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Using a set of Zeeman sublevels of an alkali atom such as rubidium or sodium, we propose to construct a pair of coherentc population trap (CPT) states, which can be individually addressed and populated at will. The system can be arranged such that there exists a dressed state that is a linear combination of these two CPT states. We have earlier shown the capability of forming discrete quantum gates using this configuration [J.Phys. B, (2006),

Quantum information deals with representing bits “0” and “1” by objects that obey the rules of quantum mechanics. The fact that quantum objects can be prepared in a superposition of |0⟩ and |1⟩ opens a new facet wherein operations hitherto not possible in the realm of classical mechanics can be performed. Although quantum mechanics deals with discrete states, it is possible to use “continuous functions” for quantum information in three different methods. In the first method, we use parameters that take up continuous values, such as voltage pulse, etc., as given by Lloyd and Braunstein (

CPT states are a special form of dressed states, which are eigenstates of the total atom + field Hamiltonian (

The configuration considered here consists of two hyperfine levels _{1/2}|_{1/2}|_{1/2}, _{−}⟩, |_{0}⟩, and |_{+}⟩, respectively. Similarly, sublevels of the excited state _{1/2} are labeled |_{−}⟩, |_{0}⟩, and |_{+}⟩. The transitions |_{−}⟩↔|_{−}⟩ and |_{+}⟩ ↔|_{+}⟩ are coupled by light polarized in the “z” direction with respect to the axis of quantization (labeled as Π polarization in the figure). Due to the symmetry of the corresponding wavefunctions, the transition |_{0}⟩ ↔|_{0}⟩ turns out to be forbidden. Transitions |_{−}⟩↔|_{0}⟩ and |_{0}⟩ ↔|_{+}⟩ are coupled by circularly polarized light that is _{+} helicity, and similarly, _{−} polarized light couples transitions |_{+}⟩ ↔|_{0}⟩ and |_{0}⟩ ↔|_{−}⟩, as shown in

Energy levels of an alkali atom with relevant couplings.

If only the Π polarized light is present, then the atoms in states |_{±}⟩ are excited to states |_{±}⟩, respectively, and subsequently decay to any one of the three ground states with equal probability, as shown in _{+}⟩ and |_{−}⟩ are again excited by the light, whereas atoms that reach |_{0}⟩ are left behind. This results in all atoms being emptied from |_{±}⟩ and “pumped” into state |_{0}⟩ in a finite time.

Pathways of spontaneous emission decay, leading to optical pumping.

When only the circularly polarized light is present, the pair of _{+} and _{−} forms a Λ configuration as shown in

Λ configuration, leading to CPT. See text for description.

This gives us two independent trap states to work with, which are orthogonal to each other—the application of _{0}⟩ = |_{0}⟩, and the pair of _{±} beams will prepare the atoms in the

Therefore, the states can be mapped to qubits such that |_{0}⟩ → |0⟩ and |_{−}⟩ → |1⟩. Such state preparation is robust and reliable.

The robustness of any CPT configurations can be understood by looking at the effect of decoherence on a generic CPT system. A typical system is shown in _{
eg
} and Ω_{
ef
} couple the states as shown in the figure.

Three-level schematic to explore CPT.

The corresponding Hamiltonian, in units of

where _{
e,g,f
}‘ are the eigen energies of the levels, scaled to

where _{0}⟩ is a CPT state (also called a “dark state”) because any atoms prepared in this state will remain there due to coherent cancellation of absorption probability. The population in state |

However, collisions will cause a decoherence effect, due to which the cancellation of absorption will be incomplete (

_{
gf
} will take the following form

indicating the role of collision decay _{
c
}.

The Liouville Eq. _{
c
} that affects only the ground state coherence term _{
gf
}. A few typical values of _{
c
} were chosen for the simulation to study its effect on coherence. _{
gg
}, _{
ee
}, and _{
ff
}, for two different collision rates (a) _{
c
} = 0.1 and (b)_{
c
} = 0.4. Under ideal conditions, the dark state should have _{
gg
} = _{
ff
} = 0.5 and coherence term _{
gf
} = −0.5.

Populations of all three levels as a function of time for _{
c
} = 0.0 and _{
c
} =0.1, respectively. _{
ee
} becomes zero after some time in case _{
c
} is a finite value.

It is evident that at a long time limit, the populations of ground states _{
gg
} and _{
ff
} reach 0.5 at a long time limit, in the absence of any collisions (_{
ee
} acquires a finite population, indicating that the dressed state Ψ_{0} is a leaky, rather than perfect, dark state (

_{
c
} increases. _{
gf
} between two ground states decreases and deviates from the ideal value of −0.5 as collisional damping increases.

