A Connection between V^{\beta\,\gamma}_{\alpha} and W^{\beta\,\gamma}_{\alpha}

Damping of quasiparticles in a Bose-Einstein condensate coupled to an optical cavity


We present a general theory for calculating the damping rate of elementary density wave excitations in a Bose-Einstein condensate strongly coupled to a single radiation field mode of an optical cavity. Thereby we give a detailed derivation of the huge resonant enhancement in the Beliaev damping of a density wave mode, predicted recently by Kónya et al., Phys. Rev. A 89, 051601(R) (2014). The given density-wave mode constitutes the polariton-like soft mode of the self-organization phase transition. The resonant enhancement takes place, both in the normal and ordered phases, outside the critical region. We show that the large damping rate is accompanied by a significant frequency shift of this polariton mode. Going beyond the Born-Markov approximation and determining the poles of the retarded Green’s function of the polariton, we reveal a strong coupling between the polariton and a collective mode in the phonon bath formed by the other density wave modes.

03.75.Hh, 37.30.+i, 05.30.Rt, 31.15.xm

I Introduction

Well-established properties of ultracold atoms are drastically altered when the atoms are coupled to the radiation field of an optical resonator Ritsch et al. (2013). Even if the absorption is suppressed by using only far detuned laser sources, the ensemble of atoms can represent a significant optical density which leads to a strong effect on the field of a high-finesse resonator. The back-action of the cavity field onto the atom cloud is the origin of various novel features or even phenomena. For example, the optical dipole potential exerted dynamically by the cavity field can vary considerably over the kinetic energy scale of the ultracold gas. In this limit, the phase diagram of strongly localized particles is greatly enriched with respect to the one obtained from the Bose-Hubbard model for an inert external potential Maschler et al. (2008); Larson et al. (2008); Vukics et al. (2009); Vidal et al. (2010); Li et al. (2013). In the opposite limit, i.e., when the optical dipole potential is negligible and the ultracold atoms form a Bose-Einstein condensate (BEC) which is homogeneous on the optical wavelength scale, the cavity field can still give rise to a significant effect on the elementary excitations, or as often termed “quasiparticles”. Quasiparticle features are of central importance in general for the description of dynamical many-body phenomena. A prominent example is the critical mode softening which accompanies the recently observed self-organization phase transition Nagy et al. (2010); Baumann et al. (2010).

The system of BEC in an optical resonator proved to be suitable for the quantum simulation of the Dicke model by representing the spin of the original formulation by two collective motional modes of the cloud Nagy et al. (2010); Baumann et al. (2010). The Dicke model predicts a critical point when the coupling strength reaches the geometric mean of the frequencies characteristic to the spin and to the boson mode Emary and Brandes (2003). This quantum criticality is the zero temperature limit of the spatial self-organization phase transition of atoms in a cavity Domokos and Ritsch (2002) that has been observed in experiments Black et al. (2003); Arnold et al. (2012). Quantum criticality has been observed also in other closely related experiments Schmidt et al. (2014); Keßler et al. (2014) where one can invoke a variant of the Dicke model as a few-mode, simplified model to interpret the observations. There are also many theoretical generalizations to describe other exotic phases Baksic and Ciuti (2014), such as magnetism Safaei et al. (2013), glassiness Gopalakrishnan et al. (2009); Strack and Sachdev (2011); Jing et al. (2011); Buchhold et al. (2013); Habibian et al. (2013), or related self-ordering criticality with fermionic atoms Piazza and Strack (2014); Keeling et al. (2014); Chen et al. (2014).

Critical behaviour in quantum phase transitions is determined by the dynamical features of the soft mode. In an open system the set of relevant parameters is expanded by the properties of the driving and dissipation channels. The system of laser-illuminated atoms coupled to a cavity mode realize, in fact, an open system variant of the Dicke model Dimer et al. (2007); Baden et al. (2014); Buchhold et al. (2013). Indeed, as it has been predicted Nagy et al. (2011); Öztop et al. (2012) and recent experiments have shown Brennecke et al. (2013), dissipation and the accompanying quantum fluctuations substantially modify the correlation functions and the critical exponents Dalla Torre et al. (2010); Bastidas et al. (2012); Dalla Torre et al. (2013). Dissipation is thus a key player in quantum phase transitions Morrison and Parkins (2008); Kessler et al. (2012); Grimsmo and Parkins (2013); Keeling et al. (2010); Bhaseen et al. (2012); Kopylov et al. (2013); Piazza et al. (2013); Eleuch and Rotter (2014); Xu et al. (2013).

