Collective decoherence of cold atoms coupled to a Bose-Einstein condensate
We examine the time evolution of cold atoms (impurities) interacting with an environment consisting of a degenerate bosonic quantum gas. The impurity atoms differ from the environment atoms, being of a different species. This allows one to superimpose two independent trapping potentials, each being effective only on one atomic kind, while transparent to the other. When the environment is homogeneous and the impurities are confined in a potential consisting of a set of double wells, the system can be described in terms of an effective spin-boson model, where the occupation of the left or right well of each site represents the two (pseudo)-spin states. The irreversible dynamics of such system is here studied exactly, i.e., not in terms of a Markovian master equation. The dynamics of one and two impurities is remarkably different in respect of the standard decoherence of the spin - boson system. In particular we show: (i) the appearance of coherence oscillations, (ii) the presence of super and sub decoherent states which differ from the standard ones of the spin boson model, and (iii) the persistence of coherence in the system at long times. We show that this behaviour is due to the fact that the pseudospins have an internal spatial structure. We argue that collective decoherence also prompts information about the correlation length of the environment. In a one dimensional configuration one can change even stronger the qualitative behaviour of the dephasing just by tuning the interaction of the bath.
The reasons of the great interest for the physics of ultracold atoms in recent years are manifold. On the one hand experimentalists have reached an unprecedented control over the many-body atomic state with very stable optical potentials and by the use of Feshbach resonances which allow one to change the scattering length of the atoms . In this context the tremendous experimental results that have been achieved include: the observation of the superfluid-Mott insulator transition for bosons , one dimensional strongly interacting bosons in the Tonks-Girardeau regime  and Anderson localization [4, 5]. On the other hand new experimental challenges come from different theoretical proposals for using this system for quantum information processing  and as a quantum simulator of condensed matter models (see for example [7, 8, 9] and references therein).
Not only can ultracold atoms simulate Hamiltonian systems, but such systems also offer a way to engineer non classical environments. Thanks to the flexibility of quantum gases, a broad range of regimes of irreversible dynamics of open quantum systems and in particular of spin-boson systems can be explored [10, 11, 12, 13, 14].
In the present paper we propose a new way in which an instance of the spin - boson model  can be realized with a suitable arrangement of interacting cold atoms. In particular we analyse a system consisting of cold impurity atoms interacting with a degenerate quantum gas of a different atomic species. This setup makes possible the superposition of two independent trapping potentials, each being effective on one atomic species only, while transparent to the other. When the quantum gas is homogeneous and the impurities are confined in a potential composed of double wells, the system can be described in terms of an effective spin-boson model, where occupations of the left or right well represent the two (pseudo)-spin states. At variance with other setups, where the role of the pseudospin is played by the presence or absence of one particle in a trapping well , by the vibrational modes of a single well  or by internal electronic levels , in our case each pseudospin has a spatial dimension, namely the separation between the two minima of the impurity double well. This introduces an effective suppression of the decoherence due to low frequency modes of the environment and leads to unusual and interesting phenomena, like oscillations of coherence at finite times and the survival of coherence at long times. Further novel features appear when one considers the irreversible collective decoherence of a systems of two impurities. In this case we still predict the existence of subdecoherent and of superdecoherent state, but with the interesting fact that their role is exactly the opposite from what one observes in conventional spin-boson systems. Further interesting features appear when one considers how the collective decoherence rates change as a function of the impurities’ separation and the effects of dimensionality of the system.
In discussing our investigations, for the sake of simplicity we shall consider an experimental setup where the impurity atoms are trapped by a periodic (optical) lattice. We like to stress, however, that our findings do not depend on the lattice properties (e.g., periodicity) but for the numerical results. Other setups, such as microtraps on atom chips or quantum dots, just to mention a few, can be equally envisaged.
2 The Hamiltonian
Our system is composed of a cold quantum gas of bosonic atoms and a sample of cold atoms separated from each other and immersed in the quantum gas. In presenting our investigations, we shall use the words ‘reservoir’, ‘bath’ and ‘environment’ as synonyms to indicate the quantum gas, since its properties are not the focus of the present paper.
The second-quantized form of the Hamiltonian of the impurities+bath system takes the form (see also Ref.)
is the Hamiltonian of atomic impurities, described by the field operator in the trapping potential which creates a set of double wells of size and separated by a distance , see Fig. 1,
is the Hamiltonian of the bath, composed of bosons, represented by the field operator and confined by a trapping potential and is the boson-boson coupling constant, with the scattering length of the condensate atoms, and
describes the interactions between the impurities and the bath; here is the coupling constant of impurities-gas interaction, with the scattering length of the impurities-gas collisions and their reduced mass. Both impurity and bath atoms are described in the second-quantized formalism. The field operator of the atomic impurities
can be decomposed in terms of the real eigenstates of impurity atoms localized on the double well of the potential in the state, with energy and the corresponding annihilation operator . We assume that the wavefunctions of different double wells have a negligible common support, i.e., at any position .
