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Motional Quantum States of Surface Electrons on Liquid Helium in a Tilted Magnetic Field

A. A. Zadorozhko,1, ∗ J. Chen,1, ∗ A. D. Chepelianskii,2 and D. Konstantinov1, †

1Quantum Dynamics Unit, Okinawa Institute of Science and Technology (OIST) Graduate University, Onna, 904-0495 Okinawa, Japan2LPS, Univ. Paris-Sud, CNRS, UMR 8502, F-91405, Orsay, France

(Dated: November 11, 2020)

The Jaynes-Cummings model (JCM), one of the paradigms of quantum electrodynamics, was introduced

to describe interaction between light and a fictitious two-level atom. Recently it was suggested that the JCM

Hamiltonian can be invoked to describe the motional states of electrons trapped on the surface of liquid helium

and subjected to a constant uniform magnetic field tilted with respect to the surface [Yunusova et al. Phys.

Rev. Lett. 122, 176802 (2019)]. In this case, the surface-bound (Rydberg) states of an electron are coupled to

the electron cyclotron motion by the in-plane component of tilted field. Here we investigate, both theoretically

and experimentally, the spectroscopic properties of surface electrons in a tilted magnetic field and demonstrate

that such a system exhibits a variety of phenomena common to the light dressed states of atomic and molecular

systems. This shows that electrons on helium realize a prototypical atomic system where interaction between

components can be engineered and controlled by simple means and with high accuracy, and which therefore can

be potentially used as a new flexible platform for quantum experiments. Our work introduces a pure condensed-

matter system of electrons on helium into the context of atomic, molecular and optical physics.

Keywords: Jaynes-Cummings model, cavity quantum electrodynamics, electrons on helium

I. INTRODUCTION

The Jaynes-Cummings model (JCM) describes the interac-tion between a single two-level system and a quantum har-monic oscillator [1]. Originally formulated to account forthe interaction between an atom and a single mode of thefree-space electromagnetic radiation [2], this model becameextensively used in quantum atomic, molecular and optical(AMO) physics. Recent developments in the experimentalrealization of this model, in particular within the setting ofcavity quantum electrodynamics (CQED) which exploits theradiative coupling between a two-level atom and a cavitymode [3–9], have opened a pathway for fundamental tests ofquantum theory, such as non-destructive detection of quantumstates, [10, 11] realization of nonclassical states and observa-tion of their decoherence [12, 13], and creation of many-bodyentanglement [14–16]. The main feature of CQED exploitedin the experiments is a reversible Rabi oscillation betweenthe coupled atom-photon states. These oscillations occur ata frequency determined by the rate of atom-photon couplingg explicitly appearing in the JCM Hamiltonian. The ’strongcoupling’ regime of CQED is realized when the value of g ex-ceeds the decay rates for the atomic population and the cavityfield.

Under certain conditions, the JCM Hamiltonian of CQEDcan be formally applied to describe other systems. In par-ticular, the harmonic motion of a trapped ion can representa single mode of a cavity, while the ion’s internal states canserve as a two-level atom. In this case, the coupling be-tween internal and motional states is accomplished by theion’s motion through the spatially inhomogenious laser beamswhich are used to excite transitions between internal ionic

∗These authors contributed equally to this work.†Email: denis@oist.jp

states [17, 18]. The coherence of quantum states of laser-cooled ions can be preserved for many cycles of Rabi oscil-lations. Thus the strong coupling regime of CQED can berealized in experiments, which presents the cold ion systemas another attractive platform for fundamental tests [19, 20]and quantum information purposes [21].

Free electrons trapped on the surface of liquid heliumpresent a unique, extremely clean condensed-matter systemwhich shares some striking similarities with atomic systemsstudied in AMO physics. The surface bound states of an elec-tron on liquid helium are formed due to, on the one hand, anattraction to a weak image charge inside the liquid and, onthe other hand, a repulsion from helium atoms, which pre-vents an electron to enter the liquid. So called Rydberg statesof confined motion of such an electron perpendicular to theliquid surface have the energy spectrum similar to that of anelectron in the hydrogen atom and can be spectroscopicallystudied by using microwave light [22, 23]. At temperaturesbelow 1 K, the dissipative decay rates for the excited Rydbergstates are very low because they are limited only by the in-teraction of an electron with the capillary surface waves (rip-plons) [24, 25]. Surface electrons (SE) on liquid helium showinteresting similarities with some well-known phenomena inRydberg atoms, [26] such as the Coulomb shift of the transi-tion frequency due to dipolar interaction between neighboringelectrons [27, 28] and the Lamb shift of the Rydberg transitionfrequency due to interaction of SE with the quantum field ofripplons [29, 30].

The electron motion parallel to the surface of liquid is free.However, it can be confined to the quantized cyclotron orbitsby applying a sufficiently strong magnetic field perpendicularto the surface [31]. Similar to an ion in a trap, the in-planemotion of an electron becomes harmonic, with an energyspectrum consisting of equidistant Landau levels tuned by thevalue of the applied magnetic field. Usually, the electron mo-tions parallel and perpendicular to the surface are only weaklycoupled via scattering of an electron from ripplons. However,

http://arxiv.org/abs/2011.04968v1mailto:denis@oist.jp

2

a strong coupling can be induced by applying a magnetic fieldparallel to the surface [32]. Physically, the coupling is via theLorentz force acting on an electron due to its in-plane motionand the parallel magnetic field. Recently, it was shown thatthe Hamiltonian of a surface electron in a magnetic field tiltedwith respect to the surface is formally equivalent to the Hamil-tonian of an atom coupled to the quantum field of electromag-netic radiation, with the coupling constant g proportional tothe magnitude of the in-plane component of the applied mag-netic field [33]. In such a case, the Rydberg orbital states serveas an atom, while the in-plane cyclotron motion represents asingle mode of the electromagnetic field. The tunable mix-ing between the motional states of SE can be manifested bythe shift in the transition frequency for the dressed Rydbergstates, as indeed was observed in the experiment [33].

As was seen in the past, ability to engineer and controlstates of a quantum system weakly coupled to the environ-ment can provide new platforms for fundamental studies andapplications [10–16, 19–21]. For this reason, electrons on he-lium in a tilted magnetic field can be a promising system toexplore. In this paper, we theoretically and experimentallystudy the spectroscopic properties of such a system and showthat it exhibits a variety of phenomena common to atomic andmolecular systems interacting with a laser light, such as theeigenstate mixing, the light shift of energy levels, the Autler-Townes splitting, and the electromagnetically induced trans-parency. The predicted spectroscopic properties based on theJCM Hamiltonian contain no adjustable variables, while allthe parameters which enter the theory can be readily con-trolled by simple means of constant electric and magneticfields, which allows for a detail comparison with the exper-iment. We also show that the coherent dynamics of coupledmotional states can dominate over the dissipative processes inthe system. This presents electrons on helium as a new flexi-ble platform for quantum experiments.

This paper is organized as follows. Section II introducesthe Hamiltonian of an electron on liquid helium in a tiltedmagnetic field and analyzes its eigenenergy spectrum. For thesake of clarity, the analysis is done analytically using appro-priate approximations, as well as by numerical calculations.Section III presents an experiment where the spectroscopicproperties of SE in tilted magnetic fields are studied by theStark spectroscopy method. Also, a detailed comparison ofthe experimental results with the calculations is given. Thediscussion of our results and their implications for the futureexperiments are given in Section IV.

