For a one-dimensional (1D) waveguide coupled to two or three qubits, we show that the photon-photon correlations have a wide variety of behavior, with structure that depends sensitively on the frequency and on the qubit-qubit separation L. We study the correlations by calculating the second-order correlation function ${g}_{2}(t)$ in which the interference among the photons multiply scattered from the qubits causes rich structure. In one case, for example, transmitted and reflected photons are both bunched initially, but then become strongly anti-bunched for a long time interval. We first calculate the correlation function ${g}_{2}(t)$ including non-Markovian effects and then show that a much simpler Markovian treatment, which can be solved analytically, is accurate for small qubit separation. As a result, the non-classical properties of microwaves in a 1D waveguide coupled to many superconducting qubits with experimentally accessible separation L could be readily explored with our approach.

One-dimensional (1D) waveguide-QED systems are currently generating increasing interest - systems in which photons confined in one-dimension interact with one or several two-level systems (qubits). Part of the motivation comes from the striking quantum optics effects that can be seen in these strongly coupled systems [1–19]. Another motivating factor is the promise of waveguide-QED systems for quantum information processing [20–27]. Finally, a key driver of the interest in waveguide-QED systems is the tremendous experimental progress that has been made recently in a number of systems [28–41]. Perhaps the leading system for waveguide-QED investigations and applications is an open microwave transmission line coupled to superconducting qubits [34–37, 42, 43]. While much of the work to date has focused on systems in which there is a single qubit, and there is a growing literature on the case of two qubits [10, 17, 19, 41, 44–49], an important future direction for both fundamental effects and possible applications is to study a waveguide coupled to multiple (or many) qubits. As a step in this direction, here we compare and contrast results for one, two, and three qubits coupled to a waveguide [see Figure 1(a)], focusing in particular on the generation of photon-photon correlations.

Correlations between photons are a key signature of non-classical light. They are often characterized by the second-order correlation function (photon-photon correlation function) ${g}_{2}(t)$ where t is the observation time between the two photons (see below for precise definition) [50]. The uncorrelated, classical value is ${g}_{2}=1$ (obtained, for example, for a coherent state). Bunching of photons, ${g}_{2}>1$, often occurs due to the bosonic nature of photons but anti-bunching, ${g}_{2}<1$, also occurs [50]. In recent experiments, ${g}_{2}(t)$ of microwave photons coupled to superconducting qubits was measured, and both bunching and anti-bunching were observed [38, 51]. In a multi-qubit situation, one expects to have interference between the various scattered partial waves; interference effects in the photon-photon correlations ${g}_{2}(t)$ are known as ‘quantum beats’ [52].

In this paper, we first present our method of calculation, which exploits a bosonic representation of the qubits in the rotating wave approximation. We obtain a complicated yet analytic result for ${g}_{2}(t)$ in the Markovian limit and show, by comparison with the full numerical result, that it is adequate for small, experimentally accessible separations between the qubits. In presenting results, we focus on an off-resonant case in which single photons have equal probability of being transmitted or reflected, and take the separation between qubits, denoted L, to be either ${\lambda}_{0}/4$ or ${\lambda}_{0}/8$ where ${\lambda}_{0}$ is the wavelength of a photon at the qubit resonant frequency. We find several striking features in ${g}_{2}(t)$: First, for $L={\lambda}_{0}/4$, the transmitted photons are largely bunched for all times and become more strongly bunched as the number of qubits increases, while the reflected photons oscillate between strong bunching and anti-bunching, showing particularly strong quantum beats in the three qubit case. Second, for $L={\lambda}_{0}/8$, we find the surprising situation that both transmitted and reflected photons are bunched at $t=0$ but then become anti-bunched for a large time interval. This suggests that the photons in this case become organized into bursts.

2 Method

The Hamiltonian describing N identical qubits coupled to a 1D waveguide [see Figure 1(a)] is, in the rotating wave approximation,

where ${\sigma}_{i}^{\pm}$ are the raising/lowering operators for i th qubit, ${l}_{i}$ is its position which is fixed by $L={l}_{i+1}-{l}_{i}$, ${\omega}_{0}$ is the transition frequency of the qubit, and ${\mathrm{\Gamma}}^{\prime}$ is the decay rate to channels other than the waveguide. The spontaneous decay rate to the waveguide continuum is given by $\mathrm{\Gamma}=2{V}^{2}/c$. In the waveguide QED context, ‘strong coupling’ signifies that the spontaneous decay rate to the waveguide is much faster than the decay to all other modes, namely that the Purcell factor is large, $P\equiv \mathrm{\Gamma}/{\mathrm{\Gamma}}^{\prime}\gg 1$.

