Shape fluctuations and optical transition of ${\mathrm{He}}_2^{*}$ excimer tracers in superfluid ${}^4\mathrm{He}$

Metastable ${\mathrm{He}}_2^{*}$ excimer molecules have been utilized as tracer particles of the normal component in superfluid ${}^4\mathrm{He}$ (He II) which can be imaged via laser-induced fluorescence. These excimer molecules form tiny bubbles in He II and can bind to quantized vortices at sufficiently low temperatures, thereby allowing for direct visualization of vortex dynamics in an inviscid superfluid. However, the $a^3{\mathrm{\Sigma }}_u^{+}\to c^3{\mathrm{\Sigma }}_g^{+}$ optical absorption line, which is responsible for the fluorescence imaging of the ${\mathrm{He}}_2^{*}$ molecules, is controlled by fluctuations on the bubble shape, and its exact line profile is not known at low temperatures. In this paper, we present a bubble model for evaluating the surface fluctuation eigenmodes of the excimers in He II. The line profile of the $a^3{\mathrm{\Sigma }}_u^{+}\to c^3{\mathrm{\Sigma }}_g^{+}$ transition is calculated at different temperatures by considering both the zero-point and thermal fluctuations on the bubble shape. We show that as the temperature drops from 2 K to 20 mK, the peak absorption strength is enhanced by a factor of about five, accompanying a blueshift of the peak location by about 2 nm. A double-peak line profile due to the rotational levels of the molecular core can be resolved. This bubble model also allows us to evaluate the stiffness of the ${\mathrm{He}}_2^{*}$ bubbles and hence their diffusion constant in He II due to scattering off thermal phonons. Our results will aid the design of future experiments on imaging quantized vortices in He II using ${\mathrm{He}}_2^{*}$ tracers.

Figure 1

(a) Schematic diagram showing the cycling transitions for imaging the ${\mathrm{He}}_2^{*}$ triplet molecules. The levels labeled 0, 1, 2 are the vibrational levels for each corresponding electronic state. (b) Schematic diagram showing the $a\left(0\right)\to c\left(0\right)$ and $c\left(0\right)\to d\left(0\right)$ absorption transition lines.

Figure 2

Interaction potential between a ${\mathrm{He}}_2^{*}$ excimer molecule and a ground state helium atom for the molecule (a) in the $a^3{\mathrm{\Sigma }}_u^{+}$ state; (b) in the $c^3{\mathrm{\Sigma }}_g^{+}$ state along the collinear direction of the molecular core; and (c) in the $c^3{\mathrm{\Sigma }}_g^{+}$ state along the “T” direction as defined in Ref. [43]. Red dots are the data listed in Table II of Ref. [43]. Solid curves are our fits.

Figure 3

Calculated frequencies of the $l=0$ and $l=2$ surface modes of the ${\mathrm{He}}_2^{*}(a^3{\mathrm{\Sigma }}_u^{+}$) bubble as a function of helium pressure at 0.5 and 2 K.

Figure 4

$a\left(0\right)\to c\left(0\right)$ absorption line at 2 K under zero applied pressure normalized by $I_0$. The blue dashed curve and the red dotted curve are the absorption lines due to the $P$ branch transition $J$:$1\to 0$ and $R$ branch transition $J$:$1\to 2$. The black solid curve is the overall absorption line with the total area normalized to unity.

Figure 5

(a) The peak location and (b) the FWHM of the overall $a\left(0\right)\to c\left(0\right)$ absorption line as a function of pressure at 2 K. The red curves represent the calculated results. The crosses and circles are experimental data of Eltsov et al. [40], taken in the temperature range of 1.76 to 2.05 K.

Figure 6

Calculated overall $a\left(0\right)\to c\left(0\right)$ absorption lines at various temperatures normalized by $I_0$. The total area below each curve is normalized to unity

Figure 7

The momentum-transfer cross section ${\sigma }_T\left(k\right)$ calculated using the $l=0,1,2$ modes. The dotted and dashed curves represent the thermal factor $-k^4\partial n/\partial k$, which are scaled vertically for visibility.

Figure 8

Calculated diffusion coefficient $D$ of ${\mathrm{He}}_2^{*}(a^3{\mathrm{\Sigma }}_u^{+}$) molecules in He II. The triangles represent the data extracted from Zmeev et al.'s experiment [67]. The red solid line is the diffusion curve calculated using the rigid-sphere model as discussed in the text.