Nonlinear Spectroscopic Effects in Quantum Gases Induced by Atom-Atom Interactions

0.36 MB·English

by  Alexander Safonov

Nonlinear Spectroscopic Effects in Quantum Gases
Induced by Atom–Atom Interactions
A.I.Safonova,b,∗ I.I.Safonovaa, and I.S.Yasnikovc
a
National Research Centre Kurchatov Institute, 123182 Moscow, Russia
b
Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Moscow region, Russia
c
Togliatti State University, 445667 Togliatti, Russia
(Dated: August 28, 2012; in final form, October 23, 2012)
Weconsidernonlinearspectroscopiceffects–interaction-enhanceddoubleresonanceandspectrum
3
instability − that appear in ultracold quantum gases owing to collisional frequency shift of atomic
1
0 transitionsand,consequently,duetothedependenceofthefrequenciesonthepopulationofvarious
2 internal states of the particles. Special emphasis is put to two simplest cases, (a) the gas of two-
level atoms and (b) double resonance in a gas of three-level bosons, in which the probe transition
n
frequency remains constant.
a
J

9 I. INTRODUCTION real physical systems. Spatial inhomogeneity is included

onlyinthe analysisofapossibleINEDORspectrumline
]
s shape in a linear gradient of the external field.
a As is well-known, interaction of a multilevel quan-
g tum system simultaneously with several resonance fields
t- is, under certain conditions, accompanied by nonlinear II. NONLINEAR DYNAMICS OF A
n spectroscopic phenomena, coherent population trapping THREE-LEVEL SYSTEM
a
(CPN) [1]and electromagneticallyinduced transparency
u
(EIT) [2], which are caused by the formation of a spe-
q In general, the evolution of a quantum system inter-
. cial“dark”superposition state immune to the resonance actingwithresonancefieldsisdescribedbyLiouville–von
t
a fields. These effects, however, are still linear in a sense Neumanequationforthe componentsofthe spindensity
m that the resonance transition frequencies in the quan-
matrix ρ [4]
tum system are independent of the population of vari-
-
d ous states. In this work, we consider another, generally ∂ρ
n speaking, a more generalclass of phenomena induced by i~∂t =[Hˆ(t),ρ]−iLˆ(t)ρ, (2.1)
o
exactly such a dependence exemplified by collisional or
[c contact frequency shift of intra-atomic (e.g., hyperfine) whereHˆ(t)=Hˆ0+Uˆ(t)istheHamiltonianincludingthe
transitionsinquantumgasesowingtotheinteractionbe- unperturbed term Hˆ and the time-dependent perturba-
0
1 tween of the gas particles. In our previous work [3], we tion Uˆ(t) due to the external ac field, ˆ is the Lindblad
v L
showedthatinteractionofagasofthree-levelatomswith superoperator [5] responsible for dissipation and square
6
two lightfields results,due to the contactshift, ina spe- brackets, as usually, denote quantum-mechanical com-
6
7 cifickindofdoubleresonance,interaction-enhanceddou- mutation.
