Measuring molecular electric dipoles using trapped atomic ions and ultrafast laser pulses

Jordi Mur-Petit and Juan José García-Ripoll

Phys. Rev. A 91, 012504 – Published 12 January 2015

Abstract

We study a hybrid quantum system composed of an ion and an electric dipole. We show how a trapped ion can be used to measure the small electric field generated by a classical dipole. We discuss the application of this scheme to measure the electric dipole moment of cold polar molecules, whose internal state can be controlled with ultrafast laser pulses, by trapping them in the vicinity of a trapped ion.

Received 6 June 2013
Revised 22 August 2014

DOI:https://doi.org/10.1103/PhysRevA.91.012504

Authors & Affiliations

Jordi Mur-Petit^*

Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA

Juan José García-Ripoll

Instituto de Física Fundamental, IFF-CSIC, Serrano 113 bis, E-28006 Madrid, Spain

^*Corresponding author: jordi.mur@csic.es

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Vol. 91, Iss. 1 — January 2015

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Images

Figure 1
Scheme of the system. An ion of mass $m$ and charge $q$ (green dot with “+”) is confined in an ion trap with a “stylus electrode” geometry [29] (gray cylinder) while a polar molecule (orange oval) of mass $M$ and EDM $μ$ (thick arrow) is trapped by a focused laser (blue shaded area) a distance $z_{0}$ below. $ω$ and $Ω$ stand for the corresponding trapping frequencies.
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Figure 2
(a) Accumulated phase for a $^{40} {Ca}^{+} + CaH (X^{2} Σ)$ system with $ω / 2 π = 1$ MHz, $Ω / 2 π = 1$ kHz, $T = 150$ ns, $Ω_{Rabi} / 2 π = 300$ MHz, $Δ_{Ca} / 2 π = 0.8$ GHz, and $Δ_{CaH} / 2 π = 330$ MHz. Blue filled circles are calculated for $z_{0}^{'} = 20 μ m$ , red squares for $z_{0}^{'} = 30 μ m$ ; solid lines are sinusoidal fits. (b) As (a), with $μ = μ_{CaH}$ , as a function of ion-dipole distance. (c) The same for $^{40} {Ca}^{+} +^{40} K^{87} Rb (X^{1} Σ^{+})$ with $T = 100$ ns, $Ω_{Rabi} / 2 π = 500$ MHz, $Δ_{Ca} / 2 π = 1$ GHz, $Δ_{KRb} / 2 π = 250$ MHz, and traps as in (a). (d) As (c) as a function of pulse duration with $μ = μ_{KRb}$ and $z_{0}^{'} = 20 μ m$ . Here, $z_{0}^{'}$ is the actual ion-dipole distance including the displacement due to the IDC discussed in Appendix pp1.
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Figure 3
Instantaneous EDM $μ (t)$ for a free molecule (dashed blue line) and for a molecule subject to $π$ pulses at times $t = {1 / 5, 3 / 5, 1} \times (2 π / 6 B_{rot})$ (solid red). The horizontal dotted line indicates the average value $\bar{μ} \approx 0.76 μ$ .
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Figure 4
Time evolution of the initial state $| χ (t = 0) 〉 = | + σ_{x} 〉$ represented on the Bloch sphere (yellow) in spin space, under the sequence of pulses described in Appendix pp2. The position of the state vector is indicated each time by a thick red line. (a) Initial state; (b) state just before the first $π$ pulse at time $t = T_{rot} / 5 - δ (0 < δ ≪ T_{rot})$ ; (c) just after the first $π$ pulse at time $t = T_{rot} / 5 + δ$ ; (d) at time $t = 2 T_{rot} / 5$ : it has come back to $| + σ_{x} 〉$ ; (e) at time $t = 3 T_{rot} / 5 - δ$ , just before the next $π$ pulse. $T_{rot} = h / E_{rot}$ is the time for a full rotation in the absence of pulses; cf. Eq. (A3).
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Figure 5
Model five-level system. The two rotational states used to construct a state with nonzero EDM are $| X, J = 0 〉$ and $| X, J = 2 〉$ . These states are coupled by two lasers in a Raman scheme (colored arrows) via $| A, J^{'} = 1 〉$ . These three states form the Hilbert space of interest, indicated by the shadowed box. In the electric-dipole approximation, state $| X, J = 2 〉$ can leak outside this space to $| A, J^{'} = 3 〉$ , and this to $| X, J = 4 〉$ . Relevant laser frequencies $ω_{1 (2)}$ , Rabi frequencies $Ω_{m k}$ [Eq. (C2)], and detunings $Δ_{1 (2)}$ are also identified. Note that the nonharmonic character of the rotational spectrum renders the detunings and, hence, Rabi frequencies for transitions to and from $| A, J^{'} = 3 〉$ generally different from those to and from $| A, J^{'} = 1 〉$ .
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Figure 6
Numerical evolution of the five-level system described by Eq. (C1) starting with all population in state $| 0 〉$ (solid blue line) using the parameters in Table 2. At the end of the simulation, $\approx 92 %$ population is in $| 2 〉$ (dashed green line), while less than $\approx 3.7 %$ has remained in $| 0 〉$ and $\approx 3.9 %$ has leaked to $| 4 〉$ (solid purple line). After the pulses, population in states $| 1, 3 〉$ (solid red and orange lines) is negligible due to the large value of the detuning. The gray line with small wiggles shows $10^{3} | 1 - N (t) |$ , with $N (t)$ the total population at time $t$ . The dashed Gaussian profiles centered around $t \approx 4$ stand for the laser intensity profiles (in arbitrary units).
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