Is it Possible to do Experimental Cosmology Using Cold Atoms?

Is it Possible to do Experimental Cosmology Using Cold Atoms?

Renato Ejnisman^* and Nicholas P.Bigelow

Department of Physics and Astronomy and Laboratory for Laser Energetics

University of Rochester, Rochester NY 14627 USA

Received December 10, 1997

We analyze the possibility of using cold atoms to experimentally investigate the cosmic relic neutrinos and photons that were created after the Big Bang and presently constitute a background at temperatures of a few degress Kelvin.

1 Introduction

Since the discovery of the cosmic microwave background (CMB) by Penzias and Wilson [1], cosmology departed from the status of a philosophical discipline to that of a modern experimental branch of science. The CMB consists of photons at a temperature T_g = 2.7 K that were originally created at about 10⁵ s after the Big Bang. More recent observations have shown an anisotropy in the background at the level DT_g/T_g» 10^-5 that gives hints on how the process of galaxy formation took place [2].

An even more interesting and challenging achievement would be the detection of the relic cosmic neutrino background (CNB). The temperature of such neutrinos, T_n = 1.9 K, implies a relatively sparse density (r_n » 300 cm^-3 for zero chemical potential), which together with their extremely weak interaction with matter have eliminated any chances of direct observation up to date. Since the creation of the CNB occurred about 1s after the Big Bang, it serves as a probe to look that far back in the past. Its observation would therefore provide a very stringent test of the Big Bang theory and cosmological models, not to mention information about the physics of the neutrinos themselves.

The observation of the anisotropy of the CMB and especially the detection of the CNB comprise very difficult tasks, thus requiring extremely accurate experimental techniques. Recently, there have appeared in the literature many works and proposals of other precision experiments that make use of cold atoms (T £ 100 mK). Such experiments and proposals entail a wide range of applications in physics with extremely high accuracy and relatively low cost. Examples are: atomic clocks (with expected precision of 1 part in 10¹⁶) [3], achievement of Bose-Einstein condensation [4], laser trapping of radioactive atoms [5,6] and proposals for experiments to measure atomic parity violation [7], permanent electric dipole moments [8] and studies of b-decay [6]. Motivated by these results and proposals, we performed a thorough study to check the feasibility of employing cold atoms as a tool to observe the CMB and the CNB, whose main results are reported in the present work. As will be shown, the conclusions for the CNB case are rather discouraging and for the CMB case quite tempting.

2 Prospects for the CNB

Since the typical cross-sections for interaction with neutrinos goes as s_n » G_F²E_n², one would obtain s_n » 10^-62cm² for massless relic neutrinos or a factor of a few orders of magnitude higher for massive ones. Such a weak interaction deny any possibilities to detect the CNB by conventional methods.

In order to circumvent this problem, a number of experiments making use of coherent effects were proposed [9,10,11]. They rely on the fact that one can think of the neutrinos as de Broglie waves of wavelength much longer than the average size and spacing of the scatterers. However, Langacker et al [12] and Cabibbo and Maiani [13] showed that refs. [10] overestimated this effect. Also, Tupper et al [11] arrived at negative conclusions for their proposal. This leaves only the work of Stodolsky [9] as a possible (but improbable for the cases studied by him) route to measure the CNB. Other proposals to use resonant absorption of cosmic neutrinos [14] will not be discussed here (all proposals are reviewed in ref. [15]).

The idea of Opher and Lewis [10] relied on measurements of mechanical effects due to the total reflection of the neutrinos on a target of refraction index n_n (for iron, e.g., assuming a density of 10²³cm^-3, |n_n-1| @ 10^-10). Here forth, n represents only the electron neutrino; the effects of the other neutrino flavors will be disregarded. Total reflection would occur for incidence angles below . However, as shown in refs. [12,13], there was a flaw in the calculation. The main problem was that the incident wave would have a penetration depth D = l/2pq_c. This was by far larger than the size of the detectors considered, so most of the wave would actually be transmitted without transferring momentum to the target. Since |n_n -1| goes linearly with density, things would be even worse for cold atoms. Typical densities of cold atomic samples are r » 10¹⁰-10¹³cm^-3, so |n_n -1| would be smaller by at least a factor of about 10¹⁰ comparing to the case of iron.

Another scheme proposed by Stodolsky deserves a closer analysis. In his proposal [9], one would make use of the parity violating term of the V-A Hamiltonian of the weak interactions. For a target consisting of a single electron, the term goes as , where is the spin of the electron and its velocity. Therefore, the two helicity states of the electron, corresponding to = ±1, are split in energy. In fact, one can think of the neutrino background as having a velocity with respect to an electron at rest on earth due to the motion of the solar system through the galaxy. For a target consisting of N particles, this would give rise to an energy shift between the case when the particles are polarized parallel or anti-parallel to the velocity of [12]:

where G_F is the weak coupling constant, ásñ is the average spin of the target, is the number density of neutrinos (anti-neutrinos) and x is a factor that equals 1 for massless neutrinos and 1 / 2 for m_n >> k_BT_n.

