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Abstract
Organic chemistry offers the potential to synthesize individual molecules for specific tasks in quantum information processing. One of their striking properties is the emission of single photons with nearly ideal coherence. Here, we implement Ramsey spectroscopy to measure the electronic state coherence of a single molecule. Conventionally, the emitter’s coherence is characterized by saturation or intensity auto-correlation measurements. However, both methods are under the steady influence of continuous interaction with the excitation laser. This influence can be eliminated by using a pump–probe sequence of two optical pulses to observe the decoherence. We have measured a near-transform-limited decoherence time of 17.6 ns, while the radiative lifetime is 10.1 ns. We also perform frequency-detuned excitation, gaining richer insights into the dephasing behavior of the molecule. The experiments exhibit that optical Ramsey spectroscopy is a promising tool for measuring the emitter’s coherence properties.
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1. INTRODUCTION
The fundamental property in the quantum information processing of a single emitter and its emitted photons is coherence. While coherence has often been observed in emitted photons [1,2], the coherence of the photons’ emitters themselves is rarely studied. Nevertheless, much can be learned from studying the emitter instead of only the photons: a coherence analysis in the solid state reveals insights into the emitters’ relaxation processes and environmental influences [3]. This information can be used for characterization of synthesized molecules, or even as input to a synthetic design process that then optimizes the molecules’ coherence [4].
Coupling to the environment usually affects the effective oscillator strength of the emitter [5]. Coupling to the environment can, on the one hand, be used for the nanoscale sensing of electric fields [6,7], pressure [8], mechanical stress [9], or of environmental quantum effects [10]. On the other hand, tailored and reproducible coupling becomes important when emitters are used in optical quantum information technology, where the control of their photon emission and absorption is a prerequisite for the required fidelity of operations.
At cryogenic temperatures, where phonon contributions are frozen out, several emitters such as molecules [11], quantum dots [12,13], or defect centers in diamond [14] exhibit line narrowing down to a few GHz or even MHz until the theoretical limit for the spectral linewidth is reached. This allows such emitters to be isolated by spectroscopic means [11]. Such a lifetime limited transition is solely determined by the time the emitter spends in the excited state due to the energy–time uncertainty relation. This decay time from the excited state is of the order of several nanoseconds. It is called the radiative lifetime or the “longitudinal relaxation time” [15], and is denoted as ${T_1}$.
When the system’s temperature is increased, the longitudinal relaxation time is usually not affected. On the other hand, the coherence of the emitter is strongly reduced. This coherence time ${T_2}$ is also known as the “transversal relaxation time” [15]. It is the time in which the system is able to undergo coherent oscillations [16]. A reduced coherence time broadens the spectral response of the emitter. An ideal two-level system, isolated from all external influences, follows the ratio $1/{T_2} = 1/(2{T_1})$, which is known as the Fourier limit.
Decoherence introduced by the environment can be observed in a pump–probe sequence of two optical pulses [17]. The first pulse prepares the system in a coherent superposition between the ground state and excited state. The second pulse probes the remaining coherence of the system. This technique is known as Ramsey spectroscopy. It consists of two resonant $\pi /2$ pulses and a variable delay time, $\tau$, between them. This method is well established in atomic spectroscopy [18–20]. Ramsey spectroscopy is often performed on ensembles, while single atoms and ions have also been researched in the past [21,22]. Most experiments deal with ground state transitions and long coherence times, such that they are common for the use in atomic clocks [23,24]. While probing the long lived ground state transitions is most common for frequency metrology, some experiments have been implemented with short interactions of atoms or ions with a coherent laser field to probe electronic coherence [21,25]. Only a few experiments have addressed single emitters in the solid state and their optical transitions [26–28]. Since for many solid-state emitters such as quantum dots, their ${T_2}$ times are found to be in the picosecond range, it is hard to experimentally probe the coherence on these systems. This is because short and coherent optical pulses are required [29].
Fig. 1. Single molecule spectroscopy. (a) Molecular structure of 2.3,8.9-dibenzanthanthrene (DBATT). (b) Simplified level scheme. (c) Concept of fluorescence excitation spectroscopy. When the laser frequency is scanned, the number of Stokes-shifted photons indicates the population of the excited state. By fitting its Lorentzian-like curve, we can get the corresponding linewidth and count rate (background subtracted). The saturation parameter $S$ is defined as the ratio of excitation intensity and saturation intensity, and is used here to represent the excitation power. A linewidth of 18.1 MHz and a count rate of 65 kcnts/s are measured for the saturation parameter $S = {0.2}$.