1 ble resonance (INEDOR). In addition, as will be shown To obtain the general evolution equation for the com-
. below, a gas of two-level atoms with a nonzero contact ponentsofthedensitymatrixofagasofthree-levelatoms
1
shift can exhibit spectrum instability of the resonance interacting with two resonance fields we use the previ-
0
3 transition, namely, the dependence of the resonance line ously derived relation for the contact shift of the tran-
1 shape and the final population of the levels on sweep di- sition 1 2 in a spatially homogeneous gas in the
| i → | i
: rection,as wellasonthe relationbetweenthe amplitude presence of the third state 3 [6]. We will be interested
v | i
i of the probe field, sweep rate and the magnitude of the in coherent population of the states. In this case, the
X
contact shift. frequency shift ∆ (ρ) ω (ρ) ω (0) (i,j =1,2,3) of
ij ij ij
≡ −
r the transition between the states i and j at a certain
a The behavior of an arbitrary two-level system is com- | i | i
nonzero gas density n Tr(ρ ) vanishes for fermions,
monly described in terms of effective spin (pseudo-spin) ≡ ij
whereas for bosons in the absence of a Bose Einstein
s = 1/2. We will also follow this representation, in each −
condensate it is
case attributing spin to a particular pair of quantum
statescoupledbyaresonancetransition. Inthiswork,we ~∆ = (ρ +ρ )δλ +
12 11 22 12
disregard spin waves and related effects associated with
+ (ρ ρ )∆λ +2ρ (λ+ λ+), (2.2)
spatialtransportofspinpolarization,whichactuallycor- 22− 11 12 33 23− 13
responds to zero spin diffusion constant. Below we will ~∆13 = (ρ11+ρ33)δλ13+
discuss to what extent this assumption corresponds to + (ρ ρ )∆λ +2ρ (λ+ λ+). (2.3)
33− 11 13 22 23− 12
Here, ρ are the components of the density matrix in
ij
terms of the eigen wavefunctions of the unperturbed
∗Electronicaddress: [email protected] HamiltonianHˆ ,δλ =λ λ ,∆λ =λ +λ 2λ+,
0 ij jj− ii ij ii jj− ij
2
∂ρ Ω
12 p
i = (ρ ρ )exp[i∆ω (ρ)t]+
22 11 12
∂t 2 −
Ω
d
+ ρ exp[i∆ω (ρ)t] iγ ρ , (2.8)
32 13 12 12
2 −
∂ρ Ω
32 d
i = ρ exp[i∆ω (ρ)t]
12 13
∂t − 2 −
Ω
p
ρ exp[i∆ω (ρ)t] iγ ρ , (2.9)
31 12 32 32
− 2 −
where Ω and ∆ω (ρ) ω ω (ρ) are the
p(d) 12(13) p(d) 12(13)
≡ −
Rabi frequency and density-dependent frequency detun-
ing of the probe (drive) field and γ are the transverse
ij
relaxation rates. The light fields are thought to be spa-
tially homogeneous, which usually holds for hyperfine
Figure 1: Scheme of (left) a three-level system and (right)
transitions, whose wavelengths are much larger that the
influenceoftheRabioscillationsbetweenthestates|1iand|3i
geometrical size of the sample.
onthe|1i−|2itransitionfrequencyandintensity. Forclarity,
theRabiperiod2πΩ−1ofthedrivetransitionisassumedtobe If,asalreadymentionedabove,theprobefieldismuch
d weaker than the drive field, Ω Ω , the second term
muchlongerthanthedetectiontimeofthe|1i−|2iresonance p d
≪
line. in Eqs. (2.4) and (2.6) can be neglected. In this case,
Eqs. (2.4)–(2.6) can be written in a more compact form
of a usual Bloch equation for the precession of the spin
tλh±ije≡inhtiejr|aλc|tiijoin±iinsttehnesistpyinλp=art4oπf~t2hae/mm,atwrihxicehlemisecnotmo-f pspoalacreizoaftitohne(s“tmataegsne1tizaantidon”3) v(Mectzor=M(0,in0,tρh3e3 Hilρb1e1r)t,
monly used to describe cold collisions, when the par- M⊥ =2(Reρ13,Imρ1|3i,0)): | i −
tial amplitudes of scatteringwith a nonzero angularmo- ∂M ∆λ
mentum of the relative motion of the colliding particles ∂t =M× Ω˜ + ~13Mz −γ13M⊥, (2.10)
“freeze out”, m is the atomic mass, a is the respective (cid:18) (cid:19)
s-wave scattering length. The superscript “+” denotes whereΩ˜ =(Ω ,0,ω (0) ω +~−1δλ (ρ +ρ )). For
d 13 d 13 11 33
thatthewavefunctionoftwocollidingbosonsissymmet- −
clarity, we separate in the precession frequency Ω˜ the
ricwithrespecttopermutationofboththeirpseudo-spin
componentthatremainsnearlyconstantinaweakprobe
andspatialcoordinates. Thedoublyantisymmetriccom-
field, when ρ +ρ const. The second term in the
ponent,obviously,donotcontributetos-wavescattering. 11 33 ≈
parenthesisintheright-handsideofEq. (2.10)expresses
Below we restrict ourselves to the typical situation in
explicitly the dependence of the precession frequency on
double-resonance experiments, when the state |2i is ini- the current value of the magnetization vector M. In the
tially unpopulated and its population changes insignifi-
subsequent sections, we discuss the direct consequences
cantlyduringtheinteractionwithaweakprobefieldwith
of this, generally speaking, nonlinear precession in two
the frequency ω , whereas the populations of the states
p limiting cases.