¿From the above, it is clear that this scheme would only yield non-zero results if there is an asymmetry between the number of neutrinos and anti-neutrinos. These particles should possess a Fermi distribution at a temperature T_n = 1.9 K, such that the number density of neutrinos at energies corresponding to momenta between q and q+dq is given by [16]:

with an analogous expression for anti-neutrinos (with an opposite sign for the chemical potential m).

Although it is generally assumed that m_n = 0, it is not possible to rule out a non-zero value for the chemical potential. In fact, in the case of complete degeneracy, |m_n| >> k_BT_n, the density of neutrinos (in natural units) is given by:

for the case of m_n > 0. The case of negative m_n swaps the role of neutrinos and anti- neutrinos.

The best current bounds on the chemical potentials come from indirect evidences. The fact that the neutrino energy should not be larger than the critical density, r_c = 2 ×10^-29 g cm^-3, implies an upper limit on |m_n| of about 6 ×10^-3 eV. Big Bang nucleosynthesis sets a more stringent limit of -0.5 £ m_n £ 3 in units of 10^-4 eV [17], which is however not as trustworthy as the bounds from the critical density [14]. Also of interest is the fact that if the sterile neutrino solution to the solar and atmospheric neutrino anomalies is true, then a non-zero value for m_n is required to still make this solution compatible with Big Bang nucleosynthesis [18], which might prove to be a lower bound for |m_n|.

Assuming the quite unlikely case of complete degeneracy [19] and a target of N particles, the energy shift between spin parallel and anti-parallel to the earth's motion can be obtained by using (1) and (2):

DE @ 6×10^-28N ásñm_n³ (3)

where both DE and m_n are in units of eV.

The cases analyzed by Stodolsky included a single trapped electron (for which DE @ 6 ×10^-35 eV, using m_n = 6×10^-3 eV) and a large ferromagnet (for which DE @ 6 ×10^-8 eV, for the same condition and N = 10²⁷). In both cases, the numbers turn out to be very disappointing for any reasonable value of m_n.

One could think of using cold atoms in two ways. The first one is to take advantage of the extremely low temperatures that can be obtained with these atoms and try to verify a temperature difference between a sample polarized parallel or anti-parallel to the velocity of the solar system. In a recent experiment [20], about 10⁹ atoms of sodium of ásñ = 1 were initially captured in a magnetic trap and, by means of evaporative cooling, about 10⁷ atoms attained a temperature of about 1 mK. Later, these atoms suffered adiabatic expansion which brought them to T » 0.1 nK. Using N = 10⁷ and DE = 9 ×10^-15 eV (or T » 10^-10 K), one could obtain an upper limit of |m_n| £ 100 eV. This bound is still rather weak when compared to the indirect limits mentioned above.

In a more optimistic scenario, one could use Cs¹ which can be trapped at the F=4 state (increasing the average spin) and can have about an order of magnitude more atoms than sodium captured in the magneto-optical trap (which is used as a first stage to load the magnetic trap [20]). Also, one could take advantage of the adiabatic expansion to decrease the temperature of the cloud even further. The temperature of an ideal gas in adiabatic expansion goes with the radius R as T µ R^-3(g-1). For a monoatomic gas, g = 5/3 and one obtains T µ R^-2. In fact, this is an oversimplification of the problem because at these temperatures and densities collective effects should be taken into account. Even so, this simplified picture yields a temperature of 10^-10 K, which is close to the observed [20] for a cloud initially at T = 1 mK and R = 10 mm and final size of about 1mm. If we push this limit a little further, we can start with a cloud of radius 1 mm and expand it to 1mm to obtain a final temperature of T = 1 pK. For such temperatures, and assuming N = 10⁸ and ásñ = 4, it would be possible to obtain |m_n| £ 3 eV. Even though this limit is better than direct bounds obtained from measurements of b-decay spectra [21], indirect evidences still provide more stringent cut-offs for m_n. In fact, to achieve the limit |m_n| < 6 ×10^-3 eV, it would be necessary to have a sample with N @ 10¹⁶ atoms at a hypothetical temperature of T = 1 pK. It is hard to foresee the achievemet of such conditions in a near future, even though similar numbers were used in recent experimental proposals [22,23]. The requirements for each of the scenarios considered above are summarized in table 1.

An alternative route would be to try to measure the energy of the system directly. In atomic clocks, the transition 6S_1/2 (F = 4) ® 6S_1/2 (F = 5) of Cs is expected to yield results with a precision of 1/10¹⁶ using cold atoms [3]. This corresponds to an accuracy in energy of DE @ 4 ×10^-21 eV. Since this kind of measurement yields the energy of the transition of individual atoms, the energy shift from having the atoms polarized parallel or anti-parallel to the motion of the earth would not reflect that of the whole system. Therefore, eq. (1) would not contain the factor N in this case. Hence, one would obtain the weak bound |m_n| < 120 eV.