Fig. 2. Molecule characterization. (a) Saturation behavior of the 2.3,8.9-dibenzanthanthrene (DBATT) molecule. The plot in purple shows how the count rate saturates at high excitation powers. The plot in dark blue shows how the linewidth of the single molecule is affected by power broadening. (b) Intensity auto-correlation histograms (all curves have an offset of one for better visibility). Rabi oscillations are clearly visible. With increasing continuous-wave (cw) excitation power, the Rabi frequency ${\varOmega _{{\rm{Rabi}}}}$ increases. Values are given in units of MHz.
Here, we analyze the coherence properties of a single molecule in a cryogenic environment by optical Ramsey spectroscopy. To fully investigate the coherence, we are not limited to resonant Ramsey pulses; in particular, we are able to include spectral detuning. To date, the number of detuning-dependent Ramsey spectroscopic studies on single emitter systems has been limited [30,31], although detuning is at the heart of the frequency metrological nature of Ramsey studies [24,32]. In this paper, we present detuning-dependent pump–probe spectroscopy for one of the most fundamental emitters: a single molecule.
The experiment consists of a cryogenic confocal microscope, described elsewhere [1,33,34]. In summary, a single 2.3,8.9-dibenzanthanthrene (DBATT) molecule at $T = 1.4 \,{\rm{K}}$ [Fig. 1(a)] is selected microscopically and spectroscopically. The relevant electronic transitions for this study are depicted in Fig. 1(b). We utilize fluorescence excitation spectroscopy and collect the Stokes-shifted photons on single photon detectors.
2. CONVENTIONAL METHODS
Before performing pump–probe spectroscopy, we estimate the transversal lifetime ${T_2}$ of the studied molecule by employing saturation and intensity auto-correlation measurements. The almost Lorentzian line [35] of the single molecule under continuous-wave (cw) excitation [see Fig. 1(c)] is affected by power broadening. The resulting linewidth is modeled as a usual two-level system according to
Figure 2(a) shows that, as expected for a two-level system, the linewidth increases with excitation power, while the emitted intensity asymptotically approaches a certain value. We fit the data according to Eq. (1) and find a natural linewidth of $2\pi \times (25.3 \pm 0.3)\;{\rm{MHz}}$ at the low excitation limit. This corresponds to an effective coherence time of ${T_2} = 12.6 \pm 0.2 \;{\rm{ns}}$, which is likely affected by the jitter of the laser frequency.
Since this method is highly susceptible to experimental subtleties, such as noise on the laser frequency control, we now turn to an analysis of the second order correlation function. Figure 2(b) shows the ${g^{(2)}}(\tau)$ function of the same individual molecule. At the low excitation limit, the recorded clicks show the typical anti-bunching behavior for a single emitter. When the excitation intensity is increased, Rabi oscillations occur. The decay of these coherent oscillations in the timing histogram reveals the system’s ${T_2}$ time as outlined in the literature [5,16,36]. To extract the ${T_2}$ time from the raw data, we fit ${g^{(2)}}(\tau)$ with its analytical expression (see [5] for an in-depth discussion):
Here, ${\varOmega _{{\rm{Rabi}}}}$ is the Rabi frequency, and ${\varGamma _1}$ and ${\varGamma _2}$ correspond to $1/{T_1}$ and $1/{T_2}$, respectively. The values ${T_1} = 10.1 \pm 0.1 \;{\rm{ns}}$ and ${T_2} = 15.3 \pm 0.4 \;{\rm{ns}}$ fit the data well throughout all excitation powers. The found coherence time is compatible with the prior acquired value above, when a small frequency drift of the laser is considered.
Fig. 3. Longitudinal and transversal lifetime measurements. (a) Pulse sequence for Ramsey spectroscopy. It starts with a 30 ns long pulse, followed by 201 pairs of $\pi /2$ pulses (Ramsey sequence). The free-evolution time $\tau$ for each Ramsey pair gradually increases from 0 ns to 60 ns. These pairs are separated by a blank time ($90\;{\rm{ns}} \gg {{{T}}_1}$) to reinitialize the molecule in the ground state. The full sequence is 25 µs long. (b) Corresponding molecule’s response, which reflects the excited state population dynamics. The long pulse in the sequence causes Rabi oscillation, yielding the Rabi frequency of the molecule. (c) Molecular spontaneous emission can be observed when the light is switched off. By fitting it with an exponential decay, we obtain the longitudinal lifetime ${T_1} = 10.1 \pm 0.1 \;{\rm{ns}}$. The inset demonstrates the longitudinal relaxation of a molecule excited with a $\pi /2$ pulse. (d) Intensity decay of molecular fluorescence after the Ramsey sequences. The transversal lifetime amounts to ${T_2} = 17.6 \pm 0.8 \;{\rm{ns}}$. The inset schematically shows the process of decoherence of a molecule excited by a $\pi /2$ pulse; the orange arrow corresponds to the second $\pi /2$ pulse of the Ramsey pair, which allows us to project the current state of the molecule onto the excited state. The decay time of this projection amplitude corresponds to the ${T_2}$ time.
