Quantum bistability in the hyperfine ground state of atoms

First order phase transitions are ubiquitous in nature, however, this notion is ambiguous and highly debated in the case of quantum systems out of thermal equilibrium. We construct a paradigmatic example which allows for elucidating the key concepts. We show that atoms in an optical cavity can manifest a first-order dissipative phase transition where the stable co-existing phases are quantum states with high quantum purity. These states include hyperfine ground states of atoms and coherent states of electromagnetic field modes. The scheme benefits from the collective enhancement of the coupling between the atoms and the cavity field. Thereby we propose a readily feasible experimental scheme to study the dissipative phase transition phenomenology in the quantum limit, allowing for, in particular, performing a finite-size scaling to the thermodynamic limit.

equilibrium.In these steady states there is a steady current, e.g.photon flux through the system which provides for a measurable macroscopic signature of the steady states.Measurement is a substantial component for using the phase transition terminology.A typical experimental setting for this phenomenology is cavity quantum electrodynamics (QED).In the optical domain of cavity QED, atoms are loaded into a Fabry-Pérot-type optical resonator which is driven by a laser source.The light partially transmitted through the mirrors carries information about the intra-cavity field created in an interaction with the atoms.Continuous monitoring of the outcoupled light by a photodetector results in a measurement signal of the intensity, which serves as an order parameter of the intra-cavity system.Even if the system has only a few degrees of freedom, i.e., it is not macroscopic, the steady states can be considered phases because of the macroscopic measurement signal.
Optical bistability [19,20,21,22,23] is a paradigmatic example of a first-order phase transition in cavity QED.In its common form [20] it satisfies the first two conditions, (i) and (ii), but not the third one (iii).The transmission of a laser-driven optical resonator mode can be suppressed or allowed at the same drive intensity, depending on the state of the atoms loaded into the mode volume.The nonlinearity originates from the saturation effect of two-level atoms.When fully saturated, the atomic medium is transparent for the light, which can then fill the resonantly driven cavity mode in a selfconsistent manner.A very different solution can co-exist in a given range of the drive intensity, namely, that the atoms are close to their ground state and amount to a very absorptive medium.Light intensity is then suppressed securing the low excitation of the atoms.In this phase the state of the atoms is close to a pure quantum state, i.e., the ground state.However, in the bright phase the saturated atoms correspond to a high-entropy mixed state, breaking condition (iii) above.
All the three conditions are met in the case of the photon-blockade breakdown (PBB) phase transition [24,25,26].The experimental configuration is very similar, however, the atomic medium comprises only a single two-level atom and its electric dipole coupling with the mode has to be very large.This can be achieved in superconducting circuit QED systems in the microwave frequency regime where the signatures of a closely related bistability effect have been observed with threelevel atoms [16].
In this paper we present an optical cavity QED scheme which manifests a first-order dissipative quantum phase transition in an intrinsic form and makes use of the collective enhancement of the coupling between the mode and an ensemble of atoms.Here, reaching the critical domain also necessitates large cooperativity, but in contrast to PBB, this can be achieved not only by increasing the (single-atom) coupling constant -a feat that appears impossible in the optical domain -but the atom number.However, the increase of the atom number is not accompanied by an increase of the volume of the atom cloud, spatial dimensions do not play an inherent role.
The proposed scheme is the extension of a recent experiment in which we observed transmission blockade breakdown [27,28], cf. also [29] for a related scenario.The extension consists of the use of two cavity modes instead of just one as in the experiment [28].The modes interact with an ensemble of atoms modelled by a fourlevel scheme.The phases correspond to steady states with high quantum purity.The two modes are either in vacuum or in a high-intensity coherent state, meanwhile the atoms are in one of their hyperfine ground states.Because of this simplicity of the steady states, a semiclassical theory suffices to describe the phase diagram and to calculate the order parameter as a function of the control parameters of the system.
Consider a cold ensemble of N atoms interacting with two modes of a cavity with frequencies ω Ci and linewidths κ i (i = 1, 2 for the two modes, respectively).The modes are externally driven by coherent laser lights at frequencies ω i , with driving amplitudes √ N η i , respectively.The scaling factor √ N is introduced for later convenience.The cavity modes couple to the electric dipole transitions |g 1 ↔ |e 1 and |g 2 ↔ |e 2 , respectively.These transitions have resonance frequencies ω Ai , linewidths γ i , and the electric dipole interaction strength is expressed in terms of the single-photon Rabi frequency, with d i being the atomic dipole moment, and the mode volume is V.Note that by introducing N into the definition of the g i , these parameters describe the collective coupling between the modes and the cold atomic ensemble.
Importantly for the scheme, atoms from the excited levels may decay not only to the cavity-coupled ground state, i.e. |e i → |g i , but cross decays |e 1 → |g 2 , |e 2 → |g 1 are also possible with rates Γ 1 and Γ 2 , respectively.This is the mechanism that couples the subspaces 1 and 2.
The time evolution of the dynamical variables can be described by a semiclassical mean-field model using Maxwell-Bloch equations.Let α i = Tr(a i )/ √ N denote the complex amplitude of the cavity field modes, where a i is the annihilation operator of the respective cavity mode.The atomic polarization of the corresponding transition is the quantum average m i = |g i e i | .In the spirit of the mean-field approximation, the atoms are not distinguished according to their coordinates or velocities in the cavity mode, so we do not need to add ensemble averaging to the collective variables.The filling ratios of the states |g i are denoted by n gi = |g i g i | -they are numbers between 0 and 1 -and n ei for the states |e i are defined analogously.The mean-field equations of motion read The equations of the field mode α i and the polarization m i are written in a frame rotating at ω i (i = 1, 2).Without loss of generality, for simplicity, we will consider a symmetric case that the parameters with index i = 1 and 2 are equal pairwise, γ i = γ, Γ i = Γ, κ i = κ, and g i = g for i = 1, 2. We consider resonant driving of the cavity modes, ∆ C1 = ∆ C2 = 0.The cavity linewidth κ = 1.32γ and the atom-cavity coupling are taken from the experiment [28], for this latter the single atom coupling g(N = 1) = 0.1γ.Without loss of generality we chose Γ = γ.The drive amplitudes η 1 and η 2 are left to be the control parameters of the system which can be tuned to explore different phases and transitions between them.
