Phase noise measurement of external cavity diode lasers and implications for optomechanical sideband cooling of GHz mechanical modes

Cavity opto-mechanical cooling via radiation pressure dynamical backaction enables ground state cooling of mechanical oscillators, provided the laser exhibits sufficiently low phase noise. Here, we investigate and measure the excess phase noise of widely tunable external cavity diode lasers, which have been used in a range of recent nano-optomechanical experiments, including ground-state cooling. We report significant excess frequency noise, with peak values on the order of 10^7 rad^2 Hz near 3.5 GHz, attributed to the diode lasers' relaxation oscillations. The measurements reveal that even at GHz frequencies diode lasers do not exhibit quantum limited performance. The associated excess backaction can preclude ground-state cooling even in state-of-the-art nano-optomechanical systems.

Introduction: In recent years the mutual coupling of optical and mechanical degree of freedom has been observed in a plethora of systems and gives rise to a variety of phenomena [1][2][3][4]. This parametric radiation pressure coupling [5] enables sensitive measurements of the mechanical oscillator's position, amplification and cooling of mechanical motion via dynamical backaction, optomechanical normal mode splitting, optomechanically induced transparency quantum coherent coupling of optical and mechanical degrees of freedom, and optomechanical entanglement. Of particular attention has been the objective to achieve ground state cooling of a macroscopic mechanical oscillator using the technique of optomechanical resolved sideband cooling [6][7][8].
Previous experiments and theoretical analysis [6,[9][10][11][12][13][14] have shown, however, that optomechanical experiments in general, and sideband cooling in particular, are sensitive to excess phase noise of the employed laser. This necessitates the use of filtering cavities [15] or low-noise solid-state lasers [6] such as Ti:Sa and YAG lasers, which offer quantum-limited performance for sufficiently high Fourier frequencies (typically > 10 MHz). Diode lasers, in contrast, exhibit significant excess phase noise in this frequency range [16] and its impact has been observed in optomechanical cooling experiments of a 75 MHz radial breathing mode [6]. Moreover, there exists an additional, well-known contribution [17] to the excess phase and amplitude noise at high Fourier frequencies (> 1 GHz), which is fundamentally linked to damped relaxation oscillations caused by the carrier population dynamics [18][19][20][21]. These relaxation oscillations cause primarily excess phase noise, whose magnitude is in close agreement with theoretical modeling [22,23]. Interestingly, optical feedback (such as provided by an external cavity)-while reducing noise at low frequency-can even lead to an enhancement of this relaxation oscillation noise [24]. * Electronic address: tobias.kippenberg@epfl.ch Quantitative measurements of the high frequency excess phase noise (at GHz frequencies) for modern widely tunable external cavity diode lasers, however, are scarce. Such studies have become increasingly important as novel nano-optomechanical systems such as 1-D nanobeams [25] and 2-D photonic crystals [26] operate in this GHz frequency range, and quantitative knowledge of the phase noise is therefore relevant to quantum cavity optomechanical experiments. In particular ground state cooling of a nanomechanical oscillator has been reported with an unfiltered external cavity diode laser [27]. As such, characterization of extended cavity diode laser phase noise in the GHz domain and evaluation of its impact on quantum optomechanical experiments is highly desirable. Here we present such a characterization of widely tunable external cavity diode lasers as used in recent optomechanical experiments [27]. Our results indicate that as expected, significant excess phase noise is indeed present in such lasers at GHz frequencies whose magnitude can impact optomechanical sideband cooling of nano-optomechanical systems.
Theory: Radiation pressure optomechanical sideband cooling allows cooling of a mechanical oscillator to a minimum occupation [7,8] where Ω m is the mechanical frequency and κ is the optical energy decay rate. This limit arises from the quantum fluctuation of the cooling laser field. Excess classical phase or amplitude noise causes a fluctuating radiation pres- sure force noise that increases this residual occupancy. Of particular relevance is phase noise, whose heating effect has been observed in experiments employing toroidal opto-mechanical resonators [6].