3. RAMSEY METHOD
In the previous section, we analyzed the photon emission properties under cw excitation. In other words, the excitation laser on the molecule was switched on for the entire experiment. Thus, the photonic coherence was being probed while the molecule was under the steady influence of the excitation laser. In the following, we analyze the coherence properties of the single molecule without such a constant illumination using optical pump–probe spectroscopy. The system is brought into a coherent superposition state by the pumping $\pi /2$ pulse followed by some interaction-free time $\tau$ in darkness. It is then probed by another $\pi /2$ pulse [23]. In the absence of detuning and dephasing, a measurement of the excited state population after the second pulse reveals the coherence decay time ${T_2}$ [see Fig. 3(d)].
The optical pulses for pump–probe spectroscopy are created using an electro-optical modulator, which rapidly switches the excitation laser beam (see Supplement 1). Such a pulse generation procedure enables us to implement an arbitrarily timed sequence. In contrast to pulse generation with optical delay lines or pulsed lasers, a fixed phase relation between the first and second pulses is given for times below 200 ns, constrained by the excitation laser’s coherence times. The implemented pulse sequence is depicted in Fig. 3(a) (see Supplement 1 for details). As before, the molecule is simplified as a two-level system with the ground state $|g\rangle$ and excited state $|e\rangle$ along with a level splitting of $\hbar {\omega _0}$. The molecule is illuminated by single-mode laser light with a base frequency of $\omega$. The detuning between the laser frequency and level splitting is labeled as $\varDelta = \omega - {\omega _0}$. The incoming laser field drives Rabi oscillations with a characteristic Rabi frequency $\varOmega$. To describe the evolution of the system, the Lindblad master equation is utilized (see Supplement 1).
The molecule’s response to the pump–probe pulse sequence in Fig. 3(a) is sketched in Fig. 3(b). To experimentally analyze the ${T_2}$ time, we integrate the number of clicks that stem from the excited state’s decay after the second Ramsey pulse. First, we will focus on the case without detuning, $\varDelta = 0$. A plot of this integrated fluorescence is shown in Fig. 3(d). For short delays, the second pulse increases the excited state population, as two subsequent and phase-coherent $\pi$/2 pulses correspond to a $\pi$ pulse. The longer the time delay, the more the emitter loses its coherence, and for long delays, it reaches a level that corresponds to the excitation of a single, independent $\pi /2$ pulse.
To obtain a reliable value for the ${T_2}$ time, we take the experimental imperfections in detuning and the pulse area into account. Specifically, we first extract the Rabi frequency and ${T_1}$ time as fitting parameters from a separate measurement with 80 ns pulse (see Supplement 1 Fig. S2). The resulting ${T_1}$ time is $10.1 \pm 0.1\;{\rm{ns}}$, matching the reference value found in the literature [37].
From the data shown in Fig. 3(d), the value of ${T_2}$ is found to be $17.6 \pm 0.8\;{\rm{ns}}$. This value deviates from the theoretically predicted value for the Fourier limitation of $2 \cdot {T_1} = {T_2} \approx 20 \,{\rm{ns}}$. In principle, this indicates the presence of additional broadening mechanisms in the vicinity of the emitter, for example, coupling to the host matrix. Commonly, this is described with an effective ${T_2}$ time, defined as $1/{T_2} = 1/(2{T_1}) + 1/T_2^*$.
As our experimental implementation allows us to set all parameters independently, we now turn towards detuning-dependent experiments. We illuminate the molecule with the Ramsey sequence while scanning the laser frequency in the range of ${-}2\pi \times 165\;{\rm{MHz}}$ to $2\pi \times 152\;{\rm{MHz}}$ in 50 frequency steps. The total recording time for each step amounts to around 7 s.
For a deeper understanding of the frequency-detuned excited state dynamics, we analyze data with a defined free-evolution time. Figure 4(a) shows a detuning-dependent photon arrival histogram for a fixed free-evolution time $\tau$ of 8.1 ns. Due to the coherent phase relation between the Ramsey pulses, the second pulse further enhances the molecular excitation. At the same time, when the laser is detuned, the laser frequency and free precession of the quantum emitter do not match. Hence, the observed molecular fluorescence after the second pulse has an oscillatory dependence on detuning $\varDelta$. In the case of $\varDelta \cdot \tau = 2\pi n$, the photon count reaches a maximum value (where $n \in {\mathbb Z}$). In contrast, when $\varDelta \cdot \tau = (2n - 1)\pi$, the photon count is minimal, meaning that the effective population is reduced compared to the level when the second pulse is not present.