1 and 3 can vary quite arbitrarily under the action of
| i | i
a strong drive field with the frequency ω . In this case,
d
ρ ρ ,ρ , and therefore the last term in the right-
22 11 33 III. INTERACTION-ENHANCED DOUBLE
≪
hand side of Eq.(2.3) canbe omitted. Thus, the general RESONANCE
equation(2.1)fortheevolutionofthe componentsofthe
density matrix of the gas of three-level bosons with a
As follows from Eq. (2.2), excitation of the transition
ladder (Ξ) level scheme (Fig. 1) in the absence of spon-
from the state 1 to 3 in a Bose gas induces frequency
taneous longitudinal relaxation becomes (cp. [1]) | i | i
modulationofthe transition 1 2 associatedwiththe
| i−| i
Rabi oscillations of the populations of the states 1 and
i∂ρ11 = Ωdρ exp[i∆ω (ρ)t] 3 [6]. If both transitions are excited simultan|eoiusly,
∂t 2 13 13 − | i
thismodulationleadstoanewphenomenon,interaction-
Ω
pρ exp[i∆ω (ρ)t] c.c., (2.4) enhanced double resonance.
21 12
− 2 − Here, for simplicity and clarity, we restrict ourselves
i∂ρ33 = Ωdρ13exp[i∆ω13(ρ)t]+c.c., (2.5) to the case λ11 = λ+12, when the contact shift of the
∂t − 2 transition 1 2 vanishes at ρ = ρ = 0. Another
22 33
| i−| i
∂ρ31 Ωd advantage of this case is that it allows direct compari-
i = (ρ ρ )exp[i∆ω (ρ)t]+
∂t 2 33− 11 13 sonwith the experimentsonelectron-nucleardouble res-
Ω onance (ENDOR) in atomic hydrogen [9, 13]. To sim-
p
+ ρ exp[i∆ω (ρ)t] iγ ρ , (2.6)
2 32 12 − 13 31 plify our consideration even further, we assume that the
∂ρ Ω frequency of the drive transition 1 3 remains con-
i ∂t22 = 2pρ21exp[i∆ω12(ρ)t]−c.c., (2.7) stant, which requires ∆λ13 to be|suiffi−ci|enitly small. An
3
opposite case is consideredin Sec. IV. Despite the above Thenthedeviationb=B B ofthefieldfromthisvalue
0
−
assumptions,ageneralanalyticsolutionofnonlinearsys- is equivalent to the frequency detuning ∆ω (ρ,b)=γ b
13 d
tem(2.7)–(2.10)is hardlypossible. The approachimple- and∆ω (ρ,b)=γ bofthedriveandprobefield,respec-
12 p
mented below does not feature such generality but pro- tively (here, γ is the corresponding gyromagnetic ra-
d(p)
videsquantitativeandphysicallytransparentdescription tio). OwingtothisZeemancontributiontothefrequency
in the case of interest. Ω˜ (ρ,b)=γ B2+b2 of the magnetization precession
13 d d
Physicsofthe novelkindofdouble resonancebecomes forcedbythedrivefieldB (t)=B exp(iω t),theampli-
d d d
p
clear from Fig. 1. In the absence of relaxation, the pop- tudeoftheoscillatingcomponentofω inEq.(3.1),i.e.,
12
ulations of the states 1 and 3 oscillate under con- the amplitude of the frequency modulation of the probe
| i | i
tinuous excitation of the transition 1 3 in the ini- transition, is a Lorentzian function of the static field b.
