Hysteresis in a Superfluid Atom Circuit
We study a hysteresis phenomenon in a rotating BEC with a weak link in a quasi-one-dimensional torus by proposing a microscopic theoretical model including a dissipation bath. By analyzing the role of dissipation and the decay rates of all the energy levels, we are able to give a microscopic interpretation of hysteresis recently observed in the experiment and confirm that the hysteresis is the result of the presence of metastable state. In particular, we obtain the hysteresis loops in a quench process just as that in the experiment. We also find that the shape and size of the hysteresis loop change drastically with the strength of the link.
Atomtronics BTSe (); RAPe (); LAmic () is an emerging interdisciplinary field that focuses on ultracold atom analogs of electronic circuits and devices. A series of theoretical demonstrations LAmi (); ACMa (); LAmico (); Dagha () and impressive experiments ARama (); CRyu () with BECs, a testing bed for atomtronics, have established this analogy, especially for Josephson effects BPAn (); SLev (); FCat (), Bloch Oscillations OMor (). Nevertheless, hysteresis in a superfluid atomtronic gas, which is considered to be essential to the realization of an atomic-gas superconducting quantum interference device (SQUID), has not been directly observed until recently SEck (). In this experiment, both hysteresis and the quantization of flow have been observed in an atomtronic circuit formed from a ring of superfluid BEC obstructed by a rotating weak link. Just as the essential role it plays in the electronic circuit, the realization of hysteresis in atomtronic circuit will greatly accelerate the development of atomtronics because the controllability of hysteresis is crucial for the requirements of practical devices.
Hysteresis, widely used in electronic circuits, is the phenomenon where the state of a physical system depends upon its history. The canonical example of hysteresis in a classical system is that in ferromagnetism. According to the Landau theory of phase transitions LDLa (): the energy landscape, which changes with the applied magnetic field, is produced by calculating the energy of the system as a function of magnetization. Hysteresis occurs when the energy landscape has two local minima separated by an energy barrier. At some critical field, the barrier disappears and the system has only a global minimum.
At a rudimentary level, hysteresis manifests itself as the competition between experimental time scales and internal time scales, the former are determined by the frequencies of the applied perturbation and the latter are governed by relaxation, decay and so on VBan (). To observe a hysteresis in a quantum system, we require that some eigenstates of the system are metastable, namely, we can safely neglect the decays of those states in the experimental time scale. In this paper, we investigate the essential role that the metastable states play in the formation of hysteresis.
The swallowtail energy loop is a generic feature of hysteresis in an atomtronic circuit BWuQ (); EJMu (); SBah (); Dmit (). However, much of the study of swallowtails is rooted in the exact solutions to the Gross-Pitaevskii equation JCBr (); JCBro (). Recently, the dynamics simulation in a toroidal BEC AIYa (); AIYak () is also confined to the mean-field approximation.
In our study we introduce a microscopic model for the dissipation, and thus we can associate the swallowtail energy loop with the existence of metastable states. By this microscopic model, we obtain the relative decay rates of all many-body states, which decides whether a metastable state exists in our system or not, and leads to a quantitative calculation of the hysteresis in our system. Our calculations show that the interaction tends to increase the size of the hysteresis loop, while the strength of the link tends to decrease it, and confirm that there exists a metastable state which results in the hysteresis loops.
Ii Theoretical Model
ii.1 Two-mode approximation
We consider a quasi-one-dimensional dilute gas, containing bosonic atoms in a thin annulus of radius and cross-sectional radius , which rotates at frequency driven by a rotating repulsive potential JDal (). The Hamiltonian of this system in the rotating frame with frequency is given by
where is the atomic mass, is the strength of contact interaction, is the angular momentum operator, and the potential takes the form
which depletes the density in a small portion of the ring and thereby creates a weak link.
In terms of single-particle eigenstates of with angular momentum , the Hamiltonian can be written as
where , , and () is the creation (annihilation) operator of boson with angular momentum .
In the experiment SEck (), a two-step sequence is used to observe hysteresis in a BEC of Na atoms. The BEC is firstly prepared in either the or the circulation state by either not rotating the weak link or by rotating it at . Then the weak link is rotated at various angular velocities for a while. is ramped to a certain value in step 1 while to a chosen in step 2. The transitions of average angular momentum and occur at different values of and form hysteresis loops.
