# Manipulating Quantum States of Molecules Created via Photoassociation of Bose-Einstein Condensates

###### Abstract

We show the quantum state transfer technique in two-color photoassociation (PA) of a Bose-Einstein condensate, where a quantized field is used to couple the free-bound transition from atom state to excited molecular state. Under the weak excitation condition, we find that quantum states of the quantized field can be transferred to the created molecular condensate. The feasibility of this technique is confirmed by considering the atomic and molecular decays discovered in the current PA experiments. The present results allow us to manipulate quantum states of molecules in the photoassociation of a Bose-Einstein condensate.

###### pacs:

34.50.Rk, 03.75.Nt, 32.80.Pj, 33.80.PsOver the last few years, a particularly important development in ultracold quantum systems has been the ability to form and manipulate quantum degenerate molecular gases via photoassociation (PA) in Bose-Einstein Condensates (BECs). Besides the creation of molecular quantum gases with Feshbach resonances Feshbach , a more general method, say, a stimulated optical Raman transition (STIRAP) has been employed to directly produce deeply bound molecules in recent years optical1 ; optical2 . STIRAP was proposed as a promising way for a fast, efficient and robust process to convert a BEC of atoms into a molecular condensate molecular1 ; molecular2 ; molecular3 . The central idea for this kind of STIRAP is the realization of the dark superposition state of a BEC of atoms and a BEC of molecules, which has been observed in recent experiments experiment .

Although PA process has been widely studied in both theoretical and experimental aspects, the possibility of manipulating quantum states of the created cold molecular gas is not well discovered in such process. Control of quantum states of molecular gas may be achieved by realizing quantum state transfer from photons to molecules formed in the PA process. Most recently, one of us studied the possibility of quantum conversion between light and molecules via coherent two-color PA by using a single-mode associating light Hui . Here we shall study further the validity of quantum state transfer from a quantized associating light to molecular gas in the PA process.

In this letter, we identify the quantum state transfer process in two-color PA of a Bose-Einstein condensate by quantizing associating light. Under the weak excitation condition, we show quantum states of the quantized field can be fully transferred to the created molecular condensate. The multi-mode associating light is applied and the effect of atomic/molecular decays is discussed in detail in this model. The present result confirms the possibility of quantum state transfer in the current PA experiments, and will have wide applications to, e.g. quantum information science with cold molecules computing .

Turning to the situation of Fig. 1, we consider the quasi one-dimensional model of N identical atoms that have condensed into the same one-particle state with wave vector . Through a quasi-one dimensional quantized associating field

(1) | |||||

where is the coupling coefficient between the atoms and quantized field, and represent the one- and two-photon detunings, respectively. We note here has the units of sm, since here we consider quasi-one dimensional system molecular3 ; experiment .

The evolution of the quantum field can be described in the slowly varying amplitude approximation by the propagation equation:

(2) |

On the other hand, the evolution of atomic field operators is governed by a set of Heisenberg equations

(3) | |||||

where , and denote the decay rates of corresponding atomic or molecular states. With these formulas we find the atom-molecule evolutions can be described by

(4) | |||||

(5) | |||||

It should be noted that we have disregarded the motion of the trapped atoms and molecules. This is valid since the time scale for coherent STIRAP is shorter than the time scale for the motion of the atoms and/or molecules in the trap molecular1 ; molecular2 ; molecular3 . Also, as long as laser intensities permit STIRAP during a time interval much shorter than the time scales for collisions between atoms and molecules, collisions are negligible as well. The above two Eqs. can be recast into

(6) | |||||

(7) | |||||

Here and are transversal decay rates. From Eqs. (6) and (7), we further obtain

(8) | |||||

The above equation is hard to tackle. However, according to the present experiments experiment , we can simplify it with several approximations. Firstly, we shall consider the weak-excitation condition, i.e. only a small ratio of atoms are converted into molecules, thus the last term in the right-hand side (r.h.s.) of above formula is very small and can be safely neglected. Secondly, we assume changes sufficiently slowly with time so that adiabatic condition is fulfilled. For this only the first term in the r.h.s. of (8) plays important role in the dynamical evolution of atomic fields. On the other hand, in this paper we shall first neglect the effect of two-photon detuning and atomic-molecular decay that will be discussed later in detail. By introducing a normalized time , where is a characteristic time scale, and expanding the r.h.s. of (8) in powers of , we find in lowest nonvanishing order,

(9) |

Thus Eq. (6) can be recast into

(10) |

Here is the atomic BEC density that is assumed to be constant in weak excitation case. Substituting this result into Eq. (2) yields

