Dynamics, dephasing and clustering of impurity atoms in Bose-Einstein condensates

We investigate the influence of a Bose-Einstein condensate (BEC) on the properties of immersed impurity atoms, which are trapped in an optical lattice. Assuming a weak coupling of the impurity atoms to the BEC, we derive a quantum master equation for the lattice system. In the special case of fixed impurities with two internal states the atoms represent a quantum register and the quantum master equation reproduces the exact evolution of the qubits. We characterise the qubit dephasing which is caused by the interspecies coupling and show that the effect of sub- and superdecoherence is observable for realistic experimental parameters. Furthermore, the BEC phonons mediate an attractive interaction between the impurities, which has an important impact on their spatial distribution. If the lattice atoms are allowed to move, there occurs a sharp transition with the impurities aggregating in a macroscopic cluster at experimentally achievable temperatures. We also investigate the impact of the BEC on the transport properties of the impurity atoms and show that a crossover from coherent to diffusive behaviour occurs with increasing interaction strength.


Introduction
Ultracold atoms in optical lattices have attracted considerable interest during the last few years. Theoretical investigations showed that ultracold atoms in optical lattices can be used for mimicking a wide range of models encountered in condensed matter physics [1]. These models include the Hubbard Hamiltonian [2,3], spin-spin interactions [4,5], high-temperature superconductivity [6,7], effective magnetic fields [8]- [13], even with non-abelian gauge potentials [14,15], and the fractional quantum Hall effect [13,16], to name but a few. Experimental efforts have led to an unprecedented control over the properties of optical lattice systems [17], including such milestones as the Mott insulator to superfluid transition [18,19], investigations of Bose-Fermi [20,21] as well as Bose-Bose mixtures [22], vortex pinning [23], the creation of repulsively bound atom pairs [24], and cold controlled collisions between atoms in optical lattices [25,26]. Especially for the latter experiments the absence of lattice phonons and thus the suppression of decoherence mechanisms was crucial. However, when mimicking the behaviour of electrons in crystals this lack of phonons might lead to an oversimplification of the underlying model and it is desirable to introduce phonons in a controlled manner into the optical lattice system.
One way of achieving this goal is to immerse the optical lattice system into a Bose-Einstein condensate (BEC). Experiments where both atom species are trapped by the optical lattice are common and decoherence effects in such systems have already been observed [20,21]. Furthermore, these Bose-Fermi mixtures promise rich phase diagrams including charge and spin density wave phases [27,28], pairing of fermions with bosons [29] and a supersolid phase [30]. We, however, focus on the case where only one species is trapped by the optical lattice. This can be achieved by a suitable choice of the laser wavelengths and atomic species such that the BEC is not affected by the optical lattice [31]. The lattice atoms interact with the condensate via density-density interaction, which can be described by Bogoliubov phonons coupling to the impurities. Earlier studies on such systems have shown that this coupling can be exploited to cool the lattice atoms to extremely low temperatures [32,33], which are otherwise very difficult to achieve. In [34], the present authors have derived a model in which lattice atoms dressed by a coherent state of Bogoliubov phonons constitute polarons [35]. The model exhibits an attractive interaction potential between the lattice atoms [36,37] and allows a generalized master equation to be deduced which shows that the system exhibits a crossover from coherent to diffusive dynamics.
Instead, in this work, we concentrate on describing the lattice atoms by a quantum master equation (QME), which is derived in section 3. We show in section 4 that for the case of fixed impurities the system can be solved exactly and after tracing out the BEC degrees of freedom the QME reproduces this exact solution. Due to the coupling to the phonons the lattice atoms experience dephasing, which can be used to demonstrate the effects of sub-and super-decoherence [38] and to probe spatial properties of the BEC analogous to [39]. We also investigate the severe effect this decoherence has if the atoms in the lattice are used as a quantum register. In section 5, we show that the attractive interaction mediated between the lattice atoms leads to the formation of atom clusters, which should be observable for typical experimental parameter regimes. This effect is reminiscent of the clustering of ad-atoms on crystal surfaces [40]. The influence of the phonon coupling on the transport properties of the lattice atoms is investigated in section 6. We show that the crossover from coherent to diffusive transport, which was observed in [34], can also be described by the QME. In contrast to the treatment in [34], the QME in addition gives access to the off-diagonal elements of the density operator. We also show the limitations of the QME by applying it to a tilted lattice system and comparing the results to a near-exact numerical solution for the time evolution.

