36.1 Introduction

Laser spectroscopy (LS) has a long history and many applications in fundamental and applied science. Determining the transition frequency between electronic states in isolated atoms using LS represents the most accurate measurement of a natural constant we can currently perform. Today, these measurements approach 18 significant digits in optical clocks and can no longer be stated in hertz, since the unit hertz (based on a hyperfine transition in atomic cesium) can only be realized with an uncertainty at the 16th digit by the best Cs primary standards. Why is it interesting to perform spectroscopy with such a high resolution? In the past, theoretical breakthroughs have often been triggered by experimental observations that were incompatible with the understanding of nature at the time. A prime example is the development of quantum mechanics and quantum electrodynamic (QED) theory that goes hand in hand with the refinement of spectroscopic resolution. This led to the discovery of the Fraunhofer lines in the spectrum of the Sun and the Lamb shift in hydrogen. Today, improved spectroscopic resolution and access to previously inaccessible systems probe our understanding of nature through comparison with theoretical predictions. Examples include tests of QED theory, Einstein's theory of relativity, and the Standard Model of particle physics. Any experimentally observed deviation from theoretical predictions may shed light on some of the “grand questions” in physics: the apparent incompatibility of gravity with the other three fundamental forces, the observed asymmetry between matter and antimatter, and the origin of dark energy and dark matter. Spectroscopic tests at the highest level can provide guidance toward answers to these questions. Examples include the search for dark matter and dark energy candidates that couple weakly to normal matter and change the energy levels, tests of parity violation and the search for an electric dipole moment of the electron through precision spectroscopy of molecules, or probing for a violation of Einstein's equivalence principle through spectroscopic tests of local Lorentz and local position invariance.

What does all of this have to do with this book on quantum information? LS and quantum information processing (QIP) actually have a lot in common. Both fields require isolation of the system of interest (may it be atoms, ions, or molecules) from environmental perturbations, such as electromagnetic fields, in order to achieve long coherence times (QIP) or to measure the unperturbed transition frequency (LS). Excellent control over the internal (electronic) and external (motional) degrees of freedom of the system using laser cooling and Ramsey or Rabi pulse sequences is at the heart of both fields. More recently, advanced quantum control techniques that have been developed in the context of QIP with trapped ions have been transferred to precision spectroscopy. A prominent example is quantum logic spectroscopy (QLS) that enabled the investigation of previously inaccessible ion species. By trapping a spectroscopy ion simultaneously with a so‐called logic ion, the latter provides sympathetic laser cooling and internal state readout mediated through the strong Coulomb interaction between the ions and controlled via laser pulses. This approach eliminates the need for a laser cooling and detection transition in the spectroscopy ion and thus extends the list of possible candidates for precision spectroscopy by many species with interesting properties. Two ions form a compound system that combines the advantages of both species. A similar approach has been proposed for large‐scale quantum information processors, where “memory” and “computational” qubits can be realized by two different ion species.

We will start in Sections 36.2 and 36.3 with a brief introduction to the physics of two‐ion crystals and their manipulation using lasers. In Sections we will present different implementations of QLS for narrow optical clock transitions, broad dipole‐allowed transitions, and internal state detection of a molecular ion, respectively. Another example of the fruitful synergy between LS and QIP is the advent of employing entangled or other nonclassical states to improve resolution or accuracy in spectroscopy. We will discuss the elimination of magnetic field shifts by entangling two ions and the potential use of nonclassical motional states and entangled spin states for enhancing the resolution of optical spectroscopy in Section 36.7. Finally, future directions for quantum logic‐enabled spectroscopy will be briefly discussed in Section 36.8.

36.2 Trapping and Doppler Cooling of a Two‐Ion Crystal

For most of this chapter we consider a crystal composed of two singly‐charged ions of different species, confined in a linear Paul trap as shown in Figure 36.1 (1–3). In this context “crystal” refers to the fact that the ions are well localized and aligned along the axial direction. Localization is achieved through Doppler‐ or even ground‐state cooling (GSC) with laser light interacting with at least one of the two ions. An oscillating radial electric quadrupole field provides radial confinement along the directions, and a static 3D static quadrupole field confines the ions axially ( , corresponding to the symmetry axis of the linear Paul trap). A single trapped ion performs motion at two different timescales: a fast micromotion with small amplitude, driven by the oscillating radial quadrupole field, and a slower secular motion. We assume that the micromotion amplitude is significantly smaller than the secular motion's amplitude and that the frequencies are sufficiently spectrally separated. The distance between the ions is only determined by their charge and is thus independent of their mass. For typical trap parameters with secular frequencies on the order of a few megahertz, the distance between the ions is a few micrometers. If we assume small secular oscillation amplitudes (compared to the ion's separation), the nonlinear Coulomb interaction can be linearized. In such a situation micromotion can be largely neglected, and the trap provides harmonic confinement in all three directions with trap frequencies for a single ion. For two ions, their motion is strongly coupled, and a normal mode description for the motion of ion around its equilibrium position applies (4,5). Each of the six normal modes of a two‐ion crystal is described by a mode frequency and an eigenvector with the normalization condition . The factors determine the oscillation amplitude for ion . The quantized motion of each ion with mass is therefore described by a superposition of all modes:

