Quantum State Transfer via Noisy Photonic and Phononic Waveguides

Quantum State Transfer via Noisy Photonic and Phononic Waveguides

B. Vermersch Institute for Theoretical Physics, University of Innsbruck, A-6020, Innsbruck, Austria Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria    P.-O. Guimond Institute for Theoretical Physics, University of Innsbruck, A-6020, Innsbruck, Austria Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria    H. Pichler ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA    P. Zoller Institute for Theoretical Physics, University of Innsbruck, A-6020, Innsbruck, Austria Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
July 30, 2019
Abstract

We describe a quantum state transfer protocol, where a quantum state of photons stored in a first cavity can be faithfully transferred to a second distant cavity via an infinite 1D waveguide, while being immune to arbitrary noise (e.g. thermal noise) injected into the waveguide. We extend the model and protocol to a cavity QED setup, where atomic ensembles, or single atoms representing quantum memory, are coupled to a cavity mode. We present a detailed study of sensitivity to imperfections, and apply a quantum error correction protocol to account for random losses (or additions) of photons in the waveguide. Our numerical analysis is enabled by matrix product state techniques to simulate the complete quantum circuit, which we generalize to include thermal input fields. Our discussion applies both to photonic and phononic quantum networks.

thanks: These two authors contributed equally.thanks: These two authors contributed equally.

Introduction.— The ability to transfer quantum states between distant nodes of a quantum network via a quantum channel is a basic task in quantum information processing Kimble (2008); Northup and Blatt (2014); Hammerer et al. (2010); Reiserer and Rempe (2015). An outstanding challenge is to achieve quantum state transfer Cirac et al. (1997); Nikolopoulos and Jex (2013) (QST) with high fidelity despite the presence of noise and decoherence in the quantum channel. In a quantum optical setup the quantum channels are realized as 1D waveguides, where quantum information is carried by ‘flying qubits’ implemented either by photons in the optical Ritter et al. (2012); Tiecke et al. (2014); Goban et al. (2015) or microwave regime Eichler et al. (2012); van Loo et al. (2013); Wenner et al. (2014); Grezes et al. (2014), or phonons Eisert et al. (2004); Hatanaka et al. (2014). Imperfections in the quantum channel thus include photon or phonon loss, and, in particular for microwave photons and phonons, a (thermal) noise background Habraken et al. (2012). In this Letter we propose a QST protocol and a corresponding quantum optical setup which allow for state transfer with high fidelity, undeterred by these imperfections. A key feature is that our protocol and setup are a priori immune to quantum or classical noise injected into the 1D waveguide, while imperfections such as random generation and loss of photons or phonons during transmission can be naturally corrected with an appropriate quantum error correction (QEC) scheme Michael et al. (2016).

Figure 1: Quantum state transfer (QST) via a noisy waveguide. (a) QST where qubits are coupled directly with chiral coupling to a waveguide representing the quantum channel. (b) QST in a cavity QED setup, where atoms representing qubits are coupled to the waveguide with a cavity as mediator. (c) Fidelity for QST of a qubit as a function of photon occupation representing a thermal noise injected into the waveguide for setups (a) and (b). For the protocol described in the text, setup (b) is robust to injected noise. (d) ‘Write’ of a quantum state from cavity  to a temporal mode in the (noisy) waveguide, and ‘Read’ back to cavity  as a linear multimode encoder and decoder with encoding [decoding] functions [] (see text).

The generic setup for QST in a quantum optical network is illustrated in Fig. 1 as transmission of a qubit state from a first to a second distant two-level atom via an infinite 1D bosonic open waveguide. The scheme of Fig. 1(a) assumes a chiral coupling of the two-level atoms to the waveguide Lodahl et al. (2017); Bliokh et al. (2015), as demonstrated in recent experiments with atoms Mitsch et al. (2014) and quantum dots Söllner et al. (2015). The atomic qubit is transferred in a decay process with a time-varying coupling to a right-moving photonic (or phononic) wavepacket propagating in the waveguide, i.e.  where and denote the atomic and channel states. The transfer of the qubit state is then completed by reabsorbing the photon (or phonon) in the second atom via the inverse operation, essentially mimicking the time-reversed process of the initial decay. Such transfer protocols have been discussed in the theoretical literature Cirac et al. (1997); Stannigel et al. (2010); Nikolopoulos and Jex (2013); Ramos et al. (2016); Yao et al. (2013); Dlaska et al. (2017), and demonstrated in recent experiments Ritter et al. (2012). A central assumption underlying these studies is, however, that the waveguide is initially prepared in the vacuum state, i.e. at zero temperature, and – as shown in Fig. 1(c) – the fidelity for QST (formally defined in App. A) will degrade significantly in the presence of noise, e.g. thermal Habraken et al. (2012). Below we show that a simple variant of the setup with a cavity as mediator makes the QST protocol immune against arbitrary injected noise Cirac et al. (1997); Clark et al. (2003) [cf. Fig. 1(b)]. Robust QST also provides the basis for distribution of entanglement in a quantum network.