To find ${g}_{2}(t)$, we first obtain the two-photon eigenstate of ${H}_{0}$. As discussed in Ref. [17], it is convenient to use a bosonic representation of the qubits that includes an on-site interaction,

The raising/lowering operators ${\sigma}_{i}^{\pm}$ in ${H}_{0}$ are replaced by the bosonic creation/annihilation operators ${d}_{i}^{\u2020}$ and ${d}_{i}$, respectively. One then takes $U\to \mathrm{\infty}$ in the end to project out occupations greater than 1. In this bosonic representation, the $U=0$ case corresponds to a non-interacting Hamiltonian and can readily be solved. In terms of the non-interacting wavefunctions and Green functions, a formal expression for the two-photon ‘interacting’ wavefunction in the $U\to \mathrm{\infty}$ limit can be obtained; this then is the solution to the waveguide QED problem in which we are interested. Finally, the two-photon wavefunction together with the one-photon wavefunction yields ${g}_{2}(t)$ for a weak incident coherent state. More details of this procedure are given in the appendices.

The Markovian approximation allows a considerable simplification of the final result [17]. In the present context, the Markovian approximation consists of an approximate treatment of certain interference terms valid for small separation between the qubits. In the formal expression for the two-photon wavefunction discussed above, there is an integral over the non-interacting wavefunctions which generally must be performed numerically. The non-interacting wavefunctions naturally involve interference factors ${e}^{i2kL}$ that make this integral difficult. However, if the qubits are close enough, k may be replaced by ${k}_{0}={\omega}_{0}/c$, allowing the integral to be performed analytically using contour integration (the analytic expression of the final result is lengthy, so we just give the steps of the derivation in the appendices as well as the $N=2$ result as an example). All of the results in this paper are obtained in the regime where this is valid. An example of the checks we have made is shown in Figure 1(b): the full numerical result is in good agreement with that from the small separation approximation.

We compare the one, two, and three qubit cases: $N=1,2,\text{or}3$. In order to make a fair comparison, the typical transmission through the system in the three cases should be the same; otherwise, the lower probability of finding a photon in one case compared to another will affect ${g}_{2}$. We therefore consider off-resonance cases (i.e.$\omega \ne {\omega}_{0}$ where ω is the incoming photon frequency) in which the single-photon transmission probability T is fixed. Because the single-photon transmission spectrum depends on the number of qubits, the frequency used is different in the three cases $N=1\text{-}3$. Due to the asymmetry of the single-photon transmission spectrum in certain cases, the criterion used throughout this work is to pick up the frequency closest to ${\omega}_{0}$ so that ${g}_{2}(0)$ is the largest.

In the following results, we consider $N=1,2,\text{or}3$; ${k}_{0}L=\pi /4\text{or}\pi /2$; and $T=50\mathrm{\%}$. The single photon transmission curves used to choose the photon frequency ω are shown in Figure 2. We use Γ as our unit of frequency, take ${\omega}_{0}=100\mathrm{\Gamma}$, and consider the lossless case, ${\mathrm{\Gamma}}^{\prime}=0$.

3 Results

The results for a single qubit, shown in Figure 3 panels (a) and (d), provide a point of comparison for the two and three qubit cases discussed below; throughout we consider the response to an incident weak coherent state. Non-classical light in a waveguide produced by a single qubit has been extensively investigated theoretically [1, 2, 4–6, 8, 11–13] as well as experimentally with microwave photons [38]. We see that for our chosen detuning such that $T=50\mathrm{\%}$, the transmitted field shows bunching while the reflected field is anti-bunched. The correlation decays to its classical value (namely, 1) quickly and with little structure. For this reason the single value ${g}_{2}(0)$ is a good indication of the nature of the correlations overall. Note that in panel (d), ${g}_{2}(0)=0$ due to the inability of a single excited qubit to release two photons at the same time.