36.1

where denotes the lowering (raising) operator of mode , and is the extent of the ground state of the ion's wavepacket for this mode. Explicit expressions for the mode frequencies and amplitudes for a two‐ion crystal are given by Wübbena et al. (6). The two modes along each direction can be separated into an in‐phase (IP) ( , ions moving in the same direction) and an out‐of‐phase (OP) ( , ions moving in opposite directions) mode. The oscillation amplitudes for each ion depend on the mass ratio . In the case of equal masses ( ), the two mode vectors corresponding to the IP and OP mode are and . For large deviations from , the ions behave as almost independent oscillators, so that for each of the two modes in a given direction only one of the two ions has a large motional amplitude. This effect is particularly pronounced for the radial modes and results in an elevated steady‐state temperature in the presence of additional heating when only one of the two ions in a two‐ion crystal with unequal mass is laser cooled (6). The physics behind this effect can be understood from the following argument. In standard Doppler cooling theory, the cooling and heating rates from photon absorption and emission both scale with the square of the amplitude of the cooling ion's motion. Therefore, the final temperature of the modes in a two‐ion crystal is always the Doppler cooling temperature. However, reaching this equilibrium temperature takes longer for . Fluctuating electric fields can lead to external heating. Since they are believed to originate from the trap electrodes, which are far away compared to the inter‐ion distance, the electric field of this noise can be assumed to be homogeneous across the ion crystal. Therefore, modes with a dominant IP character are heated stronger than modes with an OP character. The fluctuating fields add to the rate equation for cooling with a heating rate independent of . This results in an elevated steady‐state temperature for the modes for which the cooling ion has a small amplitude (weakly cooled modes). As a rule of thumb, sympathetic cooling for a two‐ion crystal works for mass ratios between 3 and . It should be noted that it is advantageous to have the lighter ion as the cooling ion, since in this case the modes with the strongest susceptibility to heating (most IP character) are the ones with the largest amplitude for the cooling ion.

Nonlinearities in the Coulomb interaction can lead to coupling, for example, between radial and axial OP modes (7). Strong radial OP motion modulates the axial distance between the ions, resulting in increased heating and loss of coherence for a cold axial OP mode. More details on mixed‐species ion crystals can be found in the reviews (2, 3).

36.3 Coherent Atom–Light Interaction and State Manipulation

Since the seminal idea of Cirac and Zoller (8), coherent manipulation of internal and motional degrees of freedom, is at the heart of QIP and has been extended by Wineland and others toward quantum logic‐enabled spectroscopy with trapped ions ( 2,9).

Following standard textbooks and reviews ( 2,10), let us consider a single trapped two‐level atom with long‐lived internal states connected through a transition with linewidth and energy . The ion's motion for a selected mode with frequency is described by the quantum harmonic oscillator with eigenstate and raising operator . The full Hamiltonian of the system interacting with a monochromatic light field of frequency , phase , and projection along is given by the sum of the atomic, motional, and interaction Hamiltonian: . Using Pauli spin matrices and neglecting the zero point energy of the motion, the individual components can be written as:

where and are the position operator and the size of the ground‐state wavefunction of the ion in the mode, respectively. The coupling of the light field to the internal states is mediated by the atomic raising/lowering operators with a strength characterized by the Rabi frequency . In the interaction picture with respect to the atom and motion, and after a rotating wave approximation (RWA) with respect to the detuning , we get for the interaction Hamiltonian:

36.2

where we have introduced the Lamb‐Dicke factor that relates the wavelength of the laser with the spatial extent of the ion. For optical radiation and typical ion wavepacket sizes of tens of nanometers for laser‐cooled ions, . The coupling between motional states and is given by the matrix element

36.3

where ( ) is the lesser (greater) of and , the change in motional quantum number, and the generalized Laguerre polynomial.

The time evolution under this Hamiltonian can be analytically solved when we assume resonant interactions ( ) and by neglecting coupling to other levels ( ), which is also called the strong‐binding or resolved sideband regime (see Section 36.5.1). The interaction results in Rabi flopping between the states . Writing the state as a vector with amplitudes , the interaction implements rotations of the state vector according to the rotation matrix

36.4

with , where .