To simulate the hysterisis in the experiment, we consider two quench processes of parameter : is ramped suddenly from (or ) to a chosen and then the system is rotated at for a time period such that it arrives at its steady state. The quantity we describe the hysterisis is the average angular momentum
To simplify the discussion, we consider just two single-particle eigenstates: the nonrotating state and the state with azimuthal angular momentum . In fact, since the energy of atoms is proportional to , as long as , atoms in either the or the state will have lower energies than those in the other states. According to the processes we consider, it is reasonable to assume that most atoms will stay in these two levels during the whole process. Then the Hilbert subspace is given by where and , denoting the numbers of bosons that occupy the states and , satisfy . The Hamiltonian (3) can be approximated as
where , , , , , and .
ii.2 Model of dissipation
In the phenomenon of hysteresis, dissipation plays an essential role. Here we introduce a simple model to describe the dissipation of our system:
where is the Hamiltonian of the reservoir , and is the interaction between the reservoir and the system . It is obvious that the energy of can be dissipated into while the particle number of is conserved.
It is worthy to point out that the dissipation model given by Eq. (6) and Eq. (7) is a direct generalization of the Caldeira-Leggett model AJLg (). In fact, our dissipation model can be regarded as bosonic two-level atoms interact with an ensemble of bosons, which becomes the standard Calderira-Leggett model when . As argued by Caldeira and Leggett, their model is generally applicable when the coupling between the two-level system and the environment is sufficient weak, which is the main reason why we choose the dissipation model given in Eq. (6) and Eq. (7). In our case, for example, the main physical source of the environment may be a small number of atoms in the other modes except the two modes we consider.
Let , be the eigenstates of and , having eigenvalues and respectively. Let be the density operator at time of the global system , and be the reduced density operator at time of the system , which is defined by . In the basis of eigenstates of , the master equation is written as CCoh ():
where is the population of the energy level of , and is the probability per unit time for the system to make a transition from level to level as a result of its coupling with for , which is given by
where the initial state of is in the vacuum state , , and is the energy spectrum at . In the last line of Eq. (9), we assume that does not depend on , and let .
According to Eq. (9), the conditions for the transition rate are
It means that the dissipation from to will occur if and only if the eigenenergy of is above that of , and the transition amplitude is not zero.
If an eigenstate of that is not a ground state does not significantly dissipate to any state with lower energy in the time period we consider, then it is called a metastable state. We will show that the existence of metastable states is essential to the formation of hysterisis, which is discussed in Sec. III.3. It needs to be stressed that the decay rate for the metastable state into the other lower levels may not equal to zero exactly, but it is far less than those of other excited levels. In the time period we consider, we can safely neglect the decay of the metastable state. Hence, the final state of our system is a probability distribution of the ground state and the metastable state, and the probability distribution depends on the history of our system, thus leading to the hysteresis, which is shown in Sec. III.4.
Iii Analysis of hysterisis
In this section, we first establish a general formalism for the analysis of hysteresis in our system. Then we use it to describe four different cases to explore the microscopic mechanism of hysterisis in our system by steps.
Since the hysteresis is the property of the steady state of a system, it is necessary to determine the steady state of our system by solving the master equation (8).
Let , and assume that if . Then Eq. (8) can be rewritten as
where the total decay rate of the state is
If , , then the steady state will be the ground state , and our system shows no hysteresis in the plane of -.
According to Eq. (12), the population in the steady state is not zero if and only if . Obviously, the decay rate of the ground state . In addition, if the state is a metastable state in our system, then also we have . In the steady state derived from Eq. (12), the population of a state without decay is
where the probability of decay from to is
with . In other words, is the decay probability from to over all possible decay paths.
The average angular momentum in the steady state is
In particular, when there does not exist metastable states in our system, Eq. (16) becomes
which implies that does not depend on the initial state, and there can not exist any hysteresis in the - plane. Otherwise, Eq. (16) implies that depends on the initial condition, and it usually forms a hysteresis in the plane.
iii.1 The case of
In this case, the eigenstates of are
for with eigenenergies
When , , and is the ground state. Because , the amplitude , which implies that is not a metastable state. Hence the steady state at () is , and then the average angular momentum .
When , , and is the ground state. Because , the amplitude , which implies that is not a metastable state. Hence the steady state at () is , and then the average angular momentum . Because at any , the steady state does not depend on its history, and there will not exist hysteresis, but the average momentum will make a transition between and at , see a demonstration with in Fig. 1.
iii.2 The case of
In this case, the Hamiltonian can be diagonalized as
The eigenvectors and eigenvalues of are
Because , we have , and is the ground state.
According to Eq. (22), we obtain ,
which implies that is not metastable, i.e., there does not exist metastable states in our system. Thus the steady state is the ground state , which does not depend on the initial conditions. Then the average angular momentum in the ground state is
which is numerically demonstrated in Fig. 2 with parameters and .
iii.3 The case of
In this case, the eigenstates and eigenenergies of are
Let us analyze the condition for the appearance of a metastable state in our system. Because
the state is not a metastable state if and only if or . In other words, is a metastable state if and only if is a local minimum of the function . Since 0, will take a local maximum when . Hence only the two ends of , i.e. and/or , may be a local minimum of . Therefore the condition for the existence of a metastable state is and , which implies that
When , , and then is the ground state and is the metastable state. When , , and then is the ground state and is the metastable state.
