Spinons and Holons with Polarized Photons in a Nonlinear Waveguide

We show that the spin-charge separation predicted for correlated fermions in one dimension, could be observed using polarized photons propagating in a nonlinear optical waveguide. Using coherent control techniques and employing a cold atom ensemble interacting with the photons, large nonlinearities in the single photon level can be achieved. We show that the latter can allow for the simulation of a strongly interacting gas, which is made of stationary dark-state polaritons of two species and then shown to form a Luttinger liquid of effective fermions for the right regime of interactions. The system can be tuned optically to the relevant regime where the spin-charge separation is expected to occur. The characteristic features of the separation as demonstrated in the different spin and charge densities and velocities can be efficiently detected via optical measurements of the emitted photons with current optical technologies.


Spin-charge separation and quantum simulators
One-dimensional (1D) physical systems have attracted much attention due to their novel and sometimes spectacular features. Unlike two-or three-dimensional systems, the physics of 1D Lieb-Liniger model [1] is well captured by the Luttinger liquid theory in the low-energy domain [2]. In 1D Luttinger liquid theory, collective excitations rather than single excitations appear due to the tight transverse confinement forcing particles to move along one direction and thus converting any individual motion into a collective one. The collective modes of spin and charge in the 1D electronic gases surprisingly can be shown to propagate with different velocities, i.e., the spin and the charge separate into spinons and holons [2,3,4,5]. In experiments, the observation of spin-charge separation is however challenging -the control of interactions is still elusive and no distinct features of separation are obtained, although several seminal works have been made using copperoxide chain compounds SrCuO 2 [6], metallic chains [7], superconductors [8], and more recently, GaAs/AlGaAs heterostructures [9,10,11].
At the same time, works on artificially engineered quantum optical systems in which many-body effects could be reproduced in well controlled environments have recently emerged. To observe spin-charge separation in cold atoms, experimental proposals involving bosonic and fermionic species have been suggested [12,13,14,15]. However, the challenges in the trapping and cooling of fermionic gases, especially the lack of necessary individual accessibility and measurement of correlations in general, make current results inconclusive. Strongly correlated photons and polaritons, as hybrid light-matter quantum simulators, have been recently proposed. Initially using coupled resonator implementations [16], Mott transitions [17,18,19] and then effective spin models and Fractional Hall states of light were shown to be possible [20,21,22,23,24,25]. More recently, by employing hollow-core optical waveguides filled with cold atom ensembles and using slow-light techiques [26,27,28], it was shown that it is possible to prepare a Tonks gas of photons [29]. Using two atomic species, a two-component Lieb-Liniger model has also been recently suggested [30]. Compared to cold-atom proposals, photonic proposals should allow for more direct measurements of local observables and correlation functions of the emitted photon states.
In this article, we consider a novel scheme involving two oppositely circularly polarized quantum beams and single atomic species to simulate a two-component interacting gas in 1D. We note the difference to an earlier scheme, where two quantum fields with different frequencies interacting with two species of four-level atoms were employed [30]. The current proposal has the distinct advantage over in two major ways. Firstly in terms of more efficient detection of the correlation in output polarized states and secondly in terms of the loading and preparation process into the fiber which requires handling a single atomic species rather than two.
The paper is organized as follows. We first review the basics of the single-and twocomponent bosonic Lieb-Liniger model [1], and especially its mapping to Luttinger liquid theory. We then describe in detail the preparation of a polaritonic Lieb-Liniger model with two bosonic components in the waveguide employing slow-light techniques. Finally we discuss the necessary range of optical parameters in order to drive the system to the relevant spin-charge separation regime. We conclude by analyzing how the effective photonic spin and charge densities and velocities can be probed. This is done by releasing the trapped polaritons into outgoing photons where optical measurements can reconstruct the characteristic functions of the effect.