3 Prospects for the CMB

In the case of the cosmic microwave background, one has to check not only for the feasibility of schemes for observation, but also for how they compare to existing methods. A possible way to use cold atoms to detect the CMB can be the following: start with a sample of atoms in a magnetic trap with all the atoms in the lower hyperfine state (e.g., F=4 for Cs); then, photons from the CMB with energies close to that of the 6S_1/2 (F = 4) ® 6S_1/2 (F = 3) transition (f=9.2 GHz) could promote some of the trapped atoms to the lower untrapped state which would create a loss mechanism for such traps.

The loss rate from the trap could be measured with the trap shielded from the CMB and then compared with the unshielded case. From the difference in loss rate, the contribution due to loss from CMB absorption could be inferred. A somewhat similar experiment was performed by Schappe and coworkers to measure the cross-section for collisions between electrons and atoms [24]. In our case, instead of the cross-section, one would be interested in obtaining the flux of photons and, from that, the temperature of this background radiation.

The work with electrons yielded cross-sections with uncertainties of about 10% [24]. Assuming that for the case of CMB the uncertainty on the photon flux obtained by this method is of 10%, one would obtain an uncertainty in the temperature of 7%. This is to be compared with observations of the CMB at comparable wavelengths that have a precision on absolute temperatures of about 2% [25]. Optimized traps should be able to yield results of such precision for reasonable loss rates.

Before we move on to obtain the loss rate of atoms due to the interaction with the CMB, it is important to consider a technical difficulty that would have to be faced. It comes from the fact that at 9.2 GHz, the number of photons from the low energy tail of the blackbody radiation at room temperature is about two orders of magnitude higher than the one due to the CMB. A way around that is to use ⁴He cryostats that can cool the system down to about 2 K at the lowest temperatures achievable [26], which is sufficiently low for the blackbody radiation due to ambient temperatures not to pose major difficulties.

The cross-section for a F = 4 ® F = 3 transition can be calculated by standard equations [27]. From them, a CMB photon flux of approximately 20 cm^-2s^-1Hz^-1 and a 10⁴ enhancement due to the use of a horn with an area of the order of 1 m², one obtains a loss rate of approximately 10⁴ atoms per second for a trap of 10¹⁰ atoms. This is certainly unacceptable compared to a typical lifetime of about one minute for these traps [20].

A way to improve this scenario is to employ high-Q cavities to increase the absorption of photons and therefore the loss rate due to the interaction with the CMB. Cavities of Q up to 10⁶ have been used with magnetic fields inside for experiments with Rydberg atoms [26]. The loss rate would then be increased by a factor Q l³ /V, where l » 3 cm for 9.2 GHz and V is the cavity mode volume. If the loss rate is increased in this way by 10⁵, it will become approximately 10⁹s^-1 for a trap with 10¹⁰ atoms, which is now a measurable rate. As stated before, the challenge then would be to make sure that a 2% precision is achieved as in the current existing methods.

It is important to notice that a nice feature of this kind of measurement is that the "antena" we use (i.e., the atoms) have a narrow band for absorption. This might be interesting for studies of the CMB that envolve detailed knowledge of the microwave frequencies. Another good facet of the present technique is that the part of the CMB spectrum where it would enable investigations (low energy tail of the blackbody spectrum) is more subject to eventual distortions of a Plack distribution [25].

Of course, other ways can be thought of to approach this problem. For example, one could perform experiments with superconducting cavities (of Q » 10⁹ - 10¹¹) in a microgravity environment where the atoms are initially prepared in a certain state (e.g., by optical pumping). A subsequent observation would reveal if they had absorbed photons and changed to another state. In fact, for this case, the use of cold Rydberg atoms would be even more appropriate since their large dipole moments enable them to interact more strongly with microwave radiation. Even though the interaction of blackbody radiation with Rydberg atoms has been observed for temperatures down to T=50 K [26,28], it is not clear how it would be possible to obtain a precision of 2% at the even lower temperatures of the CMB. Also, the fact that a microgravity environment is wanted would make this experiment much more costly.

4 Conclusions

We have analyzed the possibility to use cold atoms as a tool to detect the cosmic neutrino background and the cosmic microwave background. In the former case, although it would be possible to obtain upper limits on the neutrino chemical potential, these limits would not be competitive with other bounds estimated through indirect evidences. In the latter case, the method considered to measure the flux of photons from the CMB may be able to yield results with levels of precision close to that of other existing methods and should therefore be further explored.

To conclude, we would like to state that we are confident that techniques similar to the ones considered here and presented in the references of the introduction are a good alternative to make precise and fundamental physics at relatively low costs. In particular, we believe that analyses such as the one performed in this work will eventually lead to a feasible way to perform the important task of detecting the cosmic neutrino background.

We are very grateful to A.C.Melissinos for essential discussions. R.E. was supported by a CNPq fellowship (Brazil). This work was supported by the NSF and the David and Lucile Packard Foundation.

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Publication Dates

• Publication in this collection
  28 Apr 1999
• Date of issue
  Mar 1998

History

• Received
  10 Dec 1997