Dispersion enhanced tunability of -frequency response to its cavity-length change

Savannah L. Cuozzo1 and Eugeniy E. Mikhailov1, ∗ 1Department of Physics, College of William & Mary, Williamsburg, Virginia 23187, USA (Dated: September 7, 2019) We report on the controllable response of the lasing frequency to the cavity round-trip path change. This is achieved by modifying the dispersion of the intracavity medium in the four-wave mixing regime in Rb. We can either increase the response by at least a factor of 2.7 or drastically reduce it. The former regime is useful for sensitive measurements tracking the cavity round trip length and the latter regime is useful for precision metrology.

PACS numbers: 42.50.Lc, 42.50.Nn

We control the response of the lasing frequency to the laser cavity length change on demand — allowing for ei- ther dramatic enhancement or suppression. The reso- 100 nant frequency link to the cavity round trip path is the (a) foundation for optical precision measurements such as displacement tracking, temperature sensing, optical ro- 50 tation tracking [1], gravitational wave sensing [2], and refractive index change sensing [3]. In other applica- (b) tions, the laser provides a stable frequency reference, such 0 as precision interferometry [4], optical atomic clocks [5], and distance ranging [6], where the response of the las- (c) -50 ing frequency to the cavity path length change should (i)

be reduced. Our findings allow for improved laser as- (MHz) Change Frequency Laser 0 50 100 150 sisted precision metrology and potentially make less bulky and immune to the environmental changes in (a) real world applications. 105 The addition of a dispersive medium to a cavity mod- ifies its frequency response [7] to the geometrical path

change (dp) according to |PF| (b) 0 n dp 10 dfd = − f0 , (1) ng ptot (c) (ii) where f is the original resonant frequency, p = p + 0 tot e 0 50 100 150 p n is the total optical round-trip path of the cavity, p d d Empty Cavity Detuning (MHz) is the length of the dispersive element, pe is the length of the empty (non-dispersive) part of the cavity, n is the re- FIG. 1. (Color online) (i) is the experimental lasing fre- fractive index, and ng is the generalized refractive group index given by quency dependence on empty cavity detuning (round trip path change) in (a) bifurcating regime with estimated ultra np ∂n high PF> 108, (b) high pulling regime with PF= 2.7±0.4, (c) n = n + d f . (2) g p 0 ∂f enhanced stability regime where |PF| < 0.2 crossing 0. The tot solid lines (a and b) show our best fits of the laser frequency arXiv:1812.08260v3 [quant-ph] 25 Jul 2019 We define the pulling factor (PF) as the ratio of disper- dependence using the model described by Eq.4; the (c) line is the polynomial fit of the 5th degree. The straight dashed sive to empty (non-dispersive, ng = n) cavity response for the same path change line shows the PF= 1 dependence (i.e. for an empty cavity). (ii) is the PF calculated based on the fits presented in (i). df n PF ≡ d = . (3) dfe ng We tune the PF in the range from -0.3 to at least The PF is the figure of merit for the enhancement of the 2.7 ± 0.4 (see Fig.1), by tailoring the refractive index cavity response relative to canonical lasers or passive cav- of our lasing medium. This is the first demonstration of ities operating in the weak dispersion regime with n = n. g high and tunable PF in the laser. We can also push our system response further and reach the bifurcating regime. Due to Kramers-Kronig relationship the negative dis- ∗ [email protected] persion is accompanied by local absorption, so it is not 2

10-7 1.5 / = 0 / = 0 th is the bifurcating threshold strength. th / = 0.5 / = 0.5 1 th th The analysis of the dispersion (Eq.4) and its influence / = 1 50 / = 1 th th / = 2 / = 2 0.5 th th on the resonant frequency of the cavity and PF is shown 0 0 in Fig.2. As expected, the amplification line has positive n-1 dispersion on resonance (see Fig.2a). Positive disper- -0.5 -50 sion is associated with a large and positive group index, -1 (c) which results in weak dependence (low pulling factor) of (a) (MHz) Change Frequency Laser -100 -1.5 -100 -50 0 50 100 -50 0 50 the lasing frequency on the cavity path change (empty Frequency detuning (MHz) Empty Cavity Detuning (MHz) cavity detuning), as shown in Fig.2b. Away from reso- 5 100 10 Bistable, / = 2 th nance, the dispersion is negative leading to high PFs, as PF max Monotonic PF 50 shown in Fig.2b. The stronger the amplification ( ) the min Bifurcation Region smaller the PFmin is at the center of the resonance, as 0 shown in Fig.2b. Consequently, the PF max continuously grows and reaches infinity at  =  where the resonant 100 th Pulling Factor Pulling -50 frequency bifurcates (see Fig.2b). (b) (d) To track dependence of the cavity resonant frequency