The experiment performed by Brennecke et al. Brennecke et al. (2013) revealed that the interaction between the quasiparticles in a BEC is relevant to quantitatively interpret measurement data on the superradiant phase transition of the Dicke-model. Motivated by this observation we generalized the previous models so that to include other dissipation channels that can play a non-negligible role. In the special case under consideration, the soft mode consists dominantly of a collective density wave excitation of the BEC Mottl et al. (2012). Therefore, the friction of a density wave quasiparticle in a superfluid of weakly interacting bosonic atoms has to be reconsidered.

There are basically two collisional mechanisms responsible for the decay of a density wave in a BEC Ozeri et al. (2005). The first one is Landau damping Jackson and Zaremba (2003); Guilleumas and Pitaevskii (1999, 2003); Tsuchiya and Griffin (2005); Jackson and Zaremba (2002), in which the given quasiparticle and another one combine into a third quasiparticle. This mechanism needs a thermal occupation of the other excitation, therefore it vanishes at zero temperature. On the other hand, it exists also in non-superfluid systems. The second mechanism, characteristic only to superfluids, is Beliaev damping Hodby et al. (2001); Katz et al. (2002). In this case, stimulated by the superfluid background, the selected quasiparticle decays into two lower energy excitations. This process occurs even at zero temperature Kagan and Maksimov (2001).

In general, the damping rate of quasiparticles that constitute the soft mode is expected to depend on the control parameter of the phase transition. This is simply because the frequency of the soft mode varies over a large range before it vanishes at the critical point. However, the monotonous variation of the frequency as approaching to the critical point is accompanied, unexpectedly, by a drastic, resonance-like enhancement in the damping rate Kónya et al. (2014). Although the mode softening, as we will show, is a necessary ingredient for the effect, the resonant peak is clearly outside the critical region.

In this paper we will present a detailed derivation of this effect that has already been briefly reported in Ref. Kónya et al. (2014). The damping rate enhancement can be attributed to the interaction with the other density wave modes of the condensate via s-wave collision. These density waves are associated with quasi-momentum modes that form a continuum bath for a large BEC, hence we can evaluate its effect within the Born–Markov approximation. However, it turns out that the interaction between the soft mode and the other quasiparticles is not so weak and we need to resort to a more accurate analysis which is exempt from the Born approximation underlying the results of Ref. Kónya et al. (2014). The presented calculation reveals that the soft mode has a non-negligible influence back on the spectrum of the bath of quasi-momentum modes. That is, the nonlinear s-wave scattering couples significantly other modes into the dynamics, thus the soft mode is one component in a set of interacting bosonic modes.

The rest of the paper is structured as follows. In Sec. II, we will introduce the model for the BEC-cavity system which includes many degrees of freedom of the ultracold atom gas. We will present the equations of motion which allow for describing the system beyond the standard Bogoliubov-type mean field approach. This latter, limited to a linearized treatment of quantum fluctuations, is used in Sec. III to determine the polariton and phonon degrees of freedom which are cross-coupled through the terms higher than first order in quantum fluctuations. The effect of phonons on the polaritons is taken into account by means of a bosonization approximation given in Sec. IV. In Sec. V, the Beliaev and Landau damping rates are evaluated first within Born–Markov approximation for the phonon bath, and then the Markov approximation is carried out also non-perturbatively by means of the Green’s function method. Finally, we summarize the results in Sec. VI.

Ii Ultracold atoms in an optical resonator

We consider a Bose-Einstein condensate of ultracold alkali atoms loaded in the volume of a high-finesse, single-mode, optical resonator. The atoms are illuminated by a far-detuned laser from a direction perpendicular to the cavity axis. The detuning between the laser and the atomic transition frequency is large enough so that the atoms behave as linear scatterers and their internal dynamics can be adiabatically eliminated. At the same time, the scattering is enhanced in the cavity mode since the driving frequency is close to that of the selected single cavity mode, i.e., the detuning is on the order of the cavity linewidth .