Model
The system under consideration is composed of a BEC and an ultracold gas of impurity atoms trapped in an optical lattice giving a set-up like that shown in figure 1. The impurities can be either bosonic or fermionic atoms. For most of the results derived in this work, the statistics of the lattice atoms is unimportant, but for concreteness we will assume bosonic impurities in the following. The dynamics of the whole system is governed by the Hamiltonian H =Ĥ L +Ĥ B +Ĥ I , wherê describes the BEC, withφ † the condensate field operator, m b the mass of the BEC atoms, V ext (r) an external potential confining the BEC, and g > 0 the coupling constant between the BEC atoms. Here, we have assumed that the optical lattice potential does not affect the BEC atoms, which can be achieved by choosing the laser wavelengths and atom species accordingly [31]. The atoms in the optical lattice, which are distinguishable from the atoms in the BEC, are described by the HamiltonianĤ L , whereas the interaction between the two sub-systems is given byĤ Here,χ † is the field operator of the lattice atoms, and κ is the density-density coupling constant between the BEC and the impurities. It was shown in [34] that in the tight-binding limit and under the condition |κ|/gn 0 ξ D 1 (where D is the dimensionality of the system, n 0 the density of the BEC in the trap centre, and ξ =h/ √ m b gn 0 the healing length) the total Hamiltonian reduces to a Hubbard-Holstein model, given bŷ Here, the prime at the sum indicates that zero energy modes have been excluded,b † ν creates a Bogoliubov phonon in mode ν with energyhω ν ,â † j creates a lattice atom in site j,n j =â † jâ j is the number operator on site j, J describes the hopping of the impurities, U is their on-site interaction, and the coupling between the phonons and the lattice atoms is given by In this equation, η j is a Wannier function describing an atom in lattice site j, φ 0 is the solution of the Gross-Pitaevskii equation and u ν and v ν solve the Bogoliubov-de Gennes equations, see also [41]. For the case of a homogenous condensate, which we consider in the remainder of the paper, the expressions for the atom-phonon coupling is given by where q is the phonon quasi-momentum, ε q = (hq) 2 /2m b the free particle energy,hω q = ε q (ε q + 2gn 0 ) the Bogoliubov dispersion relation and f j (q) = −1/2 dr|η j (r)| 2 exp(iq · r), with the quantization volume. The latter integral can, in general, not be solved analytically. However, for sufficiently deep lattices [42], the Wannier functions can be approximated by Gaussians, yielding 5 where ω t is the trapping frequency of a harmonic trap approximating the lattice potential at a given lattice site.
An experimental realization of this set-up is achievable with present techniques as follows. The creation of Rb condensates with 10 6 atoms has been demonstrated leading to the desired BEC densities of about 10 20 m −3 in three-dimensions (3D), see for instance [43]. By choosing a sufficiently flat trapping potential of a few Hertz the BEC can be assumed to be homogenous to a good approximation in the centre of the trap, extending over a few micrometres. Furthermore, in [20,21] a mixture of Rb and K atoms has been created and trapped in a 3D optical lattice, where several tens of thousands of lattice sites have been occupied. The filling for the K atoms ranged between 0 and 1 atoms per lattice site, whereas the filling of the Rb atoms could exceed 5 atoms per site in the centre of the trap. Although in these experiments both atom species were trapped by the optical lattice, techniques to trap only one of the two species, for example K, have been studied extensively in [31]. Applying these techniques leaves the Rb atoms virtually unaffected and enables the creation of a nearly homogenous BEC.