| i−| i
tially pure state 1 sample at the frequency (see Fig. 1) As a result, ω (t) oscillates between the Zeeman-only
| i 12
Ω˜ = Ω2 +∆ω2 . This results in the frequency mod- zero-density lower bound (dashed line in Fig. 2a)
13 d 13
ulation of the probe transition 1 2 [6]
p | i−| i ω (0,B +b)=ω (0,B )+γ b (3.2)
12 0 12 0 p
Ω 2 Ω˜ t
~ω =~ω (0)+2n∆λ d sin2 13 , (3.1) andtheupperbound,whichisthesumoftheZeemanand
12 12 (cid:18)Ω˜13(cid:19) 2 ! contact-shift contributions (thick solid line in Fig. 2a),
wmhoedruela∆tiλon=cλan+23e−asλil+1y3.beTchoemapmapralibtuledewiotfhtohrisefvreenqumenucchy ω12(ρ,B0+b)=ω12(0,B0)+γpb+(cid:18)2n~∆λ(cid:19)Bd2B+d2b2,
greater than the linewidth of the probe transition. In (3.3)
this case, excitation of the transition 1 3 periodi- at the field-dependent frequency Ω˜13(b) [3]. According
| i−| i
cally drives the probe transition out of resonance. For to Eq. (3.3), the upper bound of the probe frequency is
clarity, Fig. 1 corresponds to slow driving in a sense generally a nonmonotonic function of the external static
that the Rabi period 2πΩ−1 of the drive transition is field.
d
thought to be much longer than the time needed to de- Thetime-averageabsorptionamplitudeA(b,ωp)atthe
tect the 1 2 resonance line and the entire 1 2 probefrequencywithinthebounds(3.2)and(3.3)ispro-
| i−| i | i−| i
spectrummovesperiodicallyforeandbackalongthe fre- portional to the average population of the initial state
quency axis and simultaneously changes in amplitude. and the probability density to find the system at given
The waveformsshownin Fig.1 are the schematic “snap- values of b and ωp. The probability density, in turn,
shots” of the spectrum at different phases of the Rabi peaks at the lower and upper bound of the probe transi-
cycle. Obviously,electromagneticabsorptionatthefixed tionfrequency,wherethesystemspendsmoretime,since
frequency ωp of the probe field also changes periodically in these cases, dω12(b,ωp)/dt=0 [6].