II. BACKGROUND AND MODEL

Free electrons can be trapped near the surface of liquid he-lium due to, on the one hand, a weak attraction to the liquiddue to polarizability of helium atoms and, on the other hand,a potential barrier at the vapor-liquid interface due to hard-core repulsion from the helium atoms arising from the Pauliexclusion principle. According to quantum mechanical prin-ciples, this allows such electrons to hover above the surfaceof liquid helium at a distance of about 10 nm, thus forming

a two-dimensional (2D) electron system [34, 35]. The basicquantum-mechanical Hamiltonian for a single electron aboveliquid helium is given by

H =P2

2me+V (R), (1)

where me is the bare electron mass. Assuming an infinitelyextended flat surface of liquid, the potential energy of an elec-tron can be written as

V (R) =V0Θ(−z)−Λ

zΘ(z). (2)

Here z is the electron coordinate in the direction perpendicularto the surface, V0 ∼ 1 eV is the height of the repulsive potentialbarrier at the vapor-liquid interface located at z=0, and Θ(z)is the Heaviside (step) function. The last term in (2) describesattraction of an electron to a weak image charge inside theliquid, where Λ is determined by the dielectric constant ofliquid helium ε (for vapor we assume ε = 1) as

Λ =e2

16πε0

(

ε − 1ε + 1

)

. (3)

Here, ε0 is the vacuum permittivity and e> 0 is the elementarycharge. The Hamiltonian (1) can be separated into two partscorresponding to the orbital motion of an electron in the direc-tion perpendicular to the surface (Hz) and parallel to the sur-face. In z-direction, the electron motion is quantized into thesurface bound states which are the eigenstates of the Hamilto-nian

Hz =p2z

2me+V0Θ(−z)−

Λ

zΘ(z). (4)

The energy spectrum of this motion can be easily found bymaking a reasonable assumption of a rigid-wall repulsive bar-rier, that is V0 → +∞. In this case it coincides with the en-ergy spectrum of an electron in the hydrogen atom −Re/n2,n = 1,2, .. , where Re = meΛ

2/(2h̄2) is the effective Rydbergconstant. This constant is about 63 meV (36 meV) for an elec-tron above liquid 4He (3He). An electron in the ground Ryd-berg state localizes above the surface of liquid at an averagedistance 〈z〉 ∼ rB, where rB = h̄2/(Λme) is the effective Bohrradius. This radius is about 7.8 nm (10.3 nm) for an electronabove liquid 4He (3He).

In experiments, there is always a static electric field E⊥ ap-plied perpendicular to the liquid surface. Such a field servesas an effective positive charge background needed to neutral-ize the Coulomb repulsion between electrons. In addition, thedc Stark shift induced by E⊥ provides a very convenient wayto tune energy difference between the Rydberg states for theirspectroscopic studies, as will be described in Section III. Suchan applied field adds an additional term eE⊥z to the Hamil-tonian (4) and changes its eigenenergy spectrum. For suffi-ciently small values of E⊥, the energy shift for n-th Rydberg

3

state is given by eE⊥znn, where znn is the mean value of thecoordinate operator z for this state. It is clear that the dcStark shift is linear due to the inversion symmetry breakingin z direction imposed by the repulsive barrier at the liquidsurface. Already for moderate fields E⊥ ∼ 10 V/cm the per-turbation theory does not provide accurate estimates for theshifts, therefore one has to numerically solve the 1D eigen-value problem with the Hamiltonian (4).

The electron motion parallel to the surface is free witha continuous parabolic energy spectrum p2/(2me), wherep = pxex + pyey is the electron in-plane momentum and ei,i = x,y,z is the unit vector in i direction. When SE are sub-ject to a static magnetic field B = Bzez applied perpendicularto the surface, the electron in-plane motion is quantized intothe states with an equidistant energy spectrum h̄ωc(l + 1/2)(the Landau levels), where ωc = eBz/me is the cyclotron fre-quency and l = 0,1, .. is the quantum number. The verticaland in-plane motions of an electron are uncoupled, and thefull Hamiltonian describing the electron’s orbital motion canbe represented as

H0 = Hz +(p+ eA)2

2me= Hz + h̄ωc

(

a†a+1

2

)

, (5)

where A is the vector potential. Choosing the Landau gaugeA = Bzxey, for which the eigenvalue of the momentum op-erator py is a good quantum number, we define the opera-

tor a = (√

2lB)−1 (pxl2B/h̄− i(x+ x0)

)

, where x0 = py/(eBz)

and lB =√

h̄/eBz. The operator a satisfies the commuta-

tion relation [a,a†]=1. Each electron eigenstate is the prod-uct of a Rydberg state |n〉 of vertical motion correspondingto the eigenenergy En of the Hamiltonian Hz and a state |l〉of in-plane cyclotron motion, where a†a|l〉 = l|l〉. Through-out this paper we disregard the spin state of electron becausethe spin-orbit interaction for SE on liquid helium is negligiblysmall [36], so the spin degree of freedom is always uncoupledfrom the orbital motion.

When an additional component of static magnetic field isapplied parallel to the liquid surface, in other words when themagnetic field B is tilted with respect to z axis, the vertical andin-plane motions are coupled and the electron eigenstates areno longer the simple product states. For certainty, we considerthe non-zero components of the field only in the y and z direc-tions and use the Landau gauge for the corresponding vectorpotential A = Byzex + Bzxey. The electron full Hamiltonianbecomes

H = Ha + h̄ωc

(

a†a+1

2

)

+h̄ωy√

2lB

(

a† + a)

z, (6)

where we define ωy = eBy/me. The Hamiltonian for the or-bital motion in z direction includes now an additional (dia-magnetic) term due to the parallel component of the magneticfield, that is

Ha = Hz +meω

2y z

2

2= ∑

n

εα |α〉〈α|. (7)

Here, α and εα , α = 1,2, .. , are the Rydberg states and cor-responding eigenenergies of the vertical motion renormilizeddue to the diamagnetic term [33]. In what follows, it willbe more convenient to work in the product basis |n, l〉 ofthe Hamiltonian H0 given Eq. (5), with corresponding energyeigenvalues En + h̄ωc(l + 1/2), rather than using the renor-malized product basis |α, l〉.

The last term in the Hamiltonian (6) presents the coupling(paramagnetic) term due to the parallel component of the mag-netic field. It arises because of the interaction between themagnetic dipole moment of an electron due to the orbital an-gular momentum pxz and the magnetic field By. The Hamil-tonian (6) is reminiscent of that of an atom which interactswith a quantum field a†a of an electromagnetic mode via theelectrical dipole moment ez of the atom [37]. Thus, our sys-tem can be thought of as an atom with the Rydberg states ofthe renormalized Hamiltonian (7) coupled to the bosonic fieldof the electron cyclotron motion, and with the coupling rateg tuned by the value of the in-plane field By [33]. However,we note that the inversion symmetry breaking for z directionin the SE system makes the situation somewhat different. Inparticular, the non-zero diagonal matrix elements znn makecontributions to the eigenvalues and eigenstates of the cou-pled system described by Eq. (6), while it is usually not thecase for an atom-in-cavity system.

