Ramsey interferometry of non-Hermitian quantum impurities

We introduce a Ramsey pulse scheme which extracts the non-Hermitian Hamiltonian associated to an arbitrary Lindblad dynamics. We propose a realted protocol to measure via interferometry a generalised Loschmidt echo of a generic state evolving in time with the non-Hermitian Hamiltonian itself, and we apply the scheme to a one-dimensional weakly interacting Bose gas coupled to a stochastic atomic impurity. The Loschmidt echo is mapped into a functional integral from which we calculate the long-time decohering dynamics at arbitrary impurity strengths. For strong dissipation we uncover the phenomenology of a quantum many-body Zeno effect: corrections to the decoherence exponent resulting from the impurity self-energy becomes purely imaginary, in contrast to the regime of small dissipation where they instead enhance the decay of quantum coherences. Our results illustrate the prospects for experiments employing Ramsey interferometry to study dissipative quantum impurities in condensed matter and cold atoms systems.

An atomic wire subject to localised particle losses, or a quantum spin chain subject to on-site dephasing, are instances of 'dissipative' impurity problems.The nomenclature is borrowed from the traditional research field of equilibrium quantum impurities in many-body systems, which comprises archetypical cases ranging from X-ray edge singularities to magnetic impurities embedded in fermions [1,2].Systems with quantum impurities have represented important stepping stones in understanding the physics of strongly correlated systems, and one could similarly argue that adding localised dissipation on an otherwise extended system, could shed light on unveiling the mechanisms intertwining incoherent processes and quantum correlations in many-body systems.The surge of interest in this modern area of research has been ignited by few recent experiments in cold gases: shining an electron beam on a localised spatial region of an atomic BEC of 87 Rb atoms [3][4][5][6][7][8], induces a Zeno effect which dictates the counterintuitive decrease of atom loss rate at strong dissipation.Dissipative impurities can also constitute a resource in quantum many body engineering, as they are employed to implement scanning gate microscopes of ultra-cold bosons [9,10].
In its conventional formulation, the quantum Zeno effect predicts the 'freezing' of the wavefunction when frequent measurements exceed a rate threshold [11][12][13][14].The phenomenon extends beyond the theory of quantum measurement, and it comprises the generic arrest of quantum evolution provided by stochastic fields [15], including the decoupling of a system from its decohering environment by the application of a sequence of fast pulses [16].The connection between Zeno effect and many-body physics has been first drawn in the context of quantum circuits where unitary gates and random-in-space and time projective measurements compete, inducing a transition in the entanglement entropy from volume to area law when measurements become frequent [17].Primarily motivated by the experiments in Refs.[5][6][7], the physics of the Zeno effect has also entered the field of dissipative impurities.The effect of 1/ω noise on the transport properties of Kane-Fisher barriers [18,19] have been studied with non-equilibrium Luttinger liquids [20], while a series of related works have shown that strong local losses can inhibit particles' emission at the Fermi surface [21][22][23]; the interplay of Zeno physics with many-body correlations has the potential to promote lossy mobile impurities into a novel class of Fermi-polarons [24] and the list of examples could continue [25,26].
In general, the dynamics induced by a Markovian quantum master equation comprise an 'imaginary' or non-Hermitian Hamiltonian (which is quadratic in the Lindblad operator) in combination with a term describing stochastic 'jumps' driven by quantum noise [27][28][29].
In the following, we shall show that a sequence of Ramsey pulses can experimentally decouple these two contributions, and can be employed to measure via interferometry a generalised form of Loschmidt echo, which evolves solely with the former, representing an operative route to define non-Hermitian Hamiltonians.We apply this scheme to the problem of a dissipative impurity in a weakly interacting Bose gas, and we predict, with functional integrals techniques, a Zeno effect for the decohering exponent of the echo, which is non-perturbative in the impurity strength.This is at variance with previous works aiming at studying noise averaged Loschmidt echoes in locally dephasing spin models [30,31], which re-FIG. 1.An atomic impurity (grey dot) embedded in a one dimensional wire of cold bosons (in red).The impurity has three internal atomic levels: the level |0 is inert to scattering with the bosonic cloud, and is used as a control state for Ramsey interferometry which is employed to measure the Loschmidt echo for the non-Hermitian Hamiltonian associated to stochastic dynamics.The latter is generated by driving with a sequence of Markovian pulses the levels |1a and |1b which are density coupled to the background Bose gas.
quire the full Lindblad evolution, and it represents an alternative pathway to the study of dissipative impurities, since it does not rely on the measurement of transport properties across noisy barriers.