Such a transverse pumping geometry is known to exhibit a critical point, as illustrated in Fig. 1. Below a threshold pump power, a homogeneous Bose-Einstein condensate together with no coherent photons in the cavity remains a stable solution. This is interesting since the collisional properties and damping of quasiparticles can be studied for the elementary case of a homogeneous superfluid. When the intensity of the driving laser exceeds a critical value, the condensate density is spatially modulated according to the cavity mode function, and the condensate atoms can coherently scatter photons into the cavity. There appears two stable self-organized solutions connected by a symmetry, which is spontaneously broken in the high intensity phase. The theory we will develop below applies, of course, also to this inhomogeneous situation.

Figure 1: (Color online) Schematic representation of the self-organization phase transition. Left: below a threshold value of the transverse laser pump power, the BEC fills the cavity homogeneously on the wavelength scale, and there is no light scattering from the pump laser into the cavity. Above threshold (right panel), the condensate self-organizes into a wavelength periodic pattern and Bragg-scatters into the cavity. The field building up in the cavity traps the atoms in the patterned spatial structure thereby stabilizing the ordered phase.

The essentials of the self-organization phase transition can be seized by a two-mode approximation, which can be mapped to the Dicke model Nagy et al. (2010); Baumann et al. (2010). The measured phase diagram as well as the spectrum of fluctuations can be interpreted by means of a single motional mode coupled to the cavity photon mode. Such a simplified approach has been thus verified, although the experiment included effectively a two-dimensional geometry for the cloud. The parameters of the two-mode model, of course, depend on the geometric factors and the dimension of the problem. In the following, we have to resort to a multimode model for describing higher-order than usual mean field effects. However, similarly to the mean-field description of the self-organization phase transition, we will stick to considering only one-dimensional motion of the atoms, which offers the most transparent presentation of the effect of the coupling to photons on the damping properties of superfluid quasiparticles. Later, when certain results are of interest also quantitatively, we will consider the question of dimensionality.

ii.1 Hamiltonian in Bloch-state basis

The single-mode cavity field is described by the mode function , where is the wave number, and is associated with the bosonic annihilation and creation operators and . The atomic motion is represented by the second-quantized wavefunction and its hermitian conjugate . The grand canonical Hamiltonian of the system, in units of , in a frame rotating at the laser frequency is given by


The first term is the photon energy in the rotating frame, the detuning must be negative (“red”) in order to have a well defined ground state. Next, the spatial integral contains the kinetic energy for particles with mass and the chemical potential . There are three kinds of interaction in the system. The first is connected to the scattering between the laser drive and the cavity mode which is described by the effective amplitude . The spatial dependence of this interaction inherits the cavity mode function. Note that the time-dependent driving is removed from this term by going to the rotating frame. The second is the dispersive phase shift exerted by the atoms on the cavity mode resonance, and is characterized by being the resonance shift by a single atom at an antinode. This interaction involves a cavity photon absorption and emission, thus the spatial dependence is . Both of these interactions is proportional simply to the matter-wave field density . Finally, the last term is nonlinear in the atom density and accounts for the s-wave collisions between the atoms, the strength is given by .

The periodicity of the atom-field interaction terms with the wavenumber suggests that we introduce the Bloch-state basis for the atomic field operator


where the quasi-momentum is in the interval . The lowest band is with homogeneous wavefunction. The first and second excited bands are expanded by combinations of the and modes having and wave functions, which are coupled by the kinetic energy term. Modes in these bands carry, beside the quasi-momentum , a momentum equivalent of the photon wave number. Higher bands are neglected in this study, which is exactly valid below the critical point and is a good approximation above, but still in the vicinity of the critical point Kónya et al. (2011). In brief, the matter-wave field is treated in a three-band approximation Nagy et al. (2013) instead of the previously used two-mode description Nagy et al. (2010, 2011); Kónya et al. (2012).

The grand canonical Hamiltonian written in Bloch basis reads as


The cavity Hamiltonian remains the same,


The atomic Hamiltonian is given by


Note that for the and modes are coupled. As a result of scattering a laser photon into the cavity, or reversely, atoms are transfered between the and modes


The next dispersive interaction term is proportional to the product of the photon number and the atomic occupation numbers


The collision term consists of two parts,


For normal collisions, the quasi-momentum is conserved


where is the difference between the total incoming and outgoing quasi-momenta. For umklapp processes, the value of the total quasi-momentum changes with or :


As we will see later, umklapp processes are negligible, so we don’t give the detailed expression here.

ii.2 Bose-Einstein condensate in the cavity

All the system variables can be split to the sum of their expectation values and quantum fluctuations,


We assume that the condensate is formed at the center of the lowest band.

