Microscopic physics of quantum self-organisation of optical lattices in cavities

We study quantum particles at zero temperature in an optical lattice coupled to a resonant cavity mode. The cavity field substantially modifies the particle dynamics in the lattice, and for strong particle-field coupling leads to a quantum phase with only every second site occupied. We study the growth of this new order out of a homogeneous initial distribution for few particles as the microscopic physics underlying a quantum phase transition. Simulations reveal that the growth dynamics crucially depends on the initial quantum many-body state of the particles and can be monitored via the cavity fluorescence. Studying the relaxation time of the ordering reveals inhibited tunnelling, which indicates that the effective mass of the particles is increased by the interaction with the cavity field. However, the relaxation becomes very quick for large coupling.


Introduction
Ultracold atoms in an optical lattice formed by a far detuned laser field constitute an ideal system to study quantum phase transitions, ie phase transitions at zero temperature [1]. In the most prominent example first predicted theoretically [2] and confirmed experimentally [3] it was found that the particle ground state changes from a superfluid state where all atoms are delocalised to a perfectly ordered Mott insulator state for increasing lattice depth. More complex phases as supersolids, etc. were predicted if long range interactions or mixed species setups [4] are used, but these are harder to realize and measure experimentally. While the final states are well understood, these phase transitions require the buildup or decay of long range correlations, the mechanism and time scale of which is not fully understood.
In a parallel development a dynamical transition to a self-organised phase in optical lattices was found for classical particles, when the lattice is placed inside an optical resonator. It originates from interference of the resonantly enhanced light field scattered by the atoms into the cavity mode with the lattice light itself and leads to a preferred occupation of every second site [5]. For a finite temperature cloud thermal density fluctuations are amplified and lead to a runaway self-organisation by feedback from the cavity field. However, for a BEC (T ≈ 0) the initial density is perfectly homogeneous and only quantum fluctuations which go beyond a mean field description of the cold gas can start the self-organising process when the cavity interaction is switched on. Tunnelling results in a dynamical change of the atomic phase at T = 0 which gets irreversible only if cavity decay is included. In this work we study this quantum dynamics on the microscopic level and show how it depends on the precise quantum properties of the initial atomic and field state beyond any mean field density.
The paper is also intended to show the limitations of the effective Bose-Hubbard type model developed in [6] for atom-cavity systems. We demonstrate that in the regime of moderate coupling the simple Bose-Hubbard approach reproduces very well the results of a full Monte Carlo wave-function simulation, while it breaks down in the regime of stronger coupling. Even in this regime, however, it predicts the steady state surprisingly well, whereas the relaxation time to this state is predicted wrong. We show that the relaxation time of the system exhibits a highly non-trivial behaviour. In the regime of moderate atom-cavity coupling the relaxation time is composed of the timescale of photon counts and that of tunnelling. The combination of these two time scales leads to a minimum behaviour in the relaxation time, while for stronger coupling the relaxation becomes very quick, a behaviour observed in the full simulations but not reproducible with the Bose-Hubbard approach.
We first describe our system, the model, and solution methods applied. Afterwards, we go on investigating the dynamics of a single atom in the system. Although selforganisation cannot be defined in this case, we demonstrate that already here the relaxation time exhibits the same behaviour we find later for two atoms, and here it is easier to give a qualitative picture of this behaviour. We finally turn to the case of two atoms and show how the increasing coupling results in a transition from a T = 0 homogeneous initial condition into a self-organised configuration in steady state.

System, models
The proposed setup is depicted in Figure 1. It consists of a one dimensional optical lattice within a cavity sustaining a single mode with its axis aligned orthogonally to the lattice axis -such systems have been studied in diverse theoretical contexts before [7,8,9,10], and is available experimentally [11,12]. We assume that the cavity mode function is constant along the lattice direction, still, as we will see below, it modifies significantly the dynamics of atoms in the lattice.