What we achieved with the scaling with N of the dynamical variables and parameters introduced above is that N does not appear in the system (1), not even as the upper limit of the range of the population variables.Moreover, g with the above definition (incorporating a factor of √ N ) makes g 2 proportional to the ensemble cooperativity . This latter quantity is a measure of nonlinearity as attested by that optical bistability in a system of twolevel atoms coupled to a cavity mode becomes possible in the C ∼ 1 regime.Note that for Γ i = 0 the system (1) separates to two uncoupled two-level systems, where bistability would originate from the saturation of the atoms.Although we will consider large atomic detuning ∆ A with respect to the linewidth γ (for numerical calculations ∆ A = −12γ was chosen, where the negative sign stands for red detuning), significant excited state population n ei can occur for large intensities.This possibility is taken into account in these equations.Nevertheless, in the following we will study another solution of the mean-field equations which is bound to the cross-coupling decay terms, and takes place in the low-excitation limit of the atoms.
The steady state solution of Eqs.(1) can be obtained by setting the temporal derivatives on the left hand side to zero.The remaining system of algebraic equations can be transformed into a single, 7th order polynomial equation with real coefficients for the variable n e1 − n g1 .Such an equation can have 1, 3, 5, or 7 real solutions out of which respectively 1, 2, 3, or 4 are stable, the rest are unstable.The number of stable solutions depends on the control parameters η 1 and η 2 , and domains with different numbers are depicted as a phase diagram in Figure 2.
The phase diagram depends on the cooperativity that in the present setup can be changed by the atom number N .Whereas N = 5 • 10 3 (panel (a)) allows for 1 or 3 solutions only, the large atom numbers N = 10 5 and 10 6 (panels (c) and (d)) give rise to domains with 5 (orange edge of the bright yellow domain) or even 7 solutions  Closer look at the concrete solutions in the domains with 5 and 7 solutions (not shown here) reveal that the excited states n ei are significantly populated, while the polarizations m i have low values.This means that the steady states correspond to statistical mixtures, i.e., the quantum purity of the state is low.In the following we will focus on the bottom left corner of the phase diagrams where only one or two stable solutions exist.The different solutions in a given domain of the phase diagram are distinct in a macroscopic observable, which is the transmitted power κ|α i | 2 in our case (i = 1, 2).This is a suitable order parameter of phases and is readily obtained from the mean-field model.The solution (valid for our case of ∆ C = 0 and highlighting the role of the cooperativity as a measure of nonlinearity.The factor η 2 i /κ 2 is simply the num- ber of photons in the resonantly driven empty cavity and will be used as a normalization factor.The second factor above can be identified as transmittance.Equation ( 2) is not an explicit solution as the population difference n ei − n gi depends on the intracavity intensity |α i | 2 .However, this form allows for getting insight to the phases.
If the population n g1 1 and n e1 n g1 , the transmittance through the mode 1 is suppressed for large cooperativity C 1.As there is no field in the cavity mode 1, all the atoms being in state |g 1 is a stable solution.On the other hand, according to the solution above with n e2 ≈ 0 and n g2 ≈ 0, mode 2 is closely resonantly excited, which leads to transmittance 1. Reversely, there is also a stable solution in which all the atoms are in |g 2 , i.e. n g2 1, and the transmittance of mode 1 is close to unity.The domain with 3 solutions in Fig. 2 corresponds to the case when these stable steady states coexist.This will be further investigated along circular sections of the phase diagram, i.e., where η 2 1 + η 2 2 ≡ η 2 is constant.As the total input power per atom is proportional to η 2 1 +η 2 2 , this section represents a fixed total drive intensity per atom, and increasing the angle ϕ = arctan η 2 /η 1 from 0 to π/2 corresponds to a continuous switching from driving mode 1 to 2.
Figure 3 shows cavity transmittance (a), ground state (b) and total excited state populations (c) as a function of the angle measured from axis η 1 for case N = 5000 along circular arcs of various radii plotted in Figure 2 (a) with the same colours.Because of the 1 ↔ 2 symmetry of the scheme, the plot of the same quantities with index 2 are just the mirror images of the ones with index 1, therefore only the latter is shown.The red and the green arcs do not cross the bistable region, hence there is only one real solution along those, which is, of course, stable.The rest of the curves, both for the transmittance and the ground state population show a characteristic S-shaped form of a bistability with overlapping stable solutions (solid line) connected by an unstable one (dashed line).High (low) transmittance corresponds to low (high) relative ground state population.For decreasing the total input power, the Sshaped curves show convergence in panels (a) and (b), whereas a gradual decrease of the population in the excited states is shown in (c).In this limit the bistability is formed between the two hyperfine ground states, the excited states being only virtually populated underway the two-photon transition between the ground states.One can identify thus a dissipative quantum phase transition in the spirit of the three conditions given in the introduction, where in particular, the phases correspond to quantum states of high purity.
The thermodynamic limit, where the bistability becomes a phase transition can be defined as C → ∞ while η is kept constant.In a practical case, the cooperativity can be increased by the atom number, hence the N → ∞ implies that the actual drive power N η 2 has to go to infinity.The axes of the phase diagrams in Figure 2 already used this scaling.Therefore the circular arcs of a radius η/γ = 0.29, plotted in each phase diagram (white), are fixed in the finite-size scaling.While the boundaries of the multivalued domain vary slightly, the phase diagram is qualitatively the same.For increasing cooperativity (via atom number), Fig. 4 (a) and (b) show that the S-shaped curve tends to a sharper Z-shaped one (mirrored).Interestingly, when going towards the thermodynamic limit, both solutions become stable in almost the total range of the control parameter η 2 /η 1 .Simultaneously, as shown in Fig. 4 (c), the population in the excited states tends to completely vanish in this limit.Thus, in the thermodynamic limit, the proposed system has two stable solutions with the atoms being in one of the ground states |g 1(2) and the other mode 2 (1) being populated by a coherent state, meaning that in this limit the perfect quantum purity of the phases of the system is achieved.
In conclusion, the proposed scheme opens the way to an experimental investigation of a first-order dissipative quantum phase transition.The four-level atomic scheme can be realized to a good approximation within the hyperfine structure of e.g. the D2 line of rubidium-87, as it has been discussed in [28].The atom number can be varied in a controlled way over many orders of magnitude in an experiment, allowing thus for a finite-size scaling to the thermodynamic limit.Beside the possibility of investigating fundamental concepts of phase transitions in mesoscopic quantum systems, the bistability between long-lived ground states holds prospects for new atomic memory architectures.