The QME
In order to investigate the behaviour of the impurity atoms, we derive a QME for the lattice system by tracing out the surrounding BEC. The details are given in appendix A. Here, we note that the main condition of deriving the QME is that the sound velocity of the condensate, c ∼ √ gn 0 /m b , is larger than the typical hopping speed of the atoms, i.e. c J a/h, where a = λ/2 is the distance between two lattice sites. With this assumption we get Here, N q (T ) = 1/(exp(hω q /k B T ) − 1) is the number of thermal phonons at temperature T , and L (t) = Tr Bˆ (t), where Tr B denotes the trace over the condensate. Furthermore, the Hamiltonian describes the coherent evolution of the lattice atoms, most notably an off-site interaction which is mediated by the phonons of the BEC. The interaction term includes a transient behaviour described by the cosine functions, which accounts for suddenly turning on the interaction between lattice atoms and BEC. The sum over all these cosine functions vanishes in the limit t → ∞. The Born approximation used in deriving the QME is valid if the perturbation caused by the BEC is small compared to the typical energy scales given by the lattice system. If only the hopping term is important for the lattice atoms, this energy scale is determined by J . The energy scale of the perturbation is given by the so-called polaron energy E p = q (F q,l F * q,l + F * q,l F q,l )/2hω q ∼ κ 2 /gξ D , which leads to the condition J E p . This parameter regime is complementary to the one considered in [34], where the opposite limit was investigated. As we will show in the following section, the condition J E p is not required for the case of J = 0, where an analytical solution of the dynamics of the whole system can be derived. After tracing out the BEC degrees of freedom this solution agrees with the one given by the QME.

Fixed impurities and the quantum register
For very deep optical lattices the hopping constant J is essentially zero and the impurities cannot leave their site. If there is a maximum of one atom in each lattice site, the set-up can be used as a quantum register [44]. The lattice atoms represent qubits on which single qubit rotations can be implemented via external laser pulses. For the manipulation of the atoms single site addressability is necessary, which may be achieved by using infrared lattices [45], by leaving empty sites between the atoms [46,47], by exploiting the properties of marker atoms [48], or by additional external fields [49,50]. An entangling two-qubit gate can be implemented by using the interaction which is mediated by the condensate [37]. For this, it is necessary to turn the interaction on and off, which can be done by using different internal states of the lattice atoms, some of which couple to the BEC, and others that do not couple. However, due to the coupling to the BEC the qubits also experience dephasing, which will be investigated in this section.

Analytical solution of the time evolution
We first calculate the time evolution of the atoms subject to the BEC coupling. For easier notation, we focus on the case where the lattice atoms have only one internal state and we assume that there is a maximum of one atom per lattice site. In the regime where our model introduced in section 2 is valid, i.e. |κ|/gn 0 ξ D 1, and for J = 0, the Hamiltonian of the whole system simplifies toĤ 0 =Ĥ BEC +Ĥ I , whereĤ I is given in equation (4) and This operator can be decomposed using the methods described in [51], yieldinĝ Here, we have definedF l, The last term in equation (12) describes the off-site interaction between two atoms confined in lattice sites l and l , which is mediated by a phonon with quasi-momentum q. The total interaction potential is derived by adding all phonon contributions together. For t → ∞, the oscillations caused by the sine function cancel each other and the off-site interaction potential is given by 4 This interaction potential was already found earlier [34,37], however, the exact solution in addition reveals the transient behaviour after suddenly turning on the interaction, which is described by the sine functions.

7
The coupling to the BEC leads to dephasing of the lattice atoms as becomes apparent after tracing out the condensate degrees of freedom. There are in essence two different types of dephasing. One can be observed when a lattice atom is driven into a superposition of two different states which couple differently to the condensate. This situation has been discussed in [39]. The other type of dephasing is given when comparing the phase of two atoms in different lattice sites. We illustrate this effect of dephasing by calculating the correlation A combination of the two dephasing effects is discussed in section 4.2.
If we choose the same initial conditions as in appendix A, namelyˆ (0) =ˆ L (0) ⊗ˆ B (0), andˆ B (0) describes a thermal state of the BEC, we get Here, Tr L denotes the trace over the lattice system,Ĥ g was introduced in equation (10), and we made use of the identity for thermally distributedˆ B and some complex number α q [52]. The first part of equation (14) describes the correlations between lattice sites γ and β which are induced by the dynamics of the lattice atoms. In our case, their time evolution only contributes a phase term. The second part describes the dephasing which is caused by the coupling to the BEC atoms. Let us denote the dephasing term by . In the thermodynamic limit of the Bogoliubov modes we can replace the sum by an integral and get where d q = κ 2 n 0 ε q exp(−q 2 x 2 0 /2)/ h ω q , F q, j = d q exp(iqr j ), and D is the number of spatial dimensions. For zero temperature, D = 3, and ξ x 0 the integral can be approximated by neglecting the exponential function which gives ( In both cases the factor describing the dephasing fulfils > ς > 0 for a suitable real number ς . This is independent of time or of the distance between the two atoms, which is different from the 1D case, where numerical calculations show that decays to 0 in the thermodynamic limit for t → ∞ as well as for |r γ − r β | → ∞, indicating that no correlations survive.
