Spectrum, Landau-Zener theory and driven-dissipative dynamics
of a staircase of photons
We study the production of photons in a model of three bosonic atomic modes non-linearly coupled to a cavity mode. In absence of external driving and dissipation, the energy levels at different photon numbers assemble into the steps of an energy staircase which can be employed as guidance for preparing multi-photon states. We consider adiabatic photon production, driving the system through a sequence of Landau-Zener transitions in the presence of external coherent light pumping. We also analyse the non-equilibrium dynamics of the system in the presence of competing coherent drive and cavity photon losses, and we find that the number of produced photons relaxes to a well-resolved metastable plateau before a dynamical instability, inherent to the type of light-matter coupling considered in the system, takes over, signalling a departure from the photons’ steady state attained at intermediate times. We discuss the sensitivity of the time scales for the onset of this instability to system parameters and predict the metastable value of photons produced, solving the driven-dissipative dynamics including three-body correlations between light and matter degrees of freedom.
pacs:05.30.Rt, 64.60.Ht , 75.10.Jm
The last ten years have witnessed swift progress in quantum optics platforms where light and matter are strongly coupled and interplay to simulate archetypal as well as novel phases of matter hemm. Examples range from Bose-Einstein condensates coupled to optical cavity photons, where the Dicke transition is engineered ess, to the recent demonstration of supersolids leonard and phases with competing order parameters in a condensate trapped at the intersection of two optical cavities leonard2; leonard3; gopal. Most of these models realise scenarios where matter and light collectively interact as in the case of superradiant phase transitions or in ’Dicke-Hubbard’ systems characterised by critical points separating a superfluid from a Mott insulating phase hemm2. Strong light-matter coupling regimes have also enabled the preparation of highly squeezed states of atomic ensembles by quantum non-demolition measurements host; bohnet, photon-mediated spin interactions lero; hosten2016quantum in optical cavities, photon blockade effects hartman; hamsen; cirac; rein; ima or non-classical light in cavity optomechanics platforms brooks. Recent experiments have extended photon-mediated interactions to optical-clock atoms norcia, to spin-1 atoms Zhiqiang:17; davis2018photon, and to multi-mode cavities vaidya2018tunable to enable further advances in quantum metrology masson and quantum simulation.
The control and the preparation of multi-photon states is of paramount importance for a progress towards a many-body physics of coupled light and matter in these platforms. Single- and multi-photon preparation has a long history in cavity QED brattke; bertet; uren; sayrin; cooper, including photon generation in high quality cavities varcoe, the control of single-photon states emitted by polaritons stan, as well as the conversion of collective atomic excitations into single photonic states within optical resonators vuletic, encompassing quantum homodyne tomography lv01.
In this work we consider a novel cavity-QED platform composed of collective atomic degrees of freedom strongly non-linearly coupled to a cavity photon: this system can be employed to engineer multi-photon states out of an empty cavity via adiabatic as well as with far-from-equilibrium driving protocols.
Specifically, we study an effective model of two bosonic atomic modes interacting with a photon; the model results from a two-photon resonant process occurring in a cavity hosting an ensemble of spin-1 atoms Rabi-coupled to the cavity mode. In equilibrium conditions, the eigenstates of this system compose a ‘staircase’ structure: each level, or ‘step’, is characterised by a different photon number which is a conserved quantity in undriven conditions. In the presence of weak coherent photonic pumping, an adiabatic variation in time of the energy levels of the atomic degrees of freedom allows for climbing the staircase, transiting across a sequence of level crossings, and thus preparing a desired number of photons out of an initially empty cavity.
Complementarily, we also consider the far-from-equilibrium dynamical preparation of photons in the system, suddenly switching the coherent photon pumping as well as including natural sources of dissipation, such as incoherent cavity-photon losses.
We highlight the formation of a metastable steady state in the late time driven-dissipative dynamics of photons, and discuss the dependence of its life-time on system parameters, solving the dynamics with the inclusion of three-body correlations between light and atomic degrees of freedom.