We treat the gas of bosons following Bogoliubov’s approach (see, for instance, Ref.) and assuming a very shallow trapping potential , such that the bosonic gas can be considered homogeneous. In the degenerate regime the bosonic field can be decomposed as
where is the condensate wave function (or order parameter), is the number of atoms in the condensate and , are the annihilation and creation operators of the Bogoliubov modes with momentum . For a homogeneous condensate , being the volume. Its Bogoliubov modes
where and is the condensate density. As one can see from (9), low-energy excitations have phonon-like (wave-like) spectrum, whereas high-energy excitations have particle-like spectrum. The condition for wave-like excitations is , i.e., , or equivalently , where is the speed of sound at zero temperature. Note that and describe the limiting case of non-interacting bosons, each with energy .
for the impurities,
for the quantum gas, with
for the condensate and
for the collective excitations (Bogoliubov modes) of energy in the condensate, and
for the interaction Hamiltonian; the terms which are quadratic in the Bogoliubov excitation operators give negligible contributions and have been omitted. The first term in (14) describes transitions between impurities’ vibrational states due to the condensate, whereas the remaining terms describe similar transitions induced by the collective excitations in the condensate. In a homogeneous condensate, transitions between different vibrational eigenstates of the impurities induced by the condensate are suppressed, while all vibrational states get an energy shift ,
so the contribution of the first term in (14) can be included in the definition of .
In the limit of deep, symmetric wells in each double well and separated by a high energy barrier, the tunneling between adjacent wells is suppressed. In this regime the ground states and of, respectively, the left and right well of double well are well separated in space with vanishing spatial overlap, their coupling to the excited states becomes negligible and the total Hamiltonian further simplifies into
where we have defined the coupling frequencies
and is the number operator of impurities in the double well in the well .
We consider the case where each double well is occupied by at most one impurity atom. This allows us to describe the occupation of the left and right well of each site in terms of pseudospin states. Introducing the Pauli operators as , , the Hamiltonian (16) takes the form of the independent boson model 
where a constant energy shift has been omitted. We note that spin-boson systems with larger spin values can be realized in the same way with higher occupation of the double wells.
The effects due to quantum noise on coherent superpositions of states of a double well spin-boson hamiltonian have been analyzed in the markovian regime. In [21, 22, 23] the effects of a cold atom reservoir has been analyzed, while  has considered the effects of scattered photons, taking into account also the role of the inter-well separation. As we will show in the following section, for our system it is possible to carry a full analysis of the impurity dynamics, going beyond the Markov approximation.
3 Exact reduced impurities dynamics
The dynamics due to the spin-boson Hamiltonian (18) is amenable of an exact analytical solution and is characterized by decoherence without dissipation [25, 26, 27]. The time-evolution operator corresponding to the Hamiltonian (18) can be factorized into a product of simpler exponential operators,
where the functions
have been introduced for ease of notation. Details of the derivation of (19) for the time evolution operator are given in Appendix A. As in this paper we are interested in the irreversible collective decoherence of the impurities we will focus our attention on the conditional displacement operator
Indeed this operator is the one responsible of the decoherence of impurities as it induces entanglement between them and the reservoir. Labeling the state of the impurities as with denoting the presence of the atom, respectively, in the left or right well, the matrix elements of reduced density operator of the impurities are
Assuming that each mode of the bosonic environment is in a mixed state at equilibrium at temperature the decay exponent contains all the information concerning the time dependence of the decoherence process and takes the form
with . The phase factors , and , whose specific form is given in appendix B, do not play any role in the decoherence . They contain however interesting information on the effective coupling between the pseudospins induced by the consensate and will be analysed in a future paper  .
4 Results for the decoherence
As mentioned in the Introduction, we shall assume that the impurity atoms are trapped by an optical (super)lattice, whose form can be controlled and varied in time with great accuracy [30, 31]. The coupling frequencies are accordingly evaluated in Appendix C assuming an optical lattice, with identical, double wells in each site, and deep trapping of impurity atoms in their wells, with identical confinement in each direction. Atomic wave functions can then be approximated by harmonic oscillator ground states of variance parameter , where is the corresponding harmonic frequency. As will be clear shortly, acts as a natural cutoff parameter, quenching the coupling with high frequency modes.
