Abstract
Single experimental shots of ultracold quantum gases sample the manyparticle probability distribution. In a few cases such single shots could be successfully simulated from a given manybody wavefunction^{1,2,3,4}, but for realistic timedependent manybody dynamics this has been difficult to achieve. Here, we show how single shots can be simulated from numerical solutions of the timedependent manybody Schrödinger equation. Using this approach, we provide firstprinciple explanations for fluctuations in the collision of attractive Bose–Einstein condensates (BECs), for the appearance of randomly fluctuating vortices and for the centreofmass fluctuations of attractive BECs in a harmonic trap. We also show how such simulations provide full counting distributions and correlation functions of any order. Such calculations have not been previously possible and our method is broadly applicable to manybody systems whose phenomenology is driven by information beyond what is typically available in loworder correlation functions.
Main
A postulate of quantum mechanics states that the positions r_{1}, …, r_{N} of N particles measured in an experiment are distributed according to the Nparticle probability density P(r_{1}, …, r_{N}) = Ψ(r_{1}, …, r_{N})^{2}, where Ψ(r_{1}, …, r_{N}) is the manybody wavefunction of the system. In many experiments the positions of individual particles cannot be measured directly. Ultracold atom experiments provide a rare exception to this rule, which is why we focus on ultracold atoms in the following, but the concept is completely general. If Ψ(r_{1}, …, r_{N}) is known, single experimental shots can be simulated by drawing the positions of all particles from P(r_{1}, …, r_{N}), which results in a vector of positions (r_{1}′, …, r_{N}′) that we refer to as a single shot. This has been realized for timeinvariant manybody systems^{1,2,3,4}. However, for timedependent manybody systems it has remained a challenge. The difficulty stems from the fact that the functional form of the wavefunction is generally not known in manybody dynamics.
Attempts at simulating single shots have been reported in the context of semiclassical dynamics: several authors have interpreted classical trajectories obtained within the truncated Wigner approximation as individual realizations of experiments^{5,6}. Under the strict condition that the Wigner function is nonnegative, some authors consider this interpretation plausible^{7} or have fewer objections to it^{8}. Here we show that this interpretation must also be dismissed for positive Wigner functions (see Supplementary Information). Although quantum Monte Carlo algorithms^{9} sample the Nparticle probability to obtain lower ground state energies, for timedependent manybody systems not even the nodal structure of the wavefunction is known in advance, and hence quantum Monte Carlo methods are less suited. For further details see Supplementary Information.
For sampling P(r_{1}, …, r_{N}) it helps to realize that
where, for example, P(r_{2}r_{1}) denotes the conditional probability to find a particle at r_{2} given that another one is at r_{1}. First, r_{1}′ is drawn from P(r_{1}), then r_{2}′ from P(r_{2}r_{1}′), then r_{3}′ from P(r_{3}r_{2}′, r_{1}′) and so on. Note that a histogram of a single shot (r_{1}′, …, r_{N}′) is analogous to an image obtained in an experiment. Here we provide an algorithm to simulate single shots from any Nboson wavefunction Ψ〉 = ∑ _{n}C_{n}n〉, where n〉 = n_{1}, …, n_{M}〉 are configurations constructed by distributing N bosons over M orbitals φ_{i} (see Methods and Supplementary Information). In the following we show how singleshot simulations provide insights into strongly correlated manybody systems. The wavefunctions are obtained by numerically solving the timedependent manybody Schrödinger equation using the multiconfigurational timedependent Hartree method for bosons (MCTDHB; refs 10,11,12). Here,
denotes a general manybody Hamiltonian in D dimensions with an external potential V (r) and a regularized contact interaction δ_{ε}(r) = (2πε^{2})^{−D/2}${\text{e}}^{{\text{r}}^{2}/2{\epsilon}^{2}}$. The meanfield parameter λ = λ_{0}(N − 1) characterizes the interaction strength (see Methods).