Quantum logic operations are typically performed near the ground state of motion, for which the size of the wavepacket is small:

36.5

In this case we can apply the so‐called Lamb‐Dicke approximation in which the exponential in Eq. 36.3 is replaced by the first two terms of its series expansion:

By tuning the frequency of the laser, the interaction can become resonant with either of the three terms in curly braces. The first term, (after RWA with ), corresponds to the so‐called carrier transitions (CAR) that change the electronic state of the ion without affecting the motion ( ). By tuning to , the second term in the Hamiltonian becomes resonant, (after RWA), and we get red sideband transitions (RSB) that remove a quantum of motion ( ) when changing the internal state from . Similarly, we get blue sideband transitions (BSB) that add a quantum of motion ( ) when changing the internal state by tuning the laser to , resulting in the Hamiltonian (after RWA). The Rabi frequencies of sideband transitions and are reduced compared to the Rabi frequency of carrier transitions by the Lamb‐Dicke factor and enhanced by the initial motional state . In cases for which the Lamb‐Dicke approximation is not strictly fulfilled, this simplified scaling is modified, and higher order sidebands become possible according to the matrix element in Eq. 36.3. Deep inside the Lamb‐Dicke regime ( ), the recoil of a single photon does not change the motional state of the ion, since the recoil energy is smaller than the energy of a harmonic oscillator excitation. The absence of recoil shifts is one of the main motivations for performing precision spectroscopy of trapped atoms. As we will see in Section 36.5, the detection of residual recoil can be used for a novel spectroscopy technique.

Pulse sequences can be constructed by multiplying the appropriately extended rotation matrices that operate in a Hilbert space including the atomic spin and a truncated Fock state basis. It is worthwhile noting that the phase depends on the position of the ion within the light field. It is therefore different for different ions and can be arbitrarily chosen (e.g., to be zero) for the first pulse but needs to be kept track of for all ions and subsequent pulses.

The coupling between light and atoms can deviate in real systems from this simplified picture in many ways. Changes to the spatial distribution of the ion's wavepacket in the trap can reduce the coupling to light. Common examples are micromotion and the other so‐called spectator modes that are not involved in the coupling (2). In both cases, the reduced coupling can be understood as either a reduction of the carrier from sideband modulation by the spectator mode or a smearing out of the ion's wavepacket, similar to the Debye–Waller effect in X‐ray scattering of solids.

36.4 Quantum Logic Spectroscopy for Optical Clocks

36.4.1 Introduction to Optical Clocks

Clocks or absolute frequency measurements in general are characterized by two properties: stability and accuracy. Stability is a measure of the statistical uncertainty in determining the transition frequency. Most clocks are operated in a regime where they are quantum projection noise limited. In this case the two‐sample variance (the so‐called Allan deviation) of the instability for a given number of atoms and averaging time is given by (11)

36.6

It is clear from this equation that high (optical) frequencies , long interrogation times , and short cycle times are desirable to reach a low‐frequency uncertainty as fast as possible. While the interrogation time is fundamentally limited by the lifetime of the excited clock state, the practical limitation in optical clocks is given by the coherence time of the clock laser (12).

Accuracy determines how well the unperturbed transition frequency is realized. The accuracy of high‐performance optical clocks is based on estimating all possible line‐shifting effects and the associated uncertainty of these shifts. It is plausible that clock candidates in which all major shifts are small to begin with offer great potential for highest accuracy.

The transition in is such a case: its clock transition at 267 nm has a narrow linewidth of 8 mHz and is free of electric quadrupole shifts. It has only a weak nuclear Zeeman shift (13) and features the smallest black‐body radiation (BBR) shift of all investigated atomic species (14). These properties make an excellent candidate for a high‐accuracy optical clock. However, the cooling transition in is at a wavelength of 167 nm and thus not yet accessible with commercial cw laser systems. Therefore, QLS based on coherent manipulation of long‐lived states was developed to probe transitions in . Sympathetic cooling and internal state preparation and readout are provided through a co‐trapped logic ion (15–18). The same technique is applicable to all trapped ions with narrow spectroscopy transitions, such as highly charged ions, molecular ions, or even (anti‐)protons. In the following section we will describe the main interrogation protocol and address state preparation techniques before providing a brief summary of the clock features.

36.4.2 Quantum Logic State Mapping

The simplest quantum logic interrogation sequence is shown in Figure 36.2 and starts with a two‐ion crystal cooled to the ground state of motion in one of the motional modes (index ) and clock or spectroscopy (index ) as well as logic (index ) ion prepared in their electronic ground state . The clock transition is typically probed by Rabi spectroscopy using a single long pulse that implements, for example, a rotation,⁰¹ leaving the clock ion in general in an electronic superposition state

where describes a rotation of the spectroscopy ion's spin around the ‐axis ( ) with an angle , given by the Rabi frequency and interrogation time according to the expressions following Eq. 36.4. For clock operation the length of the pulse is chosen such that the rotation is near , where the signal‐to‐noise ratio (SNR) is close to its maximum in the presence of technical noise. In conventional clocks, internal state detection at this point would provide the excitation probability. In QLS, a RSB ‐pulse converts the electronic superposition into a motional superposition

In quantum information language this corresponds to a SWAP operation between electronic and motional states on the spectroscopy ion for the prepared state. A similar SWAP operation on the logic ion maps the motional superposition back into an electronic superposition on the logic ion

Detection of the logic ion's internal state using state‐selective fluorescence provides the excitation probability of the clock ion during the first pulse. Figure 36.3 shows a QLS resonance scan and resonant Rabi flopping on the transition in . A similar sequence is used for the clock transition (see below). While internal and motional states are entangled during the RSB transfer pulses, the coupled quantum system factorizes at the end of the pulses. Although not relevant for QLS, the entire state transfer sequence is phase coherent. This has been demonstrated in a Ramsey experiment in which the first pulse is applied to the spectroscopy ion. The state is then mapped onto the logic ion where the second Ramsey pulse is applied. The full sequence reads (from right to left) . Interestingly, no phase coherence between the logic and spectroscopy laser is required, since the laser phase of the two pulses on each ions cancels. The proof is left as an exercise for the reader.

