Deterministic quantum state transfer between remote qubits in cavities
Performing a faithful transfer of an unknown quantum state is a key challenge for enabling quantum networks. The realization of networks with a small number of quantum links is now actively pursued, which calls for an assessment of different state transfer methods to guide future design decisions. Here, we theoretically investigate quantum state transfer between two distant qubits, each in a cavity, connected by a waveguide, e.g., an optical fiber. We evaluate the achievable success probabilities of state transfer for two different protocols: standard wave packet shaping and adiabatic passage. The main loss sources are transmission losses in the waveguide and absorption losses in the cavities. While special cases studied in the literature indicate that adiabatic passages may be beneficial in this context, it remained an open question under which conditions this is the case and whether their use will be advantageous in practice. We answer these questions by providing a full analysis, showing that state transfer by adiabatic passage – in contrast to wave packet shaping – can mitigate the effects of undesired cavity losses, far beyond the regime of coupling to a single waveguide mode and the regime of lossless waveguides, as was proposed so far. Furthermore, we show that the photon arrival probability is in fact bounded in a trade-off between losses due to non-adiabaticity and due to coupling to off-resonant waveguide modes. We clarify that neither protocol can avoid transmission losses and discuss how the cavity parameters should be chosen to achieve an optimal state transfer.
The ability to faithfully transmit an unknown quantum state between remote locations is a key primitive for the development of various quantum technologies. The quest to create long-distance links that can connect multiple nodes into a quantum internet Kimble (2008); Northup and Blatt (2014); Gibney (2016) is motivated by applications such as unconditionally secure communication Bruß and Lütkenhaus (2000); Lo et al. (2014), distributed quantum computing Beals et al. (2013), quantum fingerprinting Buhrman et al. (2001); Xu et al. (2015), quantum credit cards Goorden et al. (2014), quantum secret voting Hillery et al. (2006), quantum secret sharing Hillery et al. (1999), secure quantum cloud computing Broadbent et al. (2009); Barz et al. (2012), quantum time and frequency metrology Komar et al. (2014), and tests of the foundations of quantum physics Bell (2004); Hensen et al. (2015). Short-distance links acting as ‘quantum USB cables’, on the other hand, allow the connection of different types of quantum hardware and are a promising approach to scalable quantum computing architectures Monroe et al. (2014).
A quantum state transfer can be accomplished probabilistically or deterministically. Probabilistic state transfer protocols use a two-step procedure: First, an entangled state between two network nodes is generated in a probabilistic and heralded fashion Cabrillo et al. (1999); Chou et al. (2005); Moehring et al. (2007); Hofmann et al. (2012); Bernien et al. (2013). Second, this entangled link is used for transferring a quantum state by teleportation Olmschenk et al. (2009); Bao et al. (2012); Krauter et al. (2013); Nölleke et al. (2013); Pfaff et al. (2014). For deterministic state transfer det (), a qubit state at one node is mapped onto a photon wave packet, which propagates across the desired distance and is then absorbed by the qubit in the receiving cavity Cirac et al. (1997); Ritter et al. (2012). Deterministic approaches, as discussed here, do not rely on the availability of entangled resource states. Hence, such protocols are particularly well suited for the implementation of time-continuous schemes for quantum information processing Muschik et al. (2013); Hofer et al. (2013); Vollbrecht et al. (2011) and also dispense with the photon counters typically required for probabilistic approaches, e.g., Duan and Monroe (2010).
We are interested in the simple scenario depicted in Fig. 1a, in which two nodes are connected by a waveguide. Each node consists of a qubit placed in a cavity. In this context, relevant to, e.g., atoms, ions, and superconducting circuits coupled by waveguides, we study the task of transmitting quantum information deterministically between the two nodes. We focus on evaluating the performance of two deterministic protocols. First, we consider the standard approach based on wave packet shaping Cirac et al. (1997), in which a classical qubit drive, as depicted in Fig.1b, is designed such that the photon emitted by the first qubit is entirely captured by the second qubit, without reflections. The second approach uses the techniques of adiabatic passage to perform a quantum state transfer Pellizzari (1997) in the same setup but with classical driving fields in a counterintuitive order, in which the receiving drive is turned on before the emitting drive, as shown in Fig. 1c. While experiments performing state transfer by wave packet shaping have already been carried out Ritter et al. (2012), state transfer between remote nodes by adiabatic passage has yet to be realized experimentally.
