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 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 Kippenberg2008 (); Marquardt2009 (); Favero2009a (); Aspelmeyer2010 (). This parametric radiation pressure coupling Braginsky1977 () 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 Schliesser2008 (); Marquardt2007 (); Wilson-Rae2007 ().
Previous experiments and theoretical analysis Schliesser2008 (); Diosi2008 (); Rabl2009 (); Yin2009 (); Phelps2011 (); Abdi2011 (); Ghobadi2011 () 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 Arcizet2006a () or low-noise solid-state lasers Schliesser2008 () 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 Zhang1995 () and its impact has been observed in optomechanical cooling experiments of a 75 MHz radial breathing mode Schliesser2008 (). Moreover, there exists an additional, well-known contribution Wieman1991 () to the excess phase and amplitude noise at high Fourier frequencies (), which is fundamentally linked to damped relaxation oscillations caused by the carrier population dynamics Piazzolla1982 (); Vahala1983 (); Vahala1983b (); Yamamoto1983 (). These relaxation oscillations cause primarily excess phase noise, whose magnitude is in close agreement with theoretical modeling Ritter1993 (); Ahmed2004 (). 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 Chen1984 ().
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 Eichenfield2009 () and 2-D photonic crystals Gavartin2011 () 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 Chan2011 (). 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 Chan2011 (). 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 Wilson-Rae2007 (); Marquardt2007 () of , where is the mechanical frequency and 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 pressure 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 Schliesser2008 ().
It is instructive to first consider a coherent phase modulation at a frequency of the lasers input field, . Pumping an opto-mechanical system residing in the resolved-sideband regime at the lower sideband () yields an intracavity field of , where denotes the ratio of cavity coupling to its feeding mode compared to total cavity losses . 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 , where is a (for the present analysis irrelevant) static force, the launched input power and 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 density . Alternatively, such fluctuations may be described in terms of laser frequency noise with a spectrum , and we use both descriptions interchangeably. The resulting force fluctuation spectrum is given by Schliesser2008 ()
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 occupancy of the cold bath that the laser field is providing, , by comparing it to the Langevin force fluctuations of the thermal bath with . The final occupancy of the oscillator in the presence of sideband cooling is then given by with in the resolved-sideband regime. This yields an excess occupancy due to frequency noise of Schliesser2008 (); Rabl2009 ()
where is the intracavity photon number in the resolved-sideband regime. For an optimized power, the lowest occupancy that can be reached is given by
where we have used the vacuum optomechanical coupling rate Gorodetsky2010 () and neglected quantum backaction \theendnote\theendnoteendnote: \theendnoteFor comparison with ref. Rabl2009 () note that we use energy decay rates instead of field decay rates , and we denote with the mechanical energy dissipation rate instead of the mechanical decoherence rate , which we refer to as here. .
Measurement of the diode laser phase noise: Laser phase noise is frequently modeled by a (Gaussian) random phase which obeys the simple noise model , where is the laser linewidth, and is a correlation time, leading to a low-pass-type frequency noise spectrum
with a white noise model in the limit Diosi2008 (); Rabl2009 (); Phelps2011 (); Ghobadi2011 (). 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 for unstabilized external-cavity diode lasers. Moreover, relaxation oscillations occur at high () Fourier frequencies, which are not described by the above model. Therefore, it is important to measure the frequency-dependent phase noise spectrum .
To this end, an optical cavity is employed for quadrature rotation Zhang1995 (); Riehle2004 (), 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. Ritter1993 (); Riehle2004 ()).
The devices under test are three extended cavity diode lasers in Littman-Metcalf configuration of the most commonly used models \theendnote\theendnoteendnote: \theendnoteNew Focus TLB-6328 (serial numbers 008 and 280) and TLB-6330 (serial number 043).. 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 ) and the transmission is detected by a fast photodetector (New Focus) whose photocurrent is fed into a spectrum analyzer (ESA). The transduction of frequency noise into power fluctuations at the output of the cavity ( denotes photon flux) is given by:
Here, is the laser detuning from the cavity resonance, and the analysis frequency.