0 0.5 1 1.5 2 (MHz) Change Frequency Laser -100 -100 -50 0 50 100 on the cavity path length, we solve: (units of ) th Empty Cavity Detuning (MHz) c ptot = m , (8) FIG. 2. (Color online) (a) Refractive index change (n − fd 1); (b) dependence of maximum and minimal achievable PF where m is the fixed mode number and c is the speed of on resonance strength; (c) laser frequency change and (d) light in vacuum. In experiment, it is easier to track the bifurcating behavior as functions of detuning (or cavity path empty cavity detuning (i.e. resonance frequency change, change). For all figures γ is set to 6 MHz. ∆fe), which is directly linked to the cavity path change via Eq.1 with ng = n. The resulting dependencies are surprising that so far the P F > 1 regime was experi- shown in Fig.2c. mentally demonstrated only in passive, non-lasing cav- If the negative dispersion is strong enough, the group ities [8–10] with PF= 363. For active cavities, Yablon index could be negative. This would lead to negative et al. [11] inferred a PF∼ 190 via analysis of the las- PF and to negative dependence of the lasing frequency ing linewidth. The increased stability regime (PF< 1) on the cavity detuning (see line corresponding  = 2th was demonstrated in lasing [12] cavities with the small- in Fig.2c). This behavior is nonphysical, since it corre- est being PF= 1/663 [13]. Superradiant (“bad-cavity”) sponds to a bifurcation [16]: multiple lasing frequencies lasers, where an atomic gain line is much narrower than for the same cavity detuning. Consequently, the laser a cavity linewidth, exhibit ultra low PF< 10−6 [14, 15]. would ‘jump’ to avoid negative PF region and preserve Our empty cavity linewidth is about 13 MHz which is the monotonic behavior, as shown in Fig.2d and exper- larger than any atomic decoherence time, so we operate imentally in Fig.1(i)a. in the “bad-cavity” regime. However, unlike previously The most important conclusion from the amplifying reported work in [14, 15], we can also achieve higher than line analysis is that high pulling (response enhancement) one PF. regions exist slightly away from the gain resonance. The Similar to [16], we present a simple model of the trans- precursor of such regime is a reduced PF region in close mission or amplification spectral line where the index of vicinity to the resonance. The off resonance behavior was refraction has the dependence: overlooked in the literature, while it actually provides the road to high PF. Away from the amplification resonance, γ∆f n(f) = 1 +  , (4) the system still has enough gain to sustain lasing, and ∆f 2 + γ2 yet it still has large negative dispersion (see Fig.2a). where  is the resonance strength, ∆f is the detuning As detuning from the resonance increases, the dispersion becomes negligible, PF approaches unity (see Fig.2c and from the medium resonance frequency (fm), and γ is the resonance width, since n(f) − 1  10−5 for a vapor filled experimental data in Fig.1 a and b), amplification drops cavity. For transmission or gain with  > 0, and eventually lasing ceases. the minimum and maximum PF are To experimentally demonstrate the modified lasing re- sponse to the cavity path change, one needs a narrow 1 √ PFmax = at ∆f = ± 3γ, (5) gain line to achieve the highest positive dispersion. We 1 − /th utilized the N-level pumping scheme depicted in Fig.3. 1 The theory and preliminary experimental study of this PFmin = at ∆f = 0 (6) 1 + 8/th arrangement are covered in references [12, 17, 18]. The strong pumping field Ω creates a transmission line for where 1 the field α due to electromagnetically induced trans- 8γ ptot parency. However, the Ω field alone is not enough to cre- th = (7) 1 fm pd ate the amplification. To create the gain for the α field, 3

