State engineering of impurities in a lattice by coupling to a Bose gas

In the following, we explore the ground state of the system with periodic boundary conditions for varying V₀ and g_AB, fixing the intraspecies interaction strength to g_AA/E_R λ = 0.0236. It turns out, that this interaction strength lies within the range of possible values which lead to the manifestation of regimes in which the ground states differ substantially. An analysis of the dependence on g_AA is performed in section 3.2. In order to extract information out of the complete many-body wave function, we project onto the above-mentioned number states $| {\vec{n}}^{A}\rangle \otimes | {\vec{n}}^{B}\rangle$ and determine the probability of being in the number state $| {\vec{n}}^{A}\rangle$ for the impurity A species, irrespective of the number state configurations of the B species, namely

$$\begin{eqnarray}&&P(| {\vec{n}}^{A}\rangle )=\displaystyle \sum _{i}| \langle {\vec{n}}_{i}^{B}| \,\otimes \,\langle {\vec{n}}^{A}| {\rm{\Psi }}\rangle {| }^{2},\end{eqnarray} \tag{ 4 }$$

where $\{| {\vec{n}}_{i}^{B}\rangle \}$ could be any number state basis set of the Bose gas with fixed particle number and $| {\rm{\Psi }}\rangle$ is the total many-body ground state wave function. In order to associate the impurity state $| {n}_{1}^{A},{n}_{2}^{A},{n}_{3}^{A}\rangle$ with a spatial distribution we construct the number states either with a generalized Wannier basis of the lowest band or the corresponding Bloch basis set, indicated in the following by the subscript W or Bl, respectively. In principle it is necessary to consider also Wannier states of higher bands, but it turns out that the many-body ground state wave function is approximately well described by a superposition of tensor products of number states $| {\vec{n}}^{A}\rangle \otimes | {\vec{n}}^{B}\rangle$ with the Fock space of the A species restricted to the lowest band. Number states spanned by Wannier or Bloch states of higher bands do not contribute. The order of the entries in the number state, built of Wannier states, is connected to the localization of the Wannier states in the wells from left to right, e.g. n₁ describes the number of A atoms in the left localized Wannier state of the lowest band. In contrast to that, the ordering for the number states, built of Bloch states, follows the energy of the Bloch states of the lowest band, e.g. n₁ corresponds to the energetically lowest Bloch state. The transformation between Bloch and Wannier states is given by

$$\begin{eqnarray}&&{w}_{R}^{b}({\rm{x}})=\displaystyle \frac{1}{\sqrt{k}}\displaystyle \sum _{p}\exp (-{\rm{i}}{p}{R}){\phi }_{p}^{b}({\rm{x}}),\end{eqnarray} \tag{ 5 }$$

where ${w}_{R}({\rm{x}})$ is the Wannier state associated with the position R of the corresponding well, k the number of lattice sites, p the momentum of the Bloch states ${\phi }_{p}({\rm{x}})$ and b the index of the band.

Figure 2 shows that the many-body ground state can be divided into four different regions. For small lattice depths and interspecies interaction strengths all four particles of the A species populate the energetically lowest Bloch state (figure 2(c)). Increasing the lattice depth for small g_AB three lattice atoms will localize separately in different wells with the one extra particle being delocalized over all the wells (figure 2(a)). This can be seen by the fact that the corresponding probability is given by $P(| 2,1,1{\rangle }_{W})=\tfrac{1}{3}$, meaning that the remainder of the probability is equally distributed over the states $| 1,2,1{ \rangle }_{W}$ and $| 1,1,2{ \rangle }_{W}$, resulting in the state

$$\begin{eqnarray}&&| {\rm{\Psi }}{\rangle }_{M}=\displaystyle \frac{1}{\sqrt{3}}[| 2,1,1{\rangle }_{W}\otimes | {\psi }_{B}^{1}\rangle +| 1,2,1{\rangle }_{W}\otimes | {\psi }_{B}^{2}\rangle +| 1,1,2{\rangle }_{W}\otimes | {\psi }_{B}^{3}\rangle ].\,\end{eqnarray} \tag{ 6 }$$

Due to translational symmetry each of the states contributes equally with a probability of $\tfrac{1}{3}$. The reader should note that the structure of the wave function, given in equation (6), can only be uncovered by the procedure of projection onto number states $| {\vec{n}}^{A}\rangle \otimes | {\vec{n}}^{B}\rangle$ and is not explicitly given by the numerical simulation. Therefore, $| {\rm{\Psi }}{\rangle }_{M}$ is not the exact result of ML-MCTDHX but an approximation to it. Our approach is to perform the correlation-including ML-MCTDHX calculations and to subsequently analyse them.