A general Markovian quantum master equation with one Lindblad channel reads with non-Hermitian Hamiltonian and quantum jump term being respectively The Ramsey protocol we propose to probe Ĥeff relies on the assumption that the strength of the interaction between the system and the environment can be controlled by an additional, discrete degree of freedom of the system.For simplicity, we consider a generic system with (at least) two internal states |0 and |1 , the Lindblad dynamics being active only when the system is in the state |1 .This is expressed by the following replacement, valid in the extended Hilbert space that includes the internal level: In cold atomic systems, the latter can be realized with spin-dependent interactions with the bath [32,33] as employed in studies of Bose polarons [34][35][36], see Fig. 1.
The density matrix of the system is prepared in a pure, factorized state ρ0 = |ψ ψ|⊗|0 0|.A π/2 pulse flips the internal state |0 → (|0 + |1 )/ √ 2, and the full density matrix changes accordingly: At this stage, the crucial observation is that, for a Lindblad operator as in Eq. ( 3), the quantum jump term acts only on the right-bottom element of the density matrix, i.e., on 1| ρ(t)|1 .The off-diagonal elements evolve only under the action of the non-Hermitian Hamiltonian, as shown in the Supplemental Material.The third and final step is a second Ramsey π/2 pulse and a subsequent readout measurement of the time-dependent contrast, σz (t) .In analogy with the purely coherent case (see Supplemental Material (SM) and Refs.[37][38][39][40]), such observable probes only the offdiagonal elements and yields, then, the overlap of the the initial state with the one evolved with Ĥeff In the following, we take |ψ as the ground state of Ĥ0 and we assume that its energy is zero, ψ|e iH0t = ψ|.We stress how this scheme for extracting non-Hermitian evolutions is completely general, not restricted to cold atomic systems, and, most importantly, it does not rely on the necessity to detect quantum jumps in order to extract dynamics.Therefore, it can offer a systematic advantage over methods requiring averages over stochastic realisations.
The model setup we consider is an ultracold onedimensional Bose gas interacting with a localised atomic impurity.The discrete degree of freedom used for interferometry will be an internal state of the atom.To realize Eq. ( 3) and the Lindblad dynamics, the atom has a non-interacting internal level |0 , and at least two additional levels, labelled as |1a and |1b , density-coupled to bosons via the interaction Hamiltonian where x 0 denotes the position of the atom, ψσ its spinor wavefunction, nσ ≡ ψ † σ (x 0 ) ψσ (x 0 ) the occupation of the σ level and nB (x 0 ) the bosonic number operator at x = x 0 ; we assume in the following x 0 = 0. Eq. ( 6) describes a strength controlled by the states of the impurity.The Lindblad dynamics can be engineered by acting within the subspace {|1a , |1b } with an additional external field, different from the one employed in the Ramsey protocol.With a sequence of π pulses flipping between the two states, the coupling can be promoted to a timedependent quantity g(t), oscillating between g a and g b , as depicted in 1.For a suitably chosen fast sequence of random pulses, g(t) is a stochastic variable with first and second moments where • • • denotes the temporal average over several π-pulses.Higher order moments are assumed to be negligible.The conditions on second and higher moments are equivalent to assume that the autocorrelation time of the density operator of the impurity is the smallest scale in the problem.The temporal average over mutliple autocorrelation times yields an equation of motion for the density matrix equal to Eq. ( 1), with Lindblad operator L = nB (0) ⊗ |1 1|.Expanding the Bose field in terms of Bogolyubov excitations, the Lindblad operator and the non-Hermitian Hamiltonian (2) become respectively and where we have defined k ≡ Λτ −Λτ dk.The cut-off Λ τ is a consequence of the Markov approximation (cf.Eq. ( 7)) on the statistics of the π-pulses, which are assumed to evolve on the shortest time scale τ in the model.However, this assumption is no longer valid when the dispersion relation ω k enters the particle-like regime and momenta are of the order of k Λ τ ∝ 1/τ , thus requiring to cut off momentum modes beyond this UV scale.The parameter Λτ in Eq. ( 9) comes from the normal ordering of Ĥeff , and it is related to the cutoff Λ τ via Λτ = 2 + Λ 2 τ − √ 2. In the expression (9), bk are the Bogolyubov annihilation operators in the BEC, n 0 is the density of the condensate, and we have defined where g expresses the dissipation strength in Ĥeff and replaces the microscopic coupling constant (we have used units where = c = m = ξ = 1).The contrast (5) and the related Loschmidt amplitude G(t) are now expressed in terms of a functional integral with fixed boundary conditions in time, following the standard coherent state Trotter decomposition.The derivation follows Ref. [41], and it is discussed in detail in the Supplemental Material.Specifically, from Eq. ( 9) we find (11) where We remark that the functional integral formula ( 12) is suited to describe the outcome of the interferometric measurement discussed above for any choice of Lindblad operator L, which can be local or extended in space.Eq. ( 12) is analogous to a Matsubara functional integral in imaginary time, as it can be readily seen from the similarity between the time evolution operator, exp (−it Ĥ), and the Boltzmann weight, exp (−β Ĥ); accordingly, we define the real time Matsubara frequencies ω n = 2πn/t, with n ∈ Z. Implementing the boundary conditions requires however an additional Lagrange multiplier, as discussed in the Supplemental Material.