From Lieb-Liniger bosons to Luttinger liquids: The basics of Spin-Charge Separation revisited
One of the most famous 1D models is the Lieb-Liniger model [1], which describes N bosons with a Dirac-delta interaction as where m is the mass and g is the interaction strength. Although the Lieb-Liniger model is exactly solvable through the Bethe ansatz approach, it is still generally very difficult to extract correlation functions from the solutions at any interaction regime. Luttinger liquid theory on the other hand can give the low energy universal description of the Lieb-Liniger model at low temperatures. In the Luttinger liquid phase, the low energy excitations are no longer single but collective modes with linear dispersion. The confinement of interactions between particles in 1D forces any individual motion to affect the collective system. The description of the dynamics in terms of collective bosonic fields is called 'bosonization' approach and we briefly present it in the following. Assume N particles described by ψ(r) = ψ(z)ϕ 0 (r ⊥ ), where the particles move along one direction, say z direction. Strong confinement is applied to the transverse direction r ⊥ = {x, y}, allowing only for the lowest energy transverse quantum state ϕ 0 (r ⊥ ) to be considered. The Lieb-Liniger model of the parallel component reads [2,3,4,5] where the collective bosonic fields ψ (z) and ψ † (z) can be expressed as with ρ (z) = ψ † ψ the particle density and θ (z) the phase operator. As described in [2], the phase and density fields are canonically conjugated fields, and the collective density operator can be expressed as If ρ(z) varies slowly with z, we can retain the lowest frequency component for m = 0 and write where the fields θ(z) and 1 π ∂ z φ(z ) are canonically conjugated. The Lieb-Liniger Hamiltonian (2) for the lowest component ψ(z) e −iθ(z) ρ 1/2 0 can be mapped to a Luttinger liquid with Hamiltonian (see [2,3,4,5]): where all the interaction effects are encoded into two effective parameters: the propagation velocity of density disturbances υ and the so-called Luttinger parameter K sl controlling the long-distance decay of correlations.
Going beyond the simple case with single-component bosonic system, interesting physics can be obtained by mixing two bosonic components or by considering two internal degrees of freedom of bosons, which is analogous to assigning a "spin"1/2 to bosons. The two-component Lieb-Liniger model in this case would read: where m s is the mass and χ s and χ ↑↓ are the intra-and interspecies interaction with s =↑, ↓ representing the two spins. Following the literature [2,3,4,5], the charge-and spin-related fields can be defined as: The Hamiltonian separates into two parts (more details in [2]) and reads as: The charge part is given by and the spin part is defined as under the separation conditions Here ρ 0 = ρ 0,↑ + ρ 0,↓ and ρ 0,s are the densities for two species. With χ = χ s , u = u s = ρ 0,s χ s /m s , and K = K s = π 2 ρ 0,s /(m s χ s ), the charge and spin velocities and Luttinger parameters are u charge,spin = u 1 ± χ ↑↓ /χ, K charge,spin = K/ 1 ± χ ↑↓ /χ.

The Optical Waveguide Setup
Our proposal is based on exploiting the available huge photonic nonlinearities possible to generate in specific quantum optical setups. More specifically, we envisage the use of a highly nonlinear waveguide where the necessary nonlinearity will emerge through the strong interaction of the propagating photons to existing emitters in the waveguide. Recent experiments have developed two similar setups in this direction, both capable of implementing our proposal with either current or near future platforms. In these experiments, cold atomic ensembles are brought close to the surface of a tapered fiber [31,32] or are loaded inside the core of a hollow-core waveguide [33,34,35,36] as shown in figure 1 (a). The available optical nonlinearity based on the Electromagnetically Induced Transparency (EIT) effect can be used as we will show to create situations where the trapped photons obey Lieb-Liniger physics. The process to generate the strongly correlated states of photons is as follows: First, laser-cooled atoms exhibiting a multiple atomic-level structure shown in figure 1 (b) are moved into position so they will interact strongly with incident quantum light fields. Initially, resonant to the corresponding transitions, two optical pulses with opposite polarizationsÊ ↑,+ (z, t) andÊ ↓,+ (z, t) are sent in from one direction, say the left side. They are injected into the waveguide with the co-propagating classical control fields Ω ↑,+ (t) and Ω ↓,+ (t) initially turned on. As soon as the two quantum pulses completely enter into the waveguide, the classical fields Ω s,+ are adiabatically turned off, convertinĝ E s,+ into coherent atomic excitations as in usual slow-light experiments for s =↑, ↓. We then adiabatically switch on both Ω s,+ and Ω s,− from two sides. The probe pulses become trapped due to the effective Bragg scattering from the stationary classical waves as analyzed in [26,27,28]. At this stage the pulses are noninteracting with the photons expanding freely due to the dispersion. By slowly shifting the d-levels, the effective masses can be kept constant whereas the effective intra-and interspecies repulsions are increased. This drives the system into a strongly interacting regime. This dynamic evolution is possible by keeping for example the corresponding photon detunings ∆ s constant while shifting the d-level. Once this correlated state is achieved, the fieldsfor example Ω s,+ -from the pair of control fields that trap polaritons, are slowly turned off. This will release the corresponding quasi-particles by turning them to propagating photons which will then exit the fiber. As all correlations established in the previous step are retained, these wave packets comprise of two separated effective charge and spin density waves.