Photonic quantum network model.— We consider the setup illustrated in Fig. 1(b), where each ‘node’ consists of a two-level atom as qubit coupled to a cavity mode. We assume that the cavity QED setup is designed with a chiral light-matter interface with coupling to right-moving modes of the waveguide 111The use of cavities also allows to efficiently decouple the atoms from unwanted emission into non-guided modes Sayrin et al. (2015). In the language of quantum optics the setup of Fig. 1(b) is a cascaded quantum system Gardiner and Zoller (2015), where the first node is unidirectionally coupled to the second one. The dynamics is described by a quantum stochastic Schrödinger equation (QSSE) Gardiner and Zoller (2015) for the composite system of nodes and waveguide as (). The Hamiltonian is with the Jaynes-Cummings Hamiltonian for node in the rotating wave approximation (RWA). Here are annihilation operators for the cavity modes and ’s are Pauli operators for the two-level atoms with levels . We assume that the cavities are tuned to resonance with the two-level atoms (), and the Hamiltonian is written in the rotating frame. The coupling of the first and second cavity (located at ) to the right-moving modes of the channel is described by the interaction Hamiltonian

(1)

in the RWA. Here denotes the annihilation operators of the continuum of right-moving modes with frequency within a bandwidth around the atomic transition frequency, is the velocity of light, and is a decay rate to the waveguide. In the second line of Eq. (1) we have rewritten this interaction in terms of quantum noise operators satisfying white noise commutation relations . The parameter , with , denotes the time delay of the propagation between the two nodes, and is the propagation phase. For a cascaded quantum system with purely unidirectional couplings, and can always be absorbed in a redefinition of the time and phase of the second node. Noise injected into the waveguide is specified by the hierarchy of normally ordered correlation functions of . In particular the Fourier transform of the correlation function provides the spectrum of the incident noise , which for white (thermal) noise corresponds to with occupation number and flat spectrum .

Quantum state transfer protocol.— To illustrate immunity to injected noise in QST we consider first a minimal model of a pair of cavities coupled to the waveguide. The quantum Langevin equations (QLEs) for the annihilation operators of the two cavity modes in the Heisenberg picture read [cf. App. B]

(2)

These equations describe the driving of the first cavity by an input noise field 222 is always written in the interaction picture and acts as a source term for the nodes in (2)., while the second cavity is driven by both and the first cavity. We can always find a family of coupling functions , satisfying the time-reversal condition [see inset Fig. 1(a)], which achieves a mapping

(3)

i.e. the operator of the first cavity mode at initial time is mapped to the second cavity mode at final time , with no admixture from [cf. App. C]. In other words, an arbitrary photon superposition state prepared initially in the first cavity can be faithfully transferred to the second distant cavity without being contaminated by incident noise. This result holds without any assumption on the noise statistics. It is intrinsically related to the linearity of the above QLEs, which allows the effect of noise acting equally on both cavities to drop out by quantum interference. The setup can thus be combined with other elements of linear optics, such as beamsplitters [cf. App. C].

Robustness of QST to injected noise generalizes immediately to more complex systems representing effective ‘coupled harmonic oscillators’. We can then add atomic ensembles of two-level atoms represented by atomic hyperfine states Julsgaard et al. (2004); Reimann et al. (2015); Hammerer et al. (2010) to the first and second cavities (). Spin-excitations in atomic ensembles Colombe et al. (2007); Brennecke et al. (2007), generated by the collective spin operator with the sum over atomic spin-operators of node , are again harmonic for low densities. Moreover, they can be coupled in a Raman process to the cavity mode, , as familiar from the read and write of photonic quantum states to atomic ensembles as quantum memory 333Adiabatic elimination of the cavity provides QLEs for analogous to (2).. This provides a way of getting an effective time-dependent coupling to the waveguide in a setup with constant cavity decay. Our protocol thus generalizes to transfer of quantum states stored as long-lived spin excitation in a first atomic ensemble to a second remote ensemble [cf. App. D].