The coherent electromagnetic field amplitude in the resonator is . The total number of condensate atoms is , which is distributed according to the amplitudes and between the homogeneous and the cosine-like modes, respectively. The normalization condition is then , which allows for determining the chemical potential . The condensate does not extend into the sine-like mode because it is not coupled to the and modes by the coherent atom-photon interactions. This follows simply from the parity conservation of the interaction (1). The operators denoted by tilde correspond to the fluctuations.

The threshold for the self-organization phase transition is at . Below the critical driving, the system is in the normal phase corresponding to the simple solution , , and Nagy et al. (2010); Baumann et al. (2010). Above threshold, gradually increases, and far above threshold the approximation of restricting the atomic wavefunction into three bands is no longer valid.

The excitations of the system can be grouped into two sets. For , the laser pump couples to the operators , and , and these form the polariton excitations of the system. The remaining modes, , and , form the phonon excitations.

It is useful to introduce new parameters for the coupling strengths,


which have well defined values in the thermodynamic limit, defined as . Accordingly, the critical coupling is


which we will use in the following for scaling the driving strength.

ii.3 Equations of motion beyond the Bogoliubov approximation

The dynamics of the system is given by the Heisenberg equation of motion:


After we substitute Eq. (11) into this formula, we obtain a hierarchy of terms. In the standard Bogoliubov approximation, only the zeroth and the first order terms are kept. The mean-field equations are given by the zeroth order terms and the dynamics of the excitations is determined by the first order terms. Since we aim at describing the polariton-phonon interaction in our model, we have to go one step further and include the second order terms into our description.

The mean-field equations now read


where the back-action of the fluctuations through the expectation value of the second order terms were omitted. Numerically, we can search for the steady state solution of these equations, where the left hand side is set to zero.

Now, we give the equations of the fluctuations. Let us introduce the compact vector notation for the polariton and phonon variables


respectively. The operators in each of these vectors are linearly coupled among each other, and there is a non-linear cross-coupling between the elements of the different vectors


These equations establish the basis of our calculations in the rest of the paper. The linear part, represented by the matrices and , are treated usually in the Bogoliubov-type mean field descriptions. The additional terms have not yet been investigated in the context of coupled BEC and optical cavity systems.

Furthermore, we note that there is also a nonlinear polariton-polariton and phonon-phonon interaction in the system, but these effects are neglected in (17). The reason behind this approximation is that (i) the polariton-polariton interaction turns out to be nonresonant, and (ii) the phonon-phonon interaction does not give a contribution to the polariton damping rate, which we aim to calculate. In fact, the phonon-phonon interaction determines the damping rate of the phonons. Later on, we will introduce this phonon damping as a phenomenological parameter.

Iii Polaritons and phonons

In the previous section, we separated the elementary excitations of the system to polariton and phonon sets. There is a linear coupling among the variables within each of these sets in (17). In the following, we will perform a Bogoliubov-type diagonalization in order to determine the polariton and phonon eigenmodes which are then coupled in higher order interaction terms.

iii.1 Bogoliubov normal modes

The matrices and representing the linear coupling among the polariton-type and the phonon-type modes, respectively, have left and right eigenvectors


The polariton and the phonon normal modes are defined then by


where the indexes the polariton eigenfrequencies and the phonon bands. As usual for the general Bogoliubov transformation, the normal modes mix the creation and annihilation operators. In order to be able to separately deal with the annihilation and creation processes for polariton and phonon elementary excitations in the following, we make use of the symmetries of the system of equations.

Let us introduce the matrix


which simply swaps the creation and the annihilation operators


and where the quasi-momentum is also reflected in the second case. It follows that the matrices and have the symmetry,


The symmetry ensures that the eigenvalues and eigenvectors come in pairs,


Note that the phonon spectrum is symmetric in the quasi-momentum: . The phonon spectrum for is plotted in Fig. 2.

Figure 2: (Color online) Phonon spectrum. The laser pump picks the polariton excitation of which frequency is denoted by a red dot. The scattering processes underlying the Landau and Beliaev damping processes are schematically represented on the left and right sides, respectively.

The symmetry guarantees that in a pair of complex eigenvalues the imaginary parts are the same, whereas the real parts have equal magnitude but opposite sign. We can thus refer to positive and negative frequency modes, according to the sign of the real part of the complex eigenfrequency. For the corresponding eigenvectors, one can prove that and that . The normal mode expansion can be expressed in terms of only the positive frequency modes,


where means that we are summing over only the positive frequency modes. The negative modes are automatically included by the second term. By means of using the symmetry, the annihilation and the creation of quasiparticles is manifestly separated in this form.