_{
ee
} increases with _{
c
}. The ground state coherence, as shown in

However, increasing the intensity of lasers results in an increase of the Rabi frequency, which can overcome the effect of collisional damping, as shown in

_{
gf
} for _{
c
} = 0.4 shifts towards the ideal value when the Rabi frequency Ω_{
eg
} = Ω_{
ef
} = Ω is increased, indicating that higher intensity can overcome the loss of coherence due to collisions. The inset shows an expanded part of a long time limit.

When both the

and obtains the dressed state under the condition Ω_{
σ+} = Ω_{
σ−} = Ω_{
σ
}

Here, Ω_{
σ
} and Ω_{
π
} are the Rabi frequencies associated with the

or, with a few rearrangements, as

where

and states |_{0}⟩ = |0⟩ and |_{−}⟩ = |1⟩ as defined earlier. Exp(_{
π
} and Ω_{
σ
}.

The dressed state shown in Eq. 6 is a combination of two trap states |_{0}⟩ and |_{−}⟩. This combined state is also a trap state and will not evolve further (_{
π
} and Ω_{
σ
}. When this change is brought about adiabatically and slowly, then the state is continuously and adiabatically varied. This transition does not populate the excited states and hence is a coherent transformation. iv) Varying angle

At first look, Eq. _{
π
} = 0 gives |_{0}⟩ and setting Ω_{
σ
} = 0 gives |_{−}⟩, which is the opposite of the effect created by individual lasers. However, this must be seen in the context of properties of coherent superpositions of states that lead to the coherence-induced phenomena. For instance, for the dark state |Ψ_{0}⟩ in Eq. 2 in _{
ef
}, and for state |_{
eg
} (

In the case of the stimulated Raman adiabatic passage (STIRAP) (

An experimental setup is proposed to achieve states that continuously span the vector space. A pair of magnetic coils (labeled Mg) will help define the “z” axis of quantization for the rubidium atoms (labeled Rb). A combination of a half-wave plate (HWP1) and a polarizing beamsplitter (PBS) will divide the laser light into vertical and horizontally polarized lights, whose ratio can be very precisely controlled. The vertically polarized light forms the _{+} and _{−} lights. The second half-wave plate (HWP2) helps align the second beam in a proper polarization, overcoming any changes that arise due to reflections.

Precise positioning of the HWP1 will enable varying Rabi frequencies Ω_{
π
} and Ω_{
σ
}, such that sin^{2}(^{2}(_{
π,σ
} when used in Eq.

By rotating HWP1, the laser light can be precisely divided into the

that acts on column vectors

We propose a setup as follows. A pair of beams with _{+} and _{−} polarizations in the yz plane, coming from opposite directions, +

The two beams will add up to give (

This results in a light beam propagating in the _{
x
} = 2

Polarization of light as a function of distance “x.”

Wherever the light beam is z-polarized, it is equivalent to the _{0}⟩ = |_{0}⟩, equivalent to _{+} and _{−} fields, and hence, the atoms will get into the CPT state of |_{−}⟩. In the region between, the light is in different polarizations, and hence, the atoms will be in state |

There are now two methods of obtaining continuous variable states—in the first option, consider a slow atom moving along the “z” direction. This atom encounters light with a continuously varying polarization, which results in the atom adiabatically changing its state as per Eq.

The second option is to take a collection of cold atoms and have the _{±} beams incident on them as described earlier. Atoms at different spatial points experience different polarizations and therefore are prepared in different states, in a continuous distribution of states, the state vector depending upon its “z” position.

In the presence of an external magnetic field, each of these atoms aligns at an angle with respect to the magnetic field direction, depending upon its position. The |_{0}⟩ component of the state aligns perpendicular to the field, and the |_{−}⟩ component aligns parallel to the field due to their spin values. For the combination of these two states in the form given in Eq. 7, the angle of alignment will be a vector sum of these parallel and perpendicular components, with a weightage of sin(_{1} = _{1}
_{1} and _{2} = _{2}
_{2} are, respectively, the magnetic moments of the two atoms. _{1} and _{2} are the net spins of the two atoms, and

We have considered a system consisting of the two ground states of an alkali atom and their Zeeman sublevels. This makes a total of six levels, coupled by two lasers that differ in their polarizations. The interaction leads to a dressed state that is a combination of all three ground states in such a way that it can be mapped to a state of the form |

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

AV developed most calculations; BS created the numerical simulations. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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