The eigenvalues and eigenstates of the Hamiltonian (6) canbe obtained numerically, for example by the diagonalizationof the matrix representation of H constructed on a sufficientlylarge Hilbert sub-space. As mentioned earlier, we prefer touse the product basis |n, l〉 of the eigenstates of the Hamilto-nian (5). Also, for the sake of comparison with experiment(Section III), where the energy spectrum of SE is probed bylooking at the microwave-excited transitions between the low-est energy level and the higher energy levels, it is convenientto subtract the ground-state energy of the cyclotron motionh̄ωc/2 from the full Hamiltonian (6). Figure 1 shows the en-ergy eigenvalues of the Hamiltonian versus the perpendicularmagnetic field Bz for several values of By. The eigenvaluesare obtained by the numerical diagonalization of the Hamil-tonian matrix constructed in a subset of |n, l〉 with 1 ≤ n ≤ 6and 0 ≤ l ≤ 50. For By = 0, that is when there is no cou-pling between the vertical and in-plane motion of SE, there isa manifold of energy levels En,l = En+ h̄ωcl for each n-th Ry-dberg state, each shifting linearly with Bz and having a slopeproportional to l. For the sake of clarity, the lowest energylevels (l=0) for each manifold are marked in Fig. 1 by thecollective indexes (n,0). The largest effect of the non-zeroBy can be seen at the crossings of energy levels of differentmanifolds. In particular, the coupling leads to the avoidedcrossing of energy levels, which implies the mixing betweenthe corresponding product states. Fig. 2 shows a magnifiedfragment of Fig. 1 illustrating such an avoided crossing be-tween energy levels (3,0) and (2,1) corresponding to the first(l = 0) Landau level of the third (n = 3) Rydberg state mani-fold and the second (l = 1) Landau level of the second (n = 2)Rydberg state manifold, respectively. In addition, there aresignificantly weaker anti-crossings between these levels andthe energy levels of the n = 1 manifold indicated by (1, l) for

4

z

-

-

-

-By

y

y

-

FIG. 1: (color online) Energy eigenvalues (in GHz) of the Hamilto-

nian given by Eq. (6) (minus h̄ωc/2) for an electron on liquid3He

in a perpendicular electric field E⊥ = 15 V/cm versus the magneticfield Bz. Different colors correspond to the different values of the

coupling field By = 0 (black lines), 0.1 (red lines), 0.2 T (blue lines).The lowest (l = 0) energy levels for each n-th manifold are indicatedby the collective indexes (n,0).

several values of l.

Following our analogy with an atom in a cavity, we can talkabout ’dressed’ states of the electron orbital motion whose en-ergies at the crossing point are separated by a gap proportionalto the coupling strength. The dressed states are mixtures ofthe product states |n, l〉, thus in general are entangled states ofthe electron orbital motion perpendicular and parallel to theliquid surface. Of particular interest is the crossing betweentwo levels for which the quantum numbers l and l′ differ by1. The region of Bz near the crossing of such energy levelscorresponds to the resonant regime of coupling in CQED. Inthis regime, we can obtain essential results in an analyticalform by considering only a subspace of two nearly degenerateeigenstates |n, l + 1〉 and |n′, l〉 of the uncoupled system andperform Hamiltonian diagonalization in this subspace. Thetreatment is similar to the two-level atom coupled to a quan-tum electromagnetic mode, that is JCM [6]. It is convenient tointroduce the coupling constant gnn′ defined by the couplingmatrix element of the interaction Hamiltonian HI given by helast term in (6)

〈n, l|HI |n′, l′〉= gnn′√

l + 1δl+1,l′ + gnn′√

lδl−1,l′ , (8)

where δl,l′ is the Kronecker delta. Thus, we obtain

gnn′ =h̄ωy√

2

znn′

lB=

√

h̄meωcω2y2

znn′ . (9)

Under the above assumptions, the eigenstates of the Hamilto-

FIG. 2: (color online) Energy eigenvalues (in GHz) of the Hamilto-

nian given by Eq. (6) (minus h̄ωc/2) versus the magnetic field Bzillustrating the avoided crossing of energy levels (2,1) and (3,0)which happens at non-zero coupling fields By. Several energy lev-

els of the n = 1 manifold are also indicated by the collective indexes(1, l). The plots are obtained under the same conditions as for Fig. 1.

nian are given by

|+, l〉= cos(θl/2)|n′, l〉+ sin(θl/2)|n, l+ 1〉, (10a)|−, l〉=−sin(θl/2)|n′, l〉+ cos(θl/2)|n, l+ 1〉, (10b)

where the ’mixing angle’ θl is given by

θl = tan−1(

gnn′√

l + 1

2E∆

)

, (11)

and the corresponding energy eigenstates are given by

E± = EΣ ±√

E2∆ +(l+ 1)|gnn′|2. (12)

Here, EΣ(∆) =(

Ẽn,l+1 ± Ẽn′,l)

/2, Ẽn,l = En,l +meωy(z2)nn/2,

and (z2)nn is the mean value of z2 for the n-th Rydberg state.

The energy splitting between the dressed states (10) at theenergy crossing for uncoupled states (E∆ = 0) is given by2√

l + 1|gnn′ |. Note that the scaling of this splitting with lis similar to the scaling of the Rabi splitting in CQED, whereit changes with the number of photons in the cavity nph as√

nph [6]. In addition, the splitting increases linearly with thecoupling field By and the transition dipole moment znn′ .

As can be seen in Fig. 2, for the crossings between two lev-els for which the quantum numbers l and l′ differ by more than1, there are significantly weaker anti-crossings. This effect

5

0.0 0.2 0.4 0.6 0.8 1.0

By (T)

4

3

2

1

0

1

2

3

4

5

(GHz)

l=

l=

FIG. 3: (color online) The Lamb shift (∆0) and light shift (∆1) ofthe Rydberg n = 1 → 2 transition frequency versus the coupling fieldBy obtained numerically (solid lines) and using the perturbation the-

ory (dashed lines) for an electron on liquid 3He in the perpendic-

ular electric field E⊥ = 15 V/cm and perpendicular magnetic fieldBz = 0.65 T.

comes from the higher-order mixing between different eigen-states, which can be captured by the diagonalization of theHamiltonian (6) on a sufficiently large sub-space of the prod-uct basis |n, l〉.

Even far from the level crossing, which corresponds to thenon-resonant (dispersive) regime of coupling in CQED [6],there is an appreciable shift of the energy levels induced bythe state mixing. This is analogous to the ’light shift’ of theatomic transition frequency experienced by an atom in the de-tuned cavity field [1]. It is instructive to consider the casewhere the interaction term HI in (6) can be treated as a per-turbation. Also, we will assume sufficiently small values ofBy such that the diamagnetic term meω

2y z

2/2 can be treated asa perturbation as well. As before, it is convenient to choosethe eigenstates |n, l〉 of the Hamiltonian (5) as an unperturbedbasis, therefore treat H1 = HI +meω

2y z

2/2 as the perturbation.To the lowest order in By, the shift of the unperturbed energy

level En,l is the sum of the first-order correction δE(1)n,l due to

the diamagnetic term and the second-order correction δE(2)n,l

due to the interaction term

∆En,l =meω

2y (z

2)nn

2+

h̄2ω2y

2l2B∑n′,l′

|znn′ |2|〈l′|a† + a|l〉|2Enn′ + h̄ωc(l − l′)

, (13)

where Enn′ = En −En′ . Interestingly, there is a strong cancel-lation between the first and the second terms in (13), which isreminiscent of the calculations of the Lamb shift in the Hydro-gen atom [38] and the ripplonic Lamb shift in electrons on he-lium [29]. This can be seen by using the Bethe-type approach,that is expanding the mean value (z2)nn in the first term in (13)using the completeness relation for the eigenstates |n〉, that is(z2)nn = ∑n′ |znn′ |2. Plugging this expansion in (13), it is clear

that the leading term proportional to |znn|2 cancels out. Theremaining shift reads

∆En,l =meω

2y

2∑

n′ 6=n|znn′ |2

(

1+h̄ωcl

Enn′ + h̄ωc+

h̄ωc(l + 1)

Enn′ − h̄ωc

)

.