So far, the symmetry consideration was very general. It relies solely on the fact that the set of variables includes hermitian conjugate pairs of bosonic annihilation and creation operators, which is then inherited by the Bogoliubov normal modes. To be more specific, here we deal with a Hamiltonian system, which implies an additional symmetry of the polariton and phonon coupling matrices, and , respectively. This symmetry can be formulated by means of the matrix


and reads


The symmetry ensures that the eigenfrequencies are real and it also gives a relation between the left and the right eigenvectors:


where gives the sign of the argument. Since the left and right eigenvectors form a reciprocal basis with respect to each other, we obtain the normalization conditions


With the help of these conditions, one can prove that


which verifies that the positive frequency normal modes are bosonic quasiparticles.

iii.2 Polariton-phonon interaction

Let us now rewrite the coupled polariton-phonon equations of motion in (17) in terms of the positive frequency normal modes, i.e., quasiparticles, by using (24). The equation for the polaritons read ()


where the coefficients are given by


These expressions involve the components of the left- and right eigenvectors of the linear coupling matrices, and the coupling matrix appearing in the original equation (17). All these quantities depend on the mean-field solution, and can be calculated, in general, only numerically. In the first step, the mean field is determined by solving the coupled, nonlinear algebraic equations (15). Then linear matrix algebra is used in a straightforward manner.

Similarly, the phonon equations read ()


where the coefficients are


We will show in Appendix A that the connection between the coefficients and implies


This result allows us to introduce an effective Hamiltonian for the polaritons and the phonons, from which the above two equations of motion can be derived as Heisenberg-equations.

iii.3 Effective Hamiltonian

The full effective Hamiltonian corresponding to the two equations of motion, Eqs. (30) and (32), is given by


So far, we presented a theory which can generally describe the interaction of selected quasiparticles of a cavity-BEC system with the continuum of phonons. In the following we will use the main results of the theory in an interesting, highly non-trivial case. Now, without losing generality, we will consider only a certain part of the full effective Hamiltonian, which refers a selected polariton quasiparticle, which is the soft mode of the self-organization phase transition, denoted by . The frequency of the soft mode as a function of the control parameter normalized to the critical value, , is plotted in Fig. 3). Further, we denote by the lowest, and by the middle phonon branches displayed in Fig. 2.

Figure 3: (Color online) Real part of the polariton quasiparticle frequency (only the positive frequency part is shown) as a function of the external laser drive strength. This polariton is the soft mode of the self-organization phase transition, hence the frequency vanishes at a critical point. Without external driving (), the polariton frequency has just the value where the middle and the upper branches touch for in the phonon spectrum in Fig. 2.

The relevant part of the effective Hamiltonian accounting for the polariton-phonon coupling is


where is the soft mode frequency. The coefficients and describe the strengths of the so-called Landau- and Beliaev-type coupling processes (illustrated in Fig. 2). In the former, the polariton merges with a phonon from the lowest branch to create a phonon on the middle branch. In this process a condensate atom is created simultaneously. The latter, Beliaev process corresponds to the creation of two phonons, this process is stimulated by the background condensate. The energy and the quasi-momentum quantum numbers obviously need to be conserved during these processes. Furthermore the total momentum has to be conserved also, which means that one of the phonons should be in the middle and one should be in the lowest branch.

The Heisenberg equations of motion generated by this Hamiltonian are nonlinear and cannot be solved generally. In accordance with the usual treatment of open systems and Markov approximation, we will approximate the state of the phonon degrees of freedom as being close to a thermal equilibrium.

Iv Bosonization of the phonon bath

Let us introduce two operators which correspond to the Landau and Beliaev processes, respectively,


where and are unspecified normalization coefficients.The identity implies the algebraic relations


By assuming that the occupation number in the phonon modes remains close to the thermal one, we can use the following mean field approximation


where is the thermal occupation number. By setting the normalization factors as


we obtain normal bosonic commutation relations


In this approximation scheme, we have introduced new bosonic modes describing the phonons. The effective Hamiltonian can be rewritten as


where we used the eigenfrequencies of the Landau-type and Beliaev-type quasiparticles, which come from the definition (37). This is now a solvable, quadratic Hamiltonian leading to coupled, linear equations of motion