The central problem for deterministic state transfer protocols is photon loss. Photon losses mainly occur either during transmission in the waveguide or locally in the cavities. Note that even for links spanning hundreds of meters, state-of-the-art cavity setups with mirror absorption losses of only a few parts per million per round-trip nevertheless operate in a ‘cavity loss’ dominated regime (see Table 1).
In this article, we analyze the limitations and prospects for transferring quantum information in the presence of the aforementioned photon losses, leading to two main results: First, we show that neither wave packet shaping nor adiabatic passage can mitigate waveguide transmission losses. It has been stated in the literature (e.g., in Refs. Serafini et al. (2006); Yin and Li (2007); Chen et al. (2007); Ye et al. (2008); Lü et al. (2008); Zhou et al. (2009); Clader (2014); Chen et al. (2015); Hua et al. (2015); Huang et al. (2016)) that waveguide losses can be avoided in the single mode or short-fiber limit Pellizzari (1997), in which the cavities couple effectively only to a single mode of the waveguide. We show that this is incorrect since the photon arrival probability is bounded by a trade-off between losses due to non-adiabaticity and losses due to coupling to off-resonant waveguide modes. Taking this trade-off into account will be important for optimizing the experimental design parameters of future quantum networks.
Second, we derive an analytical solution of the achievable state transfer success probability for adiabatic passages and provide a full numerical analysis. With this analysis, we show that, in contrast to wave packet shaping, quantum state transfer by adiabatic passage can mitigate losses due to absorption in the cavities far beyond the regime of lossless waveguides introduced in the original proposal Pellizzari (1997) and the single mode limit introduced in Ref. van Enk et al. (1999). We show, however, that the single mode limit imposes far stronger constraints on the parameters of the system than is necessary: in order to mitigate cavity losses it is sufficient to be in the ‘long photon limit’, in which the effective length of the photon is longer than the distance between the two nodes of the setup. The long photon regime is naturally reached for short transmission links and can be realized for distances up to thousands of kilometers, using slowly varying classical driving fields, by state-of-the-art experiments.
The paper is organized as follows: First, we provide a brief overview of the setup and the main results in Sec. II. In Sec. III we describe the setup and the two quantum state transfer protocols under consideration in detail. In Sec. IV we treat the influence of waveguide losses; Sec. V then also includes the influence of cavity losses. Finally, we discuss further experimental imperfections in Sec. VI and give our conclusion and outlook in Sec. VII.
Ii Overview and main results
In the following we provide a brief overview of the setup, the two quantum state transfer protocols considered and the main results. The rest of the paper provides the detailed explanation and derivations of these main results.
Setup: We consider two emitters (matter qubits) placed in distant cavities and that are connected by a waveguide (for example, an optical fiber) of length , as displayed in Fig. 1a and detailed in Sec. III.1. We consider the transfer of a quantum state encoded in the ground state levels of the emitters from cavity to cavity ,
where and are the normalized amplitudes and the indices and refer to the qubits in cavities and , respectively. The state of the emitter qubit is mapped to flying photonic Fock states such that , (and conversely , ) Boozer et al. (2007), where the quantum state with index refers to the photonic state in both cavity and waveguide. Note that this setting is not limited to the use of photonic Fock states , . Quantum information can also be encoded in the light field using polarization qubits (our results apply to specific types of polarization encoded state transfer protocols, as explained in pol ()).
We assume here two identical cavities and of length , with . The outer mirrors (M1 and M4 in Fig. 1a), not coupled to the waveguide, have a reflectivity . The two inner mirrors (M2 and M3 in Fig. 1a), adjoined to the waveguide, have a reflectivity . The cavities are asymmetric with reflectivities , such that photons leave predominantly through the inner mirrors. The rate of this desired photon coupling between waveguide and either cavity is proportional to the transmission of the interfacing mirrors (M2 and M3).
We consider waveguide losses parameterized by a loss rate , and cavity losses at a rate . The latter rate refers to photons leaving through the outer mirrors (M1 and M4) with transmission (with ), and absorption and scattering losses in the mirrors at rate . The effect of other experimental imperfections such as spontaneous decay of the emitters, timing errors of the classical drive and in- and outcoupling losses due to imperfect coupling of cavity and waveguide will be discussed in Sec. VI.