36.4.3 Quantum Logic‐Enabled Internal State Preparation

Internal state preparation in conventional clocks is typically performed using polarization effects to induce optical pumping. In , this is in principle also possible using ( )‐polarized light on the short‐lived ( s) transition to prepare the ( ) ground state and is in fact employed in clock operation. Population accumulation always requires dissipation, which in this case is provided by spontaneous emission. However, quantum logic enables a very powerful state preparation technique in which coherent manipulation of the spectroscopy ion is combined with dissipation on the logic ion. The basic principle of quantum logic assisted state preparation is shown in Figure 36.4. The coherent manipulation on the spectroscopy ion consists of a sequence of CAR and RSB pulses, followed by GSC on the logic ion. This last step is dissipative and makes the entire sequence irreversible, since an RSB from the ground state leaves any of the electronic ground state levels untouched. By combining a suitable sequence of such pulses, any of the levels can be prepared with high probability, as shown in Figure 36.4. This is a very versatile state preparation technique that has important applications, for example, in the preparation and detection of internal states of molecular ions as discussed in Section 36.6 .

36.5 Photon Recoil Spectroscopy

The original QLS scheme requires sufficiently long‐lived excited states to implement the state mapping sequence. Therefore, only transitions with a linewidth below kHz can be investigated using this technique. However, there are many other spectroscopically interesting transitions with broader linewidths in ions that do not possess a suitable transition for laser cooling. Examples include metal ions of astrophysical interest, such as Fe , Ti , and many others. Line strengths and isotopic shifts are being used to calibrate solar cycles and in astrophysical searches for a possible variation of the fine‐structure constant (21). In nuclear physics, precision isotope shifts of dipole‐allowed transitions have been employed to reveal information about atomic and nuclear structure (22,23). The level structure of many of these ions is so dense that no fast cycling transition is known. In the past, spectroscopy has therefore been performed in gas discharge cells or collinear LS with a resolution of a few megahertz at best.

A variant of QLS has therefore been developed to investigate broad and possibly nonclosed transitions with high resolution and accuracy. Since the state amplitudes induced during the spectroscopy laser pulse are lost after spontaneous emission, state mapping can no longer be implemented. Instead, the change in the motional state from photon recoil upon absorption of a spectroscopy laser photon is used as a signal that is detected via the logic ion. In analogy to Dehmelt's “shelved optical electron amplifier” (24), one can view this technique as a “photon recoil signal amplifier,” in which the signal (change in motional state from absorbing a few photons) is amplified by mapping it onto the logic ion, where it turns into thousands of photons being scattered or not.

To understand the technique, we first need to develop a quantum mechanical picture of the motional effects during photon absorption. We will then discuss the basic principle of the technique and provide high‐resolution photon recoil spectroscopy (PRS) of the line as an example.

36.5.1 Absorption of Single Photons

In the derivation of atom‐light interaction in Section 36.3 we have made specific choices for the involved energy scales . We will now try to provide an intuitive picture for the general case of photon absorption and the different regimes. We are particularly interested in possible line‐shifting effects.

It is instructive to look at the absorption cross section of a trapped two‐level atom in motional state that absorbs photons from a light field of frequency directed along the ‐direction. In the weak excitation limit, it is given by (25)

36.7

Here, the energy levels of the harmonically bound atom are given by . The cross section is the sum over Lorentzian resonances of linewidth , spaced by the harmonic oscillator levels. Each resonance is weighted by the transition matrix element between motional levels given by Eq. 36.3. The full absorption cross section is the sum over the cross sections for individual initial motional states , weighted by their population .

We distinguish several limiting regimes that can be realized in different combinations.

• Classical regime : no ground‐state harmonic oscillator effects; shift of absorption cross section only from possible recoil effects
• Quantum regime : effects from the bounded ground state in the harmonic oscillator are observable in an asymmetry of the absorption cross section toward high‐energy values (RSB transitions from the ground state are forbidden)
• Strong‐binding or resolved sideband regime : absorption spectrum is peaked near carrier and sideband resonances; by tuning the laser frequency, a single sideband/carrier transition can be excited
• Weak‐binding regime : absorption spectrum is continuous, and several transitions may be excited simultaneously, resulting in a shifted (asymmetric) absorption profile in the free atom (Lamb‐Dicke) limit
• Lamb‐Dicke limit : sideband transitions are suppressed; photon recoil does not shift the absorption spectrum; however, the absorption lineshape may still be skewed toward higher energy in the quantum regime
• Free atom limit : the absorption spectrum is shifted by photon recoil.

