arXiv:0705.3155 · Physical Review Letters

Observation of a Topological and Parity-Dependent Phase of m=0 Spin States

Koji Usami, Mikio Kozuma

Published Oct 4, 2007• 8 citations

Abstract

A Ramsey interrogation scheme was used to measure the phase shift of laser-cooled $^{87}\mathrm{Rb}$ clock-transition pseudospins arising as a result of a reversal of a bias-magnetic field, i.e., $\mathbf{B}\ensuremath{\rightarrow}\ensuremath{-}\mathbf{B}$, during the interrogation. While no phase shift occurred when the reversal was sudden, the Ramsey fringes were shifted by a factor of $\ensuremath{\pi}$ when the reversal was adiabatic. We thus verified the prediction that the spin states $|F,m=0⟩$ acquire a purely topological and parity-dependent phase factor of $(\ensuremath{-}1{)}^{F}$ as a result of $\mathbf{B}\ensuremath{\rightarrow}\ensuremath{-}\mathbf{B}$.

DOI: 10.1103/physrevlett.99.140404

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arXiv:0705.3155 · Physical Review Letters

Observation of a Topological and Parity-Dependent Phase of m=0 Spin States

Koji Usami, Mikio Kozuma

Published Oct 4, 2007• 8 citations

arXiv HTML PDF

Abstract

A Ramsey interrogation scheme was used to measure the phase shift of laser-cooled

^{87}\mathrm{Rb}

clock-transition pseudospins arising as a result of a reversal of a bias-magnetic field, i.e.,

\mathbf{B}\ensuremath{\rightarrow}\ensuremath{-}\mathbf{B}

, during the interrogation. While no phase shift occurred when the reversal was sudden, the Ramsey fringes were shifted by a factor of

\ensuremath{\pi}

when the reversal was adiabatic. We thus verified the prediction that the spin states

|F,m=0⟩

acquire a purely topological and parity-dependent phase factor of

(\ensuremath{-}1{)}^{F}

as a result of

\mathbf{B}\ensuremath{\rightarrow}\ensuremath{-}\mathbf{B}

DOI: 10.1103/physrevlett.99.140404

Related papers

Interference fringes of m=0 spin states under the Majorana transition caused by rapid half-rotation of a magnetic field

Atsushi Takahashi, Atsuo Morinaga

The phase shift and visibility of fringes in the Ramsey atom interferometer composed of the $|F=1$ ,${m}_{F}=0\ensuremath{\rangle}$ and $|F=2$ ,${m}_{F}=0\ensuremath{\rangle}$ states were examined systematically for rapid half-rotation of the magnetic field. It was verified that the phase shifts by $\ensuremath{\pi}$ rad in the adiabatic regime, but it does not shift from the original one in the nonadiabatic regime. These results support Robbins and Berry's claim [J. M. Robbins and M. V. Berry, J. Phys. A 27, L435 (1994)]. The fact that the interference fringes disappear in the intermediate regime and reappear in the nonadiabatic regime can be explained by the Majorana transition caused by a rapid reverse of the magnetic field.The phase shift and visibility of fringes in the Ramsey atom interferometer composed of the $|F=1$,${m}_{F}=0\ensuremath{\rangle}$ and $|F=2$,${m}_{F}=0\ensuremath{\rangle}$ states were examined systematically for rapid half-rotation of the magnetic field. It was verified that the phase shifts by $\ensuremath{\pi}$ rad in the adiabatic regime, but it does not shift from the original one in the nonadiabatic regime. These results support Robbins and Berry's claim [J. M. Robbins and M. V. Berry, J. Phys. A 27, L435 (1994)]. The fact that the interference fringes disappear in the intermediate regime and reappear in the nonadiabatic regime can be explained by the Majorana transition caused by a rapid reverse of the magnetic field.