(14)As a particular example, let us consider the frequency shift

for the transition between the ground (n = 1) and the first-excited (n = 2) Rydberg states. For an electron occupying thel-th state of the cyclotron motion, the corresponding shift inthe transition frequency is given by ∆l = (∆E2,l −∆E1,l)/h. Inparticular, for l = 0 we obtain

∆0 =meω

2y

2h

(

∑n 6=2

|z2n|2En2

En2 + h̄ωc− ∑

n 6=1|z1n|2

En1

En1 + h̄ωc

)

.

(15)This shift is an analogue of the Lamb shift in an atom due toits interaction with the vacuum (nph = 0) cavity field [1]. Inthe experiment, the Lamb shift can be observed by excitingthe Rydberg n = 1 → 2 transition in SE occupying the lowest(l = 0) Landau level (Section III). Similar expressions can beobtained for the light shifts ∆l for SE occupying the higherLandau levels. Figure 3 shows the corresponding frequencyshifts ∆0 and ∆1 (in GHz) obtained using the perturbation ap-proach as described above (dashed lines). To illustrate va-lidity of this perturbation approach, the corresponding shiftsobtained from the diagonalization of the full Hamiltonian (6)are plotted by the solid lines. We note that both shifts wereexperimentally confirmed earlier [33].

To summarize this section, the calculated eigenstates andenergy eigenvalues of SE in a tilted magnetic field allows usto make certain predictions regarding the spectroscopic prop-erties of this system. In particular, we predict certain featuresarising from the mixing of the electron motional states, suchas the off-resonance shifts and resonant avoided crossings inthe energy spectrum of SE, which can be probed in an experi-ment which employs spectroscopic methods.

III. EXPERIMENTAL SETUP AND RESULTS

To check predictions of our calculations regarding the en-ergy spectrum of SE, we performed an experiment with elec-trons on liquid 3He using the Stark spectroscopy method em-ployed earlier [22, 30, 39]. The experiment is performed ina leak-tight cylindrical copper cell (see Fig. 4) cooled downto temperatures below 1 K in a dilution refrigerator. The cellis placed inside a superconducting vector magnet (not shown)which can produce a static magnetic field in both y (horizon-tal) and z (vertical) directions. The cell can be filled with the3He gas through a thin capillary tube from a room temperaturestorage tank. The cell has two side windows located oppositeto each other and fitted with home-made microwave (MW)waveguide flanges. The waveguide flanges serve as input andoutput ports for MW radiation which is used to excite tran-sitions between energy eigenstates of SE. Both windows are

6

MWin MWout

InSbLOCK-IN

REFIN

OUT

3He

VBVGu

F

R0>>R

FIG. 4: (color online) Schematic 3D view of the experimental cell.

Details are provided in the text.

sealed with the MW-transparent Kapton film using the Sty-cast epoxy to prevent leakage of helium from the cell into thevacuum space of the refrigerator.

Inside the cell, there are two round metal discs of diame-ter D = 30 mm which form a parallel-plate capacitor with thedistance between the disks d = 2 mm. This distance is de-termined by the height of four cylindrical quartz spacers (notshown in Fig. 4) placed between the disks. In addition, eachdisk is divided into three concentric electrodes by two circu-lar gaps of diameters 18 and 24 mm, each having a width ofabout 0.2 mm. 3He gas, which is introduced into the cell, iscondensed in the cooled cell until the surface of liquid he-lium covers the bottom disk and the liquid level is set approx-imately in the middle between the bottom and top disks. Freeelectrons are injected into the space above the surface of liq-uid from a tungsten filament F by thermionic emission, whileapplying a positive voltage VB to the inner circular electrodeof the bottom disk. As a result, the injected electrons are at-tracted towards the liquid surface and form a round pool withtypical areal density ns of the order 10

7 cm−2 on the surfacejust above the positively biased bottom electrode. With a pos-itive voltage applied, SE can be held on the surface of liquidhelium indefinitely long. In addition, the middle electrodesof the top and bottom disks serve as guard rings. By apply-ing a negative voltage VGu to the guard rings, electrons can bestronger confined on the liquid surface to prevent their escapeto the grounded walls of the experimental cell.

To excite transitions between the Rydberg states of SE, theMW radiation in the 100 GHz range is transmitted into the cellfrom a room-temperature source (not shown in Fig. 4) througha waveguide coupled to the input port of the cell. In the exper-iment, the frequency ω of the radiation is fixed, while the tran-

sition frequency of SE is tuned to match the MW frequencyby sweeping the perpendicular electric field E⊥ = VB/d (dcStark effect). The MW absorption due to resonant transitionsinduced in SE is measured as a change in the power of ra-diation transmitted through the cell. In order to measure thetransmitted power, the output MW port of the cell is coupledto a cryogenic InSb bolometer (QMC Instruments Ltd.) oper-ating at the temperature of the mixing chamber of the dilutionrefrigerator. The bolometer changes its resistance R when itis heated by the incident radiation. To observe change in thebolometer resistance, therefore the incident MW power, wepass a dc current I ≈ 100 µA generated by a battery and mea-sure the voltage drop IR across the bolometer. To increasesensitivity of the method, we apply a small modulating acvoltage Vac = 40 mVrms at the frequency fm ≈ 10 kHz to thecentral electrode of the top disk of the parallel-plate capaci-tor. Due to the Stark shift, this modulates the detuning of thetransition frequency of SE with respect to the MW frequency,therefore the MW power absorbed by SE and the MW powertransmitted through the cell. The corresponding modulationof the voltage across the bolometer is then detected using theconventional lock-in amplifier operated at the modulation fre-quency fm.

An example of the bolometer signal recorded by the lock-in amplifier for SE at Bz = 0.7 T, By = 1 T, and under pres-ence of MW radiation at the frequency ω/2π = 90 GHz isshown in Fig. 5. For this frequency, the observed signal cor-responds to the transition from the ground (n = 1) state to thefirst excited (n = 2) Rydberg state of SE. By keeping the am-plitude of the modulating voltage Vac to be much smaller thanthe width of the transition line, we record the derivative ofthe absorption line. Then, the absorption line can be obtainedfrom the recorded bolometer signal by numerical integration(the red line in Fig. 5). All data presented here were takenat cell temperatures T = 0.3-0.4 K, measured by a calibratedruthenium-oxide chip attached to the cell’s top. The width ofthe line is mostly determined by the inhomogeneous broad-ening due to non-uniformity of E⊥, which is much lager thanthe intrinsic linewidth of the transition line due to the scatter-ing from ripplons [40]. Note that the power of MW excitationwas kept very low in the experiment described here in orderto avoid strong heating of SE and heating-related effects asso-ciated with electron-electron interaction [28].