Specifically, we consider Na impurity atoms trapped in a far-detuned optical lattice and a Rb condensate. The condensate density is , the lattice wavelength is , and we have taken and . The depth of the optical lattice is described by the parameter , being the optical lattice potential maximum intensity and the recoil energy of impurity atoms in the lattice; in our evaluations we put . Finally, we assume , where is the Bohr radius, for the scattering length of impurities-condensate mixtures. This parameter can be modified in laboratory with the help of Feshbach resonances.
4.1 Single impurity decoherence
We first examine the decoherence exponent of a single impurity
Such quantity, that will be a useful benchmark in our analysis of the collective decoherence of impurity pairs, shows already interesting features. Assuming, from now on, that the condensate is at temperature , we obtain
We note the dependence of on the length , where is the distance between two wells within each site. The presence of the factor supresses the effect of the reservoir modes at small . This is clearly understandable: environment modes whose wavelength is longer than cannot “resolve” the spatially separated wells within each site. The consequences of this fact will be clear shortly. Replacing the sum over discrete modes to a continuum with the usual rule , choosing as azimuthal axis and using well known relations for Bogoliubov modes , we finally obtain
The superscript is to remind us that we are dealing with impurities interacting with a condensate. For the special case of a bath of noninteracting bosons is obtained from (29) simply imposing and . Let us point out that the spectral density, which reads
has a non trivial form, which at small frequencies, scales as for the interacting case, where is the dimensionality of the condensate, and as for the non interacting case. It is worth noticing that while the former case is always superohmic, the latter one is subohmic, ohmic and superohmic depending on the dimensionality of the environment. Note that the high power in is due to the fact that the bath has to “resolve” the structure of the impurity, formally again the factor . Furthermore, as already pointed out, no “ad hoc” cutoff frequency needs to be inserted but appears naturally in the decaying exponential of variance in (29).
Fig.2 shows clearly that the impurity maintains much of its coherence at long times. Such survival is due to the above mentioned suppressed effect of soft modes, which are responsible of the long time behavior of , and is more pronounced when the environment consists of a condensate than in the case of a reservoir consisting of free bosons. This can be intuitively described in terms of greater ”stiffness” of the condensate whose Bogoliubov modes are less displaced by the coupled impurity. The condensate is even able to give some coherence back to the impurity, since is not monotonic in time. Oscillations of coherence in spin-boson systems were predicted in  (and even earlier, in a different context, in ).
We can distinguish three stages in the dynamics of the ’s. In the first stage , as can be easily seen from a series expansion of (29). This very short stage, shown in the inset of Fig.2, corresponds to coherent dynamics. The second stage corresponds to a Markovian behavior, i.e., , and lasts a few tens of microseconds. Finally, in the third stage saturates to a stationary value. This behaviour calls for particular caution in treating an environment of (free or interacting) bosons as a Markovian reservoir for atomic impurities immersed in it, which is clearly not the case in the present situation.
4.2 Collective decoherence of two impurities
Decoherence of quantum systems in a common environment is characterized by collective decoherence. It is well known that two spins interacting with the same bosonic reservoir with a spin - boson interaction Hamiltonian like the one discussed in this paper show sub – and super – decoherence . In simple words the decoherence rate of the two spins is not simply but, according to the initial state of the spins, much smaller or larger. In this final section of the present manuscript we analyse the specific features of collective decoherence in our system.
For two pseudospins, three decoherence parameters appear in the density matrix elements independently of the exact form of the impurities’ state. One is and appears in elements such as , , etc., which corresponds to individual dephasing of each impurity atom; two more parameters and appear in elements such as and , and corresponds to decay of the coherences between states with the particles in the same or in the opposite side, respectively, of the double well. For two pseudospins at distance , these two parameters are
Calculations similar to those performed for give for a condensate environment
In the above equations it is easy to identify the term which quantifies the deviation to the dechoherence exponent typical of the decoherence of two impurities interacting with independent environments. Note that while depends only on i.e. on the spatial size of the double well, depends non trivially on i.e. on the distance between the impurities of different wells. As before the special case of a bath of noninteracting bosons , are obtained from the above equations (33) simply imposing and .