(11) | |||||

where is the total atomic number in the interacting region. Noting that is time dependent, the r.h.s. of above equation (11) represents an adiabatic Raman absorption or enhancement of the quantized field. The group velocity of the associating field is given by . Here is related to atomic density experiment and can be regarded as constant in the weak-excitation condition. The mixing angle is defined according to that is governed by the atomic number and the strength of external classical field. Different from the usual electromagnetically induced transparency (EIT) case where the group velocity of associating light is proportional to EIT , here the group velocity of quantized field is proportional to due to the nonlinearity in the Hamiltonian (1). When the classical field is adiabatically turned off so that the mixing angle , one has , which means the initial photon wave packet is decelerated to a full stop. Equation (11) has the following simple solution

(12) |

On the other hand, from Eq. (9), one can easily find

(13) |

With the expression (13) one can readily reach a full quantum state transfer from photons to created molecules via PA process. For example, if initially a strong classical field is applied so that or , one has . Then, after is adiabatic turned off, i.e. , from (13) we have , say, quantum states of the associating light are fully transferred into the created molecular BEC. The similar result is well-known in quantum memory technique with EIT. Nevertheless, identification of quantum state transfer in the present model suggests a novel technique to create molecules in non-classical states via a PA process.

In above discussions, we have ignored the decay of ground atomic and molecular states. However, recent experiments experiment indicate is not very small in a PA process. As a result, it is important to further verify the validity of quantum state transfer technique when decay of atomic and molecular states is included. Noting that one- and two-photon detunings can be adjusted in experiment experiment , we would like to consider here small detuning case so that and . The lowest order in Eq. (8) then reads

(14) |

Substituting this result into Eq. (6) and then into Eq. (2) yields

(15) |

Thus the quantum field propagates with the group velocity

(16) |

which approaches when . The presence of atomic decay results in a nonzero group velocity even when the classical field is turned off. For weak associating field case, typical parameter values can be taken as experiment s, s, and s. One can then evaluate the limit of group velocity by kms. Furthermore, if we consider a larger number of trapped atoms, say BEC1 , the limit of group velocity will be decreased by four orders and yields ms. The numerical result indicates the associating light can still reach an approximate “stop” for the large atomic number case.

The first term in the r.h.s. of above equation (15) represents an adiabatic Raman absorption or enhancement of the quantized field, while the second term represents a decay. The solution can be given by

(17) |

where

The above integral can be calculated straightforwardly. Similar to the former discussion, we assume initially the classical field is strong so that , while finally . We then find from Eq. (17)

(18) | |||||

where and . The above equation reduces into the formula (12) if . Based on the previous parameters used in calculating the limit of group velocity, one can see . Thus we have and , and the created molecular BEC is given by

(19) |

This formula shows quantum state transfer technique from quantized associating field to created molecules is still valid when . Therefore, to obtain a sufficient quantum state transfer, the time duration of a PA process should satisfy . For atomic-molecular BECs, the decay is about s for the weak associating case. Then the time limit is about ms, which is in the same order with storage time of EIT quantum memory EIT experiment . Such requirement can also be expressed that the propagation depth of associating light satisfy during the PA process. With typical values of parameters one can find is about mm. For BECs, the spatial scale of the interaction region is about mm BEC . Thus the requirement can be generally satisfied if only the classical field is turned down to be before the quantized field propagates through the atomic-molecular BEC in PA experiments. These results show that even for the case of nonzero atomic decays, we can still reach a novel technique to manipulate quantum states of created molecules via a quantized associating field as long as the time of PA process is small enough. Now that the mapping technique can be applied to separate Raman transitions at the same time, it is also possible to transfer entanglement from a pair of associating light beams as generated, e.g. in parametric down-conversion to a pair of created molecular BECs.

As a final remark, we note that the above analysis involves the weak-excitation approximation, valid when ratio of molecular number and atomic number is very small. Making use of Eq. (19) one finds . If the initial number density of associating photons is much less than the number density of atoms, the formed molecular number is always much smaller than the atomic number, and the weak-excitation condition is valid during the PA process. It is also noteworthy that such condition is true in the present PA experiments experiment .

In conclusion, We have shown the quantum state transfer process in two-color PA of a Bose-Einstein condensate, where a quantized field is used to couple the free-bound transition from atom state to excited molecular state. Under the weak excitation condition, we find quantum states of the quantized field can be transferred to the created molecular condensates. Effects of atomic and molecular decays in the present model are discussed in detail. Our results show that quantum state transfer proposed here is achievable in the current PA experiments, and will have wide range of applications to quantum information science. For example, if the created molecular state is untrapped, we may create molecular lasers with controllable quantum states. Besides, if considering a squeezed associating light in the PA process, one may study the molecule-light entanglement entanglement , and so on.

This work is supported by NUS academic research Grant No. WBS: R-144-000-189-305, and by NSF of China under grants No. 10575053.

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