To observe effect of the tilted magnetic field on the energyspectrum of SE, we recorded MW absorption measured by theabove method either for a fixed value of Bz and different val-ues of By, or for a fixed value of By and different values ofBz. The results are presented as 2D color maps of the inte-grated bolometer signal versus the varying component of B-field (horizontal axis) and E⊥-field (vertical axis). The toppanel of Fig. 6 shows an example of such a plot illustratingthe Lamb shift of the transition line with increasing couplingbetween the vertical motion and in-plane cyclotron motion.Here, the integrated bolometer signal taken for SE under MWexcitation at ω/2π = 90 GHz and at a fixed value of the per-pendicular magnetic field Bz = 0.584 T is plotted for differentvalues of the coupling field By. At By = 0 (no coupling), then = 1 → 2 transition occurs at E⊥ ≈ 23 V/cm, which is in a

7

14 16 18 20 22 24-2

-1

0

1

2

Bol

omet

er S

igna

l (V

)

E (V/cm)

-0.2

0.0

0.2

Inte

grat

ed S

igna

l (a.

u.)

FIG. 5: (color online) Bolometer signal recorded at T = 0.33 K forelectron density ns = 5× 106 cm−2 by sweeping the perpendicularelectric field E⊥ to tune the n= 1→ 2 transition between the Rydbergstates of SE at Bz = 0.7 T and By = 1 T in resonance with MWsat frequency ω/2π = 90 GHz. The red line is an integrated signalwhich gives the inhomogeneously broadened absorption line for the

n = 1 → 2 transition.

reasonable agreement with the expected transition frequencyω21/2π = (E2−E1)/h = 90 GHz for the Stark-shifted energylevels of electrons on liquid 3He [40]. With increasing By,the absorption line shifts towards lower values of E⊥. Thismeans that in order to have the same transition frequency ω21between n = 1 and n = 2 Rydberg states, the energy levelsmust experience an additional shift in order to compensate forthe reduction in the Stark shift due to E⊥. The correspond-ing shift in the transition frequency is plotted in the bottompanel of Fig. 6. To obtain conversion between the shift inE⊥ and the corresponding shift in ω21, we used the numericalvalue of κ = 0.74 (GHz·cm)/V for the slope of ω21 versus E⊥dependence, which was obtained in the experiment by mea-suring the absorption line for different values of the excita-tion frequency ω . For the sake of comparison, the Lamb shift∆0 and the light shift ∆1 due to the coupling of the Rydbergstates to the in-plane motion, which were calculated as de-scribed in Section II, are plotted by the solid and dashed line,respectively. It is clear that the observed shift in the transitionfrequency corresponds to the Lamb shift ∆0, which impliesthat SE mostly occupy the lowest (l = 0) Landau level. In-deed, at Bz = 0.584 T the energy splitting between the groundl = 0 state and the first excited l = 1 state of the cyclotronmotion is about 0.78 K, which results in the thermal popula-tion of the ground state of about 92 %. However, note thatthere is a barely visible splitting in the absorption line aroundBy ∼ 0.5 K. This splitting most likely appears due to transi-tion of the small fraction of electrons occupying the first ex-cited l = 1 state. These electrons experience the transitionfrequency shift ∆1 of an opposite sign to ∆0, see dashed linein Fig. 6. At By ∼ 0.5 T, the difference between ∆0 and ∆1becomes comparable to the linewidth of the absorption line,

0 0.2 0.4 0.6 0.8 B

y (T)

15

20

25

E⊥

(V

/cm

)

∆0

∆1

0 0.2 0.4 0.6 0.8 B

y (T)

-4

-2

0

2

4

6

∆ (

GH

z)

FIG. 6: (color online) (top panel) Color map of the integrated

bolometer signal versus the coupling magnetic field By and tuning

electric field E⊥ for SE on liquid 3He at T = 0.33 K and for electrondensity ns = 5× 106 cm−2. The perpendicular component of mag-netic field is fixed at Bz = 0.584 T. (bottom panel) The same datareplotted versus the detuning (in GHz) from the center of the absorp-

tion line at By = 0. For the sake of comparison, the theoretical resultsfor the transition frequency shifts ∆0 and ∆1 (see Section II) obtainedby the numerical diagonalization of the Hamiltonian (6) are shown

by the white solid line and black dashed line, respectively.

which results in its splitting. We note that this splitting wasmore clearly observed in Ref. [33], where significantly largerMW power was used in order to observe the transition line bya photo-assisted transport spectroscopy method.

We note that broadening of the transition line increases withthe increasing in-plane magnetic field By, as clearly seen inFig. 6. This broadening is due to the many-electron effectsand can be qualitatively understood using a simplified semi-classical argument. Due to the Coulomb interaction betweenelectrons each electron experiences an instantaneous fluctuat-ing electric field E f (t), with the Gaussian distribution [42]. Insuch a field the electron moves with the in-plane drift velocityu = E f /Bz, which produces the Lorentz force euBy acting onthe electron in z-direction. The broadening of the transition

line can be estimated as eBy〈E f 〉/Bz, where 〈E f 〉 ∝ n3/4e is therms Gaussian width of the fluctuating electric field [43]. Thus,the broadening is expected to increase with the in-plane mag-netic field By. A detailed quantum theory of the many-electronbroadening in tilted magnetic fields is beyond the scope of thiswork and will be reported elsewhere.

8

01,

11,

21,

31,

41,

51,

61,

71,

81,

02,

12,

22,

32,

03,

13,

23,

0,

0,

1,

1,

2,

2,

1n 2n 3n

0yB

c

c

0yB

FIG. 7: (color online) Schematic energy level diagram for three low-

est Rydberg state manifolds of an electron in a perpendicular mag-

netic field. Each manifold consists of the equidistant Landau levels

h̄ωc(l + 1/2), l = 0,1, .. . For By 6= 0, the coupling leads to mix-ing between the states of different manifolds. Arrows show possible

transitions between the dressed states, as described in the text.

So far, we described MW-induced transitions of SE forwhich the quantum number l is conserved. These transitions(for l = 0 and 1) are showed by solid arrows in Fig. 7, whichpresents the schematic diagram of energy levels for three low-est Rydberg state manifolds of an electron in a perpendicularmagnetic field. Without the in-plane component of magneticfield (no coupling between the vertical and in-plane motionalstates), resonant transitions between states which change l areforbidden by the selection rules (we disregard the weak inter-action of SE with ripplons which can lead to the second-orderphoton-assisted scattering processes). The coupling inducedby a non-zero By leads to mixing of different states, thereforea non-zero transition dipole moment for two arbitrary statesof different manifolds. It is easy to see it by considering thesituation far from the level crossing (the dispersive regime ofCQED, see Section II) and apply first-order perturbation the-ory. As earlier, it is convenient to use the eigenstates |n, l〉 ofthe Hamiltonian (5) as an unperturbed basis, therefore treatH1 = HI +meω

2y z

2/2 as the perturbation. To the lowest orderin By, the admixed (dressed) state reads

-ωc

+ωc

0 0.1 0.2 0.3 0.4 0.5 0.6

Bz (T)

15

20

25

30

35

40

E⊥

(V

/cm

)

FIG. 8: (color online) MW absorption versus the perpendicular com-

ponent of magnetic field Bz and tuning electric field E⊥ (scale re-versed) taken for SE irradiated by MWs with frequency ω/2π =90 GHz at T = 0.37 K and for electron density ns = 10

7 cm−2. Thein-plane component of magnetic field is fixed at By = 0.2 T. Twosideband transition branches are marked by ±ωc. The red dashedlines plot κ−1 (ω21 ±ωc), where κ = 0.74 (GHz·cm)/V is the con-version factor between the transition frequency ω21 and E⊥. Thewhite solid line shows a slice of the color map taken at Bz = 0.29 T.

|n, l〉ad = |n, l〉+h̄ωy√

2lB∑n′

znn′

√

l+ 12(1± 1)