While quantum logic operations and optical clocks are operated in the resolved sideband and quantum regime in the Lamb‐Dicke limit, we are now interested in the weak‐binding and quantum regime in the Lamb‐Dicke limit. In this regime, the excitation resonance for carrier and sidebands overlap, and the probability to change the motional state upon photon absorption is small, since we are in the Lamb‐Dicke regime (see Figure 36.6). As can be seen from the interaction Hamiltonian Eq. 36.2, absorption of a single photon induces a recoil kick corresponding to a displacement of the ion's wavepacket by , where is the displacement operator and the displacement phase with respect to the motion of the ion in the trap. It is easy to check that starting from the ground state of motion, the expectation values of momentum squared and energy after photon absorption are given by and , respectively. However, it should be noted that these are ensemble‐averaged values. This means that most of the time, no motional excitation will be measured, but once the motional state changes, the change is at least a gain in one quantum of excitation (which has a much larger energy than . This is a direct consequence of operating in the Lamb‐Dicke regime. Spontaneous emission will again occur dominantly on the carrier transition. Recoil kicks from emission average with a directionality corresponding to the emission pattern for this transition.

36.5.2 Principle of Photon Recoil Spectroscopy

As in the case of the original QLS technique, we start with a two‐ion crystal that has been cooled to the ground state of motion of at least one normal mode using the logic ion (Figure 36.7(a)), which we assume to be the axial IP mode with frequency . Spectroscopy is performed by applying short (compared to the oscillation time of the ion in the trap, pulses of light along the direction of the ground‐state cooled normal mode. The probability of photon absorption, and thus a change in the motional state, is proportional to the absorption cross section (Figure 36.6). However, the recoil kick is now distributed between the two axial normal modes according to the mode amplitudes and and Lamb‐Dicke factors and of the spectroscopy ion. The pulses are applied synchronously to the oscillation period of the ion in the trap. Therefore, recoil kicks are always exerted during the same phase of the oscillation of the IP mode, resulting in a motional state resembling a displaced vacuum state after absorption of photons. The corresponding population distribution is illustrated in Figure 36.7(b). The ground‐state population for such a state is given by . It is interesting to note that the ground‐state depletion is quadratic in the number of absorbed photons. Since the pulse repetition rate is not synchronized with the OP motion, the effect of the photon recoil onto this mode averages to zero. However, the variance of the momentum in this mode scales as , which results in a diffusive excitation. Similarly, recoil kicks from spontaneous emission result in an additional diffusion of the wavepacket in phase space (26).

Since the motional modes are shared between spectroscopy and logic ion, the remaining ground‐state population is detected by transferring all motionally excited population into the state of the logic ion (Figure 36.7(c)) through an RSB ‐pulse implementing . According to Eq. 36.3, the Rabi frequency for this pulse depends on the initial motional state. Therefore, a rotation cannot be implemented for all population simultaneously using Rabi excitation. However, the population can be transferred for all states simultaneously in an adiabatic scheme such as stimulated Raman adiabatic passage (STIRAP) using delayed pulses (27,28). State‐selective fluorescence detection on the logic ion as illustrated in Figure 36.7(d) provides the measurement signal.

36.5.3 Example for Photon Recoil Spectroscopy

Figure 36.8 shows the setup and results for PRS of the transition in (29) with a natural linewidth of 21.6 MHz. In the experiment, 70 spectroscopy pulses with 50‐ns duration, each followed by a repump pulse to clear out the population, are applied before the motional excitation is detected on the logic ion using an adiabatic passage pulse (28). Each frequency point is averaged around 250 times. The observed linewidth of 34 MHz is a consequence of Fourier broadening due to the short spectroscopy pulses and Zeeman shifts of the magnetic substates from the applied magnetic field of 0.584(1) mT. The red line in Figure 36.8(c) is a numerical simulation of the Master equation using a truncated motional Fock state basis, corrected for experimental imperfections, such as reduced signal contrast and offset. The agreement with experimental data demonstrates full control over all relevant degrees of freedom.

The sensitivity of the technique can be calibrated using the signal level achieved when the metastable state is not cleared out as a reference. From the branching ratio of 14.5 : 1 of the excited state into the and states (30), respectively, we know that the saturated excitation level corresponds to having absorbed 15.5 photons. This can be used to determine that an SNR of 1 is achieved after absorbing 9.5 photons for a Lamb‐Dicke parameter of . Compared to laser‐induced fluorescence spectroscopy, this is a several orders of magnitude improvement. The high photon sensitivity is a consequence of measuring absorbed instead of scattered photons in a background free implementation, the favorable scaling of motional ground‐state depletion, and the near unity detection efficiency of motional excitation on the logic ion. Even higher sensitivity can be achieved by making the Lamb‐Dicke factor larger or using nonclassical states of motion as demonstrated with motional Schrödinger‐cat states in Ref. (31) and discussed in Section 36.7 .