The bare, G 0 k,n , and impurity dressed, G ef f kq,n , Matsubara Green's functions can be derived from Eq. ( 12) after some manipulations which yield (see Supplemental Material for the details of the calculations) Before proceeding further, we observe that Eq. ( 45) carries a crucial information on the perturbative expansion of log G(t) in powers of the coupling g: all corrections corresponding to a dressing of the Green's functions can be resummed and expressed in terms of a renormalized coupling strength This parameter is small since |g| max = 1/2π 0.16 (cf.Fig. 2); it is therefore convenient to develop an expansion of log G in powers of g.The functional integral (12) can be now evaluated (see Supplemental Material), obtaining the following exact expression of the Loschmidt amplitude In Eq. ( 15) matrices act only in momentum space, and G 0 is a shorthand for G n=0 ; correspondingly, traces run only over momenta.We can observe here the role played by the renormalized coupling in the analytic expression.The first term in a naive perturbation theory corresponds to the first line, i.e., it is obtained by replacing G ef f → G 0 .Crucially, almost all corrections to naive perturbation theory are small and under perturbative control even at strong coupling, since they can be resummed and expressed in terms of g, as manifested by the presence of the dressed Green's functions in Eq. ( 15).The only possibly relevant contribution to the naive perturbation theory comes from the second line, that also contains the bare coupling g 2 : the leading term at long times can be evaluated exactly, and the sum of first and second line yields the Loschmidt echo log (G(t)/G τ (t)) −gρ 2 t/2, (16) in terms of the amplitude G τ (t) ≡ exp(−g Λτ t), which can be controlled by shaping the noise profile.The right hand side of Eq. ( 16) represents non-perturbative corrections to the leading decoherence damping, G τ (t), expected in general for a stochastic scatterer embedded in an otherwise coherent medium.Nevertheless, the renormalized coupling g, which is real for small values of the bare coupling g, becomes purely imaginary at strong bare coupling, g i(2π) −1 , as illustrated in Fig. 2. The fact that g is imaginary for large dissipative strengths, indicates that the rate decay function of the Loschmidt echo will be entirely dominated by the bare decay exponent∝ g Λ.The occurrence that all higher order corrections to decoherence are neutralised and resum to an imaginary exponent is an incarnation of the Zeno effect: for strong dissipation (large g), the incoherent scatterer perfectly 'reflects' bosons which impinge upon it, and its only effect is to imprint a phase shift on reflected wavefunctions (see for related ideas the cold-atoms experiment in Ref. [42]).
In conclusion, we have proposed how to measure, via Ramsey interferometry, the Loschmidt amplitude of an effective non-Hermitian Hamiltonian associated to a Lindbladian.The onset of a many-body Zeno effect can be directly probed by the readout of σ z without resorting to measurements of transport properties or to probing unequal time correlation functions.We have demonstrated through Eqns.( 45) and ( 14) that unitarity is restored for quasi-particle dynamics at strong dissipation strength, while Eq. ( 15) and Eq. ( 16) show that, in spite of the onset of the Zeno effect, a damping persists and becomes dominant at strong coupling.It would be interesting to study, in the future, whether the Zeno effects can manifestly similarly in the interferometric properties of other systems, or whether its imprint on the Loschmidt echo is inherently non-universal.
Our results pave the way for a number of further exploratory directions.First of all, it would be natural to study extension of our calculations in the case of a mobile impurity in view of recent connections between polarons and Zeno physics [24].Furthermore, the approach developed for extracting the leading decay rate of G(t) is completely general, and it could be, for instance, extended to more realistic dissipative impurities by taking into account the spatial profile of the impurity wavefunction or the correlation time of the noise.Finally, the short-time pattern of the generalised Loschmidt echo could serve as a mean to characterise dynamical quantum phase transitions of non-Hermitian systems [43,44].We also foresee the possibility of applying concepts developed for the study of dynamical topological phenomena [44,45] to the more recent field of non-Hermitian topology [46,47], with direct access to the echo of physically realizable non-Hermitian Hamiltonians 'defined' via the protocol discussed here.Since out-of-time order correlations can be measured via Ramsey interferometry [48], we also foresee in the future an extension our results in the direction of probing scrambling in non-hermitian quantum systems.F.T. thanks C. Mordini for useful discussions.F.G. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC-2111 -390814868.J.M. was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 745608 (QUAKE4PRELIMAT).