Classification of the relevant regimes: We distinguish between two different regimes, the single mode limit and the long photon regime. To this end, we introduce two length scales: refers to the natural spatial length of a photon that got emitted by a cavity (in the absence of an atom). refers to the spatial length of a photon that got emitted by a qubit driven by a classical field mediated by a cavity (see inset in Fig. 1a). The single mode limit refers to the parameter regime in which the cavity linewidth is much smaller than the free spectral range of the waveguide (with the speed of light in the waveguide) Pellizzari (1997). This regime is characterized by the single mode parameter
We define the long photon limit through two main conditions. First, the desired coupling rate of the cavity to the waveguide is assumed to be much larger than the effective coupling of the qubit to the cavity (defined in Eq. (18)) such that the cavities’ photon population is always much less than one. Under this assumption, the cavity can be eliminated, leading to an effective qubit-waveguide coupling rate (Sec. III.2.1). In analogy to the natural spatial length of the photon as defined above, the length of the photon is defined by . Second, the length of the photon is assumed to be larger than the link, such that .
While is a fixed quantity for a given setup (see Table 1 for typical values), can be varied via the effective coupling between qubit and cavity.
Current experiments can access the long photon limit by choosing a small amplitude of the effective coupling and applying the classical driving field for a long exposure time.
In particular, they can reach the regime ,
in which they can operate in the long photon limit but not in the single mode limit for a given fiber length .
Quantum state transfer by wave packet shaping: The standard protocol for transferring a quantum state deterministically between two cavities is based on wave packet shaping Cirac et al. (1997); Ritter et al. (2012); Stannigel et al. (2011). The main idea behind wave packet shaping is to choose a temporal variation of the classical driving field applied to the atoms in cavities and such that in the absence of losses, the photon emitted by the first cavity is perfectly absorbed by the second cavity. This approach avoids the reflection of the photon by the highly reflective mirror M3 of the second cavity due to a quantum interference effect, as studied in Gorshkov et al. (2007); Fleischhauer et al. (2000); Dilley et al. (2012). For simplicity and concreteness, we discuss a time-symmetric wave packet emitted by the first qubit Cirac et al. (1997) due to a classical coupling , which can be reabsorbed by the second qubit under a time-reversed coupling . Here is the time delay between the first and the second coupling; see Fig. 1b and Sec. III.2.1. Note that the wave packet is not required to be symmetric: any choice of shaping pulses that avoids the reflection of the wave packet from cavity yields the limitations discussed below.
Quantum state transfer by adiabatic passage: Adiabatic passage as a protocol to transfer a quantum state between two remote qubits in cavities Pellizzari (1997) uses the methods known from STIRAP in atoms Vitanov et al. (2017) within the setup shown in Fig. 1a. The principal idea is to perform a coherent transfer through a dark state with respect to the photon fields. This transfer is accomplished by temporally shaping the intensity of the classical driving fields of both atoms with a Gaussian shape in a counterintuitive order; see Fig. 1c and Sec. III.2.2. Importantly, adiabatic passage state transfer has to be performed in the long photon limit.
Limitations of wave packet shaping: We find that, by using the method of wave packet shaping, the maximal success probability (formally defined in Sec. III.1.3) of quantum state transfer is strictly limited by (below), i.e., ; see Sec. IV and Sec. V. Here, is given by
and denotes the probability of a photon to propagate through a waveguide of length
multiplied by the probability of a photon being emitted from the cavity into the desired output mode
and being absorbed by the second cavity . Due to symmetry reasons, the probability for a photon to enter the second cavity equals the emitting probability, i.e., .