with a period of the 1 3 Rabi oscillations. It should To illustrate the effect of spatial inhomogeneity, let
| i−| i
be emphasized that these changes are associated with us consider in more detail an infinite sample with a spa-
changesinboth thepopulationoftheinitialstateand the tiallyhomogeneousdensityinalineargradient B ofthe
∇
transition frequency, in contrast to conventional double static field and homogeneous light fields. In this case,
resonance, which is solely due to a change in the pop- different parts of the sample simultaneously experience
ulation of the initial state caused by the drive transi- all possible field values. The absorption amplitude at a
tions. Clearly, such interaction-induced frequency mod- given probe frequency ωp is proportional to the integral
ulation can greatly enhance the effect, which therefore I(ω ) = A(b,ω ) ∂N db along the line ω = const
p p ∂b p
can be called INteraction-Enhanced DOuble Resonance within the segments AB and CD in Fig. 2a. The result
R (cid:0) (cid:1)
(INEDOR). of numerical integration with the parameters of the EN-
Let us consider in more detail a possible line shape DORexperimentswith 2Datomichydrogenatadensity
of the INEDOR spectrum in a strong drive field when, of 3 1012 cm−2 in the strong polarizing magnetic field
·
in contrast to the case illustrated in Fig. 1, simultane- B =45 kG and the drive field B =1 mG [9, 13, 14], at
d
ous frequency and amplitude modulation of the 1 2 the maximum amplitude ∆B = 2n∆λ(~γ )−1 = 89 G
c p
| i−| i
absorption line is relatively fast and therefore is inte- of the contact shift of the hyperfine transition b c
| i−| i
grated by the detection system. This implies that the in field units (here, γ and γ are to a good accuracy
p d
1 3 Rabi frequency is much higher than the rate of equal to the gyromagnetic ratio of electron and proton,
| i−| i
field or frequency sweep through the 1 2 resonance respectively), is shown in Fig. 2b as a function of the
orthe inversetime constantτ−1 ofthe| die−te|ctiionsystem, drive frequency ω (lower horizontal axis) at constant
d
Ω τ 1. ω = ω (0,B ). Alternatively, the INEDOR spectrum
d p 12 0
≫
Generally, the energies of all three levels and, conse- can be detected by sweeping ω (upper horizontal axis)
p
quently,bothtransitionfrequenciesdependontheexter- atconstantω (inFig.2b,ω =ω (0,B )). Inthiscase,
d d 13 0
nalstaticfield. Thespecificnatureofthisfieldisinsignif- the spectrum is inverted on the frequency scale because
icant but for definiteness we shall consider the magnetic an increase in ω corresponds to an increase in the res-
d
field B. Let the field B correspond to the exact reso- onance value of the static field ω /γ and, consequently,
0 d d
nance 1 3 at a certain density ρ, ω (ρ,B ) = ω . to a positive displacement of the mean-field Lorentzian
13 0 d
| i−| i
4
Figure 2: (a) Field dependence of the |1i−|2i frequency shift (in units of γpHd) under the CW excitation of the |1i−|3i
resonance. Horizontal axis is the field detuning (ω13 −ωd)/γd from the |1i−|3i resonance in units of the excitation field
Bd. Solid and dashed line are, respectively, the upper bound of the sum of the Zeeman and mean-field contributions and the
Zeemancontributionalone. Horizontallinesindicate(dash-dottedline)thevalueofω12 atminimumand(solidline)theprobe
frequency. Verticaldashed linecorrespondstotheresonancefield forthe|1i−|3i transition. Inset istheoverviewof thesame
dependence. (b) |1i−|2i absorption amplitude as a function of thedrive frequency(the lower horizontal axis is thedetuning,
in Hertz, from the |1i−|3i resonance) for the monochromatic probe field with the fixed frequency ωp = ω12(0,B0). Sharp
peak corresponds to theminimum of theprobe frequency in (a). The spectrum that appears when ωp (upperhorizontal axis)
is swept at fixed ωd =ω13 is inverted with respect to the frequency axis. The parameters correspond to 2D atomic hydrogen
with a density of 3·1012 cm−2 in the high polarizing field B = 45 kG, except for the sign of the contact shift: Bd = 1 mG,
∆Bc =89 G.
peak(3.3)onthecurveω (b)inFig.2a. Thelatterisin ω ω (0,B ) (see Fig. 2a). It is readily shown that
12 min 12 0
−
turnequivalenttoadecreaseintheprobefrequencyω at fora sufficiently highamplitude ofthe contactshift such
p
constantω . WhichwayofobservingtheINEDORspec- that 2n∆λ ~γ B and b B [3],
d p d d
| |≫ ≫
trum is preferred depends on the details of a particular
γ 3
experiment. δω = dδω γ (2∆B B2)1/3, (3.4)
13 γ 12 ≃ 2 d c d
Theabsorptionamplitudeasafunctionofω (Fig.2b) p
d
decreases substantially within the drive resonance and where ∆B = 2n∆λ(~γ )−1 is the maximum contact
c p
has a sharp maximum at ω =ω , i.e., when the min-
p min shift of the 1 2 resonance in field units. Thus, the
imum value of Eq. (3.3) coincides with the probe fre- | i−| i
INEDORlinewidthisindependentofthe staticfieldgra-
quency. This explains the dispersion-looking ENDOR
dient. On the other hand, the spectrum intensity is in-
spectraof2Datomic hydrogen[9,13]. Qualitatively,the
versely proportional to B .