It is instructive to first consider a coherent phase modulation at a frequency Ω m of the lasers input field, Pumping an opto-mechanical system residing in the resolvedsideband regime (Ω m κ) at the lower sideband (∆ = −Ω m ) yields an intracavity field of a(t) where η = κ ex /κ denotes the ratio of cavity coupling κ ex to its feeding mode compared to total cavity losses κ. Note that the modulation sideband is resonantly enhanced by the cavity instead of being suppressed by a putative cavity filtering effect. The simultaneous presence of carrier and modulation sideband leads to a radiation-pressure force Ωm sin(Ω m t) +F , whereF is a (for the present analysis irrelevant) static force, P = ω|s in | 2 the launched input power and G = ∂ω c /∂x is the frequency pull parameter of the optomechanical system.
These considerations carry over directly to (pure) phase fluctuations of the cooling laser field described by a (symmetrized, double-sided) spectral densityS φφ (Ω). Alternatively, such fluctuations may be described in terms of laser frequency noise with a spectrumS ωω (Ω) = Sφφ(Ω) =S φφ (Ω) · Ω 2 , and we use both descriptions interchangeably. The resulting force fluctuation spectrum is given by [ in the resolved-sideband regime. It is straightforward to derive from this excess force noise the residual occupation of the mechanical oscillator by expressing it as an effective occupancyn L of the cold bath that the laser field is providing,n L ≈S L F F (Ω m )/2m eff Γ m Ω m , by comparing it to the Langevin force fluctuations of the thermal bath S th F F (Ω) = 2m eff Γ mnth Ω m withn th = k B T / Ω m . The final occupancy of the oscillator in the presence of sideband cooling is then given by n f ≈ (n L +n th ) Γ m /Γ cool , with Γ cool ≈ 2ηG 2 P/m eff Ω 3 m ω in the resolved-sideband regime. This yields an excess occupancy due to frequency noise of [6,10] wheren p ≈ ηκP/ ωΩ 2 m is the intracavity photon number in the resolved-sideband regime. For an optimized power, the lowest occupancy that can be reached is given bȳ where we have used the vacuum optomechanical coupling rate [28] g 0 = G /2m eff Ω m and neglected quantum backaction [31].
Measurement of the diode laser phase noise: Laser phase noise is frequently modeled by a (Gaussian) random phase φ(t) which obeys the simple noise model φ (t)φ(s) = γ c Γ L e −γc|t−s| , where Γ L is the laser linewidth, and γ −1 c is a correlation time, leading to a low-pass-type frequency noise spectrum with a white noise model in the limit γ c → ∞ [9,10,12,14]. In practice, the relation between the laser linewidth and the frequency noise spectrum does not follow this simple model, as there are several contributions of different physical origin to the phase noise of a diode laser: The laser's linewidth is mostly dominated by acoustic fluctuations occurring at low Fourier frequencies, leading to a typical short-term linewidth of ∼ 300 kHz for unstabilized external-cavity diode lasers. Moreover, relaxation oscillations occur at high (> 1 GHz) Fourier frequencies, which are not described by the above model. Therefore, it is important to measure the frequency-dependent phase noise spectrumS φφ (Ω).
To this end, an optical cavity is employed for quadrature rotation [16,29], converting phase to amplitude fluctuations which are measured with a photodetector (cf. Figure 1). In principle, a high-resolution spectrum of the optical field can also be used for phase noise measurement, in which case the relaxation oscillations appear as sidebands around the carrier (cf. e.g. [22,29]).