Enn′ ∓ h̄ωc|n′, l ± 1〉. (16)

It is clear that for SE occupying states |1, l〉 of the first Ry-dberg state manifold, there exists nonvanishing probabilityfor transitions to states |n, l ± 1〉ad of higher (n > 1) Rydbergstate manifolds. Two such possible transitions are indicatedin Fig. 7 by dotted arrows and marked as ±ωc. In an ex-periment, these transitions should result in two sideband ab-sorption peaks with their transition frequencies separated by±ωc from the absorption peak due to transitions which con-serves the quantum number l (solid arrows in Fig. 7). Fig-ure 8 shows a color map of the measured absorption versusthe perpendicular component of magnetic field Bz and tuningelectric field E⊥ taken for SE under MW excitation at the fre-quency ω/2π = 90 GHz and at a fixed value of the couplingfield By = 0.2 T. In addition to the strong absorption signal

centered at E(0)⊥ ≈ 23 V, which corresponds to the transition

|1, l〉→ |2, l〉, there are two sideband absorption peaks locatedon the opposite sides from the main peak. We note that suchsideband peaks has been also seeing in the early spectroscopicexperiments by Zipfel et al. [41]. The dashed lines in Fig. 8represent dependance κ−1 (ω21 ±ωc), which confirm that thefrequencies of two sideband absorption peaks are separated by±ωc from the main peak. Note that in addition to the sidebandtransitions which are accompanied by change of the quantum

9

number l by ±1, there should exist also much weaker higher-order sideband transitions which are accompanied by changeof the quantum number l by ±2, ±3, etc.

We note that at sufficiently low values of Bz . 0.2 T the ab-sorption line is smeared in the presence of non-zero By, asclearly seen in Fig. 8. This effect has been observed ear-lier by Zipfel et al. for SE in the in-plane magnetic field(Bz = 0) and can be understood using the following simpli-fied argument [32]. As an electron moves randomly alongthe surface of liquid helium with the thermal velocity vxin x-direction, it experiences the Lorentz force evxBy in z-direction due to the in-plane field By. This leads to the fluctu-ating effective electric field in z-direction with rms amplitude〈E⊥〉=(kBT/me)1/2By. This results in broadening of the tran-sition line and strong reduction of the absorption near Bz = 0,in agreement with our data shown in Fig. 8. Application of asufficiently large perpendicular magnetic field strongly affectsthe electron in-plane motion, as discussed earlier. The full ac-count of Bz-field on the transition line broadening, includingthe many-electron effects, will be given elsewhere.

To conclude this section, we consider the effect of couplingbetween the states |n, l + 1〉 and |n′, l〉 near their energy levelcrossing. As discussed in Section II, this corresponds to theresonant regime of coupling in CQED, the hallmark of whichis an avoided level crossing for the dressed eigenstates. Thiseffect is illustrated in Fig. 7 for n = 2 and n′ = 3 (see alsoFig. 2 in Section II). At By = 0, the Landau levels of n = 2and n= 3 Rydberg state manifolds are aligned, so each energyvalue is twice degenerate. At By 6= 0, the coupling leads to apair of hybridized states |±, l〉 (with corresponding energiesE+ 6= E−) for each l = 0,1, .. . Thus, in a spectroscopic ex-periment one expects to observe the absorption doublets cor-responding to the MW-induced transitions from the groundstate |1,0〉 to the pair of states |±, l〉. An example of such adoublet is shown in Fig. 7 by two double arrows marked as±. This is analogous to the Autler-Townes doublet observedin atomic and molecular systems, as well as in other quantumsystems such as quantum dots, in a pump-probe experimentwhere a strong (pump) electromagnetic field induces repul-sion between two dressed atomic levels (the Autler-Townessplitting), while another weaker field probes transitions be-tween a third levels and the dressed states [37, 44, 45]. The re-pulsion between dressed states and the Autler-Townes doubletare also observed in our experiment. Figure 9 shows an exam-ple of the measured absorption versus the perpendicular com-ponent of magnetic field Bz and tuning electric field E⊥ takenfor SE under MW excitation at frequency ω/2π = 120.5 GHzand at a fixed value of the coupling field By = 0.2 T. Thetransition |1,0〉 → |3,0〉 results in an absorption peak cen-tered at E⊥ ≈ 20 V/cm. Its position is almost independentof Bz. The position of the sideband transition |1,0〉 → |2,1〉changes linearly with Bz. The corresponding transition fre-quencies cross at Bz = 1.18 T, near which there is an avoidedcrossing between absorption peaks. To illustrate it further,Fig. 10 (top panel) shows a color map of the measured absorp-tion taken at the level crossing field Bz = 1.18 T for differentvalues of the coupling field By. As By increases from zero,the absorption peak centered at E⊥ = 20 V/cm splits into two

peaks. The top and bottom branches of this Autler-Townesdoublet correspond to the transitions of SE from the groundstate |1,0〉 to the hybridized states |+,0〉 and |−,0〉, respec-tively (see Fig. 7). As discussed in Section II, the energy split-ting between two lines of the doublet is approximately givenby 2g23 =

√2h̄ωcωyz23, which increases linearly with By, al-

though some deviations from this analytical expression comefrom the higher order mixing between eigenstates. For thesake of comparison, the position of transition lines obtainedfrom the energy spectrum calculated by diagonalization of theHamiltonian (6) are shown in the bottom panel of Fig. 10.

|1,0〉→ |3,0〉

|1,0〉→ |2,1〉

|1,0〉→ |3,0〉

|1,0〉→ |2,1〉

0.7 0.8 0.9 1 1.1 1.2 1.3

Bz (T)

15

20

25

30

35

40

45

E⊥

(V

/cm

)

FIG. 9: (color online) MW absorption versus the perpendicular com-

ponent of magnetic field Bz and tuning electric field E⊥ (scale re-versed) taken for SE irradiated by MWs with frequency ω/2π =120.5 GHz at T = 0.35 K and for electron density ns = 10

7 cm−2.The in-plane component of magnetic field is fixed at By = 0.2 T. Thedirect transition |1,0〉 → |3,0〉 and the sideband transition |1,0〉 →|2,1〉 are indicated for clarity. The white solid line shows a slice ofthe color map taken at the level crossing field Bz = 1.18 T, illustratingthe avoided crossing between two transition lines. An additional ab-

sorption peak at E⊥ ≈ 10 V/cm is due to the resonant |1,0〉 → |4,0〉transition.