hole inthe absorptionamplitude is becausethe atomsof |∇ |
the otherwise resonant regions of the sample are period-
ically driven out of the probe resonance and, as a result, IV. SPECTRUM NONLINEARITY OF A GAS
spendonly a smallfractionoftime inthe resonancecon- OF TWO-LEVEL BOSONS
ditions. On the other hand, the absorption maximum is
due to the fact that the drive resonance introduces zero
IncontrasttousualopticalBlochequations,Eq. (2.10)
gradientof the 1 2 transitionfrequency in a certain
includes an essentially nonlinear term proportional to
| i−| i
region of the sample and, therefore, much more atoms
the contact shift of the resonance frequency. To observe
become resonant.
this nonlinearity it is sufficient to have just two levels.
The widthofthe double-resonancecurveinthe probe- To avoid confusion, we keep levels 1 and 3 omitting for
frequencyunitscanbeestimatedasthedifferenceδω = brevitythe subscripts13andp(d). Thus,inthis section,
12
5
∆λ ∆λ . In addition, we set the total density con-
13
≡
stant, n ρ +ρ = const. The nonlinearity is seen
11 33
≡
mostclearlyinthespectrumatalowenoughsweeprate.
In particular, if the rate ~−1∆λdM /dt ~−1Ωn∆λ
z
∼
of the variation of the transition frequency due to a
change in the populations is greater than the frequency
sweep rate dω/dt, the Rabi oscillations occur on the
background of a relatively slow change in the magne-
tization along with the frequency of the light field, so
that the system stays all the time near the resonance
M ∆λ ~[ω(ρ) ω(0)] nδλ. Meanwhile, the popu-
z
≈ − −
lations periodically vary relatively quickly when the res-
onance conditions are fulfilled, which in turn drives the
system out of resonance for a certain time until the fre-
quency of the light field is adjusted to the new value of
the transition frequency. The picture of such a reen-
trant resonance repeats until the transition frequency
Figure 3: Evolution of the population of the state |3i during
stops changing because the limiting value of the mag-
the linear upward (indicated by the right arrow) and down-
netization (e.g., M = n) is reached. Frequency sweep
z wardfrequencysweepofthemicrowavefieldaccordingtothe
in the opposite direction is accompanied by usual be-
numericsolutionofEq.(2.10). Thetotaldensitycorrespond-
havior of the populations, since the repeated fulfillment
ing to each curve is marked in units of the critical density
of the resonance conditions is impossible. On the other nc. The values of the parameters correspond to the hyper-
hand, it is clear that the contact frequency shift affects fine transition b → a in three-dimensional atomic hydrogen:
the spectrum considerably if it drives the system out of Ω = 10 s−1, dω/dt = 2×103 s−2, nc ≈ 2.23×1018 cm−3,
the resonance conditions. This requires that the contact γ =0.3 s−1, Γ=0[14], ∆λ/~=−3×10−16 cm3/s.
