (g (k) ) + E 01 A(g (k) ) + E 00 , (12) the choice of domain Dom H (k) being the one of essentially self-adjointness for all k > 0. We denote the strongly continuous group of unitaries on h ⊗ F generated by the unique self-adjoint extension of H (k) by U (k) (t). Let the triple (S, L, H) appearing in (4) be obtained from E = (E αβ ) through (5).
The space h ⊗ F = h ⊗ Γ L 2 (R) consists of vectors Ψ = (Ψ m ) m≥0 which are sequences of symmetric h−valued functions Ψ m (t 1 , · · · , t m ) where t j ∈ R. Following Gregoratti, we define the following spaces: (for I a Borel subset of R and H a Hilbert space)

Definition 9 (The Gregoratti Hamiltonian) Define the following operator
It follows from the work of Chebotarev and Gregoratti [1,4] that the operator H is essentially self-adjoint and its unique self-adjoint extension generates the unitary group U (t) = Θ t V t where V t is the unitary solution to the following quantum stochastic differential equation (3): The main result of this section is the following theorem.
Theorem 10 Let 0 ≤ T < ∞. We have the following Before proving the theorem (see the end of this section), we make some preparations. As in the previous section, we would like to use the Trotter-Kato Theorem, therefore, for every Φ in a core for Dom(H), we need to construct an approximating sequence Φ (k) that satisfies the first condition of Theorem 3. We again employ a smearing through convolution with g (k) , this time applied as a second quantization.
Let Φ be an element in Dom(H). We define an element Φ (k) in the domain of Here Γ(G (k) ) denotes the second quantization of G (k) . Note that G (k) is a contraction ( g (k) 1 = 1, i.e. ĝ (k) ∞ ≤ 1 withĝ (k) the Fourier transform i.e. its second quantization is well-defined.
Proof. Since the linear span of exponential vectors v ⊗ e(h) is dense in h⊗ F and Γ(G (k) ) is bounded, it is enough to prove the Lemma for all vectors of the form Φ = v ⊗ e(h). We have where in the last step we used [9,Thm. 2.16].
We now recall the following result, see for instance [12].
Lemma 13 Let C : L 2 (R) → L 2 (R) be a contraction. We have for h ∈ L 2 (R) Moreover, on the domain of A(C † h), we have Note that we have the following generalization equation (10): The action of H (k) on Φ (k) can now be written as Here we have used Lemma 13 and the fact that A(G (k) † g (k) ) = A(ρ (k) ).

Lemma 14
The singular component of equation (17) converges strongly to zero as k → ∞, i.e., We defer the proof of this lemma to the next section.
Using Lemma 12, we find that the first term in Equation (17) converges to the first term in the Hamiltonian H given by Equation (14), i.e.
In the proof of the Lemma 14, we saw that A(ρ (k) )Φ converges in L 2 -norm to κ + a(0 − )Φ + κ − a(0 + )Φ. Therefore, we find for the last line of Equation (17 ) Employing the boundary condition, we have that Here we have used the algebraic identities Applying the Trotter-Kato Theorem, this completes the proof of our main result Theorem 10.

Proof of Lemma 14
Setting V (k) = ia  Φ + E 11 A(ρ (k) )Φ + E 10 Φ, we see that where in the last step we used that Γ(G (k) ) is a contraction. If we can show that V (k) goes to 0, then we will have that the first term A(g (k) )Γ(G (k) )V (k) 2 2 converges to 0. From the boundary conditions we have so that, in fact, κ + a (0 − ))Φ goes to 0 in norm with rate faster than 1 √ k . We will now establish this result below, but first we need to recall the definition of a pseudoexponential vector from [4].
Definition 15 Let : t → F t be a function from R to B (h) and define the corresponding pseudo-exponential vector Ψ (F, h) as for given h ∈ h, where T denotes chronological ordering. That is where σ is a permutation for which t σ(1) ≥ · · · ≥ t σ(m) .
We have Z m and this prefactor is clearly O 1 k from the argument used in Lemma 8.
However, we then have where σ is the chronological time ordering permutation. We note however that [F t , F s ] = 0 for all t, s, therefore we have where we used (18). From the argument in Lemma 8 again, we see that this is

Epilogue
After completion of this work, the authors became aware of the book by W. von Waldenfels [13] which gives a complete resolvent analysis of the Chebotarev-Gregoratti-von Waldenfels Hamiltonian, and in the final chapter describes a strong resolvent limit by coloured noise approximations. The convergence is comparable to the strong uniform convergence considered here, but the approach is very different.