Physical Review AApr 29, 20100 citations

Phase shift of an adiabatic rotating magnetic field in Ramsey atom interferometry for m=0 sodium-atom spin states

Atsushi Takahashi, Hiromitsu Imai, 和也沼崎 +1 more

The $\ensuremath{\pi}$-phase shift between $|F=1,{m}_{F}=0⟩$ and $|F=2,{m}_{F}=0⟩$ states for an adiabatic rotation of a magnetic field observed in Ramsey atom interferometry arises from the negative sign of the transition amplitude between the $|1,0⟩$ and $|2,0⟩$ states when the wave functions are rotated to the opposite direction. Using a two-photon Raman atom interferometer with a cold ensemble of sodium atoms, the phase shift for a half-rotation of the magnetic field was confirmed to be $\ensuremath{\pi}\text{ }\text{rad}$ with an uncertainty of 4%, and the phase shift for a noncyclic adiabatic rotation of the magnetic field was investigated.The $\ensuremath{\pi}$-phase shift between $|F=1,{m}_{F}=0⟩$ and $|F=2,{m}_{F}=0⟩$ states for an adiabatic rotation of a magnetic field observed in Ramsey atom interferometry arises from the negative sign of the transition amplitude between the $|1,0⟩$ and $|2,0⟩$ states when the wave functions are rotated to the opposite direction. Using a two-photon Raman atom interferometer with a cold ensemble of sodium atoms, the phase shift for a half-rotation of the magnetic field was confirmed to be $\ensuremath{\pi}\text{ }\text{rad}$ with an uncertainty of 4%, and the phase shift for a noncyclic adiabatic rotation of the magnetic field was investigated.

Physical Review ANov 20, 20098 citations

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Jonathan M. Robbins, Michael Berry

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Topological vortex formation in a Bose-Einstein condensate under gravitational field

Yuki Kawaguchi, Mikio Nakahara, Tetsuo Ohmi

Topological phase imprinting is a unique technique for vortex formation in a Bose-Einstein condensate (BEC) of an alkali-metal gas, in that it does not involve rotation: the BEC is trapped in a quadrupole field with a uniform bias field which is reversed adiabatically leading to vortex formation at the center of the magnetic trap. The scenario has been experimentally verified by Leanhardt et al. employing $^{23}\mathrm{Na}$ atoms. Recently similar experiments have been conducted by Hirotani et al. in which a BEC of $^{87}\mathrm{Rb}$ atoms was used. In the latter experiments the authors found that fine-tuning of the field reverse time ${T}_{\mathrm{rev}}$ is required to achieve stable vortex formation. Otherwise, they often observed vortex fragmentation or a condensate without a vortex. It is shown in this paper that this behavior can be attributed to the heavy mass of the $\mathrm{Rb}$ atom. The confining potential, which depends on the eigenvalue ${m}_{B}$ of the hyperfine spin $\mathbit{F}$ along the magnetic field, is now shifted by the gravitational field perpendicular to the vortex line. Then the positions of two weak-field-seeking states with ${m}_{B}=1$ and 2 deviate from each other. This effect is more prominent for BECs with a heavy atomic mass, for which the deviation is greater and, moreover, the Thomas-Fermi radius is smaller. We found, by solving the Gross-Pitaevskii equation numerically, that two condensates interact in a very complicated way leading to fragmentation of vortices, unless ${T}_{\mathrm{rev}}$ is properly tuned.Topological phase imprinting is a unique technique for vortex formation in a Bose-Einstein condensate (BEC) of an alkali-metal gas, in that it does not involve rotation: the BEC is trapped in a quadrupole field with a uniform bias field which is reversed adiabatically leading to vortex formation at the center of the magnetic trap. The scenario has been experimentally verified by Leanhardt et al. employing $^{23}\mathrm{Na}$ atoms. Recently similar experiments have been conducted by Hirotani et al. in which a BEC of $^{87}\mathrm{Rb}$ atoms was used. In the latter experiments the authors found that fine-tuning of the field reverse time ${T}_{\mathrm{rev}}$ is required to achieve stable vortex formation. Otherwise, they often observed vortex fragmentation or a condensate without a vortex. It is shown in this paper that this behavior can be attributed to the heavy mass of the $\mathrm{Rb}$ atom. The confining potential, which depends on the eigenvalue ${m}_{B}$ of the hyperfine spin $\mathbit{F}$ along the magnetic field, is now shifted by the gravitational field perpendicular to the vortex line. Then the positions of two weak-field-seeking states with ${m}_{B}=1$ and 2 deviate from each other. This effect is more prominent for BECs with a heavy atomic mass, for which the deviation is greater and, moreover, the Thomas-Fermi radius is smaller. We found, by solving the Gross-Pitaevskii equation numerically, that two condensates interact in a very complicated way leading to fragmentation of vortices, unless ${T}_{\mathrm{rev}}$ is properly tuned.