3 2 700 MHz below the 5S F =1 → 5P F=1 transition, 5P3/2 F= 1 1/2 g 1/2 0 and D2 is set to 500 MHz below the 5S1/2Fg = 2 → 2 87 5P F= 5P3/2F=3 Rb transition, as seen in Fig.3. They pro- 1/2 1 β vide amplification for fields α and β, which are generated orthogonal to pump fields polarization. Only the α field Ω2 resonantly circulates in the cavity, since the pumps exit

α the cavity via the second PBS and β is kept off-resonance Ω1 with the cavity. Since the D1 pump laser is fixed, the beatnote of the Fg=2 ΔHFS pump (Ω1) and the lasing field (α) with its frequency 87 Fg=1 close to Rb hyperfine splitting (∆HFS ≈ 6.8 GHz) is related to the frequency of the ring cavity laser and al- FIG. 3. (Color online) Schematic diagram of interacting lows us to monitor the dispersive laser frequency change 87 light fields and relevant Rb levels. (∆fd). We control the cavity length by locking it to an auxiliary laser (called the lock laser) with a wavelength

LOCK of 795 nm that is far detuned from any atomic resonances Spectrum LASER Analyzer and senses a “would be empty” (dispersion-free) cavity D1 LASER detuning (∆fe). This lock laser beam counter propa- gates relative the pump beams and the lasing field to FAST D2 LASER FIBER PD BEAM avoid contaminating the detectors monitoring the ring SPLITTER FIBER PD MAGNETIC LENS cavity lasing. Two wave plates (WP) are placed inside LENS WP PBS Rb Cell the cavity. One is to spoil polarization of the lock field 87 PBS Rb λ/2 and allow it to circulate in the cavity. The other rotates SHIELDING the lasing field polarization by a small amount. This al- lows it to exit the cavity and mix in with the pump field CAVITY LOCK SCOPE on the fast photodetector. The maximum response has the lower bound of WP LENS PF = 1.1 × 108 at the 90% confidence level for the PZT PD max data set (a), shown in Fig.1. The upper bound for PFmax is infinity since the data set belongs to the bi- FIG. 4. (Color online) Schematic diagram of the setup. furcating regime. However, one can smoothly approach Labels are: PD is photo detector, WP is wave plate, λ/2 is this limit by carefully controlling the cavity detuning as half wavelength wave plate, PBS is polarizing , and PZT is piezoelectric transducer. our analysis shows in Fig.1(ii)a. The PF min range is (0.08 to 0.10) for this data set. We can avoid bifurcation by increasing the pumps’ powers (i.e., we increase γ via power broadening), as we apply another strong repumping field (Ω2). There is also gain for the β field, which completes the four-wave shown in the data set (b) of Fig.1. This data demon- strates PF in the range (2.3 to 3.2). Also, the range mixing arrangement of fields Ω1,Ω2, α, and β. But the max cavity is tuned to sustain lasing only for α. of detuning with PF> 1 is wider. To estimate confi- Our lasing cavity is similar to the one used in [12]. dence bounds, we use the modified smoothed bootstrap The ring cavity is made of two polarizing beam splitters method [19]. (PBS) and two flat mirrors. The round trip path of the We are able to make our dispersive laser insensitive to cavity is 80 cm. A 22 mm long Pyrex cylindrical cell with its path change, as shown in data set (c) of Fig.1. We anti-reflection coatings on its windows is placed between tune tune the D1 laser to 400 MHz above the 5S1/2Fg=1 the two PBSs and filled with isotopically pure 87Rb. The → 5P1/2F=1 transition, and keep D2 at 500 MHz be- 87 cell is encased in a 3-layer magnetic shield and its tem- low the 5S1/2Fg = 2 → 5P3/2F=3 Rb transition, while perature is set to 100 degrees C. The optical stability of maintaining combined pump power at 95 mW. Assuming the cavity is increased by adding a 30 cm focal length lens a smooth dependence on the empty cavity detuning, the placed between the two mirrors. This lens also places the PF at the bottom of the U-like curve is exactly zero, as cavity’s mode waist inside the 87Rb cell. the laser frequency decreases and then increases, while To produce experimental data sets (a) and (b) shown the cavity path (the auxiliary laser detuning) changes in the Fig.1, two pump lasers are tuned near D1 (795 nm) monotonically. Our model governed by Eq.4 cannot ex- and D2 (780 nm) corresponding to Ω1 and Ω2 fields in plain the arching behavior, since it does not account for Fig.3. The pump fields are coupled to a fiber beam the dependence of the dispersion on the lasing power. splitter and amplified by a solid state tapered amplifier to However, a more complete model which solves density powers ranging between 100 mW for set (a) and 170 mW matrix equations of the N-level scheme predicted such a for set (b), and then injected into a ring cavity through possibility [12]. a polarizing beam splitter (PBS). The D1 laser is tuned There is an ongoing debate whether or not the modified 4 cavity response leads to improved sensitivity (signal to duce the response, making our laser vibration insensitive. noise ratio) of path-change sensitive detectors. However, These findings broadly impact the fields of laser sensing laser-based sensors in certain applications might benefit and metrology, including laser ranging, laser gyroscopes, either from enhanced PF> 1 (for example gyroscopes [7]) vibrometers, and laser frequency standards. or reduced PF< 1, since sensitivity, i.e. the ratio of the EEM and SLC are thankful to the support of Vir- response to the lasing linewidth (uncertainty), scales as ginia Space Grant Consortium provided by grants 17-225- 1/PF [14, 20, 21]. The tunability and versatility of our 100527-010 and NNX15AI20H. We would like to thank system allows to probe either case. M. Simons, O. Wolfe, and D. Kutzke for assembling the In conclusion, we achieved about 2.7 ± 0.4 increase of earlier prototype of our laser. We also thank S. Rochester the laser response to the cavity-path length change rel- and D. Budker for their earlier work on N-level theoreti- ative to canonical lasers. We also can significantly re- cal model.