Fig. 4. Ramsey sequence induced molecular fluorescence. (a) Experimental data of molecular fluorescence response induced by Ramsey sequence under different detuning conditions with the free-evolution time $\tau$ being fixed at 8.1 ns. (b) Simulation for the experiment in (a). The color bar on the right-hand side shows the population of excited states corresponding to different colors. It can be seen that the excited state population oscillates with the laser detuning with a period of $1/\tau$ after being excited by the second pulse. (c) Integrated counts after the second pulse. These data represent one line of the plots in Fig. 5.
Figure 4(b) shows the theoretical prediction for the experimental findings. The experiment is well described by the theory for a two-level system. The detuning dependence for the integrated response is shown in Fig. 4(c). The simulation using the quantum toolbox “QuTip” is shown in Code 1, Ref. [38].
Figure 4 represents only one of the detuning-dependent timing histograms, namely, for a fixed evolution time of $\tau = 8.1 \;{\rm{ns}}$. As in the case for zero detuning, we now integrate the counts after the second Ramsey pulse. Performing this integration procedure for all different evolution times $\tau$, the (detuning-dependent) histogram in Fig. 5(a) is obtained. The system shows arches that scale as $1/\varDelta$ and lose contrast over the coherence time of the emitter. For zero detuning, we obtain an exponential decay, shown in Fig. 3(d).
Fig. 5. Ramsey spectroscopy. (a) Experimental data of Ramsey spectroscopy. The molecular fluorescence after Ramsey sequence excitation is color mapped. For better visibility, the $y$ axis is cut at $\tau = 31.2 \;{\rm{ns}}$. (b) Simulation for the experiment in (a) by numerically solving the Lindblad master equation. Experimentally determined ${T_1} = 10.1 \;{\rm{ns}}$ and ${T_2} = 17.6 \;{\rm{ns}}$ are used as simulation parameters. The color bar is shown on the right-hand side. The darkest and brightest colors correspond to 0.03 and 0.83 excited state populations, and these values correspond to 257 and 4012 integrated counts, respectively. The small orange arrow indicates the free-evolution time $\tau = 8.1 \;{\rm{ns}}$ at which the data in Fig. 4 are measured.
As before, the experimental findings excellently match the theoretical model shown in Fig. 5(b). These detuning-dependent experiments underline the validity of the earlier determined coherence time of the emitter.
4. CONCLUSION AND OUTLOOK
In conclusion, we have presented a measurement method for comprehensively investigating the properties of single molecule coherence under cryogenic conditions. Specifically, we implemented coherent pump–probe spectroscopy on a single molecule. Compared to conventional techniques that use cw lasers, the short pulses used in this method avoid the influence of power broadening and also reduce heating of the sample. We also implemented detuning-dependent spectroscopy on the single molecule to show the possibility to mitigate the influence of frequency uncertainty in the excitation light field. This work extends earlier studies on the coherence properties of emitted photons [1] away from the purely optical properties and yields insight into the behavior of an extended solid-state system.
The coherence time of the emitter is found to be ${T_2} = 17.6 \pm 0.8 \;{\rm{ns}}$, which is close to the Fourier limited value and only slightly deviates from the ideal ${T_2}$ value of approximately 20 ns. Such a deviation may be caused by a small environmental reduction of the coherence time of $T_2^* = 250 \;{\rm{ns}}$, or by experimental subtleties, such as timing jitters and finite rise times. The frequency detuning-dependent experiment we introduced reconfirms that the system can be treated as a two-level system.
As the coherence is preserved inside a single molecule, we envision that a molecule might be, for example, coupled to another internal or external degree of freedom, such as available electrons [39]. In turn, this could allow the implementation of more sophisticated memory schemes or allow more complex implementations of coherent spectroscopy.
Funding
Bundesministerium für Bildung und Forschung (13N15972); Deutsche Forschungsgemeinschaft (GE2737/5-1; Germany’s Excellence Strategy (EXC-2123 QuantumFrontiers, 390837967) European Cooperation in Science and Technology (MP1403).
Acknowledgment
We acknowledge discussions with Dr. J. Becker, Oxford. We thank Dr. Jörg Wrachtrup for his continuous support. Funding was provided by the German Federal Ministry of Education and Research, funding program quantum technologies - from basic research to market [13N15972]. Further funding sources: Deutsche Forschungsgemeinschaft; Germany’s Excellence Strategy; the COST Action “Nanoscale Quantum Optics” funded by COST.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.
Supplemental document
See Supplement 1 for supporting content.
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