Figure 1 :
Figure 1: (a) Parameters of the cavity QED scheme with two laser-driven modes interacting with an ensemble of atoms.The modes are spatially separated only for illustrative purposes, in practice, two fundamental modes of the cavity can be used.(b)Relevant level scheme of the atoms with two dipole-allowed transitions cross-coupled by relaxation processes.Schematic panels at the bottom represent that the atomic ground states switch the transmission or reflection of the cavity drives.

Figure 2 :
Figure 2: Phase diagram with domains with different number of stable solutions of the system (1) on the plane of the drive amplitudes η1, η2.The cooperativity increases from (a) to (d), corresponding to atom numbers N = 5 • 10 3 , 10 4 , 10 5 , and 10 6 , respectively, with the single atom coupling g(N = 1) = 0.1γ.Relevant quantities along the coloured quarter circular arcs plotted on the phase diagrams will be shown later in Figs. 3 and 4. Radii of the coloured circular arcs are η/γ = 0.29, 1.13, 2.25, 3.38, 4.5.The white arcs in the low drive limit are of particular interest with respect to DQPT.

Figure 3 :
Figure 3: Crossing domains with multiple stable solutions.The transmittance of cavity for mode 1 (a), and relative atomic population (b-c) are shown along the circular arcs of corresponding colours plotted in Figure2 (a).Arc measure ϕ is measured from the vertical axis.Solid (dashed) lines correspond to stable (unstable) solutions.

N = 5 • 10 3 N = 10 4 N = 10 5 NFigure 4 :
Figure 4: Finite size scaling to the thermodynamic limit.The order parameter represented by the transmittance of cavity for mode 1 is plotted in (a), and relative atomic population are shown in (b-c) along the circular arcs on the phase space in Figure2(a) with a radius of η/γ = 0.29 for different atom numbers N .ϕ is measured from the vertical axis.Solid (dashed) lines correspond to stable (unstable) solutions.