[1] W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. M. S. Shahriar, Opt. Express 24, 27444 (2016). Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. [12] D. T. Kutzke, O. Wolfe, S. M. Rochester, D. Budker, 57, 61 (1985). I. Novikova, and E. E. Mikhailov, Opt. Lett. 42, 2846 [2] B. P. Abbott and et al. (LIGO Scientific Collaboration (2017). and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 [13] J. Yablon, Z. Zhou, N. Condon, D. Hileman, S. Tseng, (2016). and S. Shahriar, Opt. Express 25, 30327 (2017). [3] Y. X. N. Jiyang Li, Haisha Niu, Optical Engineering 56, [14] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. 050901 (2017). Holland, and J. K. Thompson, Nature 484, 78 (2012). [4] D. Martynov and et al, Phys. Rev. D 93, 112004 (2016). [15] M. A. Norcia, J. R. K. Cline, J. A. Muniz, J. M. Robin- [5] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. son, R. B. Hutson, A. Goban, G. E. Marti, J. Ye, and Schmidt, Rev. Mod. Phys. 87, 637 (2015). J. K. Thompson, Phys. Rev. X 8, 021036 (2018). [6] Y.-S. Jang and S.-W. Kim, Int. J. Precis. Eng. Manuf. [16] Z. Zhou, J. Yablon, M. Zhou, Y. Wang, A. Heifetz, and 18, 1881 (2017). M. Shahriar, Optics Communications 358, 6 (2016). [7] M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, [17] E. E. Mikhailov, J. Evans, D. Budker, S. M. Rochester, M. Messall, and K. Salit, Phys. Rev. A 75, 053807 and I. Novikova, Optical Engineering 53, 102709 (2014). (2007). [18] N. B. Phillips, I. Novikova, E. E. Mikhailov, D. Bud- [8] G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, Phys. ker, and S. Rochester, Journal of Modern Optics 60, 64 Rev. Lett. 99, 133601 (2007). (2013), arXiv:1205.2567. [9] D. D. Smith, H. A. Luckay, H. Chang, and K. Myneni, [19] B. Efron and R. Tibshirani, An Introduction to the Boot- Phys. Rev. A 94, 023828 (2016). strap (Chapman and Hall/CRC, 1994). [10] D. D. Smith, H. Chang, P. F. Bertone, K. Myneni, L. M. [20] S. J. M. Kuppens, M. P. van Exter, and J. P. Woerdman, Smith, and B. E. Grantham, Opt. Express 26, 14905 Phys. Rev. Lett. 72, 3815 (1994). (2018). [21] C. Henry, IEEE Journal of Quantum Electronics 18, 259 [11] J. Yablon, Z. Zhou, M. Zhou, Y. Wang, S. Tseng, and (1982).