An interesting feature of the data presented in Figs. 9 and10 is that two lines of the doublet show different values forthe absorption rate. In particular, the upper branch of thedoublet shows lower absorption, which also varies with thecoupling field By. Surprisingly, in a certain window of Byvalues around 0.4 T the absorption disappears, that is the sys-tem becomes transparent for the microwave radiation. Thiseffect appears due to quantum interference of the probabil-ity amplitudes of the dipole-allowed transitions. Accordingto Fermi’s golden rule, the probability of transitions fromthe ground state |1,0〉 to the excited dressed states |±,0〉 areproportional to |z±|2, respectively, with the transition dipolemoments z± = 〈±,0|z|1,0〉. As discussed in Section II, inthe two-level approximation the dressed states are |±,0〉 =2−1/2 (|2,1〉± |3,0〉). Beyond the two-level approximation,

10

|1,0〉→ |+,0〉

|1,0〉→ |-,0〉

|1,0〉→ |+,0〉

|1,0〉→ |-,0〉

0.2 0.4 0.6 0.8 B

y (T)

15

20

25

30

E⊥

(V

/cm

)

FIG. 10: (color online) (top panel) MW absorption versus the cou-

pling field By and tuning field E⊥ (scale reversed) taken for SE underthe same conditions as in Fig. 9. The perpendicular component of

magnetic field is fixed at Bz = 1.18 T. Two branches correspond-ing to the transitions from the ground state |1,0〉 to hybridized states|−,0〉 and |+,0〉 are indicated for clarity. The white solid line showsa slice of the color map taken at By = 0.8 T, illustrating the Autler-Townes absorption doublet. (bottom panel) The position of two tran-

sition lines of the doublet obtained from the diagonlization of the full

Hamiltonian (6). The calculated square of the dipole transition mo-

ment z± is given by the color tone for each line, as described in thetext. The darker tone corresponds to the larger value of |z±|2.

the states |2,1〉 and |3,0〉 are also admixed with other states. Inparticular, according to (16) up to the leading order in h̄ωy/Rethe admixed states read

|2,1〉ad ≈ |2,1〉+z22√2lB

(

By

Bz

)

|2,0〉− z22lB

(

By

Bz

)

|2,3〉,

|3,0〉ad ≈ |3,0〉−z33

lB

(

By

Bz

)

|3,1〉. (17)

Thus, the transition dipole moments to the leading order aregiven by

z± =z22z21√

2lB

(

By

Bz

)

± z31. (18)

It is clear that there is either a cancellation or an enhance-ment of the above dipole moments, therefore the correspond-ing transition probabilities, depending on the relative sign be-tween the matrix elements z21 and z31. Using numerical esti-mates |z31/z21| ≈ 0.5 and z22 ≈ 3.91rB valid for the conditionsof our experiment, the complete cancellation in (18) occurs atBy = 0.49 T, which agrees reasonably well with Fig.10. Thenumerically calculated squares of the transition moments z±for two branches are plotted in the bottom panel of Fig. 10 asthe color tone for each line of the doublet. The darker tonecorresponds to the larger value of |z±|2. Comparison with thetop panel of Fig. 10 shows that the variation of the measuredabsorption rate for two branches is in a very good agreementwith the corresponding variation of the transition probability.

We note that this effect is reminiscent of the electromagnet-ically induced transparency (EIT) observed in the two-photonspectroscopy of three-level systems [46, 47]. In such exper-iments, two energy levels are coupled by a resonant electro-magnetic (control) field, while another weak (probe) field istuned for the transition between one of this level and a thirdlevel. Similar to our case, significant suppression of the probefield absorption by the system in such experiments can be ex-plained by the destructive interference of the dipole-allowedtransitions between the third level and the dressed states ofthe two levels coupled by the control field [47].

IV. DISCUSSION AND CONCLUSION

We have shown that our straightforward calculations basedon the numerical diagonalization of the full Hamiltonian (6)are able to reproduce all features of the measured spectro-scopic properties of SE very well. It is important to emphasizethat the JCM Hamiltonian (6) is completely controlled onlyby the values of dc electric (via the Stark effect) and magnetic(via the Landau quantization and tunable coupling betweenthe motional states) fields used in the experiment, thereforethe calculations contain no adjustable parameters. This al-lows us to conclude that the considered system of SE in atilted magnetic field can serve as a convenient and robust plat-form complementary to atomic and molecular systems, suchas trapped ions and Rydberg atoms in cavities, to realize JCMand study related quantum phenomena.

As a particular example, let us consider the regime of res-onant coupling near the energy crossing for states |1,1〉 and

11

01,

02,

0,

0,

1n 2n

11,

11 21

12,21,

1,

1,

122g

1222 g

MW

0yB 0yB

FIG. 11: (color online) Schematic energy level diagram at the energy

crossing for states |1,1〉 and |2,0〉. Γ11 and Γ21 represent decay ratesof these states due to inelastic two-ripplon scattering, as discussed

in the text. The double arrow indicates the transition between the

ground state |1,0〉 and the hybridized state |−,0〉, which can be in-duced by the microwave electric field polarized parallel to the surface

of liquid (see further discussion in the text).

|2,0〉, see Fig. 11. For SE on liquid 3He in a perpendicularelectric field E⊥ = 15 V/cm such a level crossing can be real-ized with an applied perpendicular magnetic field Bz ≈ 2.82 T,see Fig. 1. JCM predicts that if an electron is initially pre-pared in the state |2,0〉 and is suddenly brought to the situ-ation where this state crosses levels with the state |1,1〉, forexample by applying a Stark pulse of the electric field E⊥, thestate of electron undergoes the coherent (Rabi) oscillations be-tween |2,0〉 and |1,1〉. In particular, the system oscillates be-tween zero and one quantum of the in-plane cyclotron motion.This is an analog of the vacuum Rabi oscillations in CQED.The Rabi frequency of oscillations is given by g12/h. Ac-cording to Eq. (9), it is the order of 10 GHz at the couplingfield By ≈ 1.5 T. The regime of strong coupling, where thesystem undergoes many cycles of such reversible oscillationsbetween two states, can be reached providing that the relax-ation and dephasing rates for these states are much smallerthan the Rabi frequency. Thus, it is important to provide anestimation for these rates.

At sufficiently low temperatures (below 1 K), the dissipa-tive processes in SE are mostly due to their coupling to thecapillary waves excited on the surface of liquid helium. Thesewaves introduce distortion of the surface of the liquid, there-fore modifying the electron potential energy (2). In particular,the first term in (2) becomes V0Θ(ξ −z), where ξ (r) is the sur-face displacement which is a function of the electron positionvector r in the xy-plane. The standard quantization procedureallows us to represent ξ as

ξ = ∑q

Qqeiqr(bq + b

†−q), (19)

where Qq =√

h̄q/(2Sρωq), S is the surface area, and Bose

operators bq and b†q describe ripplons with the usual capil-

lary wave dispersion ωq =√

αq3/ρ ( here α is the surfacetension of liquid and ρ is the liquid density). The secondterm in (2), which corresponds to the polarization attractionof an electron to helium atoms comprising the liquid, has anintegral form which depends on ξ . Since the mean displace-ment of surface 〈ξ 〉 . 1 Å is much smaller than the meandistance 〈z〉 between an electron and the surface, it is con-ventional to expand the potential energy of electron in ξ andapply the perturbation theory [35]. The term linear in ξ is re-sponsible for the one-ripplon scattering processes, which arealmost elastic due to the softness of ripplon modes. The dis-sipative processes in SE are mostly due to two-ripplon scat-tering, in particular the spontaneous emission of couples ofshort-wavelength ripplons. These processes are usually ac-counted for by the term quadratic in ξ in the first order ofthe perturbation theory. [48] The strongest contribution to thisprocess comes from the presence of the large repulsive barrierV0, with a corresponding term in the electron-ripplon interac-tion Hamiltonian given by

V(2)int (z,r) =

1

2

∂V0Θ(ξ − z)∂ z

∣

∣

∣

∣

∣

ξ=0

ξ (r)2. (20)

According to Fermi’s golden rule, the rate of decay from astate |n, l〉 into a lower-energy state |n′, l′〉 due to the two-ripplon emission is then given by

Γnn′ =π

h̄

∂V0Θ(ξ − z)∂ z

∣

∣

∣

∣

∣

ξ=0

2

n,n′

∑q,q′

Q2qQ2q′(Nq + 1)(Nq′ + 1)

×|(ei(q′−q)r)ll′ |2δ (En′,l′ −En,l + h̄ωq + h̄ωq′), (21)where Nq is the average ripplon occupation number. An im-portant point regarding the two-ripplon emission process isthat, because of the softness of ripplons modes, the conserva-tion of energy and momentum requires the total momentum oftwo ripplons to be small, that is q ≈−q′ [48]. This introducessignificant simplifications for the estimation of the above ex-pression for the decay rate. In addition, for practical purposesit is convenient to represent the matrix element of interactionappearing in (21) as

∂V0Θ(ξ − z)∂ z

∣

∣

∣

∣

∣

ξ=0

n,n′

=

√

2V0h̄2

meψ ′n(0)ψ

′n′(0)

=

√8meV0

h̄

√

(

∂υ

∂ z

)

nn

(

∂υ

∂ z

)

n′n′. (22)

Here, ψn(z) is the wavefunction for the motion perpendicu-lar to the surface and υ(z) = −Λ/z+ eE⊥z. Note that for thenumerical estimation of the matrix elements in (22) it is suffi-cient to use the wavefunctions corresponding to the rigid-wallrepulsive barrier.