For $N=2\text{or}3$, we start by considering the case ${k}_{0}L=\pi /2$, in which case the qubits are separated by ${\lambda}_{0}/4$; the results are shown in Figure 3. The presence of quantum beats coming from interference among the partial waves scattered by the qubits is clear, especially for three qubits. In the transmitted wave, photon bunching is considerably enhanced in magnitude and extends for a longer time (compared to a single qubit). In reflection, ${g}_{2}(t)$ develops a striking oscillation between strongly bunched and anti-bunched [panel (f)]. Such behavior in ${g}_{2}$ suggests that the photons become organized periodically in time and space.

Turning now to the case ${k}_{0}L=\pi /4$ (a separation of ${\lambda}_{0}/8$), we see in Figure 4 that the behavior is completely different. First, the quantum beats largely disappear in both transmission and reflection. Instead, for $N=3$ we see that both the reflected and transmitted photons are initially bunched, in the reflected case quite strongly bunched. The initial bunching is followed in both cases by anti-bunching. This anti-bunching is dramatic for the transmitted photons: strong anti-bunching persists for a time interval of several tens of ${\mathrm{\Gamma}}^{-1}$ (the natural unit of time in our problem). Initial bunching followed by a long interval of anti-bunching suggests that the photons are organized into bursts.

The different behavior for ${k}_{0}L=\pi /4$ compared to ${k}_{0}L=\pi /2$ can be traced to a difference in the structure of the poles of the single photon Green function (see, e.g., the discussion in Ref. [17]). For instance in the $N=2$ cases, for ${k}_{0}L=\pi /2$ there are two dominant poles that have the same decay rate but different real frequencies, leading to maximum interference effects between those two processes. In contrast, for ${k}_{0}L=\pi /4$, the poles have very different decay rates; the one decaying most rapidly yields the sharp initial bunching, while the one with the slowest decay produces the long time anti-bunching.

To study how the correlations depend on the frequency of the photons, we show the initial correlation, ${g}_{2}(0)$, in Figures 5 and 6. Because of the oscillating structure in ${g}_{2}(t)$ when there are multiple qubits, ${g}_{2}(0)$ is not necessarily a good indication of the behavior at later times; nevertheless, the degree of initial bunching or anti-bunching is a physically important and measurable quantity. For a fair comparison between the $N=1,2,\text{and}3$ cases, we first plot ${g}_{2}(0)$ as a function of the single photon transmission, T; see Figure 5. To match the desired T with an off-resonant photon frequency we follow the following procedure: Starting near the resonant frequency ${\omega}_{0}$ (where $T=0.1\mathrm{\%}$), we scan toward smaller frequencies until $T=99.9\mathrm{\%}$ is reached. We then use frequencies within the scanned range to calculate ${g}_{2}(0)$ for both transmission and reflection as a function of T at ${k}_{0}L=\pi /2$ and $\pi /4$. Another way of presenting the data is to simply plot ${g}_{2}(0)$ directly as a function of frequency, as in Figure 6. By comparing with Figure 2, we see that the method above for selecting the range of frequencies to use in making Figure 5 selects the range with largest ${g}_{2}(0)$ for a given value of T. Finally, note that for reflection from one qubit, ${g}_{2}(0)=0$ in all cases, as mentioned above, and so is not plotted in panels (b) and (d) of both figures.

Several general trends are clear from Figure 5. Bunching is favored over anti-bunching for both $N=2\text{and}3$. As the single photon transmission increases, ${g}_{2}(0)$ decreases for transmission but generally increases for reflection. Opposite trends for transmission and reflection are natural based on the simple argument that incoming uncorrelated photons divide between transmitted and reflected ones so that bunching in one implies anti-bunching in the other. Clearly, this simple argument does not apply here; indeed, it is striking and surprising that for a broad range of parameters both transmitted and reflected photons are bunched.

Trends as the number of qubits increases from 1 to 3 are also evident in Figure 5. In panels (a) and (d) the trend is monotonic: For ${k}_{0}L=\pi /4$, the reflected photons become tremendously bunched [panel (d)], whereas for ${k}_{0}L=\pi /2$ and transmitted photons [panel (a)], the curves cross at the same point indicating that the trend changes sign-increasing bunching as N increases for $T\ge 0.25$ but decreasing bunching for smaller T. In the other two cases, panels (b) and (c), the trend as N increases from 1 to 3 is not monotonic. For ${k}_{0}L=\pi /2$ and $T\le 0.65$, the reflected photons switch from being anti-bunched to bunched to anti-bunched as N changes from 1 to 3, but show increasing bunching for larger T [panel (b)]. Finally, in panel (c) [${k}_{0}L=\pi /4$ and transmitted photons], there is a monotonic trend toward less bunching for $T\le 0.25$ but non-monotonic behavior for larger transmission.

