# Images of a Bose-Einstein condensate: diagonal dynamical Bogoliubov vacuum

###### Abstract

Evolution of a Bose-Einstein condensate subject to a time-dependent external perturbation can be described by a time-dependent Bogoliubov theory: a condensate initially in its ground state Bogoliubov vacuum evolves into a time-dependent excited state which can be formally written as a time-dependent Bogoliubov vacuum annihilated by time-dependent quasiparticle annihilation operators. We prove that any Bogoliubov vacuum can be brought to a diagonal form in a time-dependent orthonormal basis. This diagonal form is tailored for simulations of quantum measurements on an excited condensate. As an example we work out phase imprinting of a dark soliton followed by a density measurement.

Quantum measurements on Bose-condensed systems can give quite unexpected results. For example, in the classic paper by Javanainen and Yoo [1] a density measurement on a Fock state with particles equally divided between two counter-propagating plane waves reveals an interference pattern with a phase chosen randomly in every realization of the experiment. The Fock state has a uniform single particle density distribution, but its measurement unexpectedly reveals interference between the two counter-propagating condensates. The Fock state is a quantum superposition over -particle condensates with different relative phases in their wave functions [2], , but every single realization of the experiment reveals such a density distribution as if the state before the density measurement were one of the condensates with a randomly chosen phase . This effect is best explained [2] when the density measurement, which is a destructive measurement of all particle positions at the same time, is replaced by an equivalent sequential measurement of one position after another. With an increasing number of measured positions a quantum state of the remaining particles gradually “collapses” from the initial uniform superposition over all phases to a state with a more and more localized phase . For a large a measurement of only a small fraction of all particles practically collapses the state of remaining particles to a condensate with phase . The lesson from this instructive example [1, 2] is that when we want to predict possible outcomes of measurements on a Bose-condensed state, then it is helpful to rewrite the state as a superposition over condensates with different condensate wave functions in a hope that will give an approximate probability for different density measurement outcomes . However, as the set of all condensates is non-orthogonal and overcomplete, the “amplitude” is not unique but depends on coordinates used to parameterize the space of condensate wave functions . In general a special care should be taken to choose the right parameterization where the simple “Born rule” gives the correct probability.

The existing literature about measurements on Bose-condensed systems [1, 2] concentrates on simple most beautiful examples of Bose-condensed states. In this letter we construct for the first time a measurement theory able to predict probability of different density measurement outcomes on a realistic Bose-Einstein condensate of interacting particles evolving under influence of external time-dependent potentials. This problem is important because in many experiments, like phase imprinting of dark solitons [3] or condensate splitting in atom interferometers [4], manipulation of the condensate generates substantial dynamical depletion which can qualitatively affect measured density patterns. In addition to being realistic our theory also reveals a beautiful diagonal structure hidden in the quantum state of a condensate excited from its ground state by a time-dependent potential.

A generic Bose-Einstein condensate consisting of weakly interacting atoms can be described by a number-conserving version of the Bogoliubov theory [5] which assumes that most particles occupy the condensate wave function. Time-dependent problems can be treated with the time-dependent version of the theory where the condensate wave function evolves according to the time-dependent Gross-Pitaevskii equation (GPE) and the Bogoliubov modes and solve the time-dependent Bogoliubov-de Gennes equations (BdGE) [5]. In this framework, starting with a condensate in the ground state, time-dependent perturbation excites the condensate to a state which is formally a time-dependent Bogoliubov vacuum annihilated by the time-dependent quasiparticle annihilation operators

(1) |

Here the operator annihilates an atom in a condensate wave function . The modes and are solutions of the time-dependent BdGE projected on the subspace orthogonal to , i.e. , and normalized so that . In the time-dependent vacuum the reduced single particle density matrix

(2) |

has common eigenstates with the operator

(3) |

A mode is a non-condensate eigenmode of the density matrix (2) occupied on average by particles. Thanks to the completeness relation

(4) |

valid in the subspace orthogonal to , the operator has common eigenstates with the

(5) |

Furthermore ’s allow us to get a diagonal form of

(6) |

which is a map between the subspace orthogonal to and the subspace orthogonal to . Indeed, the property of the Bogoliubov modes and Eq.(4) imply a quasi-commutator from which follows the diagonal form in Eq.(6). What is more, owing to , one gets In Ref. [6] we have shown that in the stationary case the Bogoliubov vacuum has the following form

(7) |

where ’s annihilate atoms in single particle states orthogonal to the condensate wave function . Matrix fulfills the equation

(8) |

which is equivalent to the set of the conditions . In the stationary and time reversal invariant case all modes can be chosen as real functions so that the matrix is real symmetric and can be diagonalized by an orthogonal transformation [6, 7]. In a generic time-dependent case, or when a stationary breaks the -invariance like in e.g. vortex state, the matrix is complex and symmetric. However, using only the general properties of and it is easy to show that the choice of ’s for the modes in Eq.(7) makes the time-dependent Bogoliubov vacuum diagonal

(9) |

where the eigenvalues of the matrix are

(10) |

Indeed, an action of on both sides of Eq. (8) yields equivalent to Eq.(10).