The devices under test are three 1550 nm extended cavity diode lasers in Littman-Metcalf configuration of the most commonly used models [32]. Care is taken to introduce proper optical isolation of the laser diode to avoid optical feedback. The quadrature rotating cavity is a fiber coupled silica microcavity (linewidth of κ/2π ≈ 2 GHz) and the transmission is detected by a fast photodetector (New Focus) whose photocurrent is fed into a spectrum analyzer (ESA). The transduction of frequency noiseS ωω (Ω) into power fluctuations at the output of the cavity (I denotes photon flux) is given by: Here, ∆ is the laser detuning from the cavity resonance, and Ω the analysis frequency. The noise equivalent power of the employed photodetector is ∼ 24 pW / √ Hz and therefore not sufficient to detect the quantum phase/amplitude noise for the power levels used in this work (< 1 mW), but does allow to detect excess noise. Indeed, as shown in Fig. 2, we observe a peak at ca. 3.5 GHz in the detected photocurrent fluctuations when the laser is detuned from the cavity resonance. This noise has been reported previously [18][19][20][21][22][23] and is attributed to relaxation oscillations, which due to the short carrier lifetime exhibit high frequencies well FIG. 2: Noise of a semiconductor laser with weak optical feedback from a grating (Littman configuration). Shown is the power spectral density (PSD) of photocurrent fluctuations, normalized to total photocurrent, when laser light is directly detected (red), or tuned to the side-of-the-fringe of a ca. 4 GHz-wide optical cavity (blue). The yellow trace is the background signal in the same normalization, which was subtracted from all traces.
into GHz range. We confirmed that the noise is indeed predominantly phase noise, by scanning the laser across the cavity resonance while keeping the analysis frequency fixed (Fig. 3). The pronounced double-peak structure follows eq. (5) and reveals that the noise is predominantly in the phase quadrature.
To calibrate the measured noise spectra, we imprint onto the diode laser a known phase modulation using an external (fiber based) phase modulator following the method of ref. [28]. In brief, the V π of the phase modulator is determined in independent measurements by scanning a second diode laser over the phase modulated laser, in order to determine the strength of the modulation sidebands. The measured V π and the manufacturers specifications differed by typically less than 10%. In a second and independent measurement (to characterize the noise level of a third laser) the phase modulator was characterized by scanning a phase modulated laser over a narrow cavity resonance and recording the transmission spectrum. Calibration via the modulation peak proceeds by using the relationS cal ωω (Ω) ≡ Ω 2 δφ 2 /(4 · RSB), where RSB is the resolution bandwidth of the recording with the electronic spectrum analyzer, δφ the modulation index and Ω the modulation frequency. Figure 4 shows this calibration procedure applied to the three lasers. The level of frequency fluctuations was measured for a total of three devices and found to vary only slightly between the lasers (despite their differing by 10 years in manufacturing date). The maximum frequency noise was in the range ofS max ωω ≈ O(10 7 ) rad 2 Hz, corresponding to phase fluctuations about 30 dB above the quantum noise limit S φφ (Ω) = ω/4P of a P = 1 mW beam. This level of phase noise agrees well with theoretical predictions [23]. Ground-state cooling limitations: In order to achieve ground state cooling, only a certain amount of laser phase noise can be tolerated, as the presence of the cooling light in the cavity leads to additional fluctuating forces and an excess phonon number according to eq. (1). For an optimum cooling laser power the residual thermal occupancy and the excess occupancy caused by radiation pressure fluctuations are equal, and their sum can be below unity only ifS ωω (Ω m ) < constituting a necessary condition for ground state cooling (n min f < 1) [10]. Evidently, systems that exhibit large optomechanical coupling g 0 and low mechanical decoherence rate γ = k B T / Q m (that is, low bath temperature T and high mechanical quality factor Q m ) can tolerate larger amounts of laser frequency noise. However, even the recently reported nano-optomechanical system [27] with the record-high g 0 /2π = 0.91 MHz as well as T ≈ 30 K and Q m ≈ 50, 000 requiresS ωω (2π 3.68 GHz) < 4 · 10 5 rad 2 Hz, a value reached by neither of the three lasers we have tested.
We conclude that widely employed frequency-tunable external cavity diode lasers should not be expected to be quantum limited, but exhibit significant excess phase noise up to very high Fourier frequencies. We have observed a peak in this noise at a frequency around 3.5 GHz.
This observation implies important limitations to optomechanical sideband cooling also for systems based on microwave-frequency mechanical oscillators if the laser noise is not suppressed, e.g. by external cavity filters [30].