From the explicit dependence on frequency shown in Figure 6, we see that bunching is generally favored even outside the frequency range chosen in Figure 5 (which in the ${k}_{0}L=\pi /4$ case is quite small (<Γ)). Comparing to the single photon transmission spectrum (see Figure 2), we point out two features: First, in the ${k}_{0}L=\pi /4$ case, ${g}_{2}(0)$ shows the asymmetry with respect to ${\omega}_{0}$ [panel (c) and (d)] seen in $T(\omega )$; this again can be traced to the asymmetric pole structure of the Green functions mentioned above. ${g}_{2}(0)$ is larger (for reflection) and varies more rapidly on the red-detuned side ($\omega <{\omega}_{0}$), which explains why we chose the frequency range use in Figure 5. In fact, on the blue-detuned side ($\omega >{\omega}_{0}$) the structure in ${g}_{2}(t)$ is less dramatic, and it returns to 1 faster (data not shown). Second, as also shown in Figure 5, the peaks of ${g}_{2}(0)$ for transmission are located where $T=0$, while the peaks of reflected ${g}_{2}(0)$ are located where $T=1$. Note that the leftmost peak of $N=3$ in Figure 6(d) is completely due to the small denominator ($R=1-T\approx 0$) at that point.

4 Conclusion

In this work, we have calculated the second-order correlation function, ${g}_{2}(t)$, for photons in a one-dimensional waveguide interacting with one, two or three qubits. By taking the separation between the qubits small, we are able to make a Markovian approximation which then allows an analytic solution. The small separation and small N on which we focus means that these systems are within the range of current experimental capability [19].

The interference among the partial waves scattered from the qubits leads to a variety of behavior in ${g}_{2}$ that is sensitive to both the separation between the qubits (L) and the frequency of the incoming photons. As examples of the rich variety accessible in these waveguide QED structures, we mention three here in conclusion: (i) For a wide range of parameters, both transmitted and reflected photons are initially bunched. (ii) For reflected photons with $N=3$ and ${k}_{0}L=\pi /2$, ${g}_{2}(t)$ oscillates between bunching and anti-bunching [Figure 3(f)]. (iii) For transmitted photons with $N=3$ and ${k}_{0}L=\pi /4$, initial strong bunching is followed by a long (i.e.$\sim 30{\mathrm{\Gamma}}^{-1}$) interval of antibunching [Figure 4(c)]. These last two observations suggest that some nascent organization of the photons may be occurring, providing an interesting direction for future research.

The single photon eigenstate $|{\varphi}_{1}(k){\u3009}_{\alpha}$ with $\alpha =\mathrm{L},\mathrm{R}$ is by definition the eigenstate of ${H}_{0}$, i.e., ${H}_{0}|{\varphi}_{1}(k){\u3009}_{\alpha}=\u0127ck|{\varphi}_{1}(k){\u3009}_{\alpha}$, where

and the incoming photon travels in the α-direction with wavevector k. The single photon transmission amplitude is given by ${t}_{N}(k)$ and the reflection amplitude by ${r}_{1}(k)$. Note that the positions of the qubits are chosen to be symmetric with respect to the origin, i.e., ${l}_{N-i+1}=-{l}_{i}$, in order to take advantage of parity symmetry. Setting $\u0127=c=1$ from now on, we have for $N=2$[17]

with ${\eta}_{\pm}\equiv 2k-2{\omega}_{0}\pm i\mathrm{\Gamma}$. Note that we do not need the other amplitudes for the rest of this section. For $N=1$ results see, e.g., Ref. [8].

We can now construct the two-photon ‘non-interacting’ eigenstate

where V is given in Eq. (2) and E is the two photon energy, one can derive the two-photon interacting eigenstate in the coordinate representation in the $U\to \mathrm{\infty}$ limit:

the whole prescription is complete and in principle one may numerically compute the two-photon interacting eigenstate Eq. (20) for any N.