A standard quantum optical model for the system with a single atom moving in one dimension along the lattice axis is obtained by adiabatic elimination of the atomic excited state(s). This is justified in the regime where the driving -in our case, the laser generating the lattice (pump) -is far detuned from the atomic transition frequency [13,14]. The Hamiltonian for a single atom and the cavity then reads ( = 1): Here x, p, and µ are the atomic position and momentum operators and the mass, respectively; a is the cavity field operator, ∆ C = ω − ω C is the cavity detuning -ω is the laser, ω C is the cavity frequency -, and K is the lattice field wave number. The first two terms describe the atomic motion in the lattice, which appears as a classical potential after the elimination of the atomic internal dynamics. The potential depth for a two-level atom V 0 = η 2 /∆ A where η is the pump Rabi frequency and ∆ A = ω − ω A , ω A is the atomic transition frequency. The third term is the free Hamiltonian of the cavity mode with the detuning shifted by U 0 = g 2 /∆ A where g is the atom-cavity interaction coupling constant. The last term describes the atom-mode interaction which stems from stimulated absorption from the pump followed by stimulated emission into the cavity mode and the reverse process. Atomic spontaneous emission is strongly suppressed due to the large atom-pump detuning and therefore neglected. The cavity mode is, however, coupled to the surrounding EM modes, resulting in the decay of cavity photons (escape through the mirrors). The process is described by the Liouvillean dynamics [15] Lρ = κ 2aρa † − a † a, ρ + .
Note that model (1a-1b) is not specific to a two-level atom but generally applicable to linearly polarisable particles, atoms and molecules alike. In this case the parameter U 0 is proportional to the susceptibility of the particles [14]. In the following we shall hence speak about particles without any further specification.
As described theoretically [5,16] and observed experimentally [17] the model (1a-1b) features a phase transition termed self-organisation for a finite temperature classical gas. As this occurs for red-detuned driving, ie high field seeking particles with U 0 , ∆ C < 0 we will restrict ourselves to this case. Self-organisation can be qualitatively understood as follows: The system has three steady-state configurations in a mean field description: (i) an "unorganised" configuration where the particles are equally distributed at all lattice sites and scatter no light into the cavity due to destructive interference and (ii) two "organised" configurations in which the particles occupy either only the even or only the odd sites of the lattice and scatter superradiantly into the cavity. In the latter case the last term in the Hamiltonian (1a) deepens the lattice potential at the positions of the particles, so that they are self-trapped or "self-organised".
Configurations (ii) have lower energy and entropy (lower symmetry) than configuration (i). At a given temperature the system chooses between configuration (i) and one of configurations (ii) so as to minimise the free energy. Lowering the temperature results in a phase transition when the symmetry of configuration (i) is spontaneously broken into the lower symmetry of one of configurations (ii). ‡ The above qualitative picture is modified by the fact that the system never reaches thermal equilibrium with some external heat bath as energy is continuously flowing through the system from the pump via scattering on the particles into the cavity field and then out via the cavity loss channel. Self-organisation is therefore a dynamical phase transition for which the above mentioned configurations are steady-state patterns. In steady state the particles have a momentum distribution determined by the cavity field fluctuations, which, in most cases of physical interest, resembles very much a thermal distribution [18]. In this sense it is justified to speak about an effective temperature of the particles and use the picture of an equilibrium phase transition as we did above.
Let us now turn to the case of zero temperature and envisage a fixed number of classical point particles at each lattice site. In contrast to above, no matter whether we are above or below the threshold, no dynamics will arise because a homogeneous ‡ Note that self-organisation is a phase transition without any direct particle-particle interaction: only an effective interaction exists generated by the single cavity mode with which all particles interact. gas scatters no field into the cavity due to destructive interference. If no photons are present initially, such a classical gas cannot break the symmetry and is unable to escape the initial homogeneous configuration (i).
In the following we show that this is quite different for a quantum Bose gas at T = 0, which can be prepared by loading a BEC into the lattice [19]. Interestingly in this case self-organisation and the superradiant build-up of the cavity photon number starts immediately, but depends crucially on the quantum fluctuations of the gas at T = 0. Hence both for a Mott insulator and superfluid state (BEC) as initial condition, quantum fluctuations and the possibility of tunnelling between lattice sites immediately start selforganisation. This is combined with an intricate measurement-induced dynamics related to the information gained via the dissipation channel of the photon-loss.
Let us emphasise that there are two main differences as compared to the abovedescribed classical self-organisation: the initial temperature of the particles is zero and the particles in the lattice are confined strongly enough so that hopping between the lattice sites is due solely to tunnelling. Redistribution thus is a coherent quantum process and requires no direct inter-particle interaction.