Interestingly, the increase of g_AB leads to two different regions in the configuration space, in contrast to the case of g_AA = 0 [59]. For g_AA = 0, increasing the lattice depth and the interspecies interaction strength, the ground state wave function undergoes a transition from an uncorrelated to a highly correlated state, which manifests itself in the localization of the lattice atoms in the latter regime of large lattice depths and interspecies interaction strengths. This means that all A atoms cluster in a single well, while the B atoms are expelled from it. This clustering can be understood in terms of an attractive induced impurity–impurity interaction, which is mediated by the B species (see figure 1). In contrast to that, allowing for a repulsive intraspecies interaction of strength g_AA between the A atoms counteracts the induced interaction. However, these two types of interactions do not simply add up, since the induced interaction is of long-range type, whereas the intraspecies interaction is of contact type. As a result, depending on the choice of the lattice depth V₀ and the interspecies interaction strength g_AB the impurities of species A either accumulate all in one well (figure 2(b)), which is already happening in the case of g_AA = 0, or pairwise in two different wells (figure 2(d)). Apparently, in the case of pairwise localization of the lattice atoms (figure 2(d)) the repulsive interaction counteracts an accumulation of all lattice atoms due to the induced interaction. Performing a thorough analysis of the excitation spectrum of the many-body system, we find that the ground state for large g_AB is in both regions threefold degenerate (triplet). This essentially means that in the corresponding region the ground states are either given by

$$\begin{eqnarray}&&| 2,2,0{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{1}\rangle ,\,| 2,0,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{2}\rangle \,\mathrm{and}\,| 0,2,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{3}\rangle \quad \mathrm{or}\ \mathrm{by}\,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&| 4,0,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{1}\rangle ,\,| 0,4,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{2}\rangle \,\mathrm{and}\,| 0,0,4{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{3}\rangle .\end{eqnarray} \tag{ 8 }$$

The states $\{| {{\rm{\Psi }}}_{B}^{i}\rangle \}$ and $\{| {\bar{{\rm{\Psi }}}}_{B}^{i}\rangle \}$ ⁴ are each normalized to unity and incorporate the localization effect of the A species by e.g. spatially avoiding the impurities correspondingly, which can be seen in the one-body density (see section 3.3, figure 5). Due to the degeneracy of the ground state one can choose such superpositions of the states in equations (7) and (8), which preserve the translational invariance of the total Hamiltonian. For this reason the probabilities in figures 2(a), (b), (d) are bounded by a maximum value of 1/3⁵ . Furthermore, it is now possible to use this degeneracy in order to select any of the states in the respective degenerate manifold. Technically, this is simply done by applying a small asymmetry to the lattice potential, thereby energetically favouring one of the above-mentioned states. For example, marginally increasing the depth of the left well of the lattice potential will break the translational symmetry and thereby lift the degeneracy, such that the ground state is solely given by $| 4,0,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{1}\rangle$ in one regime. In this sense, with the lattice depth of individual wells we have introduced an additional control parameter for the manipulation of impurity configurations.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

