Recently we have shown that an arbitrarily prescribed quantum state of a single-mode cavity field can be generated deterministically by a sequence of Raman interactions [1]. The amplitudes and phases of classical Raman pulses control the quantum evolution of the state of an atom and a cavity field, and any desired cavity field state can be reached at a given time. Unlike other proposals [2, 3, 4, 5, 6], our method can be translated directly to trapped-ion systems which share a similar Hamiltonian structure [7, 8]. In fact, the essential idea of our scheme can be applied to other Raman related systems which could include the coherent manipulation of momentum states of neutral atoms [9].

In this paper, we show how Raman pulses can synthesize an arbitrary prescribed superposition of hyperfine magnetic states in an atom. The ability of controlling internal states of atoms could open doors for interesting applications [2, 10, 11]. Although the method of adiabatic passage is a powerful technique to transfer populations [12], the question how to achieve a superposition of hyperfine states with any desired values of complex amplitudes remains a challenge [13]. In this paper we shall present a solution with the hyperfine ground state of sodium as an example.

To begin, let us review the key idea of our scheme of arbitrary state preparation. Consider an N’state system described by a complete basis {|n〉}, where n = 1,2, ..N. Our goal is to create a superposition state

with prescribed complex amplitudes c_n. We assume that the initial state is a simplest state |1〉 which can be easily prepared. Since the quantum evolution is time reversible, the state preparation problem is equivalent to finding a way to evolve |Ψ〉 back to |1〉 unitarily. Quite generally, if the system allows the two types of external interaction indicated in Fig. 1, we can always find a solution. In Fig. 1 we label the states |n〉 in a two-row format, and the two types of interaction are classified by the vertical (Fig. 1a) and diagonal (Fig. 1b) flows. For each type of interaction, each state couples with one state only. Such a two-state interaction structure is the key because populations can always be transferred completely from one state to another. Therefore by applying the two interactions alternatively, we can sweep all the populations down to |1〉, i.e.,

where U_V and U_D, are the evolution operators for the vertical and diagonal flows respectively. For each evolution, λ_j characterizes the set of parameters (e.g. the duration, phases, amplitudes and polarizations of external fields) of the corresponding interaction. We choose λ_j in such a way that all the populations in the state |j + 1〉 are transferred to |j〉. Since U_a(λ_j) (a = V, D) also affects lower states |n < j〉, λ_j is determined by the history of the system. In Ref. [1] we have shown that a solution for {λ_j} always exists.

 

Fig. 1. Two types of interactions connecting different states. (a) Vertical (b) Diagonal

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Quite remarkably, there are several interesting quantum systems that indeed allow the two types of interactions as depicted in Fig. 1. For example, a two-level atom in a quantized cavity field [1], a trapped ion [8], and the manifold of hyperfine levels. In the following we describe in detail the last case. Let us consider the hyperfine levels |F, m_F〉 of 3² S _1/2 in sodium, where m_F = -2, -1, 0,1, 2 for F = 2 and m_F = -1, 0,1 for F = 1. We assume that the initial state is |F = 2, m_F = -2) which can be prepared by optical pumping.

To generate an arbitrary superposition state of all hyperfine levels in the 3² S _1/2 manifold, we have to find a way to evolve the prescribed state back to |F = 2, m_F = -2). This can be done by a Raman pulse sequence as shown in Fig. 2. There are two types of Raman transition. The ‘vertical’ transitions (∆m_F = 0, ∆F = ±1) are driven by two π-polarized pulses. The ‘diagonal’ transitions (∆m_F = -1, ∆F = ±1) are driven by a π-polarized pulse and a circularly polarized pulse with + or - helicity. We assume that all Raman pulses are far-detuned from the P–states so that effects of spontaneous decay of P states can be suppressed. In addition we assume that Raman pulses satisfy the resonance condition, ∆ω = ω ₀, where ∆ω is the frequency difference in each pulse pair and ω ₀ is the frequency difference between the F = 2 and F = 1 levels. There could be intensity-dependent level shifts during the Raman process. If these level shifts become significant (i.e., comparable with the effective Rabi-frequency), ∆ω should be adjusted in order to maintain the resonance condition. We should point out that at each step resonance is needed only for the two levels between which we want to transfer population completely from one level to another.

 

Fig. 2. An 8-pulse sequence that force any given state (upper left) in the hyperfine manifold of 3² S _1/2 of sodium to evolve into |F = 2, m_F = -2〉 (bottom right). The solid and empty circles represent occupied and unoccupied states respectively

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In Fig. 2, the first Raman pulse brings all the population in |F = 2, m_F = 2〉 to |F = 1,m_F = 1〉. We have to keep track of the transitions in lower levels (dotted lines) in order to determine the correct parameters for the later pulses. The second Raman pulses do a similar job as the first one in the vertical direction. Because the selection rule does not allow a vertical transition between 0–0 states, we have to get around it by applying four diagonal interactions through the steps (3)–(6). After the final two pulses (7) and (8), the state |F = 2, m_F = -2) is reached. It is straight forward but tedious to write down the equations that govern the required pulse area and phase at each step, the readers may find more details in a related systems in [1]. The 8-pulse sequence given in Fig. 2 is good for all superposition states. However, a shorter pulse sequence could be used for some special states. For example, the state

can be obtained without applying the pulses (3) and (4). In this simple case, all the required Raman pulses are π pulses, except that the pulse areas in the steps (7) and (8) should be adjusted according to the Clebsch-Gordan coefficients.

In conclusion, we have discussed a method to generate an arbitrary prescribed quantum state that is suitable for general systems that allow interactions defined in Fig. 1. As an example, we have described how a general superposition of hyperfine levels in sodium can be achieved by Raman pulses. The method can obviously be extended for other atoms with a similar structure of hyperfine states.