Finally, to proceed with the Markovian approximation, we explicitly write down the integrands in Eqs. (21) and (23), replace the factors $exp(2ikL)$ by $exp(2i{k}_{0}L)$ therein, and do the double integral by standard contour integral techniques enclosing the poles in the upper half complex plane (for the $N=3$ case, for example, the denominator of each transmission amplitude is a cubic polynomial in k, so there are three roots). The $N=2$ case [17] could serve as an illustrative example owing to its relatively simple polynomial structure: For ${k}_{0}L=A\pi $ with $0\le A\le 1/2$, we have

where $\eta \equiv E-2{\omega}_{0}+i\mathrm{\Gamma}$, $\gamma \equiv ({e}^{i\pi A}+1)\mathrm{\Gamma}+2i{\omega}_{0}$, and ${\beta}_{\pm}\equiv {e}^{i\pi A}\pm 1$. During the two contour integrations, ${x}_{1}>{l}_{2}$ and ${x}_{2}={x}_{1}+t$ (with $t>0$) are used. Due to parity symmetry, ${G}_{21}={G}_{12}$, ${G}_{22}={G}_{11}$, ${G}_{1}^{\mathrm{L},\mathrm{L}}(-{x}_{1},-{x}_{2})={G}_{2}^{\mathrm{R},\mathrm{R}}({x}_{1},{x}_{2})$ and ${G}_{2}^{\mathrm{L},\mathrm{L}}(-{x}_{1},-{x}_{2})={G}_{1}^{\mathrm{R},\mathrm{R}}({x}_{1},{x}_{2})$.

Appendix 2: Two-photon correlation function ${g}_{2}(t)$

For a non-dispersive photonic field operator in the Heisenberg picture which satisfies ${a}^{\u2020}(x,t)={a}^{\u2020}(x-ct)$, the two-photon correlation function ${g}_{2}(t)$ can be rewritten in the Schrödinger picture as

where $|\psi \u3009$ is the asymptotic output state and $\alpha ={\alpha}^{\prime}=R$ for transmitted photons or $\alpha ={\alpha}^{\prime}=L$ for reflected photons. The second equality holds if a weak incident coherent state (mean photon number $\overline{n}\ll 1$) with right-going photons is assumed - as is appropriate for comparison with an eventual experiment - such that we consider only two-photon states in the numerator and one-photon states in the denominator. The justification for the latter is twofold: (i) In the numerator, the 0- and 1-photon states are eliminated by the annihilation operators, leaving the 2-photon sector untouched which, then, can be described by ${}_{\alpha ,{\alpha}^{\prime}}\u3008x,x+ct|{\psi}_{2}({k}_{1},{k}_{2})\u3009_{\mathrm{R},\mathrm{R}}$. (ii) In the denominator, the probability of having only one photon is much larger then having two, so that the factors ${|}_{\mathrm{R},\mathrm{R}}{\u3008{\psi}_{2}({k}_{1},{k}_{2})|{a}_{\alpha}^{\u2020}(x){a}_{\alpha}(x)|{\psi}_{2}({k}_{1},{k}_{2})\u3009}_{\mathrm{R},\mathrm{R}}{|}^{2}$ can be replaced by the single photon eigenstate ${|}_{\alpha}{\u3008x|{\varphi}_{1}({k}_{1})\u3009}_{\mathrm{R}}{|}^{2}$ given that ${k}_{1}={k}_{2}=E/2$ (i.e. two identical incident photons). We are thus lead to an explicit expression for the photon-photon correlations in terms of the 1- and 2-photon states found using the method outlined in Appendix 1.

Declarations

Acknowledgements

This work was supported by U.S. NSF Grant No. PHY-10-68698. HZ was supported by a John T. Chambers Fellowship from the Fitzpatrick Institute for Photonics at Duke University.