Returning to the setup of Fig. 1(b) with a single atom as qubit coupled to a cavity mode, we achieve – in contrast to the setup of Fig. 1(a) – QST immune to injected noise in a three step process. (i) We first map the atomic qubit state to the cavity mode with the cavity decoupled from the waveguide 444The cavity modes can be prepared initially in a vacuum state via a dissipative optical pumping process with atoms, analogous to Ref. Habraken et al. (2012). (ii) With atomic qubits decoupled from cavities we transfer the photon superposition state to the second cavity as above 555If there is an imperfection in step (i), the resulting mixed state is transferred without additional error to the second cavity.. (iii) We perform the time-inverse of step (i) in the second node. This QST protocol generalizes to several atoms as a quantum register representing an entangled state of qubits, which can either be transferred sequentially, or mapped collectively to a multiphoton superposition state in the cavity, to be transferred to the second node 666This is achieved, for example, with quantum logic operations available with trapped ions stored in a cavity Casabone et al. (2015).. As depicted in Fig. 1(d), we can understand our QST protocol in the chiral cavity setup [Fig. 1(b)], consisting of a write operation of the qubit in the first cavity to the waveguide as a quantum data bus, followed by a read into the second cavity. This ‘write’ and ‘read’ are both linear operations on the set of operators consisting of cavity and waveguide modes, or as an encoder and decoder into temporal modes specified by , and physically implemented by the chiral cavity-waveguide interface.

Numerical techniques.— We now study the sensitivity of the above protocol to errors. Imperfections may arise from inexact external control parameters including timing and deviations from perfect chirality. Moreover, loss or addition of photons can occur during propagation. We describe below a QEC scheme which corrects for such single photon errors.

Figure 2: Role of imperfections. (a) Effect on the fidelity of a finite transfer time , and (b) of an imperfect timing of . (c) Fidelity as a function of for different factors and (see text). Solid lines: . Dashed lines: . The fidelity is maximal when is multiple of . (d) By increasing for the fidelity decreases. (e) QEC with the setup subject to waveguide and cavity losses. (f) QEC with the setup coupled to a reservoir with photon occupation . Black: no error correction. Red: correction against single photon losses. Blue: correction against single photon losses or additions. Solid lines: , . Full circles: , . Empty circles: , .

A study of imperfections in QST will necessarily be numerical in nature, as it requires solution of the QSSE with injected noise accounting for nonlinearities in atom-light coupling. Beyond Eq. (1), the Hamiltonian must include coupling to both right- and left-propagating modes in the waveguide, and should account for possible couplings of waveguide and cavities to additional reservoirs representing decoherence [cf. App. B]. We have developed and employed three techniques to simulate the complete dynamics of the quantum circuits as depicted in Figs. 1(a) and 1(b). First, we use matrix product states (MPS) techniques to integrate the QSSE discretized in time steps, as developed in Ref. Pichler and Zoller (2016), which we generalize to include injected quantum noise. Our method allows a general input field to be simulated using purification techniques, by entangling time-bins of the photonic field with ancilla copies in the initial state (for related techniques developed in condensed matter physics see Ref. Schollwöck (2011)). This method also allows the study of non-Markovian effects (i.e. for finite retardation ) in the case of imperfect chiral couplings, and is well suited to represent various kinds of noise. Second, we solve the master equation describing the nodes, which allows for efficient simulations valid in the Markovian limit. Finally, to simulate the QST in non-chiral setups as described at the end of this Letter, we solve the dynamics of the nodes and of a discrete set of waveguide modes, following Ref. Ramos et al. (2016). For a detailed description of the complete model and numerical methods we refer to Apps. B and E, and present below our main results assuming thermal injected noise .

Sensitivity to coupling functions .— In Figs. 2(a) and 2(b) we study the sensitivity of QST to the functions for the minimal model of nodes represented by cavities. Figure 2(a) shows the effect of the protocol duration which in the ideal case is required to fulfill , with the maximum value of . For finite durations, the effect on the fidelity scales linearly with the noise intensity but quickly vanishes for , above which . In all other figures of this work we use . In Fig. 2(b), we show the effect of an imperfection in the timing of the coupling functions, namely . The digression from unity is quadratic in but linear in noise intensity. This result illustrates that only the proper decoding function allows one to unravel the quantum state emitted by the first cavity on top of the injected noise. Note that in addition to errors in the coupling functions, the fidelity is also sensitive to the frequency matching of the cavities Korotkov (2011), which we discuss in App. F.