We briefly recall that a BEC is condensed if its reduced singleparticle density matrix has one nonzero eigenvalue of order N (ref. 13). The eigenvalues ρ_{1} ≥ ρ_{2} ≥ … are known as natural occupations, the eigenvectors as natural orbitals. A BEC is fragmented if more than one natural occupation is of order N (ref. 14) (see Methods). Fully condensed states, that is, states with ρ_{1} = N, are of the form φ(r_{1})φ(r_{2}) × ⋯ × φ(r_{N}). All particle detections are then independent of each other, because every conditional probability in equation (1) is given by φ(r)^{2}. Single shots of such states merely reproduce the singleparticle density ρ(r) = Nφ(r)^{2}. Gross–Pitaevskii (GP) meanfield states are of this form. For any other type of state, each conditional probability in (1) depends on the values of the previously detected particles and single shots do not reproduce the singleparticle density.
As a first example we investigate a collision between independent, attractively interacting condensates that collide in D = 2 spatial dimensions—that is, r = (x, y) in an elongated trap V (r) with a flat bottom (see Methods for details of the trap). On the GP meanfield level, such BECs are known to pass through or bounce off each other, depending on the value of the relative phase^{15}. GP meanfield theory assumes the manybody state to be fully condensed at all times. However, for experimentally relevant strongly interacting states, the meanfield description is fundamentally inconsistent: interactions erode the structure of the ansatz state on a timescale which is fast compared to the collision time; fully condensed attractive BECs that are spread out over large distances are not stable and fragment quickly^{16}. We therefore choose a more stable initial state consisting of two independently created attractive BECs of 50 bosons each. It is impossible to define a relative phase for such BECs (ref. 17). Specifically, we use λ = −5.94 as an interaction strength, which is about 2% above the threshold for collapse of the ground state of all N = 100 bosons. The initial state is prepared by computing the manybody ground state of 50 bosons in the trap using M = 2 orbitals and imaginary time propagation. A copy of the ground state is placed offcentre at r = (−19.8,0). The resulting initial state is fragmented with natural occupations ρ_{1}/N = ρ_{2}/N = 49.4%, ρ_{3}/N = ρ_{4}/N = 0.6%, and is subsequently propagated in time.
Figure 1a shows the singleparticle density at different times. The condensates approach each other without spreading significantly, collide and separate again. During the collision the singleparticle density exhibits two maxima, such that the condensates seem to bounce off each other. However, single shots at the time of the collision reveal a different result (see Fig. 1b). In about half of all shots a strongly localized density maximum is visible, whereas in the other half two smaller wellseparated maxima appear. We stress that at no point was a (possibly random) phase relationship between the colliding parts assumed. In fact, such an assumption would be at variance with quantum mechanics^{17}.
Figure 1c shows the natural occupations of the system. Until the collision the natural occupations remain close to their initial values, which reflects the stability of the initial state. However, during the collision, two additional natural orbitals become occupied, indicating a buildup of even stronger correlations. As a consequence, after the collision the system can not be separated into two independent condensates.
Note that in a recent experiment similar fluctuations during the collision of two attractive BECs were observed^{18}. However, the initial state was prepared very differently, namely by splitting a single BEC in two instead of preparing two independent BECs. Whether the BEC was fragmented or not in that experiment is not clear. Moreover, the BEC was then imaged multiple times in a partially destructive way during the dynamics. The impact that partially destructive measurements have on a BEC depend strongly on the quantum state the atoms are in. We discuss such partially destructive measurements in the Supplementary Information.
In the previous example already the initial state required going beyond GP meanfield theory. We now demonstrate how singleshot simulations explain fluctuating manybody vortices that emerge dynamically from a GP meanfield initial state. Quantized vortices are a hallmark of GP meanfield theory and typically exhibit a density node at the centre of the vortex^{19}. Moreover, they appear from some critical rotation velocity of the condensate onwards^{19}. However, it recently became clear that stirring a BEC can also lead to manybody vortices far below the meanfield critical velocity. These manybody vortices typically have a finite singleparticle density at the centre, such that the vortex is barely visible^{4,20,21}. We now demonstrate how such vortices form dynamically and, similar to their meanfield counterparts, exhibit a vanishing density in single shots of an experiment.