arXiv:cond-mat/0405354Physical Review AOct 5, 200410 citations

Imaging the topological sign of a Dirac mass with spin-polarized tunneling

Payman Kazemikhah, Ruolun Zhang, Byunghoon Kim +5 more

A Dirac mass is characterized not only by its magnitude but also by its sign, which influences the Berry curvature, Hall response, and the propagation direction of chiral boundary modes. However, local spectroscopy of magnetic topological surfaces has so far accessed only the gap magnitude $|M|$ . Here we show how the topological sign of $M$ can be determined locally. We first establish the sign information that spin-resolved tunneling can access and the reference required to obtain an absolute Hall sign. Although spin--orbital entanglement generally obscures the relation between measured spin and Dirac pseudospin, crystalline rotation symmetry alleviates this ambiguity at the band edge. On a $C_3$ axis, an axial tip couples to a fully spin-polarized state whose tunneling asymmetry directly reveals the sign of the occupied-band Berry curvature, independent of orbital composition. We further show that the zeroth Landau level provides an independent mass-sign measurement with an unpolarized tip, while opposite domains and their chiral domain-wall modes provide self-consistency checks. These results pave the way for imaging the locations where the topological character changes within a material, rather than merely identifying where a spectral gap appears.A Dirac mass is characterized not only by its magnitude but also by its sign, which influences the Berry curvature, Hall response, and the propagation direction of chiral boundary modes. However, local spectroscopy of magnetic topological surfaces has so far accessed only the gap magnitude $|M|$. Here we show how the topological sign of $M$ can be determined locally. We first establish the sign information that spin-resolved tunneling can access and the reference required to obtain an absolute Hall sign. Although spin--orbital entanglement generally obscures the relation between measured spin and Dirac pseudospin, crystalline rotation symmetry alleviates this ambiguity at the band edge. On a $C_3$ axis, an axial tip couples to a fully spin-polarized state whose tunneling asymmetry directly reveals the sign of the occupied-band Berry curvature, independent of orbital composition. We further show that the zeroth Landau level provides an independent mass-sign measurement with an unpolarized tip, while opposite domains and their chiral domain-wall modes provide self-consistency checks. These results pave the way for imaging the locations where the topological character changes within a material, rather than merely identifying where a spectral gap appears.

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Preliminary results of the trapped atom clock on a chip

Clément Lacroûte, Friedemann Reinhard, F. Ramírez-Martínez +4 more

We present an atomic clock based on the interrogation of magnetically trapped (87)Rb atoms. Two photons, in the microwave and radiofrequency domain, excite the clock transition. At a magnetic field of 3.23 G the clock transition from |F = 1, m(F) = -1> to |F = 2, m(F) = 1> is 1st-order insensitive to magnetic field variations. Ramsey interrogation times longer than 2 s can be achieved, leading to a projected clock stability in the low 10(-13) at 1 s for a cloud of 10(5) atoms. We use an atom chip to cool and trap the atoms. A coplanar waveguide is integrated to the chip to carry the Ramsey interrogation signal, making the physics package as small as (5 cm)(3). We describe the experimental setup and show preliminary Ramsey fringes of line width 1.25 Hz.We present an atomic clock based on the interrogation of magnetically trapped (87)Rb atoms. Two photons, in the microwave and radiofrequency domain, excite the clock transition. At a magnetic field of 3.23 G the clock transition from |F = 1, m(F) = -1> to |F = 2, m(F) = 1> is 1st-order insensitive to magnetic field variations. Ramsey interrogation times longer than 2 s can be achieved, leading to a projected clock stability in the low 10(-13) at 1 s for a cloud of 10(5) atoms. We use an atom chip to cool and trap the atoms. A coplanar waveguide is integrated to the chip to carry the Ramsey interrogation signal, making the physics package as small as (5 cm)(3). We describe the experimental setup and show preliminary Ramsey fringes of line width 1.25 Hz.