Ramsey interferometry for Hermitian Hamiltonians
The same interferometric protocol discussed in the main text can be applied to Hermitian Hamiltonians.We consider the state-dependent interaction encoded in and prepare the initial state in the tensor product of the many-body wave function |ψ with the control state |0 of the auxiliary spin After a π/2 pulse this state is mapped into the superposition whose time evolution reads because of the conditioned activation of the perturbation V upon occupation of the level |1 in the Hamiltonian (17).By letting the system evolve for the time t and applying a second π/2 pulse to the state (20), we find Measuring the imbalance between the |1 and |0 states, we recover the standard Loschmidt amplitude σ z = Re ψ|e iHt e −i(H+V )t |ψ .
Non-Hermitian evolution from contrast measurement We show that a contrast measurement performed after the protocol explained in the main text probes the Loschmidt amplitude of the non-Hermitian Hamiltonian reported in Eq. ( 2) in the main text.
We consider the action of the Liouvillian on product states of the form For Lindblad operators as in Eq. () in the main text, the action from the left is nontrivial only on components with n = 1: A similar equation holds for the action of L † from the right, the role of n and m being exchanged.The quantum jump term affects therefore only the component of ρ with n = 1, m = 1: The action of the full quantum master equation ( 1) on the components of the density matrix is therefore The time evolution of the off-diagonal components reads instead ρ10 (t) = p 10 e −iH eff t ρe iH0t , ρ01 (t) = p 01 e −iH0t ρe iH † eff t .
We can now conclude by showing that the combination of the second π/2 pulse and the contrast measurement probe only such components.In fact, a combination of a π/2 pulse and a measurement of σ z is equivalent to a measurement of σ x , since the matrix of a π/2-pulse is and therefore R † π/2 σ z Π π/2 = σ x .In conclusion, denoting by tr the trace acting only on the many-body degrees of freedom of the system and not on the internal one used for interferometry, which yields = 2Re tr[p 10 e iH0t e −iH eff t ρ], which gives Eq. ( 5) in the main text once ρ = |ψ ψ| and p 10 = 1/2 are substituted from Eq. ( 4).

Matsubara functional integral
We express G(t) it in terms of a functional integral with fixed boundary conditions.For illustrative purposes, we reproduce here such derivation in the case of a single bosonic mode with Hamiltonian ĥ The precise form of the above Hamiltonian is chosen in analogy to Eq. ( 9) in the main text.We begin by identifying the proper boundary conditions which should be imposed on the bosonic fields: and Eq. ( 34) is a periodicity condition α(0) = α(t) and it can be taken into account following the formalism for finite temperature functional integrals, i.e. by introducing Fourier-transformed fields to take advantage of periodicity in time.
This action for φ can either be derived by writing the complex field b in terms of real and imaginary parts, and by integrating out the latter, or can be derived by noticing that the Green's functions G 0 n,k of the scalar field can be computed from those of the complex field.In the following, we will choose the latter approach.
From Eq. ( 40), we find the complex boson Green's function from which the bare scalar Green's function reads .
From z n = ω n e −iδωn and from ω −n = −ω n one sees that the sum z n + z −n is a negative infinitesimal −2iδω 2 n .We neglect such infinitesimal at the numerator since it does not influence the location of poles in the complex plane.Its influence in the denominator is to shift its zeros away from the real axis.The denominator equals ω 2 k − ω 2 n (1 − 2iδω k ), and, to first order in δ, it vanishes for ω n ±(ω k + iδω 2 k ).We will label them in the following as ω + k ≡ ω k + iδ.We therefore obtain for the bare Green's function Impurity dressed Green's function The impurity dressed Green's function is obtained after inversion of the quadratic kernel in Eq. ( 42).The inversion is trivial in Matsubara frequency indices (the kernel is diagonal) but not in momentum space.To make progress, we write the Dyson equation for each component of the Green's function (we suppress the index n in the following few lines, and restore it at the end of the calculation): Since V p1p2 = 2ig does not depend on momentum indices, we find To gain further insight, we perform some algebra in the denominator of Eq. ( 45):

FIG. 2 .
FIG.2.Plot of the real and imaginary parts of the renormalised impurity strength, g, as a function of the bare dissipation strength g.For g1 the real part vanishes, while the imaginary part reaches an asymptote at 1/2π, indicating the onset of a quantum Zeno regime.