Interestingly, the solution of the QME introduced in section 3 gives the same correlation functions, which also holds for the time evolution of the operatorsâ j after tracing out the condensate. This is due to the fact that the interaction commutes with the lattice Hamiltonian for J = 0 and that the BEC is initially in a Gaussian, namely a thermal, state [53]. Hence, for the case J = 0 the QME describes the lattice atoms exactly after the trace over the BEC is taken. For J = 0 the interaction Hamiltonian and the one describing the lattice atoms do no longer commute, and the QME is only valid for a large hopping with J E p .

Sub-and super-decoherence and probing the BEC
It has been predicted in [38] that decoherence effects caused by a qubit-bath coupling can be enhanced (super-decoherence) or suppressed (sub-decoherence) for certain cases 5 . Here, we show that these effects are indeed observable in a BEC for realistic experimental parameters. The effect can moreover be used in order to probe such BEC properties as the temperature with different sensitivity, similar to the method introduced by Bruderer and Jaksch [39]. Let us assume that the atoms have two internal states |0 and |1 which couple to the BEC with different coupling constants κ 0 = 0 and κ 1 = κ. This case can be easily generalized to a non-zero κ 0 . The different coupling strengths can, for example, be realized in a 40 K-87 Rb mixture. The positions of the Feshbach resonances for the scattering with the rubidium atoms depends on the fine structure levels |F, m F of the potassium atoms [54]. By choosing the external magnetic field close to one Feshbach resonance considerable differences between the scattering lengths and hence the coupling constants for different internal states can be achieved, and it is even possible to tune one scattering length to zero.
Initially, the total state of the two atoms is described by the density matrixˆ 2 (0) = i jkl=0,1 i jkl (0)|i j kl|. After evolving for a certain time t and tracing out the BEC the density matrix changes toˆ 2 (t) =Û coh i jkl=0,1 i jkl (t)|i j kl|Û † coh , whereÛ coh describes the coherent evolution of the atoms induced by the off-site interaction term, which for our case only changes the phase of the matrix elements, but not their absolute value. The elements i jkl (t) are given by 0110 and 1001 and i jkl (t) = + i jkl (0) for 1100 and 0011 . The functions +,0,− are calculated analogously to the dephasing term in section 4.1 and are given by where r γ and r β denote the positions of the two atoms. Although the overall structure of these three decoherence terms looks very similar they behave quite differently. Let us for the moment assume that only such momenta q contribute considerably to the sum for which the condition q(r γ − r β ) 1 holds. This is typically the case for a temperature of T ≈ 0 and the atoms trapped in two neighbouring lattice sites. Then, − ≈ 1, which means that the decoherence for an initial state |ψ = (|10 ± |01 )/ √ 2 is strongly suppressed, because the fluctuations of the condensate happen on a length scale which is too large to resolve the distance between the two atoms. In contrast, the decay of the off-diagonal elements of the initial state |ψ = (|11 ± |00 )/ predicted in [38] for a general type of qubit-environment coupling and are similar to the Dicke effect [55] well known in quantum optics. However, in our case it is not the decay rate of the excited state which is suppressed or enhanced. Instead, only the off-diagonal elements of the density matrix are affected leaving the occupation numbers unchanged.
As depicted in figure 2, these effects can indeed be observed for realistic experimental parameters. Figure 2(a) shows the time dependence of the three functions. − reaches its stationary state very quickly, whereas the two other functions drop exponentially to zero. The temperature dependence is shown in figure 2(b). Since for higher temperatures phonons with shorter wavelengths become more important, the condition q(r γ − r β ) 1 is no longer fulfilled and − decreases, however considerably slower than + . The condition is also violated when the distance between the two atoms is increased, as shown in figure 2(c). Interestingly, the function + shows exactly the opposite behaviour: with increasing distance, the value of the function increases as well, until it reaches the same value as − .