So far, we gained insight into the state configurations that are populated by the impurities. While this finding itself allows for a systematic control of the impurities in the lattice, it is nevertheless of interest in which way the correlation with the Bose gas impacts this very species. Therefore, we also analyse the probability distribution $P(| {\vec{n}}^{B}\rangle )$ of the number states $| {\vec{n}}^{B}\rangle$ that build the corresponding B species states $\{| {{\rm{\Psi }}}_{B}^{i}\rangle \}$ and $\{| {\bar{{\rm{\Psi }}}}_{B}^{i}\rangle \}$. We find that for each of the strongly coupled degenerate ground states the B species states $\{| {{\rm{\Psi }}}_{B}^{i}\rangle \}$ and $\{| {\bar{{\rm{\Psi }}}}_{B}^{i}\rangle \}$ each populate the same number states with equal probability, i.e. $| \langle {\vec{n}}^{B}| {{\rm{\Psi }}}_{B}^{i}\rangle {| }^{2}=| \langle {\vec{n}}^{B}| {{\rm{\Psi }}}_{B}^{j}\rangle {| }^{2}$ with $i,j\in \{1,2,3\}$, while differing only in a relative phase for each coefficient of the number state. This is why it is sufficient to show only the probability distribution of one B species state for the states in equations (7) and (8). As a basis for the number states $| {n}_{1}^{B},{n}_{2}^{B},{n}_{3}^{B},{n}_{4}^{B},{n}_{5}^{B}\rangle$ we choose plane waves $(1/\sqrt{L})\exp ({\rm{i}}\kappa {\rm{x}})$, with wave vectors $\kappa =2\pi z/L$ (z = 0, ±1, ±2, ...), where n₁ corresponds to the κ = 0 mode, n₂, n₃ correspond to z = ±1 and n₄, n₅ to z = ±2. It turns out that the population of all higher momentum states is negligible, which we checked explicitly by projecting onto the corresponding number states. In figure 3, we see that due to the correlation with the impurity species the Bose gas can no longer be described by a single number state with all particles occupying the $\kappa =0$ mode⁶ . Interestingly, it is also not sufficient to consider only single particle excitations. It is rather necessary to consider up to four particle excitations for the B species states in both regimes. Comparing the B species states in the two regimes, one finds that they strongly populate the same number states, but differ w.r.t. the quantitative distribution among those number states. For example, in figure 3(a) the number states $| 9,1,0,0,0\rangle$ and $| 9,0,1,0,0\rangle$ are less populated than the number state $| 10,0,0,0,0\rangle$, whereas their probability exceeds that of $| 10,0,0,0,0\rangle$ in figure 3(b). These findings support the choice of our treatment of the many-body problem using a method that is in particular capable of taking all necessary correlations into account. An approximation of the many-body Hamiltonian which relies on few-particle excitations, will not capture the localization pattern presented in figure 2.

Zoom In Zoom Out Reset image size

In the previous subsection, we have identified four different number state configurations for the impurity species, while assuming a fixed intraspecies interaction strength of ${g}_{{AA}}/{E}_{R}\lambda =0.023\,6$. However, it is not clear whether this holds for a broad regime of couplings g_AA. In order to explore the range of validity of this crossover, we fix the lattice depth as well as the interspecies interaction strength such that for g_AA = 0 we arrive at a degenerate subspace of ground states given by equation (8), instead of the one given by equation (7) for g_AA/E_R λ = 0.0236. The reader should note that for g_AA = 0 the correlated region, which is split into two sub-regions for g_AA/E_R λ = 0.0236, is solely described by the degenerate manifold in equation (8) (see [59]). Again, we calculate the probability $P(| {\vec{n}}^{A}\rangle )$, while varying the coupling strength g_AA for fixed V₀ and g_AB. In figure 4, we see that for small g_AA the correlated region is well described by a single triplet of ground states, as in the case of ${g}_{{AA}}=0$, where all impurities accumulate in a single well. A further increase of the intraspecies interaction strength leads to a break-up of the cluster into two pairs (equation (7)) and finally results in a ground state with all A atoms localized separately in different wells with the one extra particle being delocalized over all the wells (see equation (6)). Essentially, this means that one needs a certain intraspecies interaction strength g_AA between the impurities in order to arrive at a crossover diagram as in figure 2. Below that critical value the crossover is well captured by the g_AA = 0 case, including, if ${g}_{{AA}}\ne 0$, a regime for small g_AB and large V₀ where the ground state is given by $| {\rm{\Psi }}{\rangle }_{M}$ (equation (6)). In other words, for small g_AA the crossover diagram in figure 2 will consist of three different regimes, where the two regimes in figures 2(b) and (d) will merge into a single one, describing complete localization of the impurities in a single well (equation (8)). Thus, the ground state comprising pairwise localization of the A atoms will not exist in this case.