shift were at least comparable with the Rabi frequency
(thebandwidthofthemicrowavegeneratorisassumedto
be sufficiently narrow;forexample,the spectralwidthof nal population of the state 3 after sweeping through
the generator in the experiments with atomic hydrogen the resonance conditions shar|pily increases from the low
was δf/f .10−10 [14]) Thus, under the conditions valueρ Ω2 dω/dt−1givenbytheproductoftheRabi
33
frequency∼Ω an|d the|duration Ωdω/dt−1 of the sweep
Ωn∆λ &~ dω ; n∆λ &~Ω (4.1) through the resonance line, to n|early u|nity. A further
| | dt | |
(cid:12) (cid:12) increase in density is accompanied by a mere increase in
(cid:12) (cid:12)
thereappearsaspectrum(cid:12)“hys(cid:12)teresis”(Fig.3). Ifinaddi- thefrequencydetuning,atwhichthelimitingpopulation
(cid:12) (cid:12)
tionn∆λ ~Ω,adrasticchangeinthespectrumupon is reached. As is easily seen, this detuning is exactly the
| |≫ maximum possible contact frequency shift n∆λ. At the
reaching a certain critical value of the total gas density
oppositesweepdirection,thefinalpopulationofthestate
n or any other parameter entering the first condition of
c
3 ,onthecontrary,decreaseswithanincreaseindensity.
(4.1) occurs almost abruptly (Fig. 4).
| i
Asonemightexpect,thefinalpopulationisindependent
Figure 3 shows the evolution of the population of the
onthesweepdirectioninthe low-densitylimit,whenthe
state 3 during the upward and downward (indicated
| i contact shift vanishes. Thus, the shape of the spectrum
by the arrows) linear frequency sweep of the microwave
dependsontherateanddirectionofthefrequencysweep
field,accordingtothenumericsolutionofEq.(2.10)with
of the light field, as well as on the field amplitude.
theparameterscorrespondingtothe hyperfinetransition
b a in three-dimensional atomic hydrogen, namely, AsisseeninFig.4,thecriticaldensitync (determined
|atit→he|Riabi frequency Ω = 10 s−1, sweep rate dω/dt = as an abscissa of the steepest slope) does not exactly
2 103 s−2 and the total density n 2 1018 cm−3 [14], conformto the condition Ωnc∆λ ~dω/dt, whichfol-
∆×λ/~= 3 10−16cm3/s(seeAppen∼dix·). Thecharacter lows from (4.1), and rath|er incre|as∝es s|uperli|nearly with
of the sp−ect×rum is almost insensitive to the transverse dω/dt. The origin of this behavior has to be clarifies.
| |
relaxation rate (see below). In the calculations, we used Note only, that the period 2π/Ω of the Rabi oscillations
γ = 0.3 s−1. The longitudinal relaxation rate in atomic atalowrateofthemicrowavefrequencysweep(leftcurve
hydrogen is quite low and was therefore neglected. The inFig.4)becomescomparablewiththedurationofpass-
total gas density corresponding to each curve is given in ing through the resonance line. Damping oscillations of
unTithseofcanlccu≈la2t.e2d3d×ep10en18decnmc−e3o.f the final population of tshuemfiabnlayl poroipguinlaattieonfroomf ththeestfaatcet |t3hiatsetehneastysnte>mnficnparlley-
the state 3 on the total gas density at various sweep gets out of the resonance conditions in this or that par-
rates dω/|dti and the same values of the other parame- ticular phase of effective Rabi oscillations (waves on the
ters a|s in Fi|g. 3 is shown in Fig. 4. When the density slanted parts of curves in Fig. 3 at n>nc).
increases from a subcritical to supercritical value, the fi- Equation(2.10)holdsforaspatiallyhomogeneoussys-
6
inatwo-levelsystemisindependentofthemutualcoher-
ence of the single-particle states [19].
We are truly grateful to S. A. Vasiliev for fruitful dis-
cussionsandprovidinguswiththedataofatomichydro-
gen experiments the University of Turku, Finland. This
work was supported by the Human Capital Foundation,
contract no. 211).
