Few-fermion thermometry

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1 Few-fermion thermometry Phys. Rev. A 97, (2018) Tomasz Sowiński Institute of Physics of the Polish Academy of Sciences Co-authors: Marcin Płodzień Rafał Demkowicz-Dobrzański FEW-BODY PROBLEMS FewBody.ifpan.edu.pl

2 motivation Selim Jochim s experimental group in Heidelberg 6Li Atomic number (Z) 3 Nucleons (Z+N) 6 Total electronic spin (S) 1/2 Total nuclear spin (I) 1 message from experiments two flavours of spinless fermions well controlled number of particles one-dimensional confinement strong interactions between flavours Hyperfine states (F=S+I) 1/2 or 3/2 "i = F =3/2,m F = 3/2i #i = F =1/2,m F =1/2i what for??? extreme control on the system extreme precision of measurements from one' to 'many' crossover

3 message from experiments F. Serwane et al., Science 332, 6027 (2011)

4 message from experiments F. Serwane et al., Science 332, 6027 (2011)

5 the model two distinguishable flavours of fermions ( and ) both flavours may have different masses both flavours confined in one-dimensional trap opposite spins do interact via sort range δ-like potential Ĥ = X Z dx ˆ (x) + g Z apple ~ 2 2m d 2 dx 2 + V (x) dx ˆ # (x) ˆ " (x) ˆ "(x) ˆ #(x) ˆ (x) (anti-)commutation relations n ˆ (x), ˆ (x )o 0 = (x x 0 ) n ˆ (x), ˆ (x )o 0 =0 h Ĥi h Ĥi ˆN", = ˆN#, =0 Z ˆN = dx ˆ (x) ˆ (x)

6 many-body spectrum Notice the quasi-degeneracy of the spectrum for strong repulsions! Utilise it for precise thermometry!

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sha1_base64="qpm0mkrgdw77y38keznfacvy8au=">aaaco3icbzbnbxmxeizny1cbphqgny4wevkrosgbwprcikciy5gatliciq93dmvv611sl1jk7d/g18ar/gnnbogj3he2lvoqr2tp1tszmvgtvepar+m3aopa9rs3b21udw7fuxtvu3v/wbetaynwkkpvmtoew1rs49rjp/c0msilrofjcv5yvt/5gmbkuh+5zyxzgudazljwf6jfl7jxqbwnr++ghox4nrdmcohjxmvwz33pbffkdbplflwqrhnallo9oqctybwzvzemq0rii6q32yfwh4vul5awoi5qo6g4tboyvm7uuxfskgw6rlzycxhoc5wfq3mbdu7bnzxksuhskpumpo1im16d8lywdlkkobpg7sz+w1uf/6vnapen5l7qqnaoxxprvivisrlcrfjpudi1diyli8otrjzxgmyfmb2wyhaat+f4ndeiuvemtxpp+3rw0kcrtpelnff4nhw+as1+/ows0/fwennb/havn3mx5gwb8agewy7ecaateaohmaubh+etfigv0efoe/qj+rlu3yguzh7cx4p+/wf6b62q</latexit> <latexit sha1_base64="wfdgubqrdcqbckq9yt5kqmegvzq=">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</latexit> thermal sensitivity ˆ T = 1 Z e Ĥ/k BT = X i P i E i ihe i Measurements: Some set of probabilities as functions of T. The limit is determined by the Fisher information. {p i (T )}! F({p i })= X i T 2 1 n F({p i }) 1 p i dpi dt 2 IF: One can measure Gibbs probabilities {p i (T )} = {P i (T )} then the Fisher information is the largest possible! F Q (ˆ T )= Ĥ2 T 4 = Tr[ˆ T Ĥ2 ] Tr[ˆ T Ĥ] 2 T 4

9 quantum Fisher information Fisher Information N = N = 2 g=14 F Q (ˆ T )= F Q (ˆ T )= Temperature T Ĥ2 T 4 = Tr[ˆ T Ĥ2 ] Tr[ˆ T Ĥ] 2 T 4

10 the simplest scenario Fisher Information N = N = 2 g=14 F Q (ˆ T )= Temperature T The probability of finding a particle just above the Fermi level

11 Occupations of the lowest orbitals Fisher Information N = N = 2 g=14 F Q (ˆ T )= Temperature T Probabilities p 0, p 1, p 2, p 3, p 4 that in two the lowest orbitals one can find 0, 1, 2, 3, or 4 particles. {

12 All occupations are known Fisher Information N = N = 2 g=14 F Q (ˆ T )= Temperature T When we know all probabilities of finding four particles in different configurations

13 the fundamental bound Quantum Fisher Information g = 14 g = 10 Fundamental bound g = 5 Fundamental bound when level spacings can be engineered N = N = 2 L. A. Correa et al. Phys. Rev. Lett. 114, (2015) g = Temperature T

14 Few-fermion thermometry Phys. Rev. A 97, (2018) Tomasz Sowiński Institute of Physics of the Polish Academy of Sciences Co-authors: Marcin Płodzień Rafał Demkowicz-Dobrzański FEW-BODY PROBLEMS FewBody.ifpan.edu.pl