Waveguide losses: It has been stated Serafini et al. (2006); Yin and Li (2007); Chen et al. (2007); Ye et al. (2008); Lü et al. (2008); Zhou et al. (2009); Clader (2014); Chen et al. (2015); Hua et al. (2015); Huang et al. (2016) that limitations due to waveguide losses can be overcome in the single mode limit. These results are based on a description that takes only a single waveguide mode into account and in which, in analogy to stimulated Raman adiabatic passages (STIRAP), the corresponding success probability of state transfer is given by
as detailed in Appendix B. The effective atom-waveguide coupling is denoted by (see Sec. IV.2) and the pulse width of the driving laser by (see Sec. III.2.2). In the adiabatic limit the success probability reaches unity, corresponding to a perfect state transfer. We provide an analytical example that demonstrates why the coupling to far-detuned waveguide modes can in fact not be neglected. As explained in Sec. IV.2, including three waveguide modes already leads to non-negligible effects, even deep in the single mode limit. The corresponding amended success probability of state transfer is given by
revealing a clear trade-off (see Sec. IV.2 for details). While the first summand in Eq. (7) recovers the dependency seen in previous work Serafini et al. (2006); Yin and Li (2007); Chen et al. (2007); Ye et al. (2008); Lü et al. (2008); Zhou et al. (2009); Clader (2014); Chen et al. (2015); Hua et al. (2015); Huang et al. (2016), the second summand in Eq. (7) arises due to the coupling to detuned waveguide modes. As a result, choosing the adiabatic limit as done in previous work is in fact incompatible with obtaining a high success probability of state transfer. Optimizing with respect to leads to
These results are also shown numerically for an even larger parameter space, taking a large number of waveguide modes into account (see Sec. IV.1). We show that, in the absence of cavity losses (), the success probability of state transfer is strictly limited by .
Cavity losses: In contrast to wave packet shaping, quantum state transfer by adiabatic passage allows one to outperform limitations due to cavity losses imposed by . Note that for high-finesse cavities, cavity losses can play a significant role due to the high number of round-trips of the photon. Experimental values of for current optical setups are given in Table 1.
It has already been shown that limitations due to can be mitigated for perfect waveguides () Pellizzari (1997) and in the single mode limit for imperfect waveguides van Enk et al. (1999). In this article, we show that quantum state transfer by adiabatic passage can in fact mitigate cavity losses for both and well beyond the single mode limit; see Sec. V and Fig. 7. We find that the parameter regime over which cavity losses can be mitigated is determined by the long photon limit. The figure of merit determining the maximal success probability of state transfer for a given waveguide with length and loss rate is the probability of the photon to leave the cavity , as demonstrated in Sec. V. Extending the analytics for waveguide losses only, we show that the success probability of state transfer in the presence of both cavity and waveguide losses is given by
where (see Appendix A). As a result, we find (Sec. V.2 and Fig. 7) that the success probability of state transfer by adiabatic passage exceeds , i.e., , and thus adiabatic passage allows for better state transfer performance than wave packet shaping, which is limited by , cf. last two columns of Table 1.
For adiabatic passages, the same state transfer success probability can therefore be obtained using a cm-long slowly emitting cavity with a linewidth of tens of kHz or a m-short fast emitting cavity with a linewidth of tens of MHz, as long as their probabilities of emitting the photon into the desired output mode are equal. Experimental values for the state transfer probability that can be reached by adiabatic passages are given in Table 1.
|Mainz Pfister et al. (2016)|
|Innsbruck Stute et al. (2012)|
|Paris Hunger et al. (2010)|
|Bonn K Steiner et al. (2014)|
|Caltech Hood et al. (1998); van Enk et al. (1999)|
|Hamsen et al. (2017); Chibani (2016)|
|Bonn M Gallego et al. (2016); WAl ()|
|Aarhus Herskind et al. (2008)|
|Sussex Begley et al. (2016); MKe ()|
Iii Setup and Transfer Protocols
In this section, we provide a detailed description of the setup (Sec. III.1) and the state transfer protocols (Sec. III.2) under consideration. In the following, we use the language of optical platforms, considering atoms as matter qubits and an optical fiber as a waveguide. Note that our derivations also apply to other platforms such as, e.g., superconducting qubits.
iii.1 Basic Model
We model the cavity-fiber-cavity system as three linearly coupled cavities with the field modes being represented by independent annihilation and creation operators of the corresponding cavity or fiber mode. As explained in Appendix D, we also employed an alternative description for our numerical simulations in which the system is described by the eigenmodes of the cavity-fiber-cavity system. Both descriptions yield the same results in the regime of high finesse cavities and for time scales that are long compared to the round-trip time of a photon. Throughout the main text, we will use the former choice of basis states.
The full Hamiltonian for the setup under consideration in Fig. 2 is given by
The Hamiltonian describes the bare evolution of both cavities and and is given by
where the annihilation operator () refers to the cavity mode of cavity (). In Eq. (12) we consider only a single cavity resonance for each cavity, with frequency for both cavities. The restriction to a single cavity mode is well justified in the limit in which the cavity length is much smaller than the fiber length, .