We are using two approaches to the problem. The first one is the direct simulation of the system (1a-1b) by the Monte Carlo wave-function (MCWF) method, which unravels the corresponding Master equation in terms of individual quantum trajectories. This approach takes into account the full particle and cavity dynamics, with the cavity decay accounted for by quantum jumps, and can be practically pursued up to a few particles with the help of a new simulation framework [20]. The second approach analogous to standard Bose Hubbard models is based on a second-quantised form of the Hamiltonian (1a): dx Ψ † (x) H Ψ(x) where the field operator Ψ(x) is restricted to the lowest vibrational band of the lattice.
To obtain the smallest possible system useful for studying self-organisation, we restrict the dynamics to only one lattice wavelength, that is, two lattice sites (cf Figure 1) with periodic boundary condition. This is the smallest system which can seize the difference between the configurations described above and contains all the essential physics.
With two lattice sites the lowest vibrational band constitutes a two-dimensional Hilbert space, for which the localised Wannier basis with state |l localised at the left and |r at the right lattice site can be used. (In the MCWFS there are several left and right states corresponding to a large number of vibrational bands.) Hence Ψ(x) = x | l b l + x | r b r , where b l and b r are the corresponding bosonic annihilation operators. Putting the restricted field operator back into the second-quantised Hamiltonian we obtain the Bose-Hubbard type Hamiltonian: where J ≡ l| (p 2 /(2µ) The dynamics of particles in cavities as described by such Hamiltonians in system configurations different from the one investigated here has been discussed in Refs. [6,21,22]. A very attractive feature of Hamiltonian (2) is that it is simple enough so that together with the Liouvillean (1b) the full time-dependent Master equation can be solved even for several particles.

Single-particle dynamics
We first consider the dynamics of a single particle initially prepared in one of the localised states (say, |r ). Without coupling to the cavity (U 0 = 0) the particle moves unperturbed in the lattice via tunnelling, which, in the case of two sites corresponds to an oscillation between states |r and |l . This can be monitored via the expectation value Kx , which, as displayed in Figure 2(a) (red line), oscillates accordingly between ±π/2 (cf also Figure 1). This simple behaviour is significantly modified in the presence of even a weakly coupled cavity, U 0 = 0. Now the particle scatters photons from the lattice field into the cavity mode, depending on its state. Photons can decay according to the Liouvillean (1b) and allow to monitor the particle motion. The decay of a cavity photon can be modelled by a quantum jump, which is mathematically described by the application of the cavity field operator a on the state vector of the system. This, in turn, changes the whole particle-field wave function and thus gives feedback on the particle localisation.
When the coupling is weak enough, the field in the cavity will be small, and the contribution of the last term of Hamiltonian (1a) to the potential felt by the particle (second term in the same Hamiltonian) is negligible, so that it still makes sense to define the localised particle states solely from the lattice potential.
We assume that these states are well localised. When a point-like particle is placed into the lattice at position x in steady state it radiates a coherent field |α(x) into the cavity where the amplitude is determined by the Liouvillean dynamics (1a-1b) and reads where the second equality holds under the resonance condition ∆ C − NU 0 = −κ (N is the particle number), to which we restrict ourselves in the following. This makes that the cavity field increases monotonically with increasing coupling. Accordingly, in state |r the particle will radiate an approximately coherent state |α , while in state |l a coherent state with opposite phase |−α , where α = α(x = π/(2K)).
If we assume that tunnelling is much slower than cavity field evolution, then the latter will follow adiabatically the former. Without cavity jumps the system evolves coherently and since the back action of the cavity field on the particle motion is negligible by our assumption, this evolution amounts to an oscillation between states |r, α and |l, −α , hence at a given time instant t the overall state of the particle-cavity system reads approximately  Figure 2. Simulated data for a single particle in the lattice-cavity system. Parameters: V 0 = −10 ω rec , κ = 10 ω rec , ∆ C − U 0 = −κ, with the recoil frequency ω rec ≡ K 2 /(2µ). The colour code: red corresponds to U 0 = 0, green to U 0 = −0.005ω rec , blue to U 0 = −0.5ω rec , and magenta to U 0 = −10 ω rec ; with maximal cavity photon numbers amounting to 0, 2.5 · 10 −4 , 2.5 · 10 −2 , and 0. Now imagine that at time t a jump happens: Immediately after the jump the state of the system reads that is, the cavity jump is reflected back onto the particle motion and results in a jump of the phase of the tunnelling oscillation. This behaviour is verified by the simulations, an example trajectory is displayed in Figure 2(a) (green line). Here the parameters were chosen such that the maximal expectation value of the cavity photon number is only 2.5 · 10 −4 -this maximum is achieved when the particle is prepared perfectly localised at a lattice site.