Ii The model
We consider an optical cavity supporting a photonic mode ( in Fig. 1) of frequency , Rabi coupled via the interaction coupling to the atoms. The three ground energy levels, e.g. Zeeman states in an atom of hyperfine spin (Zhiqiang:17; masson; davis2018photon), are denoted with , , , and the first two are detuned upwards and downwards, by and , with respect to the latter, which is assumed to have a macroscopic occupation, . A couple of external lasers of frequency can assist transitions from the two levels and to two auxiliary levels and respectively, with amplitudes . The lasers are far detuned from the transitions to the auxiliary levels by . Because of the detunings , single-atom transitions assisted by the laser (or ) and by a cavity photon, transferring population from the states to (through levels and ) are off-resonant. However, transitions from the atomic state to , involving two atoms and assisted by a virtual photon emitted into the cavity and then rescattered, are resonant if the two detunings compensate each other (a schematic of the energy levels and of the transitions is provided in Fig. 1). This resonant transition is pivotal for the realization of the photonic staircase at the core of this work, and the associated effective Hamiltonian reads
The last term embodies the photon-assisted resonance process changing simultaneously the population of the two atomic levels . In Eq. (1) we have reabsorbed the large occupation, , of the level in the coupling ; the mode can therefore be treated classically, while we assume that the occupation of the levels remains small. The frequency stands for the cavity mode frequency relative to the frequency of the lasers.
This derivation follows the lines of Ref. borregard17, considering
three-level atoms Rabi coupled to a single mode optical cavity of frequency . Each atom has an internal structure consisting of the three states , , (ordered with decreasing energy), and the two states have a different Rabi coupling constant, , with the photonic cavity mode. With the energies of detunings, and , larger than all the other energy scales involved in the system, one can microscopically derive, via adiabatic elimination, the hamiltonian (1), and find that , , .
Following this procedure, one finds that and are collective operators summing over all the single-particle excitations of the atoms, and they therefore have bosonic commutation relations.
Iii A staircase of photons
The hamiltonian (1) conserves the number of photons in the system, , allowing for diagonalization in sectors of the Hilbert space with fixed number of photons, . We describe, in each of these sectors, the lowest energy state using the variational ansatz state
where is the vacuum state simultaneously annihilated by . The unitary map transforms the operators and as
yielding the following expectation values
The latter expressions allow to compute the energy of the system, , on the ground-state variational ansatz, , and accordingly to find the value of the parameter, , yielding the minimum of the energy,
Using this equation we can, for instance, evaluate the population (and the coherences) of the level on the ground state
From Eq. (5), it follows that a real solution exists if
for parameters not satisfying this relation, the system exhibits an unstable behavior. The physical interpretation of Eq. (7) is that the strength of the photon-mediated interaction must be less than the quadratic Zeeman shift for the system to be stable.
Furthermore, a stability condition akin to (7) was already recognized in the context of coherent dissociation of a molecular condensate into a multiple-mode atomic one vardi; yuro; anglin; kayali; if the molecular mode is highly occupied, one can linearize the Hamiltonian around the latter, and describe the process of dissociation with an Hamiltonian formally equivalent to (1) (in our system, the role of the highly occupied mode is taken by the level ). As a result of this, the coupling term does not conserve the number of particles created (annihilated) by () and for couplings violating (7), the eigenvalues of (1) becomes complex vardi; yuro; anglin; kayali signalling an instable character of the modes diagonalising the hamiltonian of the system (cf. with Eq. (9) below).
with and ; the angle is, as usual, determined by requiring that off-diagonal terms proportional, for instance, to and its hermitian conjugate vanish (the result coincides with Eq. (7)). We can therefore write the diagonal form of (1) in a sector with fixed number of photons, ,
where the ground state energy of the system reads
The energy draws (as a function of ) a staircase in which each step is associated to a different value of . This is plotted in Fig. 2 together with few excited states energies (computed from Eq. (9)) plotted as dashed lines.