The fiber modes are described by the Hamiltonian
where the annihilation operator denotes the th fiber mode with frequency . We assume the fiber mode with frequency to be resonant with the cavity modes and , which translates into the condition with integer . Note that the fiber can alternatively be modeled by using spatially localized modes, allowing for a more intuitive representation of a travelling photon Ramos et al. (2016).
The interaction Hamiltonian is given by the coupling between cavity and fiber modes Pellizzari (1997)
where h.c. is the Hermitian conjugate. The coupling strengths and of the cavity modes and to the fiber modes are related to the effective decay rates of the cavities or coupled to the fiber given by Vermersch et al. (2017). For optical implementations, the cavity emission rate into the desired output mode is given by van Enk et al. (1999). Therefore, the coupling strengths between cavity and fiber modes are given by
where is the transmission coefficient of the identical inner mirrors (M2 and M3 in Fig. 1) and is the speed of light in the fiber. Note that the coupling is equally strong for all fiber modes. The phase factor in Eq. (14) introduces alternating signs for the coupling to even or odd modes in the fiber. As illustrated in Fig. 3c, even and odd fiber modes correspond to wave functions with an even or odd number of nodes in the intensity profile.
Each atom is modeled as a three-level system with degenerate ground states and of equal energy and an excited state with energy with respect to the ground states (see Fig. 2). The bare atomic Hamiltonian is hence given by
The transition between the atomic ground state and the excited state is coupled to the cavity field () in cavity () with coupling strength (). The transition between the atomic ground state and the excited state is driven by a time-dependent classical field with Rabi frequency and frequency . Note that the atom-cavity coupling depends on the mode volume of the cavity and on the position and dipole moment of the atom Claude Cohen-Tannoudji (2004). In order to avoid populating the excited state , which suffers from spontaneous emission at rate , the classical drive and cavity are strongly detuned from the atomic transition, i.e., such that photon loss due to atomic decay is strongly suppressed. The effect of spontaneous emission is discussed in Sec. VI. Due to the strongly detuned laser drive, the excited state can be eliminated such that the effective atom-cavity interaction Hamiltonian Cirac et al. (1997) is given by
where () is the raising (lowering) operator for the qubit in cavity . The effective atom-cavity coupling is given by
After the excited state is eliminated, the bare atomic Hamiltonian in Eq. (16) vanishes. Note that eliminating the excited state also results in effective Stark shifts for both ground states and , which however can be compensated; see Ref. Pellizzari (1997).
The full Hamiltonian given in Eq. (11) can be expressed in an interaction picture with respect to such that
Here, we discuss the two main sources of imperfection in deterministic state transfer: fiber and cavity losses. The influence of other imperfections will be discussed in Sec. VI.
The loss Hamiltonian is given by . To model losses in the fiber, we consider each fiber mode to couple in a Markovian way to a bath of bosonic modes with annihilation (creation) operators () described by the Hamiltonian
The fiber loss rate , where is the absorption coefficient in the fiber van Enk et al. (1999). The absorption coefficient is defined by the fraction absorbed inside a fiber of length
where is the attenuation coefficient of the fiber in decibels per kilometer. For telecom wavelength fibers, a typical attenuation is dB/km, yielding a fiber loss rate of kHz, and for optical wavelengths, a typical attenuation is dB/km, with rate kHz. Note that frequency conversion from optical to telecom wavelengths has been achieved with efficiencies up to , e.g., Pelc et al. (2011). The probability of a photon to propagate through a fiber of length is given by , as defined in Eq. (4).
Equivalently, we model cavity losses by considering each cavity and to decay to free space, with the interaction given by the coupling of cavity modes and to a frequency bath with annihilation (creation) operator () and ():
Here, cavity losses at rate include the losses through the outer mirrors (M1 and M4 in Fig. 1a) with transmission as well as absorption losses in the cavities. The total linewidth of the cavity consists of the rate of coupling into the fiber as well as the total loss rate such that
where contains both transmission and absorption losses: . In Table 1 we summarize the cavity losses of a selection of experiments. The probability of a photon to be emitted into the desired output mode as defined in Eq. (5) can be rephrased as
which is equivalent to the probability of the photon being absorbed by the (second) cavity.