The jump is a stochastic event and in ensemble average the jumps of the phases on individual trajectories result in a dephasing and hence damping of the oscillation. This behaviour, as displayed in Figure 2(b) (green lines) is verified by both the MCWFS and the simulation of the time-dependent Master equation based on the Bose-Hubbard Hamiltonian (2). In this regime of very low cavity photon number, the correspondence between the two models is very good. Increasing the photon number results in several jumps happening in one tunnelling cycle: in ensemble average this corresponds to the over-damped regime of the tunnelling oscillation (cf Figure 2(a-b) blue line).
The above picture of the dynamics on one Monte Carlo trajectory is not valid in the regime of stronger coupling where the photon number is higher. Here the cavity field modifies significantly the potential felt by the particle and hence the states |r and |l defined solely by the lattice potential lose their significance because many other particle spatial states enter the dynamics. Cavity decays are much more frequent, and the stronger field fluctuations are reflected back onto the potential. Accordingly, as we observe in Figure 2 (magenta line), the Bose-Hubbard approach being based on those two states breaks down in this regime.
The steady state of the dynamics is, quite independently of the coupling, the mixed state ρ ss ≈ (|r, α r, α| + |l, −α l, −α|)/2. As displayed in Figure 3, the relaxation time to this state, that is, the time scale of the damping of the tunnelling oscillation exhibits a non-trivial behaviour. For moderate coupling, according to the discussion above, this is composed of two time scales: the time scale of cavity photon decay, and that of tunnelling. The former being inversely proportional to the photon number gets faster with increasing coupling.
On the other hand the latter, for far less obvious reasons, gets slower. The discussion of the exact mechanism of this slowing down of the tunnelling is not in the scope of the present paper, and must be a subject of further study. Here we merely note that this behaviour is verified by both the MCWF and the Bose-Hubbard approach, and can phenomenologically be considered as an increase of the particle's effective mass. It is then analogous with the effective mass of electrons in crystals: there, when tunnelling between lattice sites, not only the electron has to tunnel, but a quasi particle called polaron composed of the electron and phonons. (The polaron problem has huge literature, [23] is one of the earliest references, for a quite comprehensive review see eg [24]). Here, roughly speaking, when the particle tunnels from |r to |l , the cavity field also has to tunnel from |α to |−α , which is inhibited by increasing α. Note that here it is a single EM mode that generates the effective mass, instead of several phonon modes as in the above polaron analogue. Here we see an example of the combination of optical lattices and CQED being able to reproduce a wider range of solid state physics phenomena (in particular the existence of phonons) than optical lattices alone. The combination of one accelerating and one slowing time scale results in the minimum behaviour of the relaxation time in Figure 3 around U 0 = −0.05 ω rec . Ultimately, with high enough coupling the cavity-generated potential starts to dominate the lattice potential, in which regime strong fluctuations and self-trapping lead to fast relaxation as observed in the MCWFS (red line |U 0 | ω rec ). Obviously, the Bose-Hubbard approach is unable to reproduce the behaviour in this regime.

Two-particle dynamics
Having understood the dissipative quantum dynamics of a single particle in our latticecavity system, we now turn to the case of two particles. Two particles on two lattice sites with periodic boundary conditions is the minimal system that can exhibit the difference between the configurations described above for self-organisation. In the Bose-Hubbard approach where there is only one state at each lattice site the Hilbert space for the particles is spanned by only three states: |1, 1 ≡ |0 -the Mott insulator (MI) state, which corresponds to the homogeneous distribution or unorganised configuration -, and |2, 0 ≡ |− and |0, 2 ≡ |+ corresponding to the two organised configurations.
|0 scatters no field (and no photons) into the cavity due to destructive interference between the fields scattered by the two particles, while |− and |+ scatter |−2α and |2α , respectively, the factor 2 being due to constructive interference. The difference between the two configurations can be monitored via the density correlation n l n r , which is 1 for the MI state and 0 in the subspace spanned by |± .