Iv Landau-Zener theory
of the photons’ staircase
The staircase structure facilitates the preparation of a desired number of photons. In order to illustrate this aspect, we add to the hamiltonian, , a term, , accounting for coherent pumping of photons into the system at rate
Corrections to the spectrum are plotted in Fig. 3, and they can be evaluated exactly diagonalizing for few energy levels, using as basis the eigenstates of the unperturbed Hamiltonian (from now on we have dropped the dependence from in to lighten the notation). Our goal is to study Landau-Zener (LZ) transitions among ground state levels with different number of photons, as induced by a time-dependent control parameter . For this LZ analysis, transitions involving excited states dot not play a significant role and they have not been included (e.g. from a ground state with photons into the first excited states of the next photonic manifold, such as a transition from to ), since the matrix elements of connecting two ground states are always larger than those involving excited ones in the parameter regime of large (see Appendix). A special case of study is represented by the transition pertaining the manifolds with zero and one photons: see right panel of Fig. 3.
Accordingly, we write the Hamiltonian in the basis , where it looks tri-diagonal since the perturbation couples states differing only by one photon; these are represented by off-diagonal terms in the following matrix representation of
The overlap is calculated following Ref. perel as (see also Appendix)
In order to gain intuition for the perturbative corrections induced by a weak photon pumping on the photon staircase spectrum, we first consider a simple perturbative analysis in the parameters’ regime , . In this limit, the overlap reads , and the ground state energies, . Let us now consider an energy level , with photon number , crossing with a level with energy and ; at their intersection, occurring at , a straightforward application of degenerate perturbation theory in , yields an energy splitting
Beyond this simple analysis, an exact numerical evaluation of the eigenvalues of the energy matrix (12), as a function of , provides the energy level structure portrayed in the left panel of Fig. 3. A small pumping rate, , is sufficient to induce an effective energy level repulsion reshaping the staircase structure into a sequence of avoided crossings among ground states with different photon numbers. We remark that, although the coherent photon pumping does not commute with the unperturbed Hamiltonian (1), , its effect, for small , is negligible for values of away from the crossing points , and in these regions we can still effectively consider a good quantum number.
According to this structure, an adiabatic climbing of the staircase from a state with zero photons (blue line in the left panel of Fig. 3) to a state with a certain photonic population, can be designed as follows: We start from the ground state with zero photons and given initial at time , and we drive linearly in time the control parameter , with rate ; following the argument presented above, the drive can induce a transition to the ground state with one photon as approaches the crossing point located at (see the zoom of the crossing among and in the region : inset of left panel of Fig. 3).
The right panel of Fig. 3 shows that a slow ramp would favour the transition to the first excited state of the manifold with one photon, but at intermediate ramp speeds, instead, the probability to transit into the ground state is the dominant one. As increases further, the subsequent transitions will basically occur among ground states of manifolds with different photon numbers (if the ramp is moderately slow), since, as discussed at the beginning of this section, for the only sizeable matrix elements of the operator controlling the transition, , are those connecting ground states and (see also the explicit expressions of these overlaps in the Appendix). Therefore, a sequence of LZ-like transitions allows to climb the staircase and to achieve a target number of photons.
Before discussing the driven-dissipative dynamics of the model, we observe that the jumps between the steps of the staircase characterised by different integer values of photons (see Fig. 4), and explored as is increased in time, recalls the current-voltage staircase profile observed in the phenomenon of Coulomb blockade Alha.
Although the underlying mechanism is different, the two cases share the feature that every step of the staircase corresponds to a state with distinctly resolved physical properties: in our quantum optics set-up, for intervals of away from avoided crossings, each step of the staircase is associated to a fixed and quantised number of photons with negligible fluctuations.