When our system is initially prepared in the ground state by choosing , the average angular momentum . As is quenched to the region , is still the ground state with . As is quenched to the region , is not a ground state but it is metastable, and we still have . As is quenched to the region , then is not stable or metastable, and the system evolves into the ground state with . Similarly, we can analyze the case when is quenched back from to , and we find that when , and when . Therefore it forms the hysteresis in the plane, see Fig. 3 for a demonstration with and .
iii.4 The case of
In this general case, we can not solve the eigen problem of analytically, and must resort to the numerical method.
First, we need to calculate all the eigenenergies and eigenstates of , and determine whether there is a metastable state in our system, which can be obtained from the decay rates of excited states of . If there is a metastable state, then there is an excited state with near zero decay rate. Here we give an example on how to numerically confirm the existence of metastable states. When , , and , the decay rates of excited states with and are shown in Fig. (a)a and Fig. (b)b respectively. In Fig. (a)a the two smallest decay rates are and , which implies that there does not exist metastable states in this case. In Fig. (b)b, the four smallest decay rates are given by , , , and , which implies that the -th eigenstate is metastable.
Then we calculate the average momentum as a function of to show the hysterisis in our system. As shown in Sec. III.3, the hysterisis will appear when and demonstrated in Fig. 3. Now we examine how the hysterisis changes with the increasing of for given parameters and , which is demonstrated in Fig. 5 with different . In the case of and , there is a regular hysteresis in the plane of when shown in Fig. (a)a, which is smoother than that in Fig. 3. When , the hysterisis becomes narrower demonstrated in Fig. (b)b. When in Fig. (c)c, the hysterisis almost disappears. When shown in Fig. (d)d, the behavior of becomes similar as that in Fig. 2, and the hysteresis completely disappears.
As shown above, the appearance of metastable state is essential to the formation of hysterisis. Thus it is natural to characterize the degree of hysterisis in our system with the width of the region with metastable states. Denote the region of with metastable states with , and the degree of hysterisis becomes
Based on the concept of the degree of hysteresis, we investigate how changes the degree of hysteresis for given and , which is given in Fig. 6. In particular, it defines a critical point such that is a condition for the appearance of hysteresis. For example, when and , the critical point .
Furthermore, we present the degree of hysteresis for different parameters when . Fig. 7 suggests that is determined by the competition between and : the former tends to increase while the latter decrease it. Also, we notice that ( the boundary between the parameter region where and the blank space in Fig. 7). In fact, for , if is set to be , , it means that is determined by the energy band structure of Hamiltonian (5) for , i.e.,, it’s obvious that the energy band structure is determined by , leading to .
Iv Discussion and summary
In our present treatment of hysteresis, we focuses on the instantaneous change of , which simplifies our calculation of hysteresis and simulates well with the present experiment. However, it is worthy to point out that this is not the unique process to observe the hysterisis in our system. For example, another interesting process for this purpose is the adiabatic change of with a slow variant rate. In this adiabatic process, the many-body Landau-Zener tunneling CZen (); LDLan (); CWit () may be involved to affect the hysteresis, which needs to be investigated further in future.
In summary, we have presented a microscopic theory of the hysteresis in an atomic BEC, in which a two-mode model is used to describe the BEC system in a ring with an external potential, and an ensemble of bosons is introduced to act as the environment of the BEC system. We find that such a simple two-mode model captures the essence of hysteresis, an effect from the interactions between atoms and the dissipation induced by its environment. In particular, the existence of an metastable state is essential for the formation of hysteresis in our system. More precisely, the steady state depends on the initial state when a metastable state appears, which leads to the hysteresis in the average momentum as a function of the rotation frequency . We also find that the parameter of the external potential , which controls the tunneling between the two modes we consider, weakens the effect of hysterisis in our system. In particular, for a given particle number and the interaction constant , there is a critical potential such that if the potential is beyond the critical value, the hysteresis will be completely destroyed. We numerically give the critical potential parameter for different interaction parameter when , and observe that , which is also proved theoretically. It is worthy to point out that the predictions of our microscopic theory are consistent with the observations in the present experiment. We hope that our theory of hysteresis will promote our understandings of hysteresis at the microscopic level, and gives more rich and precise predictions beyond the mean field theory.
Acknowledgements.This work is supported by NSF of China (Grant Nos. 11475254 and 11175247) and NKBRSF of China (Grant Nos. 2012CB922104 and 2014CB921202).
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