PRS has been used to perform several absolute frequency measurements on different isotopes and transitions ( 22,32) using spectroscopy lasers locked to a frequency comb that was referenced to a calibrated hydrogen maser at PTB. As in the case of optical clocks, the accuracy of these measurements is determined by all possible line‐shifting effects. In the described experiment, all relevant shifts, such as AC‐Stark or Zeeman shifts, should be ideally zero and have been measured to be well below 100 kHz. Care needs to be taken to avoid introducing a frequency shift when generating the short pulses with, for example, an acousto‐optic modulator (AOM) with a limited bandwidth. In case the AOM bandwidth is small and not centered with respect to the radio‐frequency pulse bandwidth used to drive the AOM, shifts on the order of 1 MHz can be induced. For unpolarized atoms and linear spectroscopy and repump light polarization, the lineshape of the resonance deep in the Lamb‐Dicke and classical regime is symmetric. However, since we start from the motional ground state, the lineshape is asymmetric for the first absorbed photon due to truncation effects as shown in Figure 36.6. The situation is even more complicated, since the absorption cross section changes for subsequent photons. In this situation heating and cooling effects need to be considered. This is a problematic systematic effect for laser‐induced fluorescence spectroscopy (33). In our situation, a theoretical analysis shows that heating effects for a blue‐detuned spectroscopy laser generate a slightly stronger motional excitation compared to a red‐detuned spectroscopy laser (29). The resulting center frequency shift is on the order of and can be subtracted from the measured frequency with high accuracy, resulting in a total inaccuracy below 100 kHz. This measurement resolution is achieved after averaging typically between 10 and 15 min using the two‐point sampling technique introduced in Section 36.4.4.

36.6 Quantum Logic with Molecular Ions

While PRS relies on scattering of a few photons, there are many other systems with complex level structure in which spontaneous scattering of even single photons is detrimental. This is especially true for molecules that have rovibrational structure. Spontaneous emission from an electronic excited state typically changes the rovibrational state. While molecular spectroscopy in beams or colliding molecules in a gas cell has been very successful and found widespread applications, high‐resolution molecular spectroscopy of a trapped sample remains a challenge. QLS of molecular ions promises to achieve this goal by implementing controlled coherent operations on a trapped molecular ion and operations involving the scattering of photons, such as cooling, state preparation, and detection, on the co‐trapped logic ion. In the following, we will present a scheme for nondestructive detection of the internal state of a molecular ion and the deterministic preparation of a particular quantum state.

36.6.1 Nondestructive State Detection

Spontaneous scattering during coherent interaction between laser light and a trapped molecule can be avoided in the dispersive regime using off‐resonant laser beams. While the off‐resonant scattering rate in a two‐level system with linewidth scales as , where is the resonant Rabi frequency and the detuning from resonance, the effective Rabi frequency describing the strength of coherent interaction scales as . The ratio of the two can be made arbitrarily small while maintaining the same by increasing and . The remaining effect on the molecule in this situation is an AC‐Stark shift, which can be turned into an optical dipole force by introducing a gradient. In particular, we can implement an oscillating dipole force through a moving optical lattice, implemented by two counterpropagating laser fields with a large common detuning with respect to an electronic resonance and a small relative detuning , corresponding to the oscillation frequency. When tuned to one of the motional modes of the two‐ion crystal with Lamb‐Dicke factor , the interaction Hamiltonian after a RWA is given by . In an alternative picture, the two laser fields drive Raman transitions with a common detuning from an excited state, starting and ending at the same electronic state, but detuned by a motional state separation. If the detuning is chosen such that for only one selected initial state , while for all other states , motional excitation only occurs when the molecule is in the selected state (34,35). From the time‐evolution operator , we see that the Hamiltonian implements a displacement operator exciting coherent motion when applied to the motional ground state. This motional excitation can be detected through the logic ion using the techniques developed for PRS (see Section 36.5 ). In the absence of imperfections, this approach allows single‐shot molecular state detection, since the molecule being in the selected state or not is directly mapped to a spin flip on the logic ion, which can be detected with near unit fidelity.

This idea has been implemented to detect the rotational state of the molecular ion with as the logic ion (36). This state has been chosen, since its resonance is spectrally separated from all other molecular resonances. In addition, the average dwell time of the molecule in this state is on the order of a few seconds, before it is driven to other states via BBR. Since the optical resonance of the atomic ion is only 1.5 THz away from the molecular resonance, and coupling to the atom is much stronger compared to the molecular coupling between the two rotational states, we get an appreciable dipole force also from . This results in a partial spin rotation on the logic ion, which introduces quantum projection noise and thus requires averaging for state discrimination. In principle, this can be avoided by implementing a CNOT gate between the molecule and the atom that takes into account the nonresonant light interaction contributions from both. One possibility to implement an interaction that realizes a CNOT operation (at least for some initial states) is based on a motional qubit that allows Rabi flopping between two motional Fock states without changing the electronic state. We use a single excitation of either the IP or OP axial mode as our motional qubit basis . A lattice field (oscillating dipole force) that is tuned to the difference frequency of the two modes induces transitions between the qubit states. This interaction restricts time evolution to these two states that act as a spin‐ system and enables coherent qubit manipulations. The interaction Hamiltonian for the motion in the RWA in this situation is given by