V Driven-dissipative dynamics
We now consider the competition between coherent pumping and photon losses, occurring at rate . We assume to suddenly switch at times the coherent pump to counterbalance cavity losses. We prepare the system in the ground state of (1) with zero photons, , and we consider the time evolution ruled by the following set of equations of motion for the expectation values of the ’molecular’ degrees of freedom: atomic coherences, , and populations, ,
coupled to the dynamics of photons
where we have assumed the atomic and photonic degrees of freedom to be in a Gaussian state, and we have included terms describing three-body correlations between light and matter degrees of freedom, such as . The equation of motion for the the latter reads
During driven-dissipative dynamics the magnitude of three-body correlations between light and matter in Eqs. (15) and (16) remains small, with the consequence that the two sectors are almost dynamically decoupled. This allows the number of photons to relax towards a steady state value, , while the atomic excitations, , are still slowly growing as a consequence of the pumping. The steady-state number of photons generated is predicted by the formula
where is the asymptotic steady state value of , attained long before the system enters into the instability region delimited by the bound (7) (the critical value of the interaction strength for the onset of the instability is renormalised to a lower value after the inclusion of three-body light-matter correlations). The time-scales separation in the dynamics of light and matter degrees of freedom is at the origin of the metastability of the plateau reached at long times by . In fact, a slow growth of , occurring while the photons’ steady state is already established, provokes a growth in as well, which acts as a source in Eq. (17) ruling the dynamics of . The latter determines at late times an increase in the number of photons produced, which eventually renders the system unstable (cf. again with Eq. (7)). At the timescales for the onset of this dynamical instability, the quantities evolving under Eqs. (15) and (16) blow up exponentially, signalling a departure from the plateau in , which is accordingly a metastable phenomenon. The characteristic time, , for the departure from this metastable photonic steady-state is proportional to : in Fig. 5 we portrait time-resolved profiles of at increasing pumping rates, , and we illustrate the dynamical instability as becomes sufficiently large. Fitting the breakdown time of the photonic plateau as a function of and , and taking the limit , one can see that , which suggests that can be delayed increasing the number of atoms in the mode .
The inset of Fig. 5, displaying the average number of photons as function of , demonstrates instead that the quantization of typical of the staircase structure (still present when the photonic pumping is adiabatically switched and photons generated via slow LZ transitions, cf. Fig. 3 and related discussion) is lost when pumping and dissipation are suddenly turned on, since ground and excited states of the staircase are strongly mixed in this case.
Photon generation via LZ transitions represents also a more convenient route in view of preparing quantum light in our system.
In each one of the plateaux of Fig. 4 the photonic degree of freedom is in a state with fixed and quantised number of photons, , provided dissipation, , is weak enough to affect the dynamics of the system only at late times.
On the contrary, suddenly switching the photon pumping results in a transient dynamics with no quantised photon number (cf. Fig. 5), which asymptotes to a plateau where quantum features have been erased.
Specifically, we have resolved the dynamics of our model combining a Gaussian ansatz for the atoms, as done in Eqs. (15), with a truncated ansatz for the density matrix of the light, , with and (see Appendix).
Although in the plateaux shown in Fig. 5 the presence of the atomic degrees of freedom sizeably enhances the asymptotic expectation value of compared to the decoupled () case (and therefore the system is in a state where light and matter are hybridised), the light generated is classical, as we have checked by calculating the photonic variance, , from the density matrix ansatz, , found always very close to – a signature of the classical nature of the light produced in the cavity.
This occurs for times , when the system has reached the steady state and at the same time the dissipation has washed out any quantum feature present at short times (see Appendix).
Naturally, also in the case of LZ photon preparation we expect that dissipation will classicalise the state of light at late times, but the staircase structure of the photonic response (see Fig 4) guarantees that, at intermediate times, the light degree of freedom will be found in a quantum state with a well defined number of photons and few fluctuations on the top of them.
Conveniently for experimental realizations, the speed at which the staircase can be climbed is collectively enhanced by the large atomic population in the state. The reason for this is the scaling with of the characteristic energy scale , which sets an ultimate limit on the gap. A limit on the coupling strength , in turn, is set by decoherence from atomic spontaneous emission at rate , where is the single-atom cooperativity given the atomic excited-state linewidth . Thus, climbing the staircase requires large collective cooperativity , in addition to the requirement to avoid photon loss. Collective cooperativities are routinely achieved with atomic ensembles in optical cavities, making the staircase accessible to current experiments.