IEEE Transactions on Ultrasonics Ferroelectrics and Frequency ControlJan 1, 201020 citations

A Ramsey interrogation scheme was used to measure the phase shift of laser-cooled

^{87}\mathrm{Rb}

clock-transition pseudospins arising as a result of a reversal of a bias-magnetic field, i.e.,

\mathbf{B}\ensuremath{\rightarrow}\ensuremath{-}\mathbf{B}

, during the interrogation. While no phase shift occurred when the reversal was sudden, the Ramsey fringes were shifted by a factor of

\ensuremath{\pi}

when the reversal was adiabatic. We thus verified the prediction that the spin states

|F,m=0⟩

acquire a purely topological and parity-dependent phase factor of

(\ensuremath{-}1{)}^{F}

as a result of

\mathbf{B}\ensuremath{\rightarrow}\ensuremath{-}\mathbf{B}

The phase shift and visibility of fringes in the Ramsey atom interferometer composed of the

|F=1

,${m}_{F}=0\ensuremath{\rangle}$ and

|F=2

,${m}_{F}=0\ensuremath{\rangle}$ states were examined systematically for rapid half-rotation of the magnetic field. It was verified that the phase shifts by $\ensuremath{\pi}$ rad in the adiabatic regime, but it does not shift from the original one in the nonadiabatic regime. These results support Robbins and Berry's claim [J. M. Robbins and M. V. Berry, J. Phys. A 27, L435 (1994)]. The fact that the interference fringes disappear in the intermediate regime and reappear in the nonadiabatic regime can be explained by the Majorana transition caused by a rapid reverse of the magnetic field.

The $\ensuremath{\pi}$-phase shift between

|F=1,{m}_{F}=0⟩

and

|F=2,{m}_{F}=0⟩

states for an adiabatic rotation of a magnetic field observed in Ramsey atom interferometry arises from the negative sign of the transition amplitude between the

|1,0⟩

and

|2,0⟩

states when the wave functions are rotated to the opposite direction. Using a two-photon Raman atom interferometer with a cold ensemble of sodium atoms, the phase shift for a half-rotation of the magnetic field was confirmed to be $\ensuremath{\pi}\text{ }\text{rad}$ with an uncertainty of 4%, and the phase shift for a noncyclic adiabatic rotation of the magnetic field was investigated.

^{23}\mathrm{Na}

atoms. Recently similar experiments have been conducted by Hirotani et al. in which a BEC of

^{87}\mathrm{Rb}

atoms was used. In the latter experiments the authors found that fine-tuning of the field reverse time

{T}_{\mathrm{rev}}

is required to achieve stable vortex formation. Otherwise, they often observed vortex fragmentation or a condensate without a vortex. It is shown in this paper that this behavior can be attributed to the heavy mass of the

\mathrm{Rb}

atom. The confining potential, which depends on the eigenvalue

{m}_{B}

of the hyperfine spin $\mathbit{F}$ along the magnetic field, is now shifted by the gravitational field perpendicular to the vortex line. Then the positions of two weak-field-seeking states with

{m}_{B}=1

and 2 deviate from each other. This effect is more prominent for BECs with a heavy atomic mass, for which the deviation is greater and, moreover, the Thomas-Fermi radius is smaller. We found, by solving the Gross-Pitaevskii equation numerically, that two condensates interact in a very complicated way leading to fragmentation of vortices, unless

{T}_{\mathrm{rev}}

is properly tuned.

|M|

. Here we show how the topological sign of

M

can be determined locally. We first establish the sign information that spin-resolved tunneling can access and the reference required to obtain an absolute Hall sign. Although spin--orbital entanglement generally obscures the relation between measured spin and Dirac pseudospin, crystalline rotation symmetry alleviates this ambiguity at the band edge. On a

C_3

axis, an axial tip couples to a fully spin-polarized state whose tunneling asymmetry directly reveals the sign of the occupied-band Berry curvature, independent of orbital composition. We further show that the zeroth Landau level provides an independent mass-sign measurement with an unpolarized tip, while opposite domains and their chiral domain-wall modes provide self-consistency checks. These results pave the way for imaging the locations where the topological character changes within a material, rather than merely identifying where a spectral gap appears.