Implications for the quantum register
The dephasing investigated in the previous section has severe implications on the performance of a quantum register. In order to store the information two internal states |0 and |1 of the lattice atoms are needed, and we assume that these states do not couple to the BEC, in which case there is also no decoherence caused by the condensate coupling. However, the BEC can be used to perform a two-qubit gate between two atoms submerged into it as detailed in [37]. In short, if an atom is in a state, say, |1 , it will be driven by a laser pulse to a state |1 which couples to the condensate, whereas the state |0 is either unaffected or driven to a state |0 which does not couple to the BEC. If both atoms involved are in state |1 they can exchange BEC phonons which causes an additional energy shift. In the previous section, this evolution was described by the unitaryÛ coh , and by appropriately chosen laser pulses this operator results in a controlled-phase gate. However, due to the coupling the atoms also get entangled with the BEC, which results in dephasing after tracing out the condensate.
The influence of the dephasing on the density matrixˆ 2 is suitably expressed by using Kraus operators E j . The super-operator giving the effect of the dephasing (excluding the controlled-phase gate) can be expressed as where d is the dimensionality of the quantum system, here d = 4 for two qubits. Since the dephasing commutes with the controlled-phase operation, the effect of the noisy gate is described by . The Kraus operators for are given in appendix B, where we also show how the average fidelity F of the noisy gate can be calculated using the explicit form of these operators. We find From the definition of the dephasing terms it is evident that the fidelity is worse for higher interactions κ between the BEC and the lattice atoms. On the other hand, a lower interaction κ means a lower interaction strength between the two atoms and hence it takes a longer time to perform the gate. In the 3D case, by choosing an arbitrary low interaction strength κ, the average gate fidelity can be brought arbitrarily close to 1, however the time to perform the gate gets arbitrarily long as well. Since there also exist other decoherence mechanisms in cold atom systems, this will eventually decrease the performance of the set-up. For concreteness, we consider the 3D case where two 133 Cs atoms are placed in two neighbouring lattice sites with wavelength λ = 790 nm. They are surrounded by a 23 Na BEC with a number density of (5 × 10 6 ) 3 m −3 and an interaction strength of g = 1.5 × 10 −2 E R λ 3 .
Here, E R = (2πh) 2 /2m l λ 2 is the recoil energy, with m l the mass of the lattice atoms and λ the wavelength of the laser creating the lattice. The interspecies coupling is given by κ = 1.1 × 10 −2 E R λ 3 . In the thermodynamic limit, for i = j and ξ x 0 the mediated interaction is where a is the distance between two neighbouring lattice sites. This leads to the (minimal) gate time t g =hπ/V 1,2 = 40 ms [37], where the gate fidelity is given by F = 0.99. For getting the last result, we used the approximations x exp(−c x κ 2 √ 2π 2 g 2 n 0 ξ 3 ), which hold in the thermodynamic limit, T = 0, ξ x 0 , and where c 0 = 1, c − = 2 and c + = 4.
For the case of two or one spatial dimensions, the interaction is, in general, larger and thus the expected gate times shorter. However, the fidelities decrease much faster than for 3D, and all of our numerical tests showed that for the same gate times the 3D fidelities were always better than the ones achieved in lower dimensions. Although this restricts the applicability of this setup for quantum information purposes, the scheme can still be used for probing the interaction strength which is mediated by the condensate, simply by measuring the phase differences for varying interaction times.

Clustering of the lattice atoms
If the lattice is not uniformly filled with atoms but the filling ranges somewhere between zero and one atom per lattice site, the mediated interaction has an important impact on the spatial distribution of the atoms. For low enough temperatures, it will lead to atom clusters. To observe the clustering of the atoms we have to allow for a weak hopping J |V 1,2 |, |U |, 6 because otherwise the atom distribution remains stationary. We can assume that the perturbation due to the hopping does not change the interaction potential V i, j derived from the exact solution, which is in agreement with our earlier findings [34]. The hopping energy can furthermore be neglected compared to the interaction potential, such that the weak hopping leads only to a rearrangement of the atoms, which is similar to the treatment of ad-atoms on crystal surfaces [40]. The Hamiltonian for this situation is given bŷ The mediated on-site interaction V 1,1 , which in our case is always attractive, cf equation (13), might overcompensate the on-site interaction U , i.e. U + V 1,1 < 0. If this happens, all the atoms can aggregate in a single lattice site, which will inevitably lead to three-body losses. We therefore assume that the repulsive on-site interaction U is large enough such that states with more than one atom in a single lattice site can be neglected, i.e. U + V 1,1 J . For the parameters used below (see captions of figures 3 and 4) the mediated on-site interaction is given by V 1,1 = −0.03E R in 1D and V 1,1 = −0.08E R in 2D. Since we require a deep lattice for the hopping J to be small, a sufficiently high U can easily be achieved. For a reasonably deep lattice of 15E R and perpendicular confinement of 35E R , the interaction strength is on the order of U ≈ 0.4E R for a 1D lattice and U ≈ 0.3E R for 2D and thus sufficiently high to overcome the induced on-site attractive interaction. The repulsive interaction strength corresponds to a temperature of U/k B = 150 nK in 1D (U/k B = 130 nK in 2D), such that neglecting states with more than one atom per lattice site is well-justified for the temperature regime considered, see below. With these assumptions, the ground state for the lattice system consists of a cluster where all the atoms are located in neighbouring lattice sites. We note that this also holds if the lattice is loaded with spin-polarized fermions, in which case Pauli's exclusion principle ensures that there is never more than one atom in each lattice site. The restriction to a maximum of one atom per lattice site and ignoring interactions beyond nearest neighbours makes it also possible to map the system to an Ising model by using the correspondencen j = (1 +ŝ j )/2, which giveŝ where a constant term has been neglected. We will make use of this correspondence shortly when we compare our results to analytical ones found for the Ising model in 2D.