The noise equivalent power of the employed photodetector is p and therefore not sufficient to detect the quantum phase/amplitude noise for the power levels used in this work (), but does allow to detect excess noise. Indeed, as shown in Fig. 2, we observe a peak at ca. in the detected photocurrent fluctuations when the laser is detuned from the cavity resonance. This noise has been reported previously Piazzolla1982 (); Vahala1983 (); Vahala1983b (); Yamamoto1983 (); Ritter1993 (); Ahmed2004 () and is attributed to relaxation oscillations, which due to the short carrier lifetime exhibit high frequencies well 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. Gorodetsky2010 (). In brief, the 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 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 relation , where RSB is the resolution bandwidth of the recording with the electronic spectrum analyzer, the modulation index and 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 of , corresponding to phase fluctuations about above the quantum noise limit of a beam. This level of phase noise agrees well with theoretical predictions Ahmed2004 ().
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 if
constituting a necessary condition for ground state cooling () Rabl2009 (). Evidently, systems that exhibit large optomechanical coupling and low mechanical decoherence rate (that is, low bath temperature and high mechanical quality factor ) can tolerate larger amounts of laser frequency noise. However, even the recently reported nano-optomechanical system Chan2011 () with the record-high as well as and requires , 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 . 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 Hald2005 ().
Acknowledgements: We are grateful to V. L. Velichansky and P. Poizat for stimulating discussions. This work is supported by the NCCR of Quantum Engineering.
- (1) T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
- (2) F. Marquardt and S. M. Girvin, Physics 2, 40 (2009).
- (3) I. Favero and K. Karrai, Nature Photonics 3, 201 (2009).
- (4) M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, Journal of the Optical Society of America B 27, A189 (2010).
- (5) V. B. Braginsky and A. B. Manukin, Measurement of Weak Forces in Physics Experiments (University of Chicago Press, 1977).
- (6) A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. Kippenberg, Nature Physics 4, 415 (2008).
- (7) F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Physical Review Letters 99, 093902 (2007).
- (8) I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, Physical Review Letters 99, 093901 (2007).
- (9) L. Diósi, Physical Review A 78, 021801 (2008).
- (10) P. Rabl, C. Genes, K. Hammerer, and M. Aspelmeyer, Physical Review A 80, 063819 (2009).
- (11) Z. Yin, Physical Review A 80, 033821 (2009).
- (12) G. A. Phelps and P. Meystre, Physical Review A 83, 063838 (2011).
- (13) M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, Physical Review A 84, 032325 (2011).
- (14) R. Ghobadi, A. R. Bahrampour, and C. Simon, Physical Review A 84, 063827 (2011).
- (15) O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature 444, 71 (2006).
- (16) T. C. Zhang, P. Poizat, J. Ph. aand Grelu, J.-F. Roch, P. Garngier, F. Marin, A. Bramati, V. Jost, M. D. Levenson, and E. Giacobino, Quantum and semiclassical optics 7, 601 (1995).
- (17) C. E. Wieman and L. Hollberg, Review of Scientific Instruments 62, 1 (1991).
- (18) S. Piazzolla, P. Spano, and M. Tamburini, Applied Physics Letters 41, 695 (1982).
- (19) K. Vahala, C. Harder, and A. Yariv, Applied Physics Letters 42, 211 (1983).
- (20) K. Vahala and A. Yariv, IEEE Journal of Quantum Electronics 19, 1102 (1983).
- (21) Y. Yamamoto, S. Saito, and T. Mukaim, IEEE Journal of Quantum Electronics 19, 47 (1983).
- (22) A. Ritter and H. Haug, J. Opt. Soc. Am. B 10, 130 (1993).
- (23) M. Ahmed and M. Yamada, J. Appl. Phys. 95, 7573 (2004).
- (24) Y. C. Chen, Applied Physics Letters 44, 10 (1984).
- (25) M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
- (26) E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, Physical Review Letters 106, 203902 (2011).
- (27) J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, Nature 478, 8992 (2011).
- (28) M. Gorodetsky, A. Schliesser, G. Anetsberger, S. Deleglise, and T. J. Kippenberg, Optics Express 18, 23236 (2010).
- (29) F. Riehle, Frequency Standards (Wiley-VCH, 2004).
- (30) J. Hald and V. Ruseva, Journal of the Optical Society of America B 22, 2338 (2005).