As in the case of single impurity decoherence the impurities do not loose all their coherence: and saturate to a stationary value that can be varied with the help of Feshbach resonances. Furthermore Fig.3 shows that in a system of two impurities coherence oscillations appear, both for interacting and non–interacting bosons in the environment (even more pronounced oscillation are shown in Fig. 5). Such coherence revival is due to the collective nature of the coupling, as quantified by ( for free bosons). As shown in Fig.4 also the ’s are characterized by three different time scales comparable to those analysed for . In the first stage the difference is negligible, since the presence of each impurity cannot have modified yet the environment seen by the other one; in the second stage, corresponding to the Markovian dynamics, the difference steadily grows up; and in the third stage it decreases, reaching a stationary value.
For a pair of impurities we observe super – and sub – decoherence, however with a peculiarity which is characteristic of the system here considered. Indeed we observe sub – decoherence in and super - decoherence with , at variance with what one observes in a standard spin boson model, where their role would be exchanged . This different behaviour is due to the particular configuration of our system: gets contribution from superpositions of the states and , where the atoms sit in wells with identical distance, whereas the states and , contributing to , correspond to atoms sitting in wells with different separations.
Further insight on the features of the collective decoherence is gained by considering the decoherence of impurities sitting in sites which are at a larger distance than . In Fig.5 we plot the decoherence exponents for impurities trapped in lattice sites at distances , and respectively. These plots suggest the following picture: initially the impurities decohere independently, as if they were each immersed in its own environment; at some later time, the environment correlations due to the impurities act back on them and give rise to oscillating deviations from . The onset time of these oscillations depends on the separation: the larger the separation, the later the onset. On the other hand, the correlations become weaker as the distance increases and the oscillations become consequently smaller in amplitude. At large separation (here, approximately ), the parameters and are hardly discernible from , since the environment correlations induced by the impurities vanish. Similar features in a related context are reported in . In summary and also prompt information about the correlation length of the environment.
4.3 Decoherence in one dimension
Finally, we examine the decoherence process in a one-dimensional condensate. Since, as previously discussed, the spectral density (30) is superohmic for an interacting gas, but subohmic for a free Bose gas, we expect qualitative different results for the two cases, in contrast to the three-dimensional case. The decay exponents in one dimension become
for one impurity and
for two impurities in a condensate. The behaviour of these parameters critically depends on the nature of the environment, see Fig.6. In particular, decoherence in a one-dimensional sample of free bosons results Markovian, in agreement with the naive expectation, due to its subohmic spectral density.
We have shown how a system of impurity atoms trapped in an array of double wells, interacting with a cold atomic gas, is described, in a suitable regime, by a spin - boson hamiltonian. The specific nature of our system, in which the pseudospins, associated with the presence of an impurity in the right/left well of each site, have a spatial dimension introduces peculiar features in the decoherence of a single impurity as well as in the collective decoherence, with the persistence of coherence at long times, the presence of coherence oscillations and counterintuitive super / sub decoherent states.
We have shown in particular that for a three-dimensional bath one never has a Markovian behaviour. A one-dimensional bath is in this respect more interesting since one can go from a non-Markovian to a Markovian behaviour just by tuning the interaction of the bath.
As a final comment we would like to say a few words about the role of the quadratic terms in the Bogoliubov operators which we have neglected in our derivation of Hamiltonian (14). Although a detailed study of their effects is beyond the scope of the present article, we would like to point out that their effects are negligible with respect to the linear terms we have analyzed in the present manuscript. One can show that their inclusion amounts to taking into account elastic scattering of Bogoliubov particles, which is simply responsible of an energy shift, inelastic scattering processes and Bogoliubov pair creation and annihilation. In these two latter additional terms the length of wave vectors that can play some role in the impurities’ dynamics is limited from below by the finite size of the condensate and from above by cutoff parameter . It can be shown that, in this frequency range, the coupling constants of the neglected processes are, for the values of parameters assumed in our analysis, three orders of magnitude smaller than the coupling constants of the linear terms. As a consequence, a rough estimate leads us to suppose that any possible relevant effect of the quadratic terms in the Hamiltonian would become apparent at time scales that are three orders of magnitude larger than those examined in this article.
Appendix A Disentangling the time evolution operator
The factorization of the time evolution operator is often an impossible task. When the Hamiltonian contains operators forming a Lie algebra the transformation of into a product of simpler exponential operators is however possible in some cases . Here we show a practical way to transform , which we write as
where is to be determined, as well as the quantities and . Since at the time evolution operator reduces to the identity operator, . All unknown quantities can be found with the help of the relation
which holds for any time-independent Hamiltonian and of the relation
for , , and , and to the differential equation
for the unknown exponential operator , which we write as
A comparison with (43) gives
Moreover, using Glauber’s relation
the two exponentials linear in Bogoliubov operators can be merged into