Realizing a two-component Lieb-Liniger model of polarized photons
The system described above and shown in figure 1 obeys the Hamiltonian: where s, s =↑, ↓ and |d, ↑, ↓ = |d, ↓, ↑ . The continuous collective atomic spin operators σ µ;ν ≡ σ µ;ν (z, t) describe the averages of the flip operators |µ ν| over atoms in a small region around z. The density of atoms is n s z and g s , g ss are the coupling strengths between the quantum fields and atoms, while ∆ s and ∆ ss are one-photon detunings from the corresponding transitions. For simplicity, we assume that g s = g ss = g. Furthermore, we label the two quantum and two classical fields with frequencies ω Q,s and ω C,s and wave vectors k Q,s and k C,s , respectively. Both quantum fieldsÊ ↑,+ (z, t) andÊ ↓,+ (z, t) drive four possible atomic transitions. The fieldsÊ s,± (z, t) are detuned by ∆ s from the transition |a → |b, s and by ∆ ss from |c, s → |d, s, s .Ê s,± (z, t) also drive the transitions from |c, s → |d, s,s with detuning ∆ ss . Heres =↑, ↓ ands = s. Finally, the applied classical control beams with Rabi frequencies Ω s,± (t) couple to both atoms and drive the transitions |b, s → |c, s .
The evolution of the slowly-varying quantum operatorsÊ s,± are given by four Maxwell-Bloch (MB) equations (∂ t + ν∂ z )Ê s,± = i √ 2πn s z g (σ a;b,s,± + σ c,s;d,s,s,± + σ c,s;d,s,s,± ) with four levels of the s-th atoms denoted as |a, s , |b, s , |c, s and |d, s, s . When writing down the MB equations (14), we have introduced the slowly-varying collective operators σ µ;ν = σ µ;ν,+ (z, t)e ik Q,s z + σ µ;ν,− (z, t)e −ik Q,s z , and ν is the velocity of quantum fields in an empty waveguide §. In the derivation of equations of motion we assume the Rabi frequencies of the control fields to be slowly varied. The slow-light polariton operators are defined as Ψ s,± = cos θ sÊs,± − sin θ s 2πn s z σ c,s;a , where tan θ s = g √ 2πn s z /Ω s . For stationary polaritons we have assumed that the amplitudes of the counterpropagating classical fields are equal, i.e., Ω s,± ≡ Ω s . In the limit when the excitations are mostly in spin-wave form, i.e., sin θ s 1, and since σ c,s;a = −gÊ s,± /Ω s , the polariton operators become Ψ s,± = 2πn s z g Ω sÊ s,± .
Setting Ψ s = (Ψ s,+ + Ψ s,− )/2 and A s = (Ψ s,+ − Ψ s,− )/2 as the symmetric and antisymmetric combinations of the two polaritons, we derive the equations of motion § We would like to highlight here the relative simplicity of the above evolution equation compared to the one we considered in [30] where extra phase terms have to be involved due to the existence of two-atomic-species different frequencies on the incident quantum fields. for the polariton combinations Ψ s , A s : The noise terms in Eq. (18) account for the dissipative processes that take place during the evolution. Fortunately, for the dark-state polaritons under consideration, as long as the spontaneous emission rates Γ from the states |c, s and |d, s, s are much less than the detunings |∆ ss |, the losses in the timescales of interest are not significant and thus can be neglected as discussed in [26,27,28,29,30]. Assuming optical depth of a few thousand and a large ratio between the density of atoms to the density of photons n s z /n s ph ∼ 10 4 , the antisymmetric combinations A ↑ and A ↓ can be adiabatically eliminated from the equations of motion for the polaritons and moreover, the nonlinear terms like Ψ † s Ψ s A s and A † s A s Ψ s are negligible. In this regime, Eq. (18) simplifies to a nonlinear Schrödinger Eq. (19) for polaritons which reads: which is related to an effective two-component Lieb-Liniger model of polaritons Here m s = −Γ s 1D n s z /(4∆ s ν s ) is the effective mass for s-th polaritons with Γ s 1D = 4πg 2 /ν the spontaneous emission rate of a single atom into the waveguide modes and ν s = νΩ 2 s / (πg 2 n s z ) the group velocity of the propagating polaritons. The intraspecies repulsion is characterized by χ s = Γ s 1D ν s /∆ ss and the interspecies repulsion by χ ↑↓ = s=↑,↓ Γ s 1D ν s /∆ ss . To reach the spin-charge separation regime, we employ the mapping to a Luttinger liquid model described in the previous section. For this to be possible, we should first of all check the tunability of the relevant parameters to the repulsive regime. This in our case (see Eq. (19)) implies tuning ∆ s ∆ ss < 0 which leads to m s χ s > 0. Similarly ∆ ↑ ∆ ↓ > 0 forces the effective masses m ↑ m ↓ > 0 and ∆ ↑↑ ∆ ↓↓ > 0 tunes χ ↑ χ ↓ > 0. These conditions are satisfied by tuning the lasers such that: ∆ ↑ and ∆ ↓ are negative (positive) while at the same time ∆ ↑↑ and ∆ ↓↓ are positive (negative).
Apart from the repulsive interaction regime, the separation condition Eq. (12) needs to be satisfied as well, which in our case means setting: where the polariton density ρ 0,s equals to the photon density n s ph . The effective charge and spin densities are the sum and difference of the two-species polaritonic densities, which read as with ρ s = ρ 0,s − 1 π ∇φ s m exp [i2m (πρ 0,s z − φ s )]. Keeping only the lowest components with m = 0, ±1, the charge and spin density operators in the bosonic language can be represented as Here the first term in ρ charge is the average density ρ 0 = ρ 0,↑ + ρ 0,↓ . In our two-species photonic system, we set ρ 0,↑ = ρ 0,↓ = 1 2 ρ 0 for each polarization component. The second gradient term in ρ charge and ρ spin are the density oscillations with zero momentum. The third term in ρ charge and second term in ρ spin are the density fluctuations of the 2k F components [37]. We label γ s as the ratio of the interaction to the kinetic energies for each polariton species γ s = m s χ s /ρ 0,s . Combining the two separation conditions in Eq. (21) together, one gets γ ↑ = γ ↓ . For χ = χ s and γ = γ s , the velocities and Luttinger parameters can be expressed as u = χ/ √ γ and K = π/ √ γ. As also demonstrated for a similar system albeit with one quantum field [29], γ here can also be tuned from zero to finite to extremely large, corresponding to non-, weak-and strong-correlated regimes, which implies a wide tunable range for u and K.