Authors’ Affiliations

(1)

Department of Physics, Duke University

References

Rupasov VI, Yudson VI: Rigorous theory of cooperative spontaneous emission of radiation from a lumped system of two-level atoms: Bethe ansatz method. Zh Èksp Teor Fiz 1984, 87: 1617–1630. . [Sov Phys JETP 1984, 60:927–934] . [Sov Phys JETP 1984, 60:927-934]ADS

Chang DE, Sørensen AS, Hemmer PR, Lukin MD: Quantum optics with surface plasmons. Phys Rev Lett 2006., 97(5): Article ID 053002 Article ID 053002 10.1103/PhysRevLett.97.053002

Shen J-T, Fan S: Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a two-level system. Phys Rev Lett 2007., 98(15): Article ID 153003 Article ID 153003 10.1103/PhysRevLett.98.153003

Shen J-T, Fan S: Strongly correlated multiparticle transport in one dimension through a quantum impurity. Phys Rev A 2007., 76(6): Article ID 062709 Article ID 062709 10.1103/PhysRevA.76.062709

Yudson VI, Reineker P: Multiphoton scattering in a one-dimensional waveguide with resonant atoms. Phys Rev A 2008., 78: Article ID 052713 Article ID 052713 10.1103/PhysRevA.78.052713

Witthaut D, Sørensen AS: Photon scattering by a three-level emitter in a one-dimensional waveguide. New J Phys 2010., 12(4): Article ID 043052 Article ID 043052 10.1088/1367-2630/12/4/043052

Zheng H, Gauthier DJ, Baranger HU: Waveguide QED: many-body bound-state effects in coherent and Fock-state scattering from a two-level system. Phys Rev A 2010., 82(6): Article ID 063816 Article ID 063816 10.1103/PhysRevA.82.063816

Ian H, Liu Y-X, Nori F: Tunable electromagnetically induced transparency and absorption with dressed superconducting qubits. Phys Rev A 2010., 81: Article ID 063823 Article ID 063823 10.1103/PhysRevA.81.063823

Rephaeli E, Kocabaş ŞE, Fan S: Few-photon transport in a waveguide coupled to a pair of colocated two-level atoms. Phys Rev A 2011., 84: Article ID 063832 Article ID 063832 10.1103/PhysRevA.84.063832

Roy D: Two-photon scattering by a driven three-level emitter in a one-dimensional waveguide and electromagnetically induced transparency. Phys Rev Lett 2011., 106(5): Article ID 053601 Article ID 053601 10.1103/PhysRevLett.106.053601

Roy D: Correlated few-photon transport in one-dimensional waveguides: linear and nonlinear dispersions. Phys Rev A 2011., 83: Article ID 043823 Article ID 043823 10.1103/PhysRevA.83.043823

Shi T, Fan S, Sun CP: Two-photon transport in a waveguide coupled to a cavity in a two-level system. Phys Rev A 2011., 84: Article ID 063803 Article ID 063803 10.1103/PhysRevA.84.063803

Zheng H, Gauthier DJ, Baranger HU: Cavity-free photon blockade induced by many-body bound states. Phys Rev Lett 2011., 107: Article ID 223601 Article ID 223601 10.1103/PhysRevLett.107.223601

Zheng H, Gauthier DJ, Baranger HU: Strongly correlated photons generated by coupling a three- or four-level system to a waveguide. Phys Rev A 2012., 85: Article ID 043832 Article ID 043832 10.1103/PhysRevA.85.043832

Rephaeli E, Fan S: Stimulated emission from a single excited atom in a waveguide. Phys Rev Lett 2012., 108: Article ID 143602 Article ID 143602 10.1103/PhysRevLett.108.143602

Zheng H, Baranger HU: Persistent quantum beats and long-distance entanglement from waveguide-mediated interactions. Phys Rev Lett 2013., 110(11): Article ID 113601 Article ID 113601 10.1103/PhysRevLett.110.113601

Moeferdt M, Schmitteckert P, Busch K: Correlated photons in one-dimensional waveguides. Opt. Lett. 2013, 38: 3693. 10.1364/OL.38.003693View ArticleADS

Lalumière K, Sanders BC, van Loo AF, Fedorov A, Wallraff A, Blais A: Input-output theory for waveguide QED with an ensemble of inhomogeneous atoms. Phys Rev A 2013., 88: Article ID 043806 Article ID 043806 10.1103/PhysRevA.88.043806

Shen J-T, Fan S: Coherent single photon transport in a one-dimensional waveguide coupled with superconducting quantum bits. Phys Rev Lett 2005., 95(21): Article ID 213001 Article ID 213001 10.1103/PhysRevLett.95.213001