Appendix A: Contact Shift in Atomic Hydrogen
The magnitude of the contact shift of the hyperfine
transition b a in atomic hydrogen can be found as
| i↔ | i
follows. Thediatomicstatesinthebasis S,m ;I,m of
S I
| i
the total electron and nuclear spins of the pair of atoms
take the form
Figure 4: Calculated total-density dependence of the final
bb = 1, 1;1, 1 (A1)
population of the state |3i at various frequency sweep rates | i | − − i
1
|dω/dt|. Otherparameters are thesame as in Fig. 3. ab (ab + ba )=
+
| i ≡ √2 | i | i
= cosθ 1, 1;1,0 sinθ 1,0;1, 1 , (A2)
| − i− | − i
tem. In the case of spatial inhomogeneity, there appears aa = cos2θ 1, 1;1,1 +sin2θ 1,1;1, 1
magnetization transport owing to exchange and dipole– | i | − i | − i−
sin2θ sin2θ
dipole interactions of the atomic pseudo-spins, which 1,0;1,0 0,0;0,0 (A3)
− 2 | i− 2 | i
leads to the emergence of spin waves [10, 12]. The effect
of the contact shift of the spin-wave spectrum requires where tan(2θ) = A[(γ +γ )hB]−1, γ (γ ) is the gyro-
separate consideration, which lies beyond the scope of e p e p
magnetic ratio of electron (proton), A/h = 1420 MHz
the present work.
is the hyperfine constant of hydrogen). That is, the
The effect described in this section can occur not only
states bb and ab are pure electronic and nuclear
in atomic hydrogen but also in ultracold alkali vapors | i | i+
triplets irrespective of the value of magnetic field. Con-
and a number of other systems. In any case, the partic- sequently, λ+ = λ = 4π~2a , where a is the triplet
ular characterof interaction, which results in the depen- ab bb m t t
s-wave scattering length. Thus, according to Eq. (2.3),
denceofthetransitionfrequencyonthepopulationofthe
the contact shift of the transition b a vanishes at
statesinvolvedisinsignificant. Thereasonwhythespec- | i−| i
ρ = 0 in an arbitrary field. On the other had, the
trum nonlinearity was not observed in the experiments aa
state aa contains the singlet component (the last term
on the contact shift in 87Rb vapor [15] is that the sec- | i
in the hight-hand side of Eq. (A3)). Consequently,
ond condition of (4.1) was violated. In fact, the s-wave
scatteringlengths of 87Rb invarioushyperfine statesare ∆λab = λbb + λaa − 2λ+ab = πm~2(as − at)sin22θ 6= 0,
such that the maximum differential contact shift at the at as = 30(5) pm [11]. In the field B = 4.5 T,
density n 1013 cm−3 was as small as n∆λ/h 0.2 Hz ∆λ−ab/~ = 3 10−16 cm3/s, and the frequency shift
(∆ω/ω ∼3 10−11), whereas the Rabi frequenc∼y of the at the dens−ity ×ρaa = 2 1018 cm−3 is about 100 Hz,
two-pho∼ton ·transition in the pulsed microwave/RF field which is two orders of m·agnitude greater than−the Rabi
was about 2.5 kHz. frequency.
The dependence of the resonance field on the sample Replacing the electron spin by the nuclear spin all
magnetization knowingly leads to a so-called ferromag- the aforesaid is automatically generalized to the tran-
netic instability [16], e.g., ofthe FMR spectrumof ferro- sitions b c , a d and c d , wherefrom
magnetic films, ESR in atomic hydrogen [17] and NMR ∆λ =|∆iλ↔ |=i∆|λi ↔= |π~i2(a |ai)↔sin|22iθ. However,
in 3He [18]. In contrastto this type of instability, whose thebpcopulatiaodnsofthcedstatemswitsh−opptositeelectronspins
condition is determined the transverse relaxation rate, cannotbesimultaneouslyhighowingtoafastrecombina-
the effect considered in this section does not depend di- tionofsuchpairs;thus,thesiftofthetransitions b c
| i↔| i
rectly on the transverse relaxation, as the contact shift and a d is hardly detectable.
| i↔| i
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