Hence the procedure of time evolution of the Bogoliubov vacuum state in the particle representation becomes extremely simple: i) First one has to evolve Bogoliubov modes together with the condensate wave function which is easy because different modes evolve independently from each other. ii) Next one has to diagonalize the operator (diagonalization of turns out to be better convergent numerically) in order to get eigenvalues and eigenvectors . iii) Finally one calculates and from Eq. (10) obtains values of . Note that there is a one to one correspondence between moduli of and the eigenvalues of the single particle density matrix :

(11) |

However, the phases of ’s cannot be determined from the single particle matrix only, but they also depend on the eigenvalues , see Eq.(10). Finally, once the phases are known it is convenient to make the transformation

(12) |

In the following we skip the primes.

For large the diagonal state (9) can be rewritten as a superposition over -particle condensates

(13) |

with the normalized condensate wave functions

(14) |

The gaussian amplitude gives an accurate vacuum state when . The gaussian square of the amplitude

(15) |

is a candidate for the probability distribution for different . If this choice is correct, then simulation of the density measurement amounts to choosing a random set of ’s which determines the measured density

(16) |

To show the correctness of Eq.(15), we average over and for we get the density

(17) |

averaged over different realizations of . This average density should be equal to the expectation value of the density operator in the state (13)

(18) |

with . These two densities are the same for the highly occupied modes with or . There are discrepancies for poorly occupied modes but those modes give negligible contribution to the total density. The probability (15) accurately reproduces the single particle density matrix. This accuracy can be best explained when we look at the overlap between condensates

(19) |

Condensates become orthogonal on the length scale of . For the highly occupied modes the fluctuations of are much greater than the width of the overlap and for these modes the overlap can be replaced by a delta function . With the delta overlap the condensates are orthogonal and the probability is simply given by the Born rule in Eq.(15).

To summarize, a density measurement, which is a measurement of all atomic positions, can be approximately simulated in two steps. In the first step a condensate wave function (14) is chosen randomly from the gaussian distribution . In the second step atomic positions are measured in the chosen condensate. The first step already gives a smoothed density distribution on top of which the second step only adds statistical fluctuations. In most applications one is not interested in the statistical fluctuations but in the smoothed density obtained after filtering out the statistical noise, compare the histogram and the solid line in Fig.2. Thus for most applications the density measurement can be very efficiently and accurately simulated with only the first step of the procedure which immediately gives a smoothed density with randomly chosen ’s. In the following we give an example of this procedure: density measurement after phase imprinting of a dark soliton.

To describe evolution of the condensate wave function we solve the time-dependent GPE that in harmonic oscillator trap units reads

(20) |

Here is an external potential created by the laser beam. As the effective 1D interaction strength we choose corresponding to parameters of the Hannover experiment [3]. The dark soliton is a kink with the wave function winding a phase of as goes from negative to positive. At the same time the density drops to zero at explaining the name dark soliton. Experimentally the soliton is excited by a short laser pulse which imprints on the wave function a phase that changes by when one goes from negative to positive . To simulate the phase imprinting we started with the ground state of the stationary GPE for a condensate in the harmonic trap, , and applied (similarly as in Ref.[8] where they use rather small ) the perturbation (where , ) that lasted for . Then was evolved up to (or ms). The density of the resulting wave function is plotted in Fig 1(a) where the dark soliton is clearly visible in the center of the trap.

The Bogoliubov modes solve the time-dependent BdGE

(21) | |||||

(22) |

where . We have evolved the Bogoliubov modes, starting with the eigenmodes of the stationary BdGE for a condensate in the ground state, up to . The single particle density is plotted in Fig. 1(b). In Fig. 1(c) the densities of condensate, noncondensate atoms and the total single particle density are plotted. At both values of (the realistic here and the in Ref.[8]) the single particle density shows that minimum at the soliton location is gradually filled with particles depleted from the condensate. These dynamical studies confirm earlier predictions that a static soliton is going to fill up after a few milliseconds [9].

After the evolution of the Bogoliubov modes was completed we calculated the eigenvalues and the non-condensate eigenmodes at the final time. It turns out that the soliton notch is filled up with atoms depleted to only one mode: the with the largest . In Fig.2(d) the total depletion density (solid line) is plotted together with a barely visible dashed plot of the . As the depletion in the soliton notch comes only from and because we would like to know what will be seen in the soliton notch in a single experiment we truncate the final time-dependent Bogoliubov vacuum to . We made exact simulation of density measurement in the truncated vacuum using the algorithm of Ref.[1] and Ref.[9]. A typical outcome is shown as the histogram in Fig.2. It turns out that every observed histogram can be very well fitted with the density where the only free parameter is . The histogram in Fig.2 is fitted by the density shown in the figure as the solid line. This is in agreement with our measurement theory predicting the smoothed histograms to be with a random .

In conclusion, we derived a convenient diagonal form of the time-dependent Bogoliubov vacuum which greatly facilitates simulations of quantum measurements on Bose-condensed systems.

## Acknowledgements

We thank Zbyszek Karkuszewski for stimulating discussions. KS was supported by the KBN grant PBZ-MIN-008/P03/2003. The work of JD was supported by Polish Goverment scientific funds (2005-2008) as a research project.

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