Consider the ground state of a repulsively interacting BEC of N = 10, 000 bosons in a twodimensional (2D) harmonic trap with ω_{x} = ω_{y} = 1 at an interaction strength λ = 17. The manybody ground state using M = 2 orbitals is practically fully condensed, with ρ_{1}/N = 99.99%, and therefore described well by GP meanfield theory. We then switch on a timedependent stirring potential V_{s}(r, t) = (1/2)η(t)[x(t)^{2} − y(t)^{2}] that imparts angular momentum onto the BEC. Here x(t) and y(t) vary harmonically and the amplitude η(t) is linearly ramped up from zero to a finite value, kept there for some time and ramped back down again to zero (see Methods).
Figure 2a shows the density together with single shots at different times. The evolution of the natural occupations is shown in Fig. 2b. While the system is condensed, single shots reproduce the singleparticle density, as expected for a condensed state. However, over the course of time an additional natural orbital becomes occupied and the BEC becomes fragmented. The outcome of single shots fluctuates more and more, and vortices appear at random locations, with no significant density at the vortex core.
The previous examples have demonstrated that loworder correlation functions, such as the singleparticle density, can provide an inappropriate picture of the physical outcomes of single shots of an experiment. Even secondorder correlations would not have been sufficient to predict the outcomes in the examples above. As a last example we demonstrate the importance of Nth order correlations and the possibility to obtain full distribution functions using singleshot simulations. For this purpose we consider a seemingly simple system consisting of N attractively interacting bosons in a harmonic trap in one dimension—that is, D = 1 and r = x.
Independent of the type of the interaction between the bosons and its strength λ, the exact wavefunction of the centreofmass coordinate X = (1/N)∑ _{i}x_{i} of the manybody ground state is given by a Gaussian , with (ref. 22). For increasingly strong attractive interaction one expects the bosons to localize near the centre of the trap. However, note that Ψ_{mb}(X) is independent of the interaction and delocalizes entirely in the limit ω_{x} → 0. On the other hand, if one calculates the ground state using GP meanfield theory, the variance of the centreofmass coordinate is simply X_{mf}^{2} = (1/N^{2})∑ _{i}^{N}σ_{mf} = σ_{mf}/N, where σ_{mf} = 〈φ_{mf}x^{2}φ_{mf}〉^{1/2} and φ_{mf}(x) is the meanfield orbital.
The widths X_{mb} and X_{mf} of the centreofmass distribution are generally very different. Even going to large particle numbers does not change this discrepancy: for small values of ω_{x} or for strong attractive interaction, the meanfield orbital φ_{mf}(x) approaches a soliton and —that is, the width of the meanfield centreofmass wavefunction becomes , whereas . The width X_{mb} can then exceed X_{mf} regardless of N by any amount. The difference between the two is entirely due to correlation effects that are not captured by meanfield theory.
To illustrate this point we compute the manybody ground state of N = 10 attractively interacting bosons in a harmonic trap, ω_{x} = 1/100, in one dimension at an interaction strength λ = −0.423 using imaginary time propagation for different numbers of orbitals. From the obtained ground states we generate 10,000 single shots to obtain the full distribution function of the centreofmass coordinate. Figure 3 shows fits to the respective histograms of the centreofmass distributions together with the exact analytical result. The manybody result for M = 10 orbitals is indistinguishable from the exact result and significantly broader than the meanfield (M = 1) result. The full counting distribution of the centre of mass can thus be obtained by means of singleshot simulations. In the present example the manybody correlations are the cause of the onset of the delocalization of the ground state.
Methods
Bose–Einstein condensation.