Probing of the photonic spinons and holons
The typical detection of spin-charge separation can occur through dynamically probing the time evolution of a single excitation as in cold-atom proposals [12,13,14,15], or by measuring the corresponding single-particle spectral function as in condensed matter experiments [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In our case, we propose to extract the charge and spin velocities by measuring the Fourier transform of density-density correlations D(ω, q) for energy ω and momentum q. As derived for the two-component system in [37], for the 2k F component of density operator ρ 2k F = 2ρ 0 cos 2k F z − √ 2φ charge cos √ 2φ spin (the last term in Eq. (23)), the Fourier transform D(ω, q) of the density-density operator T τ ρ 2k F (z, τ ) ρ 2k F (0, 0) is given by where Γ is the gamma function, Θ is the step function, F 1 is the Appell's hypergeometric function, and α is a short-distance cutoff. D(ω, q) depends on the velocity ω/q and should exhibit two peaks centered around u charge q and u spin q [2,3,4,5]. In our photonic system, probing of the spinon and holon branches can be done by measuring the correlation functions of densities of the fields as they exit, for a specific quasi-momenta q. For a clear distinction between the two effective spin and charge peaks, we should set our optical detectors around q = 2π/z 0 , i.e., z 0 apart (z 0 here corresponds to roughly the length of the fiber). To give an illustration of the expected behavior, D(ω, q) in the unit of ρ 2 0 α is plotted in figure 2 with the intra-and interspecies repulsion ratio χ ↑↓ /χ = 0.6, which in turn tunes the charge and spin velocities to u charge = 2u spin = u 1 + χ ↑↓ /χ = 2 2 5 u, and Luttinger parameters K charge = 1 2 K spin = 1 2 5 2 K. We choose u charge = 1, u spin = 0.5 and K charge = 0.55, K spin = 1.1 via tuning u and K wihch require γ ∼ 20. This as shown in [29] is achieved at optical depths OD = 2000 and roughly N ↑,↓ ph = 10 photons initially in each pulse and single-atom co-operativity of η = 0.4 . These values are for the moment out of the current experimental range where optical depths of a few hundred have been achieved, but should not be out of the question in the near to midterm future [31,32,38]. In calculating the optical interaction parameters appearing in the Hamiltonian Eq. (20), we haven take into account both the linear and nonlinear loss mechanisms as layed out in [29].
Co-operativity here is the ratio of spontaneous emission into the waveguide to total spontaneous emission.

Conclusion
We have described in detail a strongly correlated photonic scheme to simulate a purely fermionic effect, spin-charge separation. In more detail, we have shown that polarized photons interacting with a cold atomic ensemble can be made to obey twocomponent Lieb-Liniger physics and even behave as a quantum Luttinger liquid. The relevant interactions exhibit the necessary tunability for steering the photons to the effective spin-charge separation regime. Efficient observations of the characteristic features of the separation using standard quantum optical methods should be feasible based on correlations measurements of the outgoing photons which here carry opposite polarizations. The current proposal is different from a similar scheme proposed earlier by some of us, where two species of atoms were coupled to two quantum fields of two different frequencies but of same polarization [30]. Here a single species of atoms is shown to suffice in order to induce the required intra-and interspecies interactions, which combined with the easier detection of the polarized output states makes this approach more feasible.