iii.1.3 Equations of Motion
As we are interested in performing a quantum state transfer, we solve the dynamics of the full system according to the Hamiltonian in Eq. (19), taking into account the loss mechanisms described in Eqs. (20), (22) and (23) using a single-excitation Wigner Weisskopf ansatz. The wave function of the full model in this single excitation ansatz is given by
where denotes the state of system with the excitation in the atom in cavity with amplitude , the state with the excitation in the cavity with amplitude and the state with the excitation in the th fiber mode with amplitude . The sixth and seventh term in Eq. (27) describe the baths associated with the cavity and fiber losses as modeled in the previous section, where is the vacuum state of light field and are the amplitudes of the baths . Lastly, the amplitude denotes the state of the system without an excitation, i.e., corresponds to the state in which both atoms are in the ground state , while the cavities and the fiber are empty.
Starting from the Schrödinger equation , we obtain the time evolution of the amplitudes of the system
where the amplitudes of the lossy channels have been intregrated out Dorner and Zoller (2002). Finally, we solve Eq. (28) for the initial state to obtain the success probability of the state transfer, which we define as the probability
This probability provides a measure for a successful transmission of the photonic excitation through the setup. The error denotes the probability to emit a photon into an undesired channel , or . Fig. 2b illustrates the coupling scheme corresponding to Eq. (28).
iii.2 Quantum State Transfer Protocols
iii.2.1 Wave Packet Shaping
As explained in Sec. II, one possibility to realize quantum state transfer by wave packet shaping is to produce a time-symmetric photon wave packet inside the fiber by the first combined atom-cavity system such that the back reflection of the wave packet at the inner mirror of the second cavity (M3 in Fig. 2) is prevented Cirac et al. (1997).
For the time-symmetric wave packet shaping we consider here, it is essential to use a temporal profile for the classical drive of the first atom that produces a time-symmetric wave packet in the fiber. The classical drive for the second atom is then given by the time-reversed temporal profile with time delay . We consider the regime in which the maximal coupling between atom and cavity is much smaller than the cavity decay rate (and equal for both cavities). In this regime, we can effectively eliminate the cavity, which results in an effective coupling rate between the atoms and the fiber modes given by Habraken et al. (2012).
In this case, a possible classical drive sequence for the atom-fiber coupling rate is given by Ref. Stannigel et al. (2011)
where the maximal atom-fiber coupling rate is given by . This drive sequence generates a time-symmetric wave packet with exponential shape and of length . Experimentally, the relevant parameter to vary the coupling rate is given by the classical laser drive , as shown in Fig. 1b. This classical drive relates to the effective atom-fiber drive sequence in Eq. (30) through the expression
Note that the wave packet shaping approach works for both limits, the long photon limit () and also the limit of short wave packets with .
iii.2.2 Adiabatic Passage
The general idea of performing quantum state transfer by adiabatic passage is to use the methods known from STIRAP Vitanov et al. (2017) for atoms to perform a coherent transfer by using a dark state with respect to the photon fields Pellizzari (1997); van Enk et al. (1999).
The time-dependent coupling of the atoms to the cavity modes in Eq. (19) is varied via the classical laser drive of the atoms. The classical laser drive of both atoms realizes a counterintuitive pulse sequence Vitanov and Stenholm (1997a), in which the classical field in the receiving cavity is switched on before the driving field of the sending cavity :
We choose the temporal profiles of both pulses to be Gaussian functions of equal maximal strength with a retardation between them
where is the pulse width, cf. Fig. 1c. In general, the temporal separation of the pulses is on the order of the pulse width Vitanov and Stenholm (1997a), such that we introduce the relative temporal separation .
Performing a quantum state transfer by adiabatic passage requires the optimization of three parameters: the coupling strength , the temporal width of the classical drive and the relative temporal separation of the two pulses . The maximal coupling strength is optimized for fixed atom-cavity coupling .
In contrast to wave packet shaping, quantum state transfer by adiabatic passage only works in the long photon limit. The reason lies in the mechanism of adiabatic passage, for which a standing wave of the photon field is required to perform the transfer.