For an Nboson state Ψ〉 = ∑ _{n}C_{n}(t)n〉 and a bosonic field operator the reduced singleparticle density matrix is defined as
with . By diagonalizing ρ_{ij} one obtains ρ^{(1)}(rr′) = ∑ _{i}ρ_{i}φ_{i}^{NO}(r)φ_{i}^{NO∗}(r′). The eigenvalues ρ_{1} ≥ ρ_{2} ≥ … satisfy ∑ ρ_{i} = N and are known as natural occupations, the eigenvectors φ_{i}^{NO}(r) as natural orbitals. If there is only one eigenvalue the BEC is condensed^{13}, if more than one the BEC is fragmented^{14}. The diagonal ρ(r) ≡ ρ^{(1)}(rr′ = r) is the singleparticle density of the Nboson wavefunction.
Singleshot algorithm.
Here we show how single shots can be simulated from a general Nboson wavefunction expanded in M orbitals Ψ〉 = ∑ _{n}C_{n}n〉, where n〉 = n_{1}, …, n_{M}〉 and ∑ _{i=1}^{M}n_{i} = N. The special case M = 2 has been treated in earlier works^{1,3,4}. The goal is to draw the positions r_{1}′, …, r_{N}′ of N bosons from the probability distribution P(r_{1}, …, r_{N}). We achieve this by evaluating the conditional probabilities in (1).
For this purpose we define reduced wavefunctions
of n = N − k bosons with normalization constants . The respective singleparticle densities are given by and . The first position r_{1}′ is drawn randomly from the distribution P(r) = ρ_{0}(r)/N. Assuming that positions r_{k}′, …, r_{1}′ have already been drawn, the conditional probability density for the next particle P(rr_{k}′, …, r_{1}′) = P(r, r_{k}′, …, r_{1}′)/P(r_{k}′, …, r_{1}′) is given by
because P(r_{k}′, …, r_{1}′) is a constant. The problem is thus reduced to obtaining the wavefunction Ψ^{(k)}〉 = ∑ _{n}C_{n}^{(k)}n〉 from the wavefunction Ψ^{(k−1)}〉 = ∑ _{n}C_{n}^{(k−1)}n〉, where the sums over run over all configurations of n and n + 1 bosons, respectively. Defining n^{q} = (n_{1}, …, n_{q} + 1, …, n_{M}) one finds from (4) that
Using (6) in a general Morbital algorithm requires an ordering of the configurations n〉 for all particle numbers n = 1, …, N. Combinadics^{11} provide such an ordering by assigning the index
to each configuration n〉. Using (7) all coefficients C_{n}^{(k)} can then be obtained by evaluating the sums in (6) and is determined by normalization. Using the coefficients C_{n}^{(k)} we evaluate ρ_{k}(r) and by means of (5) we then draw r_{k+1}′ from P(rr_{k}′, …, r_{1}′). This concludes the algorithm to simulate single shots. It is now easy to see that also correlation functions of arbitrary order can be evaluated. By realizing that
the kth order correlation function is evaluated at r_{1}, …, r_{k} as the product of the reduced densities ρ_{j−1}(r_{j}). Thus, to evaluate the correlation function the only modification to the singleshot algorithm above consists in choosing the positions r_{1}′, …, r_{k}′ rather than drawing them randomly.
MCTDHB.
In the MCTDHB (refs 10,11,12) method the manyboson wavefunction is expanded in all configurations that can be constructed by distributing N bosons over M timedependent orbitals φ_{i}(r, t). The ansatz for the timedependent manyboson wavefunction reads:
In (9) the C_{n}(t) are timedependent expansion coefficients and the n; t〉 are timedependent permanents built from the orbitals φ_{i}(r, t):
The MCTDHB equations of motion are derived by requiring stationarity of the manybody Schrödinger action functional
with respect to variations of the coefficients and the orbitals. The μ_{kj}(t) are timedependent Lagrange multipliers that ensure the orthonormality of the orbitals. With increasing M the solution of the MCTDHB equations converges to an exact solution of the timedependent manybody Schrödinger equation and numerically exact results have been obtained previously^{23,24}. For bosons interacting via a deltafunction interaction and M = 1, the MCTDHB equations of motion reduce to the timedependent Gross–Pitaevskii equation. For more information see the literature^{10,11,12}.