Imperfect chirality.— For an optical fiber with chirally coupled resonators Sayrin et al. (2015), the nodes emit only a fraction of their excitations in the right direction. The dynamics then also depends on the propagation phase  Lodahl et al. (2017) and on the time delay . As illustrated in Fig. 2(c), the effect of imperfect chirality in the Markovian regime () crucially depends on , as a consequence of interferences between the photon emissions of the two cavities in the left direction. In particular, for , they interfere destructively, leading to a higher fidelity. This interference decreases for finite values of , as shown in Fig. 2(d).

Quantum error correction.— In contrast to ‘injected’ noise, loss and injection of photons occurring during propagation between the two cavities represent decoherence mechanisms, which affect the fidelity of the protocol Northup and Blatt (2014). Such errors can be corrected in the framework of QEC. Instead of encoding the qubits in the Fock states and , we use multiphoton states, with the requirement that the loss or addition of a photon projects them onto a new subspace where the error can be detected and corrected. A possibility is to use a basis of cat states, i.e. superposition of coherent states Haroche and Raimond (2006); Ourjoumtsev et al. (2006), where a photon loss only induces a change of parity of the photon number Ofek et al. (2016). While we present the efficiency of QST with cat states in App. G, we use here a basis of orthogonal photonic states for the qubit encoding Michael et al. (2016).

We first consider a protocol protecting against single photon losses. Here, the state of the first qubit is mapped to the first cavity as , where the cavity logical basis has even photon parity. This unitary transformation can be realized with optimal control pulses driving the qubit and the cavity while using the dispersive shift between the qubit and the cavity mode as nonlinear element Ofek et al. (2016). Waveguide losses, with rates , can be modeled with a beamsplitter with transmission probability , whereas the rate of cavity losses is denoted . The single photon loss probability is then . The density matrix of the second cavity at the end of the protocol reads

(4)

where the unnormalized states and , written explicitly in App. G, have even or odd parity, respectively, and satisfy . The state corresponds to the case where no stochastic photon loss occurred, whereas the state is obtained if one photon was lost in the process. The last step of the protocol consists in measuring the photon number parity in the second cavity, and – conditional on the outcome – apply unitary operations transferring the photon state to qubit . As shown in Fig. 2(e), this encoding improves significantly the fidelity for small losses , up to a threshold value . Note that both protocols are insensitive to injected noise.

We consider now a situation where the waveguide is coupled to a finite temperature reservoir with thermal occupation number which stochastically adds and absorbs photons. Here the qubit state is encoded as , where have photon number modulo . The state after the transfer is a mixture of with corresponding to the cases of a single photon loss, of no photon loss or addition, and of a single photon addition. These states satisfy and and are distinguishable by measurement of the photon number modulo . In the limit of small error probabilities, one retrieves the original qubit state by applying a unitary operation conditioned on the measurement outcome. In Fig. 2(f) we show that this protocol corrects the errors for independently of injected noise intensity. This approach extends to arbitrary number of photon losses and additions, although at the cost of a lower range of achievable  Michael et al. (2016).

Closed systems.—

Figure 3: QST in non-cascaded systems. (a) QST in a closed system with two cavities coupled to a finite waveguide. (b) Fidelity as a function of the cavity nonlinearities and for different initial occupation of the waveguide. The fidelity approaches unity in the linear limit .

Our results can also be observed in closed systems [cf. Fig. 3(a)], where two cavities are coupled, for instance, via a finite optical fiber or a microwave transmission line Blais et al. (2004). Note that in circuit QED setups, time-dependent couplings can be realized via tunable couplers Korotkov (2011); Wenner et al. (2014); Srinivasan et al. (2014). This system is not chiral, as the dynamics of the first cavity can be perturbed by reflections from the second one. We numerically demonstrate robustness against noise, which is here represented as initial occupation of the waveguide. In addition, we consider the effect of Kerr nonlinearities, i.e. we add terms [cf. App. H], which are relevant for circuit QED setups Ofek et al. (2016), to the Hamiltonian. The results are presented in Fig. 3(b) with each (discrete) waveguide mode initially in a coherent state . QST becomes robust against noise in the transition from the cavity as an effective two-level system () to perfect harmonic oscillator ().