As a future direction, it would be interesting to study a many-body version of the problem analysed in this work, which can be realised, for instance, considering a one-dimensional lattice of several cavities (modelled as in Fig. 1), connected one to each other by next-neighbour photonic hoppings (in the spirit of an Hubbard model; see for similar ideas in quantum optics the review in houck12).
Studying the competition of this kinetic term with the
driving and dissipation discussed in this work, would pave the way to a quantum many body simulator for the preparation of multi-photon states, which would benefit of the tunability properties of the photons’ staircase as a leverage for experimental implementations. It would be, for instance, intriguing to look for driven-dissipative phase transitions in this many-body version of our system following the directions mentioned in the introduction.
We acknowledge discussions with F. Reiter. JM acknowledges support by the EU Horizon 2020 research and innovation program under Marie Sklodowska-Curie Grant Agreement No. 745608 (MC). YS and ED acknowledge support from Harvard-MIT CUA, NSF Grant No. DMR-1308435, AFOSR-MURI Quantum Phases of Matter (grant FA9550-14-1-0035), AFOSR-MURI Photonic Quantum Matter (award FA95501610323).
Appendix A Matrix elements for the Landau-Zener transition
In this section of the Appendix we detail the calculation of the matrix elements of the perturbation connecting ground and excited states of the staircase, involved in the study of the transitions in Fig. 3. The squeezed ground state (2) can be represented in the Fock basis of the occupation numbers of the modes, , as (see for instance Ref. perel)
where is the squeezing angle for a fixed number of photons, . We describe the excited states of the system using the quasiparticle creation operators and introduced in Eq. (8). We represent these operators inverting the Bogolyubov rotation (8):
Any excited state of the system can be represented as
with and .
In Eq. (11) we introduced a term accounting for coherent pumping of photons into the system at rate , . This term introduces mixing between the ground state in the sector with photons and excited states in the sector with photons. In order to account for this effect, we calculate the overlap of the excited states in the sector with photons with the ground state of the manifold with photons, .
First of all, we notice that the overlap between the excited states with quasiparticle excitations of only one type, and the ground state in the neighbouring sector, is equal to zero for any number of excitations ():
This is a consequence of the fact that the ground state (19) can be written as a superposition of states with the same number of excitations in the () sectors of the Fock space of the original degrees of freedom of the model.
The non-zero overlaps with excited states induced by the perturbation operator is between the state with quasiparticles in both () modes, . The first non-trivial overlap is . We calculate this overlap using the representation (19),
Analogously, we calculate the overlap
In these expressions one can recognise the overlap between the ground states of adjacent photonic manifolds, given by Eq. (13).
These overlaps allow to solve the LZ problem reported in Fig. 3b, and to check that adding the next excited state , the feature of an intermediate window of ramp speeds where the transition occurs only involving ground states, remains substantially unaffected. This is reported in Fig. 6 of this Appendix.
Appendix B Combined Gutzwiller and Gaussian ansätze for driven-dissipative dynamics
In this section of the Appendix we summarise the calculation of the photon variance , for which we resort to the following ansatz for the system density matrix:
where is a Gaussian ansatz density matrix for the atomic degrees of freedom, while for the photonic degree of freedom we write a density matrix in a bosonic Hilbert space truncated up to bosons:
with (we use in the following calculations). This is in spirit similar to the Gutzwiller ansatz employed in Ref. walt for the dissipative dynamics of bosons. Inserting the ansatz (25) in the Lindblad equation
with incoherent photon losses at rate , we find that the equation of motions for the two-point functions of the atomic degrees of freedom follow Eqs. (15) (with three-body correlations set to zero), with the difference that now , while the equations (16) for the one and two-point functions of the photon are replaced by the linear system of equations of motion for the matrix elements of . As initial conditions, we consider . These equations appear cumbersome for large , but they can be readily derived. Here, for illustrative purposes, we write down the equations of motion for the population of the mode, , and for the first coherence, :
As a sanity check we benchmarked our predictions for in the exactly solvable case, .
The variance is then straightforwardly written in terms of the matrix elements of ,
An instance of the dynamics of is reported in Fig. 7, showing that at late times, . This circumstance is independent from the specific choice of parameters adopted.