Parameters.
For all D = 2 dimensional simulations in this work we assume tight harmonic confinement with a frequency ω_{z} and a harmonic oscillator length along the zdirection. The bosons interact via a 2D regularized contact interaction potential (ℏ^{2}λ_{0}/m)δ_{ε}(r), with δ_{ε}(r) = (2πε^{2})^{−1}${\text{e}}^{{\text{r}}^{2}/2{\epsilon}^{2}}$, r = (x, y) and a dimensionless interaction strength , where a is the scattering length and m the mass of boson. We note that it is important to regularize contact interaction potentials for D > 1 (refs 25,26). For the collision of attractively interacting BECs the external potential used for the simulations is given by V (r) = V_{x}(x) + V_{g}(x) + V_{y}(y), with V_{x}(x) = (1/2)mω_{x}^{2}x^{2}, V_{y}(y) = (1/2)mω_{y}^{2}y^{2}, and V_{g}(x) = $C{\text{e}}^{{x}^{2}/2{\sigma}^{2}}$, where C = mσ^{2}ω_{x}^{2}. We obtain dimensionless units ℏ = m = 1 and the Hamiltonian (2) by measuring energy in units of ℏω_{y}, length in units of and time in units of 1/ω_{y}. We use a planewave discrete variable representation to represent all orbitals and operators. The width of the contact interaction is ε = 0.15 and the grid spacing is Δx = Δy = ε/2 unless stated otherwise. For the elongated trap the parameter values are ω_{x} = 0.07, ω_{y} = 1 and σ = 10 on a grid [−43.2,43.2] × [−3.6,3.6], which creates a trap with a flat bottom at the centre. For the rotating BEC the parameter values are ω_{x} = ω_{y} = 1. η(t) is linearly ramped up from zero to η_{max} = 0.1 over a time span t_{r} = 80. η(t) is then kept constant for t_{up} = 220 and ramped back down to zero over a time span t_{r}. The potential V_{s}(r, t) = (1/2)η(t)[x(t)^{2} − y(t)^{2}] rotates harmonically with x(t) = xcos(Ωt) + ysin(Ωt) and y(t) = −xsin(Ωt) + ycos(Ωt), where Ω = π/4. The grid size is [−8,8] × [−8,8] with 214 grid points in each direction.
For the D = 1 dimensional simulations we assume tight harmonic confinement along the y and zdirections with a radial frequency ω_{⊥} = ω_{y} = ω_{z} and an oscillator length . The contact interaction potential is then given by (2ℏ^{2}a/ml_{⊥}^{2})δ_{ε}(x), with δ_{ε}(x) = (2πε^{2})^{−1/2}${\text{e}}^{{x}^{2}/2{\epsilon}^{2}}$. We use ℏω_{⊥} as the unit of energy and l_{⊥} as the unit of length. The dimensionless interaction strength is then given by λ_{0} = 2a/l_{⊥}. The harmonic potential along the xdirection ω_{x} = 1/100 is much weaker than the radial confinement ω_{⊥} = 1. The grid size is [−90,90].
Image processing.
The histograms of the positions of particles obtained using the singleshot algorithm have a resolution that is determined by the grid spacing. For better visibility and in analogy to a realistic imaging system we convolved the data points of each histogram with a pointspread function (PSF). As a PSF we used a Gaussian of width 3 × 3 pixels.
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Acknowledgements
K.S. acknowledges financial support through the Karel Urbanek Postdoctoral Research Fellowship. Computing time was provided by the High Performance Computing Center (HLRS) in Stuttgart, Germany. We would like to thank J. Schmiedmayer for interesting discussions about partially destructive measurements.
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K.S. and M.K. conceived the ideas and designed the study. K.S. developed the algorithm and carried out the simulations. K.S. and M.K. wrote the paper.
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Sakmann, K., Kasevich, M. Singleshot simulations of dynamic quantum manybody systems. Nature Phys 12, 451–454 (2016). https://doi.org/10.1038/nphys3631
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