Conclusion.— Robustness to arbitrary injected noise in transferring a quantum state between two cavities relies on the linearity of the write and read into temporal modes [cf. Fig. 1(d)], with quantum noise canceled by quantum interference. While here we have focused on QST between two distant cavity modes, our approach generalizes to a setup involving many nonlocal bosonic resonator modes, which can be realized with various physical platforms, and as hybrid systems.

Note added— A related setup and protocol have been proposed in an independent work by Z. L. Xiang et al. Xiang et al. ().

Acknowledgements.
BV and POG contributed equally to this work. HP provided the matrix product state code to integrate the QSSE, which was extended by POG to noisy input. We thank C. Muschik, M. Leib, K. Mølmer, B. Vogell and G. Kirchmair for discussions. Simulations of the master equation were performed with the QuTiP toolbox Johansson et al. (2013). Work at Innsbruck is supported by the EU projects UQUAM and RYSQ, the SFB FOQUS of the Austrian Science Fund (FWF Project No. F4016-N23) and the Army Research Laboratory Center for Distributed Quantum Information via the project Scalable Ion-Trap Quantum Network (SciNet).

Appendix A Calculation of average fidelities for quantum state transfer

Throughout this work the fidelity of quantum state transfer (QST) is defined as the overlap between the final state of the second node at the end of the protocol and the state obtained in an ideal transfer, averaged over all initial qubit states for the first node. Formally, one can access this value by making use of a Choi-Jamiolkowski isomorphism between quantum processes and quantum states Nielsen (2002), where the first node is prepared initially in an entangled state with an ancilla copy of itself which is otherwise completely decoupled from the dynamics.

For the situation depicted in Fig. 1(b) of the main text, we focus on the transfer between the cavities via noisy waveguide. The initial state of the first node thus reads

(5)

where the index denotes the cavity and the ancilla, while the atom is decoupled. The average fidelity is then obtained numerically by simulating the QST from cavity to cavity , including imperfections. We denote the density matrix of second cavity and ancilla at the end of the protocol and its ideal value with up to propagation phase factors for the component. The fidelity then reads

(6)

For nodes realized with atomic ensembles of atoms as two-level systems, we apply a similar procedure, where the Fock states and in Eq. (5) are replaced with the collective ground state of the ensemble and the excited state , with the creation operator for excitation of atom in the node . In the quantum error correction (QEC) protocols, these states are replaced with the multiphoton states as defined in the main text.

Appendix B Dynamics of cavities chirally coupled to a waveguide

Here we provide details on the model of two cavities coupled to a waveguide with time-dependent decay rates, including imperfections such as deviation from unidirectionality, propagation losses and cavity decays. We present the corresponding form of the quantum stochastic Schrödinger equation (QSSE), the quantum Langevin equations (QLEs) and the master equations supporting the analytical and numerical study of the QST dynamics.

b.1 Model

Our model consists of cavities as harmonic oscillators coupled on resonance to atomic two-level systems as qubits, and with chiral coupling to a waveguide. The dynamics is described by the Hamiltonian , with the node Hamiltonians as given in the main text, and couplings to the waveguide:

(7)

The Hamiltonian is written in a frame where node operators and rotate with the cavity frequency , and in an interaction picture with the waveguide bare Hamiltonian . Here and are left- and right-moving waveguide modes satisfying , and we assumed a linear dispersion relation around . Moreover, in writing Eq. (7), we have assumed under the Born-Markov approximation that the decay rates can be considered constant over the bandwidth . Here, the parameter defines the chirality of the coupling with rates , respectively , to the right- and left-moving modes. For the perfectly chiral case (), correspond to the interaction Hamiltonian as written in Eq. (1) of the main text.