12

Using the above simplifications, it is straightforward to per-form summations in (21) and estimate the decay rate. In par-ticular, the rates Γ21 and Γ11 for the decay of the excited states|2,0〉 and |1,1〉 (see Fig. 11), respectively, can be representedas [25, 48]

Γ21 =meV0

4π l2Bρ2h̄2

(

∂υ

∂ z

)

11

(

∂υ

∂ z

)

22

q̃3

ω2q̃ |∂ωq̃/∂ q̃|,(23)

Γ11 =meV0

4π l2Bρ2h̄2

(

∂υ

∂ z

)2

11

q̃3

ω2q̃ |∂ωq̃/∂ q̃|. (24)

Here, q̃ is the ripplon wave number which satisfies the en-ergy conservation relation 2h̄ωq̃ = E2 − E1. For SE on liq-uid 3He in the perpendicular electric field 15 V/cm we haveq̃ ≈ 3× 107 cm−1. Then, numerical evaluation of the aboveexpressions give Γ21 ≈ 6× 105 s−1 and Γ11 ≈ 1.4× 106 s−1,which corresponds to the lifetime of an excited state of the or-der 1 µs. We note that this estimate agrees very well with theexperimentally observed relaxation time of the excited Ryd-berg states of SE in zero magnetic field [49].

The first-order elastic one-ripplon processes lead to thestronger scattering of SE between the degenerate states ofeach energy level. The rate of this process can be roughly es-timated using the self-consistent Born approximation (SCBA)and the known rate of the elastic one-ripplon scattering ν0 inzero magnetic field. According to SCBA, the elastic scat-tering rate νB in non-zero magnetic field is enhanced by afactor of h̄ωc/∆c, where ∆c is the collision-broadened widthof the Landau levels. This leads to a simple relation νB =√

2ωcν0/π [50]. The typical one-ripplon scattering rate at

T = 0.1 K is about 106 s−1, which gives νB ≈ 5× 108 s−1.This process will lead to the dephasing. For a many-electronsystem, the electron-electron interaction can lead to signifi-cant broadening of the Landau levels, therefore to reductionof the above elastic scattering rate νB [51].

Thus, a single surface-bound electron on liquid helium ina tilted magnetic field realizes a quantum system with on-siteJCM-type interaction where the coherent part of evolution candominates over the dissipative processes. Another remark-able fact regarding electrons on helium is that at sufficientlylow temperatures T ≤ 1 K and moderate electron densitiesns ≤ 109 cm−2, SE crystallize into a triangular Wigner lat-tice [35]. In addition, the Coulomb interaction introduces aneffective interaction between electrons associated with theirvertical quantized motion. To the lowest order in rB/a, where

a ≈ n−1/2s is the inter-electron distance, this interaction is de-scribed by [24, 52]

Hee =e2n

3/2s

16πε0∑j 6= j′

(z j − z j′)2 (25)

≈ e2n

3/2s

16πε0∑j 6= j′

[

(z22 − z11)2σ jz σ j′

z + 4|z12|2(σ j+σ j′

− + h.c.)]

,

where j and j′ are the lattice sites and the effective spin op-erators for each site are defined as σ jz = |2〉 j j〈2| − |1〉 j j〈1|

and σ j+ = (σj−)

† = |2〉 j j〈1| (note that we neglected excitationfor the n > 2 Rydberg states). The two terms in the aboveHamiltonian describe the static dipolar interaction and the ex-citation hopping between the lattice sites. Thus, SE systemon liquid helium subject to a tilted magnetic field realizes aquantum many-body system with on-site interaction of JCM-type and strong interaction between the sites. This is an inter-esting complement to the Jaynes-Cummings lattice models,which has recently attracted attention due to the possibilityto study quantum phase transitions in strongly-correlated sys-tems [53]. The system described here, where the on-site in-teraction can be easily adjusted by the value of the appliedmagnetic field, while the inter-site interaction can be readilyvaried by changing the density of SE, presents a promisingflexible platform to study such many-body models. In addi-tion to the conventional absorption measurements describedhere, the image-charge detection method can provide a con-venient way to measure the quantum state of SE [54]. As wasshown, this method potentially can be scaled to detect excita-tion of a single lattice site.

Another interesting point about SE in a tilted magnetic fieldis that the coupling introduces a strong nonlinearity in the har-monic spectrum of the in-plane cyclotron motion, in particu-lar when the Landau levels of two Rydberg manifolds align.Under this condition, the energy levels of the hybridized in-plane motion become strongly non-equidistant, and the res-onant transition between the ground state |1,0〉 and one ofthe hybridized state, e.g. |−,0〉 (see Fig. 11), can be excitedby the MW radiation with electric field polarized parallel tothe surface. This presents an opportunity for anther interest-ing CQED-type experiment with a many-electron ensembleon liquid helium coupled to a single-mode cavity resonator.Recently, the strong coupling of such an ensemble to a high-quality single-mode Fabry-Perot resonator has been demon-strated [56]. In this experiment, the resonator geometry favorsthe polarization of the microwave electric field being parallelto the surface, therefore the cyclotron motion of SE in a per-pendicular magnetic field was excited. However, the linearityof such a coupled system precludes to observe differences be-tween purely classical behaviour and any predictions basedon quantum mechanics [56]. By introducing the parallel com-ponent of magnetic field, the strong coupling between an en-semble of the highly nonlinear two-level systems and a singlemode of a cavity field can be readily realized and studied.

In conclusion, we study the motional quantum states of thesurface-bound electrons on liquid helium subjected to a tiltedmagnetic field and show that they are described by the JCMHamiltonian. The predictions of theory regarding spectro-scopic properties of such a system show complete agreementwith the experimental results, without using any adjustable pa-rameters. The system shows many similarities with the quan-tum systems interacting with light, in particular a number ofphenomena related to the ac Stark effect in atomic and molec-ular systems. Also, we predict that for moderately low tem-peratures and values of the coupling magnetic field the coher-ent evolution of coupled states dominates over the dissipativeprocesses in the system. Thus, this system potentially presentsa new robust and flexible platform for quantum experiments.

13

Interestingly, our work introduces a pure condensed-mattersystem of electrons on helium into the context of atomic andoptical physics, which might provide an opportunity to bridgedifferent fields, for example many-body physics and quantum

optics.Acknowledgements The work was supported by an inter-

nal grant from Okinawa Institute of Science and Technology(OIST) Graduate University.

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