In an experiment, the lattice atoms will have a finite temperature T , which leads to a breaking up of the ground state into smaller clusters, reflecting the increased average energy. It is therefore important to investigate at which temperatures the clustering can be observed. To this end, we simulated the lattice system using the well-established Metropolis algorithm [58,59], which is often used in the simulation of classical lattice spin and lattice gas models when the kinetic energy can be neglected compared to the interaction energies. For a 1D lattice, the averaged number of clusters 7 for different temperatures is shown in figure 3(a). For temperatures above T = 7 nK, the number of clusters does not change very much with the temperature and roughly 110 clusters exist, which shows that most of the atoms do not have nearest neighbours. A drastic reduction of the cluster number only occurs for T < 7 nK, such that the number of clusters finally reaches N c = 1 for T ≈ 0. A method to achieve such low temperatures of below 7 nK has been proposed, where the surrounding BEC is first used for dark-state cooling [32,33]. The numerical data in figure 3 is compared to an analytical result derived in the thermodynamic limit, where the number of lattice sites M goes to infinity whilst keeping the filling fraction N /M constant, and only taking nearest-neighbour interactions into account [60]. In this case, the normalized number of clusters N c /N is given by where k B is Boltzmann's constant. Our calculations show that the numerical results taking only the nearest neighbour term of the interaction into account is in excellent agreement with the result for the thermodynamic limit. We furthermore observe that the number of clusters for the full interaction potential is for low temperatures considerably smaller than for the truncated one, which indicates that the interaction terms beyond nearest neighbours make the clusters more stable. This also becomes apparent when the temperature is fixed and the number of clusters is calculated for different coupling strengths κ, as shown in figure 3(b). For small κ the interaction beyond nearest neighbours is still quite weak and does not change the number of clusters significantly, whereas for stronger coupling κ, interaction terms beyond nearest neighbours are considerable, reflected in a smaller number of clusters or, equivalently, in a larger average cluster size.