Chang DE, Sørensen AS, Demler EA, Lukin MD: A single-photon transistor using nanoscale surface plasmons. Nat Phys 2007, 3: 807–812. 10.1038/nphys708View Article

Zhou L, Gong ZR, Liu Y-X, Sun CP, Nori F: Controllable scattering of a single photon inside a one-dimensional resonator waveguide. Phys Rev Lett 2008., 101(10): Article ID 100501 Article ID 100501 10.1103/PhysRevLett.101.100501

Longo P, Schmitteckert P, Busch K: Few-photon transport in low-dimensional systems: interaction-induced radiation trapping. Phys Rev Lett 2010., 104(2): Article ID 023602 Article ID 023602 10.1103/PhysRevLett.104.023602

Kolchin P, Oulton RF, Zhang X: Nonlinear quantum optics in a waveguide: distinct single photons strongly interacting at the single atom level. Phys Rev Lett 2011., 106(11): Article ID 113601 Article ID 113601 10.1103/PhysRevLett.106.113601

Eichler C, Bozyigit D, Wallraff A: Characterizing quantum microwave radiation and its entanglement with superconducting qubits using linear detectors. Phys Rev A 2012., 86: Article ID 032106 Article ID 032106 10.1103/PhysRevA.86.032106

Zheng H, Gauthier DJ, Baranger HU: Decoy-state quantum key distribution with nonclassical light generated in a one-dimensional waveguide. Opt Lett 2013, 38(5):622–624. 10.1364/OL.38.000622View ArticleADS

Zheng H, Gauthier DJ, Baranger HU: Waveguide-QED-based photonic quantum computation. Phys Rev Lett 2013., 111: Article ID 090502 Article ID 090502 10.1103/PhysRevLett.111.090502

Akimov AV, Mukherjee A, Yu CL, Chang DE, Zibrov AS, Hemmer PR, Park H, Lukin MD: Generation of single optical plasmons in metallic nanowires coupled to quantum dots. Nature 2007., 450: Article ID 402 Article ID 402 10.1038/nature06230

Bajcsy M, Hofferberth S, Balic V, Peyronel T, Hafezi M, Zibrov AS, Vuletic V, Lukin MD: Efficient all-optical switching using slow light within a hollow fiber. Phys Rev Lett 2009., 102(20): Article ID 203902 Article ID 203902 10.1103/PhysRevLett.102.203902

Babinec TM, Hausmann BJM, Khan M, Zhang Y, Maze JR, Hemmer PR, Lončar M: A diamond nanowire single-photon source. Nat Nanotechnol 2010., 5(1038): Article ID 195 Article ID 195 10.1038/nnano.2010.6

Claudon J, Bleuse J, Malik NS, Bazin M, Jaffrennou P, Gregersen N, Sauvan C, Lalanne P, Gérard J-M: A highly efficient single-photon source based on a quantum dot in a photonic nanowire. Nat Photonics 2010., 4(1038): Article ID 174 Article ID 174 10.1038/nphoton.2009.287

Bleuse J, Claudon J, Creasey M, Malik NS, Gérard J-M, Maksymov I, Hugonin J-P, Lalanne P: Inhibition, enhancement, and control of spontaneous emission in photonic nanowires. Phys Rev Lett 2011., 106: Article ID 103601 Article ID 103601 10.1103/PhysRevLett.106.103601

Laucht A, Pütz S, Günthner T, Hauke N, Saive R, Frédérick S, Bichler M, Amann M-C, Holleitner AW, Kaniber M, Finley JJ: A waveguide-coupled on-chip single-photon source. Phys Rev X 2012., 2: Article ID 011014 Article ID 011014 10.1103/PhysRevX.2.011014

Astafiev O, Zagoskin AM, Abdumalikov AA, Pashkin YA, Yamamoto T, Inomata K, Nakamura Y, Tsai JS: Resonance fluorescence of a single artificial atom. Science 2010, 327(5967):840–843. 10.1126/science.1181918View ArticleADS

Astafiev OV, Abdumalikov AA, Zagoskin AM, Pashkin YA, Nakamura Y, Tsai JS: Ultimate on-chip quantum amplifier. Phys Rev Lett 2010., 104: Article ID 183603 Article ID 183603 10.1103/PhysRevLett.104.183603