When the particles are initially in the MI state, then for U 0 = 0 at time t the particle state is With U 0 = 0 under the simplifying conditions we discussed above for the single particle case we have for the joint system If a jump happens in this state (application of a), then the state immediately after the jump reads There are two points worth noting here: Firstly, the quantum jump in the photon number erases all information about the phase of the tunnelling oscillation in the particle Hilbert space. Secondly, after the escape of one single photon from the cavity tunnelling stops immediately. Indeed, in the state (8) both |−, −2α and |+, 2α tunnels to |0, 0 (note that we assume again the cavity field following adiabatically the tunnelling), but their phases are opposite, and hence the two paths interfere destructively. A second jump at t ′ > t, however, makes the phases match again, and puts the state |Ψ ′′ (t ′ ) ∝ a |Ψ ′ (t) ∝ |−, −2α + |+, 2α back into the tunnelling cycle (7). An example MCWF trajectory exhibiting this behaviour is plotted in Figure 4(a) (green line). We observe that a quantum jump brings the system into the state (8), signalled by n l n r = 0, and it remains there until the next jump, when it starts to oscillate anew. In ensemble average these stochastic "dark" periods of the tunnelling oscillation lead to damping just as in the single-particle case. The final steady state is always a mixture At this point it becomes clear that any mean-field description of this system is bound to fail: a mean cavity field description would prohibit the possibility of a coherent superposition of different particle configurations radiating different fields as in (7), which is essential for the onset of the dynamics from a homogeneous zero-temperature initial condition (see also [25]). On the other hand, a particle mean field cannot capture the difference between states (9) with different w, because this appears only in the density correlation.
As displayed in Figure 4(b), our two approaches for simulating the damping agree well in the regime of moderate coupling. However, just as in the single-particle case,  strong coupling -in the regime where the cavity-generated potential dominates the lattice one -results in extremely quick damping, which cannot be reproduced by the Bose-Hubbard approach. Here the relaxation time exhibits the same behaviour as we have seen in the single-particle case (cf Figure 5(a)): with increasing coupling it has a minimum, after which the Bose-Hubbard approach gives a monotonic increase of the relaxation time, while the MCWFS gives a peak, and for even stronger coupling very quick relaxation. For the interpretation of this behaviour the same discussion applies as above for the single-particle case.
Increasing coupling results in a decrease of the weight w of the MI component in steady state, cf Figure 5 case) the steady state is confined solely into the self-organised subspace. This proves our initial assertion that even when the system is started from a T = 0 homogeneous state (here the MI state) self-organisation can occur. On the same Figure we also see that this is confirmed by both approaches, only the relaxation time of the process is predicted wrongly by the Bose-Hubbard approach for strong coupling. It is easy to see that starting the system from the superfluid (SF) state instead of the MI as above does not change the steady state since already the first quantum jump erases the information about the initial condition completely. The process of relaxation will, however, be different. This process can be monitored by a time resolved analysis of the intensity escaping the cavity, which is proportional to the cavity photon number: an example is displayed in Figure 6. Here, we are in the over-damped regime of the tunnelling oscillation. When prepared in the MI state, the particles do not scatter initially, and the buildup of the cavity field occurs on the time scale of the self-organisation process. With the SF initial state, on the other hand, some field is built up almost instantly (on the time scale of κ −1 ), because in the SF the states |± have finite weight; while the rest of the field is built up on the longer time scale.

Conclusions
In summary, we have seen that coupling an optical lattice gas at T = 0 to a cavity induces an irreversible reorganisation of the particles, a process which can be monitored directly in an experiment by the time-resolved analysis of the intensity escaping the cavity. We have shown that no classical or mean-field description of either the particles or the cavity field can account for the phenomenon. For strong enough coupling the process leads to a fast self-organisation of the particles, a phase in which they occupy every second site in the lattice, and scatter superradiantly into the cavity. An important conclusion of the work is that while an effective low-dimensional Bose-Hubbard type model cannot reproduce the time evolution in the strong-coupling regime as observed in more detailed simulations, it can still predict the steady state remarkably well. This model can therefore be used in the future for a high number of particles to study possible quantum phase transitions occurring in the steady state of this dissipative system.