The interaction Hamiltonian can be rewritten as

(8)

where we have set , . Here is the propagation phase and is the time delay. The quantum noise operators

(9)

satisfy and represent the incoming or ‘injected’ light field interacting with the cavities. Finally, we include in our description the effect of cavity and propagation losses by adding to the interaction Hamiltonian coupling terms to additional channels Gardiner and Zoller (2004).

b.2 Quantum stochastic Schrödinger equation

The quantum stochastic Schrödinger equation (QSSE) describes the stochastic evolution of the system as

(10)

where represents the wavefunction of the system of nodes and quantum channels, which can be interpreted within the framework of Ito or Stratonovich calculus Gardiner and Zoller (2015). As detailed in App. E, the numerical simulation of QSSE can be performed using a matrix product state (MPS) ansatz for the wavefunction , where the state of the waveguide (e.g. vacuum, thermal or coherent) can be efficiently represented. It is particularly well-suited in the non-cascaded case and for long delay times () where non-Markovian effects arise Pichler et al. (2015); Ramos et al. (2016).

b.3 Quantum Langevin Equations

The quantum Langevin equations (QLEs), describing the dynamics of an arbitrary operator acting on the nodes in the Heisenberg picture, is the starting point of our analytical study of the QST dynamics. In the following, we present their explicit form for the various situations addressed in the main text.

b.3.1 Ideal setup

In the ideal setup the system has no (additional) losses and the coupling between cavities and waveguide is perfectly chiral with the two nodes decoupled from left-moving modes. In the Heisenberg picture the dynamics of an arbitrary operator acting on the nodes can be obtained by formal integration of the Heisenberg equation for right-moving modes around the resonant frequency (see for instance Ref. Gardiner and Zoller (2015)), which leads to

(11)

where we redefined the time of the second cavity to eliminate the time delay and absorbed the propagation phase in the definition of the cavity operators . Here, the notation refers to taking the complex conjugate of the total expression, except for operator . Note that the operator is expressed as in Eq. (9), i.e. in the interaction picture, and represents here a source term driving the cavities. For and , we obtain Eqs. (2) of the main text when atoms and cavities are decoupled [].

b.3.2 Imperfect chirality

We now consider the case where the chiral coupling between nodes and waveguide is not perfect (i.e. unidirectional), that is the nodes also couple to left-moving modes . In order to obtain a system of coupled local differential equations, we neglect the time delay assuming the Born-Markov approximation , with the maximum value of . The QLEs then read Pichler et al. (2015)

where we redefined and assumed . Note the importance of the propagation phase in the above equation, which governs the interference between the emission of the two cavities.

b.3.3 Additional losses

In the case of cavity and waveguide losses, the QLEs can be written as Gardiner and Zoller (2004)

(13)

with , and where we have assumed perfect chirality (). Here is the coupling of each cavity to unwanted (non-guided) modes with input fields , and the waveguide losses with rates are modelled by a beamsplitter mixing to an additional waveguide with and input field .

b.4 Master equation

The master equation allows to perform numerical simulations of the evolution of the node reduced density matrix , where denotes the trace over propagating modes. The mapping from the QLEs to the master equation can be realized using different techniques Gardiner and Zoller (2004); Vogel and Welsch (2006); Gardiner and Zoller (2015). To do so, one writes the QLE in the following form

(14)

where the index refers to interactions involving node operators and the index to input fields . For instance, in the case of Eq. (11), we have , , , , , , , .

In the case where each channel is in a thermal state with average photons, we obtain the master equation Gardiner and Zoller (2004); Vogel and Welsch (2006); Gardiner and Zoller (2015)

(15)

where

(16)
(17)

with .

Appendix C Operator mapping

Here we show that the QST protocol between two nodes as linear harmonic oscillators performs the operator mapping of Eq. (3) of the main text, assuming the ideal scenario where couplings to the environment are negligible () and couplings to the waveguide are perfectly chiral (). The system is described by Eqs. (2) of the main text.

c.1 Integration of the quantum Langevin equations

In the input-output formalism, the output field of the first node in the waveguide is given by , with the contribution from the first node . This variable represents the emission of the first node, containing the information about its initial quantum state. Conversely, the output field of the two cavities is given by , where the contribution of the two nodes is . Using the Langevin equation (11), these two variables evolve according to

(18)
(19)

where . We require that these functions satisfy the condition

(20)

so that the equations for and decouple. In the standard QST description, i.e. without injected noise, this condition implies that the output field vanishes, or, in other words, that the second cavity absorbs all the emission of the first one.

The general solution of Eqs. (18),(19) reads

The second node operator can then be written in the form

(23)

where the node propagators are expressed as

(24)

and the noise propagator as

(25)

In the limit of transfer times large compared to the typical emission rates , the node propagators satisfy (see for example Ref. Stannigel et al. (2011))

(26)
(27)

with effects of finite pulse duration depending on the specific shape of the functions .

c.2 Vanishing noise contribution

We now show that the noise contribution at the end of the QST vanishes due to opposite contributions from and in Eq. (25), leading to the operator mapping . We first note that the state of the injected noise field can always be decomposed as a distribution of classical states by expressing the injected noise field with a P-representation Gardiner and Zoller (2004). If we consider any particular component with , we can bound the noise contribution using the Cauchy-Schwarz inequality:

where is the integrated noise intensity. Below we provide two examples of coupling functions satisfying Eq. (20), and we show that the integral of in the last equation vanishes.