Iv Fiber Losses
In this section, we evaluate and discuss the influence of fiber losses on the achievable quantum state transfer success probability by means of a numerical analysis (Sec. IV.1) and an analytical example (Sec. IV.2). The effect of cavity losses will be addressed later in Sec. V.
iv.1 Numerical Treatment
We numerically study a wide parameter range, including the concrete regime (single mode limit) that has been identified in the literature Serafini et al. (2006); Yin and Li (2007); Chen et al. (2007); Ye et al. (2008); Lü et al. (2008); Zhou et al. (2009); Clader (2014); Chen et al. (2015); Hua et al. (2015); Huang et al. (2016) as the regime in which limitations can be overcome (see Sec. IV.2 and Appendix B). As a result, we find that even deep in the single mode limit, the success probability of the state transfer is always limited by (which is equal to for ). This result holds for both state transfer methods.
Fig. 4 provides an example of the state transfer success probability , defined in Eq. (29), as a function of the fiber length based on the experimental parameters in Ref. Stute et al. (2012) for (the effect of cavity losses will be included in Sec. V below). In this example, the identical cavities and have a length of m and an inner mirror (M2 & M3 in Fig. 1) with transmissivity of ppm.
We consider two fibers with different loss rates : first, absorption losses of dB/km corresponding to fibers at telecom wavelengths and second, absorption losses of dB/km corresponding to optical wavelengths.
In addition, in Fig. 4 we compare both state transfer methods. For every simulation, we ensure that we consider sufficiently many fiber modes in Eq. (19) that our results converge with respect to the number of fiber modes included.
For wave packet shaping, the state transfer success probability is optimized in the regime , in which the cavity can be eliminated WPS (). In the case of adiabatic passage, every plot point in Fig. 4 is optimized with respect to the pulse length , the relative temporal separation of the pulses and the pulse area . In particular, the optimized values for adiabatic passage lie in the same regime as those for wave packet shaping, i.e., , in which the cavity is barely populated. In order to reach high success probabilities, the pulse area must be large (), and due to the necessity of a weak coupling , this is achieved for long pulse durations . The pulse length is varied in the range , which translates for the chosen parameters into pulse lengths of s. The relative separation of the pulses and the coupling strength ratio are optimized in the ranges and . As a benchmark for the state transfer we plot the expected survival probability of a photon through a lossy fiber as given in Eq. (26).
We find that the numerically optimized success probability for both state transfer methods and both fiber absorption losses is in excellent agreement with the limit given by (see Fig. 4). For wave packet shaping, this agreement seems natural because, in the optimal case, the photon only passes through the fiber once. In the case of adiabatic passages, the agreement implies that the state transfer success probability is limited by and therefore fiber losses cannot be overcome. Note that the numerical results shown in Fig. 4 are obtained for a parameter set deep in the single mode limit (see Fig. 3), for which the single mode parameter (defined in Eq. (2)) for m is . Numerical results were however also derived for different parameter sets () beyond the single mode limit. We find that for all regimes the quantum state transfer is strictly limited by .
iv.2 Analytical Example
It has been stated in the literature Serafini et al. (2006); Yin and Li (2007); Chen et al. (2007); Ye et al. (2008); Lü et al. (2008); Zhou et al. (2009); Clader (2014); Chen et al. (2015); Hua et al. (2015); Huang et al. (2016) that the Hamiltonian in Eq. (19) can be simplified in the single mode limit by neglecting all fiber modes but the resonant mode . This argument is based on the weak coupling assumption , under which the fiber modes are far detuned from the cavities. However, as we show below, these off-resonant contributions integrated over long times lead to non-negligible effects.
As detailed in Appendix B, the simplified Hamiltonian that includes only a single fiber mode as used in Refs. Serafini et al. (2006); Yin and Li (2007); Chen et al. (2007); Ye et al. (2008); Lü et al. (2008); Zhou et al. (2009); Clader (2014); Chen et al. (2015); Hua et al. (2015); Huang et al. (2016) leads, in complete analogy to STIRAP in a three-level atomic system, to a success probability of state transfer by adiabatic passage that reaches unity in the adiabatic limit ,
where denotes the maximal coupling of atom and fiber.
In the following, we give an illustrative analytical example that points out a crucial difference between adiabatic passages in atoms compared to adiabatic passage state transfer in a fiber.