Eichler C, Bozyigit D, Lang C, Steffen L, Fink J, Wallraff A: Experimental state tomography of itinerant single microwave photons. Phys Rev Lett 2011., 106: Article ID 220503 Article ID 220503 10.1103/PhysRevLett.106.220503

Hoi I-C, Wilson CM, Johansson G, Palomaki T, Peropadre B, Delsing P: Demonstration of a single-photon router in the microwave regime. Phys Rev Lett 2011., 107: Article ID 073601 Article ID 073601 10.1103/PhysRevLett.107.073601

Hoi I-C, Palomaki T, Lindkvist J, Johansson G, Delsing P, Wilson CM: Generation of nonclassical microwave states using an artificial atom in 1D open space. Phys Rev Lett 2012., 108: Article ID 263601 Article ID 263601 10.1103/PhysRevLett.108.263601

Eichler C, Lang C, Fink JM, Govenius J, Filipp S, Wallraff A: Observation of entanglement between itinerant microwave photons and a superconducting qubit. Phys Rev Lett 2012., 109: Article ID 240501 Article ID 240501 10.1103/PhysRevLett.109.240501

Hoi I-C, Kockum AF, Palomaki T, Stace TM, Fan B, Tornberg L, Sathyamoorthy SR, Johansson G, Delsing P, Wilson CM: Giant cross-Kerr effect for propagating microwaves induced by an artificial atom. Phys Rev Lett 2013., 111: Article ID 053601 Article ID 053601 10.1103/PhysRevLett.111.053601

van Loo AF, Fedorov A, Lalumiére K, Sanders BC, Blais A, Wallraff A: Photon-mediated interactions between distant artificial atoms. Science 2013, 342: 1494. 10.1126/science.1244324View ArticleADS

Wallraff A, Schuster DI, Blais A, Frunzio L, Huang R-S, Majer J, Kumar S, Girvin SM, Schoelkopf RJ: Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 2004., 431: Article ID 162 Article ID 162 10.1038/nature02851

Schoelkopf RJ, Girvin SM: Wiring up quantum systems. Nature 2008., 451: Article ID 664 Article ID 664 10.1038/451664a

Dzsotjan D, Sørensen AS, Fleischhauer M: Quantum emitters coupled to surface plasmons of a nanowire: a Green’s function approach. Phys Rev B 2010., 82: Article ID 075427 Article ID 075427 10.1103/PhysRevB.82.075427

Gonzalez-Tudela A, Martin-Cano D, Moreno E, Martin-Moreno L, Tejedor C, Garcia-Vidal FJ: Entanglement of two qubits mediated by one-dimensional plasmonic waveguides. Phys Rev Lett 2011., 106: Article ID 020501 Article ID 020501 10.1103/PhysRevLett.106.020501

Dzsotjan D, Kästel J, Fleischhauer M: Dipole-dipole shift of quantum emitters coupled to surface plasmons of a nanowire. Phys Rev B 2011., 84: Article ID 075419 Article ID 075419 10.1103/PhysRevB.84.075419

González-Tudela A, Porras D: Mesoscopic entanglement induced by spontaneous emission in solid-state quantum optics. Phys Rev Lett 2013., 110: Article ID 080502 Article ID 080502 10.1103/PhysRevLett.110.080502

Gonzalez-Ballestero C, Garcia-Vidal FJ, Moreno E: Non-Markovian effects in waveguide-mediated entanglement. New J Phys 2013., 15(7): Article ID 073015 Article ID 073015 10.1088/1367-2630/15/7/073015

Roy D: Cascaded two-photon nonlinearity in a one-dimensional waveguide with multiple two-level emitters. Sci Rep 2013., 3: Article ID 2337 Article ID 2337 10.1038/srep02337

Loudon R: The Quantum Theory of Light. 3rd edition. Oxford University Press, New York; 2003.

Lang C, Bozyigit D, Eichler C, Steffen L, Fink JM, Abdumalikov AA, Baur M, Filipp S, da Silva MP, Blais A, Wallraff A: Observation of resonant photon blockade at microwave frequencies using correlation function measurements. Phys Rev Lett 2011., 106: Article ID 243601 Article ID 243601 10.1103/PhysRevLett.106.243601

Ficek Z, Sanders BC: Quantum beats in two-atom resonance fluorescence. Phys Rev A 1990, 41: 359–368. 10.1103/PhysRevA.41.359View ArticleADS

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.