(i) We first consider the functions

(28)
(29)

where we set for convenience and is the maximum decay rate. These are the coupling functions used in our numerical simulations. The integral reads

(30)

which vanishes like in the limit . This implies that, provided the integrated noise intensity does not grow exponentially with (typically the growth is linear), the noise contribution vanishes. Unless stated otherwise, we use .

(ii) This cancellation of noise is not restricted to coupling functions satisfying the time-reverse property . For example if one considers

(31)
(32)

where we set and , then the integral reads

(33)

Note however that in practice one needs to introduce a cut-off for to avoid the divergence at time .

c.3 Extension: beamsplitter operation

Figure 4: Operator mapping between four cavities. We consider four cavities in the beamsplitter configuration and subject to injected noise.

The operator mapping between two nodes can be extended to the beamsplitter mixing between four nodes, as represented in Fig. 4 where two pairs of cascaded nodes as harmonic oscillators undergo the QST protocol along waveguides mixed by a beamsplitter.

If we assume no imperfections in the couplings, the QLEs read

(34)
(36)

Here nodes and are coupled only to the modes propagating upwards in the vertical waveguide, and denotes the corresponding injected noise. In particular, if the coupling functions satisfy and , the equations can be decoupled using the linear combinations

(38)
(39)

whose evolution maps directly to Eq. (2) of the main text with the set () replaced by the sets () or (). The operator mapping can thus be applied to these sets and we obtain the mixed QST

(40)

This result shows that one can realize beamsplitter mappings between four distant cavities, where cavities and imprint their quantum states onto noisy temporal modes of the two quantum channels. These modes interfere at the beamsplitter, similarly to single photon wave-packets, before being reabsorbed perfectly by cavities and .

Appendix D Effective tunable cavity couplings via atomic ensembles

d.1 Model

Here we provide details on the realization of QST between two nodes () using atomic ensembles as harmonic oscillators. As depicted in Fig. 5(a), we consider two nodes consisting of an atomic qubit, a cavity, and an ensemble of two-level atoms with ground states and excited states . The Hamiltonian governing the dynamics of the system can be written in the form of (see main text) where the node Hamiltonian is now given by

(41)

and, having in mind a typical quantum optical setup, the coupling rates between cavities and waveguide are considered to be constant over time. The node Hamiltonian can be realized via a laser-assisted Raman transition Hammerer et al. (2010) with the assumption that additional Stark-shifts terms can be cancelled using for instance another laser coupling. Here are the collective spin operators associated to the total angular momentum , with spin and where denotes the set of Pauli matrices of the -th atom in node . In the following we will use the ground states and first excited states of the atomic ensemble to mediate the QST between the two nodes, with effective time-dependent coupling to the waveguide governed by the functions .

Figure 5: QST via atomic ensembles. (a) Setup including atomic ensembles as effective harmonic oscillators in the nodes. (b) Fidelity of QST as a function of and for different atom numbers. (c) Error as a function of the scaling parameter . The dashed line represents .

The protocol, which becomes robust against noise in the limit of large atomic ensembles, is very similar to the one described in the main text in Fig. 1(b). The first step (i) now consists in mapping the qubit state of the atom to a collective state of the atomic ensemble . This can be done for instance by detuning the atom and atomic ensemble with respect to the cavity in the Raman transitions, hence realizing an effective coupling between qubit and atomic ensemble with the cavity adiabatically eliminated. The second step (ii) consists in realizing the operator mapping in analogy to the operator mapping between cavities, with the atomic ensemble now resonantly coupled to the cavity. Finally, (iii) the atomic ensemble state is mapped to the qubit state of the second node in the reverse process of step (i).

d.2 Adiabatic elimination of the cavities

We now describe in details step (ii) where the atomic ensemble realizes the QST in presence of injected noise. During this operation, the qubits are excluded from the dynamics (). The Langevin equations associated with can be written in the form (see for instance Ref. Habraken et al. (2012) for the single atom case)

with