where the index indicates the interaction with the molecular spectroscopy ion (logic ion). Since and have the same mass, the Lamb‐Dicke factors are symmetric, that is, and . We define a relative phase between molecule and atom, corresponding to their distance and scaled with the wavevector difference between the two light fields . This definition assumes that the dipole force fields are applied along the axial direction and that the detuning of the light field with respect to the atom and molecule has the same sign. The relative phase determines how the forces from the atom and molecule are added up; , where is an integer that results in a coherent addition of the two forces, while the forces subtract for . Neglecting a global phase, the interaction Hamiltonian becomes

where we have defined an effective Rabi frequency

Here, is the Rabi frequency ratio between the molecule and the atom. This Hamiltonian induces Rabi flopping with Rabi frequency between the states and by having a single motional excitation oscillate between the IP and OP mode. The detected spin‐down population on the logic ion follows

36.8

It is a function of the Rabi frequency ratio and the relative phase between the atomic and molecular ion. In a Bloch sphere picture, the optical lattice‐induced dipole forces from the atom and the molecule act as torques on the motional qubit state vector that have to be added vectorially to yield (see Figure 36.9(b,c)).

The circuit diagram for the gate‐based quantum algorithm for internal state detection is shown in Figure 36.9(a). It starts with both axial modes cooled to the ground state via the logic ion. The state is initialized by driving a BSB pulse addressing the IP mode on the logic ion (step (i), . The Rabi frequency and interaction time of the moving optical lattice are chosen to implement a rotation in the absence of coupling to the molecule (Figure 36.9(a), step (ii), and (b)). The sequence ends by applying a second BSB pulse addressing the IP mode on the logic ion (step (iii), , followed by internal state detection (step (iv)). In the situation described, the final atomic state is , since the last BSB pulse cannot change any states. In case the molecular ion is in the selected rotational state and couples to the moving optical lattice, the interaction changes the rotation of the motional qubit as shown in Figure 36.9(c). For single‐shot state detection it can either be chosen to cancel the atomic interaction or to induce a rotation . Now the final BSB pulse changes the internal state of the logic ion . As is the case for a traditional CNOT gate, the internal state of the atom is changed depending on the state of the molecule.

The detected signal on the logic ion depends on the relative Rabi frequencies between atomic and molecular ions. The strong dependence of the molecular ion's Rabi frequency on the detuning from resonance can be used to perform QLS of a broad optical transition in a molecular ion. The atomic Rabi frequency can be considered constant over the range of relevant detunings. For each detuning the signal height on the logic ion is recorded and Eq. 36.8 is solved for . The result of several such measurements is shown in Figure 36.10. The inset shows a typical time trace of the molecular ion entering the rotational state and leaving it again. The dynamics of entering and leaving the target state is given by the interaction of BBR with the rotational level structure and residual off‐resonant excitation from the laser interaction. For performing high‐resolution spectroscopy, it would be desirable to have a deterministic state preparation procedure for the molecular quantum state available.

36.7 Nonclassical States for Spectroscopy

In the previous sections it was shown how quantum logic techniques for coherent manipulation of internal and external degrees of freedom can be used to implement new spectroscopy schemes. In these schemes, entanglement between spin and motion is present only as a transient effect during sideband pulses. From optical interferometry it has been known since the 1970s that correlations between photons, for example, in the form of squeezed (39) or NOON (40) states can improve the SNR beyond the classical limit (defined as the absence of correlations). In close analogy to these concepts for photons, correlated spin states have been proposed and implemented to measure inertia, external fields, and frequencies. See Ref. (41) for a recent review. Here, we want to discuss a few specific examples related to frequency metrology, some of which are discussed in more detail in (11).

Let us consider a Ramsey‐type spectroscopy experiment, in which two pulses on a clock transition are separated by a waiting time . In a Bloch sphere picture in a rotating frame with the transition frequency , the first pulse prepares the spin state of the atom to point along the equator. During the free precession time, a differential phase evolution between the excitation laser with frequency and the internal spin evolution results in a state . After the second pulse, the final state reads . Internal state detection provides information about the differential phase evolution modulo , from which the frequency error of the laser can be estimated. The longer the probe time , the larger is the slope. If uncorrelated atoms are interrogated simultaneously, the SNR improves by . By employing generalized Ramsey pulses in the form of phase gates that entangle all atoms into a so‐called Greenberger–Horne–Zeilinger (GHZ) state, the phase evolution becomes . The final state after the Ramsey sequence reads and exhibits an ‐times faster phase evolution. This means that for an ideal experiment, the SNR is enhanced by (compared to a single particle measurement), reaching the Heisenberg limit. Unfortunately, optical clocks are currently limited in their interrogation time by the nonwhite frequency noise of the interrogation laser. The frequency noise results in phase deviations that are larger than for probe times longer than the optimal time. GHZ states accelerate the phase evolution, thus reducing the optimal probe time, eliminating the entire quantum advantage (12). However, other forms of correlated states, such as spin squeezed states, reduce quantum projection noise and can in principle provide Heisenberg‐limited instability that scales with (see Eq. 36.6). While GHZ states of optical qubits have been created, for example, in trapped ions, squeezing has only been implemented on hyperfine clock states (11).