Zoom In Zoom Out Reset image size

Qualitatively, the crossover in figure 4 can be understood again in terms of a competition between the attractive induced interaction and the repulsive intraspecies contact interaction. For small g_AA the induced interaction is dominating the behaviour of the A species atoms, leading to their complete localization in a single well. At a certain interaction strength g_AA this is no longer the case, resulting in a pairwise accumulation of A atoms. Apparently, the different nature of the competing interactions (long-range and contact) does not lead to a trivial reduction of the four-impurity cluster to a three-impurity cluster, but rather leaves this out as a possibility and directly favours a two-impurity cluster. Astonishingly, the crossover between the different configurations is very sharp, such that the system occupies only one of the triplets without superposing them. Furthermore, it is possible to control the width of the plateau of the pairwise localization by adjusting the interspecies coupling strength g_AB correspondingly. The plateau (red) corresponding to a degenerate manifold of ground states with pairwise localization of A atoms is much broader for a larger value of g_AB (see figure 4(b)). This also means that a smaller value of the interspecies interaction strength g_AB leads to a smaller critical value of g_AA (see figure 4(a)) at which the transition to the ground state $| {\rm{\Psi }}{\rangle }_{M}$ takes place (black plateau, equation (6)). In essence, we find that by tuning the intraspecies interaction strength, we are able to control and engineer the localization of the A atoms in the lattice. The sharpness of the crossovers allows for a clear and systematic way of choosing the ground states, while the broad plateaus make the triplets robust with respect to fluctuations of g_AA. In this sense, the coupled system serves as a transistor-like switching device for number state preparation of impurities in a lattice.

Our findings in the previous sections so far relied on the fact that we assumed periodic boundary conditions. It is therefore of immediate interest in which way the localization pattern in figures 2(b) and (d) depends on the choice of the boundary conditions. For this reason, we consider solely values of the lattice depth V₀ and interspecies interaction strength g_AB of the crossover diagram such that we arrive at the ground state configurations in equations (7) and (8). The corresponding values are given in table 1. Subsequently, for those two regimes we change the boundary conditions to hard wall boundary conditions. Obviously, this change will break the translational symmetry of the Hamiltonian. Instead, the Hamiltonian now obeys parity symmetry, suggesting that the former ground state degeneracy of a triplet might now be given by a doublet. Indeed, for the states comprising complete localization of the A atoms in a single well (equation (8)) we arrive at the subspace of degenerate ground states, where the atoms of species A localize in the outer wells, i.e. $| 4,0,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{1}\rangle$ and $| 0,0,4{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{2}\rangle$ ⁷ . However, the degenerate ground state with pairwise localization for periodic boundary conditions does not exhibit any degeneracy for hard wall boundary conditions anymore. In this sense, the ground state in that regime is given by a non-degenerate parity symmetric ground state of the form $| 2,0,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}\rangle$ (singlet).

Table 1.  Degenerate subspaces of the ground state for different boundary conditions for g_AB/E_R λ = 0.135 and g_AA/E_R λ = 0.0236. The third row incorporates a repulsive Gaussian potential for the B species in the centre of the box in addition to the hard wall boundary conditions. The different values of the lattice depth V₀ lead either to a complete localization of A atoms in a single well or to a pairwise localization (see footnote 7).

Boundary conditions	V0/ER = 18	V0/ER = 22.5
Periodic	$| 4,0,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{1}\rangle$	$| 2,2,0{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{1}\rangle$
	$| 0,4,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{2}\rangle$	$| 2,0,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{2}\rangle$
	$| 0,0,4{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{3}\rangle$	$| 0,2,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{3}\rangle$
Hard walls	$| 4,0,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{1}\rangle$	$| 2,0,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}\rangle$
	$| 0,0,4{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{2}\rangle$
Hard walls and repulsive Gaussian potential	$| 0,4,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}\rangle$	$| 2,2,0{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{1}\rangle$
		$| 0,2,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{2}\rangle$

Investigating the one-body density of the Bose gas of species B, one gets an intuition for the reason why in one region the ground state triplet becomes a doublet and in the other it becomes a singlet for hard wall boundary conditions. The one-body density shows that the Bose gas accumulates in the centre of the box potential⁸ (not shown here), because it is energetically favourable for the majority of the bosons of the Bose gas to occupy the energetically lowest eigenmode of the box potential and not accumulate to either side of the box. As a consequence, the A atoms localize in the outer wells due to the repulsive coupling to the B species, thereby avoiding occupation of the middle well. For the ground states comprising complete localization of A atoms in a single well, the restriction of the A atoms to the outer wells allows for two possible many-body states out of three in equation (8), whereas for the ground states with pairwise localization of A atoms only a single state in equation (7) obeys this restriction.