We consider the regime in which the decay of the cavity is much larger than the atom-cavity coupling . We focus here on fiber losses and assume therefore throughout this section. In this regime, we can adiabatically eliminate the cavity and obtain an effectively coupled atom-fiber-atom system; see Fig. 5a. In Appendix C we discuss the very similar case of a purely photonic model in which the state of the atom is mapped rapidly onto the cavity, followed by a state transfer in the coupled cavity-fiber-cavity system.
A full description of the problem would include all fiber modes with . However, the importance of going beyond the single mode description (Appendix B), even in the regime in which and , is well illustrated by including the first pair of far detuned fiber modes in our analytical derivation of the success probability of the adiabatic state transfer. The equations of motion for the wave function can be derived from Eq. (28) by eliminating the cavity modes. We consider here three fiber modes such that the atom-fiber-atom system is described by
where is the effective atom-to-fiber coupling strength (see Fig. 2b). The level scheme analogue for this specific example is given in Fig. 5a. As introduced in Sec. III.1 even and odd modes (Fig. 3c) couple in Eq. (35) with a different sign to the cavity due to the factor in Eq. (19). This sign is the crucial difference with respect to STIRAP in atoms. In atoms, where all excited states couple with the same strength and phase to the ground states, there exists a dark state with respect to the whole manifold of excited states. In our case, however, due to the alternating sign of the coupling in Eq. (19), it is impossible to find a dark state with respect to the whole manifold of excited states because a superposition state that is dark with respect to the even modes couples to the odd modes and vice versa van Enk et al. (1999). It is thus apparent that some fiber modes (even or odd) have to be populated during the state transfer, and that consequently losses due to absorption in the fiber are unavoidable and affect the success probability of the transfer.
We consider here the particular case in which the temporal profiles of the classical driving fields in cavities and are given by a sine and a cosine function respectively (see Appendix A for details). This specific choice allows us to derive a simple analytical expression for the state transfer success probability as defined in Eq. (29). To this end, we adiabatically eliminate the far detuned modes in Eq. (35) under the assumption . The presence of leads to effective dynamics of the modes , , and that include loss terms acting on the qubits in cavities and (Fig. 5b), resulting in
where the maximal values of the coupling strengths are chosen to be equal: . The first summand in Eq. (36) yields the result as obtained from STIRAP and hence the naively truncated Hamiltonian (cf. Eq. (34) and Appendix B), resulting in a success probability of unity in the limit . However, the second summand, which is due to the presence of the far detuned fiber modes , compensates for this effect. More specifically, increasing also increases the effect of the effective decay terms on the qubit ground states. Due to this trade-off, there is an optimal value which balances the effects of non-adiabaticity (first summand) and the effects due to the coupling to the off-resonant fiber modes (second summand), leading to
which coincides with the transmission probability of a photon through a fiber of length . This result also agrees with our numerical simulations in Sec. IV.1.
V Cavity Losses
In this section, we show that in contrast to the restrictions due to fiber losses, the problem of cavity losses, which limits wave packet shaping, can be overcome to a significant extent in current experimental settings by using adiabatic passages. We derive an approximate analytical solution for the state transfer success probability that can be achieved by performing adiabatic state transfers (Sec. V.1) and provide a numerical analysis for both methods (Sec. V.2).
v.1 Approximate Analytical Treatment for Adiabatic State Transfers
In the following, we extend the analytical example provided in Sec. IV.2, which models a quantum state transfer by adiabatic passage for , to cover the effect of both cavity and fiber losses. To this end, we adiabatically eliminate the cavity modes in Eq. (28) in the limit . As in the previous section, we only include the fiber mode resonant with the cavity and the first pair of detuned fiber modes . The resulting equations of motion describe the interaction between the qubits and the fiber modes with an effective coupling strength as shown in Fig. 2b and are given by Eq. (35), with the first and last equation modified to
In this description, cavity losses lead to an effective decay that acts on the qubits with rate . Using these equations of motion, and assuming classical driving fields of sine- and cosine shape (see Sec. IV.2 and Appendix A), the initially complex problem involving time-dependent couplings and decay rates can be cast into a simpler form. This simplified description allows us to derive an approximate solution of the success probability of state transfer by adiabatic passage. As detailed in Appendix A, this solution takes the form
We find that this analytical expression agrees well with the full numerical simulation of the achievable state transfer success probability presented in the following Sec. V.2 (see Fig. 7 and Fig. 8). Note that in the case of vanishing cavity losses (, i.e., ), Eq. (40) reduces to Eq. (37).