The situation gets more interesting if we consider a 2D lattice. For T ≈ 0, the atoms aggregate in an 'island' and for increasing temperature parts of this island break away, see figure 4(a). We have calculated the size of the largest cluster (i.e. the number N I of atoms constituting the island) numerically, only taking nearest-neighbour interactions into account. This will give a lower bound for the temperature below which the formation of the island can be observed. The results are shown in figure 4(b). For very low temperatures, all the atoms are contained in a single cluster. This largest cluster gets smaller with increasing temperature and shows a pronounced transition at a temperature of T ≈ 2 nK, above which the system mainly consists out of many small clusters. It has been shown in [61], that the normalized size N I /N of the island in the thermodynamic limit and only taking nearest-neighbour interactions into account behaves as Here, N 0 = {1 − [sinh(2J s /k B T )] −4 } 1/8 , and J s = |V 1,2 |/4. For this formula to be valid, we further require a filling of N /M 1/2, such that N r = 1 − 2N /M 0, and above the transition temperature T I where N I /M hits zero for the first time, the function has to be set to zero, since for this temperature the normalized size of the largest island vanishes in the thermodynamic limit. Our Monte Carlo results are in excellent agreement with equation (24), deviations above the transition temperature are due to finite size effects. We note that a higher transition temperature T I can be achieved by increasing the interaction strength V i, j or by loading more atoms into the

Transport properties of the impurity atoms
In this section, we use the QME to investigate the behaviour of the lattice system for a hopping J E p . In general, an analytical solution of the QME is no longer possible, and the dynamics has to be calculated numerically. Let us consider the simple case of one atom initially localized in a single lattice site. With vanishing coupling to the condensate, the atom will spread across the lattice coherently in a wave-like motion, as shown in figure 5(a). The interference fringes between the two wavefronts are clearly visible. The coherent nature of the evolution is also reflected in the density matrix, where all off-diagonal elements have their maximum possible value, compare figure 5(b). The situation changes when the coupling to the condensate is increased, as shown in figure 5(c). The wavefronts still exist, however the region in between no longer exhibits clear interference patterns, which are washed out instead. This implies that the coherent nature of the evolution is impaired, which is also supported by the density matrix, The motion of the atom in the lattice can be explained by the coexistence of wave-like, coherent evolution responsible for the two wavefronts, and a diffusive, incoherent evolution. The diffusive motion stems from the decay of the off-site elements in the density matrix, which destroys the memory of the atom and causes it to perform a random walk, leading to the washing out of the interference effects between the two wavefronts. To investigate the time evolution more quantitatively, we calculate the standard deviation σ d of the atom density distribution at a time t, given by σ d = j p j ( j − j 0 ) 2 , where j labels the lattice sites, p j is the probability of finding the atom at lattice site j, and j 0 is the initial lattice site of the atom at t = 0. Figure 6(a) shows that for an increasing coupling to the condensate this standard deviation σ d decreases. This stems from the fact that the hopping is reduced due to the coupling between the lattice atoms and the phonons of the BEC, which has also been observed in [34].
Taking one standard deviation to either side of j 0 defines a suitable interval I between the two wavefronts to investigate the diffusive character of the motion. For this interval, we calculate the average atom densityp = j∈I p j /2σ d and the standard deviation of the density distribution p d = j∈I ( p j −p) 2 /2σ d . For the coherent case, we expect that this standard deviation is of the order of the average densityp due to the interference patterns, whereas for the incoherent case the standard deviation should be much lower than the average density due to the more homogenous spread of the atom between the two wavefronts. These expectations are confirmed by our findings shown in figure 6(b). We observe that the average density between the two wavefrontsp moderately increases with increasing interspecies coupling. The standard deviation of this density p d stays first approximately constant with increasing κ, indicating a regime where the evolution can still be considered to be coherent, but then for κ > 5 × 10 −3 E R λ drops off considerably, caused by the increased loss of coherence.
The QME describes the lattice atoms well as long as only dephasing has to be taken into account and energy exchange with the BEC can be neglected. This is consistent with the approximations we have used to derive the QME: it was assumed that J a/h c, which means that the atoms move slower than the critical velocity of the BEC. Then, according to the Landau criterion for superfluidity, no energy exchange is possible with a single phonon process, and higher-order phonon processes are not included into the QME. This becomes especially apparent when we try to describe the decay of Bloch oscillations with the QME. Let us assume a Stark potentialĤ s = K j jn j is applied to the 1D optical lattice. For one particle initially located in a single lattice site and not coupling to the BEC, we expect breathing oscillations with frequency ω B = K /h and width 4J a |sin(ω B t/2)|/K [63]. This is shown in figure 7(a). When the atom couples to the BEC, it can dissipate energy into the condensate modes and one expects that the atom can 'fall down' the tilted lattice, causing a net current. When using the QME to describe the dynamics this is, however, not the case. Instead, the Bloch oscillations only get washed out, but the mean position of the atom remains unchanged, as shown in figure 7(b). This is in agreement with our expectations and with other attempts to describe the system with a simple master equation approach [64]. An alternative way based on a generalized master equation, which includes the coupling of phonons to the hopping term, has been devised to describe the occurrence of an atomic net current. Details will be given elsewhere [65].