Following the above line of argumentation, one might now ask whether it is possible to change the singlet ground state into a doublet and vice versa by forcing the Bose gas out of the centre and thereby to either side of the box. In order to achieve this we implement a repulsive Gaussian potential for the B species in the middle of the box which is localized in the middle well of the lattice and has an amplitude A₀ that is approximately twice as large as V₀, i.e. A₀/E_R = 50. It turns out that this procedure has the effect we aim for, leaving an imprint on the one-body density of the corresponding B species states (see figures 5(e) and (f)), which is defined as

$$\begin{eqnarray}&&{\rho }_{B}^{(1),i}({\rm{x}})=\int {\mathrm{dx}}_{2}...{\mathrm{dx}}_{{N}_{B}}| {{\rm{\Psi }}}_{B}^{i}({\rm{x}},{{\rm{x}}}_{2},\,\ldots ,\,{{\rm{x}}}_{{N}_{B}}){| }^{2},\end{eqnarray} \tag{ 9 }$$

integrating over all B atoms except for one.

Zoom In Zoom Out Reset image size

In figure 5, we show the one-body density ${\rho }_{B}^{(1),i}$ ($i\in \{1,2,3\}$) of the states of the B species, following the nomenclature in table 1. Figure 5(e) shows that the one-body density of the B species indeed exhibits a minimum in the centre of the box, leading to an accumulation to both sides. Consequently, the A atoms accumulate in the middle well, such that only one many-body ground state out of equation (8) fulfills this restriction, namely $| 0,4,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}\rangle$. In contrast to that, for pairwise impurity localization forcing the B species out of the centre of the box allows for two possible ground states from equation (7) (see figure 5(f)), namely $| 2,2,0{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{1}\rangle$ and $| 0,2,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}^{2}\rangle$. Table 1 summarizes the engineering of the degenerate subspace of the ground state in dependence of the boundary conditions for pairwise and complete localization of the A atoms in a single well. Figure 5(a) resembles the case of g_AA = 0, where the particles of species B are expelled from the well where all the A atoms fully localize, leaving an imprint on the one-body density. In figure 5(b) the B species atoms need to be expelled from two wells due to the pairwise localization of A atoms. Because of the fact that there are less impurities per well the hereby reduced interspecies interaction allows for a larger one-body density of species B in the region of the pairwise occupied wells. Figures 5(c) and (d) are similar to (a) and (b) except for a shifting of the density closer to the centre of the box potential. This is simply an effect of the change to hard wall boundary conditions. Apart from two specific ground state configurations (figure 5(b) red and 5(d) green) it is possible to identify any of the many-body ground states in table 1 just by analysing the position of the one-body density of the B species states with respect to the lattice potential—irrespective of the boundary conditions.

Thus, we are able to engineer the character of degeneracy of the ground state by choosing the boundary conditions correspondingly (in combination with a Gaussian potential). Combining this with the fact that it is possible to switch between differently localized configurations of the impurities by tuning V₀, g_AB and g_AA, one might think of an (adiabatic) particle transfer of the following type. Initially, we prepare the ground state in one of the doublet states (using hard wall boundary conditions), e.g. $| 4,0,0{\rangle }_{W}\otimes | {{\rm{\Psi }}}_{B}^{1}\rangle$. Increasing the intraspecies interaction strength g_AA adiabatically the ground state will reconfigure to the singlet $| 2,0,2{\rangle }_{W}\otimes | {\bar{{\rm{\Psi }}}}_{B}\rangle$. Essentially, this can be interpreted as a transfer of two impurities from the left to the right well.