Apart from improving optical clocks, entangled states can be very useful in differential frequency measurements, in which the laser noise drops out. A beautiful example is the “designer atom” spectroscopy (42) in which two ions are entangled to create a state that is free of the linear Zeeman effect, but has a phase evolution corresponding to the electric quadrupole shift. This is possible, since the linear Zeeman effect shifts magnetic sublevels according to their projection along the quantization axis, while the quadrupole shift scales with (see Figure 36.11). For example, the entangled state evolves according to the phase . Here, is the electric quadrupole‐induced frequency shift between the magnetic substates and and the indices 1 and 2 are the numbers of the ions. This state is free of the linear Zeeman effect, since the corresponding phase evolution of the constituent states of the two parts of the entangled state exactly cancel. Note the factor of two in the phase evolution, which is the acceleration from entangling two atoms. Using states of this kind, the electric quadrupole moment of ions was measured with unprecedented accuracy (42).

Another field of application for correlated or nonclassical states is in PRS and quantum logic‐enabled molecular state detection. Both are based on detecting small forces through a change in the motional state of a two‐ion crystal. By employing correlated states of phonons, the sensitivity can be significantly improved. For example, motional Schrödinger‐cat states have been implemented to detect the recoil from scattering a single photon off a trapped ion (31). The interferometric sequence shown in Figure 36.12 starts by producing an equal electronic superposition , where the states and are eigenstates of the spin operator of the logic ion. By applying a state‐dependent optical dipole force onto this state, a motional Schrödinger‐cat state of the form with coherent state amplitudes is created (step (1) in Figure 36.12). After the creation of the cat state, spectroscopy light is applied (step (2)), followed by the inverse of the cat creation step (step (3)). If no spectroscopy light was absorbed by the spectroscopy ion, the initial state is recovered. Absorption of a single spectroscopy photon displaces both wavefunction components of the motional state by (see Section 36.5 ). By timing the Schrödinger‐cat state creation and photon absorption appropriately, the displacement from photon absorption is orthogonal to the Schrödinger‐cat displacement as shown in Figure 36.12. When closing the interferometer through the inverse cat creation step, the wavefunctions have enclosed an area in phase space, resulting in a geometric phase that scales with the original cat displacement . Since in principle can be made arbitrarily large, the change in phase upon photon absorption can reach . This way, the absorption of a single photon was detected (31).

Besides Schrödinger‐cat states of motion, squeezed states or Fock states can be employed to enhance force sensitivity and thus enable new applications in PRS and molecular state detection.

36.8 Future Directions

Quantum logic‐enabled spectroscopy has already enabled a number of exciting applications as described in the previous sections. Several groups worldwide are working on quantum logic optical clocks and the implementation of quantum logic techniques for molecular ions. In the future, even more atomic and molecular species with interesting spectroscopic features will become accessible through this technique. Precision spectroscopy of molecular ions may allow to improve bounds on a possible variation of the electron‐to‐proton mass ratio (43) or observe any energy differences between molecules of different chirality arising from parity violation (44). Another exciting application is spectroscopy of highly charged ions. While their strong dipole‐allowed transitions are shifted into the kiloelectron volt regime, fine‐ and hyperfine transitions are shifted into the optical regime. Level crossings between different electronic configurations can also lead to narrow optical transitions suitable for optical clocks with a very high sensitivity to a possible change in the fine‐structure constant (45). First steps toward sympathetic cooling and the preparation of a two‐ion crystal have been achieved (46). Another exciting prospect is QLS of subatomic particles such as (anti‐)protons (47). Sympathetic cooling and quantum logic state readout of protons and their antiparticle improves localization and thus reduces the uncertainty in frequency shifts. At the same time quantum logic spin flip detection after spectroscopy can be significantly faster compared to traditional detection techniques. This may lead to significantly improved bounds on matter/antimatter asymmetry in the baryonic sector.

The quantum technologies developed for quantum computing and simulations will have a direct impact on improvements of quantum logic spectroscopy. The few examples outlined in Section 36.7 are just the beginning. As quantum control improves, we will see more and more examples of quantum logic‐enabled spectroscopy that employs entangled or other nonclassical states to improve SNR beyond the classical limits. In fact, QLS represents one of the first applications of quantum technologies, with a bright future ahead.

Acknowledgments

This work has been partially supported by DFG through project SCHM2678/3‐1, CRC 1128 (geo‐Q), project A03, CRC 1227 (DQ‐mat), projects B03 and B05, by ESA, and by the State of Lower‐Saxony, Hannover, Germany under contract VWZN2927. I would like to thank F. Wolf and S. King for comments on the manuscript.

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