Another way to observe the net current is given by solving the complete dynamics of the atom and the surrounding BEC numerically. For this, the BEC has to be discretized, which is equivalent to assuming that it is trapped in an optical lattice with appropriately chosen hopping and interaction constants [66]. The Hamiltonian describing the system is then given byĤ withβ † j (β j ) an operator that creates (annihilates) a boson in lattice site j, U b the on-site interaction strength and J b the hopping matrix element of the BEC atoms, and U i the interspecies interaction strength. The time evolution of this Hamiltonian including the BEC and a single atom in the lattice is performed numerically by using the time evolving block decimation (TEBD) algorithm [67]- [69]. The resulting density distribution of the lattice atom initially placed in a single lattice site and evolved for one Bloch period is shown in figure 8(a). For a weak interaction strength U i J b the surrounding BEC atoms hardly affect the dynamics of the lattice atom and after one cycle it ends up in its original lattice site again. For increased interaction the lattice atom is able to dissipate more and more energy to the BEC and to drift towards lattice sites with a lower potential energy, which leads to a net current. This is illustrated in figure 8(b), where a competition between Bloch oscillation and net current is clearly visible. The current can be measured either directly or after a time-of-flight expansion of the lattice atoms as detailed in [70].

Conclusion
In the present paper, we have investigated the behaviour of an optical lattice system immersed into a BEC. To this end, we have derived a QME which describes the time evolution of the lattice atoms. For the case of fixed impurities, the lattice system represents a quantum register and the internal states of the atoms represent the states of the qubit. We have derived an exact solution for this system which is reproduced by the QME. We found that an interaction between the lattice atoms is mediated by the condensate and that the coupling to the phonon modes of the BEC causes a dephasing of the lattice atoms, i.e. of the qubits they represent. This dephasing can be used to probe the properties of the BEC with different sensitivity. The interaction between the lattice atoms leads to a clustering process at low enough temperatures. In contrast to on-site clusters, which are prone to three-body losses, the clustering in our case is caused by the off-site interaction terms. We have simulated the clustering process using the Metropolis algorithm and compared our results with analytical ones originally found for spin systems or ad-atoms on a crystal surface. Our findings indicate that for realistic experimental parameters the clustering process occurs at temperatures of about 5 nK and is observable with near-future experimental techniques.
We also investigated the transport properties of the lattice atoms subject to the BEC coupling. We found that the lattice atoms undergo a crossover from coherent to incoherent evolution when the interaction to the condensate is increased. This process is indicated by a washing out of the typical interference fringes in the density distribution and by vanishing off-diagonal elements in the one-particle density matrix. However, due to the approximations necessary to derive the QME, it does not include dissipation of energy to the BEC. This process is a vital effect to describe the decay of Bloch oscillations in a tilted lattice. More work is necessary to include this effect into a master equation description.
Under the assumption that the sound velocity of the condensate, c ∼ √ gn 0 /m b , is larger than the typical hopping speed of the atoms, c J a/h, the dynamics of the BEC is much faster than the typical dynamics of the lattice atoms and we perform the Markovapproximation by replacing˜ L (t ) →˜ L (t) andñ l (t ) →ñ l (t). After calculating the time integral and transforming back into the Schrödinger picture we finally arrive at equation (9).

Appendix B. Kraus operators for the two-qubit dephasing
Here, we give explicit formulae for the Kraus operators describing the operator given by (ˆ 2 ) =Û † cZ g (ˆ 2 )Û cZ . We find and all other Kraus operators are zero. Here, 1 n is the n × n unit matrix, σ z is the Pauli z matrix, and diag is a diagonal matrix with the entries on the diagonal given as the argument.
With the knowledge of the Kraus operators the average fidelity F of the gate is calculated as follows. We first note that the fidelity F of the noisy operation g with respect to the perfect (i.e. noise-free) operationÛ cZ on a special input state |ψ is defined by F(ψ) = ψ|Û † cZ g {|ψ ψ|}Û cZ |ψ . (B.7) By taking the average over all possible input states the overall performance of the gate operation can be estimated. It has been shown [71,72] that the average fidelity is given by where the integral is taken over the unit sphere S 2d−2 embedded in 2d − 1-dimensional real space, which is isomorphic to the d-dimensional complex space after eliminating a global phase, and dψ is the normalized measure over the sphere, also know as Haar measure. With this formula and the explicit form of the Kraus operators the calculation of the average fidelity equation (20) is straightforward. We note that the Kraus operators for a coupling to independent reservoirs are given by replacing − = + = ( 0 ) 2 . This gives an average fidelity of F ind = (2 + 2 0 + ( 0 ) 2 )/5. Hence, the coupling to the BEC reservoir is worse than the one to independent reservoirs if + + − < 2( 0 ) 2 .